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NA AS Seres
Se *es • n si al Scierces -VI. 490

Microlocal Analysis and Spectral Theory

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cJJpJrp
Series C: Mathematical and Physical Sciences -Vol. 490

Microlocal Analysis and
Spectral Theory
edited by
Luigi Rodino
Department of Mathematics,
University of Torino,
Torino, Italy
W
Springer Science+Business Media, B.V.

Proceedings of the NATO Advanced Study Institute on
Microlocal Analysis and Spectral Theory
II Ciocco, Castelvecchio Pascoli (Lucca), Italy
23 September - 3 October 1996
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-94-010-6371-5 ISBN 978-94-011-5626-4 (eBook)
DOI 10.1007/978-94-011-5626-4
Printed on add-free paper
All Rights Reserved
© 1997 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1997
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical, including
photocopying, recording or by any information storage and retrieval system, without written
permission from the copyright owner.

Contents
Preface vii
Linear Partial Differential Equations with Multiple Involutive Characteristics
by 0. Liess and L. Rodino 1
Gevrey and Analytic Hypoellipticity
by D. S. Tartakoff 39
Higher Microlocalization and Propagation of Singularities
by 0. Liess 61
Conormality and Lagrangian Properties in Diffractive Boundary Value Problems
by P. Laubin 91
Parametrized Pseudodifferential Operators and Geometric Invariants
by G. Grubb 115
Boundary Value Problems and Edge Pseudo-differential Operators
by B.-W. Schulze 165
Wodzicki's Noncommutative Residue and Traces for Operator Algebras on
Manifolds with Conical Singularities
by E. Schrohe 227
Lower Bounds for Pseudodifferential Operators
by C. Parenti and A. Parmeggiani 251
Weyl Formula For Globally Hypoelliptic Operator in Rn
by E. Buzano 263
Splitting in large dimension and infrared estimates
by B. Helffer 307
Microlocal Exponential Estimates and Applications to Tunneling
by A. Martinez 349

VI
A trace formula and review of some estimates for resonances
by J. Sjostrand 377
Index
439

Preface
The NATO Advanced Study Institute "Microlocal Analysis and Spectral
Theory" was held in Tuscany (Italy) at Castelvecchio Pascoli, in the district of
Lucca, hosted by the international vacation center "II Ciocco", from September
23 to October 3, 1996.
The Institute recorded the considerable progress realized recently in the field
of Microlocal Analysis. In a broad sense, Microlocal Analysis is the modern
version of the classical Fourier technique in solving partial differential
equations, where now the localization proceeding takes place with respect to the
dual variables too. Precisely, through the tools of pseudo-differential operators,
wave-front sets and Fourier integral operators, the general theory of the
linear partial differential equations is now reaching a mature form, in the frame
of Schwartz distributions or other generalized functions. At the same time,
Microlocal Analysis has grown up into a definite and independent part of
Mathematical Analysis, with other applications all around Mathematics and Physics,
one major theme being Spectral Theory for Schrodinger equation in Quantum
Mechanics.
Concerning general theory of linear PDE, contributions were presented in
the following directions:
- discussion of new topics in the Gevrey-analytic category, as propagation
of singularities and hypoellipticity in the case of multiple characteristics,
higher analytic microlocalization and applications, diffractive boundary
value problems, CR manifolds;
- advances in elliptic boundary value problems, in particular geometric
invariants associated to them, the case of the manifolds with singularities
and the corresponding edge-pseudo-differential calculus;
- new results on lower bounds for pseudo-differential operators with
multiple characteristics.
Concerning Spectral Theory, different problems for the Scrodinger equation
were discussed, in particular:
- asymptotic behaviour of the eigenvalues in the case of a polynomial
potential;

vm
- semiclassical analysis in large dimension and statistical mechanics;
- microlocal tunneling and adiabatic theory;
- asymptotic of resonances.
On the whole, the Institute was able to cover these topics in Microlocal
Analysis, which have pre-eminence because of their novelty or importance in
the applications, in the field of partial differential equations as well as in other
areas of Mathematics and Theoretical Physics.
The Institute was attended by 82 participants: 72 from NATO countries
(Belgium: 1, Denmark: 5, France: 17, Germany: 16, Italy: 22, Turkey: 2, U.K.:
1, U.S.A.: 8) and 10 from other countries (Armenia: 1, Bulgaria: 2, India: 1,
Romania: 1, Russia: 5).
The lectures were held by 13 lecturers; moreover 24 advanced seminars were
organized by the participants and devoted to the discussion of their contribution
in the field.
These Proceedings, aiming at a state-of-the-art volume, present a selection
of the aforesaid lectures. Lack of space does not allow publishing the texts of the
seminars; an excuse to the editor is that they are addressed to more experienced
readers, who would not find difficulty to find them in various journals.
We want to express our gratitude to NATO which was the main sponsor
of this meeting. Our thanks go to the Scientific Affairs Division and specially
to the NATO Science Committee, to Dr. L. Veiga da Cunha, Director of the
ASI Programme, and to Barbara and Tilo Kester, of the NATO ASI series
Publication Coordination Office.
It is our pleasure to mention other Institutions which supported financially
the meeting: Dipartimento di Matematica, Universita di Torino, covering in
part the organization expenses; Politecnico di Torino, host of the home page of
this NATO ASI; all the Institutions providing suppport to the travel expenses
of the students, among them: National Science Foundation U.S.A., TUBITAK
of Turkey, MURST of Italy, Max-Planck Institute of Germany.
On the behalf of all participants, we express our gratitute to Prof. M.
Mascarello and Eng. B. Monastero, of the Politecnico di Torino, who performed
smoothly and efficiently the work of secretary.
Finally, we wish to express our warmest thanks to Dr. S. Coriasco, Ph.D.
student at the Universita di Torino, who elaborated by computer this volume
to its final form; his help has been invaluable for us.
Luigi Rodino
Director of the Institute

LINEAR PARTIAL DIFFERENTIAL EQUATIONS WITH
MULTIPLE INVOLUTIVE CHARACTERISTICS
0. LIESS
Universitd di Bologna
Dipartimento di Matematica
Piazza di Porta S. Donato 5
1-40126 Bologna (ITALIA)
AND
L. RODINO
Universitd di Torino
Dipartimento di Matematica
V. C. Alberto 10
1-10129 Torino (ITALIA)
0. Introduction
1. Foreword. In this paper we consider linear partial differential operators
with involutive characteristics of high multiplicity in the case when no
assumptions of Levi-type is made on lower order terms; for such operators
we shall prove results on propagation of singularities, local solvability and
hypoellipticity in the frame of analytic, Gevrey and C°°-classes.
The arguments in the proofs are based mainly on the machinery
developed in [13], [15] and are given in the sections following this one. The present
introduction, however, corresponds to part of the introductory lecture to
the school, given by one of us (L.R.) and meant to help the non-specialized
participant to become familiar with some of the basic notions in the general
theory of linear partial differential operators.
The notations which we use are standard in the general theory of linear
PDE; in particular, given a multi-index a = (ai,...,an) £ Z+ we write
Da = D^1 .. .D£n, where Dj = -id/dxj.
i
L. Rodino (ed.), Microlocal Analysis and Spectral Theory, 1-38.
© 1997 Kluwer Academic Publishers.

2
Moreover, we expect the reader is familiar with the spaces Cq0^),
C°°(fi), Q being an open subset of i?n, S(Rn) and their topological duals
Vf(Q), £'(Q) and <S'(i?n). The generic linear partial differential operator P
can be written in the form
P= £ aa(o:)Da, (0.1)
\a\<m
where we assume initially the coefficients aa(x) be given in C°°(fi); then
P : C°°(Sl) ->■ C°°(ft) can be extended to a linear map
P:V'(Q)-+V'(Q). (0.2)
We call the symbol of P the function m Q X Rn
p(*f0 = E M*)?*. (0-3)
|a|<m
The principal symbol of P is
PmM)= E ««(*)£", (°-4)
|a|=m
homogeneous function of order m with respect to the dual variables £. The
characteristic manifold of P is the subset of Q X Rn given by
£ = {foOIM*,0 = <U#o}. (0.5)
An operator P is said to be elliptic in Q if E = 0, i.e. Pm(x,£) ^ 0 for
all x € Q and all £ ^ 0, whereas P is said to be of principal type if the
characteristic manifold E is non-empty and
V(z,OeS:4,^m(*,O^0. (0.6)
The theory of elliptic operators and operators of the principal type is
nowadays well developed; in this paper we shall consider operators with multiple
characteristics, i.e. satisfying
3(xlOGS:dXf€jim(xlO = 0l (0.7)
where we understand E is non-empty.
When dealing with multiple characteristics operators, it is convenient
to enlarge the Schwartz frame (0.2). Precisely, let us first introduce the
Gevrey space GS(Q), 1 < s < oo, consisting of all / £ C°°(Q) satisfying in
every K CC 0,

3
snp\Daf{x)\<CM+1(a\)s, (0.8)
xeK
with a constant C independent of a. Observe that Gl(Q) is the class A(£l)
of all the analytic functions in ft. When s > 1, we write Gq(£1) for Gs(ft) fl
Co00 (ft).
The spaces of s-ultradistributions 2?s(ft), £$(ft) are the duals of G^ft),
Gs(ft). We have the inclusions £>'(ft) C £>J(ft) C V'S(Q) for 5 < t. To have
uniform notations, we shall also write G^(ft)),£>^(ft) for C°°(ft),:D'(ft),
and 7?i (ft) for the space of generalized functions of Sato, including all the
preceding classes.
From now on, we shall assume that the coefficients aa(x) of P in (0.1)
belong to A(Q)] then for 1 < s < oo
P:Gs(ft)->Gs(ft) (0.9)
extends to a map
P:V'M^V'M (0.10)
that gives the expected more general frame to the study of the equation
Pu = f.
2. Pseudo-differential operators. In the analysis of (0.10) one is led in
a natural way to consider a larger class of operators, namely the pseudo-
differential operators, which are defined in a broad sense by
Pu{x)=p(x,D) u(x) = (2tt)-" Je'xtp(x,0m<%, (0.11)
where u — T(u) is the Fourier transform of u and p(x,£) is the so-called
symbol of P. Homogeneous classical symbols are functions in C°°(ft X Rn)
admitting an asymptotic expansion
oo
p(*,0~X>m-iM)> (0-12)
where m £ R and pTO_j(x,£) is positively homogeneous of degree m — j
with respect to £. We may obviously refer the preceding definitions (0.5),
(0.6), (0.7) to the leading term pm(x,£) in (0.12). Of course, a linear partial
differential operator is a pseudo-differential operator, with symbol given by
(0.3). We have from (0.11), (0.12)
P = p(x,D):C^(n)->C00(n) (0.13)

4
with extension
P = p{x, D) : £'(Q) -> V'(Q). (0.14)
As generalization of the linear partial differential operators with
analytic coefficients, we may consider (homogeneous classical) analytic pseudo-
differential operators. Their symbols can be expanded as in (0.12), with
pm-j(x,£) admitting holomorphic extension in a complex neighborhood of
fixi?n, satisfying there suitable uniform estimates. Beside (0.13), (0.14),
we have in this case for 1 < s < oo
P = p{x,D) : GS0{Q)^GS{Q) (0.15)
P = p(x,D) : S'a(Q)-+V'M (0.16)
with continuous action also on the Sato space V[(Q).
3. Wave front sets and microfunctions. Basic ingredients of the microlo-
cal analysis are the wave front sets, defined in the following way. Let us
begin with the Gevrey case.
Definition 0.1 Fix (x0,£°) G ft X Rn. For u e X>i(!2),l < s < oo, we
say that (x°,£°) £ WFsu if there exist <p £ Gs(£l) with <p(x) = 1 in a
neighborhood of xq, and positive constants C, e such that
|^(vti)(0l<C«p(-^l1/-) (0-17)
for all £ in a conic neighborhood of f°.
The projection on Q, of the s-wave front set WFsu is the s-singular
support of u, defined as the complement of the largest open subset of Q
where u is of the class Gs.
The oo-wave front set WF°°u of a Schwartz distribution u is defined in
the same way, by replacing (0.17) with
l^(H(0l< Cm (1 + KI)-" (0-18)
for arbitrary M = 1,2,..., where now we allow <p £ Co°(ft). In the analytic
case test-functions with compact support do not exist; we may however
say that (x°,£°) £ WFlu if there exists a sequence un, N = 1, 2,..., with
compact support, and u^ = u in a neighborhood of a:0, such that
Mflisc^^a+Kir*. iv=i,2,..., (o.i9)

5
for all £ in a neighborhood of £°. Projections of WF°°u and W^F1^ in Q
are the respective singular supports.
Next step is to consider microfunctions. Precisely, let A be an open
subset of Q X Rn conic with respect to £, with relatively compact projection
on Q. For u, v £ ^(^) we shall write u ~ v to mean that AnM^Fs(^- v) =
0, and we shall denote by MS(A) the factor space Vfs(Q)/ ~, 1 < 5 < oo.
The 5-wave front set W^F^ of a microfunction u £ Ms(A)\s a> well defined
conic closed subset of A. If P is an analytic pseudo-differential operator,
then
Vu G #(R), 1 < s < oo : WFsPu C WFsu, (0.20)
and therefore, by factorization
P : M*(A) -+ M'(A), 1 < 5 < oo, (0.21)
the inclusion (0.20) keeping valid for all u £ Ms(A). As for (0.21), we
may actually assume that p(x,£), symbol of P, is defined only in a conic
neighborhood of A. The theorems of the next sections will be stated in the
frame of the micro-operators (0.21). Note that results in the more standard
setting (0.10) are hence easily deduced, by covering Q X Rn with conic sets
A and projecting WFsu into s-singsuppu. So, for example, if P is s-micro-
hypoelliptic in the sense that WFsu = WFsPu for all u £ MS(A) in any
A, then P is s-hypoelliptic in the standard sense, i.e. s-singsuppu = s-
singsuppPu for all u G £fs{ty-
4. Fourier integral operators. The setting (0.21) has at least two
advantages, with respect to the standard local point of view. First, the classical
theorem of regularity for the solutions of the elliptic equations can be
refined by means of the formula
WFsuC WFsPuUZ, (0.22)
where u £ £s{®>) or u £ MS(A), 1 < s < oo, and E is the characteristic
manifold in (0.5). So, if we are concerned with the singularities of the
solutions u when f = Puis smooth, it will be actually sufficient to study (0.21)
in an arbitrarily small neighborhood A of the points (#0,£0) £ S.
Moreover, the machinery of the Fourier integral operators may lead to
relevant simplifications in the study of the micro-operator P = p(x,D) in
(0.21). Precisely, let x be a homogeneous analytic canonical transformation
acting from the conic neighborhood A of the point po = (£°,£0) to a conic
neighborhood T of the point x{Po) = (J/0?7/0); that x is canonical means
that it preserves the symplectic two-form a — ]C?=i d^jAdxj. Then we may

6
consider the Fourier integral operator F with phase function corresponding
to x; this is a map F : MS(A) -* MS(T), 1 < s < oo, with inverse
F~l : MS(T) -> MS{A), such that
WFs(Fu) = x{WFsu), WFa(F~lv) = X~\WFsv). (0.23)
Moreover
P = FPF~l : MS{T) -+ MS{T) (0.24)
is a (micro) pseudo-differential operator, with homogeneous classical
analytic symbol p(j/, 77) having principal part
Pm{y, V) = Pm{x~l{y, V))- (0.25)
In particular, if we assume po G E and denote by E the characteristic
manifold of P, then xO^o) G E and E = x(E) in T.
In this way, by fixing a suitable canonical transformation x> we may
reduce ourselves to the study of operators P of a truly elementary form.
Additional simplifications in the expression of P can be obtained by means
of composition with elliptic pseudo-differential factors.
5. Other geometric invariants. In the following of the paper we shall
suppose that E is an analytic regular manifold of codimension n' > 1 (more
restrictively, E will be assumed involutive, cf. the next definition 1.1) and
pm(x, £) vanishes exactly to the order k > 2 on E, i.e. there exists a constant
C > 0 such that
C-^eW)* ^ \PmM/\t\)\ < Cdx(x,Z)k, (s,0 G A, (0.26)
where d^(x^) is the distance from (#,£/|£|) to E.
We have already observed that pm (a:, £) has a geometric invariant
meaning, i.e. (0.25) is valid after conjugation by Fourier integral operators.
Consequently, the assumption (0.26) is also invariant. It is interesting, in the
case of the multiple characteristics, to relate the second term in the
asymptotic expansion
p(y, v) ~ Pm(y, v) + Pm-i(y, */) + ••• (0.27)
to the expression of pm(x,t;) and pm-i{%,Q- To this end, define

7
which also has an invariant meaning, if we limit ourselves to the points
(#,£) where pm(x^) vanishes at least to the order 2. In fact, if we consider
Pm-i(j/^)from (0.27), we may recognize that p'm-Mx,Q)-p'm-i(x,Q =
AoPm(#,£), where Xq is a vector field on T with homogeneous analytic
coefficients depending on x; therefore plm_x and pfm_i coincide at the double
characteristics set. In the present case, under the assumption (0.26) with
k > 2, we shall write
/oM) = p;»-iM)|e (o-29)
and call it sub-principal symbol of P = p(x, D), as standard.
Consider now X\,..., Xj G Tp(A),p £ E, and let Xj,j = 1,..., J, be
analytic vector fields on A such that Xj(p) — Xj. We define
$(pm(x,0,P,Xi,---,Xj) = (X1...Xjpm)(p) (0.30)
which is also invariant, in the sense that
$(Pm{x, 0) P. xu ■ ■ •, Xj) = $(pm(y, ri),x(p),dx^)Xu ..., dx(p)Xj)
(0.31)
It is clear that such invariance fails if in (0.30) we replace pm with p'm_i-
In fact, the derivatives of Xopm do not vanish in general, even if evaluated
at the double characteristics set.
However, if we assume pm(x,C) has a zero of order at least k > 3 at />,
then if J < k - 2
$(#n-i(y. v),x(p),dx(P)Xi,..., dx(p)Xj) =
= (dx o Xx...dx o Xjp'm_{){x{p))
= (*!. ..Xj&.Jip) + (Xl.. .XjXoPm)(p)
since X\.. .XjXopm vanishes at p under our assumptions. Summing up,
under our hypothesis (0.26) the following functions have an invariant meaning:
$(pm,p,Xh...,Xj) : (TP(A))J^C, pGS; (0.32)
^(p'm_up,Xh...,Xj) : (TP(A))J^C, pGE,J<*-2.(0.33)
We may go further, considering iV(E) = T(A)/T(E), the normal bundle of
E; for every (/>, X) € iV(E), with pGEjG TP(A)/TP{E) we take 7 in the
equivalence class of X and define

8
Tl{p,X) = ^(pm,p,Y11^Y) : iV(E) -+ C (0.34)
k times
Ij(p,X)=-^(p'm_l,p,Y,...X) : JV(£)-»C,l<j<*-2 (0.35)
J. ^ y
j times
In view of (0.26), we have n(/>, X) ^ 0 for X ^ 0. We reserve the notation
h{p) for the sub-principal symbol, according to (0.29).
Our results, in the next sections, will be formulated in terms of the
invariants pm, n, Jo, Ij-> 1 < J < A; — 2; the evident advantage will be that in
the proofs we shall be free to change symplectic coordinates.
1. A result on propagation of singularities
1. Statements. We shall first give a result on propagation of singularities in
the case when the characteristic manifold E is regular involutive of arbitrary
codimension n', 1 < nf < n. We recall the following definition:
Definition 1.1. A homogeneous submanifold E C T*Rn is said to be
regular involutive if:
i) for every p £ E the restriction of the two-form a — Y^j=\ d£j A dxj to
Tp(E)-1 is identically zero
and
ii) the radial vector field 2j=i£jd/d£j does not belong to T^E)1 for any
/9GE.
(T^E)1 denotes the orthogonal complement of TP(E) with respect to a.)
It is standard to observe that locally on conic sets a homogeneous
analytic submanifold is regular involutive if and only it is of form E =
{ui(x,£) = • • • = unt(x,£) = 0} where the functions Uj, j — 1,.. . ,n' are
analytic real-valued in A, homogeneous of order 1 in £, the forms du\,..., dun>
and £j=i ijdxj are linearly independent and
{uj,uk} = 0on E ,j,fc= l,...,n'. (1.1)
(Here we have written, as is standard, {/, g} = Yl]=i (dXjfdXjg-dXjgdXjf) =
Hjg.)
This definition has the following important consequence: in view of (1.1),
the vector fields HUj are tangent to E, and their restriction to E satisfies

9
the integrability condition of Frobenius, i.e., we have [HUj,HUi] = H{u.jtAj
and since {u^U{] is a linear combination with analytic coefficients of the
Uk, then [Hu., HUi] is, on E , a linear combination with analytic coefficients
of the HUk. The manifold E can be therefore endowed with a canonical
foliation T, whose leaves L are the integral manifolds of the vector fields
Hu , j = 1,..., n'. In the case of codimension 1, these leaves are of course
just the bicharacteristic strips.
For p £ E, we shall denote by Lp the leaf through p.
Our condition
C-ldx(x,Ok < \Pm{x,t/\t\)\ < Cdz(x,Ok (1-2)
implies that we may write in A
Pm(&,0= Yl M^O^foOi (1-3)
for some analytic symbols a7(a:,^) of order ra-A;, where the symbols Uj(x,£)
are defined as above. Note that one may assume without loss of
generality that (1.1) are satisfied in the whole A, what we shall suppose in the
following.
Consider now the function U(p,X) : iV(E) ->■ C defined in (0.34); in
terms of (1.3), we have
U(p,X)= £ ai(p)du\X) (1.4)
Note that the map iV(E) -+ ^(E)1]^ defined by (p,X) -+ ix° is an
isomorphism. Moreover we can identify T(E)-1- with the cotangent space
to the leaves of the foliation, \JLerT*(L). Therefore, Il(p,X) defines for
each leaf a function
nL : T*(L)/0-> C (1.5)
which in terms of (1.3) can be seen as the principal symbol of the operator
on L:
ft = H)h,|EM»,0^) (i-6)
We now begin by recalling some known results.

10
Theorem 1.2 Let pm(x,£) satisfy (1.2), and E be regular involutive. Con-
sider po £ E and write Lpo for the leaf through p0. For 1 < s < k/(k-l) we
have: if u £ MS(A) with Pu = 0, then p0 € WFsu implies Lpo C WFsu.
The theorem was proved by Bony-Shapira [3] for s = 1, for 1 < s <
k/(k - 1) by Kessab [9]. The study of P for k/(k - 1) < s < oo involves the
sub-principal symbol Jo. The following result of hypoellipticity was proved
in Liess-Rodino [14].
Theorem 1.3 Let pm(x^) satisfy (1.2), and E be regular involutive. As-
sume
n(Plz) + /0(p)#o (i.7)
for all (p,X) e iV(E). Then for k/(k - 1) < s < oo:
i) P is s-micro-hypoelliptic, i.e. WFsPu = WFsu for all u e MS(A);
ii) For every v £ MS(A) there exists u G A4S(A) such that Pu = v.
We want to give results for s > kj (k - 1) in the case when the invariant
in (1.7) vanishes. We shall limit ourselves to a special case, precisely we
assume that, possibly after multiplying P by an elliptic factor:
Pm(*,0>0 forfoOGA; (1.8)
/o(*,O<0 forfoOGE; (1-9)
Ij(p, X) is real valued on iV(E) for 0 < j < k - 2 (1.10)
where Ij is defined as in (0.35). An equivalent expression of (1.10), whose
invariant meaning is less transparent, is the following, in terms of plm_i (x, £)
from (0.28):
d"dfp'm_i(z,£) is real valued on E, for 0 < |a + /?| < k - 2 (1.11)
Of course the conditions (1.10), (1.11) are empty if k = 2.
Once (1.8), (1.9), (1.10) are assumed to be satisfied, we can multiply by
a positive elliptic factor and suppose without further loss of generality
Io(x,£) has constant value along each leaf L 6 T (1.12)
For example, multiplying by q(x,D) with q(x,£) > 0, q{x,£) = -l//o(z,£)
on E, we get as new sub-principal symbol the constant -1. The setting
(1.12) simplifies the following statement.

11
For p £ S, let us denote by Klp the Hamiltonian vector field of Ulp
in (1.5). The related integral curves run in T*^). Let us call geodesies
through p the projections on Lp of such curves; it will be easily seen from
(1.2) that no one of the geodesies reduces to the point p.
Theorem 1.4 Let pm(x,£) satisfy (1.2), and E be regular involutive. As-
sume further (1.8),(1.9),(1.10),(1.12). For k/(k - 1) < s < oo,p0 £ S we
have: if u G MS(A) with Pu — 0 and po G WFsu, then one at least of the
geodesies through po is included in WFsu.
For k = 2, the result was proved by Boutet de Monvel [4] in the case s = oo,
and stated by Lascar [10] in the case 2 < s < oo. As a model for theorem
1.4, consider in Rn = Rn X i?, x = (j/,i),£ = (r/,r):
p= E ^-^r1 (i.i3)
|7|=m
where
£ <vf > cm™ (i.i4)
H=ra
for some c > 0. The geodesies are straight lines in the leaves t = to, V — 0, r =
r0> 0.
The first step in the proof of theorem 1.4 is a standard application of
Fourier integral operators with analytic phase and amplitude functions,
cf. Liess-Rodino [14],[15], Rodino [24]. Precisely, we may define a
canonical map on A, such that the first nf new coords are given by ^i,.. .,^n/.
By conjugation with the corresponding Fourier integral operators, we are
reduced to consider an operator with principal symbol
Pm(*,0= EM^'Or (1-15)
H=fc
where £ = (£',£"),£' € Rn',£" € iK.n' + n" = n, and similarly we split
x = (a:', x"). The symbols a7(x,^) are analytic homogeneous of order m — k,
defined in a conic neig hborhood r of p0 = (x°,£>) with £Q/ = 0. Multiplying
by an elliptic factor, we may suppose m — k. The assumption (1.8) is now
that all the a1 are real valued, and for some C > 0 we have
c-'itV < E M^Of7 < err ^ (x,o € r, (l.ie)
|7|=m
which implies m is even.

12
Writing
Pm,o(*,o= E «i(M,or, (i.i7)
|7|=ra
we also have
c~lW\m < Pm,o(*,0 < c\t'\m for (*,0 e r. (l.is)
The characteristic manifold E is now given by {£' = 0} and the leaf of the
foliation L through pQ is {x" = x0",? = 0,£" = £0"}. From (0.28) we have
Pm-i — Pm-i; the sub-principal symbol Jo reduces then to
Pm-lf0(*,O =Pm-l(xl0lO» (1-19)
and (1.9), (1.10) mean that
-qrr-1 < Pm-i)0(x,n < -c-'rr-1, (1.20)
Pm-lj(a;, O = E (.dl'Prn-\){.X, 0,0 — *S real ValUed for 0 < i < fc ~ 2-
(1.21)
As for (1.12), here we shall not require it is satisfied. The function n^
in (1.5) is given, in local coordinates, by
n(*',O = pm>o(*', *°",c',e") = E M*'. *°". o,£0")O- (i-22)
Yi\=m
Consider the solutions (#'(£),£'(£)) of the system
§ = -^n^^O-^P^-Lo^'^0"^0") (i-23)
I x'(0) = z°',f(0) = V
with rf <E Rn\i # 0,pTO,o(xo,77',e0") +Pm-i>o(x0,f'') = 0. In the case
when pm-ifl does not depend on x', as we assume in (1.12), the
projections x' — x'(t) of the solutions are the geodesies through x'0 on the leaf
{a;" = x°",£' = 0,£" = £0"}. They never reduce to x' — x°\ since
dvIl(x',?) / 0 for £' ^ 0, in view of the ellipticity of U(x',^') from (1.18)
and the Euler identity. The following result gives then theorem 1.4.

13
Theorem 1.5. Letp(x,£) = pm(x^)+pm-i(x^)-\ satisfy the preceding
assumptions. In particular we assume that (1.18), (1.20), (1.21) hold in a
conic neighborhood Y of po = (#0,£0) with £0/ = 0. Let m/(m - 1) <
s < oo. Shrinking T if necessary, we can then conclude that if u G MS(T)
satisfies Pu = 0, and if on every curve j(t) = (xf(t), x0ff; 0,£°") there exists
p G 7 H T with p £ WFsu, then p0 £ WFsu. (By x'(t) we mean the first
nf components of a solution of (1.23), for rjf running through Rn , and
satisfying the condition Pm,o{%0,v'it,0") + Pm-i{x°, t,0") = O.j
Theorem 1.5 will be proved in the sections 2, 3, 4, 5. For brevity we shall
limit ourselves to argue in the Gevrey case s < oo, taking u a (Schwartz)
distribution. The extension to the case s = oo and to Gevrey (ultra-)
microfunctions do not present any additional difficulties. In the remaining
part of this section, we fix some notations and terminology to be used in
the proofs. In all what follows k > 2 is a fixed integer as in the statement
of theorem 1.5. We assume henceforth that k = m and denote (k — l)/k by
8.
2. Further notations and preliminary properties.
Definition 1.6. a) Let R^ = Rf, x Rf,'t. A C R% X R7}, is called "quasiho-
mogeneous" if (x,£) G A implies (a:, //£', uk^k~l^ff) G A for any v > 0.
b) B C R™ X R7} is called a quasihomogeneous neighborhood of A, if B
contains an open quasihomogeneous set B which contains A.
c) A function (#,£) —> /(#,£) is called a-quasihomogeneous (or
quasihomogeneous of degree a) if f{x,v£', z//^-1)^") = vaf(x,(;), Vz/ > 0.
Actually, we should call functions as in c) of the preceding definition (1, k/(k—
1))- quasihomogeneous of degree a, but since k will be kept fixed throughout
this paper, we omit explicit reference to k. An example of a
quasihomogeneous function of degree 1 is for k = 4, n — 3, n' = 2, /(#,£) = (£f -Z^)1^
in the region £f — £| > 0 • Interesting quasihomogeneous sets in the context
of this paper are sets of form U X {£ G i2n;f" G G, |£'| < c|£"|5}, where
U C i?n, G is a cone in Rn and c > 0. In principle we are interested to
work here with c large. Since we will want to remain in a conic
neighborhood of R7},, this will make sense only if we restrict our attention to sets of
form |£| > c'.
There is another way to take advantage of the product structure of i??,
when we write it as R7}, x R7},,. This is based on the following definition:
Definition 1.7. Consider x° G R71, £° G R71 with £0/ = 0, and £n G R71'.
We say that V is a bineighborhood of (a:0, £n, £°) if it contains a set of form
U x {£ G iT;£ G G,£' G G"} where U is a neighborhood of x° in Rn, G is

14
an open cone in Rn which contains £° and G' is an open cone in Rn which
contains £n.
Bineighborhoods are objects of second microlocalization. It is perhaps
illuminating to look at second microlocalization as an instance of general
high order microlocalization: see e.g. [13]. It should then be said that
definitions are chosen to be more intuitive here than are the corresponding
definitions in [13]. The reason why we could stick to simpler definitions in
this paper is of course that we do not need to microlocalize further. (In
[13] the definitions were formulated in order to allow for rather high-order
microlocalization.)
As is often the case in microlocalization, we need to distinguish for some
given point (#,£) between the points (#,£) which are "close" to (#,£) in
RnxRn and points (#,£) which we consider "far" from (#,£). Sophisticated
metrics to measure distances are sometimes considered in the literature.
Here we will work with a rather simple metric which we denote by "disW.
Definition 1.8. We denote by dzsts,((#,£), (£,£)) the expression
di8t*(M)1(x1i)) = \x-x\ + \t-t\/{\t\S + \£f).
To explain the main idea of the proof of theorem 1.5, it is useful to
introduce the following generalization of WFS.
Definition 1.9. Let U C Rn be open, consider u G V'(U) andAcUxR71.
Also consider s > 1 with \/s <6<1. We write
AflWF> = 0 (1.24)
if for any compact set K C U there is e > 0, so that if p £ Gq(x £
i?n, |a: - x°\ < e), then there are c > 0, cf > 0, c" > 0, for which
\TxM{p{x - x°)u(x))(t)\ < cexphc'KI1/*] for |£ - £°| < c"|£Y,
provided x° G K, (z°,£°) G A.
One of the main features of this definition is that the "fibers" Ax =
{£'> 0^ 0 £ A} are not necessarily conic, and indeed, a first idea which comes
to mind in the situation from theorem 1.5 is to model them on the quasi-
homogeneous structure of Rn. Let us also mention that (z0,£0) ^ WFsu,
in the sense of definition 0.1, if and only if there is c > 0 and an open cone
G which contains £° so that {(a?,0; k - x°\ < c,£ € G}f) WF^u = 0. Of
course, (1.24) is interesting only if A is unbounded in the fiber-variables,
but it may well happen that the fibers Ax considered above are non-trivial
but bounded for some x and unbounded for some other x. In particular, our

15
present definition is slightly more general than the corresponding definitions
in [15], where we only considered the situation "(a;0 xG)fl WF^u = 0"
for some set G C i?J. The reason for which we generalize the set-up in
this paper is that with our present definition it is easier to see how wave
front sets are transformed under the canonical transformations which we
have to consider later on. It is easy (and in fact standard) to show that
[Ai U A2] H WFlu = 0, if it is true that A; n WFau„ = 0 for i = 1,2. Let
pm and pm-ifi be defined by (1.17) and (1.19) respectively; we denote by
W = {(X,0 G TlPnflM) + pm-lf0(M,O = 0},
and, as before, by
6=l-l/k = (k-l)/k.
Outside any quasihomogeneous neighborhood of W1, p is essentially elliptic.
The quantitative version of this is given in the following
Proposition 1.10. Let u G V'(U) be a solution ofp(x, D)u — 0 on Y, where
T is a conic neighborhood of some point (a?°,£°). Also consider some con-
stantc and some compact set K C U. Denote A\ = {(x,£);x G K, dist^((x,
0, W) > c}. Then it follows that Ax n WF^u = 0.
The proof of this proposition is rather straightforward. We shall give it in
section 2.
In view of proposition 1.10 it is now natural to distinguish essentially
two regions, Ai and A2, in the co-tangent space of the set U. Ai will be
a small quasihomogeneous neighborhood of W\ and we shall essentially
regard p{x, D) as an operator of principal type on Ai. In the region Ai we
shall apply one more microlocalization of a form which we shall explain in
a moment. Arguing then essentially as for operators of principal type, we
shall be able to show that in the assumptions from theorem 1.5, Ai does
not intersect WF^u. On the other hand, A2 will be the complement of
some still smaller quasihomogeneous neighborhood of W. On A2 we shall
thus stay away from W' and therefore p will be essentially elliptic on A2
as a consequence of the proposition 1.10. The main thing is now that if
Ai and A2 are chosen suitably, then their union will cover a set of form
{x; \x - x°\ < c} x H, where T' is an open cone which contains £°. We want
to stress here again the fact that neither Ai nor A2 will be conic in the fiber
variables.
We next describe how one has to microlocalize further within some
fixed small quasihomogeneous neighborhood of W. The idea is to take
advantage of the bihomogeneous structure of Rn. The main step in the
proof of theorem 1.5 will then be the following result:

16
Theorem 1.11. Let the assumptions of theorem 1.5 he satisfied. Also
consider £n € Rn ■ Then we can find a bineighborhood V of (z0,^'1,^0) and
c > 0 so that
{(*,0 G V\ \x\ < c, dist„({x,£),Wf) <c}n WFZu = 0.
To see which is the relation between theorem 1.11 and theorem 1.5, we
observe that it follows from theorem 1.11, if we also use a compacity
argument, that there are constants d > 0,c" > 0 so that if A' = {(#,£); \x\ <
c', disU((z,£), W) < c"}, then A'n WF^u = 0. The proof of theorem 1.5
is then concluded as described above.
Remark 1.12. When arguing on bineighborhoodsf one will automatically
stay away from the set {(#,£);£' = 0}, which is the characteristic variety
of our operator. That this is still relevant for the Gevrey-s wave front set
of the solutions of p(x, D)u = 0 comes from the fact that the whole region
{O^O'l^l < cl£"l5} Z5 taken care of by proposition 1.10.
3. We conclude this section with some preliminary comments on the
proof of theorem 1.11. Let us then fix some f'l G Rn . Without loss of
generality we may assume that £n has the form (1,0,...,0). To simplify
notations, we shall make a renotation for variables and n. We shall then
in fact replace n', n, with n1 + 1, n + 1, respectively £ with A and £'
with V, where V G Rn'+l is written as (r,f), r G R, ? G Rn'. Also
denote A0 G i?n+1, respectively An G i?n+1, the points corresponding
to £° and £n in the new notations. In particular, we may assume that
A0 = (0, ...,0,1) and that An = (1,0,..., 0). Since pm is transversally
elliptic with respect to A' = 0, x\ — 0 will be noncharacteristic with
respect to pmjo- If V is a small bineighborhood of (0, A'1, A0), and if Wj.$ =
{(x, A) G U X i?n+1; dists,((z, A), W7) < c}, then we shall be able to write
ponVn Wlc $ in the form p ~ go (r — J2 &j)> where q is a pseudodifFerential
operator which is elliptic on V in the Gevrey-s calculus and J2 bj is a formal
sum of quasihomogeneous symbols. The precise meaning of this will be
explained in section 3. In particular, the symbols bj will not depend on r. On
V fl W'c 5 and with respect to the Gevrey-s wave front set, p(x, D)u — Q will
then be equivalent to p'(x, D)u — 0 where p\x, D) is the pseudodifFerential
operator associated with the symbol r - £6j. The main thing is now that
we can conjugate pf on aset of form {(#,£) G V\ dist^((a;,£), W") < c}nV
with some Fourier integral operator so that after conjugation we are
working with a pseudodifFerential operator with principal symbol a and which is
defined on aset of form V'n{(y,0); \y\ < s, |(<t,t/)| < c|77"|5}, where V is a
bineighborhood of (0, (1, 0,..., 0), (0,..., 0,1)). Here we have denoted the
variables of the operator after conjugation by (y, 0), where 9 = (a, rj) and
77 = (7/, 77"). The characteristic variety of the new operator is then of course

17
a — 0 and propagation of Gevrey-s wave front sets for the new operator is
easy to establish. The idea of all this is of course well understood since quite
some time. In the present situation we have a number of technical
difficulties which come from the fact that we need to localize to bineighborhoods,
while in addition we have to remain in a quasihomogeneous neighborhood
ofW7.
2. The geometry of pm + pm-i = 0
1. Let (x°, f °) eRnxRn with £'° = 0 and a conic neighborhood T of (x°, £°)
be given. We assume that (a?,£) € T implies |£'| < d|£"| for some small
constant d > 0. Also consider a classical analytic symbol p ~ £j>m pj on
T so that pm(x,£) = £|a|=A;aa(£>0£/a with aa positively homogeneous of
degree 0 for (a:, £) G T. In the present and in the next section we assume only
(1.8), (1.9), i.e. that there are strictly positive constants ct-, i = 1,2,3,4, so
that
Cl|am<PmM)<C2im (2-1)
and denoting by
Pm,o(*,0= E ««M,CKto,
|a|=ra
Pm-l,o(»,0 =Pm-l(»,0,f,,)»
that
-csirr-1 < Pm~i(x,o < -c4\am-1- (2.2)
Moreover, denote by
W = {(x,0 G rjp^x.O +Pm-i,o(ar,OlO = 0}
and by
8=l-l/k = (k-l)/k.
(It is again no loss of generality to assume that k = m, as we shall most
often do henceforth.) In particular, W is quasihomogeneous in an obvious
sense and one can use quasihomogeneity to obtain part of the results which
we need below. Since we cannot rely on quasihomogeneity alone, we shall
prefer a more direct approach. The argument is very simple anyway.
2. Our first remark is that if d (in the condition |£'| < d\£"\) is small
enough and if \^e shrink T (if necessary), then pm$ dominates pm and pm_i,o
dominates pm-i. A more precise statement about this is

18
Remark 2.1. Fix V CC T. Then there are constants c$, c&, which do not
depend on d, so that
\Pm{x,0 -Pm,o(X'0\ < C5«f|pTO(x,^)|, (2.3)
and
\Pm-l(x,0 -pm-lfl(x,Z)\ < C5d\pm-i(x,(i)\, (2.4)
if ix,Q € T, |£'| < d\£"\. In particular, if c5d < 1/2, c6d < 1/2, then
Pm(x,4) < 2pmfi(x,£), |pTO_i(a;,OI < 2|pm-i,0(a:,0li for such (x'0-
Indeed, we have e.g., pm(x,® - pmfl{x,£) = 0(K'r+7KI) < ^d\Z'\m <
&2PmM)>if M) £ T' and |f| < d|£|.
Proposition 2.2. Let c > 0 be given. Then we can find d > 0, c" > 0,
c'" > 0, f, so that
|Pm,o(&,0+Pm-l,o(»,OI > <?(\PmflM)\ + \Pm-lfl(x't)\)' (2-5)
*/ (*,0 € f, |£'| < c"|£l «^ dwfc.((x,0, W) > c.
Remark 2.3. /£ zs ftere important that we can fix c arbitrarily small As a
consequence, also cf might become in principle rather small
Proof of proposition 2.2. (Beginning.) We choose a conic neighborhood T"
of (x°,f°) in (U X Rn") and c7 so that (*,£") G r", |f'| < c7\£"\ implies
M',£") G F. Let us in fact denote T' = {(*,£); (*,£") G T", |f'| < c7|£"|}-
If we shrink T", c7, suitably, we will have Tf CC T. We fix r" with this
property, but we will further shrink c7 if necessary. In particular, we shall
work from now on in the region |£"| < c7|£'| with c7 as small as needed.
Let us next fix (#,£") G Y" and denote
W{Xtill) = W G IT'lPrnfiMt") + Pm-l,oM,£") = 0}.
The greater part of our argument will be in the variables rjf G Rn and will
refer to the set W! t„y Let us in fact consider £' for which we assume
ti8t„{(x,t',t"),W)>C.
It follows of course that |f - rf\ > c(l + |£"|)5 for any rf <E WL^y
The following lemma is immediate:
Lemma 2.4. Iff CC T, we can find eg, eg, so that
a) \Pm,oM>Z")\ < (l/2)bm_1)0(a:,0,OI, Wl < <%\Z"\S, 1 e W^,,,,

19
b) \Pm-lfi(x,V',Z")\ < (l/2)bTOl0(x,0^")l. *fW\ > <*I*Y, V' G W^'T
/n particular, rjf G M^/ >„n implies
c8\n5 < w\ < cd\n5
and \rf\ < c8\^"\s, respectively \rf\ > c9|£"|5, both imply (2.5).
Proof. We prove a), the other relations are similar. We have that \pmfi(x, 77',
f')l < c2\rf\m < ctcpZV-1 < C2cZci1\prn_ho(x,0,?% so it suffices to
shrink c$ until C2Cgvc31 < 1/2.
3. Proof of proposition 2.2 (End). We study the function 0 ->• F(x, £, 0) —
Pm.oCM^'/K'h £") +Pm-i,o(a;,0,C") for 0 > 0. We want to show that
F(x,t,\?\) > c|£"\m&. We recall that (x,£") had been fixed and that £'
was chosen so that |£' - t/| > c(l + |£"|)5 for any 7/ G wL^"y We also
observe that in view of lemma 2.4 we are left with the case eg|^"|5 <
W\ < c9|£"|5, and in this region \pm_h0(x,£)\ < c4\Z"\m < cl3\?\m-\
so it suffices to show that F(z,£,|£'|) > c^'l"1"1 for c6\£"\5 < |£'| <
cgl^'l5 if If - V'\ > c(l + \S"\)S for any 7/ G W{x(.,,y Here we note that
F(x,£,0) = pm-i,oM,£") < 0, whereas for 0 > c9|£'f, F(z,£,0) >
0. It follows therefore that we can find 0° so that F(x,£, 0°) — 0. In
particular, |0°£7|£'| - £'| > c(l + |£"|)5- It follows that F(x,£,|£'|) =
(d/^F^.^S)^ - *°*7K'|| =^" W^/iaOl? " ^7IC'II> CM
0m-i(l + |^/|j« with § on the segment [0«, |£'|]. (Here we have used that
(d/d0)F(x,£,0) = (d/d0)(0™pmfi(x,e/\e\,Z").) Since 0° > c8\e'\5, this
gives F(z,£, Ifl) > c15|n(—^(1 + K"|)« > Cl6(l + K"|)*m.
Corollary 2.5. If d > 0 is sufficiently small, \p{x,£)\ < c(\pmp(x,£) +
Pm-i^O&iOl) /or 0^0 G5 zn proposition 2.2.
4. Before we now state our main result, we prove
Lemma 2.6. For
dist„iM),W) > c, (x,o e r',in > cio,
we have
\d:dVpm(x,0\ < c\a\+W+la\/3m-W\(Prn,o + pm-i,o)(x,0\ (2-6)
Proof. We consider separately the cases \/3\ < m and |/?| > m.
I. In the first case we use that
\daxdlpm{x,Z)\ < JoMft+iaWlt'r-W- (2-7)

20
In the region |£'| > \£\5 it follows that |f \~W < |C|_51^'- We also have
|£'|m < Pm,o(z>£)- Together with proposition 2.2 this gives (2.6) in this
case. If, on the other hand, |£'| < |£|*, then
since #ra = m - 1. We conclude the argument once more with the aid of
proposition 2.2.
II. For |/?| > m we have instead:
l^afpmOcfll < clal+^l+1a!/?!|er-|/31, (2.8)
since if we derivate /3 times a term aa(#, £)£'a, then at least |/?| — m
derivatives will have to act on aa. Also note that
m - |/J| = m - (1 - £)|/?| - £|/J| = m - |/J|/m - £|/J| < m - 1 - *|/?|.
It follows that
Kp-pi < cKr^i^-^^.o.ni < cicr5|/?liPm(x,o+ft»-i(*,oi.
for the (#,£) as in the statement.
Theorem 2.7. If d > 0 (in the condition |f'| < d\£\) is sufficiently small
and for the (#,£) from the statement of proposition 2.2, it follows that
I3?3?P(*,0I < ^^xa\m\~W \P(X,Z)\. (2.9)
Proof. In view of corollary 2.5 and the preceding lemma it suffices to observe
that
l<£fl?(P -Pm)(*,fll < c^"^1*!/*! |fl-W (1 + Kl)™-1.
Remark 2.8. We frcwe nott; proved theorem 2.7. The fact that proposition
1.10 is valid follows now if we combine theorem 2.7 with the results proved
in mi
3. Factorization of the symbol
1. Notation for variables is as in nr. 3. from section 1 and assumptions are
as in (1.15), with k = m > 2 an arbitrary integer, and (1.16), (1.20). If / is
s-quasihomogeneous, then (d/d\j)f is (s-l)-quasihomogeneousfor j < nf
and [s - k/(k - l)]-quasihomogeneous if j > nf. The difference between
the two degrees is l/(k - 1), which is one of the reasons why in this theory

21
it is natural to work with formal symbols of form J2j <lj w^h <lj quasihomo-
geneous of degree m-j/(k - 1). To some extent, quasihomogeneity will be
mixed up with bihomogeneity. Note that the variables A' will here
practically be given the weight 1 and the variables A" the (higher) weight k/(k-l).
A typical situation is when we regard terms of form (d/d\f)Pa(x, 0, A") A'7,
for some positively homogeneous function a of degree //. It follows that such
terms are |7| + (//-|/?|)fc/(fc-l)-quasihomogeneous. If we now consider some
/(#, A) which is positively homogeneous of degree // in a conic
neighborhood of (0, A0), then we can expand it as a formal sum of quasihomogeneous
symbols:
/(*, A) = ^2(d/dXT'f(x, 0, A")Ato/a!, (3.1)
a>0
the sum being actually convergent if A'/| A"| is small. In particular, /(#, 0, A")
is the quasihomogeneous principal part of /. In order to remain rooted in
a more direct way in the quasihomogeneous theory, we will now prefer to
work in situations when f(x,\) is a polynomial of order at most m in r.
This can be achieved with the aid of a well-established variant of the Weier-
strass preparation theorem for symbols. Let us assume in fact at first that
homogeneous analytical coordinates are chosen so that (after multiplication
with an analytic symbol),
pm(x,X) = Tm+ £ aa(x,X)X'a. (3.2)
\a\=m,a\<m
With the aid of the classical Weierstrass preparation theorem, we can
rewrite pm as a (pointwise) product
Pm(x,\) = V(x,A)[rm+ J2<Pj(x,Or'}, (3.3)
where the (fj and ij) are analytic and homogeneous of degree m - j (the
<Pj) and 0 (the \j)) respectively, and i\) is elliptic in a conic neighborhood
of (0,A°). Moreover, the Weierstrass preparation theorem gives that the
(fj vanish at A' = 0, but more is true, as can be seen if we compare
(3.2) and (3.3). Indeed, it follows that we must have that (d/dX1)1^™ +
Ylj<m<Pj{xi£)T*) ~ 0 f°r ItI < m and X — 0- ft f°U°ws from this that
dZ(pj(x,£) — 0 if £' = 0 and \j\ < m - j. After composition with the
symbol tj)~l from the left, we can now assume henceforth that pm has the
form
pm(x,A)= £ a/yfoOr^, (3.4)
\P\+J=m
with the coefficient of rm in (3.4) equal to 1. All this followed from the
classical form of the Weierstrass preparation theorem and (3.4) is of course

22
valid in a conic neighborhood of (0, A0). To continue, we have now to apply
the Weierstrass preparation theorem for symbols. It follows then that p =
P° \Y?jLo9j{xi£)T*]i w^h 9j classical analytic symbols, p elliptic. Again
this is a relation in a conic neighborhood of (0, A0). The coefficient of rm is
here of form (l+r) with r of order -1 and r is a symbol which does not
depend on r. We next compose from the left with (1 + r)~x op~x. It follows
then that we can assume that p is of form p — YljyoPm-j, with pm as in
(3.4) and the pm_j, for j > 1, analytic symbols which are polynomials of
degree at most m - 1 in r (and which are defined in a conic neighborhood
of (0,£°) ). We next expand all the p\ as in (3.1). We conclude that p can
be written as
3 7>0
where we have not yet ordered terms according to their degree of quasi-
homogeneity. The quasihomogeneous principal part of p is Pm,o(#>A) +
pm_i,o(s, A) where, as above, pmfi(x, A) = E|/?|+j=m a/?j(M,Or^//? and
Pm-i,o(#, A) = pm-i(x, 0,£"). More generally, denote by
Pm-iAx, a) = £ (d/dxypn-iMewhi.
As observed above it is quasihomogeneous of degree \y\ + {m-j-\y\)k/(k-
1) = m — j — |7|/(fc — 1) and is a polynomial of order at most m — 1 in r if
j > 1. This gives then that
r>0
where
crm_r/(fc_1) = J](9/W)Vj(*.0,f)W (3-5)
i,7
where the sum is for m - j - \j\/(k - 1) = m - r/(fc - 1) and, again,
am-r/(k-i) *s a polynomial of order at most m — 1 in r if j > 1.
2. Our main result is the following
Theorem 3.1. Let A0 G i?n+1 6e given with A0/ = 0 and denote A'1 =
(1,0,..., 0). Tftere are bj, qij, j > 0, i — 0,1,..., m - 1, with the following
properties:
- the bj and the qij are defined and analytic on a set of form {(^,C) £
U x C?; |C'| < c\C'\SX" ^ £"}> where U is a complex neighborhood of
0 in C™+1 and Gn is a complex conic neighborhood of A0",

23
- the bj are quasihomogeneous of degree 1 - j/(k - 1), the qij of degree
m-1- i- j/{k- 1),
— the bj and qij do not depend explicitly on r, and we have that
m-1
p(*,t;)~[Tm-l+ E E ^o^-^^-E^M)). (3.6)
i = 1 j > 0 i>o
(3.6) is here valid for (a:, A) with (x,£) in the real part of the domain of
definition provided that we also assume that \r\ < ci|£"|5 for some constant
c\ > 0, which we have chosen suitably previously. In particular we can
assume that it is satisfied on a set of form V n Wj$ where V is a bineigh-
borhood of (0, A'1, A0) and A'1 = (1,0,..., 0). Moreover, the bj are real for
j < J if the Pm-ij ore real for j < J and bo is determined by the following
two conditions:
b0(xAn = [-Prn-l(xAn]1/m, (3.7)
Pmfo(a?,Mar,0»0 + Pm-i(«,OlO = 0. (3.8)
The m-th root [a]l/m is considered real for real positive arguments a. When
m is even, we can consider the positive m—th root, or also the negative
determination of the m-th root. Recall that -Pm-i^O,^") > 0 for real
(#,£). Finally, there is a constant cf so that the bj satisfy the inequality
|6i(«,0l<^'+1i!(|€,| + ri')1"j/(*_1) (3-9)
on their domain of definition.
3. The proof of the theorem is standard but technical. The first remark
is that (3.7) and (3.8) determine an analytic function &o which is
defined on a set of form {(*,£) e U x Q1; |C'| < c\C"\*,C" e <?"}, which
is 1-quasihomogeneous there and which is real-valued for real arguments.
This is clear from the implicit function theorem. Once &o is found, we can
find qio quasihomogeneous of degree m — 1 — i, so that
m-1
Pm,o(z,A)+pm_1>o(z,A) = (T™-1+ E fcoMK^-'Kr-froM)),
t = l
(3.10)
the multiplication between brackets in (3.10) being pointwise.That this is
possible, follows either by direct division or by an application of the Weier-
strass preparation theorem. It is also interesting to note that qio{x, 0,£") =
[-pm-ijo{x,X)]t — blo(x,0,(;"), as is clear from the formula (rm~l+arm~2 +
h otm~l){r - a) — rm - am. In particular, we obtain that
m-1
bh(x,Z)+ £ (ftoOc.O&r^O^O (3.11)
1 = 1

24
in a suitable quasihomogeneous neighborhood of (0,0,£°).
Once we have found </;o and 60, we shall now construct the qij and 6j, for
J > 1? by an iteration. We claim in fact that it is possible to find 60,..., bu,
ftd • • -10U/, so that
m — 1 v v r
{rm~l+ £ E ^rm-l->{r- £ 6,) = £ am_r/(,_1)+ £ P,;,
t = 1 j = 0 j = 0 r = 0 j > 0
where the 0"m_r/(jfe_i) are fr°m (3.5) and the p^j are quasihomogeneous of
degree m - (// + l)/(fc - 1) - j(& - 1). For v — 0 this follows immediately
from the conditions on the 60, </;o- To argue by induction, we shall now also
show how one can find fr^+i, </t>+i, i = 1,.. .,ra - 1, if one has already
found 60,..., 6^, qip, • • •, <li,w> i = 1,..., m — 1. Actually, we obtain for the
&M-i, </t,M-i> among others, the following conditions, in which the 6j, <ftj,
j > // + 1, are not yet involved:
m - 1
t=l
m - 1
( £ ft,H-irm~1_1)(r- &o) = ^m-(i/+i)/(ib-i) + Pi/0- (3.12)
1 = 1
Here we recall that the 0"m_(„+i)/(jfe-i) and /^o are polynomials of degree
m - 1 in r. We can thus write
ra - 1
0"m-(i/+i)/(ife-i) + PvO= E dir^
i = o
Identifying coefficients of rr in (3.12) now gives
-&i/+10m-l,O - &O0m-l,i/+l = ^0,
_^+l9m-2,0 - &0</ra-2,M-l + <lm-l,v+l = ^l,
-&i/+l<7l,0 ~ &0<7l,H-l + 92,i/+l — rfm-2?
-&H-1 + 01,i/+l — dm-i.

25
This is an raxra system for the m unknowns &^+i, <7i,i/+i, ..., <Zm-i,i/+i and
the determinant is nonvanishing. Moreover, degrees of quasihomogeneity
are correct. One can however also solve this m x ra-system in a more direct
way. Indeed, it is possible to eliminate at first the <fc,i,+i, calculate 6^+i and
then calculate successively </i^+i, </2,h-i> • • •> </m-i,H-i- The elimination of
the <ft,„+i is done multiplying the i—th equation by (&o)t_1 and summing.
We thus get
-K+l(qm-lfl+qm-2flt>Q+qm-3fl1>l+' ' '+b™~X) = do+dl&0 + ' • •+rfm-i6J1"1.
The coefficient of bu+i is here different from zero in view of (3.11). We have
now seen how we can determine the 6j, </t-j, iteratively. The preceding
arguments give also easily the extensions to the appropriate complex domains
and the estimates required in (3.9).
Finally, let us comment on the fact that the bj are real-valued for j <
J < m - 2 if the pm-\j are real for j < J. Indeed, it is seen from the
construction that in the calculation of the 0"m_r/(jfe_i), r < //, pu, v < m,
we do not consider but the values of pmj and of the pm-i,j and that we do
not have to consider terms in the formula of composition of symbols which
have complex coefficients.
4. In the sequel we are mainly interested in the &o,..., bj. It is
worthwhile to mention that they are related only to pm and pm-\. This is due to
the assumption that pm_i(0,£°) / 0. We shall denote &0H \-bj by 6 and
J2j>jbj by &', quasihomogeneous of degree 1 - (J + l)/(fc - 1). Also note
that on a set of form VnWj. $, the equation p(x, D)u — 0 is equivalent with
p(x1D)u — 0, where p(x1D) is the pseudodifferential operator associated
with r - &(#,£) - &'(#,£). This shows then in particular that in some sense
and in some bineighborhood of (0, An, A0), W = {(&, A);r = &(#,£)} is the
"true" characteristic variety associated with p. It is the main content of
section 4 that we can choose canonical coordinates in which W becomes
flat and in which the principal part of the operator p becomes -i(d/dt),
where we have written t for the variable x\. In particular, p will transform
after conjugation with some F.I.O.'s associated with this canonical map to
the operator -i(d/dt) + r, where r is a pseudodifferential operator with
symbol in Si-(-J+i)/(*-i).
Proposition 3.2. Any quasihomogeneous neighborhood ofW is also a quasi-
homogeneous neighborhood of W and viceversa.
Proof. This is based on the fact that W is parametrized by (a:, 6o(^»0»0»
whereas W is parametrized by (a?, &(£,£)>0- ^ P — O^&oO^OiO an(^
Q — (a:, &o(#,0»0 correspond to the same parameters (a:,£), then \P-Q\ —

26
\b(x,£) - 6o(x,OI < Ej=i Mx,t)\ < c(|e| + K"|V-2/*. The proposition
now follows easily.
4. The phase function
1. The notations for phase variables are as in the sections 2 and 3. Also
assume again that A0 = (0,.. .,0,1) G Rn+l and fix A'1 G Rn'+l.
Without loss of generality, we shall assume that A'1 = (1,0,..., 0) and we shall
work in a bineighborhood of (0, A'1, A0), while still remaining in a quasiho-
mogeneous neighborhood W'c8 of W\ where W is the quasihomogeneous
characteristic variety associated with p. Here p is of course an operator
which satisfies the assumptions of the preceding section 3. What we want
is to find a canonical transformation x, so that x_1 lives on V n Wj.$ and
so that in the new canonical variables, p has a very simple form: see (4.1)
below. It will be possible to choose the canonical transformation in such a
way that the x-variable associated with the phase-variable r remains
unchanged. We denote it by t, changing notation of the ^-variables in Rn+1
from x to (t,x). Thus x £ Rn and (t,x) G i?n+1. Let us also consider
another set of canonical variables which we denote by (t,j/, <r, 77). Sometimes
we shall also write 9 for (a, 77). What we want is then to find a canonical
map x which maps (i, j/, 8) ->■ (t, x, A), and for which
{T-b{t,x,Z))oX(t,y,6) = a, (4.1)
where b is from section 3. We thus recall that b — 60 + &i + &2 H h bj is
real-valued, and that we had
p~(rTO-1+E?«(*.«.0'-m"1'>('--E6i(*.as.O)-
hi i
As for estimates, we have that the bj satisfy for j > 1
l<&$?W*,*,OI < c|a|+l/?l+j+1«!/?!j!(|£'! + K»|y-i/<*-i>.
One of the main problems in the following, is to keep control of the domains
of definition of our canonical transformations. Since we have some modest
freedom in the choice of x, we shall at first fix conditions in the 77-variables.
We shall in fact at first fix some rj e Rn with ff = 0. We shall then start
our construction for \r] - fj\ < c\fj\s. As is standard, we shall also want to
dispose of a generating function u;(i, x, a, rj) for x- The requirements for u
are thus:
u>t(t, x, a, 77) - 6(t, x, V^(t, x, a, 77)) = a,
to which we add the initial condition
o;(0,x,<7,7j) = (a:, 77).

27
(Here ut — (d/dt)ip.) It is natural to look for u in the form u;(£, #, a, rj) —
to + u)(£, a:, 77), with u satisfying :
ut(t, a, rj) - 6(t, x, V^(i, x, V)) = 0, (4.2)
o)(0lxli/) = (a:li/>. (4.3)
As an additional condition for u we ask for u £ S*, ji{rj) — (|r/| + \r)"\5),
starting from second derivatives.
One can solve (4.2), (4.3) with the aid of the method of bicharacteristics.
The only problem is with the size of the domain on which we can solve our
equation and with the estimates. How these difficulties can be circumvented
is described e.g. in the paper of [15] (in which the authors started from [8]).
The bicharacteristic system associated with (4.2) admits t as a natural
parameter. It is
t = t,
dx' Ob
-^(t,y,TJ) = --jj-(t,x(t,y,rj),Z(t,y,rj)), (4.4)
AC fiU
dt&y,r,) = far.&x(t>y>ri)Mt>y>ri))> (4-5)
A rk\\
^■(t, y, ^ v) = ^(*. *(*, y, v),$(t, y, v)), (4-6)
to which we add the initial conditions
*(0, y, rj) = y, £(0, y, rj) = r/, r(0, y, r/) = a. (4.7)
It is of course due to the initial conditions that t —)■ (#,£, r) depends also
on (y, a, 77). Note that the equation (4.6) is not coupled with the equations
(4.4), (4.5), so we can at first study (4.4), (4.5) with their respective initial
conditions and then solve (4.6), together with the last condition in (4.7)
in the end. The first step in the argument is now to show that the system
(4.4),(4.5),(4.7), admits a solution for {t £ C, |t| < c} with c independent
of (y,a, rj). How this comes about is described e.g. in [15]. Let us then
denote by X(t, y, 77), S(t, y, rj) the solution of (4.4),(4.5),(4.7). It follows as in
[15] that \X(t, y, rj) - y\ < c\t\, |S(i, y, rj) - rj\ < c\t\\fj"\5. We want to solve
here X(t, y,rj) — x for y to get y = Y({t,x,rj). That this is possible for
small fixed t and for rj £ {rj £ Cn; \q - fj\ < c\rj\8} follows from the implicit
function theorem in view of the fact that (dX/dy)(0,y,rj) — J, together
with the fact that d2X/dy2 = 0(t). (Also cf. here section 3.4 in [15].) The
next thing is to consider the function L(t,x,rj) — E(t,Y(t, x, 77), 77), which

28
associates to the a;—component on some bicharacteristic curve the
corresponding ^-component. The Hamilton-Jacobi theory gives that L(i, a;, 77) =
Va?cD(t, a:, 77). We have thus a rather explicit formula for V^u), which shows
that V^o; G S*. We can now also recover u from the relation
u(t,x,rj) = I b(s,x,rj)ds.
Jo
We conclude thus that we can define a), and therefore also V7, for (£, a:) small
and |?7-7/| < c|f/|5. We recall here that the property of 77 was that ff — 0. The
initial condition for u at t — 0 however did not depend on 77 and u is locally
uniquely determined by the initial condition and is analytic. It follows that
actually u can be defined on a set of form {(£, a:, 77) G C™£1 X Cn; |(t, a:)| <
c, |r/| < c|r/,,|^, 77" G G"}, where G" is a complex conic neighborhood of A0//.
2. To see which is the image of the canonical transformation associated
with the generating function, let us at first analyze which is the image
of points of form (0, x, 0,77) under the map (t,a;,a, 77) —> T(£,a;,a, rj) —
{t,x,tyt{t,x,(j,rj)^\tTxu(t,x,(T,rj)). When t — 0,<r = 0, we have in fact
u>t(0,a;,0,77) = b(0,x,Vxil;(0,x,0,r])) and Vxu(0,x,(T, rj) — 77. Thus the
image point is (0, a:, 6(0, a:, 77), rj) which is, as expected, a point on the
characteristic variety W. (Also the points (t / 0, a:, a — 0,77) are mapped to the
characteristic variety, but the image point is characterized in a less direct
way.)
To obtain quantitative conclusions from this, we look at
d2u
&,. x,w x (*, x,a,rj) = I + R{t, x, a, 77),
d{t,x)d(a,rj)
(the left hand side is the mixed Hessian of ^; the first row is [d2u/(dtd(r),
d2u/(dtdrn), ..., d2u/(dtdr)n], and the first column [d2u/{dtda),d2u/
(dxida), ..., d2u/(dxnda)],) where R is a matrix with entries in 5°.
Moreover, i?(0, a:, a, 77) = 0, so we have that
d2w , 1 r , A ^
5MSM(',*''-')-5/ (4'8)
if \{t,x)\ is small and / is the identity (n + 1) x (n + l)-matrix. As a
consequence of this is also easy to see that T is injective on sets of form
A1 = {P = (£, x, a, 77); dist^(P, P) < c} if P — (£, 5, a, 77) is fixed and that
T(Af) contains Bf = {Q = (t, a, A); dist„(Q, T(P) < c'} if d is sufficiently
small.

29
We also claim that T is globally injective if \t\ is small. In fact to
begin with, let us note that T is clearly injective in the variables (t,x).
If then, in addition ^(i^tr1,!/1) = ij;t{t,x,a2,r]2), Vxu(t,x,al,r]1) =
Vxu(t,x,a2,r]2), then we have at first rjl + 0(t)\rjl\5 = rj2 + 0(t)\r]2\5,
so \rjl - rj2\ < c(\r]l\ + \rj2\)5, with c as small as we please if t is small
enough. But, as observed above, on such sets T is injective, so 771 = rj2. It
then also follows immediately that a1 — o2.
Having seen that T is injective, we want next to find a lower bound for
the image of T when applied to the set
A = {(*, z, <r, V); \(t, x)\<e, \a\ + \V'\ < c\V"\s,rj" € G"}, (4.9)
where G" is a conic neighborhood of (0,..., 0,1) G Rn . It is clear in fact
that T(A) contains a set of form
B = U[{(0, x, 6(0, x, t;); |r/| < c\r,"f, V" € G'%,S. (4.10)
On the other hand it is not difficult to see that B itself contains a set of
form V fl W'cfi with V a bineighborhood of (0, (1,0,..., 0), A0).
We also need to perform similar considerations for the map (£, a:, a, rj) —>
5(t, a:, a, r/) = (t, V^a)^, #, a, r/), a, r/), which we consider again as a map on
{|(£,&)| < £> lal + I7/'! < clr/,,|^}- Once more, S is injective on sets of type
A1 and S(Af) contains a set of type Bf. To understand the global behavior
of 5, we note that for t = 0, Vvu(t^ a:, a, 77) = #, so S is the identity when
t = 0. Arguing as above, we see that 5 is injective on the set A defined in
(4.9) and that S(A) contains a set of form
D = {{t, y, a, 77); \(t, y)\ < e\ \o\ + \v'\ < c'W'f}. (4.11)
We can now consider, finally, the canonical transformation x associated
with u. In the notations above it is x = TS~l. It is thus defined at least
on D and x(^) contains a set of form V n Wc8.
5. Fourier integral operators and proof of theorem 1.11
1. We consider in this section F.I.O.'s associated with the phase function
u constructed in section 4. They will be of the following two types:
Au(t, x)= [ eiuj^x^a(t, x, 0)u(0) dO, (5.1)
B*v(t,y)= l /e^+t(^>-^^ (5.2)

30
Following [15], we call A, respectively 5*, F.I.O.'s of the first,
respectively the second kind associated with u. The integration in the 0-variable
is here on a set of form {(a, 77); |a| + |r/| < c\r}"\8,77" G G"} where G" is a
conic neighborhood in Rn of (0,..., 0,1).
The reason why we denote operators of the second kind with a V
comes of course from the fact that the formal adjoint of an operator of
the first kind is an operator of the second kind. We are interested in the
study of the operator B*p(t, a:, Dt, DX)A, where p is the pseudodifFerential
operator associated with the symbol r - &(£,#,£), and in the study of the
mapping properties of the operators A and B*. Actually, we can obtain
the main informations which we need from [15], the only difficulty being to
describe the setting in which these results have to be applied.
2. We shall apply here the theory from [15] for the weight function
(p(a, rj) = \a\s + \rf\5 + \rf'\. Sometimes we just write <p = ^(77), since we are
working in the region \a\5 < \r]"\ where \a\s + \rf\8 + \rj"\ ~ \rf\5 + \rj"\.
Starting point in our considerations is that the phase function u is in S£.
This phase function is not homogeneous and it is not possible in general to
work with homogeneous versions of wave front sets. However, if we localize
in the phase variables to sets of form {77; \rj - fj\ < c\rj\s}, then u has the
right behavior.
We want to check next that u satisfies the conditions needed to apply
the results from [15]. We need therefore to consider the conditions called
"B,C,D" and the so called "^-compatibility" introduced in that paper.
Condition B. For the situation at hand it comes to <p(VtlXu(t, a:, a, 77)) ~
y>(a, 77) on the sets on which we work. Here Vt^u;(£,a;,a, 77) = (a + b(t,x,
Vxu(t,X,TJ)),Va£)(t,X,TJ)).
Condition C. Condition C asks for the fact that \rj - rj\ < ci\rj\s implies
\Vxu(t°, x°, 77) - Vxu(t°, x°, 77)1 > c2(\r]\5+\7j\5). Also this is clear for t = 0
and will then be valid for t small.
Condition D. For any Xf CC X and for any c > 0 there is c2 so that
\ue(z,e) - ue{z,0)\>c2 if \x - xf\ > ci, z,zf ex'je r.
Let us verify also this condition: uo(z, 9) - u>o(z!, 9) = (t - tf, x - xf +
0(t)uri(z, rj) + 0(tf)ur}(zf, rj). If here \t - tf\ > ci/C, for some fixed large C,
then we are already o.k. If \t - tf\ < c\/C and t is small, also tl is small. By
choosing C conveniently, we will have then that \x - x'\ > c\/Cf.
From [15] we then obtain
Theorem 5.1. Assume that A n WF^u = 0. Then T(A) n WFlAn = 0.

31
Theorem 5.2. Let A be such that T(A)fWF> = 0. Then Af)WFiB*v =
0.
Also note that the operator q = B*pA is pseudodifferential. The symbol
calculus gives that it is associated with the symbol pox where p is the
symbol of p and x ls the canonical transformation associated with u. After
composition with an elliptic operator, it follows that the symbol of q is
a + r, where r is a symbol in Sl~(J+l^(k~lh
3. Proof of theorem 1.11. First we apply assumption (1.21). Since the
Pm-ij are real-valued for 0 < j < J = k - 2 it follows from theorem 3.1
that the bj are also real-valued for the same j. If A and 5* are defined
as before, the operator q(x,D) = B*pA has symbol a + r, where r £
S°. Using a standard argument (see for example lemma 4.1.5 in [15]), we
may then further reduce ourselves to the case when r — 0, by composing
everything with an appropriate elliptic operator. We are in this way reduced
to propagation of singularities for the operator Dt, and that is trivial. We
obtain in particular from the theorems 5.1 and 5.2 that for every solution u
of Dtu-b(t, a;, Dx)u = 0 the set WF^u is given by x(A), where A is an union
of lines which are parallel to the t-axis in the manifold a — 0 or, equivalently,
that WF^u is invariant under the action of the Hamiltonian flow from (4.4),
..., (4.7). At this moment, to relate this with the assumptions of theorem
1.5, we introduce the map x, defined by means of the reduced Hamiltonian
system:
^ (t, y\ y", v\ v") = - |r (*, *'(*, •, v", vf, v"), y", ?(*, y', y", v\ v"), v")
(5.3)
%t, y', y", rf, r,") = |£(t,x'(t,y',y", V', n"),y",£'(t,y',y", n', n"), n")
(5.4)
where &o is the principal part of 6 and we add the initial conditions:
*$(0, y', y", r,', r,") = y],, g(0, y', y", i/, v") = rfj, (5-5)
j = 1,..., n'. The solutions x'fi, y', y", rf, i/0» £j('» V'»J/"» v\ v") are quasi-
homogeneous of degrees 0 and 1 respectively. We can now give the definition
of x:
X(t, y', y", i/f O = (t, x'(t, y', y", t/, t/'), y", f (t, y', y", ,/, ,/'),
£>,y',y'w,<W)

32
where f is quasihomogeneous of degree 1 and is computed by solving
^(', y', y", v', v") = ~(t, x'(t, y\ y", v\ ,/'), y", |'(t, y\ y", v\ */'), n")
(5.6)
f'(0,y',y",r1,,r,") = a. (5.7)
From theorem 4.2.6 of [15] we have that the preceding statement of
propagation remains valid if we replace x by X- The proof in [15] is based on
the remark that, because of the quasihomogeneity of 6, the first n+ 1
components of x - X have strictly negative degree, whereas the second n + 1
components have degree < 1. It follows then easily that WF^v D x(A) = 0
if and only if WF^v n x(A) = 0; cf. definition 1.9.
The discussion of domain and image of x is similar to that for x; we
have in particular that x is defined at least in a set D of the form (4.11),
and x(D) contains a set of form V fl Wfc$. It remains to observe that the
solutions of (5.3), ..., (5.7) are exactly the solutions of (1.23), once we
return to the initial notations, i.e., write x instead of (t,x) and impose
initial data xf0 instead of (0,j/') and rjf instead of (a, 77'), with parameters
y" = x°",r]" = £°", p° = (x° = {x0f,x°"), £° = (0,£°") as in theorem 1.5.
In fact, from (3.8) we have with the present notations
Pmfi(t, *'(«), x°", b0(t, x\t),x°", mo, T(t),m,f")+
Pm-lfi(t,x>(t),xo",e") = 0;
differentiating we obtain, with U(t, x', r,£') defined as in (1.22):
^ = -(9Tn)_1(^;n + dxrpm-ifi):
^ = -(drn)-Hw + dtPm-lfi),
|| = -(fcnr^n)-1.
Observing that (drU) ^ 0 in the domain of r-&o, we are reduced to (1.23).
The proof of theorem 1.11 is now concluded by fixing £n G Rn and
considering the corresponding curve j(t) = (t, xf(t), #°"; 0,0,£°"). We know
from the assumptions in theorem 1.5 that there is i for which p = 7(f) ^
WFsu. This means that we can take a sufficiently small conic neighborhood
f of p such that f fl WF^u = 0. We now choose a bineighborhood V of
(^°^/1^°) and c > 0 so small that if we denote (with our initial notations)
B = {(x,0 G V; \x\ < c, dist „((s,0, W") < *},

33
we have x(B) C f. Therefore x(J3) n WF^u = 0 and, by propagation,
B fl WF^u ^ 0. This concludes the proof of theorem 1.11.
6. Characteristic manifolds of codimension 1
1. In the case when the characteristic manifold E has codimension nf = 1,
we are able to give more precise results. As before we argue in a small conic
neighborhood A of a point p° in E. The assumption (0.26) reads for nf = 1:
- In A we may write
Pm{x,Q = em-k(x,Z)(a(x,Z))k, (6.1)
where em-k(x,£) is an elliptic symbol, homogeneous of order ra-A;, and
the first order term a(a?,£) is real-valued and of microlocal principal
type, i.e., dx^a(x^) never vanishes and is not parallel to Y?j=i€jdxj
onS = {(x,0€A;o(x,0 = 0}.
As for the subprincipal symbol, we shall begin by assuming
l0{p)^0 for all^GE. (6.2)
Observe that E trivially satisfies definition 1.1 in the present case. The leaf
through po reduces to the bicharacteristic strip 70, i.e. the integral curve of
Ha on E, through the same point. As for the geodesies through po defined
in section 1, they all coincide with 70. After multiplying P = p(x,D) by
an elliptic factor, which does not change the validity of (6.2) and of the
properties we want to study, we may assume that
with a(x,£) as in (6.1). In particular, we may assume without loss of
generality that
Pm{x,£) is real-valued and, when k is even, non-negative. (6.4)
Let us introduce the following condition, where K may run over 0,1,..., k —
1:
Io{p) is real-valued and, when k is even, negative for p £ E;
moreover Ij(p,X) defined in (0.35) is real-valued for (6.5)
(/>, X) e iV(E) if 1 < j < k - 2 - K.
When K = k - 1, it is understood that there is no assumption on the lower
order terms (also (6.2) can be omitted); when K = k - 2, the assumption

34
only concerns Io(p). When K = 0 all the invariants in (0.35) are supposed
to be real-valued, i.e. (1.10) is satisfied. An equivalent expression for (6.5)
in terms of p'm-i{x,fi) (see (0.28)) is the following:
dx3?Pm-i(*.0 is real-valued on E for |a| + |/?| < k - 2 - K\
in addition, when k is even p^-iO^O is negative on E. (6.6)
Theorem 6.1. Let (6.1) ,(6.2) ,(6.4) ,(6.5) be satisfied for some K. As-
sume 1 < s < k/K (ifK = 0, then 1 < s < ooj. Then, possibly after shrink-
ing A, and writing 70 for the bicharacteristic strip through po = (^°,^°)
restricted to A, we have:
i) There is v G Ms(A) with Pu = 0 and WFsu = 70.
ii) Ifu£ MS(A) satisfies Pu = 0, then p0 G WFsu implies 70 C WFsu.
Hi) For every v G MS(A) there is u G MS(A) such that Pu = v.
Remark 6.2. In the case K = k (when we have no assumptions on the
lower order terms), the theorem was already known to be valid; see [1],
[25]; obviously, ii) is a particular case of theorem 1.2. In the case K = 0
(all the Ij are real-valued) the result was proved for s = 00 by Tulovskii,
[26]; the statement was however less explicit than in our theorem 6.1; as for
ii), it corresponds to theorem I.4 when n1 — 1. Concerning the case k — 3,
related results in Gevrey classes were obtained by Bernardi-Bove, [2].
Theorem 6.1 does not give information on the behavior of P = p(x,D)
in MS(A) for k/K < s < 00. In fact, for these values of s the invariants
Ik-\-K, •••> h-2 play an important role. Let us in fact fix attention on
Ik-i-K, the first not real-valued invariant, and introduce for K — 1,..., k-
1 the new assumption:
Imh-x-K^X) ± 0 for all (p,X) G AT(E),X ^ 0. (6.7)
(In the case K = k - 1, we set Imlo(p) ^ 0 for all p G E, or else, if k is
even, I0(p) £ i?_ for all p G E.)
Theorem 6.3. Let (6.1),(6.2),(6.4),(6.5),(6.7) be satisfied. Then we have
for k/K < s < 00:
i) P is s-micro-hypoelliptic, i.e., WFsPu = WFsu for all u G MS(A).
ii) For every v G MS(A) there is u G MS(A) such that Pu = v.
In the case s = 00, the result was proved by Tulovskii, [26], under
assumptions which can be proved to be equivalent to our assumptions. For
K = k - 1, theorem 6.3 is a particular case of theorem 1.3.

35
A model operator for the theorems 6.1 and 6.3 is given by the partial
differential operator in R% with analytic coefficients:
P = D* + cd(sly)Dj-1 + ^^
+ Ck-i(x, y)Dkx~l + terms of order k - 2. (6.8)
We may fixl = ^ and then obtain Ij = Cj(x, y)rjk~l~i, 0 < j < k - 2.
The conditions (6.2), (6.5), (6.6) can then be expressed in terms of the
coefficients Cj.
2. Proof of theorem 6.1. Arguing as in the sections 2, 3, 4, 5, cf. in
particular the end of section 3, nr.4, we are reduced to prove i), ii), iii) in
a small conic set T for the pseudodifferential operator with symbol
r + r(f,x,r,£),
where the variables are here (t,x) G i?n+1, (r,£) £ i?n+1; the bicharacter-
istic 70 is now the parallel line to the f-axis through po — (£°,x°,0,£°). In
view of (6.5), the symbol r belongs to SM-J+iJA*-!) with J = k-2- K,
that is r £ SK^k~l\ More precisely, from theorem 3.1 we have that
r (t, x, r, 0 = f(t, x, 0 + r0(t, x, r, f) (6.9)
where r^ £ S° and f does not depend on r. Observe now that, under our
assumption n' = 1, all the £-variables have the same weight k/(k - 1);
this means that f can be regarded as a classical analytic symbol of order
a — K/k. As for r0, it is also a classical symbol, with asymptotic expansion
involving the r-variable. At this moment we apply the following proposition.
Proposition 6.4 With the preceding notations, let P = Dt + R, where R
is a classical analytic pseudodifferential operator in T of order < a with
0 < a < 1. Assume moreover that 1 < s < \jo. Then there are two linear
maps Q,Q': MS{T) -+ MS{T) such that :
- Q,Q' are s-microlocal, i.e., WFsQu C WFsu and WFsQ'u C WFsu
- QQ' = Q'Q is the identity in MS{T).
- Q'PQ = Dt in T.
For a proof of proposition 6.4 we refer to [25], where Q, Qf are constructed
as pseudodifferential operators with symbols of infinite order. We are hence
reduced to study the operator Z^, for which the assertions in the statement
are trivial. Theorem 6.1 is therefore proved.
Remark 6.5. A natural question is whether theorem 6.1 extends to the case
of codimension nf > 1. An essential step in a prospective proof modeled on

36
the arguments above would be a version of proposition 6.4 for quasiho-
mogeneous symbols; this in turn would depend on a generalization to the
quasihomogeneous case of the infinite order calculus in [25].
Proof of theorem 6.3. Arguing as in the proof of theorem 6.1, we are
reduced to consider in T a classical analytic pseudodifFerential operator P
with symbol of form
p(t, x,t,£) = t + rK/k(t, a, f) + r(K_l)/k(t, x, r, f), (6.10)
where r^/k IS analytic, homogeneous of order K/k, with
ImrK/k(t,x,t)j:OfoT(t,x,0,t)er (6.11)
and r(x_iyk is a classical analytic symbol of order (K - l)/k. Writing
y = (t, x), T) = (r,£), we have for large |?/| and suitable positive c,C:
\p(y,V)\>c\v\K/k, (6-12)
\DayD^p(y, 17)1 < cM+W+1aW\p(y, v)\(l+ \rj\)-K^k. (6.13)
The proof is similar to that of theorem 2.7 and we omit the details. From
(6.12), (6.13) we deduce the existence of a s-microlocal inverse of P, acting
on Ms(r) for k/K < s < oo. This gives theorem 6.3.
The theorems 6.1 and 6.3 leave open the case when ImIk-i-K{p,X)
vanishes at p = />o, but J*;-i-k(/>, X) is not real-valued nearby. Let us only
observe here that in this case much depends on the behavior of Ik-\-K along
the bicharacteristic strip 70, as indicated by some results in the literature
on (non) hypoellipticity and (non-) solvability: for k = 2, see [7], [16], [19],
[20] when s = 00 and [5] when 2 < s < 00; for k > 2 see [21] and previous
works quoted there, [22], [23] when s = 00 and [6] in the Gevrey case. A
representative model for the result of Corli, [6] is the operator of form
P = Dkx + coDj"1 + • • • + ck_2_KDkx-2-KD«+1 + ixD'-^D^ (6.14)
which is proved to be non-solvable (locally) for k/K < s < oo, under the
assumption that Co, ci,..., ck-2-K are real constants with c$ ^ 0. Note that
from theorem 6.1, iii), we have that P in (6.14) is microlocally solvable for
1 < s < k/K, since (6.5) is satisfied.
References
1. Aoki T.: Calcul exponentiel des operateurs microdifferentiels d'ordre infini, I,II.
Ann. Inst. Fourier Grenoble, 33 (1983), 227-250 and 36 (1986).

37
2. Bernardi E.-Bove A. : Propagation of Gevrey singularities for a class of operators
with triple characteristics 1,11. Duke Math.J., 60 (1989, 1990), 187-205, 207-220.
3. Bony J.M.-Schapira P. : Propagation des singularites analytiques pour les
solutions des equations aux derivees partielles. Ann. Inst. Fourier Grenoble, 26 (1976),
81-140.
4. Boutet de Monvel L. : Propagation des singularites des solutions d'equations
analogues a Vequation de Schrodinger. In "Fourier Integral Operators and Partial
Differential Equations", Lecture Notes Math. Springer Verlag, vol. 459, ed. by J.
Chazarain, (1975), 1-15.
5. Corli A.: On local solvability in Gevrey classes of linear partial differential operators
with multiple characteristics. Comm. Partial Differential Equations, 14 (1989), 1-25.
6. Corli A.: On local solvability of linear partial differential operators with multiple
characteristics. J. Differential Equations, 81 (1989), 275-293.
7. Egorov J.V.: On solvability conditions for equations with double characteristics .
Dokl. A.N. SSSR, 234 (1977), 280-282; Soviet Math. Dokl., 18 (1977), 632-639.
8. Grushin V.V. - Sananin N. A.: Some theorems on the singularities of solutions of
differential equations with weighted principal symbol. Math. U.S.S.R Sb., 32 (1977),
32-44.
9. Kessab A.: Propagation des singularites Gevrey pour des operateurs a car-
acteristiques involutives. Tese, Universite de Paris-Sud, Centre d'Orsay, 1984.
10. Lascar R.: Distributions integrales de Fourier et classes de Denjoy-Carleman.
Applications. C.R.Acad. Sc. Paris, 284, Ser. A (1977), 485-488.
11. Lascar R.: Propagation des singularites des solutions d'equations pseudodifferen-
tielles quasi-homogenes. Ann. Inst. Fourier, (Grenoble), 27 (1977), 79-123.
12. Lascar R. : Propagation des singularites des solutions d'equations pseudo differ-
entielles a caracteristiques de multiplicity variables Lecture Notes Math., vol. 856,
Springer Verlag, 1981.
13. Liess O.: Conical refraction and higher microlocalization. Lecture Notes Math., vol.
1555, Springer Verlag, 1993.
14. Liess O.-Rodino L. : Inhomogeneous Gevrey classes and related pseudodifferential
operators. Boll. Un. Mat. Ital., 3-C (1984), 233-323.
15. Liess, O.-Rodino L.: Fourier integral operators and inhomogeneous Gevrey classes.
Annali Mat. Pura ed Appl., (IV) vol. 150 (1988), 167-262.
16. Menikoff A.: On hypoelliptic operators with double characteristics. Ann. Scuola
Norm. Pisa CI. Sci, Ser. IV, (1977), 689-724.
17. Parenti C- Rodino L.: A class of pseudodifferential operators with involutive
characteristics. Unpublished manuscript.
18. Parenti C- Segala F.: Propagation and reflection of sigularities for a class of
evolution equations. Comm. Partial Differential Equations, 6 (1981), 741-782.
19. Popivanov P.R.: On the local solvability of a class of pseudodifferential equations
with double characteristics. Trudy Sem. Petrovsk., 1 (1975), 237- 278; transl. Am.
Math. Soc. Transl, 118 (1982), 51-90.
20. Popivanov P.R. : Microlocal properties of a class of pseudodifferential operators
with double involutive characteristics. Banach Center Publ. vol. 19 (1987), 213-224.
21. Popivanov P.R.- Popov G.S.: A priori estimates and some microlocal properties
of a class of pseudodifferential operators. C.R. Acad. Bulg. Sci., 33:4 (1980), 461
-463.
22. Roberts G.B.: Quasi-subelliptic estimates for operators with multiple
characteristics. Comm. Partial Differential Equations, 11 (1986), 231-230.
23. Roberts G.B.: A necessary condition for the solvability of certain operators with

38
multiple characteristics Comm. Partial Differential Equations, 14 (1989), 877-929.
24. Rodino L.: Linear partial differential operators in Gevrey spaces, World Scientific
1993, Singapore.
25. Rodino L.-Zanghirati L.: Pseudodifferential operators with multiple
characteristics and Gevrey singularities. Comm. Partial Differential Equations, 11 (1986),
673-711. .
26. Tulovskii V.N. : Propagation of singularities of operators with characteristics of
constant multiplicity. Trudy Mosc. Mat. Obsc, 39 (1979); Trans. Moscow Math.
Soc, (1981), 121-144.

GEVREY AND ANALYTIC HYPOELLIPTICITY
DAVID S. TARTAKOFF
Department of Mathematics
University of Illinois at Chicago
851 S. Morgan St., m/c 249
Chicago Illinois 60607-7045, U.S.A.
e-mail: dst@uic.edu
Abstract. In these lectures we study sharp (non-isotropic) Gevrey (and
analytic) hypoellipticity for partial differential operators P which are
constructed as variable coefficient quadratic polynomials in real vector fields
satisfying the Hormander condition and which satisfy a maximal estimate.
We also present some new sharp results obtained jointly with A. Bove.
1. Introduction
In the early 1960's, J.J. Kohn introduced the d- Neumann problem as an
important tool for solving d on strictly pseudo-convex domains in Cn [1].
The C°° regularity of the solution was shown in [26], cf. also [27], [24],
using essentially the subellipticity of the problem: for strictly pseudo-convex
domains, in the quadratic form formulation there is a loss of one half
derivative in the a priori estimate. Reduction to the boundary (e.g., via pseudo-
differential opertors,) leads to a (pseudo-)differential equation of the form
of □&, whose prototype, in turn, is the celebrated 'sum of squares' operator
Y^=\ X] where the Xj are real vector fields generally assumed to satisfy
the 'Hormander condition' that their iterated brackets span the whole
tangent space. It had been conjectured and hoped that these problems would
turn out to be analytic hypoelliptic - that with locally real analytic data,
the solutions would have to be real analytic locally as well.
The now celebrated example of Baouendi and Goulaouic, from 1971
[1], simply written as P = D2X + D2 + x2D^ which is subelliptic with loss
of 1/2 derivative but whose characteristic variety is not symplectic, was
shown not to be analytic hypoelliptic and this seemed to close the door
39
L. Rodino (ed.). Microfocal Analysis and Spectral Theory, 39-59.
© 1997 Kluwer Academic Publishers.

40
on analytic hypoellipticity for non-elliptic problems. The Gevrey regularity
of this problem and others was was studied in [19], [33], [12], with the
general result that a loss of 1 - 1/ra derivatives in these problems results
in (isotropic) Gevrey hypoellipticity for all s > m.
In the case of a symplectic characteristic manifold where first brackets
suffice to span the tangent space, the author [35] showed that one could
'break the G2 - barrier' by utilizing a 'maximal' estimate, and by focusing
less on the subellipticity. There it was proved (relatively easily) that one had
hypoellipticity in all Gevrey classes Gs for s > 1, hence in their intersection
(still a non-quasianalytic class) and also in a certain quasi-analytic class,
C{LlogL}t Thig ! t
was done by considering a larger collection of non-
quasianalytic classes which behave in some essential ways like the Gevrey
classes (essentially that they are closed under composition), and whose
intersection was quasianalytic (but not yet the analytic class) [35]. As is well
known, the local real analytic hypoellipticity for the ^-Neumann problem
and for □& even on strictly pseudo-convex domains was much harder and
was finally achieved in 1978 in [40] and [36] independently. An indication
of the subtlety of the local analyticity even in the symplectic case when the
lower (even zero!) order terms are not appropriate, cf. [32].
More recent results have been in two directions - proving analytic
hypoellipticity in more degenerate settings, globally and in some cases locally,
cf. [6], [14], [7], [15], [16], [17] & [18] and counterexamples tot analytic
regularity of solutions when the characteristic variety does not have some
particularly nice properties cf. [8], [23], [10]. Earlier work [22] and [30] pointed
in these directions, though the appropriate generalizations had not been
clear.
Here we recall these results and related ones and apply the same
methods, though simpler than those required for local real analyticity, to show
that one can often 'break the Gm barrier' in more degenerate cases as well.
Recent results [9] provide sharp isotropic results on Gevrey regularity for
certain sums of squares of vector fields, and here we apply our methods
to these cases and prove still sharper results which include partial Gevrey
hypoellipticity, where the regularity depends on the variables being
examined. These results were obtained jointly with Antonio Bove. In particular,
we obtain sharp non-isotropic Gevrey regularity results for the example of
Baouendi and Goulaouic cited above. A very general setting is then
introduced and discussed.
2. Some Definitions and Notation
Definition 1 A function h(w) belongs to the Gevrey class Gd near wq if

41
there exists a constant C such that for all multi-indices a and w near w0,
\Dah(w)\<CCWa\d.
Definition 2 A function h(x,t,s) belongs to the Gevrey class Gdud2>d3 near
(#o>*o>£o) if there exists a constant C such that for all multi-indices a and
for all (x,t,s) near (xo,to,xo),
\D°W?2D^h{x,t,s)\ < CC\aWdl(*2\d2(x?>\dz-
We note in passing that sup norm estimates will follow from L2 estimates
of a very small number of additional derivatives of a localization of the
function in question, in view of the Sobolev Lemma. In fact, it suffices,
from a result of Nelson, to bound derivatives as measured by a system of
vector fields that span the tangent space, and, for real analyticity, to bound
even just powers of a system of vector fields that generate the tangent
space by their brackets. However, for non-analytic results, such as Gevrey
hypoellipticity in other classes, this will not suffice, as has been pointed out
by [2].
3. The Elliptic Case
While the real analytic hypoellipticity for elliptic partial differential
equations has been known for many decades, we sketch here a proof that will
give the flavor of our later proofs. Garding's inequality for a second order
elliptic partial differential operator P(x,D) in Rn reads:
£ \\DXiDXiv\\l < C{\\Pv\\l + |M|?} (1)
for all v £ Co° of small support. Assuming for simplicity that Pu — feC^
with u £ C°°, we apply the 'coercive' estimate above to v = <f)Dau for
suitable a and (j) G Cq° to be specified in a moment. Commuting P past
<j)Da to obtain Pu about which we know the effect of derivatives,
£ \\DXtDXj<t>Dau\\\2 < C\\<t>DaPu\\\2 + \\[P^Da]u\\l2 + \\<f>Dau\\l
Writing P as a sum of terms of the form a7D7,7 < 2 with known real
analytic coefficients we may write [P, cj)Da] as a bounded sum of terms of
the form a(x)(j)'DDa, a(x)(j)"Da, and E (£,) terms a,W<j>LjnD"'"', the sum
being taken over 0 ^ af < a, \j\ < 2.

42
Of these terms, the last are easy to treat: since the coefficients are of
known growth, analytic in this case, their derivatives combine beautifully
with the binomial coefficients so that
(a)\a^\UIPDa-a'u\\^< sup Cfl+>|^l||0D^-^||L2,
a power of |a| appearing for each 'gain' in free derivative. When all free
derivatives have been used up, we find Clal+1|a|lal||^||L2 which is bounded,
by Stirling's formula, by CuClal+1|a|!, hence yield a 'good' term in proving
analytic (or Gevrey) regularity.
The terms with derivatives on <fi must be treated carefully by using
localizing functions due to Ehrenpreis [21]: for any u and Q with u compactly
contained in ft, dist(o;, £lc) = d, and any N and m, there exists a constant
Co independent of N and m and a function </>n = 1 on u and in C™(£1)
with
\Dp(f>N\ < Co{CoN/d)\p\ \p\ < mN. (2)
The reason these localizing functions are so useful is that one may
differentiate them a high number of times, the choice of function, but not the
constants, depending on the number of derivatives one wants to estimate.
And if, say, one is really interested in N derivatives, one merely invokes
Stirling's formula again to show that when \p\ = N,\Dp(J)n\ < C± |/>|!,
which means that </>n is 'as good as analytic up to order iV.' The sole caveat
is that these estimates do not combine easily with binomial coefficients in
inductive arguments and great care must be taken to deal with this in some
cases. Indeed, this has been the stumbling block in some proofs.
The result is that all terms that appear from the bracket in the coercive
estimate above lead to good gains. The last, as we saw, gains one power
of \a\ for a gain of one in free derivatives; the same is now seen to be true
(without binomial coefficients!) for the other terms. In all, when there are
at most one or two free derivatives, we will have C'al terms, each of analytic
growth |a|lal, as long as we choose N comparable to \a\. This finishes this
(sketch of a) proof of analytic hypoellipticity for elliptic equations.
To make this argument more formal, we produce a suitable inductive
hypothesis. To do so, we will abusively write Da+2 to denote any DaD1
with |7| < 2. Then we may start with (j) differentiated:
£ ||4fcVzrU|||2 < Cill^D^Pulll^liPA^D^llh+ll^^ulll}
\p\<2
<C{Co{CoN/dfkC^\af + j^ £ ||^^D^U|||2
£=l \p\<2

43
|/?|<20^a'
A suitable inductive hypothesis then would be that
£ \\ct§D^Dau\\L2 < C^C^N^2^ (3)
I0|<2
provided |<5| + k < |<*o| + |&o| and a < a. Then appropriate choices of the
constants, relative to one another, will complete the induction step.
4. Subelliptic Cases From Complex Analysis
Perhaps the simplest non-elliptic equations to consider are the subelliptic
ones. For these, we find the useful form of Garding-type inequalities are
formulated with quadratic forms. That is, while the coercive estimate was
stated in (1) in terms of ||Pu||L2, it could equally well have been written
x:ii^iii2<c{i(p,,,)L2|+ii<2} (4)
for v of given compact support and the analysis does not change materially
(see below). By a subelliptic operator we will mean one where the norm on
the left is replaced by a Sobolev norm of fractional degree: for some positive
€ and all v as above,
\\v\\2t<C{\Re(Pv,v)\ + \\v\\l2}. (5)
As the ^-Neumann problem on a strictly pseudo-convex domain
presented perhaps the first example of a subelliptic problem and motivated
most of the later ones; it is a boundary value problem which we outline
here for completeness, though the details will not be essential in the sequel.
Let Q, be an open, relatively compact submanifold of a complex Herrrii-
tian manifold ft', with smooth boundary T = dft. We consider
together with its adjoint d The 9-Neumann problem on Q consists in
finding a (p, q) form u on ft, in the domain, DM, of d , with du in the
domain, DM+1 of d on (p, q + 1)- forms as well, and satisfying
Q(u,w) = (du,dw)n + (d*u,d*w)n = {a,w)L2^)^w G £>M,
for a given (p, </)-form a. Thus also

44
□« = {dd* + 8*d)u = a
in fi.
□i, is analogously defined as follows: Let Y be any real 2n - 1
dimensional compact CR manifold, i.e., a Hermitian manifold for which CTT =
T}fl®Tf'1®N,dimRN = 1, T°'l{= Tp'°) is integrable, and T"'1 has trivial
intersection with T^'0. Defining Ap,° = {p-forms in CT*T which annihilate
E = {T*'° ®T°'Y z,nd T0'1}, let
mapping C00(AP,?) to C°°(Ap<q+l), where 7TP)9+i = orthogonal projection
onto AM+1. Thus (d6)2 = 0. Let d£ = the adjoint of db, and set
□6=^ + ^6-
To define the Levi form, we let T be given by r = 0, dr / 0, and
choose {£j}j<n independent, real analytic vector fields, spanning T1'0 at
each point, with Lj<n tangent to T, and Lnr = 1 on T, and finally set
T = (Ln - Ln). The Levi matrix Cj^ on T is given by [Lj,Lk] = Cj,kT
mod {Lj,Lj},j < n, T is called (strictly) pseudo-convex if Cjtk is (strictly)
definite.
Note that neither the d— Neumann problem nor □& is elliptic. Although
□ is an elliptic operator, the boundary conditions are not coercive, and □&
can never be elliptic - there are only 2(n — 1) vector fields on a 2n — 1
dimensional manifold.
Furthermore, via a (pseudodifFerential) reduction to the boundary, the
d— Neumann problem is equivalent to a (pseudo-)difFerential problem of the
same general form as □& - in particular, taking real and imaginary parts of
the tangential holomorphic vector fields, the principal part of both problems
becomes a sum of squares of real vector fields whose iterated brackets span
the whole tangent space precisely when the domain is of 'finite type'.
5. Equivalent Real Problems. Subellipticity.
To fix notations, we will consider a slight generalization of the sum of
squares of real vector fields,
M
P=Y,a^3Xk + X{) + c{x) (6)
with positive definite symmetric dj^.

45
And we shall make the assumption (#2) that the {Xj},j = l,...Af,
together with their iterated brackets of length, say, m generate the entire
tangent space. Then one has at once, for v of fixed compact support
M
Z) H^i«lli2 < C{|»(Pt7ft;)|+ ||t;||^}
i=i
and, from [31], also subellipticity since the sum of just the squares on the
left dominates the Sobolev norm with loss of 1 - 1/ra derivatives on such
v:
M
£II*;* + IMI?/m < c{\*(Pv, v)\ + \\v\\l}. (7)
Theorem 1 (Derridj-Zuily) Let P be given as in (6) and satisfy the
estimate (7). Then P is Gs hypoelliptic for any s > m.
Proof: As in the proof of analytic hypoellipticity for elliptic operators
above, we introduce v = (f)jsiDau where the solution u is (assumed for
convenience to be) in C°°. Then we have, from (7),
M
J2 \\DX3<j>ND«u\\2L2 + UNDau\\2l/m < Cm$ND°Pu,<j>ND°u)L2\+
+mP,$NDa)u,$NDau)L2\ + UNDau\\2L2}.
Again writing P as a sum of terms of the form aj^XjXk, Xo, and c(x) with
known real analytic coefficients we may write ([P, (j)NDa]u, <j)js[Dau)L2 as a
bounded sum of terms (underlining a coefficient to indicate the number of
terms of a given form that may occur) of the form
[a{x)Xj(j>,NDau,(i>NDau)L2, (a(x)<j)%Dau,<})NDau)L2,
aiaixtyNXklXj, D]Da~\ <t>NDau)L2,
and
a(a - l)(a(x)<j>NDau, <f>NDau<)L2
plus terms such as S (£) terms \\a^(l)(Xj)Da-a,u\\L2\\Xj(l)NDau\\L2, the
sum being taken over 0 / cJ < a.
All of these terms have a new look to them, since now we must
distinguish between Xj derivatives and unspecified derivatives, which we have
denoted by D.

46
The Schwarz inequality will be used on all of these terms. In the first, the
Xj is brought to the right side of the inner product and a small multiple
of the L2 norm of that part absorbed on the left hand side of the basic
inequality. What remains is a large multiple of \\(j)'NDau\\2L2.
In the second term, we split the weight between the two halves, assigning
\a\ to right and H_1 to the left.
The third term is quite benign, since we may move the Xj to the second
member modulo aji error of the form \\(j)tNDau\\L2\\a(f)^Dau\\L2,
And in the fourth term, we associate one power of a with each side.
The result, modulo a small multiple of the left hand side of the basic
inequality, is that one may bound |([P,(j)js(Da]u^(j)]S[Dau)L2\ by
U'NDau\\l2, \a\*\\<t>NDau\\l^ \a\-*\\^Dau\\l^
and
The terms where Da is still present but lack the help of an Xj, such as
(j>lNDa and acjyj^D01 are new and require making use of the subellipticity.
We write, for example,
U'ND<*u\\l2 = \\k-'lm<f>'ND"u\\\lm
where A is the pseudo-differential operator with a(A) = (l + l^l2)1^2. While
K~llm(j)tNDau no longer has compact support, we can introduce a second
cut-ofF function 4>n to the left of A_1/m with the same type of growth of
derivatives as 4>n but identically equal to one on the support of 0n, so that
except for an error (namely [0n, A~llm](f)^) of arbitrarily low order we may
proceed as if K~llm(j)'NDau still has compact support.
Thus modulo further brackets, which drop the order by one full
derivative each time, we may iterate the whole estimate with A"""1/™ and either
an extra derivative on (J>n or a factor of \a\. After a total of Cm^
iterations, all free derivatives will have been used up, and the resulting terms
will either contain Pu, which will not concern us, or have the form
H2bl\\42)u\\l,
with 61 + 62 = m\a\. Invoking the bounds on derivatives of </>n above
yields a bound on these terms of |a|mlal, which is the statement of Gm
hypoellipticity. For s> m,Gs hypoellipticity follows the same lines.

47
6. Breaking the G2 Barrier. Analyticity.
For what was perhaps the most interesting case, that of the problems from
complex analysis in the case of strictly pseudo-convex domains, and where
G2 hypoellipticity had been proven by [19], [34], [11], etc., it had been
conjectured since the end of the 1960's that real analytic hypoellipticity
might still hold despite the Baouendi-Goulaouic example. Here the situation
could be modelled by the real operator
n—1 n—1
where, in local coordinates,
Y -JL_ I
j~dXj yjdt
and
_ _d_ d_
j~ dyj+Xjdf
These simple vector fields occur in the context of the Heisenberg group,
though we make no use of the group structure. Clearly the 1/2 estimate
holds, as well as the 'maximal' estimate with which we have been dealing
all along:
E ll*i* + E WyAh + IMI?,2 < c{\(Pv, v)\ + \\v\\l2}
but every effort to bound derivatives of the form v = (j>js[Tku, with T = d/dt
more effectively than G2 had failed. Of course since P is microlocally elliptic
in directions other than T this would suffice. The problem was that the
inevitable bracket [Xj,</>NTk]u = {Xj{(j)jsi))Tku could not be effectively
majorized without loss of 1/2 derivative.
6.1. GLOBAL ANALYTICITY: THE STRONGLY PSEUDOCONVEX CASE
It became clear that for global anlyticity, such derivatives on <j> were of no
concern, and in fact one should not use the specialized <j>n at all - virtually
any partition of unity would suffice. Whenever the localizing functions were
differentiated one would bring them out of the norm and start over. What
did seem to be required was a globally defined T- vector field, but that was
not a limitation in the embedded case since the (inward pointing) normal v

48
could be globally defined and, via the complex structure, T — Jv could be
taken as a global starting point; if this vector field did not commute
perfectly with the tangential holomorphic and anti-holomorphic vector fields
then it could be modified by the addition of multiples of those vector fields
in a unique way to commute adequately. This was done independently in
1976 in [28] and [34], though had been found earlier by Tanaka.
6.2. HIGHER LOCAL REGULARITY
The crucial observation in breaking the G2 barrier locally was that, utilizing
the excellent commutation relations enjoyed by the Xj,Yk, which could
always be arranged by using the Darboux theorem, one could 'correct' the
localized vector field (j)Tk in such a way that its bracket with Xj or Yj would
not have a derivative on (f> without a decrease in power of T. In fact,
[^^-x,-OT = -(^))Ji-
is free of T. Arguing that the extra derivative on <j> is far less troublesome
than free T derivatives, the localization (jy^T of T was replaced with
3 3
and (t>NTk by (T1^)*. While the localizing functions were still required
to be able to accept large numbers of derivatives, since they would occur
embedded between many T's, whenever derivatives appeared on a </>n it
was brought out of the norm as quickly as possible. The result [35] was
hypoellipticity in all the Gevrey classes except the analytic one and in some
other non-quasi analytic classes whose intersection was quasi-analytic but,
it was not hard to realize, not the real analytic class.
The real analytic hypoellipticity required a more courageous
construction. T^ and its powers were not good enough - a construction requiring
only one <f> for the entire Th seemed needed. And the idea that worked was
an iterate of the construction of T(f>N. That is, if
[Xk, T^ = 4>NT - Y^XMnWj + £0Ww))*;] = 0
3 3
modulo {<jy$X, (jy^'Y}, (and the same bracketed with Yjt,) then could one
not correct </>nT2 in similar fashion so that the bracket of this 'correction,'
(T2) , , when bracketed with either Xj or Yk contained no free T
derivatives at all? The answer is 'no'. The best one can do is to make a choice of
the order in which the X's and Y's appear and write
M =T,NT+ £ {-l)M(X*Y%fN)) XW

49
Then it comes as no surprise that (T2), commutes beautifully with
the Xj and Yk except for the order in which the vector fields occur. But
while [Xk, (T2)^] = 0 modulo terms <j$Z2 where Z may be an X or a
y, this is not true of the bracket with Yk- However, a miracle does occur
- modulo these terms and another expression of the form T^Yjk this does
occur. An extremely brief calculation gives:
Proposition 1 [Yjt, (T2), ] = Tt<j)N oYk modulo terms (j>N Z2 where each
Z is aY or an X.
It is this miracle, that modulo terms with only free Z's, the remaining terms
combine in the perfect balance of Tj^Yk^ and which is strongly tied to the
particular commutation relations enjoyed by the X's and Y's, that permits
the
Definition 3
(Tp) = Y ( 1) '5l'Xh Yh ^ xhYs* Tp~l5'+*2'
Proposition 2
|5i+52|<P
and
both modulo terms (jy^ ' Zv where each Z is aY or an X.
There remain many long and difficult arguments before one reaches the
analytic hypoellipticity, the most difficult being the brackets of this
expression with (variable!) coefficients, but the essential result contained in this
proposition, that brackets with the vector fields occurring in the
differential operator under consideration lead to expressions which can be again
subjected to the a priori estimate (the maximal estimate, in this case),
bounding X's and y's, with no sacrifice - the remaining Yk of the
proposition gives the a priori estimate its full power and, as is quite evident, each
iteration results in trade-off of one derivative on </>n for one gain in the
number of T derivatives: I Tk~l 1 where one had started with Tk) .This
V )r<t>N m t V )4>N
'one-for-one' trade-off leads, as we saw in the elliptic argument included
above just for this point, to analyticity.
7. Weakly Pseudo-convex Cases
Much attention has been given to cases where the Levi form is merely
semi-definite, the so-called weakly pseudo-convex cases. When the domain

50
Q is bounded with real analytic boundary, we know that the d- Neumann
problem is subelliptic [20] and more generally, Catlin has shown that finite
type conditions imply subellipticity more generally [5]. But beyond the
conclusion that subellipticity implies Gevrey regularity as in the above
Theorems, most attention has gone to study real analytic regularity when
there is a maximal estimate (usually requiring a lot of symmetry - cf. [12]),
e.g., [6], [7] with a series of papers [34], [13] proving local real analytic
hypoellipticity for either the ^-Neumann problem or for □& whose Levi
forms degenerate in well-controlled ways). Semi-global regularity is also an
active area of research currently.
In addition, some mixed results, where the characteristic manifold splits
into an involutive portion and a symplectic portion were studied for pseudo-
differential operators very recently by the author and Antonio Bove [4]. The
class satisfies the hypotheses for C°° hypoellipticity with loss of one
derivative, and the results state that for 1 < s < 2 the operator P propagates
singularities, i.e., smoothness, along the leaves of the characteristic variety,
if there are any, while for s > 2, the operator is G2— hypoelliptic.
Using Metivier's technique of addition of variables and making an
analytic canonical transformation we write the operator in question as
<M*,t)
x
z
X*
z
) + (L{x,Z),
X*
Y
)+Pi{x,0 + Po(x,0, (8)
where
.-I d
Xj = liVtj-Xk+j\DXn\, 1<j<*,
Zs = 7T-£— 1 < s < £,
(9)
X = (Xi,...,Xk,Xk+i,...,X2k), Z = (Zi,...,Ze), A is a self adjoint
positive definite matrix, of size 2k +1, of pseudo-differential operators of
order 0, and L is a (complex) 2k+£ dimensional vector of pseudo-differential
operators of order 0. The characteristic variety is denoted by E, and these
results are stated micro-locally:
Theorem 2 (Bove-Tartakoff, 1996) Let P be as above. Let (a?0,fo) 6 £
and W be a neighborhood of (#o>£o)- Suppose 1 < s < 2 and that (#o,£o) ^
WFs{Pu); then ifT(x0jio)n{W\{{x0^o)})nWFs{u) = ®we have (x0,&) i
WFs(u).
Theorem 3 (Bove-Tartakoff, 1996) Under the same assumptions as in
the above Theorem, let s > 2 and (a?o,fo) t WFs(Pu). Then (z0,£o) t
WFs(u), i.e. P is microlocally Gs-hypoelliptic.

51
Remark: For s = oo, this theorem is well known by [24], [3]. For s = 1,
Theorem (2) is found in [29]. And the second of these theorems was proved
using different methods in [25].
The principal part of the simplest (partial differential) prototype in
R2k+l+l js
2k £
i=i i=i
where the {Xj} satisfy the Heisenberg commutation relations (microlo-
cally), and the Zj commute with all other vector fields, the {Xj} and {Zj}
being independent. Of course, together with the first brackets of the Xj
they do span the tangent space. Here the results may be stated as saying
that Gs- hypoellipticity for s > 2. For example, and we shall have more to
say about this kind of example later, in the most concrete situation, when
„ / d d\2 ( d d\2 (d\2
p=wrX2dt) +w2+xidt) +{Yz)
in i?4, with the obvious vector fields, the characteristic manifold consists
of E = {£ = (£1,^2) — 0?7/ — 0,a?i = £2 = 0} and its leaves are free
in y. Thus the first theorem states that if (0,0,0,0;0,0,0,1) $ WFs(Pu)
and (0,0,j/,0;0,0,0,l) ft WFs(u)0 for 0 ^ \y\ < c, for some e > 0 then
(0,0,0,0;0,0,0,l)£WFs(u) if 1 < s < 2.
The proof of this result employs the analytic hypoellipticity proof of
[38] which extended [36], [37] to prove Metivier's Theorem (analytic
hypoellipticity for higher codimension symplectic characteristic variety with
loss of 1/2 derivative) [29] together with propagation techniques which in
this simple setting are easy to state. For, localizing T = d/dt, the localizing
function 0(#i, x<i, J/, t) must behave as before in (a?i, x<i, t) since one expects
fine hypoellipticity in those variables, but in y can be quite general, since
one does not expect to have analyticity in any case in that variable. Thus
whenever a derivative in y lands on the localizing function, we have no way
to produce better than G2 results unless we make the a priori assumption
that where such a derivative is non zero, the solution is already smooth.
Thus if there is no singularity in y derivatives in a band €1 < \y\ < €2, there
will be none at y = 0 either.
8. Breaking the Gs- Barrier; Other Rational Exponents
Recently there has been study of Gevrey hypoellipticity when s takes on
values other than 1/ra. Christ [9] has very recently obtained sharp isotropic
results in rational Gevrey classes that are better than those predicted by

52
the subellipticity index. In this section and those that follow we improve on
those regularity results by considering non-isotropic classes and show that
all of these results are accessible with L2 methods alone.
We consider here the particular, though apparently fairly typical,
example
= x2 + x2 + x2.
Theorem 4 The operator P is Gd hypoelliptic for all d > 3/2.
Remark: This theorem is due to M. Christ. However our proof is both
elementary and allows more precise results about partial regularity, where
derivatives in different directions grow at different rates.
Theorem 5 (Bove-Tartakoff, 1996) The operator P is G*1**2***
hypoelliptic for d\ > 7/6, &i > 1, and d3 > 3/2, and microlocally as well
Proof: We shall use the a priori estimate, for v G C°°, any vector field W,
and the localizing function (j) — </>n, as above:
£ \\x3(j>wvv\\l2 + £ WWWvWl, + mw>v\\1/3 (ii)
j j
< C{\ (PtWvrfWv)^ | + EH^(^)WPvWh} + WWp-lv\\2L2},v € C00.
3
(Note that we distinguish the different derivatives on (f> since the Xj carry
coefficients which are powers of a:, a fact that will be crucial in the sequel.)
We have used several facts in writing down this estimate: the form of P
clearly allows us to bound the basic vector fields Xj on u, and thus the 1/3
norm since second brackets suffice to span the tangent space, we have used a
symmetric form on the left, with the Xj either before or after the localizing
functions, since both will occur as errors. It follows that if Pw = f G Gs
for some s > 3 the same is true of w locally.
We shall obtain bounds of the solution locally (in L2 norms) of the type
\\<t>D?D?D?u\\v < CC\aWdl^d2^d3A^\ < N.
and to do this it suffices, by integration by parts and a simple inductive
hypothesis, to treat pure powers of Dx, Dt, and Ds.
We start with W = D*, as this turns out to be the simplest. Then
[P, <t>NDpt] = Dx{<t>N)xDpt + (<t>N)xDxDpt + xDtx(<j)N)tDvt
+x2{(S>N)uDvt + x2Dsx2{<t>N)sDvt + x4(<t>N)ssDpt.

53
Now the (£, s) derivatives on <f>jq are serious, (recall that x- derivatives leave
us in the elliptic region where the result is known) but even these derivatives
will not be harmful to a proof even of analyticity (in the variable t) if there
is a corresponding gain in powers of Dt without losing the two good X's.
These two good X's may come either from P (one X comes from P in
the first, third and fifth terms on the right here, though in the first two
terms the presence of an x derivative on fa lands us in the elliptic region)
or are created by combining powers of the variable x with copies of Dt,
namely once in the third and fifth terms and twice in the fourth and sixth,
thus reducing the power p on Dt. Note that these (one or two) powers of Dt
must still be commuted past fa to be in a useful position, and they may
land on the localizing function. If it they do, we must try to bring another
Dt out, etc. But to simplify this exposition, we include only the principal
term, ignoring these second-order brackets. Once we have X's on the left,
one of these X's will be moved to the right (by integration by parts). We
obtain:
([P,faDp]u,faDpu)L2 = (Dx(fa)xDpu,faDpu)L2
+ {{fa)xDxDpu, faDpu)L2 + 2 (xDt{fa)tDp-lu, xDtfaDpu)L2
+ (xDt{fa)ttDp-2u,xDtfaDpu)l2+2 (x2Dt{fa)sDp-1u,x2DsfaDpu)^
+ (x2Dt{fa)ssDp-\x2DtfaDpu) l2
so that (recalling the norms are squared) for any e > 0,
| ([P,faDp]u,faDpu)L2 | < C\\faDpu\\2L2 +CC2M(2\a\)\
+e\\xDtfaDpu\\2L2 +Ct{\\^Dt(fa)tDp-lu\\2L2 + ||a:A(^)ttZ)r2^|li2}
+Ce{\\x2Dt(fa)sDp-lu\\2L2 + ||o:2Dt(^)ssDr2^|li2}.
After replacing the L2 norm first on the right by again e times the 1/3
norm modulo a large constant times the —1 norm and using this to absorb
one power of D*, this leads, upon iteration p times, starting from (11) with
W = Dt, to the bounds
£ \\X^NDMh + £ UNX^vWh + \\<f>NDptu\\l,3
i i
<CC^{2\a\)\^C"^\2\a\)}\\u\\y
= CCl\aP[(2\a\)\ + C^^(2\a\)\

54
which yields analytic growth in the Dt derivatives.
Next we tackle Ds derivatives, which are not nearly as simple:
[P, <j>NDl] = DX(<I>N)XDP + {(t>N)xDxDl + 2xDtx{</>N)tDps
+x\cl>N)uDl + 2x2Dsx\cj>N)sDl + x4{<j>N)ssDps
Again, the x derivatives on </>n are not serious, and the last two terms
may be treated precisely as we did with high powers of Dt: combining x2
with Ds 'creates' a 'good' derivative (i.e., an Xj) which may be integrated
to the right in the inner product. This exchange - a derivative on </>n for a
gain in p - i.e., a gain in power of Ds derivatives is what has led to optimal
(i.e., analytic) regularity all along.
But it is the third and fourth terms that are more troublesome; and
where the new features arise. For combining only one power of x with Ds
will not generate a 'good' vector field sufficient to balance a derivative (Dt)
falling on </>n . Two powers of a: are required. However a little patience will
produce two, even in this 'worst case scenario'. For if we are repeatedly so
unlucky (and all cases do occur) as to bracket with xDt, (and we ignore
other contributions) we find that after two such brackets we may decrease
p by one.
But something even better happens. Consider: we start with X</>nD$u
and after the first use of the basic a priori estimate we have come up with
x((j>N)tDvsu. As with the proof in the general subelliptic case above, we do
have a gain of 1/3 derivative to make use of. That is, for the next iteration
we write
\\x(cf>N)tDlu\\L2 = \\A-l^x(<l>N)tDlu\\l/3,
and treat A~ll3x((j)]S[)tD'psu as the new version of <f)j^Dvsu whose Xj
derivatives as well as whose 1/3 norm is bounded in the estimate. (See the proof of
the first Theorem for justification of using A~l^3x(<pN)tD^u as if it had
compact support in introducing it into the a priori estimate again.) In the next
iteration, we will be led to analogous 'errors', such as A"ll3x2((j)]s[)ttD'psu
with one more power of x and another (Dt) derivative of <pN.
But now something very exciting happens. We may treat the x2 together
with Ds as a good vector field, namely as one of the Xj's. After these two
iterations we have gone from ||Xj0jv-D?^||l2 to ll^j^-1^3^^)"^-1^!^2-
After three of these iterations we will find \\XjA~1(<l)N)ttttttD^~3u\\iJ2 ~
llXjiA-iDsMriunttDP-tullv - ^{Xj^^tttmD^u^ which exhibits
a trade of four Ds derivatives for six derivatives on the localizing function
<i>N-
This is the 'trade-off' that leads to the Gevrey class G3/2.

55
Finally, we turn to x derivatives, where the computations are simpler,
but seem to depend on the preceeding ones: This time W — Dx and the a
priori estimate reads essentially
£ HX^jvZ^Ill, + £ UNXiDlutfv + UnDpxu\\21/3 <
J 3
3
< C{\ {<j>NDlPu^NDlu)L2 | + £ | ([<l>NDZ,X])u,<l>NDlu)L2 \
In the bracket (second term on the right) the problem is now not keeping
good vector fields - the Dx themselves are good. The difficult term that
arises is when x2Ds[Dvx,x2Ds] (or xDt[D^xDt] or double brackets) enter.
All terms contribute p terms where one of the Dx differentiates x. Thus the
error that arises is of the form pxDsDvx~l (and pptDvx~l) after the other
Xj has been moved to the second member in the inner product. From the
double brackets we may also have p(p - \)DSDVX~2 etc. That is, starting
with X^nD? we now have essentially pXx<j)NDsDvx~2 or p^X^nDsD?'3,
and similar terms with Dt.
Now the former, pXx<f)^DsDvx~2^ turns out to be the worst in terms
of managing growth of derivatives - there has been a 'gain' of two ZVs,
but a factor of p and a new Ds derivative. When this continues, what
started as X^^D? becomes, after the next iteration, p(p - 1)x2(J)nD2sD'px~2>
or pc/)nDsD^~2 or ^X^^xD2£>£~5, etc., where, again, in the first term we
may call x2Ds an X. That is, this term is of the form ^X(I>nDsD^~3. This,
iterated p/3 times, will lead to terms such as p2p/3Ds . Since we know that
derivatives in s of the solution grow like the Gevrey class G3/2, we will get
WXfaBZuWv < cV"/3P!(3/2)(1/3) < Cpp!7/6.
This is the origin of the G7/6 behavior in x. Finally, the microlocalization
poses no additional problems.
9. Other Special Cases, Leading to a General Conjecture
An analysis of the above proof, which certainly includes the statement of
G3/2 hypoellipticity, shows that it applies equally well to the somewhat
more general operator

56
for 1 < p < q.
Theorem 6 The operator P in (12) is Gd hypoelliptic for all d > q/p.
Remark 1 This theorem is also due to M. Christ. However our proof is
both elementary and allows more precise results about partial regularity,
where derivatives in different directions grow at different rates.
Theorem 7 (Bove-Tartakoff, 1996) The operator P in (12) is Gdud^d*
hypoelliptic for any d\ > 1 + l/p - l/q, d2 > 1, and d3 > q/p.
Theorem 8 When p = q = 1 this yields analytic hypoellipticity, but in all
other cases yields new examples of Gevrey classes of solutions to subelliptic
partial differential equations.
Theorem 9 (Bove-Tartakoff, 1996) The Baouendi-Goulaouic example,
P = D2x + D2t+x2D2s
is Gdljd2jds hypoelliptic for any d\ > 3/2,^2 > 1? one? d3 > 2 but not
hypoelliptic in any smoother Gevrey class.
Proof: The first part is a special case of the above result. The sharpness
comes by studying the function
TOO
ut(x, t,s) = / exp[ip2s - tp - p2x2/2 - pe] dp
Jo
for € > 1 which solves Put = 0 yet brief calculations show that ut satisfies
POO
|a*«e(0)| = |/ e-ptpkdp\~Ckk\^,
Jo
/•CO
|d*ttc(0)l = l/ e-p€p2kdp\~CkkM€,
Jo
and
\d2xkut(0)\ = | / e-p€p2kk\dp\ - Ckk\l+2t - Ck{2k)\l/2+u,
Jo
showing that for any e > l,ut £ G1/24-1/6'1^'2^ and no better.
More general situations clearly pose no additional difficulties, such as
those studied (isotropically) in [9].
P=-^2+a1(x^s)(x^ft)2 + a2(x,t,s)(x^j-s)2 (13)
where both a\(x,t,s) and a2(x,t,s) are strictly positive and belong to the
Gevrey classes under consideration.

57
10. The General Conjecture and Result
The general conjecture suggested by these results, and voiced by Treves,
concerns the iterated brackets of a set of real vector fields in the operator
i=l
where the Xj satisfy the Hormander condition that their iterated brackets
span the tangent space. Call A\ = {X\,... JOv}, A<i — A\Vl^Xi, Xj], i / j},
...,AN = \JiL~i1 AiU{[Xii:[Xi2, [. ..,[XiN_i:XiN]...)) ]}. We agree to call
Aj, j = 1,..., N the "layers" of the Lie algebra of the tangent vector fields.
Call now Ei = {(x,£)\Xj(x,£) = 0,j = l,...,r}, the multiple
characteristic set of the operator F, E2 = {(#,£)|Y(a;,£)| = OVY G A2}, i.e.
the characteristic set of the second layer of the Lie algebra. Proceeding in
this way we get to defining En = {(x,£)\Y(x,£)\ = OVY G An}- By the
Hormander assumption Ejy coincides with the null section of the cotangent
bundle.
Of course the Ej are a decreasing finite sequence of subsets of the
cotangent. We need not to assume that they are manifolds; if they are not just
think of them as of stratified varieties, whose strata are smooth manifolds.
Tentative statement 1: If all the Ej's are symplectic—i.e. every layer of
each Ej is symplectic—then P is analytic hypoelliptic.
Tentative statement 2: Assume that E^ is the first of the Ej that is not
symplectic, i.e. there is at least one stratum with an involutive leaf. Then
the operator P is Gs hypoelliptic if s > N/L
The proofs above yield the second statement in the cases studied as well
as in others under consideration by E. Bernardi, A. Bove and the author.
The first conjecture remains open.
References
1. Baouendi, M.S. and Goulaouic, C. (1971) Analyticity for Degenerate Elliptic
Equations and Applications, Proc. Symp. in Pure Math. Vol. no. 23, pp. 79-84.
2. Bolley, C, Camus, J., & Rodino, L., (1987) Hypoellipticit analytique-Gevrey et itrs
d'oprateurs. Rend. Sem. Mat. Univ. Politec. Torino Vol. no. 45(3), pp. 1-61.
3. Boutet de Monvel, L., Grigis, A., h Helffer, B., (1976) Parametrices d'operateurs
pseudo-differentiels a caracteristiques multiples Societe Math, de France, Asterisque
Vol. no. 34-35.
4. Bove, A., &; Tartakoff, D., (1996) Propagation of Gevrey Regularity for a Class of
Hypoelliptic Equations Transactions of the A.M.S. Vol. no. 348(7), pp. 2533-2575.
5. Catlin, D. (1987) Subelliptic Estimates for the d-Neumann Problem on Pseudoconvex
Domains Annals of Math. Vol. no. 126, pp. 131-191.
6. Chen, S.C. (1988) Global Real Analyticity of Solutions to the d-Neumann Problem
on Reinhardt Domains, Indiana U. Math. J. Vol. no. 37(2), pp. 421-430.
7. Chen, S.C. (1991) Global Analytic Hypoellipticity of Db on Circular Domains., Pac.
J. Math. Vol. no. 148(2), pp. 225-235.

58
8. Christ, M. (1991)Certain Sums of Squares of Vector Fields Fail to Be Analytic
Hypoelliptic, Coram, in P.D.E. Vol. no. 10, pp. 1695-1707.
9. Christ, M. (1996) Intermediate Gevrey Exponents Occur, to appear, Cornrn. P.D.E..
10. Christ, M. &, Geller, D., (1992) Counterexamples to Analytic Hypoellipticity for
Domains of Finite Type, Annals of Math. Vol. no. 135, pp. 551-566. _
11. Derridj, M. (1975) Gevrey regularity up to the boundary for the d- Neumann
problem, Proceedings of Symposia in Pure Math. Vol. no. XXX, pp. 123-126.
12. Derridj, M. (1978) Regularite pour d dans Quelques Domaines Faiblement Pseudo-
convexes., Journal of Differential Geometry Vol. no. 13(4), pp. 559-576.
13. Derridj, M. & Tartakoff, D.S., (1976) On the Global Real Analyticity for the d-
Neumann Problem. Comm. P. D. E. Vol. 5 pp. 401-435.
14. Derridj, M. & Tartakoff, D.S., (1988) Local Analyticity for Db and the d-Neumann
Problem at Certain Weakly Pseudo-Convex Points. Comm. P. D. E. Vol. 13(12)
pp. 1521-1600.
15. Derridj, M. &; Tartakoff, D.S., (1991) Local Analyticity in the d-Neumann Problem
and for □*> in Some Model Domains Without Maximal Estimates. Duke Mathematical
Journal Vol. 64(2) pp. 377-402.
16. Derridj, M. & Tartakoff, D.S., (1993) Global Analyticity for Db on Three
Dimensional Pseudoconvex CR Manifolds Comm. P. D. E. Vol. 18(11) pp. 1847-1868.
17. Derridj, M. & Tartakoff, D.S., (1993) Local Analyticity in the d-Neumann Problem
for a Class of Totally Decoupled Weakly Pseudoconvex Domains. Journal of Geometric
Analysis Vol. 3(2) pp. 141-151.
18. Derridj, M. k Tartakoff, D.S., (1994) Microlocal Analyticity for the Canonical
Solution to db on Strictly Pseudoconvex CR Manifolds of Real Dimension Three. Comm.
P. D. E. Vol. 20(9&10) pp. 1647-1667.
19. Derridj, M., and Zuily, C. (1973) Regularite analytique et Gevrey pour des
classes d'operateurs elliptiques paraboliques degenerees du second ordre, Asterisque
Vol. no. 2,3, pp. 371-381.
20. Diederich, K., and Fornaess, J.E. (1978) Pseudoconvex Domains with Real-Analytic
Boundaries Ann. of Math. Vol. no. 107, pp. 371-384.
21. Ehrenpreis, L. (1960) Solutions of some Problems of Division IV Amer. J. Math.
Vol. no. 82, pp. 522-588.
22. Grusin(1971) A certain class of elliptic pseudodifferential operators that are
degenerate on a submanifold, Mat. Sbornik (=Math. USSR Sbornik) Vol. no. 84(13),
pp. 163-195(155-185).
23. Hanges, N., &; Himonas, A., (1991) Singular Solutions for Sums of Squares of Vector
Fields., Comm. in P.D.E. Vol. no. 16(8,9), pp. 1503-1511.
24. Hormander, L.(1967) Hypoelliptic Second Order Differential Equations Acta Math.
Vol. no. 119 pp. 147-171.
25. Kajitani, K. and Wakabayashi, S. (1988) Hypoelliptic Operators in Gevrey Classes
Recent Developments in Hyperbolic Equations, L. Cattabriga et al. Ed.s, Pitman
Research Notes in Math. Vol. no. 183, pp. 115-134.
26. Kohn,J.J. (1963) Harmonic Integrals on Strongly Pseudo-Convex Manifolds, I and
II. Ann. of Math Vol. no. 78 pp. 112-148 and Vol. no. 79 pp. 450-472.
27. Kohn,J.J. and Nirenberg, L.(1967) Non-Coercive Boundary Value Problems Comm.
Pure Appl. Math. Vol. no. 18 pp. 443-492.
28. Komatsu, G. (1976) Global Analytic Hypoellipticity of the d-Neumann Problem
Tohoku Math. J. Vol. no. 28, pp. 145-156.
29. Metivier, G. (1981) Analytic hypoellipticity for operators with multiple
characteristics Comm. in P.D.E. Vol. no. 6, pp. 1-90.
30. Oleinik, O., & Radkevich, R., (1974) Conditions for the analyticity of all solutions
of a second order linear equation. (Russian) no. 3(177), 221-222. Uspehi Mat. Nauk
Vol. no. 177(3), pp. 221-222.
31. Rothschild, L.P. and Stein, E.M. (1977) Hypoelliptic Differential Operators and

59
Nilpotent Groups Acta Math. Vol. no. 137, pp. 248-315.
32. Stein, E.M., (1982) An Example on the Heisenberg Group Related to the Lewy
Operator Inventiones rnathernaticae Vol. no. 69, pp. 209-216.
33. TartakofF D.S. (1973) Gevrey Hypoellipticity for Subelliptic Boundary Value
Problems Communications on Pure and Applied Math. Vol. no. 26, pp. 251-312.
34. Tartakoff D.S. (1976) On the Global Real Analyticity of Solutions to Db on Compact
Manifolds Comm. in P.D.E. Vol. no. 1, pp. 283-311.
35. Tartakoff D.S. (1977) On the Local Gevrey and Quasianalytic Hypoellipticity for
Db. Comm. in P.D.E. Vol. no. 2, pp. 699-712.
36. Tartakoff D.S. (1978) Local Analytic Hypoellipticity for Db on Non-Degenerate
Cauchy Riemann Manifolds Proc. Nat. Acad. Sci. U.S.A. Vol. no. 75, pp. 3027-3028.
37._ Tartakoff D.S. (1980) On the Local Real Analyticity of Solutions to Ob and the
d-Neumann Problem Acta Math. Vol. no. 145, pp. 117-204.
38. Tartakoff D.S. (1983) Operators with Multiple Characteristics - An L2 Proof of
Analytic Hypoellipticity Conference on Linear Partial and Pseudodifferential Operators,
Rend. Sem. Mat. Univ. Politec. Torino 1983., pp. 251-282.
39. Tartakoff D.S. (1996) Global (and local) analyticity for second order operators
constructed from rigid vector fields on products of tori Transactions of the A.M.S.
Vol. no. 348(7), pp. 2577-2583.
40. Treves, F. (1978) Analytic Hypo-ellipticity of a Class of Pseudo-Differential
Operators with Double Characteristics and Application to the d-Neumann Problem Comm.
in P.D.E. Vol. no. 3 (6-7), pp. 475-642.

HIGHER MICROLOCALIZATION AND PROPAGATION OF
SINGULARITIES
OTTO LIESS
Dipartimento di Matematica
Universita di Bologna
40127 Bologna, ITALIA
1. Inverse Fourier transforms for functions of infraexponential
growth.
1. A good point to start is to discuss the range of applicability of the Fourier
transform, or rather of its inverse. We define the inverse Fourier transform
formally by
h(x) = (2*)-» / e^>/(*K (!)
and the problem is to see for which classes of functions (or distributions)
we can give a reasonable meaning to (1). In the last century / would have
been assumed to be integrable; nowadays we would write the corresponding
condition in terms of Lebesgue integrals and ask for / 6 Ll(Rn). There is
a natural extension of this to / £ L2(Rn) using Plancherel's theorem. A far
reaching extension was achieved by L.Schwartz, who took / £ Sf(Rn).
Every now and then it has been attempted to extend the range of these
definitions further: cf. e.g. [6], [13]. In particular, there should be some
intersections between the results described here and the results in [13], but
the present theory is in my opinion simpler. I want to show here that (1)
has a simple natural meaning if we assume that
feLUir),\f(t)\<ce«t\ (2)
where <p is sublinear. (We call a function <p : A —> R+ sublinear if V£ > 0 3ce
so that (p(t) < e\t\ + c£.) We shall say that / has infraexponential growth
(at infinity) then.
To discuss (1), assume at first that supp /cG, for some convex cone
G C Rn. (Rn = Rn\ {0}.) Even with this additional assumption it is not
61
L. Rodino (ed.), Microhcal Analysis and Spectral Theory, 61-90.
© 1997 Kluwer Academic Publishers.

62
possible to give a direct meaning to (1) for real x, but we can give a meaning
to it when we replace x by z = x + iy £ Gn, y G G1. (G1 denotes the polar
of G.) Indeed, for y° G G1 fixed, we will have (j/°,£) > c|£|, for some c > 0
if £ G G, and in fact we can find a neighborhood V of y° so that
<y,0>c|£|, if^GGandj/GK
It follows that
|ct(*+*,0| = c-(v,0 < C-<W if ^ e G,j/ G F,
so that |/(0 exp[i(z + ty,f)]| < cexp[y>(0 - c'|£|]. Therefore /(£y(*+^>
G Ll(Rn) uniformly in (x, y) € Rn xV. We conclude that
h(z) = h(x + iy)= [ e^+^/^Jde
makes sense and defines an analytic function, at first on Rn x iV, but then of
course on RnxiGL, since j/° G G1 was arbitrary. Thus, h(z) G ^(^xiG1).
(For ft open in Gn we denote by A(Q) the holomorphic functions on ft.)
Actually, in this way we have not given a direct meaning to (1) for real
a;, but, assuming that the support of / lied in G, we were able to associate
some h(z) G A(Rn X iG1) with the integral there. Let us next drop the
assumption that supp / C G and return to (1). Let us then choose a finite
collection of open convex cones Gj C Rn, j = 1,..., &, so that UfTj = Rn.
Also fix some bounded open neighborhood Go of the origin and write / in
the form
/ = /o + E/i (3)
with supp fj C Gj, j = 0,1,..., k. We can in addition assume that the fj
have infraexponential growth and are in Lj0C(Rn) for j > 1, the /0 being
bounded. The integrals hj(z) = (27r)~n/fin exp[i(z,£)]/j(£) d£ have then a
natural meaning for Imz £ Gf when j > 1, respectively for all z £ Cn
when j = 0. At this moment we have thus associated with the right hand
side of (1) the formal collection of analytic functions hj, j = 0,1,...,k.
Let us also recall that in the theory of hyperfunctions one associates with
analytic functions defined on sets of type Rn x IT, T some open cone in Rn
a "formal boundary value" on the edge "i?n" of the wedge "Rn X iP\
(Actually the situation considered in hyperfunctions is somewhat more general
and we recall it for the convenience of the reader in section 2.) We
denote the formal boundary value of some analytic function defined on some

63
wedge by b(h). It is thus a hyperfunction and is to be regarded as an
object living on Rn. In particular, we can calculate the sum J2j=0b(hj) as a
hyperfunction on Rn and we shall set
(2*)-/ e^>/(Ode = X>(M- (4)
jRn i=o
It is not difficult to see that the hyperfunction u = J^jLo K^i) depends
only on / and not on the decomposition of / in the form (3). It is this
hyperfunction which we shall call the "inverse Fourier transform" of /. We
shall also say that this u is the regularization of the right hand side of (1).
2. It might be instructive to see what happens when rather than having
"infraexponential growth" as in (2), we assume that we have the more
restrictive assumption |/(£)| < c(l + \£\)h-> f°r some non-negative real number
6. Assume as in the beginning of our argument that supp f C G and fix
T' CC GL. (If Ti , T2 are two cones in Rn or Cn, we shall write Tx CC T2
if the closure of Ti in Rn (or in Cn) lies in the interior of T2.) We now
also want to be a little bit more careful with our estimate for the
exponential expi(z,£) and therefore observe that there is some constant d > 0,
so that (j/,0 > cr\y\\t\ if £ € G and y e V. Since \t;\bexp[-c'\y\ \£\] <
c"\y\-bexp[-c'\y\ |£|/2], it follows that h(z) = /exp[t(*, £)]/(£) d£ has tem-
perated growth when y -> 0 as long as y £ T'. (In other words, there are
constants ci, &', so that \h{z)\ < c\\y\~b if y £ T'.) We conclude from this
that b(h) makes sense in distributions. When / has polynomial growth at
infinity, the meaning we give to (1) is therefore precisely the one it has in
standard distribution theory in Sf(Rn). Indeed, with appropriate changes
in notations, we could have replaced the assumption that / be a function
with polynomial growth by the condition that / £ Sf(Rn). We do not make
this here more explicit.
2. A brief review of hyperfunctions (from a local point of view)
1. Let me make a paranthesis on what hyperfunctions "locally" are. We
want here to consider hyperfunctions as equivalence classes of formal
boundary values of analytic functions defined on tuboids. (But we shall call such
"tuboids" "wedges" henceforth.) Starting point is that distributions
themselves appear often as distributional boundary values of such analytic
functions. Before we continue, we define the notion of a wedge: let us then
consider at first U open in Rn. We call "wedge over IF , or simply "wedge", a
set of type
u = (u x %T) n u c cn

64
where T is an open connected cone in Rn and U is a complex neighborhood
of U in Cn. (Sets of this form are sometimes called "tuboids" in the
literature. I myself find the notion "wedge", although perhaps less accurate,
more intuitive. The set U is often called the "edge" of a;.)
Let us now consider a finite number of wedges uj over U and let hj
be holomorphic functions on uj. Wedges will "always" be over the same
U. With the hj we associate the formal sum ^hj. It is then clear that in
the space of such formal sums we have a natural commutative addition.
Let us denote the space of all such formal sums by M. In M we shall now
introduce an equivalence relation by saying that
for hj G A{{U + iTj) n Uj) , h'k G .4(07 + iTfk) n U'k), if we can find
fis G A((U + i co (Ti U Tfs)) n (lis), ("co" is "convex hull") such that
hj = Y, fi*> h'k = Y,flk' (5)
s I
It is not difficult to see that £/^ ~ Y.K » E// ~ E/" implies Y,h'j +
^2fl ~ E^fc + E/s'- Further, if /ii,/i2 are in A(u) for the same wedge
a;, then the pointwise sum h% G 4(a;) defined by h^(z) = h\(z) + h2(z) is
easily seen to be equivalent with hi + h2, the latter sum being performed
in M. Also note that a convenient way to characterize Y^jLi hj ~ 0 , /ij G
4((C/ + ir^) H Uj) is, that we can find fjk G 4((C/ X i co (rj U Tk)) n t/jfc)
with
fjk = ~fkj and hj = yjT,fjk-
k
We shall denote the factor space Al/~ by jB(/7). The projection X ->■
M/~ is called "hyperfunctional boundary value" and is denoted by 6. (It is
the "6" we had already had above on analytic functions defined on wedges.)
Let us also consider explicitly the case n = 1. If u is a hyperfunction,
we can write it (locally this is clear) as u — b{h\) + 6(/&2) where h\ G
A(U x i(0, e)) and h2 G A(U x i(-e, 0)), e > 0. The fact that u is zero as
a hyperfunction on an open subset U' subset of U means that there is an
open set V C C, which intersects R on U' and a holomorphic function h
defined on V so that h\ is the restriction of h to Vf n {Imz > 0} and h2
the restriction of -h to V1 n {/raz < 0}. It is then easy to believe that one
can identify for example the hyperfunctions on some interval W C R with
the factor space A(V \ W)/A(V) where V is some open set in C which

65
intersects R on W. Since A(V) is dense in A(V \ W), one sees from this
that B(W) has no natural separable topology,
3. L2 estimates on Cn rather than sup-norm estimates on Rn
1. We now return to (1). Unfortunately, it is not easy to develop a non-
trivial theory of integrals of type (1) while working in the real domain. We
shall in fact replace integration on Rn with integration on Cn. Moreover, in
order to be able to apply classical results from the theory of the ^-operator
in weighted L2-spaces, we shall replace the L°°-type estimate in (3) by some
related L2-type estimate. More precisely, we shall assume that / G Lf0C(Cn)
and that
/(C)e-^fieC)+£|/mC|ei2(cn).
We have then to give a meaning to
(l/2<)" / ^/(CKAdC, (6)
Jcn
where (l/2i)n d(Ad( is the Lebesgue measure on Cn. We can then regularize
the integral in the following way: again we consider open convex cones Gj,
j = l,...,fc, and a bounded open neighborhood of the origin Go such
that Uj>iGj = Rn. We then split / into a sum / = Ylj fj wl^ SUPP
fj G Gj x ii?n, and notice that (if the fj have the same type of behaviour
at infinity which we had for /)
hj(z) = (2n)-n(l/2i)n j e^'QfjiC) dQ A d(
JCn
is analytic for \Rez\ < e, Imz G Gj, when j > 1, respectively that
ho G A(z; \Re z\<e). It follows that the integral (6) has a natural meaning
as a hyperfunction u on \x\ < e. We denote with the letter T the map
which associates with / this hyperfunction. We shall then also say that
the hyperfunction u has been represented with the aid of / and call / a
"representation function" of u. (Since we are working on \x\ < e our
representations are only local.) In a related way we say that u is the inverse
Fourier transform of /. As in our discussion of the case of inverse Fourier
transforms of functions defined on i?n, it is useful to observe that we need
not necessarily work with "functions" on Gn, but may replace "functions"
by "measures" or "distributions", which satisfy appropriate growth
conditions at infinity. Moreover, these growth type conditions can in principle
be formulated in the form of I2-type conditions, or also in the form of
L°°-type conditions. That this is so is related to the fact that on spaces of

66
analytic functions locally all reasonable norms are equivalent. We do not
make this here more precise, but refer to [22] for explicit statements.
2. The following questions are now natural:
- which hyperfunctions on \x\ < e can be represented in the form above?
- what is the degree of non-unicity in such a representation?
- how can one recognize properties of the hyperfunction u in terms of
properties of a representation function by which it can be defined?
The answer to our first question is: any hyperunction on \x\ < e can be
represented as the inverse Fourier of some /. That this is so has been
known, in another form, since quite some time. Indeed, let us at first recall
the following result,which is a consequence of results of Ehrenpreis (cf. [3]):
Theorem 3.1. Let e > 0, d > 0 and an open convex cone T C Rn be
given. Denote by i?r,d, the support-functions of the set {y G I\ \y\ < d}.
(Recall that the support-function Hk of a compact set K in Rn is defined
by Hk{£,) = snPyeK{yiQ') Consider an analytic function h defined on
ft = {z e Cn; \Re z\<e,Imze T, \Imz\ < d}.
Also fix ef < e, Tf CC T. Then we can find a sublinear function <p and a
Radon measure v on Cn with the following properties:
I <*M<)l<oo, (7)
JCn
h(z)= I ex.p[i(z,Q + <p(-IUQ-Hr,td/2(-IUQ-e'\Im<;\]dv(Q, (8)
JCn
if \Rez\< e'} Imze F, \Im z\ < d/2.
(For a direct proof of this result we refer to [22]. Also cf. [3], [31].) To relate
this result to our problem of writing some given hyperfunction u on \x\ < e
as an inverse Fourier transform, we first extend u to some hyperfunction
v defined on \x\ < e' with e1 > s. We then write v as a sum of formal
boundary values of analytic functions hj defined on wedges over \x\ < ef
and apply the preceding result for each hj and for some e' < e which satisfy
e < ef < i < ef.
3. Before we now say a few words on non-unicity, let us introduce
a notation. For A measurable in Cn and e > 0, we shall in fact
denote by £2(A, J", -e) the space of measurable functions f on A such that
fexp[-(p(ReQ +e\Im(\] G L2(A) for some sublinear function (p.
We turn to non-unicity. In fact it is quite obvious that our representation
functions (or measures) will not be unique. Let us give two examples which

67
show what can happen. Assume then at first that // is of the form {d/dQ)u
for some j and some v in C™(Cn). Then (8) makes sense and defines the
zero hyperfunction. We shall see that non-unicity is indeed deeply related
in local representations to the fact that ji has the form ji — Y^^iid/dQ)^.
Indeed, the following theorem is one of the main results in the theory:
Theorem 3.2. (Cf. [22].) Fix 0 < e < e" < e' and let f G L2(Cn, T, -ef)
be such that T(f) = 0 on \x\ < e". Then there are gj G L2(Cn,T, -e),
j = 1,..., n, so that
f = EW^ihy (9)
j
Conversely, if (9) is valid, then T(f) = 0 on \x\ < e.
It is clear that non-uniqueness as in our first example is somehow related
to the fact that we are working with representations on Cn rather than with
representations on Rn. However, our second example is with representation
functions in Rn: consider g £ C^°(i?n), g / 0, so that g(x) = 0 for |a:| < e.
Then it follows that g is a function on Rn which defines the zero-function on
\x\ < e. Since g is a representation function for g, we have thus represented
the zero hyperfunction with the non-trivial g. We might here recall that
it is not easy to characterize holes in the support of some C^-function in
terms of its Fourier transform. Theorem 3.2 (which has to be reformulated
to include representation measures) then gives a result in this direction.
Let us finally say something on our third problem. In fact, as is the
case for the standard Fourier transform, regularity properties for a given
hyperfunction are often reflected in decay properties of the representation
function. Let us give two examples: when u is a distribution on \x\ < e and
if we fix e' < £, then we can represent u on |a:| < ef in the form:
JCn
where /exp[e'|JroC| - Mn(l + |C|)] € L2(Cn) for some suitable 6 G R.
(Thus, / has polynomial, rather than infraexponential, growth in Re(.)
Another interesting case is when /exp[e|C|] G L2(Cn). (I.e., / is
exponentially decaying in the L2-sense in |£|.) In fact, it is immediate then
that the hyperfunction u associated with this / is real-analytic near zero
and the converse is also true, as is easy to see. Indeed, this is the
simplest instance of a theorem of the type of theorem 3.1, and we give a few
details to explain how the argument goes. Let us then assume that u
admits a holomorphic extension to the set {z G Cn; \z\ < 2e} and define
a continuous linear functional L on the space ,4(Cn,£|C|) of entire
analytic functions h on Cn for which the norm \h\e = sup^ \h(Q\ exp—[e|C|] is

68
finite, by L(h) = T l(h)(u). T l is of course the inverse Fourier-Borel
transform. It is easy to see that we also have \L(h)\ < c\\h\\e/2 where
||/i||£/2 = J\ jCn \h(Q\2exp[-e\C\]d( A d(\. Applying the Hahn-Banach and
the Riesz-Fischer theorem it follows that there is a function / on Cn so that
L(h) = ff (Qh(Q dCAdC and so that /exp[(e/2)|C|] G L2(Cn). (Of course,
we have to assume here that /&exp[-(e/2)|£|] € L2(Cn).) We conclude that
u{x) = Jexp[i(xX)f{C) dCAdC if |x| < e/2.
Since we consider later on real-analytic functions which depend on an
additional parameter (this will happen when we speak about pseudodifFer-
ential operators), it is useful to have a more explicit way to obtain
representation functions in this case. We are only interested in the case of germs
of analytic functions near 0; let us then assume that u is of form
u(z) = ]Cac*2a,
a
where the aa £ C satisfy Cauchy's inequalities: there are constants c, cf so
that
\aa\ < cc'Wal (10)
Actually, we want our Fourier representation to be valid in a complex
neighborhood of 0. The construction of the representation function for u will then
be performed in the following way: we shall choose representation functions
aa for za', i.e., aa will be so that
za = (/2i)n[ exp[t<*,<>MC)dCAdC. (11)
JCn
We can then take ^a aacra as a representation function for u. Here the aa
are of course not unique, and indeed, we will have to choose them according
to the value of cf in (10). However, once c' in (10) is fixed, we can work with
the same <ra, independently of u, so that our construction is stable if we
have additional parameters. The main step in the argument is the following
easy
Lemma 3.3. There are constants c, c\} so that for any A>0«;e can find
measurable functions aa on Cn with the following properties:
a) supp aac{Ce Cn; A\a\ - 1/2 < |C| < A\a\ + 1/2},
b)\K\\LHCn)<cc[alA-\°\,
c) for any h G A(Cn) we have D%h(Q) = fCn MOMO d\(Q. In
particular, (11) isvalid. (D% stands for (\/i)\a\{d I dQa-) (Weputh(Q = e^K)

69
As a consequence, we obtain:
Theorem 3.4 Let u be of form u(z) = ^aaaza, and assume that the
coefficients aa satisfy (10). Then we can find a representation function f
for u with the following properties:
a) It is of form f = J2a aa(Ja) with aa as in lemma 3.3, for some suitable
A, which depends only on cl.
b) There is c" > 0 so that exp[c"|C|]/ G L2(Cn).
It is interesting to note that, although we are working with exponentially
decreasing Fourier representations, there is no need to work with "quasi-
analytic" cut-off functions. This is of course due to the fact that we work
with representation functions which live on Cn.
4. We have stated theorem 3.4 in terms of L2-type estimates: it is this
type of estimates which we use in almost all of the paper. In the
applications of the theorem it is however more convenient to work with L°°-type
estimates. For this purpose we consider the following strengthened form of
theorem 3.4:
Theorem 3.5. Let u be as in the assumptions of theorem 3.4- Then the
conclusion there remains valid, even if we replace the condition b) by the
following stronger condition
b)' There is c" > 0 so that exp[c"|C|]/ G L°°(Cn).
4. Microfunctions and Bony's theorem on the equivalence of
definitions of the analytic wave front set.
1. Until now, we have worked locally. One can microlocalize theorem 3.2.
This is in so far interesting as it gives a representation of microfunctions in
terms of d-cohomology classes and it permits to analyze problems on micro-
functions with the aid of methods of complex analysis. For the convenience
of the reader, we recall the following definition:
Definition 4.1. (Sato) Let u be a hyperfunction defined near x°. We shall
say that (#°,£°) is not in the analytic singular spectrum of u, and write
(a:0, £°) ^ ssau if we can find open convex cones Tj C Rn} j = 1,..., k, and
holomorphic function hj defined on {x + iy £ Cn;x £ Rn,y £ Rn,\x-x°\ <
e, y G Tj, \y\ < d} so that u — J2j KN) near x° an^ so ^l0^ £° %s n°t z'n ^e
polar Gj of Tj.
One can then characterize the analytic singular spectrum as follows:
Proposition 4.2. (z°,£°) g ssau if and only if we can find a sublinear
function <p, an open cone G C Rn which contains £°, d > 0, and a repre-

70
sentation function v for u so that
\[ \v{()\2exp[-2<p(Re() + 2d\Re(\G + 2e\Im(\]d(Ad(\<oo. (12)
Jcn
Here we have denoted for a given set A C Rn by \£\A the function defined
in the following way: \£\A = |f |, if f G A, \£\A = 0 if $$ A. Note that
definition 9.1 is "geometrical", whereas (12) is analytical. The inverse Fourier
transform thus gives a relation between an analytical and a geometrical
information for u.
A result similar to proposition 4.2 is valid also in distributions. Let us
in fact denote by WFAv the analytic wave front set (as introduced e.g. in
[7]: we shall recall the corresponding definition in a more general frame in
section 6 below) of the distribution v. Then we have:
Proposition 4.3. Let v G D'[U). Then (0,£°) g WFAv if and only if we
can find an open cone G C Rn which contains £°, d > 0, 6 G R, and a
representation function v for u so that
\[ |//(C)|2exp[2d|i?eC|G + 26|/mC| + 261n(l + |C|)]dCAdC|<oo. (13)
Jcn
To simplify the statements of the following results, we shall now introduce
some (rather heavy) notation: the first is L2^0,^7, -e) and it stands (if ^o
has been fixed in Rn) for the space UwL2(W, T, —e), where the union is
for all open convex complex neighborhoods W of £°.
Furthermore, if X is some space of distributions in Cn, we denote by
X(o,q) the space of complex (0, </)-forms with coefficients in X. Moreover, if
(a?°,£°) is fixed we shall write for two hyperfunctions u and t;, that u ~ v
if (#0,£0) ^ ssA(u - v). We denote by S^o^oj the factor space B(U)/ ~,
where U is some open set containing a:0. We call the elements in ZJ^o^o)
"microfunctions" and denote by J(xo^o) the map which maps u G B(U) in
its residual class in B(xo£oy
If now ji G L2(£°,.F, -e), we can associate with it a microfunction
T(fi) in the following way: if W is the domain of definition of //, we define
/} : Cn -+ C by setting /2(C) = //(C) for C € W and /2(C) = 0 for C i W
and then we put T(//) = J^xo^(T((!)). (More precisely, T(fl) will be a
hyperfunction on |x| < e and J(xQ£Q) maps J3({a: G Rn', \x\ < e}) onto
B^^oy) It is immediate to see that T is well-defined (i.e., T(ji) does not
depend on the choice of //, W) and it is clear that T is surjective and we
want to analyze the kernel of T. In analogy with theorem 3.2 we can prove:

71
Theorem 4.4 Let ft e L2(f,T, -e') be given so that T(fi) = 0. Then
there is e and Vj G L2^0,^7, -e) with /jl = J2j{d/d(j)uj zn a complex
neighborhood of x°. (The converse is also true.)
The result thus says, intuitively, that when we work with microfunctions,
then we can practically argue for representation functions which are defined
on a complex conic neighborhoods of £° rather than on all of Cn, and that
the space of microfunctions at (0,£°) can (roughly speaking) be identified
with
U ^(e^, -0/ U ^(Cn-D^' -£)-
That this result can be of some interest can be seen from the following
theorem which gives a formulation in terms of d-cohomology for Bony's theorem
on the equivalence of Sato's and Hormander's definitions of the analytic
singular spectrum for distributions. We denote in it by L2(A, -£, -6) the
space of measurable functions /on A so that /exp[—e|JmC| —61n(l + |£|)] £
L*(A).
Theorem 4.5. Let 6 > 3 and ji £ LhJCn, -£, -6) be given and assume
that J(T(ijl)) = 0 in Buo^oy Then there is a complex conic neighborhood
W oft0, £ > 0 andv G ^n-i) W ~£> ~b + 3) so that V = dv on W.
Thus, with somewhat sloppy, by now self-explanatory, notations, we can
write that
U«*>0 ^0,»)K°. -*, -») _ U'>0 ^0,,)«°, -^ ~b)
U>0 ^fo,n-l)(^°' F> "*) U>0 ^fo.n-1) K°. -*. ~b + 3) '
(For proofs and comments on how this result is related to Bony's theorem,
cf. [22].)
5. Representation functions and pseudodifferential operators.
l.In the sections 9, 10 below, we shall show how one can extend the
preceding results to the case of higher analytic microlocalization in hyper-
functions and discuss some related arguments. One of the advantages of
our approach is that one can develop then a calculus of
pseudodifferential operators in higher microlocalization which is completely analoguous
to the theory of pseudodifferential operators in distributions. To make the
underlying ideas more transparent, we shall describe the main steps in the
calculus of pseudodifferential operators only for standard first
microlocalization. The theory which we describe is thus parallel to any theory of
infinite order analytic pseudodifferential operators. (Cf. in particular [2] or
[26].) The case of higher microlocalization can be treated in a similar way
but involves much heavier notations.

72
2. Let U be a neighborhood of the origin and consider f £ fln. Also let
p(x,£) be some analytic symbol of infraexponential type defined in a conic
neighborhood of (0,£°) for |£| large. By this we mean that there is C > 0,
a complex neighborhood ft of the origin, a complex open cone W which
contains £° and some analytic function p on S = {(z, Q G C2n; z G ft, C £
W, |C| > C} which extends p and so that
\P(z,0\<ce^ for (z,Q€S,
for some sublinear function <p : i?+ —>■ i?+. We shall later on write "p"
also for "p". Consider next u G #(o,£0)- We can represent u in the form
T(fi) where // G £2(£°, ^", —s) for some e > 0. It is no loss of generality to
assume that in fact ji G L2(Cn,^", -e). We can now define p(z, D)u to be
the microfunction associated with
h(z) = (l/2i)n f exp[i(z, Q]p(z, CMC) dC A dC (14)
JW
In fact, when W is a convex cone, the preceding expression already defines
an analytic function when \Rez\ < e, Imz G (W n /P1)1, and when W is
not convex, we can regularize the expression in exactly the same way we
regularized (6). (The growth type of p(z,C)MC) is exactly the one we had
for / in (6), only that here we have the additional analytic parameter z in
the symbol p(zX)- It is clear that this does not make the situation more
complicated.) It is easy to see that the value of p(x, D)u in #(o,£°) does not
depend on the choice of ji to represent u (recall here theorem 4.4; actually
it is in this type of situations that we need theorem 4.4) or on the way in
which we regularized (14). In particular, pseudodifFerential operators are
pseudolocal. (We start here from an analytic information for // and get
a geometric information for h.) To study composition of operators, let us
consider one more analytic symbol </(#,£). We can apply theorem 3.5 to
write q(z, Q in the form
q(z, C) = / exp[t(*, 6)]a(6, C) d6 A d6,
Jcn
for some a which satisfies for suitable ci, c<i and some sublinear F the
inequality
|a(0,C)|<Clexp[-c2|0| + F(|C|)].
Let also [i be some representation function for u which satisfies
(l/2i)n [ |/i(C)|2 exp[-2cp(Re <) + 2e\ImC|] d(Ad£< oo.
Jw
The first result which we need to study composition of operators is then

73
Proposition 5.1. q(z, D)u admits
1/(0) = (l/2i)n / a(0 - C, CMC) dC A dC (15)
as a representation function.
3. To consider composition, let us next assume that two analytic
symbols, p and q are given on {\z\ < e} x {C G W; |CI > C}. Let also
u G #(o,£0)- Since we can calculate a representation function for q(x,D)u,
it is easy to calculate p(x, D)(q(x,D)u)). We would like to show that this
is the same with r(x,D)u for some analytic pseudodifferential operator
which has symbol r(x,Q asymptotically equal with the formal symbol:
EaPa(*,Ofe(*,OM where P(a) = (d/idOa and q(a) = (d/dx)°q. The
main step in the argument is the following result:
Theorem 5.2. Consider W CC W. Define I(z,C) by the expression
I(z, C)= [ exp[t<s, 0 - 0]p(s, 0)a(0 - C, Q dO A dO.
Also denote for a G i?+ by Int[a] the integer part of a. Then there are con-
stants ci, C2, c3, ef > 0, so that I(z, C) is an analytic symbol on {(z, £); |z| <
e',C G W) |C| > c3}, and so fftaf
/(z,0 = £ P(a)(z,Qqia)(z,<;)/<*W(eM-c2\C\])X € W, |z| < e'.
|a|<7nt[Cl|C|]
(16)
6. Higher microlocalization
1. Second analytic microlocalization has first been considered by Kashiwara-
Kawai in [11]. The theory of Kashiwara has then been developed by Laurent
and is described in [17]. A theory of analytic microlocalization of arbitrary
order was developed by Sjostrand and later on extended by Lebeau: cf.
[30], [18]. A third approach to high order analytic microlocalization has
finally been proposed in [20]. The three theories are not equivalent among
themselves, in that they refer in part to different frames. Indeed, also the
approaches used in the three competing theories are very different. For
applications of arguments in higher analytic microlocalization, cf. e.g. [15],
[18], [30], [32], [34], [35], [20]. For the present author, higher analytic
microlocalization was foremost a tool to understand phenomena of
propagation of analytic singularities for solutions of partial differential equations
with characteristics of highly changing multiplicity. Since propagation
phenomena in the analytic category are often related to some kind of partial

74
analyticity, we should perhaps point out that higher order microlocalization
is much closer related to partial analyticity than is first microlocalization.
As an illustration for this we shall state in section 7 a result which
characterizes partial analyticity with the aid of the first two analytic wave front
sets, whereas it is not possible to characterize it with the aid of the first
analytic wave front set alone.
2. To develop a full theory of higher microlocalization, we need higher
order wave front sets, higher order pseudodifFerential operators and higher
order Fourier integral operators. We shall here describe the higher order
wave front sets. Higher order pseudodifFerential operators can easily be
studied in a way parallel to what we did in section 5 in the frame of first
microlocalization. We shall not give details here and refer to [20] for results
on higher order analytic pseudodifFerential operators acting on
distributions. (With the aid of the present results one can extend the results on
pseudodifFerential operators proved in [20] for distributions to the case of
hyperfunctions.) We have not explicitly studied Fourier integral operators
in higher analytic microlocalization when we microlocalize to order higher
than 3. Unfortunately, this implies that we can not show that the results
which we obtain in the case of third microlocalization and higher, have an
invariant meaning. (In fact, what has to be done is not difficult to see, but
seems to be very technical.)
3. Let us at first recall the definition of the higher analytic wave front
set as given in [20]. We start from a finite sequence of subspaces Mj, j =
0,1,...A;, in Rn such that M0 = Rn, Mj C Mj_i, Mj ^ Mj_i, Mk = {0}.
Also denote by Uj : Rn —> Mj the orthogonal projection on Mj and by
Mj = Mj © Mj+i the orthogonal complement of Mj+i in Mj. It follows in
particular that
iT = ®j>0Mj.
Also consider & G Mj Q Mj+i.
Definition 6.1. Let U be open in Rn, consider x° €U, u € D'(U)} and let
Mj and £J be as above. We shall say that (z°,£0, £*, ...,£A;"1) is not in the
analytic k-wave front set of u and write
if we can find open conic neighborhoods G^ C Mj of &,e,c,Cj,/3 > 0,
(increasing) sublinear functions Fj : R+ —> i?+ and a bounded sequence of
distributions {ui}^ C Ef(U) such that
u = U{, for |a: - x \ < £, and all i

75
|u,-(OI<c(«7|nfc_1ci)iif* = i,2,...,nJ^eGJ',i = o,...,A:-i,
in,,- ^| > i^-dn^! ^|),i = 1,..., a - it
and |ni+1 ^ > c^j ei^Vin.-i ^, j = i,..., k - 2.
(In the oral lectures, we only stated the definitions for the cases k = 2 and
k = 3. The reader will have understood why.)
Definition 6.2. A setG is called a multineighborhood o/^0,^1, ...,£*_1) if
we can find open conic neighborhoods Gj C Mj of £j so that G = {£; IIj£ G
G„j = 0,...,A;-l}.
Remark 6.3. a) One can of course write the conditions
|nj+1 CI > cj |n,- fl'+VlIIi-i ^ , j = 1,..., A; - 2 (17)
in homogeneous form Cj \Uj £|0/|IIj_i £|^ < |IIj+i^|/|IIj £| , j = 1,..., fc -
2. Smce the quantities |IIj+i^|/|IIj £| correspond to Sjostrand's ((small
parameters", this makes comparision with the theory of Sjostrand-Lebeau
easier. We stick here to the form (17), since it is more natural in propagation
phenomena.
b) When k < 2 the conditions (17) are considered void. For k = 3, they
reduce to \U2 f | > cx |IIi f |^+1/|n0 £\p.
Remark 6.4 As stated above, we have not proved that the notion WF\
has an invariant meaning when k > 3. The situation is more favorable
for k — 2. One can then give an invariant meaning to WF\, and in fact,
the above definition just reduces to second analytic microlocalization (as
considered e.g. by [17] or [30]) with respect to the involutive submanifold
£' = 0, if we denote the variables from Mi by £''. (On the equivalence of the
various definitions cf. [5], [25], [20].)
Let us also recall for the convenience of the reader how one can characterize
the second analytic wave front set in special coordinates with the aid of the
FBI-transform: we shall assume here that M\ = {£ £ #n;£n'+i = ••• =
£n = 0} and denote (&,..., £n') by f' and (fn/+i,..., fn) by £". It can then
be shown that
(x°,e,e)twF2Au,
iff we can find c, cf, c", T C i?n, r' C Rn , ot > 0, /3 > 0, open cones and a
sublinear function F : i?+ ->• R+ such that £° 6 T, ^ G H,
L(eH<*.0 " «lf 11*' " ^f/2 " /?H I*" - *T/2])
if |z° - t\ < c' (localization in a:) £ e T, £' G T', |£'| > F(\£\). The main
reason why analytic microlocalization of some order k can be interesting if
<ce~W\.

76
one studies propagation phenomena is that the following result of "micro-
Holmgren" type (for which there is no analogue in the C°°-category, of
course,) is valid :
Theorem 6.5. Let U C Rn be open and consider a:1, x2 £ U with [a:1, x2] C
J7, xl - x2 £ Mfc_i. Here and later on we denote by [a:1,a:2] the segment
with endpoints x1 and x2. Consider £°, f1, ...,£fc~2 and assume that for any
7] £ Mk-\ and any x £ [xl,x2] it follows that
(x1?,t\...,tk-2,r,)iWFiu. (18)
Moreover, assume that (a:1,^0,^1, ...,f*~2) £ WF^~lu. Then it follows that
M0,£V..,£fc~2) i WFkA~lu, whatever x£ [x\x2] is.
It also follows from this that (a?,^0,^1,...,^-2) ^ WF^u, for all x in the
connected component of (xl + M^_i) D U which contains x1.
7. Second microlocalization and partial analyticity
We want to explain here the relation between partial analyticity and higher
analytic microlocalization. Let us at first explain what we mean by "partial
analyticity". We will say that u £ D'(U) is partially analytic in the
distribution sense in the variable X\ if we can find an open set W in Rn+1 with
W fl Rn = U and v £ D'(W) (if we only ask for v £ B(W), we shall say
that u is partially analytic in the hyperfunction sense; the two assumptions
are not completely equivalent) so that
[{d/dxi) + i(d/dxn+i)]v = 0, ^n+1=0 = u.
We can here perform the restriction in view of the fact that v satisfies an
homogeneous equation which is non-characteristic with respect to xn+\ = 0.
From general results on the wave front set of a restriction we can
conclude that
(a,O £ WFAu implies fi = 0,Vx£ U. (19)
Thus we obtain complete information on the first analytic wave front set of
u when £j ^ 0, but in fact nothing can be said on the first wave front set
of u when ^ = 0. One can however prove the following result (cf. [20]):
Proposition 7.1. Consider u £ D'(U). Then the following two conditions
are equivalent:
i) u is partially analytic in the hyperfunction sense with respect to x\.

77
ii) (x°,f°) (£ WFAu, whatever x° £ U, £° £ Rn with £ / 0 and in
addition (zW1) i WF\u, whatever x° £U, £° £ Rn with £° = 0 and
whatever £l £ Rn with # = 0 for i > 2.
Remark 7.2. A similar result is of course also valid for partial analyticity
in the distribution sense, if we replace WF\ as we defined it above with
some temperated version. (Cf again [20].)
8. Successive localizations; an example
When /(£) is a (say) C°°-function which vanishes of some order s at some
point £°, we call "localization" of/ at £° the function: £|a|=s(d/d£)a/(£°)(£
-£°)a/a\. When p is an analytic symbol whose principal part vanishes
of variable multiplicity on its characteristic variety, localizations of this
principal part to subvarieties play an important role. It may happen that
the localized symbol itself vanishes of variable multiplicty, so one will be
tempted to consider localizations of the first localized symbol, etc. In a
fully involutive setting, the first localized symbol is an object of the second
microlocal category, the second localization an object of the theory of third
order microlocalization and so on. We do not make this here more
precise, but refer to [20] for examples where this idea has been implemented.
We want to show here with a simple example how this same idea has led
to the geometry of higher order wave front sets as we have considered it
above. (A more complicated situation of this type will appear in section
11.) Let us start from the symbol p = ^i^2^3 + a(#>£)£i + b(x^)^ (plus
perhaps lower order terms,) where a, 6 are symbols of order 1,
respectively 0. Thus, from the point of view of the classical theory, the principal
part is ^i^2^3 + a{x,Q£i + fcO&ifKi- I want to study this microlocally near
£° = (0,0,1), two-microlocally near £° = (0,0,1), f1 = (0,1,0), and 3-
microlocally near £° = (0,0,1), f1 = (0,1,0), £2 = (1,0,0).
The classical, i.e. 1-microlocal principal part is of course
ZiS& + a(x,Z)$ + b(x,Z)$.
For 2-microlocalization, we need the localization of the principal symbol to
the subspace £i = £2 = 0. The lowest order of vanishing is two, therefore the
localization p\ of p to this subspace is p\ — ^i^2^3+^(^? 0£i • 3-microlocally,
we calculate the localization to £1 = 0. The lowest order of vanishing is one,
the symbol of the localized becomes p<i — ^i^2^3-
This should be elliptic 3-microlocally and we want ^i^2^3 to domiate
|fi| is dominated by £2- all the other terms. We must then in particular
have that b(x^)^l can be dominated by £i£3, and this is true only if we

78
restrict our attention to regions of form £1 >> £f/£3- This is of the form
161 > cfoP-WVl^ with /? = 1.
9. Higher order microfunctions
1. Let Mj and (f°,f\.. .,f*_1) be given as in section 6. Also fix /? >
0, Cj, some multineighborhood G of (f0,^1,. ..,f*-1) and some sublinear
functions F, : R+ -+ R+. We denote by Y = Y(^°^\.. .^k~1) the set of
£ £ Rn which satisfy the following conditions:
- |ni+1*| > Cj|n^|^V|n,_^|^ i = i *-2,
- |n^| > i^dn^^i), i = i * - i.
We shall often say that Y is of type T^f1,.. .,^-1)" then. Moreover,
if Y is given of type Y^0,^1,.. . ,£fc_1), then we denote by Yc the set
{C G Cn]Re(£ y, |/mC| < c|IIjb_i(/2c C)|> f°r some suitable c> 0.
Consider now a hyperfunction u defined near a:0. For notational
simplicity we shall assume that a:0 = 0. We want to extend the definitions of
higher order analytic wave front sets to the case of hyperfunctions. This is
in so far new that in [20] higher order wave front sets were considered only
for the case of distributions.
Definition 9.1. We shall say that (O,^0,^1,.. .,£*-1) £WF\u if there are
y, d, e, a sublinear function ip and a representation function ji for u so that
^Q^-MReO+dlTl^ReQW-ellmtl] £ L2(YC). (20)
Next we note that if // is fixed and if we define ft by jl(Q = fi(Q for ( £
YC and by /2(C) = 0 if < * lfc, then (O,^1,. ..,£fc"1) $ WFkAT~\ii-fi).
Thus the microfunction associated with u at (0,£°,£\... ,£k x) depends
only on the restriction of ji to Yc-
2. In this subsection we consider a hyperfunction u defined in a
neighborhood of 0 and assume that (O,^0,^1,.. .,f*-1) £ WF^u. Let // be a
representation function of u which satisfies (20) and consider an additional
representation function //' for u. Since ji and //' both represent u, we can
write
3
with |x(C)l exP — [^>(i?eC) - e|/mC|] G L2(Cn). Moreover, since on Yc,
-(f(ReC) +e\Im(\ is dominated by |IIjfe_i.ReC| and since — |IIjb_i(i?eC)|
is plurisubharmonic on Yc, we can write
3

79
on yCl with x'j satisfying |x^(C)lexp^n^^eC)!] G £2(>c). (We use
here Hormander's theory on the ^-operator in weighted L2-spaces.) We
conclude that if (O,^0,^1, ... ,£fc_1) ^ WF\u, and if // is a representation
measure of u, then there are y/, e1, &', Y'c so that
i
with Ix^Olexp-HiieC) -£'|/mC|] G L2(Y>,).
3. Conversely, let // be a representation function of u and assume that
ji = J2jdjXj, on Yc- We can cut off Xj near Y'c CC Yc and therefore
conclude that u admits a representation measure which vanishes on Y'c. In
particular this implies that (0,£°,£\.. . ,£fc_1) ^ WF\u. Microhyperfunc-
tions at (O,^0,^1,.. . ,£fc_1) can in particular therefore be identified with
Li0tn)(e,e,...,e-\^-e,b)/dLi0tn_l)(e,e,---^-\^-e,by
4. Microregularity for hyperfunctions can be tested by duality. What we
can prove is the following result:
Theorem 9.2. Let u be given. Then there are equivalent:
i)%e,z\...,e-x)iwF\u.
ii) Let ji be a representation function for u onY^. Then there are Y CC
y;, c, d,£, 6 and a sublinear function <p so that | fy, /i(£)//(£) d£ A d(\ < c
for any h e A{Yq) which satisfies
\h{Q\ < exphV(fleC) + dlUk.^ReOlY+ellmCl + 61n(l + |C|)],
\h(Q\<Cexp[-^(ReQ+e\ImC\ + bln(l+\C\)l
for some constant C which may depend on h. (The second condition is
needed to give a meaning to f h(Q/ji(() d( A d(.)
This result is in so far interesting that the space of hyperfunctions does
not admit a natural topology. Nevertheless, we see here that for microfunc-
tions it is possible to work with duality arguments. (This will be further
developed in a future paper, which is in preparation.) With the aid of the
preceding result, we can now repeat the arguments from [20] and prove the
full analogue in hyperfunctions of theorem 6.5.
10. Restriction of microfunctions
One can use the theory of Fourier-inverse transforms and of higher
order wave front sets to define and study restrictions of higher order micro-
functions to subvarieties. Since our theory is not invariant for high order

80
microlocalization and in order to avoid notational complications, we shall
only consider the case when we want to restrict 1-microlocal microfunc-
tions. Rather than working in Rn we shall now work in Rn+l. We denote
the variables there by (t,x), t G i?, x G i?n, and want to consider
restrictions to t = 0. The Fourier-dual variables will be denoted by A = (r, £).
Let us also assume that we are given a microfunction u defined near (0, A0),
where the r component of A0 is zero. We shall therefore have A0 = (0, £°) for
some £° G i?n. For notational simplicity we assume that £° = (0,..., 0,1)
and denote by III the map rii(A) = (£i,.. . ,£n_i). To define the restriction
of u to t = 0, we can argue locally and then work with the Fourier
transform, i.e. representation functions. In fact, on the Fourier side, we have just
to integrate the r-variable away in //(A), if // is a suitable representation
function on Cn+1 for u. A problem could be that only the values of ji in
a complex conic neighborhood of A0 can have a real meaning for u. Let us
then fix a small complex conic neighborhood V C Cn of £° and consider for
some small constant c > 0 the set W = {A G Cn+1;C € V, \r\ < c\Re(\}.
We tentatively define a function v : V —> C by
»(Q = h f »(TX)dTAdf. (21)
It is easy to see that the function v satisfies the estimate
I / |//(C)|V2^fieO-|/rnC|+61n(l-f|C|)]dCA^| < Q (22)
JV
where
V'(0= sup V(r,0. (23)
\r\<c\t\
((ff is obviously sublinear.) The estimate for v shows in particular that
v can play the role of a representation function of some microfunction v
defined near (0,£°). Note that we have not used any additional assumption
on //. We however recall from the classical theory, that in order to define
restrictions of hyperfunctions we need some additional assumptions on the
hyperfunctions and that a good choice for such an assumption is that the
co-normal directions to t = 0 do not belong to the analytic wave front set
of the distribution under consideration. In fact, although we have not used
any assumption in order to define u, we will not obtain a reasonable theory
unless we impose some additional assumptions on ii. Indeed, if we do not
make additional assumptions, i) is not correctly defined as a microfunction,
in that it depends on the choice of the representation function ji choosen to
define u and on the choice of c which we had to make to define W in terms
of V. We shall therefore introduce the following additional assumption:
(0,\°,±N)(£WFlu, where N= (1,0,...,0). (24)

81
To define {?, we shall use the assumption (24) in a rather strong way. In
fact, one can show that the assumption implies that there is a constant
d > 0 and a representation function ji for ii so that
^Xy\Rer\+e\Irn\\+b\n{l+\\\) £ ^2^ (25)
where
A = {Re X e G; \Re r\ > d^ReC|, \Im X\ < c"\Re A|}, (26)
where G is some open cone in Rn+1 which contains A0.
It can now be shown that v defined with the aid of v depends only on
it, provided we argue on representation functions for u which satisfy the
condition (25). Let us also mention the following result:
Theorem 10.1. Let A0 and ii be as above, in particular we assume that
(25) is valid. Also assume that for some A1 with r1 = 0, ££ = 0, we
have (0, X°,6N + A1) g WF\ii whatever 9 e R is. Then it follows that
(O,^0,^1) i WF2Au\t=0. (Here ^ is the ^-component ofX1.)
The present considerations will be used in a forthcoming paper.
11. The Kawai-Kashiwara theorem.
Let us at first recall the Kawai-Kashiwara theorem in its classical form:
Theorem 11.1 (Kawai-Kashiwara, cf [12].) Let p(x,D) be an analytic
pseudodifferential operator defined in a conic neighborhood W of (#°,£°),
let ip : W -* R be a real analytic function such that il>(x°,£0) = 0, and
assume that p is microhyperbolic at (x°,£°) in the direction {—{dip/d^)
{x°i£0)-> {d/ip/dx)(x0^0)). Let u be a hyperfunction such that p(x, D)u = 0
on W (in the sense that WFap{x, D)u n W = 0) and assume that
WFAu n {(x,0 G W; j>(x,S) < 0} = 0.
Then it follows that (z°,£°) i WFAu.
(A real analytic function / defined in a neighborhood U of y° in Rm is
called "microhyperbolic with respect to the direction 0" if there is e > 0
and a neighborhood Uf of y° such that
/(y + tf0)/O, if0<t<e,y€Uf.)
The Kawai-Kashiwara theorem is one of the most important theorems on
propagation of analytic singularities. Many proofs, variants, extensions and
applications (to results on propagation of singularities or conical refraction)
of this result have been considered in the literature: [9], [12], [29], [36] [37],

82
[23]. Our aim in this section is to state a higher order variant of this theorem.
We expect this higher order version to have applications which are similar
in spirit to the ones of the original theorem and to help to understand
propagation phenomena in a systematic way. Actually, the original Kawai-
Kashiwara theorem consists of a "regularity part", which is theorem 11.1
above, and an existence part. Much of what we have said in the previous
sections is related to an attempt to prove higher order existence results of
Kawai-Kashiwara type. In the sequel we shall stick to the case of third wave
front sets. The reason is that the value of high order variants becomes less
and less clear if the calculus is not invariant and that for third order wave
front sets we can understand at least the meaning of the main conditions
which one imposes from an invariant point of view. To state our result, it
will be convenient to work in Rn+1 rather than in Rn. This is due to the fact
that theorems of Kawai-Kashiwara type are deeply related to the Cauchy
problem in which one has a distinguished variable in i?, which we shall call
"£", reserving the notation "a:" for the remaining variables, which will then
be variables in i?n, as above. These notations are thus in fact the same with
the ones in section 10. As for the Fourier-dual variables, they are denoted,
again as in section 10, by A = (r,£) with £ G Rn. It is convenient to write
A as (Ao, Ai,.. .An), so that r = Ao and £ = (Ai,..., An). We now start
from a classical analytic symbol p of order fi defined on U X G where U is
a neighborhood of 0 G Rn+1 and G is an open cone in Rn+1. Let p^ be its
principal part. We assume that p^ vanishes of some order s on an analytic
homogenous regular involutive variety E in T*U which contains the point
(0,A°), A0 = (0, ...,0,1). It will be no loss of generality in applications
to assume that E = {(z, A); A' = 0} for some group of variables of type
A' = (Ao, Ai,..., A^, A = (A', \d+u " "> ^n)«
Let us also compute the localization p^\ of p^ along E. It is in general
a function on the co-normal bundle to E, but in our special coordinates
above, we may just write (for A in a conic neighborhood of A0) that
Pm,i(*. A) = £ (9/dXTp^ 0, Arf+i,..., \n)\'a/al.
\a\ = s
In particular it is clear from this that p^\ is positively homogenous of order
ji in the variables A, and, in addition, homogenous of order s in the variables
A'. It also follows that
P,(z, A) = PM(z, A) + OflA'I^IAr-1).
If P/i,i were elliptic, its order of magnitude were |A,|S|A|/1~S. Thus in a small
conic neighborhood, p^\ would dominate pM. Consider d! < d, denote A" =
(A0, Ai,..., \dt) and fix A1 ^ 0 with A1" = 0 and X} = 0 for i > d. Also

83
assume that p^i vanishes of some order m on {(z, A); A" = 0}. We denote
by p^2 the localization of p^\ along A" = 0. It is thus given by the relation
P„,a(*,A)= £ (^A")V1(^0,Ad(+1,...,An)A^//?!.
101 = ™
It follows that, in addition to the homogeneities inherited from p^i, p^2 is
homogeneous of order m in A". It is possible to give an invariant meaning
also to these conditions in terms of the bi-homogeneous and bi-symplectic
structures of the normal bundle to E; we refer to [17] or [20] for details.
We have not studied the invariant meaning for the statements which
follow hereafter. In any case, p^2 is of form $}7|=m »7(z, A^/+1,.. . ,An)A"7
with a7(z, Aj'+i,..., An) positively homogeneous of order ji — m in A and
homogeneous of order s — mm Xf and the relation between p^\ and p^^ is
pM,i(*.A) = P^(*,A) + 0(|AT+1|AT"m"1|Ar-).
We shall now write the variables A" as A" = (r, £"), where again r = Ao-
Similarily, A' = (r,C'). We also fix A2 / 0 in Rn+l with Ag = 0, A? = 0 for
i > df. We moreover assume that p^2 satisfies the following conditions:
a) the coefficient of rm in p^^ does not vanish at (z = 0, £2", A^,+1,...,
A^, A^+1,..., A^). Recall that this coefficient has homogeneity // - m in A
and s - mm A'. It has therefore order of magnitude 0(|A,|*~m|A|/1~*).
b) p^2 vanishes of order m at (z = 0, A2", Aj,+1, ..., A J, Ag+1,..., A°).
c) P/x,2 is micro-hyperbolic with respect to t = 0 at (z = 0, A0, A1, A2).
By this we mean that there is a real neighborhood C/7 of z = 0, a real
tri-neighborhood G" of (£°,£\£2) ("tri" ="multi'\ for k = 3.) , and c> 0
so that P/i,2(2,r,£) = 0, z £ U', £ £ Gf together with \r\ < c|£"| implies
Im r < 0.
Note that by assumption a), the coefficient of r in p^ is elliptic in
the third order microlocal calculus near (0, A0, A1, A2). 3-microlocally near
(0, A0, A1, A2), it is therefore no loss of generality to assume (if we compose
everything with the inverse of the coefficient of r) that we have
p„(z, \) = rm+ £ aQtJ(z, Ad,+1,..., AB)<"V + 0(|A"r+1/|A'|)
\a\+j=m,j<m
+0(|A'r+7|A|), (27)
with coefficients aaj which are positively homogeneous of order zero in
A and A'. This is, actually, 3-microlocally, the model on which we work.

84
Even in regions where p^2 is 3-microlocally elliptic, it will have at most the
order of magnitude |A"|m. It can therefore dominate the remainder term
0(|A'|m+7|A|) only in regions of form |A"| > c|A/|1+^/|A|" with /? < l/m.
(We shall work with (3 = l/(2ra), to make a choice.) This is the justification
why we restrict our attention to such regions in the definition of WF\.
Remark 11.2. It seems that we have lost orders of vanishing in A/', since
the model p^ vanishes only of order m when A' = 0, if it has form (27).
Fact is that we are working 3-microlocally, in a region away from A' = 0,
and that the representation (27) can be used only there.
We can now state the following result:
Theorem 11.3 Assume that under the above assumptions u is a
distribution defined in a neighborhood of U and that it satisfies the following
conditions for some tri-neighborhood W of (0, A0, A1, A2):
WF%p(z,D)unW = Q, (28)
WF%unWn{t<Q} = Q. (29)
Then it follows that (0, A0, A1, A2) g WF\u.
Remark 11.4 Although we have stated theorem 11.3 for the case of 3-
microlocalization, the argument works as well for the case of standard wave
front sets, respectively for the case of second microlocalization. In particular,
one thus obtains a new proof for theorem 11.1. As far as the case of two-
microlocalization is concerned, I was told by prof. N.Tose that he is also
aware of the fact that a result of the type of theorem 11.3 is true. I also
think that the result remains valid in arbitrary microlocalization, but I have
not checked all details. Note that the result presented here, as well as its
analogue for the case of two-microlocalization, refer to a highly involutive
setting. The proof of theorem 11.3 will be given elsewhere.
We give an example in which one sees that the theorems above give the
possibility to study questions of propagation of singularities in a rather
systematic way. Let q(x, ^1,^2^3) be an elliptic polynomial of order two in
(^1,^2^3)? with real analytic coefficients which depend on x £ R4 (much
less is needed) and denote by p(x,£) a symbol of form:
(6 + *'*foM^i,6,6) + o(l(&,fc,6)l4/fc) + 0(l£l2)
in four variables at (0,£°) where £° = (0,0,0,1). Singularities of solutions
of p(x, D)u = 0 then propagate in planes parallel to the (a?i, £2)-plane. This
can be obtained in the following way:

85
Lemma 11.5. If n' = 3, f = (6,6,6) € R3, f1 £ R3, then WF\u at
(£°,£1) i>s either void or propagates in planes parallel to the (xi,X2)-plane.
Proof. Here ji = 3 and p^ vanishes of order s = 3 on 6 — 6 = 6 = 0.
The localization polynomial on 6 = 6 = 6 — 0 ls P^i{x^) = (6 +
^16)9(^,6,6,6). Let f1 = (#,&^). If 61 ^ °> then Pm,i is elliPtic
at (a;,^0,^1) in the two-microlocal calculus, so (a;,^0,^1) ^ WF\u. The
same is true if £| 7^ 0 and ^1 7^ 0- When £{ = O,^ / 0, then pMjl is
microhyperbolic at X\ = 0. Thus we obtain once more that (a;,^0,^1) ^
WF^u. (We apply the second order version of theorem 11.3.) Thus |£|| +
l^ll / 0 implies (a;,^0,^1) £ WF\u,\/x. It remains to study the case f1 =
(0,0, ± 1). We propagate WF\u along the (a?i,a?2)-planes in this case by
showing that WF\su at (a;,^0,^1,^2) is void for any choice of £2 = (fi,^!)-
Here ^,2(2, £) — P/mO^O- If £1 / 0 or if £2 7^ 0 an^ simultaneously
xi ^ 0, this is again ellipticity. When £2 = 0, £| 7^ 0,a?i = 0, we can use
again theorem 11.3 to see that we have no WF\su at (a?,£0,£\£2).
12. The theorem of Bony-Schapira
1. We want to explain some of the ideas involved in the proof of theorem
11.3 by sketching a proof of a theorem of Bony-Schapira, which was the
first in a number of results which were generalized in [12]. We recall at
first the Cauchy-Kowalewska theorem. Let us in fact consider some linear
partial differential operator with analytic coefficients of order m defined in
a neighborhood U of 0 £ Rn+l of form
p(t, *, Du Dx) = D?+ £ aaj(t, x)D\Dax.
|a|+j<ra,j<ra
We assume that the coefficients aaj admit analytic extensions to a complex
neighborhood ft of 0 £ Cn+1 (Here t £ R, x £ Rn, (t,x) £ U.) We also
consider the Cauchy problem
p(t, x,, D)u = gon \z\ < e2, (id/dt)^t=0 = gj,j = 0,..., m - 1. (30)
Theorem 12.1. For any E\ > 0 which is sufficiently small, we can find
e2 > 0 so that if f £ A(z £ Cn+1; \z\ < Si), gj £ A{x £ Cn; \x\ < Si), then
we can find u £ f £ A(z £ Cn+1; \z\ < £2) so that (30) is valid.
By duality we obtain from this
Theorem 12.2. For any E\ > 0 which is sufficiently small, we can find
62 > 0 so that if v £ A'(z £ Cn+1]\z\ < 62) is given, then there are
w £ A!{z £ Cn+1;|z| < ei) and Wj £ Af(z £ Cn+;\z\ < Si) such that
v(u) — w(f) +Yl7jl=J wj(9j)i tf uifi9j are as in theorem 12.1. This can

86
also be written as
m—1
v = fp(z, Dz)u; + >T D^ ® wj. (31)
i=o
Here we have denoted by *p the formal adjoint of p and by S the Dirac
distribution at 0 in the variable t.
The interesting thing is now that from (31) we can obtain by direct
means rather explicit information on the map *T, in that one can obtain an
almost explicit formula which gives the Fourier-Borel transform of w. We
explain this at first in the case of constant coefficients. In fact, if we take
the Fourier-Borel transform of (31), we get
m— 1
v(\) = P(-\)w(\)+Y,wJ(<;y, (32)
i=o
which shows that w and the Wj are just the quotient and remainder terms in
a Weierstrass-type decomposition of v. (This also shows that we may view
(31) as a non-commutative version of the Weierstrass preparation theorem.)
One can compute w explicitly from (32) using contour integration formulas.
Note in fact that it follows from (32) that
1 w(r + aX) = 1 v(r + aX) 1 V ^ (C) (r + ^
2ni a 2ni ap(-r - a, Q 2ni ^ Wj^ j ap(-T - a, Q '
whenever a and p(-r - a, Q are different from zero. Integrating this in the
complex a- plane over a contour of form |cr| = c(l + |C|) for some sufficiently
large c, (contours in contour integrals are always with counterclockwise
orientation,) we obtain
<b{T,Q = ±:[ -pr+lQ-da, (33)
2th ./M=c(i+|C|) <rp[-T ~ °, -C)
since
/
J\a
i^^ -da = 0, forifc<m-l, (34)
\<t\=c(i+\(\)vp{-t-<j,0
if c is large enough. (We can apply the residuum theorem at oo.) All this
is of course standard. For the variable coefficient case, the situation is only
slightly more involved. Indeed, let us denote by Ylj Qj a formal analytic
symbol inverse to p. It is defined e.g. on the set \r\ > c(l + |C|) if c is large.
Also consider the map f which associates with v the function
2tt»A(a) ° j<J\\\

87
One can then prove with arguments which are only slightly more
complicated than in the constant coefficient case that
f(fp(z, D)w){\) ~ w{\),f{DJt6t ® wj)(\) ~ 0.
Let me in fact sketch a proof of the first assertion. We start with a
preparation
Proposition 12.3. There is x° such that for any fixed \9 X < X°> we can
find C > 0, d > 0, such that
ll-e^^piz.D)^^ J2 9jM)]|<Ce-d|A|, if\z\<e,-\£G.
j<x\M
( This is based on
e^A)p(2,D)[e-^A) £ qj(z,\)) = p(z,-\ + D) £ «,-(*,-A) =
j<x|A| j<x\M
£ (l/<*\)p(a)(z, -\)D"Z £ <U(z,-\) = S(z,\)~l,
\a\<m 3<x\M
etc.)
We return to the proof of the first assertion: we have
f(fp(z, D)w)(X) = (l/27rt>{ / (l/a)p(z, D)[e-*zMaN}>
Ja(\)
£ qi(z,-\-*N)]d*}.
j<x\M
Applying the proposition, we remain with
w[ [ e-^zM<jN\l/a) da] = w(\).
Ja(\)
To state the theorem of Bony-Schapira let us denote by pm the principal
symbol of p. Thus
Pm(t, x, r,Z) = Tm+ Y. a"i(*> x)TJZa-
\oc\+j<m,j<m
We assume that pm is hyperbolic with respect to t = c. By this we mean
that (£, x) £ U, £ £ Rn and pm(t, x, r,£) = 0 together, imply t £ R. Let us
also recall the following result of Bony-Schapira (cf. [1]):

88
Theorem 12.4 Let e be given small enough, fix e < e and Cl > 0. Then
there is C > 0 so that if
Qi e A(x e Rn\ \x\ < e), f e A{{t,x) e fln+1;C"|t| + \x\ < e),
are given, then we can find u £ A((t,x) £ Rn+l]C\t\ + \x\ < e), so that
(SO)is valid.
We want to sketch a new proof of this results which if suitably adapted will
be used to prove theorem 11.3. The proof is by duality. One of the main
preliminary ingredients which we want to mention is an inequality for the
roots of the characteristic equation if the principal part is hyperbolic. More
precisely, one has
pm(*,r,0 = 0 implies \Imr\ < c(\Imz\\Re(\ + |/mC|), (35)
if z is in a complex neighborhood of 0 G Cn+1 and £ G Cn.
Remark 12.5. (35) is often obtained from hyperbolicity using a so called
"local form of Bochner's tube theorem". (Cf again [1], or [14])- It is there-
fore perhaps interesting to mention that it is also a consequence of a suitable
form of the Phragmen-Lindelof principle applied to the plurisubharmonic
functions p(z,Q = supJmr(z,£), p(z,Q — SUP [-ImT{zi()]> where the
supremum is over all roots r{zX) of Pm{z,r,Q — 0.
(The form of the Phragmen-Lindelof principle which we have in mind is the
following:
Theorem 12.6.(Cf. e.g.[24]) Consider A = {y 6 Cs,\y\ < 1} and let
X : A -* R be a plurisubharmonic function so that x{y) < 0 if y £ An Rs
and x{y) < ci. Then there are constants C2,c3 which depend only on c\ so
that
X{y) < C2\Imy\ if y£ Cs,\y\ < c3.
By applying this for the function />, we obtain at first p(z,Q < c2(\Imz\ +
\Im C|) for small (z, £), and fr°m this we obtain (35) after homogeneization.)
We next introduce some notations. When C, C", 6, e' are given, we denote
by
K = {{t,x)eRn+1;C\t\ + \x\<e}, (36)
K' = {(t,x) € Rn+l;C'\t\ + |x| < e'}, (37)
and by Hk, Hri, the supporting functions of K and K' in Cn+1. Thus
HK(\) = sup/m(y,A),
y£K
and similarly for K'.
It is now clear that theorem 12.4 is a consequence of the following result:

89
Proposition 12.7. Consider 0 < e < e1 < e with e fixed sufficiently small,
but no other condition on s,sl. Also fix C > 0, vl > 0. Then there are
v > 0, C > 0 (with C independent ofv'l) so that ifv is a Radon measure on
{z e Cn+1;C\Ret\ + \Rex\ < e,\Imz\ < //}, which satisfies f d\v(z)\ < 1,
then we can find analytic functional (w, w0, ..., Wm-i) related to v by (31)
so that (with the notation for Kf explained in (37)),
\w(X)\ < cexp[v'\Re\\ + HK,(Im\)], (38)
|^(C)| < cexpIi/lfleCl+^'l/mCl]. (39)
How will we prove this? Well, we have a precise geometric information on
supp v and want a precise analytic information on w and wj. Here we can
calculate w approximative^ with our contour integral formulas, whereas
the Wj can be estimated with the aid of (31). The geometric information
on supp v tells us which z are to be taken into account. Hyperbolicity gives
as an estimate on where the residua of the contour integral are located.
We can now deform the integration contour in such a way as to make the
exponential harmless. (To some extent, the situation is inverse to the one
we had in proposition 4.2: we have a geometric information on t; and we
obtain an analytic information on w.)
References
1. J.M. Bony- P. Schapira: Solutions hyperfunctions du probleme de Cauchy. In "Hy-
perfunctions and pseudodifferential equations", Springer LNM vol. 287, Springer
Verlag Berlin-Heidelberg, 1973.
2. L.Boutet de Monvel: Operateurs pseudo-differentiels analytiques et operateurs
d'ordre infini. Ann. Inst. Fourier,Grenoble, 22:3, (1972), 229-268.
3. L.Ehrenpreis: Fourier analysis in several complex variables. Interscience
Publ.Comp., 1970.
4. P. Esser- P. Laubin: Second microlocalization on involutive submanifolds. Seminaire
d'analyse superieure, Univ. de Liege, Institut de Mathematique, Liege 1987.
5. Second analytic wave front set and boundary values of holomorphic functions.
Applicable Analysis, vol. 25 (1987), 1-27.
6. I.M. Gelfand-G.E.Shilov : Generalized functions. Academic Press.
7. L. Hormander: Uniqueness theorems and wave front sets for solutions of linear
differential equations with analytic coefficients. C.P.A.M., 24, (1971), 671-704.
8. The analysis of linear partial differential operators I. Springer Verlag, Grundlehren
der mathematischen Wissenschaften, vol.251, Berlin-New York, 1983.
9. K. Kajitani- S. Wakabayashi: Microhyperbolic operators in Gevrey classes. Publ.
R.LM.S. Kyoto Univ.,25(1989), 169-221.
10. The hyperbolic mixed problem in Gevrey classes. Japan J. Math. 15 (1989), 315-
383.
11. M. Kashiwara M.- T. Kawai: Deuxierne rnicrolocalisation. Proc. Conf. Les Houches
1976, Lecture Notes in Physics vol. 126, Springer Verlag, Berlin Heidelberg New
York.
12. Microhyperbolic pseudodifferential operators I. /. Math. Soc. Jap. 27 (1975), 359-
404.

90
13. T. Kawai: On the theory of Fourier hyperfunctions and its applications to partial
differential operators with constant coefficients. J. Fac. Sci Tokyo IA, 17:3 (1970),
467-519.
14. H. Komatsu: A local version of Bochner's tube theorem. J.Fac.Sci. Tokyo, IA,19
(1972), 201-214.
15. P. Laubin: Propagation of the second analytic wave front set in conical refraction.
Proc. Con}, on hyperbolic equations and related topics, Padova, 1985.
16. Etude 2-microlocale de la diffraction. Bull. Soc. Royale de Science de Liege, 56:4
(1987), 296-416.
17. Y. Laurent: Theorie de la deuxierne rnicrolocalisation dans le dornaine cornplexe.
Birkhauser Verlag, Basel, Progress in Math., vol.53, 1985.
18. G. Lebeau: Deuxierne rnicrolocalisation sur les sous-varietes isotropes. Ann. Inst.
Fourier Grenoble, XXXV:2, (1985), 145-217.
19. O. Liess: The Cauchy problem in inhomogeneous Gevrey classes. C.P.D.E., 11,
(1986), 1379-1439.
20. Conical refraction and higher microlocalization Springer LNM 1555, 1993, Springer
Verlag, Berlin Heidelberg.
21. Higher microlocalization and propagation of analytic singularities. Kokyoroku Series
of the R.I.M.S. in Kyoto, 1996:2, 60-72.
22. d — cohomology with bounds and hyperfunctions. Preprint nr.l, University of
Bologna, 1996.
23. A. Martinez: Lectures in these proceedings.
24. R. Meise- B.A.Taylor-D. Voigt: Phragmen-Lindelof principle on algebraic varieties.
To appear.
25. Y. Okada-N. Tose : FBI-transformation and microlocalization- equivalence of the
second analytic wave front sets and the second singular spectrum. Journal de Math.
Pures et AppL, t. 70:4, (1991), 427-455.
26. M. Sato- T. Kawai- M. Kashiwara: Hyperfunctions and pseudodifferential operators.
Lecture Notes in Math., vol. 287, Springer Verlag, Berlin Heidelberg New York,
1973, 265-529.
27. J. Sjostrand: Propagation of analytic singularities for second order Dirichlet
problems I. C.P.D.E., 5:1, (1980), 41-94.
28. Singularities in Boundary Value Problems. Proc. of the Nato ASI, Maratea 1980,
Ed. by H.G.Garnir, Reidel Publ. Comp., Dordrecht Boston-London 1981, 235-271.
29. Analytic singularities and microhyperbolic boundary values problems. Math. Ann.
254 (1980), 211-256.
30. Singularites analytiques microlocales. Asterisque vol.95, 1982, Soc.Math. France.
31. B.A. Taylor: Analytically uniform spaces of infinitely differentiate functions.
C.P.A.M., vol. XXIV (1971), 39-51.
32. N. Tose: On a class of microdifferential operators with involutory double
characteristics- as an application of second microlocalization. /. Fac. Sci. Univ.
Tokyo, Sect. IA, Math. 33, (1986), 619-634.
33. The 2-microlocal canonical form for a class of microdifferential equations and
propagation of singularities. Publ. R.I.M.S. Kyoto, 23-1, (1987), 101-116.
34. Second rnicrolocalisation and conical refraction. Ann. Inst. Fourier Grenoble, 37:2,
(1987), 239-260.
35. Second rnicrolocalisation and conical refraction, II. "Algebraic analysis", Vol.11.
Volumes in honour of Prof. M.Sato, edited by T.Kawai and M.Kashiwara, Academic
Press, 1989, 867-881.
36. K. Uchikoshi: Construction of the solutions of microhyperbolic pseudodifferential
equations. J.Math. Soc. Japan, vol. 40, (1988), 289-318.
37. S. Wakabayashi: A classical approach to studies on propagation of analytic
singularities. Kokyoroku Series of the R.I.M.S. in Kyoto, 1996:2, 60-72.

CONORMALITY AND LAGRANGIAN PROPERTIES
IN DIFFRACTIVE BOUNDARY VALUE PROBLEMS
P. LAUBIN
Institut de Mathematique
Universite de Liege
Avenue des Tilleuls, 15
4000 Liege, Belgique
1. Introduction
Our main purpose is to study the lagrangian structure of the solution of
a strictly difFractive boundary value problem at the transition from the
shadow to the illuminated region. If the incoming data or the boundary
data are conormal then two lagrangian submanifolds are involved there.
Because of the geometry of the difFractive rays, their intersection is not
clean. We try to describe the solution with phase functions and oscillatory
integrals.
Let M be a real manifold with boundary and P a second order
differential operator with smooth coefficients and real principal symbol p. We
assume that p is of real principal type and not characteristic on the
boundary. Let us consider the classical Dirichlet problem
Pu = 0 in M, u\dM = 0.
If the equation of the boundary is / = 0 with / > 0 in M, the difFractive
region is defined by
G+ = {peT*dM:p(p) = 0, ^A^n > 0}
{{pJiJip
and corresponds to rays tangent to the boundary. The propagation of
singularities of C°°, Gevrey and analytic singularities is known in this setting,
see [11], [15], [8], [9]. However, very few lagrangian properties are preserved
along difFractive rays. In [10], Lebeau proves that, far away from the data,
the operator mapping the Dirichlet data to the normal derivative of the
solution belongs to a class of lagrangian Gevrey 3 distributions with weight.
91
L. Rodino (ed.), Microfocal Analysis and Spectral Theory, 91-113.
© 1997 Kluwer Academic Publishers.

92
We first study the properties of the solution at the transition from the
shadow to the illuminated region in the C°° framework. Using the
canonical invariance, we prove that the solution belongs to a class of lagrangian
distributions associated to a pair of lagrangian submanifolds. As a
consequence, we see that, for a conormal data, the second wave front lies in a
lagrangian submanifold.
We next investigate the same problem in the analytic category. Here we
use the geometry of complex canonical transforms and the H^ spaces of
Sjostrand. Our main tool is the parametrix of Lebeau. We generalize the
definition of bilagrangian distributions in this framework and describe the
FBI transform of the solution of the boundary value problem.
2. Pairs of lagrangian submanifolds
2.1. MICROLOCAL PHASE
Let X be a C°° manifold of real dimension n and with local coordinates
a?i,..., xn. On the cotangent bundle T*X, we consider the canonical 2-form
n
where the dual coordinates are defined by d£j(DXk) = Sjk- This manifold
is conic for the multiplication Mt : (#,£) »->• (%,t£). We denote by T*X =
T*X \ {0} the cotangent bundle with the zero section removed.
A submanifold A of T*X of dimension n is lagrangian if a^ — 0. It is
said conic if it is invariant through Tt for every t > 0.
The classical definition of a phase function for a conic lagrangian
submanifold is the following, [2]. For simplicity, we restrict ourself to the case
of a real non-degenerate phase function.
Definition 1 Let X be a C°° manifold and <p be a C°° real valued function
in an open conic subset T of X x IR^ \ {0} which is homogeneous of degree
1. The function <p is called a local phase function of X if dtp / 0 in T and
rk(^, w'qq) = N in the set
c^{(x,o)er:^,o) = o}.
If (p is a local phase function then the differential of the map
jrC^M:(M)4(^;(M))
is of rank n. If it is an embedding then (p is called a phase function. Since
ivCT = Jtd(Zdx) = difxdx) = d(df\cv) = 0,

93
its image A^> = jv(Cv) is a lagrangian submanifold of f*X.
2.2. 2-MICR0L0CAL PHASE
The second wave front set along a lagrangian submanifold A is defined as
a subset of the cotangent bundle of A. To define lagrangian distributions
associated to this geometric setting, we introduce new phase functions.
If A is a conic lagrangian submanifold of T*X, then we have the
identification
T*A - TAf*X
where the right hand side is the normal bundle of A. Indeed, if k is a normal
to A at a point p then TPA 9 h »-* a(/i, k) is a well-defined 1-form.
Moreover this manifold has two homogeneities: one inherited from A
and another one as a cotangent bundle. A lagrangian submanifold of T*A
is said conic bilagrangian if it is conic for both homogeneities. We introduce
phase functions that parameterize such a manifold.
Let T0 be an open subset of X x WLN \ {0} x 1RM \ {0} such that
(#, 0, rj) G To and s, t > 0 imply (#, £0, strj) G r0. Such an open set is called
a profile. An open subset r of X x IR^ \ {0} x IRM \ {0} is said biconic with
profile To if
- (x, 0, rj) G T and t > 0 imply (x, £0, trj) G T,
- for each compact subset K of To, there is e > 0 such that (a:, 0, srj) £ T
if (x,0,rj) £ K and 0 < s < e.
If T is biconic with respect to a family of profiles, it is also biconic with
respect to their union. The profile of T is the largest profile r0 such that
the last condition is satisfied.
We also introduce
Ti = {(x,0):3r) such that (x,0,rj) G T}.
This is an open conic subset of X x IR^ \ {0}.
Let p, q G 1R and r G 1N0. A C°° function / : T -* IRm is said bihomo-
geneous of degree (p, </; r) if
- /(*, tO, trj) = trf(x, 0, rj) if (x, 0, ri) e r, t > 0,
- for every (xo, 0o, ?/o) G To, there is a neighborhood V of (a?o, #o> Vo) and
a C°° function F in Vx] - €, c[ satisfying
if (x,0,r/,5)G Vx]0,c[.

94
The integer r is inserted here essentially for technical reasons. In the
application, it does not affect the 2-microlocal geometry but has some
effects on the microlocal lagrangian submanifolds involved. We say that /
has the regularity r.
Definition 2 Let
- A be a conic lagrangian submanifold ofT*X,
- (f be a C°° real valued function which is homogeneous of degree 1 in
Ti,
- $ be a C°° real valued function which is bihomogeneous of degree
(1,1; r) inY
and
CW = {(*> *» V) G To : ¥>',(*, 0) = 0, ^(s, 0, r?) = 0}.
The pair (y>, V7) « a /oca/ 2-phase function of A (with regularity r) if
- (f is a local phase function that parameterizes A,
- at each point ofC^, the vector (^i x, il>[ q) is different from 0 and
V Vftc ¥& 0 /
If yj is a phase function, the last condition means that the map (/), rj) »-)■
il>i{jyl{p),v)ls a local phase function of A. This definition has the following
consequences.
a) The map
Ud : CW "> r*A : (x,M) *-> ((», Vx)» iv*((^i,x^i^)|TCv))•
JS a lagrangian immersion.
Assume that (/i, fc, u) is in the kernel of the differential of this map. It
follows that h = 0, y^.fc = 0, y^.fc = 0, ipi^-h + V*"^'^ = 0 and there is
v such that
We have k((pfQx, ^) = 0 hence fc = 0. Moreover
hence u = 0. This proves that j^ is an immersion.

95
Let a be the canonical 1-form of T*A, 7r : T*A —>■ A the projection on
the base, (x,9,r]) a point of C^ and j = j^. If (h,k,u) is tangent to
C<^ at (a:, 0, 77) and a = j(x, 0, 77) then
j*(a).(h,k,u) = a(j*(/i,fc,u)) = cr(7r1|(i1|((fc,fc,ti)) = cr( „ , „ , )
= tl)[iX.h + tl)[fi.k = dtl)i.(h,k) = Q
since ^i is equal to 0 on C^.
Following the identification T*A ~ TaT*X, ffte map j^ can be
identified with
CW -> ^aT*X : (x, 0, t?) h+ ((*, ^), (h, ^x + ^./i + ¥&.*))
wftere /i, A; satisfy
$xM + $e.k + j>'he = 0.
Indeed, if ^./i + <pfQQ.k + $[ e = 0 and (u, v) is tangent to C^ then
(u
b) Let (y>, V7) be a local 2-phase function (with regularity r) in a biconic
set T and (#o>0o>??o) £ C<^. By the definition, <p is a local phase
function in Ti and there is a biconic open subset f of T whose profile contains
(#o> #o> Vo) such that (x, (0,77)) 1-* y>(ar, 0) + ip(x, 0, 77) is a local phase
function in f. A local 2-phase function (y>, xj)) is called a 2-phase function if j^,
jv+^ and j^^ are embeddings.
One can verify that z/ (y>, ^) is a /oca/ 2-phase function in T and
(#o>0o>f?o) G C^ ^en ^er^ is 0 biconic open set T whose profile
contains (xo,0o,r)o) such that (<p,rl>) is a 2-phase function in T.
Hence, if (y>, ijj) is a 2-phase function then
is a conic bilagrangian submanifold of T*AV?. It is denoted A^.
c) // (</?, ty) is a 2-phase function, then
n - rk(7rAv?,x) = N - rk(y^) , n - rk(7rA^,AV) = M - rk(^w),

96
and
\ ° <Poo
The first two equalities are known from the study of microlocal phase
functions. Consider the map C^ 9 (#, 0,rj) \-+ x £ X. A vector (/i, fc, u) is
in the kernel of the differential of this map if and only if h = 0, y^.fc = 0
and ifriqQ.k + il>"wU = 0. This proves the equality.
2.3. PAIRS OF LAGRANGIAN SUBMANIFOLDS
We now describe the geometric setting associated to a 2-phase. If Y is a
submanifold of a C°° manifold X, the blowup of X along Y is
XY = (X\Y) UTyX.
The sets
f| ({xeu: fj{x) > 0} U {(a, h) G TyX :xeu, dfj(x).h > 0})
i<j<p
where u is an open subset of X and /j G C°°(u;), /jiyna; = 0 for all j, form
a basis of topology of Xy F°r this topology, the projection n : Xy —> X is
continuous.
Definition 3 A pair (Ao,Ai) is a 2-microlocal pair of lagrangian subman-
ifolds of f*X if
- A0 is a conic lagrangian submanifolds of T*X, A\ C (T*X)^o,
- Ai n (T*X \ Ao) is a conic lagrangian submanifold of T*X,
- for each (p,h) G Ai r\T\0T*X, there is an open neighborhood V of
(/>, h) in (f*X)^Q and a 2-phase function (</?, ^) such that
A0 fl tt(V) = Ay and Ai n V = Av+^ U A<^/,.
In this situation, we say that the 2-phase function (</?, ij)) defines (A0, Ai).
Let Ta0Ai = Ai r\T&0(T*X). This is a conic bilagrangian submanifold of
T*A0.
Example 4 In T*]Rn, consider
sn

97
where £ = (£',£n) and H is bihomogeneous of degree (1, l;r). We have
Av = {(0,O:Cn#0}
and
>W = {((-r' ^ + Hi^ tinK'tn)) : £n * 0}.
Sn sn
If H(rjf^n) = r/f/r/l in IR3, the projection of TavAv+^ on A^> is the cusp
{(O,0:(|)3 = (|)26:6#0}.
It can be shown, see [6], that the property of being a microlocal pair of
lagrangian submanifolds is preserved by an homogeneous canonical
transformation.
Let us describe the equivalence of 2-phase functions.
Two 2-phase functions (</?, ij)) and (<£, rj>) defined in biconic open subsets
r and f of X x IR^ \ {0} x IRM \ {0} are said equivalent if there is a C°°
difFeomorphism r —> T : (x,0, rj) \-+ {x, f(x,0,rj),g(x,8,r))) such that
- <p(x, f(x, 0, rj)) + ip(x, /(x, 9, rj),g(x, 0, rj)) = <p{x, 9) + $(x, 9, 77),
- f is strictly bihomogeneous of degree (1,0; r) and g is bihomogeneous
of degree (1, l;r),
- Defo and Dvgi are invertible in T0.
These two pairs define the same 2-microlocal pair.
If A is a diagonal real invertible matrix, the pair of phases
<p(x, 9) = ip(x, 9") + ^p , t/>(x, 6", rj) = tf (*, 9", rj)
defines the same lagrangian submanifolds as (p and ip. In the same way,
¥>(*, 9) = <p(x, 9) , t/>(x, 9, rj) = i/>{x, 9, r,") +
2\rf'\
defines the same lagrangian submanifolds as (p and ij).
It can be shown that the transition between two 2-phase functions
defining the same 2-microlocal pair of lagrangian submanifolds can be obtained
by a composition of the previous reductions.
3. Bilagrangian distributions
3.1. SYMBOLS
We use only classical symbols. This is enough for the applications that we
consider here.

98
Definition 5 If m,p £ H and X is an open subset of ]Rn, we denote by
Sm>p{X,RN,nM) the set of all a G C°°(X x WLN x RM) such that for
every compact subset K of X and all multiorders a, /?, 7 there is a C > 0
satisfying
\D°DPD]a(x, M)l < C(l + |0| + \v\)m-W(l + \v\)p^
for all (x, 0,77) e K x WLN x 1RM.
Write
52°° = (J 5m'p , 5m'-°° = f| Sm'p.
It is clear that Sm'v is a Frechet space with semi-norms given by the smallest
constants which can be used in the definition.
Oscillatory integrals can be defined using symbols in Sm>p and 2-phase
functions.
Theorem 6 Let (y>, \j)) be a 2-phase function in an open biconic set Y and
let F be a closed conic subset ofT such that F<T. For every u G Cq>(X),
the linear form
a ^ /// ei(vW)+W*f*))afa ^ v)u(x) dxdedr]
defined in the set of all a £ S~°°(X;]RN X 1RM) satisfying supp(a) C F9
can be extended on 5?° in a unique way such that it is continuous on the
set of a 6 Sm'p(X^ IR , HM) satisfying supp(a) c F for every m,p.
3.2. DISTRIBUTION CLASS
Let X be a C°° manifold of dimension n and let (Ao,Ai) be a 2-
microlocal pair of lagrangian submanifolds of T*X.
Definition 7 The space Im>p(X, A0, Ai) is the set of all locally finite sums
of an element of Im(X,Ao), an element of Im+p(X, Ai DT*X) and
distributions of the form
Va(it) = (27r)-(n+2(iV+M))/4 fffeW^WW'^
where (t/,x) *5 a cftarf of X, u £ Cq:>(X), (y>, VO is a 2-phase function of
(Ao, Ai) defined in an open biconic subset T ofX(U) x IR^O} xlRM\{0}
and
a g 5m+(n-2iV)/4,P-M/2(x(t/))]RiV)IRM)
satisfies supp(o) <C I\

99
It can be shown that this space is invariant by composition with a
Fourier integral operators. Moreover, any 2-phase function defining the
pair (A0,Ai) near a point p0 £ Ao can be used to define any element
of/m'P(X,A0,Ai) near/)0.
The singularities of an element of Im>p(X, A0, Ai) are included in the
lagrangian submanifolds involved, [6].
Theorem 8 If u G Im>p{X, A0, Ai) then
WF(u) C Ao U Ax , WF$(u) C TAoAx.
4. Application to diffraction
Let us consider the boundary value problem
r (-A + {i + xn)d?)u = o
I u\Xn=0 = S0 , u\t<0 = 0
where we use the decomposition (£, xf, xn) G IR x JRn_1 x IR+. This is a
model for the strictly difFractive problems in the C°° category, see [14].
Let
p{xn,T,Z) = \Z\2-(l+Xn)T2
be the principal symbol of the operator and r(r^f) = |^|2 - r2 be the
boundary hamiltonian. Two lagrangian submanifolds are involved here. On
one hand, we consider the flowout Ao = Ao,+ U Ao,_ of
{((0,0),(r,fl):r = ±K'|#0,£B = 0}
through Hr on the boundary and followed by Hp intersected with t > 0
and xn > 0. On the other hand, the flowout Ai = Ai,+ U Ai,_ of
{((0,0),(r,O):r = ±|^Un^0}
through Hp intersected with t > 0 and xn > 0. These two manifolds are
smooth but are tangent at their intersection.
It can be checked that (Ao,±, Ai,±) is a 2-microlocal pair of lagrangian
submanifolds with
2a0,±A1>± = {(((lx3n/2 + 2^,x',xn),(±\a^',T\e\V^))^
((0,0,0),(±-a,0,^<T(v/^+ -7=))) :<?,xn> 0,£' ± 0}.

100
A 2-phase function (y±,t/>±) of (A0)±, A^i UT\0t±Ait±) is given by
and
(v±+^)(^«^,0 = *'^±iri(l-^)-1/2(*-|((x»+^)3/a-(^)3/2)).
This 2-phase function has the regularity 2.
We denote by I™(X,Aq) the set of all lagrangian distributions on A0
with symbol in S™. This means that the symbol satisfies the following
inequalities
\DZD$a(x}0)\ < Ca,p{l + \0\)m-W+(l-M\a\+W).
An analysis of the solution of the initial boundary value problem given
in [3] leads to the following result.
Theorem 9 The solution u of the previous boundary value problem belongs
to
I*-l*{WL x IT"1 x 1R+, A0, Ai UTAoAi))
n 1
+ /24/3 2 (R x R71"1 xR+,A0).
5. The geometry in the complex domain
Our purpose is to define the phase functions used to characterize the bi-
lagrangian distributions in the formalism of the Fourier-Bros-Iagolnitzer
transform. In the microlocal case, we closely follow [7] and collect some
material from [10], see also [16].
As usual, we identify
- (Cn with Rn x Rn and write z = x + iy,
- C € T*zVn with (Ci,...,Cn) € (Cn using ((h) = ZjCjhj,
- T*<Vn with Tfxy)R2n by mapping the (C-linear form C G T*<£,n to the
R-linear form h 1-4 —I£(h).
This map is symplectic if T*R2n is endowed with the usual canonical 2-form
and T*(Cn with the 2-form -la defined below.
It follows that if / is a holomorphic function, df 6 T*(Dn is identified
with d{-lf) e r(;fy)R2n since d{-lf) = -l{df) = -l(df).
In the same way, if (f is a real function then d<p G TTX y\R*2n is identified
with \Dz<p £ (Cn.

101
All the constructions described in this section are local even this is not
stated explicitly.
5.1. FBI TRANSFORM
Writing z — x + iy and ( — £ + i?/, the canonical 2-form on T*(Cn is
° = J2d(jAdzr
3
Its real and imaginary parts
1Za = ^(d£j A dxj - drjj A dyj), la = ^(d?7j A dxj + d£j A dyj)
3 3
are symplectic forms on IR2n.
Let (f be a real C\ function defined in a neighborhood of z0 G (Cn and
Av = {(2r,?D^(2r)):zG(Cn}.
i
This manifold is I-lagrangian since it is identified with
{(^^)):^r}crR2n.
If jy denotes the immersion z \-± (z, ^Dz(f(z)) then
MKa) = j;(a) = j;(d((dz)) = d^dip) = ~. dd<p.
It follows that, if ddp is non degenerate, j^ is a symplectic map from
((Cn, | dd(f) onto (Av,7£a). Its inverse is the projection.
Let us remark that
2 — 2
-ddy(z)(u,v) = -J2DZjDzk(p(z)(ukVj - UjVk)
3,k
is real for every u, v G (Cn.
The tangent space to Av at the point j^(z) is given by
This shows that if dd<p is non degenerate then Av is a totally real subman-
ifold and that its complexification is T*(Cn.

102
Let us denote
2_
Cy(u,v) = -dd<p(u,iv)
= 2j2DZjDzk<p(z)(ukvj + ujvk)
= AU(H^(z)u,v)
the Levi 2-form of (p. This form defines a hermitian quadratic form Cip(u) u).
It is positive definite if <p is strictly plurisubharmonic.
We now assume that <p is strictly plurisubharmonic and endow A^ with
the symplectic form 1Za.
If z G (Cn, we denote by u »-* u the unique antilinear bijection from
Tjv(z)T*(Kn onto itself which is the identity on Tj ^A^. Let us consider
the hermitian form
q : Tu(z)Vn X Tu(z)Vn -+ <C : (tt,«) h+ ± <t(u, v).
Tfte signature of q is (n, n) and </ is negative definite on the tangent space
to the fiber.
Indeed, since a is not degenerate, the rank of q is 2n. If L is a complex
subspace of Tj^^®71 such that q(u, u) > 0 for every u £ L\ {0} then
g(TZ, tl) = - a(TZ, u) = -qiu, u) < 0
for every u G £\{0}. This shows that the signature of q is (n, n). If (0, u) ^ 0
is tangent to the fiber at jy(z) then
2 2 —
(0, u) = (fc, - D2z(p(z).h) with u = -DjDz(p(z).h.
z z
Hence
g((0,u),(0,u)) = -2(H^{z)h,h) < 0.
The following result is proven in [8], see also [5].
Theorem 10 Let <p be a strictly plurisubharmonic function near z$ G (Cn
and x ' T*IRn —> A^ a canonical transform defined near (j/o»^o) 5WC^ that
x{Voi Vo) = (^o, *Dz<p(zo)). Here A^ is endowed with the 2-form IZa. There
is a unique holomorphic function g(z)y) near (zq) j/o)> such that
— the complexification of x is
X<D . y.r ^ r<Cn . (J/| _£, (Zj y)) ^ (2) ^(Zj y)))

103
- ig{zo>yo) = <p(zq), -Dyg(z0,yo) = %,
- the function y »->■ -Tg(z)y) has a non degenerate critical point y(z)
with signature (0,n) and critical value <p(z). Moreover, we have
2
(y(z),-Dyg(z,y(z)) = x~1(z,-iDz<p{z)).
For example, if
then
9{^y) = -{z-yf.
The FBI transform associated to <p, x near the points (j/o>??o)>2o 's
rxu(z,A) = yeiA^»")o(z,y,A)tt(y)dy
where a is a classical symbol.
5.2. ISOTROPIC AND INVOLUTIVE SUBMANIFOLDS
The following result links the isotropic submanifolds of ((Cn, ? dd<p) to the
complex structure.
Proposition 11 Let <p be an analytic strictly plurisubharmonic function
near zq £ (Cn. If T is an isotropic submanifold of ((Cn, f dcfy>) then T is
totally real and there is a unique pluriharmonic function h on T® such that
(f\r<c - h > 0 and (y> - h)\r = 0.
Moreover, there is C > 0 such that
¥>|r<D -h>Cd(z,Y)2.
Proof. Let us prove that Y is totally real. If u, in £ T^T, it follows that
2- v^
0 = - dd<p(u, zu) = 2 2^ DZjDzk<p(z) u>juk.
Since <p is strictly plurisubharmonic, u is 0.
Denote by p the injection from T to T^. Since T is isotropic, we have
0 = p*(dd(<p\r*)) = dp*(d(<plrc)).

104
Hence there is an analytic function H : T —>• (C such that
P*M<P\r*)) = dH.
Let us denote 9 the holomorphic extension of H in T®. If z £ T, we have
d((f\rv)(z) = d0(z)
on TZT® since the equality holds on TZT and both sides are (C-linear. We
get
d(y>|rc) = d(<p\rc) + d{(p\rv) = 2d(TZ9)
on T^r^ at the points of T. Modifying 9 by a constant, we can take h = 2119.
The inequality follows from the fact that y>irc - h is strictly
plurisubharmonic and vanishes to the second order on the maximal totally real
submanifold r of T®.
The function h is unique. Indeed, a plurisubharmonic function vanishing
on T and having a null differential on TZY® at the points of T is equal to
0. □
Let us now consider an involutive submanifold V of ((Cn,u; = | dd<p). If
z £ V then
TZV © i{Tzvy = (Cn.
Indeed, if u G TZV and in G (TZV)LJ we have
2 —
0 = - dd(f(u, iu) = 2j2 DzJDjk(f{^) ufHk
hence u = 0.
This shows that the union of the complexifications of the bicharacteristic
leaves of V can be locally identified with (Cn. It follows from the
proposition 11 that there is a unique real analytic function <py equal to <p on V,
pluriharmonic on the complexification of the bicharacteristic leaves of V
and such that <pv < ¥>•
Proposition 12 Let (p be an analytic strictly plurisubharmonic function
near z0£(£n.IfV is an involutive submanifold of ((Cn, f ddip) then
Avv = {{z)-iDz<fv{z)):zeVn}
is the union of the complexification of the bicharacteristic leaves ofj<p(V).

105
Proof. The submanifold j<p(V) is involutive in (A^,7£a) and totally real
since it is included in Av. Let us denote by W the union of the complexifi-
cation of the bicharacteristic leaves of jy(V).
This submanifold is Z-lagrangian. Indeed, let p = jy(z) and let Y be
the leaf of V containing z. If h £ Tpj^(T) and k £ Tpjy(y), we have
Za(h,k) = 0 since Av is Z-lagrangian and also la(ih,k) = Ha(h,k) = 0
since j<p(V) is 7£-involutive.
Let us show that W is transversal to the fibers. If
u = v + iw£ TMV) © iTpJv(T),
it follows that
q(u} u) = - a(u} u) = - a(v + iw, v - iw) = 2 <j(w, v) — 0
since j^(V) is isotropic with respect to 1Za and la. We know that </ is
negative definite on the tangent to the fiber, hence W is transversal to the
fibers.
There is a real analytic function ^ such that
W = A* = {(z,-Dxtl>(z):ze1F}.
If z £ V, we have Dz(pv(z) = Dz(f(z) = Dz^(z). Hence we can
assume that ij) = (f on V. To see that ^ = ^y, we have to show that if) is
pluriharmonic on the complexification of the leaves of V.
Let T be a leaf of V and
jr :T® -+T*<£n : z v^ (z,- Dzi>{z)).
i
We have jr(T*) = j^f. Indeed, if n is the projection T*Cn ->■ C", we
have Troj'r = idrc. Moreover, n is holomorphic hence ^(jv(F)<c) = F® for
all the leaves F of V. This implies ir^®) C MTf.
On the other hand jv(F) is 7£ and I-isotropic. Its complexification
j^T)® is isotropic for <r. Hence
0 = fa = d(j*(<;dz)) = d(?d[^c]) = - 00[ifrc]-
This proves that ^ is pluriharmonic on T^. D
Remark that jr is holomorphic since it has a holomorphic inverse.
5.3. LAGRANGIAN SUBMANIFOLDS
For a lagrangian submanifold, we can use a holomorphic function.

106
Proposition 13 Let A be a lagrangian submanifold o/T* IRn, h be a phase
function of A near p0 and x be a local canonical map from T*IRn to A^
mapping po to z$. If g the FBI phase defined in theorem 10 and
<t>k{z) = cv(Xj6)(g(z,x) + h(x,e))
then (p\ = —Tcj)^. The critical points are given by
(M) = Jrc1 ° X^1 (z, Dz(j)A (z)).
Here j is the immersion (#,0) \-± (x,h'x) and j® is its complexification.
Proof. The critical points are given by g'x + h'x = 0 and h!e = 0. The
hessian matrix
\ K* hee )
is invertible since Tgxx > 0 and rk(/i^,/i^) is maximal. This implies that
the function 0a is holomorphic. Moreover, if z G n o x(A), we have
2
(z,-:Dz<p(z)) = (z,Dzg(z,y(z)) = Xv(y{z),-Dyg(z,y(z))) G x(A).
It follows that there is a 9 such that
Dyg(*> y(z)) + Dyh(y, e) = o, d*%, o) = o.
This shows that the critical point is (y(z))8) and we get —X(f)A{z) = <p(z).
Since the first derivative are also the same at these points, the proposition
follows from the uniqueness in proposition 11. □
We have
and
<Pa(z) <¥>(*)•
The equality holds if and only if (2, *Dz<p(z)) G x(A).
In this formalism, the lagrangian distributions are defined in the
following way.
Definition 14 Let u be a distribution in an open subset Q of IR/\ A a
lagrangian submanifold ofT*Q. With the notations of proposition 13, u is
said lagrangian at po if in a neighborhood of zq, we have
(Txu)(z,\) = eiX*^h(z^)
where b is a classical analytic symbol

107
This is equivalent to the fact that u can be written u = u\ + u<i with
p0 = jh(xo) 80) not in the singular spectrum of u<i and
m(x)= [ eih^x^a(x,e)de
where T is a conic neighborhood of 0q &nd a is a classical analytic symbol
near (xo,0o).
5.4. PAIRS OF LAGRANGIAN SUBMANIFOLDS
Let us consider the FBI transform of a 2-phase function. For simplicity, we
restrict ourself to the case of one 2-microlocal parameter.
Proposition 15 Let (Ao, Ai) be a 2-microlocal pair of lagrangian subman-
ifolds and (h^) be a 2-phase function for the pair (Ao,Ai) near a point
po G Ao- We assume that h is analytic and that ij) is an analytic function
o/M,*1/2),
t/>(z, 0, a) = Vi(z, 0)a + V>3/2(z, 0)a3'2 + V>2(z, 0)a2 + 0(a5/2).
If g is an FBI phase function associated to a local canonical map x such
that x(Po) = zq £ ®n, we have
<j>(z,a) = cv(Xtg)(g(z,x) + h(x,6) + il>{x,0,<T)')
= $Ao (*) + *i {z)a + $3/2(*)<73/2 + $2(2)a2 + 0(a^2).
Here $! and $3/2 are real on n o x(Ao), $i(zo) = 0, Dz$i(z0) ^ 0 and
2$2(*o) > 0.
Proof. The critical points are the solutions of
h'g + % = 0
and are given by
(x>e) = h% ° Xi1 (*, Dztho (z))
if a = 0. So the critical points are not degenerate for small a. Let a = s2.
An easy computation gives
4>'L{z,o) = V£ + « + «-
Hence
*',(z,0) = 0, CM) = 2^(x,0).

108
Moreover, if s = 0,
This implies (x's, 0's) = 0 on s = 0. Two more derivations give
*1M) = 6V3/2M)
*g)(*,0) = 24fc(M) + 6(#X+#.«0-
The critical points are real if and only if z G 7r o x(Ao). This shows that $1
and $3/2 are real on this totally real submanifold. Since the differential of
r/)i does not vanish on Ch, we have Dz$i(zq) ^ 0.
We also get
J KA + KeO'L + g'Lx'L + ty[* = o
It follows that
^^°» = 24«^»-3((Aiiti/A4:)(t)'(c
Since (/i, V?) is a 2-phase function, the matrix
(ft* Ke\
[fit Ke)
has full rank. Using the lemma 16, we obtain T$naa{z§) > 0. □
Lemma 16 Let 5, H be hermitian matrices with H semi-positive definite.
If A = S + iH is invertible and x £ (Cn does nof belong to the image by A
of the kernel of H then I (A~lx, x) < 0.
Proof We have
(A~lx, x) = (AM"1*, A"1*) = (SA~xx, A~lx) - i (HA~lx, A~lx) .
In the right hand side, the first term is real and (HA^x, A~lx) > 0. □
With the notations of the proposition 15, a distribution u is said analytic
bilagrangian at p0 with respect to (Ao, Ai) if, in a neighborhood of z0, we
have
rS
(Txu)(z,\)= el^z^a{z,a,\)da
Jo

109
where a is holomorphic in an open set of the form
{(*, a) £ (Cn x (C : \z - z0\ < 6, \la\ < cTZa}
and is bounded by CXm for A > 1.
Since X^i^o) > 0 and <&i(z0), $3/2(^0) are real, we can choose S > 0
small such that
-I<l>(zo,6) < -I(pA0(z0).
For example, if
Ao = {((0, *„), (£', 0))}, Ax = {((0,0), (£',&))}
and g(z, y) = i(z - j/)2/2, we have
and
zV2 ia2
^*' ^ = T" + ^n + 2
6. Bilagrangian structure of the parametrix
We review the construction of the parametrix in the analytic case, see [8],
and show that, at the transition of the shadow and the illuminated region,
it defines a bilagrangian distribution if the boundary data is conormal.
We consider a differential operator with real analytic coefficients
P(x,D) = D2Xn + R(x,Dx,)
Its principal symbol is
Let r0(a:',£') = r(a;', 0,£'). We assume that the point (#o,£o) *s diffractive.
This means that ro(#o>£o) = 0 and dr0 / 0, dXnr < 0.
Following [8], we first perform a complex canonical transform. We choose
the weight function y>o(z') = \Izf\2/2 and a canonical map
mapping (#o,£o) to (^) an(* ^e glancing region {r0 = 0} to {Xz\ = 0}.
To this canonical map is associated a FBI transform.

110
After this transform, we obtain a pseudodifferential operator
P(x,D,\) = D2Xn+R{x,Dx,,\)
near (0,0) on A^. Its principal symbol p(x,£) = £2+r (#,£') is real on A^
and p(x,£) = 0 is equivalent to xn + q(xf,£) = 0 with
?(*',« = 6 - el/.OeJ + 0{£), e(0,0) > 0.
In the i?^ space, the problem is reduced to find an outgoing solution to
P(a?,D,A)u(a?,A) = 0, u\Xn=0 = g. (1)
Denote by G the solution of
f ^ + r(a:',-^BG,^G) = 0
I GKb=0 = *'•£'
We have
G(*',0 = x'^' + Ui " f e(x',e) ~ |^e(x',0 + 0(£).
Let C be the canonical relation of G(a?,,^n,^/) - £/'.£'. Lebeau shows that
there is a C1,1 function ^0(2/') real analytic for Ij/i ^ 0 such that C maps
A</,0 to A^0.
For a suitable choice of the symbol a, the operator
defines asymptotic solutions to P and maps H^ to Hvo for xn > 0, see [8]
for the choice of contours. Note that £+ and £~ are respectively close to
fa-Si* I* and $gtir/2#
We have the critical values
c.v.(WG(*',*BlO = x'.£' - ^7(^0 ± §(&P(*',0)3/a
with /) = e""1/3 on £1 = 0. Introduce a new parameter t such that t2 = £1
and argt G [-7r/2,0] if £i is real. In the definition of J/, the critical value
corresponding to the critical point £n = -ie""1/2 + 0(t2) is

Ill
It is natural to invert the trace of Jf by an operator of the form
W.A) = I4 dz, [t+2tdt [ e^-^Hy"->"W>+F{z<,t,r,><))
Jz~ Jt~ JZ"
aT(z',t,r]",\)f{z',\)dz"df.
Here
F(z', t, V") = -/p(z', t\ v'f/2 + t4l(z\ t\ r,").
The symbol aT is chosen to avoid the ramification of the asymptotic
behavior of the Airy function. If r > 0, the function
r^ / xl/2/ T^ 3/2 (z(z + T)fl2(z-T/2)-Z2
f(z) = (z + t)1'2{z --)- zz'2 = V V ;; >- —
I \/Z
is holomorphic and bounded on the 2-sheets covering of [—r, 0]. For a large
but fixed r, take
aT(z', t, V", A) = (to + r)i/4e![(^)1/2(w-x)-.3/2]
with
w = e^3\2/3t2p{z',t2,r,").
It follows that the symbol is not holomorphic in a fixed neighborhood of 0
but in a set of the form
9-7T
{(*', V>) : \z'\, \rf\ < r, | argfa - rA^e2-/3) - y | > e}.
For a good choice of contours, the operator / maps Hm on H^Q. Lebeau
shows how to invert the trace of J o I by an operator that propagates the
singular support only on one side.
In xn > 0, we get the phase
G{x',Q - y'.e + xn^n + (y' - z').V' + F(z', Jm, rj").
Using the theorem of the stationary phase function, we can reduce evaluate
the integrals involving £/',£' and we get the phase
G{x', r/, £n) + xnin - z'.rj' + F{z', v^T, rj").
Since we study the solution near a point where xn > 0, the function £n »->•
G(xlirf^n) + xn£n has exactly one critical point whose argument is close
to 7r and satisfying
Xn + m-ene(x',r)') + O(en) = 0.

112
Let
H(x\ V\ V^+m) = x'.rf + \{xn + mfh-^ix', 7/0 + 0((xn + Vl)2)
be the critical value.
Define, as above, Ao as the flowout of the set of diffractive points through
the boundary hamiltonian Hr followed by Hp and Ai as the flowout of all
the characteristic points at x = 0 through Hv.
In the boundary value problem (1), we consider the boundary data
g(xl, A) = exp(i\z'2) corresponding to a Dirac mass.
Theorem 17 The function
2^n - Xnf + H(z', (7, rf\ y/xn + a)
satisfies the conditions of proposition 15. Moreover, the solution u of the
boundary value problem (1) can be written u\ + u<i where u\ is analytic
bilagrangian and
M*, A)l < C6eA^oW+^M°x(Ao))3)+eA
near 0 for every e > 0.
References
1. Delort, J-M., Deuxierne rnicrolocalisation simultanee et front d'onde de produits,
Ann. scient. Ec. Norm. Sup., 23, 1990, 257-310.
2. Hormander, L., The analysis of linear partial differential operators I-IV, Springer-
Verlag, 1983-85.
3. Friedlander, F.G. and Melrose, R.B., The wave front set of the solution of a simple
initial-boundary value problem with glancing rays II, Math. Proc. Camb. Phil. Soc,
87, 1977, 97-120.
4. Lafitte O., The kernel of the Neumann operator for a strictly diffractive analytic
problem, Comm. in Part. Diff. Eq., 20, 1995, 419-483.
5. Laubin, P., Etude 2-microlocale de la diffraction, Bull. Soc. Roy. Sc. Liege, 4, 1987,
295-416.
6. Laubin P., Willems B., Distributions associated to a 2-microlocal pair of lagrangian
manifolds, Comm. in Part. Diff. Eq., 19, 1994, 1581-1610.
7. Lebeau, G., Deuxierne rnicrolocalisation sur les sous-varietes isotropes, Ann. Inst.
Fourier, Grenoble, 35, 1985, 145-216.
8. Lebeau, G., Regularity Gevrey 3 pour la diffraction, Comm. in Part. Diff. Eq., 9(15),
1984, 1437-1494.
9. Lebeau, G., Propagation des singularites Gevrey pour le probleme de Dirichlet,
Advances in microlocal analysis, Nato ASI, C168, 1986, 203-223.
10. Lebeau, G., Scattering frequencies and Gevrey 3 singularities, Invent, math., 90,
1987, 77-114.

113
11. Melrose R.B., Sjostrand J., Singularities in boundary value problem I, Comm. Pure
Appl. Math., 1978, 31, 593-617.
12. Melrose R.B., Sa Barreto A., Zworski M., Semi-linear diffraction of conormal waves,
preprint.
13. Melrose, R. B., Local Fourier-Airy integral operators, Duke Math. J., 42, 1975, 583-
604.
14. Melrose, R. B., Transformation of boundary value problems, Acta Math. J., 147,
1981, 149-236.
15. Sjostrand, J., Propagation of analytic singularities for second order Dirichlet
problems, I-III, Comm. Part. Diff. Eq., 5(1), 1980, 41-94; 5(2), 1980, 187-207; 6(5), 1981,
499-567.
16. Sjostrand, J., Singularites analytiques microlocales, Asterisque, 95, 1982, 1-166.

PARAMETRIZED PSEUDODIFFERENTIAL OPERATORS AND
GEOMETRIC INVARIANTS
GERD GRUBB
Copenhagen University Mathematics Department
Universitetsparken 5, DK-2100 Copenhagen, Denmark
Abstract. This is based on joint work with R. T. Seeley. The
introduction presents the problem of parameter-dependent calculi for ^do's and the
question of trace asymptotics for Atiyah-Patodi-Singer operators.
Chapter 2 establishes relations between the three operator functions: resolvent,
heat operator and power operator (zeta function). Chapter 3 explains our
parameter-dependent V>do calculus with weak polyhomogeneity, showing
how logarithmic terms appear in trace formulas. In Chapter 4, the APS
problem is treated in the case with a product structure near the boundary,
where functional calculus on the cylinder leads to precise formulas for heat
trace expansions and zeta function pole structure. Finally, Chapter 5 treats
the APS problem in the non-product case where the weakly polyhomoge-
neous V>do calculus is used to get asymptotic trace expansions generalizing
those in the product case.
1. Introduction
1.1. PARAMETER-DEPENDENT CALCULI
A typical case of an interesting parameter-dependent pseudodifferential op-
erator (henceforth abbreviated to ^do) is the resolvent R\ = (P - A)""1 of
a, say, strongly elliptic operator P on a compact manifold. Let the symbol
of P (in a local coordinate system) be
p(^0=Pm(a?,0+Pm-i(a?,0 + ---»
where each term pm-j(a?,£) is homogeneous of degree m- j (for a positive
integer m), then we write -A as
-A = eV\ M=|A|1/m,tfe[0,27r]
115
L. Rodino (ed.), Microlocal Analysis and Spectral Theory, 115-164.
© 1997 Kluwer Academic Publishers.

116
(where i is the imaginary unit yf-\), and assign to P - A the principal
symbol
pm{x,t;,\)=pm{x,Z) + e{efim
(also denoted pm (#,£,#,//)), and the full symbol p + eie/jLm where the lower
order terms are the same as those for P. The inverse of this principal
symbol,
?m(a?,f,A)=pm(a:,f,A)-1
will then be the principal symbol of the resolvent.
Here ji can be considered as one more "cotangent variable" in addition
to £1, &» • • •»£m and pm is homogeneous of degree m in (£, //).
There is a marked difference between the case of a differential operator
and that of a ^do. In the first case, pm is polynomial in (£,//), hence
homogeneous and C°° in (£,//) € R+ . In the second case, the homogeneous
symbol pm(x^) usually has a lack of smoothness at £ = 0 (it has only m
bounded derivatives), so pm will have this lack of smoothness on the whole
halfline {(0,//) | /x > 0}. (Alternatively, if pm is modified in a bounded
neighborhood of 0 to be C°°, the ensuing modification of pm takes place in
an unbounded set.)
This also has an effect on the estimates of qm. Here one has (with
(x) = (|x|2 + l)1/2):
D?qm = Om^))-m-\% for \a\ < m,
D^qm = O(((Z,Li))-2m(0m-M), for \a\ > m,
where the first line extends to all a if and only if pm is polynomial in £.
In the polynomial case one can apply the usual symbolic calculus, just in
one more variable, getting simple and straightforward results, whereas in
the general case the fact that only the first m estimates are standard (the
so-called regularity number is m), gives severe trouble.
For boundary value problems there are similar phenomena. In the
differential operator case, the resolvent parameter enters as another cotangent
variable, on a par with the others, whereas for a pseudodifferential boundary
operator, a resolvent parameter, when considered as a cotangent variable,
gives symbolic estimates where only finitely many of them are "good".
Again one assigns a regularity number to the operator, this will now be
different for the different types (trace operators, Poisson operators, singular
Green operators).
This phenomenon is one of the main subjects of the book Grubb [12].
It is shown there that in the application to obtain trace formulas for
resolvents (and heat kernels), one get finitely many well-defined terms in an

117
asymptotic expansion, namely as many the regularity number indicates.
For resolvents in the case without boundary, there is a trick to extend the
analysis to get full trace expansions with infinitely many terms, some of
them logarithmic; also this is explained in [12].
More recently, we have developed a somewhat more special calculus in
collaboration with Robert Seeley [14], which allows a systematic
construction of full asymptotic expansions for a class of ^do's containing the
resolvents: the calculus of weakly polyhomogeneous operators. It is completely
described for the boundaryless case (whereas the additional details needed
for general pseudodifFerential boundary problems only exist in a sketched
form).
For differential operators with pseudodifFerential boundary conditions,
one can however use the weakly polyhomogeneous ^do calculus in cases
where the trace formula in question can be reduced to one for an operator
in the boundary of the weakly polyhomogeneous kind.
The calculus was developed for, and applies in particular to, the general
Atiyah-Patodi-Singer problem. We describe this in detail below.
1.2. THE ATIYAH-PATODI-SINGER PROBLEM
On a compact n-dimensional C°° manifold X with boundary OX = X1,
consider a first-order elliptic differential operator
P:C°°(E1)^C0O{E2)
between sections of vector bundles over X. E\ and E<i have Hermitian
metrics, and X has a smooth volume element, defining Hilbert spaces structures
on the sections (primarily the spaces of Z^-sections, denoted L<i(E>i), and
more generally the Sobolev spaces Hs(Ei), s G R).
The restrictions of the E{ to the boundary X1 are denoted E[. A
neighborhood of Xf in X has the form Xc — X1 X [0,c], and there the E{ are
isomorphic to the pull-backs of the E[. Let xn denote the coordinate in
[0, c], xf the coordinate in Xf. Then we assume that P is represented in Xc
as
P = ^+A + xnP1 + P0), (1.2)
where a is a unitary morphism from E[ to E'2, independent of a?n, and A is
a fixed elliptic first-order differential operator on C°°(E[), self adjoint with
respect to the Hermitian metric in E[ and the volume element v(xf, 0)dx'
on X1 induced by the element v(x',xn)dxfdxn on X. The Pj are smooth
differential operators (in all variables) of order < j; they can be taken
arbitrary near Xf, but for larger a?n, Pi is subject to the requirement that
P be elliptic. All morphisms are assumed C°°.

118
In comparison with completely general elliptic first-order operators, the
assumption means (modulo homotopies) that we have restricted the
attention to operators such that when the principal symbol is written near
X1 as (Ti(x\xn)(i^nI + ai(x\xn^/)) (with a bundle isomorphism G\ from
E\ to Ei in front), then ai(#',(),£') is symmetric; cf. Grubb [13], p. 2036.
The case considered by Atiyah, Patodi and Singer in [2] is the case where,
furthermore, Pi = Pq = 0 in (1.2); this is often called the product case.
Important examples are the Dirac operator and its generalizations.
We denote u\x* = 7o^ and observe the Green's formula:
(Pu, w)x - (ti, P*w)x = -(Tot*, tf*7ow)*'. (1-3)
Since P is a first-order system, it may not be possible to formulate
a well-posed boundary value problem in terms of a differential boundary
condition (a Dirichlet condition is too much, no boundary condition is too
little, and the boundary bundle structure will not in general allow putting a
Dirichlet condition on some "half" of the boundary data). But using ^do's,
one can get well-posedness:
Definition 1.1 The APS boundary problem consists of finding u G
Hl(Ei) for a given / G L2(E2), so that
Pu = / on X, Bj0u = 0 on X'. (1.4)
Here B is an orthogonal projection in £2(^1) of the form B = II > + Po>
where II > (n<, II#, ... ) denotes the orthogonal projection onto V> (V<,
Vr, ... ), the sum of eigenspaces for A with eigenvalues A > 0 (A < 0,
|A| < P, ... ), and Po commutes with A and ranges in Vr for some R > 0.
The associated realization Pb is defined as the operator from £2(^1) to
1^2(^2) acting like P and with domain
D(PB) = {ue Px(Pi) I P7ou = 0}; (1.5)
it is a Fredholm operator called the APS operator, and the APS index
problem consists of determining its index.
This type of boundary condition is often called a spectral boundary con-
dition. The Fredholm property of Pb was shown by Seeley in [23], where it
was moreover shown that the adjoint of Pb is of a related type (in view of
(1.4)):
(Pb)* = (P*)B/, with B' = Bl(7*> Bl = I-B. (1.6)
One of the ways to study the index of Pb is to consider the "Laplacians"
Al = PB*PB, A2 = PbPb*,
(1.7)

119
and search for asymptotic expansions for t —t 0 (with e > 0):
Tr e~tAi = c_niirn/2 + • • • + c-^r1'2 + c0>i + 0(f), i = 1,2.
(1.8)
When (1.8) holds, the index is determined by
index PB = Tr e"tAl - Tr e"tA2 = c0,i - c0,2. (1.9)
Remark 1.2 The systems (^ J and (5/^ ) are injectively elliptic (also
called overdetermined elliptic or left-elliptic). The operators Ai and A2 are
realizations of truly elliptic systems (two-sided elliptic) such as
U'*7o + B'10P) ' reSp' U'£'7o + BioP*) ' (L10)
(We here use that B and Bf map into complementing subspaces of L2{E[),
and we have inserted the invertible ^do A! — A + no(A) in order to make
the boundary conditions first-order. The operators are principally the same
as in the case where B = n>, discussed in detail in [13].) Another truly
elliptic system incorporating Pb and Pb* is discussed below in Section 5.1
(and in [14]).
Remark 1.3 If a* = -a and Aa = —oA, then in the product case, P is
formally selfadjoint. Then if furthermore Ba = a(I - B), Pb is selfadjoint.
This holds in many geometrically interesting cases, see e.g. Gilkey [10].
In [2] it was shown in the product case, with B = II>, that
index Pb = / Q>(x) - \t]a; Va = v{A,0) + dimkerA;
Jx (1.11)
where a(x) is a certain form defined from the symbol of F, and r/(A, 0) is
the value at s = 0 of the eta function
V(A,s) = Tt(A\A\-s- *). (1.12)
(Here A~s~x is defined as 0 on the nullspace of A, and meromorphic
extension is used for Re s < n.) Formula (1.11) was extended to the non-product
case in [13] as
index PB = / a(x) + [ /3{xf) - \riA, (1.13)
j x. j x.

120
with a boundary form /3(xf) defined from the symbols of P and B at X'.
The forms defined from the symbols are regarded as local contributions,
whereas the term tja depends on the full set-up in a global way.
Actually, [2] did not calculate the two expressions Tr e~tAl and Tr e~tA2
separately, but only their difference. They showed for the product case that
this has the same asymptotic expansion as
Tr(e-tSlU) - Tvie-^lx) + Tr(e"tA? - aVtA5<7);
(1.14)
here Ai = JP*JP and A2 = PP*, where P is a certain extension of P to
bundles E\ and E<i over the double manifold X (cf. [2], p. 55, where the roles
of Ei and E2 are switched onI\I); the A? are #n-independent extensions
of the A* on Xc to the cylinder X° = X1 X R+. The first difference is well-
known, and the second can be analyzed by use of functional calculus for the
selfadjoint operator A\ this sufficed to get the index formula in the product
case.
In [13] the separate expansions (1.8) were proved with e = | in the non-
product case with B = n>, by a combination of the general treatment of
parameter-dependent xj^do boundary problems [12] with the special results
from [2]. It was shown that the global term -\t]a enters in Co,; for both
expansions, as -\t]a for i = 1, resp. \t]a for i = 2.
Now the index is just one special geometric invariant connected with
the APS problem. More generally, one can ask about the value of the
general coefficient Cj_n^ in (1.7), and one can ask whether there is a more
detailed structure of the 0(t£) term, giving a full asymptotic expansion
E£o ci-n,tt(i~n)/2 for the trace Trexp(-tA;).
These questions have been answered in two papers written in
cooperation with Seeley, [14] and [15]. It is shown there that there does exist a
full asymptotic expansion, which however includes also logarithmic terms
ctO~n)/2logt for j - n > 0. For the product case, a precise description of
the coefficients in terms of the zeta and eta functions of A is given, when
Bo ranges in the nullspace of A.
In the following we shall give an account of these results, explaining the
highlights of the methods.
2. The three operator-functions
2.1. DEFINITION OF THE OPERATOR FUNCTIONS
One can associate several interesting operator-functions with an elliptic
operator Q. The following have been studied extensively:

121
• The resolvent (Q - A)-1 and its asymptotic behaviour for A —> oo on
rays in C.
• The heat operator e~t(^ (t £ R+) and its asymptotic behavior for
• The power operator Q~s and the pole structure of associated functions
of s€ C.
For the questions we address here, there are essentially equivalent
formulations in terms of each of the three operator functions, and one can pass
from one formulation to another by suitable transformations. Very briefly
stated, the heat operator and the resolvent are related to one another by
the Laplace transformation, and the heat operator and power operator are
related to one another by the Mellin transformation. One can also define
the heat operator and the power operator from the resolvent by suitable
Cauchy integral formulas (Dunford integrals), and there is another
complex integration formula involving a reciprocal sinus function going from
the power function to the resolvent. (In the proofs of Theorems 2.1 and 2.3
below, we also relate the formulas to the Fourier transformation.) In the
following we collect the facts on these operator functions that we need.
Much of this has been known in the literature for a long time (but not
always explained as generally as here). Applications to trace asymptotics
have been made earlier e.g. in Seeley [22], Duistermaat and Guillemin [8],
Grubb [12], Agranovic [1], Branson and Gilkey [5]. The explanation in the
following is essentially copied from [15], and is given here with full details
since it may be of interest also for other purposes.
Suppose that Q is a closed operator in a Hilbert space having a
resolvent (Q — A)-1 which is holomorphic in some sector | arg(—A)| < a, with
\\(Q — A)_1|| = 0(|A|_1), and is meromorphic at 0 (in the sense that
(Q - A)-1 - (-A)_1rio(Q) is holomorphic at 0, where Uo(Q) is the
orthogonal projection onto the nullspace of Q). Then the power function Z(Q,s)
is defined for Re s > 0 by
Z(Q,s) = ±fc\-°(Q-\)-ld\, (2.1)
where C is a curve
Co,r0 = { A = reie | oo > r > r0 } + { A = r0eie' \0>6'>-6}
+ {X = re*2*'0) | r0 < r < oo}, (2.2)
with 7r - a < 0 < 7r and ro > 0 chosen so that (Q - A)-1 is holomorphic
for 0 < |A| < ro- If Q is invertible then Z(Q, s) = Q~s (further details are
found e.g. in Seeley [22] or Shubin [24]); in any case, Z(Q, s) is zero on the
nullspace of Q, since fc A"~s-1dA = 0. We can also write
Z(Q,s) = £/c A-(Q - A)-1^) dX, (2.3)

122
where n^-(Q) = /-n0(Q).
If Z(Q, s) is trace class for some s, then Q has a zeta function
<(Q,s) = TrZ(Q,«), (2.4)
and, for appropriate operators D and values s, a "modified zeta function"
({D,Q,s) = Tr DZ{Q,s). (2.5)
Similarly, under appropriate conditions, we define
Y(Q, s) = QZ(Q*Q, *±1) = JL /c \-l+WQ(Q*Q - A)"1^
= ifc,„A-c+i)/»g(g*Q-A)-MA (26)
(since rio(Q*Q) = rio(Q) and Qrio(Q) — 0, we can leave out the nullspace
projection), and the eta functions
V{Q,8) = TrY{Q,8), V(D,Q,s) = TtDY(Q,s). (2.7)
When Q is self ad joint,
E i*r = c«M), E signA|Ar5 = THQ,5),
AGsp(Q)\{0} AGsp(Q)\{0} (2.8)
with summation over the eigenvalues, repeated according to multiplicities.
In order to move the trace inside the integral, we may represent the
power function by use of a derivative of the resolvent. Note that
d^(Q-X)-l = m\{Q-X)-m-\ (2.9)
If Q is a ^do of order r > 0 on a compact manifold M, say, then the mth
derivative of (Q — A)"1 is a ^do of order — (1 + m)r and hence is trace class
when (m+ l)r > dim M. By an integration by parts, one can replace (2.1)
by
Z(Q. *) = (.-i)-.(-m)A /c A"1-W " A)"1*A, (2.10)
whereby (2.4) can be written
C(Q,a) = TrZ(Q,s) = ^y1.^^ fc A"-'Tr0?(Q - A)^dA,
K > K > (2.11)
for sufficiently large m. Similar modifications can be made when there is a
factor D as in (2.5) and when eta functions as in (2.7) are studied; and the
integral can be replaced by an integral over Cop when Uq(Q) is inserted in

123
front of d\. There are similar formulas for the symbols and kernels of the
operators.
When Q is lower bounded selfadjoint, the heat operator e~t(^ (also called
the exponential function or the semigroup generated by -Q) can be defined
by
e-tQ = ^!c,e-t\Q-X)-U\ t>0; (2.12)
where C is a curve encircling the full spectrum in the positive direction
and such that e~tX falls off for |A| —> oo on the curve (e.g. one can let C
begin with a ray with argument 6 ]0, f [ and end with a ray with argument
£] - §,0[). This is well-known from the literature, see e.g. Hille-Phillips
[16], Friedman [9] or Kato [20].
The exponential function and the power function of an operator Q > 0
with resolvent as above are related to one another by the formulas:
Z(Q,s) ^ffctft-ie-WltiiCftdt, Res > 0,
e-tQnk(Q) = ±fRes=ct-*Z(Q,s)r(s)ds, c>0, (2.13)
that follow e.g. from Theorem 2.3 below, with e(t) = e'^Il^Q), (p(s) =
r(*)Z(Q,s).
Taking Q = S*S for suitable operators 5, we have accordingly (cf. (2.6)):
Z(S*S, s) = fa /0°° *-ie-***IIjKS) eft,
e-^SjftiS) = £ fRes=c t-Z(STS, s)T{s) ds,
Y(S,2s) = SZ(S*S,s+ J) = -JL^tff-hse-*3-3*, {2U)
Se~tS'S = ^ kes=c t~sY(S, 2s - l)T(s) ds.
(Also here we can omit mention of the nullspace projection in the last two
formulas.)
Again, these formulas can be composed with a suitable operator D.
When the expressions are trace class (usually for Res resp. c sufficiently
large) one can take the trace on both sides in (2.14) (composed with D),
obtaining the formulas relating zeta and eta functions to exponential function
traces:
C(A S*S, s) = fa J? t*-1 TrDe-tS*sIl£(S) dt,
TTDe-tStsEb(S) = ±fRes=ct-X(D,S*S,s)r(s)ds,
V(D, 5, 2s) = {{DS, S*S, a + j) = ^ /o°°«-» Tr DSe~<s*sdt, (2 15)
TvDSe-tS*s = ± IRes=ct-sV(D,S,2s-l)r(s) ds.

124
There are similar transition formulas for the symbols and kernels of the
operators.
2.2. RELATIONS BETWEEN THE RESOLVENT AND THE POWER
FUNCTION
Let us first consider the passage between properties of the resolvent and
properties of the power and zeta functions. In order to handle operator
functions defined not only as in (2.1), but also as in (2.10), we include
functions with higher order poles at 0. We denote {0,1, 2,...} = N.
Theorem 2.1 1° Suppose that f is meromorphic at 0 with Laurent
expansion
oo
/(*) = E M-*)Jil*l<Pi (2-16)
j=-k
that f is holomorphic in the open sector Ss0 = {A G C | |argA - ir\ <
So} (for some So < n), and that /(A) = 0(|A|~a) for some a G]0,1]
as A —> oo, uniformly in each sector Ss for S < So- Let C be a curve
C^ro as zn (2-2) (a Laurent loop, since 0 = n), with 0 < ro < Q. Set
fo(X) = f(X)-Tl~-1khj(-Xy>and
C(s) = £/CA-/(A)dA, Re*>l-a, (2.17)
with X~s = r~se~ls6\ r > 0 and \0\ < n. Then £ and /o are interrelated by:
C(s) = sJn^f™r-sf0(-r)dr, l-a<Re*<l, (2.18)
/o(-A) = i/Res=,A-1S^, l-a<a<l. (2.19)
The function ^S^- is meromorphic for Res > \ — a, having simple poles at
s = j + 1 with residues (-l)J'+1C(j + 1) = ~hj> J £ N.
2° Moreover, the following properties a) and b) are equivalent:
a) / has an asymptotic expansion as X goes to infinity
oo rnj
/(-A) ~ E E *iA~aj Oog A)', 0 < ctj / +oo (2.20)
(with ntj £ N), uniformly for -X in Ss, for each S < Sq.
b) ip(s) = ^r^i is meromorphic on C with the singularity structure
*fel~-f Ji-tfv ">■>"■ , (2.21)

125
(in the sense that for large N, the left hand side minus the sums for j < N
in the right hand side is holomorphic for 1 - a^ < Re s < N + 1); and for
each real C\,Ci and each S < So,
\1>(s)\ < C(Ci,C2,£)e-*lIms|, for | Ims| > 1, d < Res < C2.
(2.22)
In particular, the singularities of ip(s) in Res < 1 are determined by
the expansion (2.20) and the singular Laurent terms of /(A) at X = 0, and
vice versa.
3° Let f take values in a Banach space, and be holomorphic in S$Q, and
meromorphic at 0 in the sense that there is a function J2jl-k(~^V^J w^
bounded operators Hj such that /o(A) = /(A)~X]Ifc(~A)«7ifj is holomorphic
for \X\ < g, some g > 0. Let ||/(A)|| be 0(|A|~a) for X —> oo in subsectors
Ss with S < S0. Then with ((s) defined by (2.17), the formulas (2.18)-(2.19)
are valid.
Proof: 1°. For j < -1 and Res > 0, fc X^~sdX = 0, since the contour can
be closed at oo in {| arg A| < tt}. So the singular part of /, YlZ\ hj{~^Vi
is "killed" by the integral over C in (2.17). For the remaining part /o, the
circular part of C can be reduced to the origin if Res < 1, reducing (2.17)
to (2.18) (note that /0 is 0(|A|"a) too).
The inversion (2.19) requires growth estimates for £(s). Replacing the
integration curve by C(S) := Cn-Sp, 0 < S < Sq, we have that
ICWI = |i Sc(S) A"Vo(A) d\\ = 0(e(*-W™%
1 - a < Ci < Re s < C2 < 1. (2.23)
For, when A = re1**' ', we can use the estimate
r-.ei(1r-5)(l-S)0^1 + ryaj d\ < Ce{*-8)\lm»\^
r
Jo
(2.24)
and there is a similar estimate on the other half of C(S).
Now let
*(s)=rr-'M-r)dr=$& (2.25)
Jo sin ns
Since (sinTrs)-1 is 0(e-*\Ims\) for |Ims| > 1, we have by (2.23) that
V>(<t + ir) = 0{e~&\T\) for 1 - a < C\ < a < C2 < 1. Also, t/>(a + ir)
is the Fourier transform F(r) of the function F(x) = e^~^xfo(—ex).

126
Since /o(A) = 0((A)~a), F(x) decays exponentially as x —> ±cx>, for
1 - a < a < 1. By Fourier inversion, F(x) = ^ f^ elXTip(a + ir)dr,
giving (2.19), for A > 0. It extends to | arg A| < #0 by analytic continuation.
It is seen from (2.17) that £(s) is holomorphic for Res > 1 - a; and
since C0' + 1) = ^/|A|=roA"i_1/(A)dA = (-1)^- for j e N, i/,{8) is
meromorphic for Res > 1 — a, having simple poles with residues —hj.
2°. Now suppose that a) holds; then
N-l mj _1
/o(-A)= ££^A-^(logA)'- £ /i,V + 0(|A|-^+£)forA^oo,
j=o t=o j=-ib (2.26)
for ajv > fc» any £ > 0. Note that
/
I
fj s dr = :—- for Re s < j + 1,
o s-j-1
rP-s{logr)ldr=- ' forRes>/? + l
l [s- P - lj "t"
(the cases / > 0 follow from the case / = 0 by application of #£); the right
hand sides extend meromorphically to C. Then we get from (2.25), for
arbitrarily large N:
tl>(8) = / [£ /^H"5 + r"sO(rN)]dr
j=0
,00 r^-l mJ "I
-71 j=o (=o j=-k
N-l . N-l mj ..
where Jin is holomorphic for 1 - ajv + £ < Res < iV + 1, and the other
terms are meromorphic on C. This gives the singularities (2.21).
To show the decay, we use the integral in (2.23) and expand on each
piece of C(8):
CW = -i(/o +ir(re^-^)-sfo(re^-s))e^-s) dr)
+ i(/o1+/i0°(^i(-,r+i))-s/o(rei(-7r+5))ei(-'r+5)dr). (2.27)
The first integral from 0 to 1 is written as
2^
t /oVe^-^-Votre^-5))^-*) dr
= £ Jo EfzTo1 e^-'^-Vhjri-' dr + J* ^v^X1"*)^) dr
= Efe1 -'U+»Z~*ki +^-S){l-')fir-'0(rN)dr. (2.28)

127
Let |Ims| > 1. The sum over j extends meromorphically to C, and its
terms are 0(e^-s^hn8^) for -oo < Ci < Res < C2 < oo. The last term
exists and is 0(e^~8^lms^) when Res < N + 1. Similar considerations hold
for the other integral from 0 to 1. In the integrals from 1 to oo we expand as
in (2.26), obtaining functions that are O^*-*)!1"1'!) for Res > l-aN + e.
We conclude that the estimate in (2.23) extends to 1 - ajv < Re s < N + 1,
|Ims| > 1, for arbitrarily large N. Dividing by sin7rs we find that ^(s)
satisfies (2.22). This shows a) => b).
Conversely, assume b). Then /o(-A) is given by (2.19), and we obtain
the expansion (2.20) by shifting the contour of integration past the poles
of ip(s). The remainder after all terms up to the singularity s = 1 - ajv is
given by the integral (2.19) but with a < 1 - aN', it is 0(\\\-aN+£) on Sj.
3°. The proof under 1° is generalized straightforwardly to Banach
spaces, with the relevant estimates valid for the norms. □
In this analysis, the poles in (2.21) may very well be considered in a
general sense where we allow some of the coefficients a^i to be 0; this is
practical for the applications where vanishing coefficients often occur, and
we shall use this point of view in the following. (So we can e.g. speak of a
simple pole with residue 0 — this is usually not called a pole.)
Corollary 2.2 When /(A) and £(s) are as in Theorem 2.1 l°-2°, then
r(s)£(s) is meromorphic on C with the singularity structure
j=-k S J l j=Q 1=0 ^+ a3 l>
hj - thy a>>'- r(ajy
Thus the singularity structure (2.29) of T(s)((s) is determined from
the asymptotic expansion (2.20) of f together with the singular part of the
Laurent expansion (2.16) (the coefficients hj with —k<j< —1), and vice
versa.
\ When So > |, one has moreover, for any Sf < So - |, any real C\ and
C2:
|r(s)C(s)| < C'(Ci,C2,S)e-s'\Ims\, for \lms\ > 1, Cx < Res < C2.
(2.30)
Proof: Since Tr(sinTrs)-1 = T(s)r(l - s), (2.29) results from (2.21) by
multiplication by T(l - s)-1, whose zeros cancel the poles hj/(s — j — 1),
j > 0. If 6 - tt/2 = 6' > 0, the estimate |<(s)| < Ce^~s)\lms\ shown in the

128
proof of Theorem 2.1 (and assured by (2.22)) implies (2.30), since T(s) is
0(e(-!+£)lImsl) for |Ims| > 1, -oo < Ci < Res < C2 < oo, any e > 0.
(Cf. e.g. the assertion in Bourbaki [3], p. 182:
|r(s)| - v^| Im5|Res-2e-f lImsl for | Ims| -> oo, (2.31)
valid for fixed Re s or Re s in compact intervals of R.) □
Note in particular that a case mj = 1 in (2.20) corresponds to a double
pole of r(s)£(s) at s = 1 - aj (in the strict sense if a^m- / 0).
2.3. RELATIONS BETWEEN THE POWER FUNCTION AND THE
EXPONENTIAL FUNCTION
Now we shall investigate the relation between properties of exponential
functions and of power and zeta functions. The general transition goes as
follows:
Theorem 2.3 1° Let e(t) be a function holomorphic in a sector Vq0 (for
some 0O G]0,f[^
V6o = {t = re[e \ r > 0, |0| < 60}, (2.32)
such that e(t) decreases exponentially for \t\ —> oo and is 0(\t\a) for t —>■ 0
in Vs, any S < 9q} for some a G R. Let <p be the Mellin transform of e,
roo
<p(s) = {Me)(s) := / ts~le(t) dt, (2.33)
Jo
for Res > -a. Then <p(s) is holomorphic for Res > -a and <p(c + if) is
0(e~8\t\) for |f | —> oo, when c > -a (uniformly for c in compact intervals
of ] - a, oo[^; and e(t) is recovered from <p(s) by the formula
<t) = £{fRes=ct-s<p(s)ds. (2.34)
2° Moreover) the following properties a) and b) are equivalent:
a) e(t) has an asymptotic expansion for t —> 0,
OO mJ
<t) ~ EE6i/J"0og0'. & / +°°> ™i G N> (2-35)
j=0 /=0
uniformly for t £Vs, for each S < 6$.
b) (f(s) is meromorphic on C with the singularity structure

129
and for each real Ci, Ci and each S < 0o,
Ms)\ < C{Cl,C2^)e-s\lms\ | Ims| > 1, d < Res < C2.
(2.37)
3° Let /(A) take values in a Banach space, and be holomorphic in
S$Q = {\tt - argA| < So} for some S0 G]§, 7r] and meromorphic at A = 0
(holomorphic for 0 < |A| < g). Assume that as A —> oo m 5$ (/or # < #o,),
some derivative d™f(\) is 0(|A|~1_e) for some e > 0 (so that /(A) is
OUXl™-1)). Let 0O and 0 be such that ]0 - 0O,0 + 0o[c]tt - £0,f[, let
C = Ce}r0 as zn (2-2) mf/i r0 G]0, g[, and /ef
<*) = i /c e"a/(A) A vw = r^i /c A-/(A) dA,
(2.38)
for t G V0O resp. Res > m-e. Then e(t) is exponentially decreasing for t —>
oo in sectors V$ with S < 0$, and is 0(\t\~m) for t —> 0, and <p(s) and e(t)
correspond to one another by (2.33), (2.34). Here, when /(A) = (Q - A)-1,
then e(t) = e'^Il^Q) and <p(s) = T(s)Z(Q,s).
Proof: 1°. Note first that replacing e(t) by tbe(t) replaces (f(s) by <p(s + b),
so we can assume that a > 0 and then consider c > 0. The function <p(s)
is holomorphic for Re 5 > 0 since the integrand ts~xe(t) is so and has an
integrable majorant there.
By a change of variables t = e*, we see that y>i(£) = <£>(i£) is the
conjugate Fourier transform of e\(x) = e(ex) G ^2(R)-
<Pi (0 = ¥>(*) = / **e(t)T = / e <e^ dx = / e ci(*)<**'
«/0 ^ «/ — OO J— OO
so by Fourier's inversion formula,
*(*) = *i(*) = £ Ho e-«Vi(0 « = 2^i /ReS=o '"VW <**•
(2.39)
Similarly, y>(c + i£) is the conjugate Fourier transform of e(ex)exc for c > 0.
The hypothesis on exponential decrease of e(t) in the sectors Vs allows
us to shift the path of integration in (2.33) from t G R+ to t G el5R+ for
H < #o (corresponding to a shift to x G R + \5); this gives:
/•oo Ar
<p(c + i£)= (reiS)c+iie(reiS) —
TOO Air*
= e~&t / r^e(rei5)(rei5)c— = e~^g(S^ c),
Jo r

130
where g is bounded as a function of £ G R, locally uniformly in c > 0.
Taking S > 0 for £ > 0 and # < 0 for £ < 0, we see that y>(c + i£) decreases
exponentially (like e~*^l) for |£| —> oo, in any vertical strip {5 = c + i£ |
Ci <c<C2,(GR} with 0 < Cx < C2. Then we can also shift the
integration path in (2.39) from Res = 0 to Res = c, c > 0. This shows 1°.
2°. Assume now in addition (2.35). Let us first write <p(s) as
pi roo
(f(s)= ts~le(t)dt + ts-le(t)dt. (2.40)
The second integral defines an entire function of s. The expansion (2.35)
means that
N-l mj
j=0 1=0
QN(t) = 0{\tfN~£) for t -> 0 in V5, (2.41)
for e > 0 and any positive integer N; we insert this in the first integral.
Observe the formulas, valid for Re s > — /?,
/V^(log«)'* = -t^,
Jo (* + />)' (2.42)
/•OO
/ ts-1+P(\ogt)1 e"* dt = #r(s + /?),
Jo
where the cases / > 0 follow from the cases / = 0 by application of dl8. The
remainder £at(£) in (2.41) gives a function holomorphic for Res > —Pn+s,
and for the powers of t we use (2.42); this shows (2.36).
To show the exponential decrease of <p(s) on general vertical strips, one
can shift the contour in (2.33) and proceed much as in the proof of Theorem
2.1. Another instructive method is to insert the expansion e* = Ylv>o ^"i
that gives
M-l
e¥^(logi)'= £ jttP>+"{logt)1 +0{fi+M-e),
i/=0
for any e > 0 and positive integer M. Then we can write
e(t) = e(*)e(e-( = ( £ £ ^^(log*)'>"< + &,(«).
Pj+v<M Krrij
with QM(t) = 0(\t\M~e) for t -»• 0 in V5,

131
where QM{i) is exponentially decreasing for \t\ —> oo in V$ since the other
terms are so, and hence
roo
/ ts-lgM{t)dt. (2.43)
Jo
p]-\-v<Ml<m]
+
The last integral defines a function that is holomorphic for Res > -M + e
and exponentially decreasing (like e~5lImsl) on strips -M + e < C\ <
Res < C<i, by 1°. For the contributions from the first integral we use the
second formula in (2.42) together with the fact that the gamma function
T(s) and its derivatives are 0(e(_2+£')lImsl), any e' > 0, for |Ims| > 1,
-oo < Cx < Res < C2 < oo, cf. e.g. [3], pp. 181-182. This gives (2.37),
completing the proof of a) =^ b).
Conversely, assume b). Then e(t) is given by (2.34), and we obtain the
expansion (2.35) by shifting the contour of integration past the poles of
(f(s). The remainder after all terms up to and including the singularity s =
-Pn is given by an integral like (2.34) but with c < -/3n; it is 0(\t\^N~£).
3°. That e(t) defined here is exponentially decreasing for \t\ —> oo in V5,
S < 0q, follows since |e~At| < e-7^ with 7 > 0 on the integration curve.
The estimate for t —> 0 follows since
J e~Xtf(X) dX = (-t)~m J {d?e-xt)f(\) dX = t~m j e-xtd?f(X) dX
for t G V5, where e~Xtd™f(\) has a fixed integrable majorant for t —> 0. The
formula (2.33) for <p is shown by a complex change of variables, where we
replace t by u/X for each A; when arg A € ]0, |[, the ray R+ is transformed
to a ray Aa with argument - arg A £ ] - |, 0[, and vice versa. The integral
of us~le~u on such a ray is again equal to T(s), as noted above. Thus (recall
that /(A) is 0(\X\m-1))
rV"1^ f e-tXf{X)d\dt=^ f f us-lX-se~uf(X)dudX
Jo JC JC JAx
= T(s)^JcX-sf(X)dX. D
3. Weakly polyhomogeneous symbols
3.1. POLYHOMOGENEOUS SYMBOL CLASSES
We here sketch the properties of the symbol class used to get trace
expansions for the general APS problem; details are given in [14].

132
Consider symbols p(z,£,/i), where x and £ G R", n G T (a sector of
C\{0». We shall say that:
p is strongly homogeneous of degree m, when
p(x,t£,tn) = rP(x,U) for Id2 + |/x|2 > M > 1,
(^)eR"x(ru{0}). (3.1)
p is weakly homogeneous of degree m, when
p(x,^,^) = tmp(x,£,n) for |£|,i > 1, (C/x) e R" x T. (3.2)
Example 3.1 Let a(a?,£) be positive and C°° on Rn x Rn, and
homogeneous in £ of degree r G N for |£| > 1. Then a(x, £) +jir and (a(z, £) +fJ>r)~l
extend to:
strongly homogeneous symbols of degree r, resp. —r, if a is
polynomial in £ (it is the symbol of a differential operator);
weakly homogeneous symbols of degree r, resp. —r, if a is not
polynomial in £ (it is the symbol of a genuine ^do).
If for example r = n = 2, a(a?,£) = £j + £2 enters in the first case, and
a(x,£) = (^1 +^2)/(^i + £2) (f°r If I > 1) enters in the second case.
Both cases can be shown to belong to the following symbol classes
(where (a(x,£) +/jf)'1 G S"r'° n S°'"r):
Definition 3.2 Sm»°(Rn, Rn, T) consists of the functions p(x, £, ji) that are
holomorphic in ji for | (£,//) | >£,// G I\ and satisfy, denoting ^ = z,
dJzp{; •, i) is in Sm+'(Rn, Rn) for \ G T, with
uniform estimates for |z| < 1, \ G closed subsectors of T. (3.3)
Moreover, we set Sm'd(Rn, Rn, T) = //d5m'°(Rn, Rn, T).
Here Sm(Rn,Rn) denotes the standard ^do symbol space consisting of
the functions p(x,£) G C°°(Rn x Rn) such that dgdgp is 0((f)m_|of|) for
all a,/3 G Nn. The rules of calculus for such symbols are well-known, see
e.g. Hormander [18], Seeley [23], Shubin [24], Hormander [19] for various
setups with local or global estimates in x. We call the symbols in 5m(Rn X Rn)
classical, when they moreover have expansions in series of homogeneous
terms (in £, |£| > 1) of degrees m — j, j G N.
When symbols p(x,£) of order m are considered as depending on one
more variable //, they lie in 5m,°:
5m(Rn,Rn) C Sm'°(Rn,Rn,r), any T. (3.4)

133
The symbols in 5m'd(Rn, Rn, V) define V>do's P = OP(p) (which depend
on the parameter /j) by the usual formula:
OP(p)f(x) = Jeix<p(x,t;,rif(0dZ, /^(R"), (3.5)
with d£ = (27r)~nd£. The definition extends to more general functions and
distributions / as in the nonparametrized case. When m < -n, OP(p) is
an integral operator with continuous kernel Kp(x, j/,//);
f (3-6)
in particular, Kv(x,x,ja) = / p(x,£,/j) d£.
The operators have good composition rules, since Sm>d • Sm ld
C Sm+m ld+d , and since one can refer to the standard rules for Sm
symbol classes, which must here hold uniformly in z as in (3.3). One finds for
example that
p g OP(sm'd), p' e 0P{sm'4') => pp' g op(sm+m'*d+d')
(3.7)
(under the usual precautions on supports or global estimates), and the
resulting symbol is described by the usual formula
aeN- (3.8)
The expansion in (3.8) is an expansion in terms with decreasing ra-
exponents m + m! - j, j —> oo (j = \a\). Such expansions enter in the
theory as follows:
When pj G Smi'd for a sequence mj \ -oo (for j —> oo, j G N), and
p G Sm°>d, we say that p ~ £jGN Pj in 5m°'d if
p - £ ^G 5mj,d for anyJ G N- (3-9)
3<J
For any given sequence pj G Smi'd with mj \ -oo, there exists a p
such that (3.9) holds.
For the present special symbols there is another type of expansion that
is of great interest:
Theorem 3.3 When p G Sm,d(Rn,Rn,r), then p has an expansion in
terms fi>d~kP(d,k){x,Q with P{d,k) £ Sm+*(Rn,Rn), such that for any N',
p(x,t,n)- £ ^kpm(x,t)€Sm+N^N(R\R\r).
o<k<N (3.10)

134
In the proof one reduces to the case d = 0 by multiplication by fi d\
then the expansion is essentially a Taylor expansion in z = - at z = 0.
Note that in (3.10), the order of P(d,k) increases with increasing fc,
whereas the power of fi decreases. A very simple example is
(1 + Kl2 + V2)-1 = /T2(l - I*-2®2 + /T4<04 -•••)•
Corollary 3.4 When p G S~°°'d, the kernel Kp(x,y,fi) of OP(p) has an
expansion
Kp(x,y^) ~ J2 /"%,*(*, y), Kp,k G C°°. (3.11)
fceN
Definition 3.2 contains no homogeneity requirements, but we now define
a polyhomogeneous subspace:
Definition 3.5 A symbol p G Sm°~d>d is called weakly
polyhomogeneous, when p ~ ^2jeNpj^ with pj G Smi~dld, mj \ -oo for j —y oo,
j G N, such that the pj are weakly homogeneous of degrees mj (cf. (3.2)).
This will be compared with:
Definition 3.6 A function p(x,£,p) G C°°(Rn xRnx QTU {0}) is called
strongly polyhomogeneous of degree m if there is a sequence of functions
Pj G C°°(Rn X Rn X (r U {0}) that are strongly homogeneous of degree
m - j (cf. (3.1)) such that
3W(P- Eft) = 0«(^|*)>m"J"|a|"fc), (3.12)
3<J
for all indices, uniformly for // in closed subsectors of V U {0}.
Then one has in fact:
Theorem 3.7 When p is strongly polyhomogeneous of degree m G Z, then
it is also weakly polyhomogeneous, with degrees w - J, J G N, and with
p G Sm'° + S°'m t/ro>0, p G Sm'° H S°'m t/ro<0,
dfdfdjp G S™-H-M H 50'm-lal-fc /or |a| + * > m, all /?. (3 13)
^45 a consequence, classical symbols of order m £ Z in n + 1 cotangent
variables give strongly polyhomogeneous symbols in n cotangent variables,
when one cotangent variable is replaced by // (here V = R+ U R-).

135
The type of parameter-dependence entering in Theorem 3.7 was used
by Agmon and by Agranovic and Vishik in resolvent studies for differential
operators; for ^do's this is the kind of parameter-dependence studied e.g. in
Shubin [24] and many other works. It is a mild generalization that does not
cover resolvents (P - A)-1 and parabolic operators such as d/dt + P when
P is truly pseudodifferential (as treated in [12]).
3.2. APPLICATIONS TO KERNEL AND TRACE EXPANSIONS
Both the expansion in Theorem 3.3 and the expansion in Definition 3.5
enter in the proof of:
Theorem 3.8 Let p be weakly polyhomogeneous as in Definition 3.5, with
thq — d < —n. Then OP(p) has a continuous kernel Kv(x,y,ii) with an
expansion on the diagonal
oo oo
Kp(x, *,//)-£ aj{x)^i^ + £[<*(*) log/* + 4(x)]/"fc ,
j=o k=o (3.14)
for \/j,\ —» oo, uniformly for \i in closed subsectors of T. The coefficients
Oj(a;) and c^_m^_n (x) are determined by Pj(x,£,ijl) for \£\ > 1 (are "local"),
while the c'k(x) are "global"
Details of proof are given in [14]. A brief explanation: One uses the
general principle that "remainders contribute to c'k terms," by Corollary
3.3. The pj contribute with (cf. (3.6))
Kp.{x,x,p)= I pj(xy£yfj,)d£
= / Pj%+ [ Pj%+ [ PjdS = h + h + h, (3.15)
where I\ gives part of the aj term, I<i gives c'k terms, and J3 gives the rest
of aj and cj-m. _n (if d-mj-n £ N) and some c'k terms. One has of course
to show that the contributions to the c'k pile up in a controlled way.
When the operator acts on a compact boundaryless manifold,
integration of Kp(x, x,ji) in x gives a similar expansion of the trace:
Corollary 3.9 Let P be a ^-dependent tydo on a compact manifold M of
dimension n, with symbol satisfying the hypotheses of Theorem 3.8 in local
coordinates. Then it is trace class, the trace satisfying
00 00
Tr P ~ £ ajfim>+n + 5>* log ix + c'k]»d-k , (3.16)
j=0 k=0

136
for \/jl\ —> oo, uniformly for ji in closed subsectors ofT. The coefficients are
derived from those in (3.14) for coordinate patches by integration over M.
The result applies in particular to expressions containing a differentiated
resolvent:
P = Sd?(Q-\)-\ (3.17)
where Q is a classical elliptic V>do of positive integer order r on a compact
boundaryless manifold M of dimension ni, with principal symbol qr{x^)
having no eigenvalues on R_, S is a classical V>do of order d, and m is
chosen so that d - r(l + m) < -n\. With ji = (-A)1/** for A in a narrow
sector T around R_, the symbol is in Sd^l+m^0nS0^^l+m^ and weakly
polyhomogeneous. Then Theorem 3.8 and its corollary lead to an expansion
of the diagonal kernel and the trace, generalizing the result of Agranovic
[1] for S = I (cf. [14] for details). The kernel K(x,y,P) satisfies on the
diagonal:
K{x,x,SdT{Q-Xyl)
,d_ . oo
~ EL^*)*2^"""1 + 2X*)logA + 4(^))A-fc-m-1, (3.18)
A;=0
for |A| —> oo, uniformly in closed subsectors of T. Consequently, one has for
the trace:
TvSd^iQ - A)"1 ~ E7o^"^1'™"1 + E(ckH^ + c'k)\-k-m~l ;
J_0 k=o (3.19)
where the coefficients are the integrals over M of the fiber traces of the
coefficients defined in (3.18).
If S is a differential operator (in particular if S = /), then co(x) = 0
and the complete coefficient of A""™""1 is locally determined.
If S and Q are both differential operators, we are in the well-known
case where no logarithms occur, and all coefficients are locally determined
(cf. [22]). This is shown by a simpler version of the above proof where
the decomposition (3.15) is not needed since the symbols are smooth and
homogeneous at £ = 0, and it gives an expansion we may write as follows
(denoting S by D):
K(x,x,Dd?(Q - A)"1) ~ £^K AQK-Af^-™-1,
TrDd?(Q - A)"1 ~ E^MAQM-A)^-™-1, (3.20)

137
for A —> oo in suitable sectors. Let r be even. Then since the symbol terms
homogeneous of odd degree satisfy p(#,-£,//) = -p(#,£,//), their
contributions to the diagonal kernel and the trace vanish (cf. (3.6)); hence
bj(x,D,Q) and bj(D,Q) are zero for d- j odd. (3.21)
For later reference we list the formula for the zeta function that follows
from (3.20) by Corollary 2.2, in the case where the differential operator Q
is selfadjoint > 0 and of order 2. We have to take the singularity resulting
from the nullspace Vo(Q) (of finite dimension vo{Q)) into account; in fact,
Dd?{Q - A)-1 - DYl0(Q)d?(-\)-1 (3.22)
is holomorphic at 0. Here Uo(Q) is an integral operator with C°° kernel
K(x,y,Uo(Q)) = J2ki<v0ui(x) ® ui(y), where the u\ are a smooth or-
thonormal basis of Vq. The kernel of DIlo(Q) is Y1ki<i/0(Dui(x)) ® **/(!/)•
Then the singularity at 0 of K(x, x} D^(g-A)^1), resp. TrD^(g-A)^1,
is
K{x,x,DIlo(Q))d?(-\)-\ resp. TrDUo(Q)&x(-\)-1.
(3.23)
In this case (3.20) is seen to correspond, by (2.10) and Corollary 2.2, to the
following pole descriptions of the diagonal kernel and trace of T(s)DZ(Q, s):
rWjr(,,,,Dzw,.))~ Ljfyp - «<*'*■ f°w)),
rw Tr,Dz(Q,.), ~ £ ^m, - msm, im
j=0 s + 2 S
with
Cj(x,D,g)= bi(x>D\Q) cj(D,Q)= [ tTCj(x}D,Q)dx}
n vy r(m + i + ™^)7 n 'v; Jm n 'vy
cj = 0 for j - d odd. (3-25)
In particular, if Q is the square of a selfadjoint first-order differential
operator A, and D is multiplication by ^0*0 > we Set f°r the pole structure of
the zeta and eta functions, taking the vanishing of coefficients into account:
rWc(*, #,.) = rW tk*zm», .» ~ £ ^4 - MSs^l.
k=08+K~ 2 S
r(S)^, A, 2s- 1) = r(S)Tr(^AZ(A2,S)) ~ £ £^^1(3.26)

138
Let us also list the consequences for the heat kernel and trace, by
Theorem 2.3:
oo ._ _d
K(x, x,De-tQIl£{Q)) ~ Yl ci(*>AQ)t2^~ ~ K(x, x, DU0(Q)),
j=0
J=° (3.27)
Tr(De^)^^cj(Ag)f'Z:!^-;
i=o
note that the effects of the nullspace projection have been cancelled out in
the second and third lines. (In Theorem 2.3, a simple pole at s = 0 for (f(s)
corresponds to a constant term for e(t).)
4. The APS resolvent in the product case
4.1. GENERALITIES ON RESOLVENTS
We now return to the APS operator on a manifold with boundary, as
described in Section 1.2.
One auxiliary tool is to consider an extension of P to a larger manifold
without boundary. As mentioned after (1.14), one can choose a specific
extension P to the double X in the product case. However, in the final
formulas, the choice of extension really plays no role, since all operators are
restricted back to X (more comments in [15], Remark 3.6), so we can let P
stand for any extension satisfying the ellipticity requirements. In the general
case, we just extend to a neighboring open manifold X = XU(X'x ] -1,0[)
preserving the ellipticity hypotheses there.
We denote the extended "Laplacians" Ai = P*P and A2 = PP*, and
set
g,-,A = (A,--A)-1. (4.1)
In the product case where A; is selfadjoint > 0 on the compact manifold X,
this is well-defined for A £ C outside a discrete subset of R+, and the zeta
and eta functions as well as the heat trace for the At- behave as described
at the end of the preceding section.
In the general case, Qiy\ is, to begin with, just defined in a parametrix
sense, but it can be modified such that for sufficiently large A
(Ait + - \)QiX+ = IonX
(4.2)

139
(as explained in detail in [14], p. 508-9). Here we use the convention of
defining, for an operator S on X, the truncation 5+ to X by
S+u = r+Se+u, (4.3)
where e+u denotes the extension of u with e+u(xf) xn) = 0 for xn < 0, and
r+ denotes restriction to {xn > 0}. We shall also write
Tr+5 = Tr[5+]. (4.4)
(The plus-subscript is often omitted when one deals with differential
operators, since they act locally.)
The Qiy\ enter as pseudodifFerential parts of the resolvents we are
looking for:
Ril\ = Qi1\J++Gijx, (4.5)
where the G^\ are singular Green operators (in the notation of Boutet de
Monvel [4]); s.g.o.s..
Remark 4.1 One of the well-known ways to describe the resolvent of a
given boundary value problem is the following: Consider a problem
(P -X)u = f on X, Tu = p on Xf, (4.6)
where P is elliptic of order d in a bundle E over X, and T is a trace operator
(from Hd(X)E) to a suitable Sobolev space Ht{X',F) over the boundary
Xf). The resolvent R\ is the solution operator R\ : / \-t u for the problem
(4.6) with (p = 0. Assume that P — A, extended to a larger manifold X,
has an inverse Q\ such that (P — A)Qa,+ = / on I, where Qa,+ maps
L,2(X)E) into Hd(X,E). Assume moreover that the problem (4.6) with
/ = 0 has a solution operator K\ : <p *-* u (such that (P - \)K\ = 0,
TK\ = /), mapping HjiX^F) into Hd(X,E). Such an operator going
from the boundary to the interior is called a Poisson operator in [4].
Then the full problem (4.6) has at most one solution for any data {/, (f}
in L,2(X)E) x Ht(X',F), since null-data give the null-solution. Moreover,
the resolvent equals
Rx = Qx,+ -K\TQx^ (4.7)
for this operator verifies (P - \)R\ = I and TR\ = 0 and is defined on all
of £2(^7 E) so it must be the unique solution operator.
In (4.7) we see the structure of the resolvent as the sum of a ^do term
and a term composed of a Poisson operator K\ and a general type of trace
operator TQa,+! here K\TQ\ + is an example of a singular Green operator.

140
Another auxiliary tool in the analysis of the inverse (4.5) is to compare
it with the inverse on the cylinder X° = X1 x R+. Define
P° = a(dn + A), P°f=(-dn + A)a*, so
P°'P° = D2n + A\ P°P°' = a(D2n + A2)a*. [ ' '
They have a meaning on X°, where P° goes from E® to 2?° > the respective
liftings of E[ and Ef2, and P° is the formal adjoint of P° with respect
to the product measure. They can be extended to bundles Ef over X° =
Xf x R; the simplest choice is to take the Ef as the liftings of E[ and
extend the formulas in (4.8). We denote the extensions P°, A° = (P°yP°,
A° = P°(P0)'.
On the cylinder X° we consider the realization JPg of P° defined by
the boundary condition £70^ = 0, with the Laplacians A? = P^P% and
A^ = PgPg*. The resolvents are:
#fA = Q?fAf+ + G?A, With
Q?fA = (^ + A2-A)-1> gJA = ^ + A2-ArV, (4-9)
the G°iX being singular Green operators (as in Remark 4.1).
In the product case one can show that the true resolvent R^\ is, near
X\ very closely related to i??A, in such a way that the singular Green
contributions to the asymptotic expansions we are looking for are essentially
the same. In the general case, R®x is a first order approximation in some
sense, so we can take it as a point of departure for the construction of the
true resolvent Riy\.
4.2. DECOMPOSITION FORMULAS IN THE PRODUCT CASE
In the product case, very precise information will be obtained for the
asymptotic expansions, on the basis of exact formulas for the operators involved.
Let
Ax = {A2 - A)1/2, for A G C \ R+; A' = A + U0{A).
(4.10)
We shall here describe the results for the case B = n> (i.e., Bo = 0) in
detail. [15] moreover treats B = n> + fl0 with fl0 ranging in Vb(A). In
a recent manucript [6], Briining and Lesch treat certain other boundary
conditions for problems as in Remark 1.3, see Remark 4.14 below.
Using the cylindrical structure, we shall write the s.g.o. terms in (4.9)
explicitly in terms of the special operator
/*oo
(Gxu)(x',xn)= e-(x^A>u(x',yn)dyn. (4.11)
./o

141
When G is an operator defined by Gu = |0°° G(xn, yn)u(xl, yn) dyn) where
Q is a function of xn) yn valued in operators on x'-space, we call G(xn,yn)
the normal kernel of G, and define its normal trace as
/»oo
trnG= / G{xn,xn)dxn, (4.12)
Jo
when it exists. The normal kernel of G\ is e~(Xn+yn)Ax, and the normal
trace is
TOO
trn Gx = / e-2x»A> dxn = (2Ax)~l. (4.13)
Jo
Example 4.2 To explain how G\ enters, consider the Dirichlet problem
for D2n + A2 - A on X°,
(D2n + A2-\)u=f, 7o« = ¥>, (4.14)
from the point of view of Remark 4.1. The Poisson operator solving (4.14)
with / = 0 is
«&,,*? = «"-%, (4-15)
and the composition 70Q? \ + acts like
1o(Dl + A2-X)-1e+f = 10 2j-xJo e-^-y^e+f(x',yn)dyn
/»oo
= Jo 2X-xe-ynAxf(x',yn)dyn, (4.16)
so the singular Green operator term as in (4.7) equals the composed
operator
GDir,A/ = -#Dir,A7oQi,A,+/
/*oo
= trxJ0 e-{Xn+yn)Axf(z',yn)dyn = £-xGxf. (4.17)
Thus the resolvent equals
(A°Dir - A)"1 = i$ir,A = Q°IX+ - ^Gx. (4.18)
For a Robin-type boundary condition 7o(#n + S)u = 0, where 5
commutes with A2 and A\ - S is invertible, one finds in a similar way that the
singular Green operator term in the resolvent is
C°RoM = ^fe^ (4-19)
In particular for the Neumann condition, G^eu x = 2A~G\.

142
The actual boundary conditions mix boundary values and normal
derivatives in a more complicated way; for example, A^ has the boundary
condition (cf. (1.6))
n>7ou = 0, n<7o(dn + A)u = 0, (4.20)
where we have used that n< = I-n> and that a*a = J. This is a Dirichlet
condition on the functions of xn valued in V> and a Robin-type condition
on the functions of xn valued in V<, so by applying Example 4.2 to each
component, we find that the singular Green term in (A^ — A)""1 has the
form
G?,A=(^n> + ^^±An<)GA (4.21)
(which has a good sense since -A is positive on V<). Along with the
corresponding formula for G^ \, this may be written as in [13], [15]:
^l,A = ^e,A + G0j\ - -0Z\G\,
Glx = *(Ge,A - G0jX + %$Gx)a*, where
G x - =14 G^ - f^L + ^)Gi
U^x - 2Ax(\A\+Ax)Ux " \2\AX T" 2\>u>"
Go,\ = 2(\A\+Ax)]Ar\Gx = ^~2A + 2A l^)^'
(4.22)
Recall (4.10); for the last expressions it is used that 1/(|A| + A\) = (|A| -
AX)/{A2 - (A2 - A)) = \A\/\ - A\/\. The indices e and o refer to the
evenness and oddness of the principal symbols with respect to £'. (The
parity alternates between even and odd in the sequences of lower order
symbols.) All the operators are defined and holomorphic for A £ C \ R+.
Moreover, Ge,A and Gq,a are holomorphic at 0 because of the factors \A\
and A that vanish on the nullspace.
From (4.13), (4.22) follow:
iv r — -A2 ■ \A\ , r _ -A ■ 1 A
trn^e,A - 4AA2 r 4i4AA> lln <^o,A — 4XAX T 4A \A'\
(4.23)
Example 4.3 The corresponding expressions for the Dirichlet and Robin-
type problems considered in Example 4.2 are
+r r<0 — -ni- — i i +r r<o _ l ax+s
^rn^Dir,A - 4,42 - ~ A A2_A, xrn <^R0b,A ~ AA\ Ax-S'
A A (4.24)
Now one can show that near Xf, the true s.g.o G^x is very similar to
the cylindrical version G?A.

143
Lemma 4.4 (Product case.) Let x € Co°(R) with x{xn) = 1 f°r \%n\ < §>
x{xn) = 0 for \xn\ > y. Tften Gi,a - X^i aX Z5 *mce c^ass zn ^2(^1)
wf/i norm OflAI-^) for |A| —> 00 with argA G [#, 27r - S\, any 8 > 0.
Tfte same is frwe of 0*[Gi,a - xG\ Ax] for A = 1,2,..., and 0/ expressions
DG\}\ - xD'G® \X> where D is a differential operator, constant in xn near
X1 and equal to Df there.
Similar estimates hold for G2,a~X^2 aX zn ^2(^2); and for the operators
(1 - x)GjA and G°a - X^JaX w ^2(fi?). Here the GjA can 6e rep/aced 6y
Ge,A orGQj\. _
All these functions are holomorphic in A £ C \ R+.
The proof is given in detail in [15], using elements of [13]. It extends to
show that the operators also map into Hs, any s, with 0(^1""^) estimates
for any N.
We now construct the zeta functions. For this, we integrate A~si^A
along an appropriate curve C as in Theorem 2.1, running along the negative
axis and around a small circle of radius
r0 < min{Ai(A1-),Ai(A1-),Ai(A2)}, (4.25)
where Ai denotes the smallest positive eigenvalue. (C could also be taken
to be a curve in Re A > 0 closer to the spectra.) Then (cf. (4.3))
Z(A,-, s) = £ /c X~'RitX d\ = ±fe X-sQitx,+ d\ + ±fc X-sGitX dX
= Z(At-, s)+ + Gz,i,s<> where we have set
In the trace calculations in Theorem 4.6 below, we shall replace G^\ by
G®x by use of Lemma 4.4. Define the transforms
Gz,z,s = ^ fc ^""S^e,A dA, Gz,o,s = 2i Ic ^""S(^o,A dA.
(4.27)
To describe the various Gz, we use the function defined for Re(-t) <
Re s < 0 by
F^) = Ue^r-^{l-r)-Hr
= Mei~S~1]i* - e(s+1)i7r) /o°° w_s_1 (1 + w)"( <*u
- ±sin7r(s + 1) rH)r(H-t) _ _r(£±!]_. (4-28)
CTiro is taken with ro G]0,1[, cf. (2.2). Ft(s) coincides with the binomial
coefficient Cjl'J'1)' a^so eclual to (sB(t, s))_1, where B is the beta function.

144
Ft(s) extends meromorphically to general s and t £ C. In particular,
F> (*) = r£*l\ = Ch, FAs) = 1,
2V ' vArr^+i) v-5y' 1V ; ' (4.29)
F0(s) = 0 if s £ 0, F,(0) = 1 if T{t) £ oo.
That Fi(s) = 1 follows directly from the first integral in (4.14), and the
formula for F0(s) follows from the fact that ^ Jc r_s_1 dr = 0 for Re s > 0.
The formulas for the singular Green operator terms are greatly
simplified when we take normal traces.
Proposition 4.5 Define Gz,e,s andGz,0,s by (4.27), cf. also (4.13), (4.10).
Then
trnGZtetS = \(Fl(s)-l)Z(A2,s),
, 2 (4-30)
tTnGz,o,s = -\Fl2(s)Y(A,2s).
Proof: Expand the operators on Xf with respect to the orthogonal eigen-
projections {n/i}/iGsp(>i) for A. Our Gz,e,s and Gz,o,s are both 0 in the zero
eigenspace. Using (4.23) we find, by replacing A by fi2r for each //,
tr„ GZte,s = trn £ Jc A-Ge,A dX = £ /c A-(^r + ife)<*A
= E iH"2'i £ r—*(li + ^t) *" • nM (4.31)
= \(-F1(s) + Fk(s))Z(A\s) = \(-l + Fk(s))Z(A\s);
)d\
tr„ (?Z)0)S - trn ^ /c A SG0,\ <*A - ^ /c A '(4X3^" + 4A |yl'|)
= E\^Ic^-H--^ + &dx.u,
= E }f W"2-1 i Ic T—1 (^r + 1) dr ■ II, (4.32)
= i(-Fi(*) + F0(5))y(yl)2s) = -iFi(*)y(A,2s). D
Note that the even part produces a function derived from the zeta
function of A, and the odd part produces a function derived from the eta
function of A. This is the fundamental observation for the following, relating
the power functions of the boundary value problem to those of A.
Now we combine this with the interior contribution, taken from the
doubled manifold X. This leads to the key result:

145
Theorem 4.6 (Product case with Bo = 0.) The zeta functions have the
following decompositions:
T(s)C(Aus) = r(s)[C+(A;,s)+±^^
+ i[Tr+(n0(At-)) - u0(Az) + (-iy\v0(A)] + hz(s), (4.33)
where the h{ are entire. Moreover, T(s)((Ai, s) isO(e(~i+€"lms\) for \ lms\
> 1, -oo < C\ < Re s < C2 < 00, any e > 0.
Here C+(A;, s) = Tr+ Z(At-, 5) (cf. (4.4)).
The basic idea in the proof goes as follows: By Lemma 4.4, the resolvent
(At- - A)-1 = (At- - A)^1 + Giy\ has the same asymptotic behavior for A
going to infinity as (At- - A)+* +x(*i\Xi an^ *he last term behaves like G?A.
Here the contribution from At- is well-known; and the contributions from
Ge,A and GQ,\ in G^A have been dealt with in Proposition 4.5; they give the
terms involving Fi(s). What remains is some adjustments due to the
Laurent expansions of the resolvents at A = 0 and the trace of G°iX restricted
to the nullspace of A, plus the contribution from an 0(|A|_j/v) term; these
adjustments yield the coefficient of -s in (4.33) and the entire function. The
explanation is slightly technical because of the need to consider
differentiated resolvents as in (2.11). We leave out further details; they are given in
[15].
Example 4.7 For the Dirichlet realization A^[r of D\ + A2, a calculation
as in (4.31) gives, by (4.24),
trnGz,Dir,5 - trn i Jc X-sG^iTjXdX = £ fc ^^rzx) d\ = -\Z{A2,s).
Then the zeta function for the Dirichlet realization Aoir of P*P has the
decomposition (with h(s) entire):
r(*K(ADir, s) = r(s)c+(Ax, *) - \r(s)C(A2, s) + h(s).
(4.34)
For the Neumann case one gets this formula with -\ replaced by +\.
A similar analysis applies to the eta functions associated with P#, and
to functions with differential operators inserted in front. Consider e.g. the
eta function T(s) Tr(</>PAj~s), where <p is a bundle morphism from E2 to
J5i, equal to (p° = ip\x> on Xf x [0,c]. (Some morphism is needed in order
to allow taking the trace in L2(Ei)] e.g., <r* can be used for (p.)

146
Theorem 4.8 (Product case with Bo = 0.) The eta function
T(s)r)((p, Pb, 2s - 1) has the following decomposition:
T(8)r,(<p, PB, 2s-1) = T(s) Ti(<pPA?)
= T(8)[Tr+(<pP&?) + i(Fk{s - 1) - l)i/(A, >4,2» - 1)]
+ ^Tr(^an0(A))(S - i)"1 + h(s), (4.35)
where hi(s) is entire. Moreover, T(s)r)((p,PB,2s-l) isO(e^~^€^Ims^) for
\lms\ > 1, -oo < C\ < Re s < C<i < oo, any e > 0.
Tftere 25 a similar result for T(s) Tr((^P*A^s), w/fcere </> 25 a morphism
from Ei to E2.
4.3. PRECISE TRACE FORMULAS IN THE PRODUCT CASE
It is shown in Theorems 4.6 and 4.8 how the zeta and eta functions of the
APS operator arise by simple addition of known zeta and eta functions with
factors defined from Fi(s) in front.
2
This makes it easy to determine the pole structure! We know the pole
structure of the zeta and eta functions of the operators P, At- and A, and we
also know the pole structure of Fi (s) from its gamma function components.
The result is that we get from each decomposition a meromorphic function
with poles where those functions have them; and the poles will be double
when there are coincidences. Accordingly, there will be heat trace
expansions with powers t& corresponding to the simple poles —/?, and powers t&
plus t& \ogt terms corresponding to double poles —/?.
We list the precise result below. An interesting aspect is that it shows
a difference between the cases n even and n odd.
In the case n even, coincidences between poles give rise to double poles
(hence log-terms in the heat operator formulation). At a double pole -/?,
the singular part consists both of a coefficient c times (s+/3)~2 and another
coefficient cf times (s + /?)_1. The first coefficient c is determined from the
symbols of the operators in a well-known local way, whereas the second
coefficient cf is usually just globally determined.
In the case n odd, there are no coincidences, hence no double poles. But
here the poles of Fi force us to evaluate the zeta and eta functions at the
2
points midways between their well-known poles; also this gives new global
coefficients. (These are the points where the poles according to (3.24) have
vanishing residue (3.25), so the value can also be regarded as the second
coefficient where the first one is 0.)
Now comes the detailed description:

147
We denote the second coefficient in the Laurent series for T(s)£(D, Q, s)
at a pole Sj = ~^+d by c'^D.Q):
c>(D,Q) = lun[T(s)«D,Q,s) - sg£l] = ^ UMMA,
3 (4.36)
here Ress=s; means the residue at s'. (In case Cj(D,Q) = 0, c'j(D, Q) is the
value of T(s)((D,Q, s) at the point.)
We also need to define some universal constants:
pm = Res,_ i m \Fx Is) = A ,1^7—7,
P'm = Res5=_i_m \F1_(s)(s + i - m)"1,
7fc = i(n(D-1) = i
(4 37)
em = Ress=_i_m iFk(8)T(s) = J.^i),
5m = Res5=i_ro ±F,(* - 1)I» = Ress=|_m ^T(s - }) = ^;
here m e N, and the k are integers avoiding negative odd numbers. (The
explicit expressions are found by use of the formula T(s) = tiT(I-s)/ sin ns.
Also /J^ can be written more explicitly, departing from the fact that -r'(l)
equals Euler's constant.)
From (3.24) we find, omitting vanishing coefficients,
r(«)c+(Ai,s) ~ ^ , , t. a . (4-38)
where cjj+(Ai) = fxtrcj(x,Ai)dx; cf. also (4.4). Since A acts on X' of
dimension n - 1, we get from (3.26):
k=0 S + ^ " 2 6
and, for example, when ^ is a morphism in E\,
_1
2 y/7TS
T(8)Fi (*)i/(tf, A, 2.) = -^-r(* + i)C(iM, A2,* + 1)
^ c2fe+i(^4,A2) */(!M,0)
> r- /tn : —; : — H II U IS Odd,
T c2k+1(^A,A2)
H W-;;—- H 1t= if n is even,

148
where c^.^A, A2) is defined as in (4.36). When tl> = I then cn-\(A, A2) =
0and<_1(^^2) = V^^,0).
Insertion of these expansions in our decompositions gives:
Corollary 4.9 The zeta function r(s)£(At-,s) is meromorphic on C, with
the following singularity structure:
For n even:
&-fc2fc(A2) A-f4(Aa) + QyM-iW)
+Jk>|(«+*-af1)2+ 5+^-^ J
,,_lVir v- c2k+1{A,A2) riJA^ + vojA),
^~ ) 4[0<_k%.1^^-k-l)(s + k + l--)+ s (J41)
For n odd:
TVcWA .\ v-£2fc±iM Tr+ n0(At) ^ 7n-i-2fcC2fcQ42)
i>0S + S 2 S k>0 S + IC 2
4o s + ™ + ^
, , lViry^ c2fc+i(A,A2) 77(A,0) + ^0(yl)1
( *r0A(!-*-i)(*+*+i-!) * J" l '
[15] moreover shows the formulas where a morphism is included. The
terms /3fm were missing in the Preprint version of [15].
Corollary 4.10 The eta function T(s)ri(<p, PB, 2s - 1) = T(s) Tr(^PAp)
is meromorphic on C, with the following singularity structure:
For n even:
r(s)T^PAr')~£c*+';+<^'>
*>0 S + k 2
v- Jn-3-2kC2k+l(faA,A2) ^ Jk+1_<lC2k+l(faA,A2)
0<fc<f-l S^K 2 fc>f-l { ^ 2 '
s + k-
"I 77T n^I J
+ ^«. (4.43,
V*(« - 2)

149
For n odd:
fc>0 * + * 2 fc>o S-tK- 2
vM(A^,-2m) Tr(An0(A))
+ 4 , + m-l + V5F(.-1) • ^^
There are similar formulas for Tr((^P*A^"s), wi£/fc <p°a replaced by a*<p°.
Let us finally list the consequences for heat traces, derived from
Corollary 4.9-4.10 by use of Theorem 2.3:
Corollary 4.11 The exponential trace Tr(e~tAt) has the following behavior
for t -> 0.
For n even:
n-l
Tr(e-tA0 ~ £ <**,+(&) ifc"? + £ 7n-i-2*c2*(A2) 4*-»r
fc>0 0<Jt<f
+ E[-^-f^(^2)log*+^-f4(^2) + (^-f - i)c2*(A2)] t*-*^
fc>f
+ (-l),-i[ E g%l(Y2)1^fe+l-?+^,0) + ,o(A)]. (4.45)
For n odd:
c2M.(A>*-*
A;>0
Tr(e-*Ai)~Ec2*.+ (^)**"»
+ E Tn-i-2fcc2fc(A2) **-*?■ + J] eroC(A2, -m - ±) tm+"
A:>0 m>0
Corollary 4.12 The associated exponential trace Tr(<^Pe~"tAl) has the fol-
lowing behavior for t —> 0.
For n even:
Tr(<pPe-**) ~ ^cjhl+^P.A!)**-*
jt>0
+ J] 7n-3-2fcC2fc+i (yPaA, A2) t*" ^ +
0<fc<f-l

150
+ (ft+i-t <W A4 a2) + (^+1_f - i)C2fc+1(A^, a2)) tfc-^]
+ ^Tr(An0(A))ri (4.47)
For n odd:
Tr(¥>Pe-«Al) ~ J] c2Jb+li+(VP, Ax)t*-t
A:>0
+ £ 7n-3-2ifeC2ib+l( A^, ^) ^"^
A:>0
+ Y, Mr0*A, -2m)r-5 + ^ Tr(An0(A)) r*. (4.48)
ra>0
Tftere are similar formulas for Tr((fP*e~tA2)f with <p°a replaced by
The proof shows the advantage of working with the power functions,
where the contributions from the boundary condition appear as simple
multiplicative formulas involving the zeta and eta functions of A; this allows
an exact analysis of the pole coefficients which can then be carried over
to the heat expansions by Theorem 2.3. If working directly in the heat
operator framework (a point of view taken up in [6]), one has to deal with
convolution-type integrals.
Gilkey and Grubb [11] show that all terms, in particular the logarithmic
ones, are nontrivial in general. Dowker, Apps, Kirsten and Bordag [7] find
no logarithms for the Dirac operator on the ball; this is due to special
symmetries and does not contradict the above since it is not a product
case.
Example 4.13 For the Dirichlet problem considered in Examples 4.2, 4.3
and 4.7, formula (4.34) implies in a similar way:
Tr(e-tA^) „ £ C2Jkf+(ai} **-? _ 1 J2 c2k(A2) tk~^;
k>o k>o (4.49)
note that all the integer and half-integer powers enter here too. There is a
similar formula for the Neumann problem, with -\ replaced by +\.
Remark 4.14 In a recent study of the gluing problem for the eta-invariant,
[6], Briining and Lesch treat boundary conditions of a somewhat different
nature than those considered here and in [14], [15]; moreover they depend

151
on a parameter and the variation in this parameter is studied. We show
below how those new boundary conditions can be handled in the present
framework: Restrict the attention to self adjoint operators P satisfying a* =
-a, a A = -Aa as in Remark 1.3. Let B be an orthogonal projection in
L2{E[) commuting with A2 and satisfying
0) °B = (I - B)a,
(ii) BAB = a\A\B for some a > -1. K ' '
([6] gives special examples of the form B = crill> + <72ll< + Bq with mor-
phisms or zero order V'do's o\ and <T2.) Because of (i), Pb is selfadjoint, and
Ab = Pb2 is the realization of P2 under the boundary condition (where
Bfo is written joB)
y0Bu = 0, y0Ba{dn + A)u = 0. (4.51)
For the second equation we note that when foBu — 0, then in view of (i),
j0Ba{dn+A)u = aj0{I-B)(dn+A){I-B)u = ay0{dn+(I-B)A)(I-B)u.
Here, by (i) and (ii), {I-B)A(I-B) = -a\A\{I-B). Thus the boundary
condition may be written:
7oBm = 0, jQ{dn-a\A\)(I-B)u = 0. (4.52)
This is a Dirichlet condition for the functions of xn valued in R(B), and a
Robin-type condition as in Example 4.2 with S = -ot\A\ for the functions
valued in R(I - B). Then by the calculations in Example 4.2, the resolvent
on X° is {A% - A)"1 = Q°1X+ + G°BX with
= (ft + 2((l4-;^l-A) + 2(A>+a|A|)(/ " 25))^- (4.53)
Now Lemma 4.4 can be extended to this case. Therefore we have as in the
proof of Theorem 4.6,
I»C(AS,,) = r(«)C+(Alf •) + T(s) Tvx £ I \-°G%xdX + h(s),
Jc (4.54)
with h(s) entire; and here
Tr* JL jf X-*GBtX d\ = Tvx, trn £ jf A"'^ dX =
^x> h I *"(sfc + 2((^I-A) + W&awP ~ ^B))^-xdX. (4.55)

152
The term 2*A L^JJ - 25)^- contributes with zero, for by (i) and the
fact that a and B commute with A2,
2(Ax+a\A\)(I ~ 2B)2A^ = 4Ax(Ax+a\A\) (7 " 5) " 4^A(^A+a|^|)a*a5
= 4^(^+c^|) (7 " 5) " a* 4Ax{Al+a\A\) (7 " ^ (4'56)
here since the trace is invariant under circular perturbations (that we can
use in a reformulation with sufficiently high A-derivatives as in (2.10)), the
contributions from these two terms will cancel each other. The remaining
terms are treated as in Proposition 4.5 (we give the details for a < 1; the
case a > 1 is similar and the case a = 1 is simpler):
_L / \-*/_=!_ . Ax-a\A\ \ ix
2tt Jc A \4A\ + 4Ax((l-a*)A*-\)) ttA
~ Z^ 4 2tt fc * S(/x2_A + (l-a*)p>-\ ~ ,4,T 2, 2 ,J rfA ' ^
M€sp(,l) (M2-A)2((l-a2)M2.A)
= i:iM-AJfer-(=i±^g-|l .' , )*■*, (4-57)
M (l-T)2(l-a2-T)
= I(-l + e-Mi-2) + Fa{s))Z{A\ s)-
with
JC (l-T)2(l-a2_T)
This is a hypergeometric function whose pole structure is easily determined
by use of Theorem 2.1. In fact, Fa(s) is of the form (2.17) with f(r) =
-a(l - r)"2"(l - a2 - r)"1. It is holomorphic on C \ [l,oo[ and has the
asymptotic expansion for -r -> oo in closed subsectors:
/(_,.) = _ar-|(l + l)-i(l + l=f£)-i
~ -ar-l £ (7)r"fc E(«2 " 1)''"' = £ ^"H- (4-59)
3
An application of Theorem 2.1 carries the terms Ujt~*~3 over into simple
poles at s — -j - \ for ^^;Fa(s) with residues Uj. The poles at integers
j + 1 stemming from the Taylor expansion at 0 are removed when we
multiply by 7T"1 sin7rs. Consequently, Fa(s) is meromorphic on C with simple
poles at the points -j - ^, j G N, with residues 7r~1(-l)J+1a;j.
Finally,
r(*K(AB,*) = r(s)c+(A1,*)
+ !(_! + g-iogd-o2) + Fa(s))r(S)C(A2, S) + />(*), (4.60)

153
which is meromorphic on C with poles at the points (n - fc)/2, k G N; here
the poles at the negative half-integers -j - \ are in general double when
n is even; otherwise the poles are simple. A heat trace expansion in terms
of t(*~n)/2 and tl+2 log* (fc, / G N) follows as usual by Theorem 2.3.
Note that (4.58) also implies: 1) Fa(s) equals 7r_1 sin its times the Mellin
transform of -a(l + r)"2 (1 - a2 + r)"1 at s - 1; cf. (2.18), (2.33).
2) (1 - a2)Fa(s) - Fa(s - 1) = -aFL(s - 1); cf. (4.28).
5. The general case
5.1. A GENERAL RESOLVENT CONSTRUCTION
In the non-product case the results will be more qualitative. A useful trick
here is to replace the separate consideration of Pb and Pb* by the study
of the skew-selfadjoint operator
?=(p T) M
this is the realization of
v-l
KP 0
under the following boundary condition on u — {^1,^2} (cf. (1.5)):
L2(E[)
Bj0u = 0, where B = (B B') : x ->• L2(E[). (5.3)
HE'2)
The advantage of taking Pb and Pb* together in this way is that Vs is
two-sided elliptic, and TZ^ = (Vs + //)_1, defined for \i G ±ro, To =
{fi G C \ {0} | | arg//| < n/2 }, satisfies
where (Ai+fj,2)'1 = i^.^ are the resolvents we are looking for (cf. (1.7)).
The diagonal terms give back the individual resolvents, and the off-diagonal
terms can be used to describe eta functions instead of zeta functions.
This allows us to stay working with first-order systems (instead of
passing to second order), at the cost of doubling up the size of the matrix.
We shall denote E1®E2 = E and E[ 0 E'2 = £".

154
We let V = f ~ ~p* J, where P is an elliptic extension of P to a bundle
E = Ei©£2 over X = IU(X'x ] -1,0[). Then V+p has a parametrix QM
(of strongly polyhomogeneous type) for \i e ±ro, and as shown in detail in
[14], p. 508-9, it can be modified such that for large fi in closed subsectors
of±r0,
(P +/i)QMf+ = J on X. (5.5)
Also here, a comparison with the cylinder case (cf. (4.8)) plays a role.
We denote (p°0 "J0') = P°, acting in E° = E%@E$. We extend V° to X°
simply by extending the formulcis (4.8) to xn e R, letting E° = E® © E°
be the lifting of E' = E[®E'2. Then the extended operator V° is skew-
self ad joint, and the resolvent is
( ^2 + ^ + ^-1 (-dn + A)(Dl + A* + ^)-i<r*\
{-a(dn + A)(Dl + A> + ^)-1 naiDl + Ai + n^a* )' ^
In particular,
(V0 + »)Ql+ = IonX0. (5.7)
Along with Vb, we study the realization V& acting like V° on X° and
with the same boundary condition (5.3) as V. With a slight abuse of
notation, we now denote
AM = (A2 + /i2)2, for/i€±r0. (5.8)
Lemma 5.1 Define the iftdo from sections of E[ to sections of E[ © E[:
and f/je Poisson operator from sections of E[ to sections of E°:
1 0\ _
^=(J a)e~XnA"S^ (5-10)
Tften K'q^ satisfies
BloK°Blli = 1 on X', (V° + /i)#gi/4 = 0 on X°. (5.11)

155
The proof is a direct verification, using that B commutes with A.
In other words, K$ : ip •->■ u solves the problem
By0u = V on X', V ;
when 5 = 0. We note that by (5.7), the full solution operator for (5.12) is
(5.13)
cf. also Remark 4.1.
Now VP is principally like the true resolvent TZ^ at Xf. However, we
prefer to use a better adapted approximate resolvent, namely
%'„, = Q,,+ - Gi with Gi = xKb^oQ^, (5.14)
where Qnt+ satisfies (5.5) and x is a cut-off function as in Lemma 4.4. By
(5.11), Tl'p maps into the domain of Vb, and by (5.5), we have for large
enough //,
(V + n)U'^ = (V + n)Q^ -(V + n)XK°BilMBl0Q^
= / " (P>, x] + X(V - V^K^BjoQ^
= I-G2, with G2 = (xnV1 + V0)KllMB1oQ^Jb-1^
the Vj denoting differential operators of order j with smooth coefficients
vanishing for xn > \c. (G\ and G2 are //-dependent, and so are many
other auxiliary operators in the following, where we do not indicate the
//-dependence explicitly.)
The exact inverse TZ^ of Vb + f1 can then be described by
Tl, = U'^I - G2)-1 = (Q„,+ - Gt)(I - G2)~\ (5.16)
whenever / - G2 is invertible. The main point is now to show that this
holds for large fi and leads to a constructive expression for Tip.
For this purpose, we analyze the various factors in (5.14) and (5.15).
Let us denote
K1 = x(l°)K(h K2 = (xnV1+V0)(10°)Ko, [' '
here K0 goes from C°°{E[) to C°°(E$), Kx and K2 go from C°°{E[ ®£{) to
C°°(E), T0 goes from C°°(£;) to C00 (£?'), and S0 goes from from C°°(£') to

156
C°°(E[(&E[). (They also define mappings beween suitable Sobolev spaces.)
Then
Gi = ffiS0T0, G2 = K2S0T0. (5.18)
In the terminology of Boutet de Monvel [4] and Grubb [12], the Kj are
parameter-dependent Poisson operators and To is a parameter-dependent
trace operator of class 0 (trace operators of class 0 are well-defined on
L2), but their usage entered elliptic theory much earlier, cf. Seeley [21],
Hormander [17]. For the considerations of these operators, we do not need
to introduce new and complicated symbol classes and composition rules for
boundary operators, for in fact they are of the strongly polyhomogeneous
type: When the parameter \i runs on a ray {// = ge*e° \ g > 0}, g enters like
another cotangent variable on a par with £i,...,£n-i, m the sense that the
standard estimates described in [4] are satisfied with {£i,.. .,£n_i,£} as
the boundary cotangent variable. This is similar to the situation described
in Theorem 3.7, now for boundary operators.
Let us refrain from further details (that presuppose a lengthy introdu-
tion to the calculi described in [4], [12], summarized in the appendix of
[14]), but just mention a consequence we need:
Lemma 5.2 With K\, K2 and To defined above, and (p a morphism in E,
the compositions To<pKj are strongly polyhomogeneous iftdo's on X' of order
-1. Moreover•, the compositions TotpQ^+Kj are strongly polyhomogeneous
ipdo's on Xf of order -2.
An important trick in the following is to reduce considerations of the
singular Green operators Gj to considerations of ^>do's in the boundary.
This is done on several levels; one is in the study of inverses that uses
Lemma 5.3 below, another is in the study of traces in Section 5.2, where a
cyclic permutation brings operators of the form TK into the picture.
First consider the problem of inversion of / - G2. Here we shall use the
elementary lemma:
Lemma 5.3 Let K : V -* W and T : W -» V be linear mappings between
vector spaces. Then I - KT : W -* W is bijective if and only if I - TK :
V —> V is bijective, and
{I - KT)-1 = 1 + K(I - TK)~lT. (5.19)
Proof: A straightforward verification. □
The lemma will be applied with K = K2 (going from sections of E[®E[
to sections of E) and T = SqTq (going the other way). This replaces the

157
construction of the inverse of / - KT = / - G2 by the construction of the
inverse of / - TK — I - S0T0K2', so that
(I-G2)'1 = I + K2{I-S1)-1S0To with S1 = S0ToK2
(5.20)
holds when / - S\ is invertible. The advantage of this reduction is that Si
is a ^>do on the boundaryless manifold Xf. The factor T0K2 is a strongly
polyhomogeneous ^>do of order -1 by Lemma 5.2, and it remains to examine
the other factor in S\ and the composition, and to apply this to construct
the inverse (/ - Si)'1.
Here we go more in details with the symbol classes introduced in Section
3.1. The following class will play a special role:
Definition 5.4 Let r be integer > 0, and let S = OP(s(z, £,//)) (or let S
have the symbol s in local coordinates). S and its symbol will be called
special parameter-dependent of order -r, when
s(x,t, p) G 5"r'°(Rn, Rn, r) n S°'-r(Rn, Rn, T) with
d?s(x,t,n) g 5-r-m'°(Rn,Rn,r)n5°'-r-m(Rn,Rn,r)
for any m, all d™s being weakly polyhomogeneous.
Example 5.5 To give examples, we first note that any strongly
polyhomogeneous symbol of degree -r satisfies Definition 5.4 by Theorem 3.7. But
there are also important weakly polyhomogeneous examples, such as the
symbol (a(x,£) + //)_1 (fi in a sector T), where a(z,£) is homogeneous of
degree r in £ for |£| > 1 and a(x,£) + ff is invertible when \i G T (by [14],
Th. 1.17).
For the operators entering in the APS problem we have:
Proposition 5.6 The ijido Sg^ on X\ with \i running in ±To, is special
parameter-dependent of order 0. So are B and the composition So = Ss^B.
Proof: (Indication.) For the proof we split Ss^ in several terms:
C = (B+ii-l{A»+A)BL\
- U-^M-^nJ + W) + V »-\a»-a)b« ) • V-Zl)
The second term has a polyhomogeneous symbol in S° C S°>° (cf. (3.4))
and is independent of //, hence is special parameter-dependent of order 0.
(This proves the statement on B.) The third term is of order -00, and its

158
boundedness in fi (with improved estimates for derivatives) is seen from
considerations on the involved eigenspaces for eigenvalues of modulus < R.
It is the first term in (5.21) that requires most of the analysis. The
crucial fact used here is that
ti-1(Ali + A)Il<=ti-1(A>i + A)(Ati-A)(A>i + \A\)-1Il<
= ?(Alt + \A\)-1n<,
^(A, - A)n> = ^{A, - A)(A, + A)(A, + lAI)"1^ (5>22)
^(A^ + IAI)-1^.
Again n< and Il> are in 5° C S°'° and independent of //, hence special
of order 0. In view of the composition rules (cf. (3.7)), it remains to prove
the statement for ^{A^ + \A\)~l. The advantage of this expression is that
A^ and \A\ are both "positive" (strongly elliptic), so that the inverse of
A^ + \A\ can be described by a natural elliptic construction. (Details are
given in [14], Proposition 3.5.) The statement on So now follows from the
composition rules. □
These operators act on Xf, of dimension n-1 (where the space variable
and cotangent variable are denoted x' and £'). For s G R we define the
space ifs,/i(Rn~1) as the Sobolev space with norm
\\4s,,= m',ri)su(Z')\\L2(R^), (5-23)
and extend the notion to sections of a Hermitian bundle E" over Xf by
use of a finite family of local coordinate systems (the space is then denoted
HS^{E")). Note that H°^{E") ~ L2{E").
We shall need
Proposition 5.7 Let S be a special parameter-dependent iftdo of order -1
in a bundle E" over Xf, with fi running in a sector T. Then for sGR,
S is continuous from HS^(E") to Hs+l^(E"), uniformly for \i in closed
subsectors Tf ofT, \fi\ > 1; and its norm as an operator in HS^{E") satisfies
\\S\\C(H^E»)) = 0(H_1) for M -> oo, fi e T'. (5.24)
For each V there is an rp/ > 0 such that I - S is invertible for \i el"
with \fi\ > rp/. The inverse equals
oo
(/ - S)"1 = I + S', S' = J2 SJ> (5-25)
i=i
where the series converges in the norm of operators in L,2(E").
Moreover, Sf is a special parameter-dependent ij)do of order -1.

159
Proof: (Indication.) By the composition rules, S composed with an in-
vertible ^>do with principal symbol ((£',//)) is special parameter-dependent
of order 0; it is not hard to show that such an operator is continuous in
Hs,/i, uniformly as stated. This implies the asserted continuity from if5,/i
to Hs+1^; and (5.24) follows since
HIM|s,/i < const. ||tt||5+ilM. (5.26)
For each sector T', take rp so large that the operator norm of S in
L2{E") is < \ for \n\ > rp; then (5.25) holds in operator norm.
The powers S^ are special parameter-dependent ^do's of order -j, by
the composition rules. Further efforts are needed to show that the sum S'
is indeed a ^>do that is special parameter-dependent of order — 1; see the
details in [14], proof of Theorem 3.8, as explained for S2 there. □
Now we use these facts to show:
Theorem 5.8 The operator S\ in (5.20) is a special parameter-dependent
tftdo of order -1 in the bundle E" — E[®E[ over Xf. Hence for each closed
subsector T of To (or -To) there is an rp > 0 such that I - S\ is invertible
for \i e T with \fi\ > rr, with inverse
oo
(7-51)-1 = / + 52, S2 = Y,si> (5'27)
52 being a special parameter-dependent iftdo of order -1 in E'{.
Furthermore, for such /j,,
(I - G2yl = / + K2(I + S2)S0To, (5.28)
and finally
HI> = (Q*+-G1)(I-G2)-1
= (QMf+ - KrfoToW + K2(I + S2)S0To)
= QMi+ - (Kx - K3)(I + S2)S0To, with K3 = QMf+ff2. (5-29)
Proof: In the formula (5.20) for Si, So is a special parameter-dependent
^>do of order 0 by Proposition 5.6, and ToK2 is a special
parameter-dependent ^>do of order -1 by Lemma 5.2 and Example 5.5, so it follows from
the composition rules (cf. (3.7)) that S\ is a special parameter-dependent
^>do of order — 1. Then Proposition 5.7 applies, showing the assertions for
s2.

160
Now the formula for (/ - G2) x follows from (5.20). The first two lines
in (5.29) then follow from (5.16) and (5.18). Consequently we have:
U, = (QMf+ - ffiS0To)(J + K2(I + S2)SoT0)
= Qv,+ +Qn,+K2(I + S2)SoTo
- IUS0T0 - KiSoT0K2(I + S2)SoT0
= Qn,+ +Qn,+K2(I + S2)SoTo [ J
-K1S0To-K1S1{I + S2)SoTo
= gMf+ - (k1-Qk+k2)(i + S2)SoTo,
using formula (5.20) for Si and the fact that / + Si (J + S2) = / + S2. This
ends the proof. □
Taking the structure of the entering Poisson and trace operators into
account, we have obtained:
Corollary 5.9 For each closed subsector T of ±To one can find rp > 0 so
that the resolvent 7£M = (Vb + //)_1 for \i e T with |//| > rr is of the form
11^ = Q^+ + KST, (5.31)
where K resp. T are a strongly polyhomogeneous Poisson resp. trace op-
erator of degree —1 and S is a special parameter-dependent iftdo on Xf of
order 0. The detailed structure is given in (5.29).
5.2. TRACE CALCULATIONS
Consider 7£M — (7^+aO-1, as described above. Since the injection of HS(X)
into L2(X) is trace class for s > n, the terms in d^TZ^ are trace class when
m > n.
Theorem 5.10 Let <p be any morphism in E = E\®E2, and let m> n =
dimX. Then
00
i=i
OO
+ Y,(cJlogli + c'J)li-m-1-j, (wH-foo, (5.32)
j=o
for fi in closed subsectors of±To. The coefficientsaj, bj andcj are integrals,
fx aj(x) dx, fx, bj(xf) dx' and fx, Cj(xf)dxf, of densities locally determined
by the symbols of P and B, while the c'j are in general globally determined.
The coefficients cq and cf0 are the same as for the case where the Pj are
zero in (1.2) (the product case).

161
Proof: We find from (5.29):
<pd?Qr,+ - <pd?[KiS0T0] - ^[(KtSi - K3(I + S2))SoT0]. (5.33)
First, Tr(<pd™QtMl+) contributes the well-known expansion
ZxT cijfin~rn~1~K For the other terms we can use the invariance of the
trace under cyclic permutation of the operators, to reduce to a study of
operators on Xf. For the middle term we find, by the Leibniz rule:
Tvxifd^KiSoTo])
£ cmi,ma,m3Trx(<pd?1K1d?>Sod?*T0)
m\+m2 +7713 =m
^7711 ,7712,7713
7711+7712 +WI3 —rn
= TrX'd?{S0To<pK1). (5.34)
By Lemma 5.2, To<f>K\ is a strongly polyhomogeneous ^>do on X' of order
-1, hence special parameter-dependent by Theorem 3.7. Then since So is
special parameter-dependent by Proposition 5.6, it follows that
d™(SoTo<pKi) is a special parameter-dependent ^>do on X' of order -ra-1.
To this we can apply our general Theorem 3.8 and its corollary, after
a reduction to local trivializations by use of a partition of unity. Since the
symbol has degrees -ra-l-j, j > 0, and //-exponent d — -m - 1, we
get an expansion in a series of locally determined terms bk
k > 0, together with a series of terms (c^i log/z + c^)//-771-1--7, j > 0, with
cJ?i locally determined.
The third term is treated similarly; here the circular permutation of the
terms resulting from the Leibniz rule gives a special parameter-dependent
^>do of order -m - 2, so Corollary 3.9 gives an expansion in a series of
locally determined terms bky2^~m~2^n~l^~k\ k > 0, together with a series
of terms (cj^ log// + c^)//-771-1--7, j > 1? with Cj^ locally determined.
Taking the contributions together we get the expansion (5.32). One
observes moreover that the terms (c0log// + Co)//~m_1 in (5.32) come only
from Tr(<pd™[KiSoTo]), which leads to the last statement in the theorem.
For, K\ and So are the same as for the case where the Pj and Pj are 0. The
third factor To = 7oQ/x,+ uses the symbol of (V + fJ,)'1 evaluated at xn = 0.
The leading term of this is the same as for the case where Pj and Pj are
0, and the lower order terms contribute ultimately with special parameter-
dependent ^do's of order -m - 2 only; the first possible nonlocal and log
contributions from this are the terms with ji~m~2 and /x~m~2 log/x. □

162
In view of (5.4), it is now easy to draw conclusions from this on
asymptotic expansions for traces of A-derivatives of <p(Ai - A)"1 = <p(Pb*Pb -
A)"1 and v>ft(Ai - A)"1 = <pPb{Pb*Pb - A)"1, etc.
Corollary 5.11 Let <pki : Ei -> Ek be morphisms, for k,l — 1,2.
The traces Tr((pnd^(Ai-A)"1) and Tr^^^-A)"1) have
asymptotic expansions (for k = 1 resp. 2^):
-m—1
oo
+ ^(c^logA + 4fcfc)(-A)i2-"1-1; (5.35)
i=o
and Tr(^i2^Ps(Ai - A)-1) and Tr^i^iV (A2 - A)-1) Aawe asj/mp-
totic expansions (for {k,l} = {1,2} resp. {2,1}J:
a0
,«(-A)^-", + EK« + ^)(-^
J=l
+ J>ifW log A + 4w)(-A)^-m; (5.36)
j=o
with coefficients described as in Theorem 5.10.
The coefficients co,jt/ and c'Q kl are the same as those for the product case.
Proof: Using (5.4), take
(<pn 0W0 0 \ (0 <p12\ /0 fl\
*={o oj'lo J'(o oj'resP-U o)« (5>37)
in Theorem 5.10, and divide by \i in the first two cases. Now replace fi by
(-A)2 and note that d\ = (2/j)"1^. □
These results yield asymptotic expansions of the traces of heat operators
¥>ne"*Al, <pnPBe~tAl, etc., and power operators </>n(Ai)~s, ¥>i2fk(Ai)~s,
etc., by use of the transition formulas in Section 2:
Theorem 5.12 There are coefficients aj^u bj,kh Cj,kh ^j,ku re^ec^ by suit-
able gamma factors to those in Corollary 5.11 (cf Theorems 2.1 and 2.3,)
such that, with v\ — Tt^hIIoCPb)), v2 — Tr(^22no(ffe*))^ the zeta and

163
eta functions have singularity structures described by:
oo - . r
T{s) Tr(v)jfcjfcZ(Ai, s)) ~ —- + —^ + 2^ — ^f"
S S~2 j=i 5-V
r(s)Tr(¥.12i^Z(A1,*)) resp. T(s)TT(<p21PB*Z(A2,s)) (5'38)
«o,fc/ . v-^ aj,kl + bj,ki , v^ ( ciM ,
n±l +Z^ w-j+1 + 2^U, 2=1^2 + „ i izl./'
and i/ie /iea£ traces ftcwe the asymptotic behavior for t —>• 0:
oo
Tr(¥5jtite-<Al) ~ ao)fcfcr ? + ^(fij.jtfc + fy,**)* *
oo
Tr(^i2PBe-tAl) resp. Tr^P^e"^2) (5-39)
oo oo
77&e c'- ^.; and ^ are m general globally defined, while the other coefficients
are local The coefficients c$^i and cf0 kl are the same as those for the product
case.
A detailed account is given in [14]. [14] and [15] also give some
information on variations of parameter-dependent situations.
Remark 5.13 Similar considerations allow the calculation ofTr(Dd™Hp)
when D is an arbitrary differential operator on X, for m > n + d, d = the
order of D. One finds that
oo
oo
+ £(<:,■ (D, Pb) log/. + c'3(D, PB))p-«*d-l-i (5.40)
i=o
(the primed coefficients global, the others local); and consequences are
drawn as above for the corresponding zeta and eta functions and
exponential traces.

164
References
1. M. S. Agranovic: Some asymptotic formulas for elliptic pseudodifferential
operators. J. Functional Analysis and Appl., 21 (1987), 63-65.
2. M. F. Atiyah, V. K. Patodi and I. M. Singer: Spectral asymmetry and
Riemannian geometry, I. Math. Proc. Camb. Phil. Soc. 77 (1975), 43-69.
3. N. Bourbaki: " Fonctions d'une variable reelle." IV, Ed. Hermann, Paris 1951.
4. L. Boutet de Monvel: Boundary problems for pseudo-differential operators. Acta
Math. 126 (1971), 11-51.
5. T. Branson and P. B. Gilkey: Residues of the eta function for an operator of
Dirac type. J. Funct. Analysis 108 (1992), 47-87.
6. J. Bruning and M. Lesch: On the eta-invariant of certain non-local boundary
value problems. To appear.
7. J. S. Dowker, J. S. Apps, K. Kirsten and M. Bordag: Spectral invariants for the
Dirac equation on the d-ball with various boundary conditions. Class. Quantum
Grav., to appear.
8. J. J. Duistermaat and V. W. Guillemin: The spectrum of positive elliptic
operators and periodic bicharacteristics. Inventiones Math. 29 (1975), 39-79.
9. A. Friedmann: " Partial Differential Equations." Holt, Rinehart and Winston,
New York 1969.
10. P. B. Gilkey: On the index of geometrical operators for Riemannian manifolds
with boundary. Adv. in Math. 102 (1993), 129-183.
11. P. B. Gilkey and G. Gnibb: Logarithmic terms in asymptotic expansions of heat
operator traces. To appear.
12. G. Gnibb: " Functional Calculus of Pseudo-Differential Boundary Problems."
Progress in Math., Vol. 65, Birkhauser, Boston 1986. Second edition 1996.
13. G. Grubb: Heat operator trace expansions and index for general Atiyah-Patodi-
Singer boundary problems. Comm. P. D. E. 17 (1992), 2031-2077.
14. G. Gnibb and R. T. Seeley: Weakly parametric pseudodifferential operators
and Atiyah-Patodi-Singer boundary problems. Inventiones Math. 121 (1995),
481-529.
15. G. Gnibb and R. T. Seeley: Zeta and eta functions for Atiyah-Patodi-Singer
operators. Journal of Geometric Analysis, to appear. Copenhagen Univ. Math.
Dept. Prepr. Ser. 11, 1993.
16. E. Hille and R. Phillips: "Functional Analysis and Semi-Groups." Amer. Math.
Soc. Colloq. Publ. 31, Providence, Rhode Island 1957.
17. L. Hormander: Pseudo-differential operators and non-elliptic boundary
problems. Ann. of Math. 83 (1966), 129-209.
18. L, Hormander: Pseudo-differential operators and hypoelliptic equations. Proc.
Symp. Pure Math. 10 (1967), 138-183.
19. L. Hormander: "The Analysis of Linear Partial Differential Operators, III."
Springer Verlag, Heidelberg 1985.
20. T. Kato: "Perturbation Theory for Linear Operators." Springer Verlag, Berlin
1966.
21. R. T. Seeley: Singular integrals and boundary value problems. Amer. J. Math.
88 (1966), 781-809.
22. R. T. Seeley: Complex powers of an elliptic operator. Amer. Math. Soc. Proc.
Symp. Pure Math. 10 (1967), 288-307.
23. R. T. Seeley: Topics in pseudo-differential operators. CIME Conference on
Pseudo-Differential Operators 1968, Edizioni Cremonese, Roma 1969, pp. 169-
305.
24. M. A. Shubin: "Pseudodifferential Operators and Spectral Theory." Nauka,
Moscow 1978.

BOUNDARY VALUE PROBLEMS AND
EDGE PSEUDO-DIFFERENTIAL OPERATORS
B.-W. SCHULZE
Institut fur Mathematik
Universitdt Potsdam
Postfach 60 15 53
14415 Potsdam
Germany
Introduction
The analysis of pseudo-differential operators on a closed compact C°°
manifold (in its standard form) allows the construction of parametrices of
elliptic operators by inverting local symbols and forming the associated
operators. Elliptic regularity and the Fredholm property of elliptic operators in
Sobolev spaces are consequences of the basic calculus of pseudo-differential
operators. It is well-known how the interplay between symbolic and
operator level, together with homotopy and operator algebra aspects, are
involved in the index theory in K-theoretic terms, cf. Atiyah, Singer [2], in
the program to express the index by analytical formulas, cf. Fedosov [9]
or in other strategies for analyzing and interpreting the index, e.g., by the
heat kernel asymptotics. For interesting classes of singular or non-compact
manifolds, essential problems like adequate operator algebras with symbolic
structures, the definition of ellipticity, and index theory, are unsolved.
For pseudo-differential boundary value problems on manifolds with C°°
boundary, Boutet de Monvel [3] found appropriate operator algebras (for
symbols with the transmission property with respect to the boundary),
Rempel, Schulze [21] (for general symbols, not necessarily having the
transmission property), cf. also the work of Vishik, Eskin [46] and Eskin [8]. Let
us mention in this context also the monographs of Rempel, Schulze [21],
Grubb [13] and of the author [34]. The present exposition will show how
boundary value problems fit into a more general class of pseudo-differential
operators on manifolds with edges. As in the (classical) C°° situation there
are to be expected new interactions to other fields of mathematics, in par-
165
L Rodino (ed.), Microlocal Analysis and Spectral Theory, 165-226.
© 1997 Kluwer Academic Publishers.

166
ticular, geometry and topology. Motivations for the analysis on singular
manifolds come from (applications in) mathematical physics and
engineering. Here, for instance, piece-wise smooth configurations in concrete models
are not less classical than smooth ones; however a transparent analysis for
higher edge and corner orders (also for non-elliptic and non-linear
equations) is still an enormous challenge. It is primarily an analytic problem to
invent manageable formalisms in terms of symbolic structures and operator
and distribution spaces. The conical and edge singularities are crucial for
understanding the hierarchy of polyhedral singularities of growing orders.
We will explain essential ideas of this theory, but we will not give the
complete calculus. The notions and results of our approach may also be regarded
as axiomatic elements for operator algebras on spaces of higher singularity
orders, e.g., (warped) polyhedra or their lower-dimensional skeletons. The
simplest non-trivial singularity is the conical one. The (infinite) cone over
a base space X is the quotient space XA — (i2+ x X)/({0} X X), where
{0} xX corresponds to the vertex (the conical singularity). In the following,
X will be a closed, compact C°° manifold. For instance, if X is embedded
in the unit sphere SN of RN+1, then
XA £ {i G RN+1 : x = 0 or x/\x\ G X}.
On the "open stretched" cone i2+ x X, different splittings of coordinates
(£, x) and (£, x) are said to be equivalent if (£, x) -> (i(t, x), x(t, x)) extends
to a diffeomorphism RxX -» iJxX; then £(0, x) — 0 for all x. For dim X =
0 we have XA = i2+ with the conical singularity t = 0 of i2+. Singularities
of cusp type as they were studied in Schulze, Shatalov, Sternin [40] and
Schulze, Tarkhanov [42] under different aspects, will not be discussed here,
though there are many links between the theories of conical singularities
and cusps.
A topological Hausdorff space B is called a "manifold with conical
singularities" if there is a finite subset S C B such that B\S is a paracompact C°°
manifold, and every v G S has a neighborhood V which is homeomorphic to
the cone XA over some closed compact C°° manifold X = X(v), such that
V \ {v} is diffeomorphic to i2+ x X. We define the "stretched" manifold
B associated with B by attaching the sets [0,1) X X(v), v G 5, to B \ S.
Then B is a C°° manifold with compact C°° boundary dB = x^X^),
and we have B\dB^B\S.
For an open set Q, C iJ9, and a cone XA we define the wedge XA x ft.
We call a topological Hausdorff space W a manifold with edges Y,Y CW,
if W\Y and Y are paracompact C°° manifolds of dimensions 1+n+q and g,
respectively, such that W is locally (near each of the j/G7) homeomorphic
to a wedge XA(y) x £l(y) with n-dimensional cone bases X(y) and open
Q(y) C iJ9, which means that there are "local coordinates" (t,x,y) G

167
i2+ xlxfi outside the edge and local coordinates y G ft on the edge. Any
pair (£, #, y) G i2+ xlxfi, (£, z, y) G i2+ X X x Q, of coordinates has to
be compatible, i.e. the diffeomorphism
(*>&,y) -> (*(*,&,»),&(*,&,y)»y(*»&,y))
is the restriction of a diffeomorphism RxXxQ,-* RxXx&to R+xXxQ,,
where £(0,z,j/) = 0 for all z,y, and j/(0,z,j/) is independent of z. The
analysis requires the stretched wedge, which locally looks like i2+ X X x
Q. The stretched manifold W associated with W, is defined by glueing
together the sets [0,1) X X(y) X Q(y) and attaching this to W \ Y. It is a
C°° manifold with C°° boundary dW which is a bundle over Y with fiber
X, and we have W \ dW £ W \Y.
A (paracompact) C°° manifold with C°° boundary can always be
regarded as a manifold W with edge Y, setting dW = Y, dimX = 0, and
the model cone of the wedge near every y G Y is i2+, the inner normal
to the boundary (with respect to some Riemannian metric). For instance,
let G C Rn be a domain with C°° boundary, W — G. Consider an elliptic
differential operator A in Rn with smooth coefficients, say the Laplacian.
Clearly A contains no specific information on the geometry of dG. The
structure of elliptic boundary conditions for A is the consequence of a
certain behaviour of A in normal direction to the boundary. We define the
so-called boundary symbol of A, which it an operator family acting on i2+,
parameterized by the points of T*dG. The operator A is reformulated as
a pseudo-differential operator along dG, with the boundary symbol as its
(operator-valued) symbol. In this sense A is expressed in anisotropic terms
relative to dG. This reformulation does not change any properties of A far
from dG.
In general, on a manifold with edges, we translate the operators near
the edges into pseudo-differential operators with specific operator-valued
symbols. These symbols take values in a pseudo-differential algebra on the
model cone. Far from the edges the operators have to remain "isotropic"
pseudo-differential operators. The calculus of pseudo-differential boundary
value problems can be formulated as one with operator-valued symbols,
cf. Schulze [34]. This permits to read off the basic structures of a
calculus also for general edge singularities, cf. Egorov, Schulze [7] and the
monograph [37]. This point of view was systematically developed in a
sequence of papers, starting with [30] and then continued under various
aspects for operator algebras with continuous and variable branching asymp-
totics in [29], [35], [36], moreover, for corner singularities in [32] and in a
joint paper with Dorschfeldt [6] and for non-compact manifolds jointly with
Dorschfeldt, Grieme in [5] and in Seiler [43]. In particular, the algebra of
pseudo-differential boundary value problems with the transmission prop-

168
erty in the sense of Boutet de Monvel [3] found a new interpretation as an
edge pseudo-differential calculus. This was elaborated in this form in detail
in the joint papers with Schrohe [24], [25] as a tool to treat boundary value
problems for conical singularities. As mentioned at the beginning, many
questions on elliptic operators in the standard calculus on a C°°
manifold are also meaningful on a manifold with singularities. This concerns,
in particular, an extension of Fedosov's analytical index formula to elliptic
operators on manifolds with edges, cf. Fedosov, Schulze, Tarkhanov [11],
Schrohe, Seiler [23], or the analysis of asymptotics of solutions, cf. Schulze,
Shatalov, Sternin [38], [39].
Let us finally note that a pseudo-differential calculus for manifolds with
higher edge and corner singularities requires parameter-dependent variants
of the already achieved operator algebras on a given manifold with
singularities. The parameter is interpreted as a additional covariable to used either
for the Mellin transform along a new corner axis or the Fourier transform
along a new edge. This iterative procedure should be based on an axiomat-
icdescription of the "higher" operator algebras. The elements of the present
exposition are chosen to be an ingredient of a future pseudo-differential
calculus on manifolds with higher singularities.
The details can be of enormous complexity unless the most efficient
strategies are discovered. This program, of course, may also be a challenge
for young mathematicians who want to be active in this field. Also in the
context of parabolic and hyperbolic operators much work is to be done.
The author thanks M. Gerisch (Max-Planck-Arbeitsgruppe "Partielle
Differentialgleichungen und Komplexe Analysis", University of Potsdam)
for valuable remarks to the manuscript.
1. Edge Sobolev spaces and operator-valued symbols
1.1. NOTATIONS AND CLASSICAL BACKGROUND
This section recalls some elementary material on Sobolev spaces and pseudo
differential operators. For more details we refer to standard monographs
such as Hormander [14], Treves [45], Kumano-go [16]. We will employ
Frechet topologies in symbol and operator spaces. The various statements
may be regarded as exercises in pseudo-differential calculus.
If ft C Rn is an open set then C°°(ft) is the space of all infinitely dif-
ferentiable functions in ft, Co°(ft) the subspace of all elements with
compact support. X>'(ft) = (Co°(ft))/ is the space of all distributions in ft,
£'(ft) = (C°°(ft))' the subspace of all distributions with compact support.
If U C Cn is an open set then A(U) is the space of all holomorphic functions
in U. We will employ the standard locally convex topologies in the spaces.
Analogous notations make sense for functions (or distributions) with values

169
in a (say Frechet) space E, namely C°°(ft, £), C°° (ft, £),..., A'(U, E). All
occurring Frechet spaces here can be written as projective limits of Banach
spaces
{Ej}jeN with continuous embeddings Ej+1 *->• Ej for all j G N. (1.1)
Here N = {0,1,2,...}. Hilbert spaces in this exposition are assumed to
be separable. Given locally convex vector spaces J5, Z?, the space of
linear continuous operators E -* E will be denoted by £(E,E) or C(E)
for E = E. If E,E are Banach spaces then £(E,E) will be considered
in the operator norm topology. S(Rn) will denote the Schwartz space in
Rn 9 x = (a?i,...,a:n), defined as the subspace of all u e C°°(i2n) for
which the semi-norms
u -» sup |xajDfw(x)|
a7GHn
are finite for all a,/? e iVn. This defines a Frechet topology in S(Rn).
If L2(Rn) is the space of all square integrable functions in iJn, i.e. the
measurable functions u on Rn with ||w||L2(Hn) — {$\u{x)\2dx}ll2 < oo,
then
for all a,/? € Afn, is an equivalent semi-norm system on S(Rn). The dual
Sf(Rn) is the space of temperate distributions in Rn. The Fourier transform
u(Q = (Fu)(t) = Je-i*tu(x)dx,
with £ = (£i,...,£n)j s£ = Si^i^t^tj induces an isomorphism T :
<S(i2n) -> <S(i2n); its inverse is given by the formula
(F-1g)(x) = Jj*g(Qdt,
where d£ = (27r)~nd£. The Fourier transform extends to an isomorphism
T : S'[Rn) -> <S'(iJn). We shall also write ^ = Tx^ and J*"1 = j*^.
The Sobolev space Hs(Rn) of smoothness s € Ris defined as the closure
of S(Rn) with respect to the norm
imh-w = {J (o2s\m\2dty. (i.2)
Here (£) = (1 + |£|2)5- The space Hs(Rn) can also be characterized as the
subspace of all u € S'{Rn) for which <0"«(0 € L2(fl£). Instead of <£) we

170
may equivalently use the function [£], defined as any element in C°°(Rn)
for which [£] > 0 and [£] = |£| for all |£| > c for a constant c> 0. Then
ci(0 < [£] < c2(0 for suitable cuc2 for all £ G iJn.
Note that [A£] = A[£] for all A > 1, |£| > c for a constant c > 0. For an
open Q C Rn we set
tffoc(ft) = {u G £>'(ft) : <pueHs(Rn) for all y> e C0°°(ft)}.
The symbols of pseudo-differential operators are defined as follows: For
\i e R and an open set U C iJm, S^t/ X iJn) is the space of all a(x,£) e
C°°(t/ x i?n) such that
sup(0-^|/?l|^^a(x,OI (L3)
teRn
is finite for all a G iVm, (3 e Nn, and arbitrary K CC U. The system
of semi-norms (1.3) defines a Frechet topology on the space S^ft X Rn).
Denote by Sfi(Rn) the subspace of ^-independent elements (symbols with
"constant coefficients"). This is a closed subspace of S^t/ X i2n), and we
have
S"(U x Rn) = C^iU.S^R71)).
Let S^(U X Rn) for fi e R be the subspace of all a{fi)(x,£) G C°°(U x
(i?n \ {0})) satisfying a^x, A£) = A/ia(/i)(x,^) for all A > 0 and x6[/,
£ G i*n \ {0}. For every excision function X(0 in Rn (i.e., x(0 e C°°(iZn),
x(£) = 0 for |£| < Co, x(0 = 1 f°r If I > ci f°r certain 0 < c0 < C\ < oo),
we have x(0^(/i)(^ X #n) C S"(tf X iJn).
The subspace of classical symbols S%(U X iJn) C S»(U X iJn) is defined
by the following condition: To a(x,£) there exists a sequence
a^-fiM € S^U X (i?n \{0})), j € N,
such that for any excision function x(Q
«(*,0 - xK)£vi)(^ e S^+V X Rn) (1.4)
i=o
for all N € N. The functions a(M_j)(:r,£), the homogeneous components
of a(x,£) of order fi - j, are uniquely determined by a(x,£). In particular,
a(ii)(xiQ ls called the homogeneous principal part of a(x,£) of order //.
By requireing continuity of the homogeneous component maps of all
orders and of the remainders we get a (nuclear) Frechet topology in the

171
space Sj{U x Rn) that is stronger than the one induced by 5M(C7X Rn). The
subspace S^(Rn) of classical symbols with constant coefficients is closed in
S»{U x Rn), and we have S${U x Rn) = C°°(t/,^(iln)).
Note that
5-°°(t/xiJn) = n^S^t/ x Rn)
= niGiV57(£7xBn)
= C°°(t/, £(#")).
In particular, S-00^") = S(Rn). Another obvious relation is S"(fl£) C
£'(fl£) for every \i 6 R, which implies T^c(S»{Rn)) C £'(££).
Theorem 1.1 Let x(C) &e an arbitrary excision function and ip(Q —
1 - x(0- Tften o(0 € ^(iJn) tmp/tes
x(0(^-»(0€5(fi?)f
M0 := (^C^(^"»)(0 € ^(«n)-
Tftere is a function h(£ + irj) € *4(Cn) st/c/& iftctf ^(0 := ^(^ + ^) satisfies
MOIt7=o = MO a«rf
MOeS^JR?) /or every r? e iJn,
w/fcere {/i^ : 77 e A } w bounded in Sfi(Rn) for every compact subset K C
Rn. Analogous relations hold for classical symbols.
The map S»(Rn) -> S"(JRn),a(£) -> /*0(0, which produces a symbol that
extends to a holomorphic function in £ + irf G Cn with the mentioned
property will also be called kernel cut-off, with the cut-off function VK0-
"We set
h(t + ir,) = (H(i,)a)(Z + iV),
where H(ip) is the continuous operator
H($): S»{Rn)^A{Cn), a(0->M^+ *»/)•
Since if (^) acts only on the £-variables, it can be extended to a continuous
operator
Hty) : S"(U x Rn) -> C°°(C/,^(Cn)),
and similarly for classical symbols.
Pseudo-differential operators based on the Fourier transform in Rn are
defined as
Op(a)u(x)= [ fe{i<x-xf^a(x,xf^)u(xf)dxfd^

172
for a(x, a?',f) £ ^(ft X ft X Rn), ft C Rn open, in the oscillatory integral
sense. If we first assume u £ Co°(ft) then
Op(a): (^(ft)-^00^)
is continuous. Denote by LM(fi) (L^(ft)) the space of all Op (a) for
arbitrary a(x, x',£) £ SM(fi x ft X iJn) (£ 55 (0 X ft X iT)). Then L-°°(ft) =
n^H-L^ft) is the space of all integral operators with kernels in C°°(ft X ft),
called the smoothing operators. In general, every Op (a) has a distributional
kernel
k(a)(x, x', x-x')= [e^x'x'K(x, x',Z)d£ £ £>'(ft x ft)
with singular support in diag (ft X ft) = {(x,x) : x £ ft}, cf. 1.1. A
closed subset K C ft X ft if called proper if ?rt"1M fl K is compact for
every M CC ft, with 7T{ : ft x ft -» ft being the projection to the i th
component, i = 1,2. Denote by L^{Q)k the subspace of all A £ LM(ft)
for which the distributional kernel is supported in a proper set K with
diag (ft x ft) C int K. Then, from the above statement on the singular
support of the distributional kernel of A we obtain that
L^(Q) = L^{n)K + L-00(n) (1.5)
in the sense of vector spaces. By definition we have L~°°(ft) = C°°(ft X ft).
Moreover, A0 £ L^{Q)k induces continuous operators
A0 : C0°° (ft) -> C0°° (ft), C°° (ft) -> C°° (ft).
Using the fact that a0(z, £) := e_^Aoe^ for e^ = eta* belongs to 5^(ft x Rn)
with Op(ao) = Aq and that the map Aq -> ao is an isomorphism
L"(n)* -> S"(Q X iJn)K := {ao(x,0 : A0 £ L"(n)*},
where 5^(ft x Ru)k is a closed subspace of 5^(ft x Rn), the space LM(fi)tf
can be equipped with a natural Frechet topology. So LM(ft) is a Frechet
space since it is a non-direct sum of Frechet spaces. In an analogous manner
we get a Frechet topology in L^(fi).
Let us recall the general definition of a non-direct sum of two Frechet
spaces E$, E\ contained as vector subspaces in a certain topological Haus-
dorff space. The non-direct sum
E0 + Ei = {e0 + ei : e0 £ £0, «i € #1}
is isomorphic to E0 © £q/A with A = {(ei - e) : e e E0f) Ei}. Both
£"0 © £q and A are Frechet spaces in a natural way. So Eq + E\ is Frechet

173
with the quotient topology. The construction can easily be generalized to
finitely many summands.
Another useful notation is the following. If E is a Frechet space which
is a left module over an algebra A, we set
[a]E = closure of {ae : e e E} in £",
for a e A. In an analogous sense we use the notation E[b] when E is a right
module over an algebra £, for b e 5, or notation like [a]J5[6]. In particular,
we will use spaces fo>]LM(fi)M or [</>]L^(ft)[^] for (p,ie Cg°(fi).
Next we remind of the invariance of pseudo-differential operators under
diffeomorphisms x : ^ -* ^ for open sets ft, ft C Rn. Denoting by x* •
Cg°(fi) -> Q°(n), C°°(ft) -> C°°(ft) the function pull-backs, to every
A e LM(fi) we can form the operator push-forward
X.A = (xTMx* : Cnft)->C°°(ft).
Then x* induces isomorphisms
X, : Z/*(Q) 4 I"(n), L$(Q) 4 Z£(ft)
This gives rise to pseudo-differential operators on C°° manifolds. For
instance, let X be a closed compact C°° manifold and k : [/ -> ft a chart on
X. Then the invariance allows us to define the spaces L^{U) — (K~1)J|cL/i(fi)
and Lj(U) = (K_1)*L^(n). Now let {[/1?..., C//v} be an open covering of
X by coordinate neighborhoods, {^i,...,^v} a subordinate partition of
unity, and {^i,.. .,^v} another system of functions ^ G Co°(t/j) with
y>j^j = y>j for all j. Then we can form
iMw = E[w]iM(^)[^]+i"~w
i=i
as a non-direct sum of Frechet spaces, where L~°°(X) is identified with
C°°(X x X) via some Riemannian metric on X. Analogously we obtain
Lj(X). This construction can easily be generalized to any paracompact
C°° manifold X. The invariance of Sobolev space distributions under
diffeomorphisms allows us the corresponding global definitions. If X is a
paracompact C°° manifold we have an evident definition of the space H*omp(X)
of compactly supported distributions of Sobolev smoothness s € R and
the space Hf0C(X) of distributions that are locally of Sobolev smoothness
s € R. The latter space is Frechet in a natural way while H*omp(X) is an
inductive limit of Hilbert spaces. If X is compact then
HS(X) := Hscomp(X) = Hf0C(X).

174
Theorem 1.2 Every Ae L»(X), A: Cg°(X) -> C°°{X), extends to
a continuous operator
A •' Hscomp(X) -» Hi0C(X)
for every s G iJ.
Remark 1.3 (ty 77&e operator M^ of multiplication by a function <p G
S(Rn) induces a continuous operator M^ : Hs(Rn) -> Hs(Rn), and
<p -> Al^, 25 continuous as operator S(Rn) -> £(Hs(Rn)) for every s € R.
(ii) The pseudo-differential operator Op(a) for a symbol a(£) G S^(Rn)
induces a continuous operator
Op{a): Hs{Rn)^Hs-^{Rn),
and Op : ^(iJ71) -> C{Hs(Rn),Hs-^(Rn)) is continuous for every s e R.
1.2. ABSTRACT WEDGE SOBOLEV SPACES
The abstract wedge Sobolev spaces were introduced in [30] for studying
pseudo-differential operators on a manifold with edges. The definition can
be motivated by an anisotropic reformulation of Hs(Rn*q) with respect
to the fictitious edge Rq 9 y in Rn x Rq 9 (z,y). Consider a group of
isomorphisms k\ : Hs(Rn) -> Hs(Rn), A £ JK+, continuous in A with
respect to the strong operator topology, given by
(kxu)(x) = X2u(Xx), XeR+, ueHs{Rn),
se R. Set
k(ti) := k^j for 7/ G iJ9
with the function 77 -> [7/] defined in the previous section. Then we have
the following elementary result:
Proposition 1.4 The space ifs(iJn+9), s e R, is the closure of
S(Rn+q) with respect to the norm
where v G S(Rn+q) is interpreted as an element v(y) G S(R^S(Rn))f
Fy-tn is the Fourier transform in Rq, applied to vector-valued functions,
and n(r)) acts on the values of v in S(R%) for every 77 G Rq.

175
Definition 1.5 Let E be a Hilbert space, and {K\}\eR+ be a strongly
continuous group of isomorphisms on E, i.e.,
(i) k\ : E -> E is an isomorphism, A e i?+,
(ii) A -» K\e G C(R+, E) for every e e E,
(Hi) k\kp = k\p for all A, p e R+, fti = id.
77*en Ws(i?9,.E), s e R, is the closure ofS(Rq,E) with respect to the
norm
^(77) = k^j. Tftts space is called an abstract wedge Sobolev space of smooth-
ness s G R with respect to {k\}\£R+.
Instead of (ii) we also write {^AJAGitf. £ C(R+,Ca(£")), c indicating
the strong operator topology.
From the properties (i), (ii), (iii) it follows that there are constants
c> 0, M > 0 such that || k{t]) \\C(e) < c[rj\M for all 77 G Rq-
Remark 1.6 (i) Equivalent norms in E give rise to equivalent norms
inWs{Rq,E).
(ii) Replacing [77] by (77) in the norm expression yields an equivalent
norm inWs{Rq,E).
(iii) The choice o/{ka}agH+ is essential for the space Ws(Rq,E); it is
fixed in concrete cases and therefore suppressed in the notation.
(iv) Definition 1.5 also makes sense for a Banach space E. Many results
on abstract wedge Sobolev spaces remain true in this case.
Example 1.7 (i) For E = Hs(Rn), {k\u)(x) = A?u(Xx), A e #+, we
have {ka}agH+ € C{R+,Ca{Hs{Rn)) and
Ws{Rq,Hs{Rn)) = Hs{Rq x Rn)
for every s G iJ.
(ii) For E = HS{R+) (= {u\R+ : u e HS{R)}), (nxu)(t) = \±u(\t),
A e R+, we have {ka}agH+ e C(^+?^(^s(^+)) anc^
W(fi«, HS(R+)) = HS(R« x JR+) = {v\RqxR+ : u € HS(R« x J*)}
/or every s £ R.
(iii) Let E be an arbitrary Hilbert space, k\ — idE for all \ € R. Then
Definition 1.5 gives us Hs(Rq,E) which is the Sobolev space of E-valued
distributions of smoothness s in the standard sense, i.e., with the norm

176
Setting T = T ln 1(r))J: for an arbitrary fixed {K\}\eR+ G C(R+, Ca{E))
we obtain for the associated space Ws(Rq, E) an isometric isomorphism
T: Ws{Rq,E)^Hs{Rq,E)
for every s € R. In particular, H°(Rq,E) = L2(Rq,E) (= the space
of square integrable E-valued functions in Rq); this is a consequence of
PlanchereVs theorem in the Hilbert space-valued case. Moreover,
W°°{Rq,E) = nseRWs(Rq,E)
is independent of the particular choice of {k,\}\£r+, i.e., yV°°(Rq,E) =
H°°(Rq,E).
(iv) For E = CN we always set k\ — ids for all X € #+. Then
Ws{Rq, CN) = Hs{Rq, CN) = Hs{Rq) ® C*.
(v) Ws(Rq, E) can be endowed with a Hilbert space scalar product that
generates the norm. The action
(xxf)(y) = ^KXf(\y) for f(y)eS(R?,E),
X e R+, extends by continuity to a group
{xxhtR+ €C(R+,£AWs(Rq,E)))
with the properties of Definition 1.5, now with respect to WS(R9, E). Then
WS{RP, Ws{Rq, E)) = Ws(Rp+q, E).
Analogously to the scalar theory we have the following characterization
of Ws{Rq, E) as a subspace of S'{Rq, E) = £{S(Rq), E):
Proposition 1.8 For every fixed s € R, \Vs(Rq, E) equals the subspace
of all u € S'(Rq, E) for which (^"^(J^u)^) € L2{Rq, E).
For an open set il C Rq we define
W°omp(n, E) = {u£ W°{Rq, E): supp uctt compact},
WfJSl, E) = {ue V(Q, E) : <pu € Wscomp{Sl, E) for every <p € C0°°(ft)},
s e R; V{Q,E) = £(q?(Q),E). The space Wscomp{Q,,E) is an
inductive limit of Hilbert spaces and Wf0C($l,E) is a Frechet space. Let E =
proj Ymij£NEi be the projective limit of Hilbert spaces {E^}j&n with
continuous embeddings E*+1 w- E^ for all j € N and an action
{KX}XeR+eC(R+,CA^))

177
with the properties in Definition 1.5 that restricts to
{/CA}A€H+€C(JR+,£ff(^'))
with the analogous properties for all j. We then obtain natural embeddings
Ws(Rq, EJ+1) ^ Ws{Rq, Ei) for all j, and we set
Ws{Rq, E) = ind l\mjeRWs{Rq, Ej).
Analogously to the scalar theory we have invariance of the wedge Sobolev
spaces under diffeomorphisms x : ft -> ft for open ft, ft C Rq:
Theorem 1.9 Let X'- ft —>- ft &e a diffeomorphism. Then the pull-back
X* : X>'(ft, E) —> £>'(ft, E) restricts to isomorphisms
X* : Kompfr E) -> Wscomp(n, E), W?0M E) -> Wfoc(fi, E)
for all s G iJ.
For a proof, cf. [34], [37] or [6].
This permits us to define the spaces
Wscomp(Y,E), W?0C(Y,E) (1.6)
on any paracompact C°° manifold Y analogously to the case of C-valued
Sobolev spaces H*omp(Y) and Hf0C(Y), respectively.
1.3. PSEUDO-DIFFERENTIAL OPERATORS WITH OPERATOR-VALUED
SYMBOLS
Let E and E be Hilbert spaces with strongly continuous groups of
isomorphisms {ka}agJI+ an(i {«a}a€.R+) respectively, cf. Definition 1.5. For \i € R
and open U C Rp, we denote S^(U x (R? \ {0}); E, E) as the subspace of
all a(M)(j/, V) € C°°(U x (Rq \ {0}), C{E, E)) satisfying
«(M)(2/> Xrl) = AM«Aa(M)(j/, J/J/cJ1 for all A e R+, y£U,neRq\ {0}
Definition 1.10 Let U C RP be open and fi € R. Then S^iU X
Rq; E, E) is the space of all a(y, n) € C°°(U x Rq,£(E, E)) such that the
semi-norms
sup (^-^Ill^^J^-^y, ri)}K{ri)\\c(BjS) (L7)
y£K
v£Rq
are finite for all a€Np, fie Nq, K CC U. Moreover, S%{U X Rq, E, E)
denotes the subspace of all
a{y,T))eS,M(UxRq;E,E)

178
for which there exists a sequence 0(/i-j)(y, v) £ S^~^(Ux (Rq\{0}); E, E),
j G N, such that for any excision function x{v) i>n Rq
N
a(y, V) ' X(V) £ Hn-i)(». 1) € 5^^+x)([/ X iJ*; E, E) (1.8)
i=o
/or all N e N. The elements of S^{U X R?]E,E) are called operator-
valued symbols, those in S^(U xRq\E, E) classical operator-valued symbols
of order \i.
The homogeneous components a>(n-j){y,r)), j € N, of an a(y,77) e
££)([/ xRq',E, E) are uniquely determined. Similarly to the scalar case (see
Section 1.1) the spaces S»{U X Rq;E,E) and S%(U X Rq\E,E) may be
endowed with natural Frechet topologies. The subspaces of y-independent
elements Sfi(Rq; £?, E) and S%(R?] E, E) are closed in the topology induced
by S^U xRq;E, E) and S%(U xRq;E,E), respectively. We have
S"(U x Rq; E, E) = C°°(U, S»{Rq; E, E)) (1.9)
and analogously for classical symbols.
Example 1.11 Let A = Yl\a+p\<naap(xiy)D%Dy be a differential
operator in Rn x Q, 9 (x,y) for an open set Q, C Rq, with aap(x,y) e
C°°(Rn X Q,). Assume that aap(x,y) is independent of x for \x\ > const
Set E = Hs(Rn), E = Hs~^(Rn), both endowed with the group actions
k\ : u(x) -> A2"w(Az), A e i2+. Then the operator family
a(y, rj) := £ aap(x, y)Dax^ € C°°(Qy X jR», £(#*(«"), ff-"(lT)))
|a+/3|<M
w an e/emen* o/5"(Q x JP; #*(«"), ff*-"(JT)) for all s £ R. If the
coefficients aap are independent of x then
a(y, rj) € S$(Q X Rq, Hs{Rn), ff»-"(JF)) for all s € R,
and the homogeneous component of order // — j is
\a+/3\=n-j
Remark 1.12 In the applications the spaces E and E run over scales
{Es}seR and {E^t^R, respectively, and then it is natural to consider
symbols of the classes
S^(UxR'';Es,Es-fi) for all seR

179
(or s in some subset of R). The above example shows that instead of (1.6)
we may expect the more precise property
D°D%Stl(U x Rq;Es,Es^) C S"-I0l(tf x Rq;Es,Es-»+W)
for all multi-indices a G Np, (3 G Nq, and all s. This will be the case in
the concrete symbol classes for manifolds with edges.
Example 1.13 The operator M^ of multiplication by a function
p(x,y)eC™({ly,S(Rnx)),
which ft C Rq is open, is a symbol in S°(ft X Rq;Hs(Rn),Hs(Rn)), and
the map
C°°{Qy,S{Rnx)) -> S°(ft x Rq;Hs{Rn),Hs{Rn)) : ^M,
is continuous for all s G iJ. Note that M^ is independent of rj.
Example 1.14 Let us set r'u :— u(0) for u G S(R). It is easy Yo see
that r': S(R) -> C extends to a continuous operator rf: HS(R) -> C for
all s > \. Then
r' eS2{Rq;Hs{R),C) for every s>^.
Also here there is no dependence on r] G Rq. Moreover,the map k{rj) : C ->
S(R) defined by k{r))c — ift(t[r)])c for any ift G S(R) is a symbol
k(r])eS-2{Rq;C,Hs{R)) for every s G R.
Ifil>{t) G Co°(JK) is identically 1 in a neighborhood oft = 0 we have rfk(r)) =
idc-
We will obtain below many other non-trivial examples of operator-
valued symbols.
Remark 1.15 The kernel cut-off construction of Theorem 1.1 has an
obvious analogue in the operator-valued case.
To every for a(y, yf, rj) G S"(« X ft X Rq, E, E) for ft C Rq and \i G R
we can form the associated pseudo-differential operator
Op(a)u(y) = j j e*y-y>a{y, y', r,)u(y')dy'ar,,
interpreted as an operator-valued analogue of an oscillatory integral. Then

180
is continuous. This gives rise to the spaces of pseudo-differential operators
with operator-valued symbols
L"(fl; E, E) = {Op(a) : a(y, y\ n) € 5"(ft x Q x R?\ E, E)}
and analogously to i^(fi; E, E). The space
L-00(n;£7,£) = nM€jiI''(Q;f?)fi)
coincides with the space of all integral operators with kernels in C°° (ft X
Q,C(E,E)).
Let x • ^ —> ^ be a diffeomorphism for open sets ft, £2 C Rq. Then the
function pull-backs
X*: CS°(n,£?)->CS°(«,£?), C~(ft,£)^C°°(ft,£)
give rise to the operator push-forward
X.A=(xT1AX*: ^(M)"^00^)-
Analogously to the scalar calculus we obtain an isomorphism
for every ji G JK, which restricts to an isomorphism between the
corresponding spaces of classical pseudo-differential operators.
On a paracompact C°° manifold Y we an define (as in Section 1.1) the
global spaces of pseudo-differential operators
L»{Y;E,E) and Z£(Y;E,£), (1.10)
respectively. The homogeneous principal symbol of order \i of an operator
A e L^(y;£?,£?) is invariantly defined as a function on the cotangent
bundle minus the zero section T*Y \ 0 with values in C(E,E). It will be
denoted by a%(A)(y, 77), where homogeneity of order \i is defines as
o»M)(yM = ^\<{A){y,n)Hl foraii \€R+.
Theorem 1.16 Every A € L»(Y;E,E), A : Cg>(Y,E) -> C°°(Y,E),
extends to a continuous operator
A: Wcomp{Y,E)^Wl;S{Y,E)
for every s G iJ.

181
The proof can be reduced to a corresponding result in local coordinates.
Let Q C Rq open. Writing S^(Qx Rq;E,E) as a projective tensor product
of C°°(ft) and S»(Rq;E,E) see (1.8), the assertion follows from the
continuity of the operator Mm of multiplication by <p G C°°(fi), cf. Example
1.13, and of Op(a) when a(rj) has constant coefficients. The tensor product
argument is based on the following theorem : Let F, F be Frechet spaces.
Every g G E®^F can be written as a convergent sum
oo
g = Y1 ^3ei ® fi ^or suitable Xj G C, ej G F, fj G F (1-H)
i=o
with £) |Aj| < °° an(^ ei an(^ /j tending to zero in E and F, respectively,
as j —> oo.
Remark 1.17 (^ The operator Mm of multiplication by a function (p G
S{Rq) induces a continuous operator
Mm: Ws{Rq,E)^Ws{Rq)E),
and the operator S(Rq) -> C(Ws(Rq,E)) : <p -> Al^ is continuous for
every s G iJ.
(it,) For a symbol a(rj) G Sfi(Rq] E,E), the pseudo-differential operator
Op{a) : W(JR*, F) -> W^JF, F),
25 continuous and
Op : 5/i(iJ(?;F,F)-^£(>Vs(iJ<?,F),>Vs-/i(iJ<?,F)),
a -> Op (a), is continuous for every s € R.
The symbol classes of Definition 1.10 are necessary also in the version
of Frechet spaces F, F that are projective limits of Hilbert spaces {E^}jeN
with continuous embeddings E*+1 <-* FJ, {Fj}jGjv with continuous embed-
dings FJ+1 «->- FJ for all j, and where the corresponding groups {^a}agH+
and {^AJAeitf.? first given on F° and F°, respectively, have restrictions to
{*\}\eR+ € C(i2+,£a(F'*)), {kaIagr, € C(«+, £,(#))
with the mentioned properties, for all j. An operator a : F -> F is
continuous if for every k £ N there exists some j (fc) G A/" such that a : _EJW -> F*
is continuous (Here a is restricted to FJW). If j : JV -» iV is a given
function we obtain the space
proj KmkeNS't(U x il9; £'<*>, £fc) (1.12)

182
and then S^t/ X Rq\E,E) is defined as the union over all (1.12) where
j runs over all j : N -» N. In an analogous manner we can proceed for
classical symbols. This yields the corresponding classes of classical pseudo-
differential operators (1.9) for Frechet spaces E, E. The simpler case when
E is a Hilbert space and E a Frechet space of the mentioned kind, is of
particular interest. Then the symbol and operator spaces are Frechet in a
natural way. The elements of the pseudo-differential calculus have
straightforward generalizations to the case of Frechet spaces J5, E (up to minor
modifications, concerning restrictions to fixed j : N -> N).
1.4. EXAMPLES: GREEN, TRACE AND POTENTIAL OPERATORS IN
BOUNDARY VALUE PROBLEMS WITH THE TRANSMISSION PROPERTY
The pseudo-differential operators with the transmission property in the
sense of the algebra [3] on a manifold with boundary contain an ideal
of operators that is responsible for the structure of Green's function and
boundary (trace) and potential conditions in elliptic boundary value
problems. We shall give a description here in terms of operator-valued symbols
in local coordinates (£, y) £ R+ x ft for an open set Q C Rq.
Set S(R+ X iJ+) = S{R X JR)|j^x;g+, S(R+) = <S(JR)|j^; these are
(nuclear) Frechet spaces. Write S(R+) as projective limit
S(R+) = pwj\imkeN(t)-kHk(R+).
On the spaces Ek = (t)~kHk(R+) we define the group of isomorphisms
{^a}agh+? acting as (K\u)(t) = \2u(\t), A £ i2+. Recall that a
continuous operator a : L2(R+) -» L2(R+) is an integral operator with kernel
9a{t, ?) £ S(R+ x iJ+) if and only if
a : L2{R+) -> <S(iJ+), a* : L2{R+) -> S{R+)
are continuous. Here a* is the L2(i2+)-adjoint.
Theorem 1.18 Let
g(y, 7?; t, t') £ C°°(^,S(R+,t X I^ft,, SS(fi*)) (1.13)
be an arbitrary function and \i £ R. Then
a(y,r,)u(t) := [#+1 / ff(y,7?;^],^])^(0^
25 a (y,rj)-dependent family of Hilbert-Schmidt operators in L2(R+), and
we have
o(y,i7) € 5^(fixiJ«;L2(iJ+),<S(fi+)),
a*(y,i7) € 5^(0 x JPjL^J^)^^)),
w/iere * denotes the point-wise adjoint in L2(R+).

183
Remark 1.19 The symbols a(y,r)) of Theorem 1.18 induce by
restriction or extension symbols
aM € S$(Q X R";HS(R+),S(R+))
with
a*{y,rj) € S&to x Ri;H°(R+),S(R+)
for all s £ R, s > -\.
More generally, for every d £ N we can form
a{y^) = 22aj(y^)Q^ (L14)
j=o
for symbols aj of order \i - j in the sense of Theorem 1.18. Then
a{y,r)) e S»{nx Rq]Hs{R+),S(R+)) for every s>d-^.
An operator Op(a) + c with an operator-valued symbol, a(y,r)) of the
form (1.13) and C = T!j=ociW> where Cj is an integral operator with
kernel in C°°(R+ x ft x i2+ x ft), is called a Green operator of order \i and
type d in the algebra of boundary value problems with the transmission
property. For ji £ N this coincides with the definition in [3], though the
equivalence is not completely obvious; it was obtained in [33].
In pseudo-differential boundary value problems it is interesting to
generate the trace and potential operators by symbols in an analogous manner.
To this end we pass to symbols of block matrix form
. N HS(R+) S(R+) ,
fl(y,«/):=( "" T )(V,»7): © -> © , s>d--
V a21 «22 / CN- CN+ *
(1.15)
for certain AL, N+ £ AT. The left upper corner fln(y, rj) is assumed as in
(1.13). For describing the structure of the remaining entries we consider for
simplicity AL = 7V+ = 1. Then ^22(2/, v) ls an element of S^(ft X Rq).
Theorem 1.20 Let
9i2(y,mt) e c™(ny,S(R+,t,s°cl(Rl)»
be an arbitrary function and n € R- Then
oi2(y,»/)c:= M"+Wy.'7;*fo])c (1.16)

184
for c € C is a symbol in S^(Q x R9;C,S(R+)). Moreover, let
92i(y,mt') € c~(^,s(i!+,t,,s°(^))
be an arbitrary function and
i f°°
«2i(j/,7?)«=[7?r+5 / g2i(y,r,;t'[ri])u(t')dt' (1.17)
JO
for u € HS(R+), s > -§. Then
a21(y,n)eS^nxW;Hs(R+),C)
for every s > -\.
More generally, if a2ij(y, v) ls °f the form (1.16) with fj, - j instead of
//, we have from the second part of Theorem 1.20
a21(y,rl) = ^a2hj(y,rl)^eS^xW]Hs(W),C) (1.18)
3=0
for every s > d - \.
If we assume an, ai2, c^i? a22 in the mentioned form where, in
particular, a2i is given by (1.17), then we obtain for (1.14) (in the case 7V+ =
AL = 1)
a(y^) e S%(Q x Rq;Hs{R+) © C,S(R+) © C), s > d - ±.
In particular, for d = 0, it follows that
fl'M € 55(« x Rq]Hs{R+) ®C,S(R+) ©C), 5 > -|.
Here a* is defined point-wise by
(w,a*u)L2(H+)eC - (au,v)v{R^)BC
for all u,ueC£°(iJ+)©C.
For arbitrary 7V+, AL the relations are analogous.
Corollary 1.21 Let a(y,r/) be given by (1.14)' Then
Op(a) : ©-*•$,
H°comv{si,cN-) Hf-»(n,cN+)
is continuous for every s > d - \. The subscript comp(y), loc(y) indicate
comp, loc with respect to y G Q.

185
The operators in Corollary 1.21 are, modulo smoothing operators, the
Green, trace and potential operators in boundary value problems, including
the right lower corners which are classical pseudo-differential operators on
Q. We omit here an explicit definition of the smoothing trace and potential
operators. More details, including a new complete description of the algebra
of boundary value problems with the transmission property in terms of
operator-valued symbols, will be given in [37]. The definitions and results
of this section play the role of examples to the general set-up of Sections
1.2,1.3. The proofs of the theorems are based on tensor product arguments
and the observation that, for instance, when we assume instead of (1.12)
g(y;t,t') € C°°(ny,S(R+,t xR+,t,)), (1.19)
the operator-valued symbol
roo
a(y,r])u(t) = [r]r+1 g(y;t[r)],t'[r}])u(t')dt' (1.20)
Jo
satisfies
a(y, A77) = \^K\a{y^ t])k^1 for all A > 1, |i/| > c
for a constant c > 0. Moreover, the correspondence g(y,t,tf) -> a(y,r/) is
continuous in the sense
C°°(n,S(R+ xR+)) ^SZl(Ri;L2(R+),S(R+)),
and the same for the adjoints. Similarly we can argue for the other entries
of the block matrix symbols.
2. Parameter-dependent pseudo-differential operators and cone
theory
2.1. PARAMETER-DEPENDENT PSEUDO-DIFFERENTIAL CALCULUS
Pseudo-differential operators on a manifold with conical singularities
require the parameter-dependent calculus on the base X of the cone. The
parameter-dependent families will be used below as operator-valued
symbols for pseudo-differential operators with respect to the Mellin transform
along jR+, the cone axis. Let us stress that symbols of this kind are not
operator-valued symbols in the sense of Section 1.2. The calculus here, also
called the order reduction approach, contains symbols acting on a space
globally without any reference to some group {k;a}a€H+-
In this section the parameter space will be A = Rl 9 A. The dependence
of symbols on the parameters will be assumed in a way (which is sufficient

186
for our purposes) that A is formally involved as an additional covariable. In
other words we consider symbols
a(x,£, A) G S[d)(U x Off) for open U C iJm, ^ e JR.
By (c/) we indicate that the notions and results make sense both for classical
and non-classical symbols in (£, A). Note that
a(x,£, A0) G S?JU x jR£) for every fixed A0.
Thus, ifQC Rn is an open set, every a{x,x',£, A) G 5fd)(n x Q x JR£j')
gives rise to a space of A-dependent pseudo-differential operators
Ifd)(n;A) = {Op(a)(A) : a € 5^,(0 x Q x JRJ+*)}.
In this calculus the space of smoothing elements is
L-°°(fi;A) = 5(A,L-00(fi)).
The invariance under diffeomorphisms x : ft -* ft holds in the parameter-
dependent case as well, here in the sense that the point-wise operator push-
forward induces an isomorphism
X.: Lfa{tt;\)-*LfafcA).
Thus, if X is a (paracompact) C°° manifold and k : U —> ft a chart, we
can introduce L^(C/;A).
The space L~°°(X) is identified with C°°(X x X) via a given Rieman-
nian metric on X. So we can talk about
L-°°(X;A):= S(A,L'°°(X))
which is the space of parameter-dependent smoothing operators on X. If
{Uj}jeN is a locally finite open covering of X by coordinate neighborhoods,
{<Pj}jeN a subordinate partition of unity, further {iftj}jeN a system of
functions iftj G C°°(Uj) with pjiftj = y>j for all j, we denote by L?dJX]A)
the space of all operators
A(\)=J£<PjAj(\)i>j + C(\) (2.1)
for arbitrary A,-(A) € Lj^ (£/.,•; A) and C(X) € L-°°(X;A). The space
Lf^JX; A) has a natural Frechet topology. In Section 1.1 we have discussed

187
in detail how to introduce adequate Frechet topologies in symbol and
operator spaces. In the present case the arguments are analogous. Also in
future, if we introduce some space and speak about its Frechet topology, it
will usually be an immediate consequence of the definition. The easy
details will be left to the reader. Every A(X) G L^(X;A) has a well-defined
parameter-dependent homogeneous principal symbol of order \i
aJ.A(A)(a:^,A)€Coo(rXxA\0),
0 indicates (£, A) = 0, and the homogeneity means
for all 5 G iJ+.
Definition 2.1 A(X) G L^{X\A) is called parameter-dependent elliptic
of order fi if a^;X(A) / 0 on T*X X A \ 0.
The parameter-dependent ellipticity can also be studied in the non-
classical case and all essential consequences hold in analogous form.
However we discuss from now on the simpler classical case which is of importance
in the applications below.
Theorem 2.2 Let A(X) G L^(X; A) be parameter-dependent elliptic of
order fi. Then there exists a B(X) £ L'^X] A) such that
A{\)B{\) - 1, B(X)A(X) - 1 G L-°°(X; A), (2.2)
where a^x(B) = (^(A))"1
An operator family B(X) G L~f(X]A) satisfying the relations (2.2) is
called a parameter-dependent parametrix of A(X). Note that the existence
of the compositions is ensured by a particular choice of 5(A) namely to
be properly supported. We will not comment on this further, since we are
mainly interested in the case that X is compact. This will be assumed in
the following.
Recall that we have on X the scale of Sobolev spaces {Hs(X)}seR. In
the above notions concerning A = Rl we may also assume / = 0, i.e., that
A disappears.
We denote the homogeneous principal symbol of A G L^(X) of order \i
by o^AA) which is an element in C°°(T*X \ 0), homogeneous of order fi in
the covariables.
The above parameter-dependent ellipticity induces for A G L^(X) the
"usual" ellipticity, which requires ^{A) ^ 0 on T*X \ 0, and we want to
mention the folloing classical result.

188
Theorem 2.3 Let A G L^(X) be given. Then the following conditions
are equivalent:
(i) A is elliptic of order \i)
(ii) the operator
A: HS(X)^HS-^{X) (2.3)
is Fredholm for an s — So G iJ.
Moreover the ellipticity of A of order \i implies the existence of a B G
L~f{X) with AB - 1, BA - 1 G L'°°(X) and (2.3) is Fredholm for all
s G iJ. Here a^(B) = (aJ(A))-1.
Remark 2.4 Another well-known property is the elliptic regularity of
solutions of An — f G Hr(X), r £ R, when A is elliptic. It says that every
solution u G H-°°(X) fre/on^s fo ifr+/i(X). /n particular, ker A C C°°(X).
Smce A* G £c/(^0 *5 a^so elliptic) we have ker A* C C°°(X) and
ind A = dim ker A - dim coker A
= dim ker A - dim ker A*
is independent of s.
If A(A) G L^(X; A) is parameter-dependent elliptic of order //, the
operator A(Ao) G ^c/(^0 *s eUiptic m the usual sense for every fixed Ao. Hence
A(A): HS(X)-+HS-»(X) (2.4)
is a A-dependent family of Fredholm operators. If we say nothing else we
will always assume in the parameter-dependent case that / > 1.
Example 2.5 Let us form an operator A(X) by the expression (2.1)
with C(X) = 0 and Aj(X) defined in local coordinates by Op((c2 + |£|2 +
|A|2)"/2) for some c> 0. Then a^x(A) = (|£|2 + |A|2)"/2, and hence A(X)
is parameter-dependent elliptic of order fi.
Remark 2.6 Let C(X) G L~°°(X] A) and assume that
1 + C(A): HS{X)-+HS{X) (2.5)
is invertible for a fixed s G R, for all X G A. Then (2.5) is invertible for
all s G R, X G A, and there exists a G(X) G L~°°(X; A) such that
(1 + C(A))-1 = 1 + G(A).
Theorem 2.7 Let A(X) G L^(X; A) 6e parameter-dependent elliptic of
order \i. Then (2.4) is a family of Fredholm operators (2.4) of index zero,
and there exists a constant c > 0 such that (2.4) is an isomorphism for all
A G A with \X\ > c and all s € R.

189
Theorem 2.8 To every fi € R there exists an R^(X) G L^(X\ A) which
is parameter-dependent elliptic of order \i such that
R»{\): HS{X)^HS-»{X)
is an isomorphism for all A G A, s G R, and (^(A))"1 G L^(X; A).
We will call any such R^(X) a parameter-dependent reduction of orders
(of order fi).
2.2. OPERATORS OF FUCHS TYPE
Let B be a manifold with conical singularities and B the associated
stretched manifold. According to the notations in the beginning we have a collar
neighborhoods V of dB = X of the form [0,1) X X with a corresponding
splitting of coordinates in (t,x). Since only a neighborhoods of t = 0 is
of interest in the specific assertions concerning the conical singularities we
may (and sometimes will) identify the neighborhoods with i2+ x X. If
M is a paracompact C°° manifold we denote by DofP(M) the space of
all differential operators on M of order \i with C°° coefficients (in local
coordinates). DifP(M) is a Frechet space in a natural way.
An operator A G DifP(int B) is said to be of Fuchs type if it has the
form
A = t-»J2a3{t){-t^ (2.6)
i=o
near dB in the splitting of coordinates (t, x) over V, where
a^JeCHflU), Diff*-'(A-)), j = 0t...,/i.
Let us consider the Mellin transform
yoo
(Mu)(*) = / t*"1^)*
JO
for u(t) G C^(iJ+,C°°(X)), * € C. Then (Afu)(z) G ^(C,C°°(X)) and
(M-lg){t) = ±-( t-zg(z)dz,
where Tp = {z : Re 2 = /?}.
In the following we will use notations like
S(TP), HS(TP), L^(X;Tp),...
in the sense of the corresponding objects with respect to iJ, identified with
Tp via r -» p + it.

190
Observe that
M~lzM = -t—.
dt
This suggests to introduce pseudo-differential operators with respect to the
Mellin transform with operator-valued symbols
h(tj,z) € C°°{R+ x JR+,l5(X;ri)), fi e R,
where z e I\ is the covariable:
2
(o?M(h)u) {t) = M74t{Mt^fc(t,t/,z)tt(t)},
w e Co°(iJ4.,C00(X)). The action of the symbol function is taken as a
(t, t\ z)-dependent pseudo-differential operator on u(tf) with respect to the
dependence of x globally along X; then the operators follows by applying
a Mellin oscillatory integral argument with operator-valued symbols.
We will also consider weighted Mellin pseudo-differential operators
(o?]li(h)u)(t) = fopM(T--'h)t-'u
for (r-U)(i, t', z) = h(t, t', z - 7), 7 € R, where here
h(t,t',z) e C™(R+ x _R+,^(X;ri_7)).
Then
opj,(*): CS°(RhC«W) "> CTiR+tCriX)) (2.7)
is continuous for every 7.
In particular, let A be of Fuchs type on XA = U+ x X, i.e., of the form
(2.7). Then we have
h(t,z) = ita3(ty e c°°(fi+,^(x;r,))
for every p € i?, and A = <~Mop^(/i) for every /3 € i?, as an operator on
cn#+,c°°(x)).
Note that when A is of Fuchs type over B the homogeneous principal
symbol of A of order fj,
^eC°°(T*(intS)\0) (2.8)
is over T*(int V)\0,V5! [0,1) x X, of the form
t""p(,0(t,Mr,O (2-9)

191
where p^ (£, z, f, £) is C°° up to t = 0. With A we also associate a principal
conormal symbol
^(^)W = E«i(0)«i (2-10)
i=o
which belongs to I^(-X"; Tp) for every /3 £ R.
Definition 2.9 An operator A € Diff^^ntB) of Fuchs type is called
elliptic of order fi, with respect to the weight 7 € R, if
(i) aJ(A) £ 0 on T*{intB) \0 and i/i^Aj^x.rV^) ^ 0 on
T*F\0
<&(>4)(*): H°{X)^H>-»(X)
is an isomorphism for every z G Tn±i , s £ R, where n = dim X.
2 '
In view of the remarks after Theorem 2.3 it suffices to require the
condition (ii) only for a particular s = so G i2. Then it is satisfied for all
se iJ.
The ellipticity should allow the parametrix construction within an
adapted algebra of pseudo-differential operators. This is just the "cone algebra"
which is then closed under parametrix construction for elliptic elements
in general. The ellipticity of Fuchs type differential operators was
studied by Kondrat'ev [15], who also has established the Fredholm property
in weighted Sobolev spaces for the case of compact B. In addition there
was characterized the elliptic regularity with asymptotics near the conical
singularities.
The weighted Sobolev space 7isn{XA) of smoothness s G R and
weight 7 G R on the stretched cone XA can be defined as closure of
C^(iJ+,C°°(X)) with respect to the norm
±-\ \\R%lmz)(Mu)(z)\\l2{x)dz\ ,
2 '
where Rs(r) G Lscl(X]R) is an order reducing family in the sense of
Theorem 2.8. Another choice of Rs(r) gives rise to an equivalent norm. There
are, of course alternative definitions of HS^(XA). In particular, for n = 0
and s G Af, 7 = 0, we have
Hs>°(R+) = lueL2{R+): (tj\ ueL2{R+) for 0 < k
<s
Then HS>°(R+) for arbitrary s G R can be defined by duality and
interpolation.

192
We have
UW+p(XA) = tpUs^{XA) for every s,f,PeR
and
US"{XA)CH?0C{XA) for every s,y e R.
It can be proved that the operator of multiplication M^ by a <p(t,x) G
Co°(i2+ X X) is continuous in the sense
Mv : Usn{XA) -> fts^(XA) (2.11)
and that (p -> A4V is continuous as Q°(i2+ x X) -> £(?^(XA)) for
every5,7 G iJ.
A cut-off function on iJ+ is any u(t) G Co°(i2+) which is real valued
and u;(£) = 1 in a neighborhoods of t = 0.
Theorem 2.10 Letu(t), u(t) be cut-off functions,
h(t,t',z)€C00(R+xR+,L»(X;rn±1_i))]
then n
uop^'{h)u : Usn{XA) -> 7/s"^(XA)
25 continuous for every s G iJ.
This is straightforward when /i is independent of £,£'. In general the
assertion follows from a tensor product argument, using (2.11). There is also
a direct proof, analogously to well-known techniques for standard pseudo-
differential operators in Sobolev spaces based on the Fourier transform.
Let us also define a space H*one(XA) 3 u(t,x), first for the case when
u(t, x) is supported in a coordinate neighborhoods U of X. Let K\ : U -> W
be a diffeomorphism to an open set W C Sn = {x G iJn+1 : \x\ = 1}
and k : iJ+ x C/ -> WA = {x G #n+1 \ {0} : x/|x| G W} defined by
*(*,&) = t*i(x). Then the condition is (l-u(t))u(t(x),x(x)) G #s(iJ£+1),
where (£(£),£(£)) = k_1(£) for an arbitrary cut-off function u(t). The
space H*one(XA) in general then follows by a partition of unity argument.
Let H*one(XA)£ for any e > 0 be the subspace of all u G H*one(XA) that
vanish for 0 < t < e. Then H*one(XA)€ is a Hilberzitable Banach space
in a natural way. Set [1 - u]Hscone(XA) = [1 - u]Hscone(XA)e for a cut-off
function u(t) such that 1 - u vanishes for 0 < t < e and define
K'«{XK) = [u]W"(XA) + [1 - u]H°cone(XA)
as non-direct sum of spaces for every 5,7 G iJ. Then, also /CS/Y(XA) is a
Hilbertizable Banach space contained in Hf0C(XA), and it is independent
of the particular choice of u. For n = dim X = 0 we simply have
/Cs'7(iJ+) - M7T'7(iJ+) + [1 - u;]tfs(iJ+),
h*(r+) = h*(r)\r+.

193
Remark 2.11 Set (K\u)(t,x) = A 2 u(\t,x) for A G i2+. Then
{K,\}\eR+ is a group of isomorphisms on the space K,S,1(XA), with the
properties required in Definition 1.5. This gives rise to a scale of weighted wedge
Sobolev spaces
Ws(Rq,ICs^(XA)) for s,yeR.
Theorem 2.12 Let A G Dijf^^ntB) be of Fuchs type and assume that
pu){0,x,f,£) / 0 for all x and (f,£) / 0 (cf. the notation (2.9)). Then
vm{A){z)\yp G Lj(X] Tp) is parameter-dependent elliptic for every f3 G R,
uniformly in c < f3 <c' for every c < cf. Moreover) there is a countable set
D C C with DDK finite for every K CC C such that
oMA)(z): H'{X)->H-»{X)
is an isomorphism for all z € C\D, s G R.
This will be explained in more detail in Section 2.3 below, where we
formulate a refinement of this result.
Corollary 2.13 Under the conditions of Theorem 2.12 there exists a
countable set E C R with Ef)K finite for every K CC R such that
o»M(A)(z): H°(X)^H°-»(X)
is an isomorphism for all z £Tp, f3 qt E, s € R.
Theorem 2.14 Let A G Diff*(intB) be of Fuchs type. Then the
following conditions are equivalent:
(i) A is elliptic of order fi, with respect to the weight 7 G R,
(ii) the operator
A: ns^{B)^ns-^-^{B) (2.12)
is Fredholm for an s = s$ G R.
This result is standard after the work of Kondrat'ev [15]. It is true
also for general elliptic operators in the cone algebra below, as well as the
elliptic regularity, which says that when A is elliptic and u G H~°°n(B)
is a solution of Au = / G US^'^{B) then u G ?f+/i'7(B), for arbitrary
r e R. The elliptic regularity also holds with respect to subspaces with
asymptotics as they will be described below.
Remark 2.15 (i) or (ii) in Theorem 2.3 imply that (2.12) is a Fred-
holm operator for every s G iJ. Then ker A and coker A are s-independent
subspaces of^°°'7(i?) and %oon~ll{B)J respectively, and hence ind A is
independent of s.

194
2.3. HOLOMORPHIC OPERATOR FUNCTIONS
The differential operators on int B of order \i of Fuchs type will be elements
of the cone algebra on B. They have the form
7--
A = ut MopM 2 (h)u0 + (1 - w)Aint(l - ui)
for an arbitrary Aint G DifP(int B), /i(£, z) = ^=0^(0^' f°r coefficients
ai(*) e C°°(i2+,DifP~J(X)), and cut-off functions a;, u;0, u\ supported in
a small collar neighborhoods V = [0,1) X X of 55 (notations for pull-
backs or push-forwards under [0,l)xl 4 B will often be suppressed
for convenience; this should not cause confusions). As mentioned above we
have a "hierarchy" of principal symbols
(ajJ(A)^a^I(A)) = (principal interior symbol, principal conormal symbol)
involved in the ellipticity. It is obvious that for two Fuchs-type differential
operators A and B of order \i and z/, respectively, AB is of Fuchs type and
of order ji + z/, and
a^(AB) = a{(AK(fl), aff^AB) - (T^(A)) (afc(fl))
(recall that, for instance, a^(A)(z) = Ei=oai(°)^i and (T^)(0 = M* +
z/)). Now in the parametrix construction for an elliptic operator of Fuchs
type we have to invert both symbol components. The discussion of the
inverse conormal symbols is particularly interesting. First observe that
h(z) = <j%j(A)(z) belongs to the space M£(X) in the sense of the following
definition.
Definition 2.16 Mq(X) for \i G R is the subspace of all h(z) G
A{C,L%(X)) such that
h(z)\rfi€L^(X]Tp) for every (3 e R,
uniformly in c < (3 < d for every c < cf.
The space Mq(X) is Frechet in a canonical way. Mq°°(X) (the
intersection over all Mq(X)) is a unclear Frechet space.
Theorem 2.17 To every f(z) G L^(X]TP) p G R fixed, there exists
an h(z) G Mq(X) such that
h(z)\rp-f(z)€L-°°(X;rp).
The proof follows by a kernel cut-off argument with respect to the
parameter.
Note that hi(z),h2(z) € M£(X) and {hx(z) - h2(z))\rp € L-°°(X;TP)
imply h1(z)-h2(z)eMoco(X).

195
Remark 2.18 Let f(z) G L^(X;TP) be parameter-dependent elliptic
of order ji. Then h(z) obtained in Theorem 2.17 has the property that
h(z)\rp G L^(X;Tp) is parameter-dependent elliptic of order fi, for all
(3 G R, uniformly in c < (3 < c for every c < cf.
Let us now mention a well-known result on holomorphic Fredholm
families (a proof may be found, for instance, in [31] or in [24]).
Proposition 2.19 LetHi, H2 be Hilbert spaces, G C C open (arc-wise
connected), and let h(z) G A(G, £(#i, H2)) be an operator function such
that h(z) : H\ -> H2 is a Fredholm operator for every z G G. Assume
that there is a Z\ G G such that h(z) : Hi -> H2 is invertible. Then there
is a countable subset D C G, D n K finite for every K CC G, such that
h(z) : Hi -> H2 is invertible for all z eG\D. In addition h~l (z) extends
from G\D, D = {pj}jez> to a meromorphic operator function with poles in
Pj G D of multiplicities mj + 1 for certain mj G N', and finite-dimensional
Laurent coefficients at (z - Pj)~^k+1\ 0 < k < mj, for all j.
An inspection of the proof of Proposition 2.19 together with Remark
2.18 and Remark 2.4 gives us the following result:
Theorem 2.20 Let A be of Fuchs type, elliptic of order \i with
respect to v!i(A) (i.e., only (i) of Definition 2.9 is required). Then h(z) :=
^(^)W = E?=ofli(o)^
h(z): HS(X)^HS-»(X),
has the property that ^(^Ir^ G L^{X\Yp) is parameter-dependent elliptic
of order fx, for every ft G R, uniformly in c < ft < c' for every c < d, and
there is a countable set D = {pj}jez C C with Df){z : c < Re z < cf} finite
for every c < cf, such that h(z) is invertible for all z G C\D. Moreover,
h~1(z) extends to a meromorphic L~^ (X)-valued function, and the Laurent-
coefficients at (z — Pj)~(k+1\ 0 < k < rrij, are finite-dimensional operators
in L~°°(X) for all 0 < k < mj, j G Z. If A C C is any subset we call
a x £ C°°(C) an A-excision function if x(z) — 0 for dist (z,A) < So,
X(z) = 1 for dist (z, A) > e, for certain 0 < £o < £\ < oo.
Definition 2.21 Let R = {p^ m^ Lj}jez be a sequence, where
(pj, mj) e C x N, \Re pj\ -» oo as \j\ ->■ oo, and Lj C L~°°(X)
is a finite-dimensional subspace of finite-dimensional operators for every
j. Such a sequence is called a discrete asymptotic type of Mellin symbols.
Moreover, M^°°(X) denotes the subspace of all L~°°(X)-valued
meromorphic functions f(z) in C such that

196
(i) f(z) has poles in pj of multiplicities irij +1 with Laurent coefficients
at (z - pj)~(k+iy> belonging to Lj for 0<k<mj,jeZ,
(ii) if x{z) is any ncR-excision function, for ncR = {Pj}jeR> then
X{z)f{z)\vp € L~°°(X]Tp) for every (3 e R, uniformly in c < (3 < d for
all c < cf.
Remark 2.22 M^°°(X) is a nuclear Frechet space in a natural way.
Let us define
as a non-direct sum of Frechet spaces.
Remark 2.23 Under the assumptions of Theorem 2.20 we have
for a certain discrete Mellin asymptotic type R.
We will call an h(z) e Mj^(X) for some discrete Mellin asymptotic type
elliptic of order p, if for any decomposition h = h0 + /&-<x>j ^o € Mq(X),
hoo G M^°°(X), ho(z)\rfi € L^(X;Tp) is parameter-dependent elliptic of
order p (then also ho(z)\rp is parameter-dependent elliptic of order p for
every p, uniformly in c < p < c', for every c < cf). This is a correct
definition, i.e., independent of the particular choice of the decomposition.
The following results are easy to verify:
Proposition 2.24 Let h(z) € Mp(X), g(z) € Mq{X) be given, with
certain discrete Mellin asymptotic types P, Q. Then the point-wise
composition h(z)g(z) belongs to M^(X) for some resulting discrete Mellin
asymptotic type R. If h and g are elliptic then so is hg.
Theorem 2.25 Let h(z) e Mj^(X) be elliptic of order p, where R is
any discrete Mellin asymptotic type. Then h~1(z) 6 Mq^{X) is elliptic of
order -p, with some resulting discrete Mellin asymptotic type Q.
2.4. ELEMENTS OF THE CONE ALGEBRA
For studying elliptic regularity of solutions on a manifold with conical
singularities it is interesting to look at subspaces of the weighted Sobolev
spaces with asymptotics. Let us fix a weight 7 e iJ, consider the
associated weight line Tn±±_ C C, n = dimX, and choose a "weight inter-
val" 9 = (#,0], -00 < # < 0, that defines for every weight 7 a strip
e7 = {z e c : 41 - 7 + 0 < Re z < H1 - t}- Let first © be finite and
call a sequence
P = {(Pi,mi,Li)}i=o,...,JV for N = N(P)

197
a discrete cisymptotic type cissociated with the weight data (7,©), if
{Pj,rrij) e C x iV, pj e 07, and Lj a finite-dimensional subspace of
C°°(X)forall j = 0,...,N.
Definition 2.26 An element u(t,x) G /CS'7(XA) has asymptotics of
type P for t -> 0 if there are coefficients Cjk — Cjk(u) e Lj, 0 < k < nij,
such that
{N rn3
£X>^log**
j=0 k=0
\ elCs^-V-(xA) (2.13)
for any cut-off function u(t). Here
Kw- (xaj = ne>0/C^-e (XA), p € #,
m ffte Frechet topology of the projective limit Denote by ^^(X^) the
subspace o//Cs'7(XA) consisting of all elements with these asymptotics.
The unique coefficients Cjk(u) define linear maps
cjk : /C£7(XA) ->/,,, 0 < ife < ro,, j - 0,..., N. (2.14)
The remainder (2.13) then gives rise to a linear map
r : /C£7(XA) -> /CS'(7"^-(XA) (2.15)
for a fixed cut-off function u. Then K^^X^) becomes a Frechet space in
the topology of the projective limit with respect to (2.14), (2.13), that is
independent of u. For infinite 0 = (-00,0] we can talk about discrete
cisymptotic types of the form P — {(pj, raj, Lj)}j€jv, with ^^ - 7 > Re pj
for all j and Re pj -» -00 for j -» 00. Then, to every k G N\ {0} we have
Pk = {(p? ra, L) e P : 2^li - 7 - ^ < ^ Pj < 2L|i - 7} in the above sense,
where Pk+1 C P* for every k. It is clear that then /C£7 t (XA) <-> /C£7(XA)
is continuous for every fc, and we can define the space
£^(XA) = lim projfceR/C^(XA)
as projective limit.
We will also set
57(XA) = [u;]/C~'7(XA) + [1 - u;]<S(iJ+ x X),
where <S(#+ x X) = S(R,C°°(X))\^. Then 5^(XA) is a nuclear Frechet
space.
Analogously to ^^(X^) we denote by %p7(B) the subspace of all
u G iffoc(int B) for which uu e /Cp7(XA) for any cut-off function u(t)
supported in a collar neighborhoods V of #B, u = [0,1) x X 9 (£, x).

198
Remark 2.27 The space ^^(X^) for every fixed s e R and P,
associated with (7,0), can he written as projective limit of Hilbert spaces
{Ei}jeN with continuous &+1 *-> E* <-> E° = ICS^(XA), where {ka}agH+
on E° induces {^a}agR+ £ C(R^^Ca(E^)) for all j, with the properties
required above in Definition 1.5. An analogous remark holds for the spaces
S]>(X*).
Denote the weighted Mellin transform with the weight p G R by Afp, i.e.,
(Mpu)(t) = M(t~pu)(z+p). Let P be a discrete asymptotic type associated
with (7,0), 0 = (tf, 0] for some 7 e R and -00 < ft < 0. Let fP(rp X X)
for 5 e i? be the image of HS(TP x X) with respect to the Fourier transform
along Tp.
Theorem 2.28 Let u(t) be an arbitrary cut-off function. Then
M1_n(ulCSp1(XA)) is a space of meromorphic HS(X)-valued functions in
Re z > IL^ + # - 7, with poles in pj of multiplicities rrij + 1, and Laurent
coefficients at (z - pi)~(k+1) in Lj for 0 < k < nij. If x{z) *5 anV ^cP-
excision function, ncP = {Pj}j=o,...,JV (N — N(P) or N = 00), then for
every f(z) G M1_n{uJCSpl(XA)) we have
(xf)\rp€Hs(TpxX)
for all p > IL^ + <p - 7, uniformly in compact p-intervals
Theorem 2.29 Let h{t,t',z) G C°°(R+ x fl+,Af£(X)), fi G R, for
a certain discrete Mellin asymptotic type R, and let u(t)f u(t) be cut-off
functions. Then, if7rcRr\Tn±± = 0,
2 '
wopj, (fc)* : /C^(XA) -> /C^(XA)
25 continuous for every s € R and every discrete asymptotic type P with
some resulting discrete asymptotic type Q, associated with given weight data
(7,e).
We now turn to an analogue of the Green operators from the calculus
of boundary value problems, where here the inner normal to the boundary
i2+ is replaced by XA.
Let us first note that
/C°'°(XA)^rfL2(iJ+xX)
and that the /C°'°(XA)-scalar product
(.,.): Cg°(XA)xC§°(XA)->C

199
extends to a non-degenerate sesquilinear pairing.
(.,.): /CS'7(XA) x K~s "7(XA) -> C
for every s,y £ R. The formal adjoint A* of an operator
A € na£RC{lC'''r(X%lC'-','s{XA))
is an element
A* e nseH£(/cs'-<5(xA),x:s-1/'-'1'(xA)).
Definition 2.30 An operator
is called a Green operator on XA if for certain (G-dependent) asymptotic
types P and Q, associated with (5,0) and (-7,©), respectively, we have
G e ns£RC(lcs«(x%ssP(x*)),
G* € n*RC{K'>-s{X%Sj(X*)).
Analogously a G G nseRC(Hs,1(B))7i00iS(B)) is called a Green operator
on B if
G € Ds£RC(n^(B)^'S(B)),
G € nseRC('Hs'-s(B),'H^(B)),
for certain (G-dependent) P,Q.
Let hj(z) G M^°°(X), j e N, for some discrete Mellin asymptotic type
Rj, 7,7i e iJ, 7rcflnrn±i_7i =0, where ~7 + J + 7j > 0, ~7j+7 > 0.
Then
Mj := ut5-^jopjf *{hj)u0 : /CS'7(XA) -> /C°°'*(XA) (2.16)
is continuous for every s e R for cut-off functions a;, a). According to
Theorem 2.29 Mj induces continuous operators between subspaces with
asymptotics.
Proposition 2.31 Let Mj = u^'^op^ 2 {hj)uo for another pair of
cut-off functions u), u0 and jj G R, ncRj fl tn±i_- = 0, -7 + j + 7j > 0,
~7j + 7 > 0. Tften Mj - Mj is a Green operator in the sense of Definition
2. SO.

200
The algebra of cone pseudo-differential operators on B with respect to
the weight data (7,5,0) for y,5 G R and 0 = (-fc,0], k e N\ {0}, is
defined as the subspace of all operators
A = u;^op^? (h)uo + (1 - u>)Aini{l -u^ + M + G, (2.17)
for arbitrary h(t, z) G C°°(B+ x X), Aint G -^(int S)> and
As—1
i=o
with Mj being of the form (2.16), and G a Green operator in the sense of
Definition 2.30. a;, uq, a>i are arbitrary cut-off functions supported in a collar
neighborhood of dB, with uuo = a;, a;a;i = u\. M is a called smoothing
Mellin operator in the cone algebra.
Remark 2.32 The operators (2.17) belong to
L'Jint B)nnseRC(n^(B),ns-^(B)).
To A we have 0^(j4), the homogeneous principal symbol of order //, and
(Jm(A), the principal conormal symbol, defined as a^I(A)(z) = /i(0,z) +
ho(z) in the notations of (2.17) and (2.16). It can be proved that
V><T${A)(t, x, t~lT, f) in a collar neighborhoods V of dB, V = [0,1) X X, is
C°° up to t = 0.
A is called elliptic if the conditions of Definition 2.9 (where o^ has to be
replaced by <r^~ ) are satisfied. Then Theorem 2.14 permits the following
refinement:
Theorem 2.33 A is elliptic of order \i, with respect to the weight 7 if
and only if
A: ns^(B)^ns-^s(B)
is a Fredholm operator for an s = So G R. The ellipticity implies that A
is Fredholm for all s € R. Moreover, there is a parametrix B in the cone
algebra of order —p, to the weight data (5,7,0) such that AB — 1 and
BA — 1 are Green operators with respect to the corresponding weight data.
Theorem 2.34 Let A be elliptic of order fi, with respect to the weight
7. Then a solution u G n'°°n(B) of Au = / G 7l8'^s(B)9 s G R, belongs
to U*"{B). In particular, for f G USq^8{B) for some discrete asymptotic
type Q we obtain u G 7/p7(S) for some resulting discrete asymptotic type
R.

201
3. Pseudo-differential calculus on manifolds with edges
3.1. EDGE-DEGENERATE DIFFERENTIAL OPERATORS
Let G be a domain in RN with piece-wise C°° geometry, locally being of
wedge type. For instance, the wedge may have the form
{(£, y) G Rn+1 x Rq : x = 0 or x/\x\ G E, y G ft},
where S C Sn = {x : |£| = 1} is an open subset with smooth boundary,
and ft C Rq open. If A(x,y,Dxiy) is an elliptic differential operator in
iJn+1 X ft with C°° coefficients we may study boundary value problems for
A with elliptic conditions along the smooth faces of dG. A common method
to do this is to introduce polar coordinates into A with respect to x. An
operator A G DifP(i2£+1 X fiy) in polar coordinates x -> (£,#), £ = |£|,
# = £/|£| G 5n, takes the form
4 = *"" £ «i-(*.»)(-*f)J(*^)a (3-1)
j+|ar|<A*
with aja(*,y) G C°°(fi+ X fi,DifP-(i+|aD(5n)). A useful model for
studying the above problems in to first consider the case when the base of the
model cone of the wedge is closed compact. This can be combined with the
methods from the standard calculus of boundary problems, say with the
transmission property, in a parameter-dependent form, cf. Schrohe, Schulze
[26], [27], [28]. We will content ourselves here with the case of closed cone
bases X. >
A wedge in general has form XA x ft for XA = (fi+ x X)/({0} x X).
Operators will be given on the open stretched wedge XA x ft. An operator
A G DifP(XA x ft) is called edge-degenerate if it has the form (3.1) for
coefficients aja(t,y) G C°°(R+ x ft,DifP-(i+'a|)(X)). For studying ellipticity
and parametrix constructions it will be necessary to pose additional
conditions along ft of trace and potential type, similarly to the case n = dim X.
Note that when A = Y^\p\<^ap{^)^ *s &Yen on R+ x ^ w^ith coefficients
ap(x) G C°°(R+ x ft) the operator A in the coordinates x = (£, y), t G i?+,
y G ft can also be written
A = r" £ aia(tfy)(-t£)''(t0,)°
J+M<M
for coefficients aja(t, y) G C°°(i2+ X ft). This shows once again that
differential operators on R+ xft in usual form are automatically edge-degenerate.

202
For instance, the Laplace operator takes the form
Among the elliptic boundary (or trace) conditions along ft are the Dirich-
let conditions. If Hf, JR+ x ft) denotes the subspace of all u(i,y) €
tff0C(iJ+ x ft) such that <p{y)u{t,y) e #s(i? x iJ^ln+xn for every y> e
Co°(ft), the local Dirichlet problem takes the form of an operator column
matrix
: H?oc{y)(R+xn)^ © , (3.2)
s > |, and a local parametrix will be an operator row matrix (P, If)
mapping in the reverse direction (properly supported in y-coordinates). For
solving (3.2) we may pass to RT instead of T for an elliptic reduction of
3
orders R e Lj(Q). It is well-known that elliptic boundary value problems
for pseudo-differential operators require both trace and potential conditions
with respect to the boundary, the latter ones mapping distributions on the
boundary into distributions in the domain, like the potential K just
mentioned, cf. [3], [20]. Analogously an edge-degenerate differential operator A
on a (stretched) wedge XA X ft, which is elliptic in the sense <r^(A) ^ 0
on T*(XA x ft) \0 and t^{A){t,x,y,t'lT^rlri) ^ 0 up to t = 0 for
(r, £, rj) ^ 0, requires for the local parametrix constructions (and globally
for the Fredholm property) additional conditions of trace and potential
type with respect to ft, including pseudo-differential operators on ft. In
other words we have to consider block matrices
© -> ©
H?omp(n,cN-) fl£"(nfc"+)
Here
WclP(y)(xA x fi) = {u € W°(R\)CS"(XA)): suPP(2/)U CC ^compact}
and W&V){X* xQ) = {u€ l^Q, £•"(**)) : ^ e Wc^p(j/)(X* x «)
for every <p 6 Co°(fi)}, cf. also Remark 2.11. An idea of the wedge pseudo-
differential calculus is to treat the problems in terms of pseudo-differential
operators along the edge Q, with operator-valued symbols operating along
the model cone XA. A starting point are the edge-degenerate symbols.
A
T
A K
T Q

203
A symbol p{t,x,y,T,£,r)) e Sj{R+ xSxfix i21+n+g) for open sets
H, C Rn, Q, C Rq,is called edge-degenerate if
p{t, s, y, r> f, *?) = p(*>«, y, fr, £, t»/)
for a p(t, x, y, r,£, r?) G 5^(5+ xExdx iJ1+n+<?).
Proposition 3.1 Lei p(i,£,y,7",£,r/) 6e an edge-degenerate symbol,
P[^){t^x^y^f^^fj) the homogeneous principal part of p{t,x,y,f,£,rj) in
(^S£) i)) / 0 o/ order // and assume that
P(M)(M,y,*S£,»/)#0 /or a// (^,!/)6fl+xEx(],(f,^)/0.
Tften £/fcere eztste an edge-degenerate symbol r(£,#,y,r,£, 77) o/ order -//
(r^)#^(^r) = 1 mod S-°°(iJ+ xExDx iJ1+n+<?),
w/fcere # means the Leibniz product between the symbols, taken with respect
to the indicated variables. The same is true for the multiplication in reverse
order or for the Leibniz product only with respect to t,x.
Similarly to the calculus on a cone it is adequate to formulate
operators globally along X and to consider corresponding (i, y, r, 7/)-dependent
operator families.
Let us fix an open covering of X by coordinate
neighborhoods {t/i,..., Un} together with a system of charts Xj : Uj -> Sj,
Ej C Rn open, a subordinate partition of unity {^i,...,^v} and
functions {^i,..., iPn}, i/>j G C°°(Uj), satisfying <pjil)j — <fj for all j.
Given a system of edge-degenerate symbols
Pj(ttx,y,T,Z,r,) € S*(R+ xS3xfix fi1+n+«), j = 1,..., AT, (3.3)
with the associated symbols pj(t,x, y,f,f, fj) which are smooth up to t = 0
we can form an operator family
£(*, y, *S v) = S w {^W^K*, y> ^> */)} Vy
where op^, denotes the pseudo-differential action in Ej C iJn with respect
to x. Then
Jfyv.M) € C°°(R+ x Q,L$(X;BH*)). (3.4)
For obtaining a pseudo-differential operator on XA x ft we have to carry
out the action also with respect to t e R+ and to y G ft. Concerning

204
the action in t we want to have a control up to t = 0. For this reason we
formulate for the edge-degenerate case a Mellin operator convention that
allows us to apply the pseudo-differential calculus along iJ+ with respect
to the Mellin transform (it is needed only for a neighborhoods of t = 0). In
other words we generalize some constructions from the cone calculus in a
suitable parameter-dependent form.
Definition 3.2 Mq(X; Rfy for fi e R is the subspace of all h(z, rj) G
A(C,L^(X;B^)) such that
h(z,r!)\r,eL"cl(X;TpxRl)
for every (5 G R, uniformly in c < (3 < cf for every c < cf.
Theorem 3.3 To every f{z,rj) G L^(X;TP X JRJJ), p G R fixed, there
exists an h(z, rj) G Mq(X\ R?) such that
M*^)lr,-/Mei-~(A-;r,xjR«).
The proof can be obtained analogously to that of Theorem 2.17 by
a kernel cut-off argument. Note also that hi(z,r]), h<i{z, r\) G Mq(X;R^)
and hx{z,rj) - h2{z,rj)\Tp G L-°°{X;TP x JRg) imply h^z.rj) - h2(z,ri) G
Mq°°(X]R^). Since the kernel cut-off only acts on covariables, we have
analogous results for the case when f(t, y, z, rj) depends on (t, y) G i?+ X ft,
up to t = 0. Then the other occurring objects also depend C°° on (£, y) G
J2+ x ft.
The Mellin operator convention for the wedge calculus consists of the
following result:
Theorem 3.4 To every P(t, y, f, r?) G C°°(#+ xfi, I^(X; iJ-J)) fftere
ext'ste an /&(£, y, z, fj) G C°° x ft, Mq(X; jR|)) st/c/i iftcrf
opl(P)(yli7) = opi(fc)(yli7) mod C00^,!"00^; JR«))
/or P(t,y,T,n) := P{t,y,tT,tr)), h(t,y,z,n) := h(t,y,z,tn), and every
such h(t,y,z,f)) is unique mod C°°(R+ x ft, Mq°°(X;.R|)). T/ie pseudo-
differential action opt a/on^f iJ+ re/ies on i/»e Fourier transform, and the
operator families are understood in the sense
Cff{R+,C00{X))->C00{R+,C00{X)).
The proof of Theorem 3.4 is technical, however the idea is easy. In a
first step we choose an f0(t, y, z, fj) e C°°(R+ x ft, L^(X; T0 x i?|)), where
To = O'r :t £ R}^R, such that
fo(t,y,-iT,fi) = P{t,y,T,fj).

205
Then a calculation shows that there is a
Pi(t,y,r, V) € C°°(fl+ x 0, ^(X;T0 x rJJ«))
such that
opt(P)(j/,»/) = opjk (/0)(y,»?) + opt(Pi)(y, /?)
C00^,!"00^;^)), where Ut,y,z,rj) := k{t,y,z,trj), Pi{t,y,T,V) :=
Pi(t,y,tr,tr]). This allows us to start an iteration which yields a sequence
h{t,y,z,~ri) € c°°(fi+ x n,^-*(x;r0 x «}))
for all S;6iV.
The asymptotic sum
oo
/>,y,M)~ £/*(*, y,M) in C~(:R+x^(X;r0xiJ?))
then gives an f(t, y, z, 7/) = f(t, y, z, £77) such that
opt(P)(yli7) = opi(/)(y,i,) mod C00^,!"00^; i*«)).
Applying Theorem 3.3 in the (t, y)-dependent form and using the fact that
op/?m(- • •) = °Pm(- • •) f°r arbitrary f3,p e R for Mellin symbols that are
holomorphic in z we obtain the assertion.
Remark 3.5 Let us set
My, r, fj) = £(0, y, f, 77), fco(y, z, fj) = P(0, y, z, 7?) and
ft(«, y, r, 7/) = P0(y, «r, t»/), fco(t, y, z, 7/) = h0(y, z, trj).
Then, 1/ we insert Po(y, ?\ f/) instead of P(t, y, f, 77) m Theorem 3.4 it
follows that
opt(P)(y,v) = op0M(ho)(y,r1) mod C~(n>I-~(*A;JR»)).
Let us now choose cut-off functions w(t), u>o(0) wi(0 satisfying wwo =
u>i, wwi = wi. Then, using Theorem 3.4 together with the pseudo-locality
of pseudo-differential operators in parameter-dependent form, we obtain
<>Pt(P)(y,»?) = w(t[//])op5f(/i)(y,7/)u;o(t[?7])
+ (l-wWi/DJofttPJCy.iyja-«!(«) (3.5)
mod C°°(ft, I-°°(XA; JJ«)), for every /} 6 R.

206
Theorem 3.6 Set
"ofay) = rMw(*W)°PAf ^^(y.^^W).
*iM = r^(l-o;(t[f/]))oft(P)(y,f/)(l-a;i(t[f/])
/or P and /i o/ Theorem 3.4, and letu(t), Uo(t) be arbitrary cut-off
functions. Then
a{y,r)) = u;(0{«o(y,^) + ai(y,7/)}a4)(0
G SM(fi x iJ*; /CS'7(XA), /Cs-/i'7"/i(XA)) (3.6)
/or every s € R. Moreover,
Opy(a) = Opy(uoPt(P)u0) mod Zr°°(XAxft), (3.7)
wftere Opy{...) = .7yiy(- • O-Ty-n-
The proof of (3.6) in Theorem 3.6 is bcised on estimates of the norms of
pseudo-differential operators by the symbols. (3.7) is a consequence of (3.5).
Remark 3.7 Let us set with the notation of Remark 3.5
rt(a)(y,ri) = rM'M)opJ7*(MM<4>(*M)
+ r«(l - w(tM))oft(fi)(y, »7)(1 - wi(«)
/or (y, rj)£Qx{Rq\ {0}). Tften
<tf («)(*/, 77) : K'«{X*) -> /C-""""^)
25 a family of continuous operators and
<r£(a)(y, At/) = A/iKA^(a)(y, r/)^1 /or a// A e R+
and all (y, 77) G ft x (JF \ {0}), s e R.
Remark 3.8 a(y, rj) in the notation of Theorem 3.6 is a parameter-
dependent family of cone pseudo-differential operators on the infinite (open
stretched) cone XA.
Remark 3.9 The relation (3.7) of Theorem 3.6 shows that for a(y, rj)
given by (3.6) we have
Op(a)GL£(XAxft).
Since (3.4) that is involved in the definition can be generated by a system
Pj of local edge-degenerate symbols (3.3) via (3.4), the correspondence
{r^i,...,r^}->Op(a)

207
may be interpreted as an operator convention for such degenerate symbols.
If we assume in addition that the symbol pj over Ej fl Xj{Uj fl Uk) is
compatible with pk over £* fl Xk{Uk n Uj) modulo symbols of order — oo, in the
sense of the rule for the symbol push-forward, (associated with the
operator push-forward to the transition diffeomorphisms {Ej fl Xj{Uj fl Uk)} ->
{H,k fl Xk{Uk n t/j)}^ /or °>tt k,j, then Op(a) /*as {t"Mpi,..., t'^pw} as the
local symbols over R+ x Ej x fi, j = 1,..., JV.
Theorem 3.10 Lei a(y,7/) 6e given 6y f&tf^. Tften Op(a) induces
continuous operators
Op(a) : VV«omp(3/)(fl,^(XA)) -> MQ)(n,/C-^-'*(JfA))
/or a// s £ R.
3.2. GREEN, TRACE AND POTENTIAL EDGE SYMBOLS
Definition 3.11 An operator-valued symbol
g(y,y\v) e n,eH55(fi x 12 x r?-,ic'«{xa) ®cN-,JC°°'s(xA) ® c*+)
/or /i, 7, (5 G JR and open ft C i29 2*5 ca//ed a Green edge symbol (with discrete
asymptotics) if
g(y,y',v) e r)seRs^(nxnxR^;ic^(xA)®cN-,ssP(xA)®cN^,
9W,V) € n.€H55(ftxftxJR«;^'-*(A'A)©CJV+I5g'Y(A'A)eCJV-)
/or g-dependent discrete asymptotic types P and Q, associated with the
corresponding weight data. Here * indicates the point-wise formal adjoint
in the sense
(ff^)/CO,o(xA)®civ+ ~ (W)ff*t;)jCO,o(XA)©cJV-
for all u e C0°°(IA)®CN-, 1; e C§°(-YA) 0 CN+.
Recall that the group action on a space E © CN equals n\ © idCN,
A G i2+, when {ka}agH+ *s the given group action on E. Here, for E we
have /CS'7(XA) with (K\u)(t,x) = A 2 w(Ai,z), or £J in the meaning of
Remark 2.18. By definition we have g(y,yf,r)) = (ffij)(y, 2/, i/)t,j=i,2-
The entry 521(2/, 2/, */) has the meaning of a vector of A/+ trace symbols,
512(2/, 2/', 7?) of AL potential symbols, whereas #22(2/, 2/???) is an AL x A/+-
matrix of symbols in 5£)(fi x ft x JK?).

208
Remark 3.12 Note that the operator families in Definition 3.11 may
be regarded as a generalization of those in Section 1.4 (for the case d —
0). The asymptotic types in Section 1.4 are the Taylor asymptotics in the
image (i.e., smoothness up to t = 0). This belongs to the motivation for
our notation to call the symbols of Definition 3.11 Green symbols.
As classical symbols the Green symbols #(y, y', 77) have a unique
sequence g(n-j)(y, yf, rj) of components of homogeneity \i - j for all j e N.
We set
< {9) (y,v) = 9M {y, y', v) \y'=y • (3-8)
This will be regarded as an operator family
JCS"{XA) £*-"»* (XA)
<{9){y,rj): e -> e
cN- cN+
for every s, where
for all A G R+ and (y, 7?) € ft x (Rq \ {0}).
Theorem 3.13 Let g(y, y', rj) be a Green symbol of order /j,. Then
Op(g) induces continuous operators
Op(y) : © -»■ ©
/or a// s £ R, with some asymptotic type P dependent on g.
For the proof it suffices to apply Theorem 1.16.
The pseudo-differential calculus on a manifold with edges gives rise
to another interesting clciss of operator-valued symbols, namely those of
smoothing Mellin type. These are (y, y', 7/)-dependent families of operators
of analogous structure as in the cone theory, cf. Section 2.4. Let us fix a
discrete Mellin asymptotic type Rja and let hja(y,yf) € C°°(Qxft,Mr*(X))
for j e iV, a G Nq, \a\ < j. Assume that for given 7,7ja G R we have
7TCRja n rn±i_7.a = 0, and -7 + j + 7ja > 0, -jja + 7 > 0. Then, for
arbitrary cut-off functions u;(f), uo(t)
mja(y, y\ V) = «(t[i7])r'*''op£a"* (fy) (y, 2/') A>o(<M)

209
is a family of continuous operators £S'7(XA) —»• /C00'7 ^(XA) for every
s € R, C°° dependent on y, y', r]. From
mja(y, j/', At?) = A'i_J+lalKAmja(j/, j/', r?)^1
for all A > 1, |»/| > const for a constant > 0 we see that
mja(y,y\v) € S^j+M(n xQx R?;IC'>''(XA),K:0O«-'>(XA)) (3.9)
for all s G iJ. Analogously we have
mjaM,v) € 53-i+|o|(n x « x i?9;^(xA),^--(xA))
for every discrete asymptotic type P with some resulting discrete
asymptotic type Q, associated with the corresponding weight data. A smoothing
Mellin edge symbol in the edge symbolic calculus is defined as
™{y, y', ^?) = EE mj<*(y, y'> v) (3.io)
j=0 |a|<j
for arbitrary mja of the form (3.9) and k e N \ {0}. Here (-fc,0] is
interpreted as weight strip for the discussion of asymptotics. As classical
elements the symbols we have unique components ^(At_/)(y,y/, rj) of order
//-/,/ G A/", and we set
°A(™){y,v) = m(ri(y,y',ri)\y'=y- (3.n)
Note that only the summands with j = |a| contribute to the term of order
\i and that in this case
*»;«,(„)(v, J/', v) = «(* M)r"+''op2a"* .(M(y. »V««>(*M)-
Remark 3.14 Analogously to Proposition 2.31 the symbol m(y,y\rj)
remains unchanged modulo some Green symbol g(y, yf, rj) (of type of a left
upper corner in the block matrices) when we change the cut-off functions
or the weight ~fja (under the mentioned assumptions).
Theorem 3.15 Let m(y, yf, rj) be a smoothing Mellin symbol of order
\i in the sense of (SAO). Then Op(ra) induces a continuous operator
OpM : Kom^^ix*)) -► wa(n,^^(xA))
and
Op(m) : Kom^lCPiX*)) -> KZ^IC^iX*))
for every s G R and every (discrete) asymptotic type P with some resulting
(discrete) asymptotic type Q.

210
3.3. CONTINUOUS ASYMPTOTICS
For higher-dimensional singularities, e.g., of edge type, the analysis of
solutions of elliptic equations with asymptotics near the singular set (say in
terms of the distance variable t e ii+, t —> 0) leads to exponents of t and of
log t that may depend on the edge variable y. The Mellin transform Mt^z of
such a y-dependent distribution with asymptotics then consists of a family
of meromorphic functions in C 9 z, where in general the poles have not
constant multiplicity under varying y. For instance, if a(y), 6(y), c(y) e C°°(Q)
for open Q C Rq, Re a(y), Re6(y), Rec(y) < \ for all ygfi, then
u(t, y) = u{t)M;\t{z - c(y)){(z - a(y))(z - 6(y))}-x
belongs to C°°(ft, L2(R+)) and we have for every fixed j/6fi
u{t, y) ~ y{y)ta(y) logf for t -+ 0
when a(y) = 6(y) and a(t/) / c(y),
w(f,y)-a(y)fa(2/)+/3(y)f6(2/) for f-> 0
for a(y) / 6(y) and a(y) / c(y), 6(y) / c(y), whereas
u(t,y)~5{y)tb{yy> for i-> 0
for a(y) = c(y), 6(y) / c(y), and similarly for a(y) / c(y), 6(y) = c(y).
Here 7(y),«(y),/3(y),5(y) are certain complex coefficients. In other words
we obtain branching discrete asymptotics, and the y-dependent behavior
can be extremely complicated.
For the systematic calculus that enables us to characterize such effects
in general, also in the vector-valued situation for weighted wedge
distributions of the classes >Vf0C(fi, /CS/Y(XA)), it is useful to extend the concept of
discrete asymptotics and to pass to the continuous asymptotics. This was
first mentioned in Rempel, Schulze [22] and then intensely studied by the
author in [29], [31], [35], [36]. We shall describe here some basic ideas.
If K CC C is a compact set we have the space A'(K) of analytic
functional carried by K in its nuclear Frechet topology, cf. Hormander [14,
Vol. 1], (for the Frechet topology cf. [31], [34]) To every p6C,m6iV,we
can define an element CG^({p}) by <C,/i> = EfeoC,ib(s) M*)I*=p> h e
A(C). Then, in particular,
m
<C,r*) = r"5>i)fccfciogfc*.

211
This shows that the discrete cisymptotics for t —> 0 can also be written as
follows. There are compact sets Kj CC C, Kj C {z : Rez < \ - 8} for
some S e iJ, j G iV, with sup{Re 2:: 2: 6 Kj} -> -00 as j -> 00, such that
there are analytic functional Q G k4'(#j) with
00
u(0~£<Cj,'~*) for *->0.
i=o
The discrete cisymptotics as a special case correspond to Kj — {pj}, where
(j is a linear combination of derivatives of the Dirac distribution in pj. Up
to now we have considered the scalar case, i.e., cisymptotics for distributions
on i2+, say in L2(R+), where 6 = 0. In general, for /CS'7(XA) with a non-
trivial cone base we take C°°(X)-valued analytic functional. If K CC C
is given then Af(K,E) = A'(K)®VE is the space of E-valued analytic
functions, carried on K, for any Frechet space E.
If a set V C C is given we denote by Vc the complement of the union
of all unbounded connected components of C \ V.
Definition 3.16 Let V C {z : *±i - 7 + & < Rez < *±i - 7} be a
compact set Vc = V, -00 < d < 0, 7 e #. 7%en u(t) G /CS'7(XA) fre/on^s to
/Cy7(XA) /or ffte continuous asymptotic type V if for some cut-off function
u(t) and an element ( G A'{V,C°°(X))
u{u-((,rz))eics^-^-{xA).
The space /Cy7(XA) can be endowed with a natural Frechet topology.
More generally if V C C is a closed set, Vc = V, V C {z : Rez <
^^ - 7}, and V 0 {z : c < Re z < c'} compact for every c < c', then we
set Vk = V H {z : Re 2 > ^±1 - 7 - (k + 1)}, hN, and define
^(XA) = nkeNK%(X%
endowed with the projective limit topology.
Theorem 3.17 For arbitrary V with the required properties, -00 <
# < 0, and every s G R we have
JCS^(XA) = JCs^+^-(XA) + /C~'7(XA). (3.12)
Moreover, for V\ + V2 '•= (Vi U V2)c
/C^7(XA) - /C^(XA) + /C^(XA), (3.13)
in the sense of non-direct sums of Frechet spaces. In (3.12), for fl = -oo,
IC°<(~t+#)-(XA) is defined as A:5'°°(XA).

212
Theorem 3.17 follows from a decomposition result for (vector-valued)
analytic functional with respect to decompositions of the carrier sets. This
employs, in particular, a Cousin problem argument.
Also operator-valued Mellin symbols with continuous asymptotic data
can be introduced:
Definition 3.18 Let V C C be a closed set, Vc = V, V n {z : c <
Rez < c'} compact for every c < cf. Then My°°(X) denotes the space of
all h(z) e A(C \ V, L-°°(X)) such that for every Vk := {z : -(* + 1) <
Rez < k + 1}, k G N, there exists a(e A'{Vk, L'°°(X)) with
h(z) - (M5^M(,t-Z))(z) € A({\Rez\ <k + l,L-°°(X)})
for some cut-off function u(t), and S e R with U - S < k + 1, and
moreover, for every V-excision function x{z) we have
X{z)h{z)\T^L-°°{X-Y0)
for every (3 G R, uniformly in c < f3 < cf for every c < cf.
My°°(X) is a (nuclear) Frechet space in a natural way. We then define
M${X) = M£{X) + My°°{X)
for fi G R as a non-direct sum of Frechet spaces. Similarly to Theorem 3.17
we have
M£(X) = M£(X) + M£(X)
for V = Vi + V2.
The pseudo-differential calculus on a manifold with conical singularities
is possible also in the framework of the continuous cisymptotics, cf. [34].
Let us mention here only some few facts.
Theorem 3.19 Leth(t,t',z) e C°°(R+ x #+,M£(X)) be given with a
V C C of the mentioned kind, where V fl Tn±i_ = 0. Then, if u(t),u(t)
7--
are cut-off functions, ^;opM 2 (h)u induces a continuous operator
for every continuous asymptotic type B with some resulting continuous
asymptotic type C, for all s € R.
The weighted Sobolev spaces 7is,1(B) on a stretched manifold B with
conical singularities also contain subspaces with continuous cisymptotics in
a natural way.

213
Theorem 3.20 The elliptic regularity of Theorem 2.34 holds in
analogous form for continuous asymptotic types.
Remark 3.21 The y-dependent discrete (branching) asymptotics of
elements in VV^/ x(fi, /CS'7(XA)) can be formulated in terms ofC°° functions
of y with values in A,(K^Cco(X)) for certain K CC C, where y-wise the
analytic functionals are finite linear combinations of derivatives of Dirac
measures in points pj e C, with coefficients in C°°(X). This concept for
the elliptic regularity for edge problems was applied in [33]. The general
calculus for variable branching asymptotics in the case of boundary value
problems may be found in [35], [36]. The edge case with non-trivial cone
base is similar.
Similarly as in Section 2.4 we define for a continuous asymptotic type P
<S^(XA) - M/C£'7(XA) + [1 -u]S(R+ x X).
For the spaces ^^(X^) and <Sp(XA) we have an analogue of Remark 2.18.
In particular, we can also form the wedge spaces
KomP(y)(n,IC^(X*)) and WL(y)(n,IC^(X*))
in the framework of the continuous asymptotics.
By replacing the discrete asymptotic types of Definition 3.11 by
continuous ones we define the Green symbols with continuous asymptotics. In
an analogous manner we introduce smoothing Mellin symbols of the form
(3.10) with continuous asymptotics in the involved hja(y,yf) e C°°(Q x
Q,Mr™(X)). This will tacitly be used below in Section 3.4.
3.4. ELLIPTICITY OF PSEUDO-DIFFERENTIAL OPERATORS ON
MANIFOLDS WITH EDGES
For discussing ellipticity of operators on a manifold with edges we want to
consider once again the edge-degenerate differential operators on a
(stretched) wedge XA x Q 9 (f, z, y):
A = r» £ «;*(', y) (-*£)'(*A,)"
for coefficients aja(t,y) € C(X'(R+ x il,DHPl-^+^{X)).
Let us assume for simplicity that a,ja(t, y) is independent of t for \t\ >
const for a constant > 0. If we set
«(y,«/):=«-" £ aUt,y)(-tj^ {tvY-.K^iX^^tC'-^iX*),

214
we have a{y,rj) G S"(fi x Rq;JCs^{XA),JCs-^-»{XA)) and A = Op(a).
Recall that a(y, 77) is classical when a,ja(t,y) is independent of £. According
to the notation in Section 3.1 the operator A has two principal symbols,
the homogeneous principal interior symbol of order //, denoted by <r^(A),
and the homogeneous principal edge symbol of order \i which is of the form
rt(A)(y,r,)=t-" £ *ia(0,v)(-£)'(tr,)°. (3.14)
j+M<m V 7
The homogeneity means a^(A){y,\rj) = A/iKA<J^(A)(y, tj)k^1 for all A e
i2+. The operators (3.14) are of Fuchs type on XA for every fixed (y, 7/).
Hence we have from the cone operator calculus a principal conormal symbol
derived from (3.14), namely
a»Mo»(A)(y,z) = £ajO(0,y)^ : Hs(X) -> ff5^(X),
i=o
5 G iJ. Here the dependence on 7/ disappears. Of course, there is also a
subordinate homogeneous principal interior symbol of (T^(A) of order \i in
the cone sense, namely (a!1(T%(A))(t,x,y,T,£) that we obtain from (Tji(A)
by inserting 77 = 0 and freezing the coefficients in t = 0, where the weight
factor £-/i remains untouched. This allows us to apply various results from
Section 2 in the present case, here in a corresponding y-dependent form.
Let us write
a%iA) (*» *> y» r> 6*?) = *"mP(m)(*i x> ?/' *r> &tf/)
for a corresponding P(M)(t, a;,y, T,£,fj) that is C°° in £ up to £ = 0 and
homogeneous of order fi in (f ,£, 7/) ^ 0.
Theorem 3.22 Lei A 6e an edge-degenerate differential operator on
XAxQ, and assume thatP(M)(0, x, y,r,£, 77) 7^ 0/or allx,y and (f,£, 7)) / 0.
Tften /or every j/6fi iftere e:nste a countable set E(y) C R with E(y) n K
/zm'ie /or every K CC R such that
*%(A)(y, rj): £^(XA) -> /CS-^"«(XA) (3.15)
is a Fredholm operator for every 7 G iJ \ £"(y), 77 / 0, for all s G iJ.
The theorem is an analogue of Theorem 2.14, where (y, rj) are fixed.
However i2+ x X is not compact with respect to t -» 00. For the Fredholm
property of operators on XA we need the conditions of Definition 2.9 that
are satisfied here for a%(A)(y,rj). In addition there is required an "exit
ellipticity" due to the exit t -» 00, cf. [7], [34]. This is automatically satisfied

215
for 77 / 0, so we drop further details here. The countable set E(y) that
determines the exceptional weights 7 = 7(2/) comes from Corollary 2.13.
The allowed weights for the ellipticity of A in the wedge sense will be those
for which 7 ^ E(y) for all a G ft. Note that the index of the Fredholm
operator (3.15) may change when we choose another admissible weight 7.
The operator family (3.15) plays an analogous role as the homogeneous
principal boundary symbol of an elliptic operator in the context of
boundary value problems, say in a bounded domain in Euclidean space, with C°°
boundary. This corresponds to the case dimX = 0. The Fredholm
property of the boundary symbol <r%(A) as a (y, 7/)-dependent operator family,
between Sobolev spaces on the half axis (the inner normal to the
boundary) is the reason for the boundary conditions (which are of trace and
potential type in the pseudo-differential case). They complete a%(A) to a
(y, 7/)-dependent family of isomorphisms. The latter property is an
ellipticity in the sense of operator-valued symbols. In boundary value problems it
is called the Shapiro-Lopatinskii condition. Together with the interior
ellipticity this finally implies the Fredholm property of a corresponding operator
block matrix on the level of global operators between Sobolev spaces.
An analogous strategy can be applied to operators on a manifold with
edges, where the contribution to the ellipticity from the edges may be
described on the wedge XA x ft. For the ellipticity of edge symbols (the
analogues of the boundary symbols) both trace and potential conditions
are necessary at the same time, even for differential operators (in contrast
to the case of boundary value problems). The minimal number of trace and
potential conditions will be dimker a%( A) (y,rj) and d\mcokeTa%(A)(y,r)),
respectively. All this is possible under some topological condition on <r§(A).
By virtue of the role of the open set ft C Rq as local coordinates on an
edge Y which is a (/-dimensional C°° manifold we may choose a smaller
fto C ft with fto CC ft and formulate everything for y G fto- Then for
simplifying notations we speak about ft, again. In addition we may assume
ft to be simply connected, with C°° boundary. Let 7 £ E(y) for all y G ft.
Then (3.15) is a family of Fredholm operators on T*ft \ 0 that is uniquely
determined by its restriction to the cosphere bundle S*Q induced by T*ft
which is compact. There is then an index element in the K theoretic sense
ind5.s<tf (A) G K(S*Q).
The condition is now that inds^a^(A) G 7r*K(ft), where n : 5*ft -» ft
is the canonical projection. This will be assumed whenever we talk about
ellipticity.

216
Now a/1(A) (y, rj) can be completed to a block matrix
Jy Jy
for vector bundles J~, J+ e Vect (Y), subscript y indicates fibres over y (in
local coordinates), where (3.16) is an isomorphism for all (y,7?) G T*fi \ 0.
The choice of <r%{T), &a{K) and ^a(Q) *s possible in such a way that when
we denote (3.16) also by <?^{A)(y,rj), we have homogeneity in the sense
-l
<{A){vM = w(*£ \)<Wm(k£ \) (3.17)
for all A G iJ+, (y,7/) G T*ft \ 0. Finally the additional entries in (3.16)
can be chosen to be the form of (3.8) for a suitable operator-valued symbol
g(y,rj) in the sense of Definition 3.11, where N± is the fibre dimension of
From (cr^(A),CT^(y4)) we can pass to an operator block matrix A =
iAij)i,j=i,2 with A = An,
A21 = OpOK(T)), A12 = Op(x°!){(K)), A22 = Op(x<tf(Q)),
where Op is the pseudo-differential action in y and x{v) an excision function
in Rq. We then also set T — A2i, K = Ai2, Q = A22 which gives us a
continuous operator
V v J H*comp(Q,J-) H£*{n,J+)
for every s e iJ, where 11*^(0,,^) are the corresponding spaces
of distributional sections of J* of Sobolev smoothness s e iJ. Now the
parametrix construction for .4 requires to invert both a^(A) and a^(A) and
to pass to a corresponding operator in reverse direction. This is possible
within the full pseudo-differential calculus for manifolds with edges as it
may be found in [7], [30], cf. also [34] for the analogue in boundary value
problems. It turns out that the inversion of a^(A) also leads to contributions
of the form (3.11) in the left upper corners, such that it is necessary to
consider such smoothing Mellin operators in the algebra from the very
beginning.
Let W be a (stretched) manifold with edges Y, cf. the notation in the
introduction. Assume for simplicity that W is compact. Then we have the
global weighted wedge Sobolev spaces
WS^(W) C #foc(int W)

217
that are locally near the edge defined by W£c/x(fi,/C5,7(XA)), and the
subspaces with asymptotics of type P (say continuous) Wp1(W) locally
being defined by Wf • )(fy£p7(XA)). The coordinate changes are
supposed to be (t, x) independent for small t. Then we have in variance and
the definitions are correct. Here and from now on we assume that dW has
a collar neighborhood V in which a global splitting of variables (£,z,y) e
[0,1) X X x Y = V is possible. In addition for every J G Vect (Y) we denote
by £TS(Y, J) the space of distributional sections in J of Sobolev smoothness
s.
An operator Q is called a global smoothing operator if it induces
continuous operators
WS^{W) W?'S(W) WS>-S(W) Wq^(W)
HS{Y,J~) tf°°(Y,J+) HS{Y,J+) J5T~(Y,J-)
for all 5 G R with (/-dependent cisymptotic types P and Q associated to
weight data (8,0) and (-7,0), respectively, 0 = (#, 0], -00 < # < 0 fixed.
The formal adjoint refers to the pairing with respect to the scalar product
in W°>°{W).
The pseudo-differential operators on W will be given near dW in terms
of the local forms on the wedges XA x ft, where Q C Rq corresponds to
local coordinates on Y.
The "algebra" of pseudo-differential operators on W with respect to
the weight data (7,7-//,©) for 7 e iJ, 0 = (-A:,0], k G N\{0}, and with
continuous asymptotics, is defined as the space of all operators
A=^A + M+Aint oy6i+g (318)
Here A is locally in XAxQ of the form Op(a) for an operator-valued symbol
a(y, 77) like (3.6), M is locally Op(ra) for m(y, 7/) like (3.10) (with continuous
asymptotics), Aint € L^(int W), £1 = Op(#), where g(y,yf,r)) is a symbol
as in Definition 3.11, also with continuous asymptotics, N± being the fibre
dimension of J*, and Q is a global smoothing operator with respect to the
weight data (7,7-//,©).
The operators of the form Gi + G will be called the Green operators in
the algebra on W.
Theorem 3.23 An operator of the form (3.18) induces continuous op-
erators
W«.t(W) W'-w-viW)
A: © -> © (3.19)
HS(Y,J~) HS~^{Y,J+)

218
and
WajP(W) Wq-^-^W)
A: e -> e
HS{Y,J~) HS-»(Y,J+)
for every s G R and arbitrary continuous asymptotic type P with some
resulting continuous asymptotic type Q.
The operators (3.18) have two principal symbols, namely <r^(*4), the
homogeneous principal interior symbol of order \i which is defined by the
left upper corner G -^(int W), and <r%{A), the homogeneous principal edge
symbol of order \i. This is an operator family
/c^(xA) /cs-^-^(xA)
<{A){y,rj): e -> e (3.20)
Jy Jy
parametrized by T*Y \ 0, with the homogeneity (3.16). The entries were
defined in Remark 3.7, (3.8) and (3.11). Thus we have a "hierarchy" of
principal symbols
WM^aM)) = (principal interior symbol, principal edge symbol).
The composition of two operators A and B of orders \i and z/, respectively
of the mentioned kind is possible when the bundles over Y and the weights
fit together. Then AB is of analogous structure, and
op"(AB) = <MK(£), of(AB) = <{A)<{B).
Definition 3.24 An operator (3.18) is called elliptic of order [i if
0) *JM) + 0 on T*(intW) \ 0 and
t"aJM)(t,x,y,r1r,^r1»/)56 0 on T*V\0,
where V = [0,1) X X x Y is a collar neighborhood of dW (in the
corresponding splitting of coordinates)
(ii) (t%(A) is an isomorphism (3.20) for some s G R and all (y, rj) G
T* Y \ 0.
Theorem 3.25 An operator (3.18) is elliptic if and only if (3.19) is a
Fredholm operator for an s = s0 G R- The ellipticity implies that (3.19)
is a Fredholm operator for all s G R. Moreover, there is a parametrix B
of analogous structure, now of order —\i and belonging to the weight data
(7 - //, 7,9), such that AB - 1 and BA - 1 are Green operators with respect
to the corresponding weight data.

219
Theorem 3.26 Let A be elliptic. Then a solution
u e w-00'7(\y)eF-00(y, J-) of Au = fe wa~™-'l(W)®H8-ti(Y,j+),
se R, belongs to Wsn{W) 0 HS(Y, J"). In particular, for
f € yvs-^-^{w) e hs-»{y, j+)
/or some continuous asymptotic type Q we obtain
ueWSjf{W)®Hs(Y,J-)
for some resulting continuous asymptotic type R.
4. Boundary value problems in the framework of the edge
calculus
As we have seen in the previous section the calculus of pseudo-differential
operators on a manifold with edges may be regarded as a generalization of
the calculus on a manifold with boundary. In the case of boundary value
problems we have certain interesting subalgebras and modifications. One
point is that the interior symbols may be the usual ones (i.e., not necessarily
edge-degenerate), with smooth dependence in the independent variables up
to the boundary. It is not evident at first glance that the corresponding
operators have some relation to the edge formalism. The following sections
will show that this is the case. Move details many be found in [34].
4.1. BOUNDARY SYMBOLS
We want to obtain operator conventions to symbols a(£, y, r, rj) e ££)(JR+ X
ft X R1^), fJ>€ R, for the "half space" iZ+ x ft, ft C Rq open.
Let us first consider a(t, y, r, rj) G Sfi(R X ft X i?1+9), and set
op(a)(y,i/)u(t) - J j'e^-^a^y^^uityt'dr,
interpreted as an operator family C™(R) -» C°°(R). By restricting the
action to C™(R+) we can form
op+(a)(y^) = r+op(a)(y^)c+: C?(R+) -+ C°°(fl+) (4.1)
with e+ : Q°(iJ+) -> C°°(R) (extension by zero) and r+ : C°°{R) -»■
C°°(R+) (restriction to R+). Then, for Op(...) = T~\y{.. )Tyi^v, we have
Op(op+(a))«(y) = J Je^-y'Kp+(a)(y,r,)v(j/)dt/dri.
Here v(y) is regarded as a Co°(.R+)-valued function of y € £2, identified
with a w(t,y) € Q°(i*+ x ft).

220
Remark 4.1 If a(t,y,r,ri) € S%(R X ft x R1+q) is a symbol with the
transmission property with respect to t = 0, fi £ Z, and if a(t,y,T,rj) is
independent of t for \t\ > const, then op+(a)(y,7/) extends to an operator
family
op+(a)(y,r,): H°(R+) -> H°-»(R+)
for s > - \, and we have
op+(a)(y,7?) € S"(« x R'-HS(R+),HS-»(R+)), (4.2)
fors > — |, where the operator-valued symbols refer to (K,\u)(t) = \2K,(\t),
AeiJ+.
A proof of this result may be found in [24], cf. also [37]. From (4.2) we
obtain an operator convention a -> Op(op+(a)) in the half space.
Many interesting symbols have not the transmission property, e.g., (1 +
|r|2 + |r/|2)2. Such symbols appear when we reduce mixed elliptic problems
to the boundary, for instance, the Zaremba problem for the Laplacian.
General operator conventions can be obtained in terms of edge-degenerate
symbols. We will often assume that the symbols are given over i2+ x ft. In
connection with the operation e+ they are to be regarded then as symbols
for t e iJ, however the subsequent restriction to i2+, cf. (4.1), makes the
operator conventions independent of the particular choice of the smooth
extension of the symbol in t e iJ.
The only specific step in operator conventions for boundary value
problems, compared with the general constructions from Section 3, is the
following observation:
Proposition 4.2 To every a(t,y,r,r]) e S%(R+ X ft X Rrp), V> € R,
there exists a 6(£,y, f, fj) € Sj(R+ X ft X R^) such that 6(£,y,r, rj) :=
b{t,y,tr,trj) satisfies
op+(a)(y, V) = r"op+ (b)(y,V) mod C00(Q,L-00(JR+;IF)),
(regarded as an operator family Cqd(R+) -> C°°(R+)).
In fact, if especially a(t, y, r, rj) satisfies a(t, y, Ar, Xrj) = AM"J'a(t, J/, r, 7/)
for |r, 7/| > cons£, A > 1, for any j € AT, then we may set
In general we obtain 6 by an asymptotic summation, using the homogeneous
components of a.
Note that the b(t,y,tr,tr)) are particular edge-degenerate symbols with
the boundary {t = 0} as the edge.

221
The version of Definition 3.2 for dim X = 0 is the following. Mq(Rq) for
\i G R is the subspace of all /i(z, 7/) G *4(C, 5^(i2-?)) such that h(z, r))\rfi G
5^(1^ X Rq) for every ft € R, uniformly in c < /? < c' for every c < c'.
Theorem 4.3 To every 6(£,y, r, 7/) in £/*e notations of Proposition 4-2
there exists an h(t, y, z, 77) G C°°(i2+ X ft, M^JR-)) such that h(t, y, z, rj) —
h(t,y,z,trj) satisfies
op+(b)(y,r)) = opf}M(h)(y,r]) mod C^L"00^; «<)),
for every (3 € R.
This is a special case of Theorem 3.4.
Corollary 4.4 To every a(t,y,r,rj) G ^(fl+xftxiJ1*9), /* € JR, there
exists an h(t, y, z, 77) G C°°(JR+ X ft, M^JR-)) sue/* tfmi /or h(t, y, z, 7/) :=
h(t,y,z,trj)
op+ (a) (y, 7/) = r^oppM{h)(y, 7?) + g(y, rj)
for every (3 G R, where op+(a)(y, rj) is regarded as an operator family
CS°(JR+) -> C°°(iJ+), andg{y,rj) G C~(ft,<S(i^L-°°(iJ+)).
Note that Op(ff) : CS°(jR+ x ft) -> C°°{R+ X ft) belongs to L-°°{R+ x
ft).
Recall that %s'7(i2+) is the weighted Mellin Sobolev space of
smoothness s G R and weight 7 G iJ on i2+. For s € N we have
W(R+) = {t*(t) G *7L2(i?+) : f^Y t*(t) G FL2{R+) for j - 0,..., 5}.
The spaces %s'7(i2+) for 5 G i£ can be defined by duality and interpolation.
The weighted Mellin transform
{M1u)(z) = M(t-^u)(z + j)
induces an isomorphism
M7 : tU2(#+) -+ L2(ri_7)
and isomophisms
M7: ^^(iJ+)^^(ri_7),
for all s € il, where Hs(ri_7) = {w € <S'(ri_7) : (Imz)su(z) € L2(ri_7)}.
Recall also that W>"i{R+) C Hf0C(R+) and'
/Cs'7(tt+) = uW<i{R+) + (1 - w)ffs(^+)

222
for any cut-off function u(t). Then /Cs/Y(i2+) is a Hilbert space, /C°'°(i2+) =
L2(R+), and (K\u)(t) = A2u(\t), A e i2+, is a strongly continuous group
of isomorphisms on /Cs/Y(i2+).
Now the wedge Sobolev spaces specialize for the set R+ x ft to
wL(y)(n,ics«(R+)), «l7en. (4.3)
and we have
#csomp(iJ+ x Q) C W?oc{y)(n,IC^(R+)) C £%,(«+ x «)•
The spaces of the type (4.3) are adequate for boundary value problems
without the transmission property.
For every pair of cut-off functions a;(i), vo{t) the operator
wr"opk(fc)0/,»?M>: £•"(«+) -> £s-^-<t(«+)
is continuous for every s e iJ. Instead of a;(i), u;o(£) we also employ o;(t[i/]),
Proposition 4.5 We have
«(*W)*"'*opJf(fc)(y,i7)Wb(t[i7]) € fif(fl x JR';/Cs^(JR+),X:s-'i^-'i(iJ+))
/or a// s £ R.
Proposition 4.6 Lei a(£, y, r, 7/) € Sfi(R^ X ft X i21+9) 6e independent
oft for t > const) and u(t),ui(t) arbitrary cut-off functions. Then
(l-«(tW))op+(a)(y,i7)(l-«1(t[i/])) € 5"(fixJR';AC^(iJ+),^-^(fl+))
/or every s € R and 7,5 6 iJ.
Now our operator convention for the half space for symbols a(£, y, r, 7/) €
SgJ(J2+ X ft X R1+q) that are independent of t for large t is
a(t,y,T,r,) -> Op(p) : Wcs0TOp(y)(ft,/C^(fl+)) -+ ^(^r^TO)
where
p(y,»/) = w(t[j/])r"opJf(/i)(yIv)wo(t[»/])
+ (1 - «(tM))op+(a)(y, »7)(1 - wi(«)
with arbitrary cut-off functions u;,u;o,u;i satisfying ujujq — u, ujuj\ = a;i, and
/i is obtained from a by Corollary 4.4. p(y, 7/) plays the role of a (complete)
boundary symbol to a(i, y, 7/, r). We shall not go into further details how to

223
specify the material of Section 4 to boundary value problems. Let us only
mention here that the concept of ellipticity including boundary symbols
applies and that we have the corresponding versions Theorem 3.25 and
Theorem 3.26.
4.2. CONORMAL SYMBOLS AND MELLIN EXPANSIONS
As noted at the beginning, the half axis i2+ may be regarded as a manifold
with conical singularities. Also here we may ask to what extent the pseudo-
differential operators op+(a) for a{t,r) e S%(R+ X R) fit into the cone
operator calculus. The answer was practically given in the preceding section
in terms of a complete boundary symbol to a, if we keep the parameters y, 77
fixed. Let us mention some further details for the special case a(r) e S^(R)
and illustrate the role of conormal symbols as it was done on the principal
symbolic level in Eskin's book [8].
We consider the operators op+(a) = r+op(a)e+ in L2(R+), where
e+ : L2(iJ+) -> L2(R) is the extension by zero, r+ : L2(R) -> L2(iJ+)
the restriction to i2+. A question is how to characterize the subalgebra of
£(L2(iJ+)) that is generated by {op+(a) : a(r) e S§(R)}. The answer is
a consequence of the following theorem and of the general results on the
cone algebra:
Theorem 4.7 Let a(r) e Sj(R). Then op+(a) can be written
op+(a) = uopM(h)u0 + (1 - u;)op+(a)(l - ui) + m + g
for arbitrary cut-off functions u(t), uo{t), vi(t)f some h(t, z) 6 C°°(R+,
Mq) and a smoothing Mellin + Green operator with discrete asymptotics,
with respect to the infinite weight integral (—oo,0].
Note that the smoothing Green operators in this case are continuous
operators
g: L2{R+)^SP{R+) with </* : L2{R+) -> 50(JR+),
for certain discrete asymptotic types P and Q (* indicates the L2(i2+)-
adjoint). The smoothing Mellin operators are asymptotic expansions of the
form
oo
m = ^uicjWoplfrihjWcjt)
i=o
for suitable hj(z) G Mr00, Rj = {(rj, nj)}jez, and jj 6 iJ, jj > 0, j+Jj >
0^7TcRj(^Ti = 0 for all j. The constants Cj in the cut-off functions a;, a)
are increcising sufficiently feist as j -> oo, and jj -> oo, j + jj -> oo as
j -> oo.

224
Corollary 4.8 The subalgebra of £(L2(iJ+)) generated by op+(a),
a(r) G S®i{R), is contained in the algebra of all operators of the form
op+(a) + m + g for arbitrary a(r) G S^(R) and smoothing Mellin -/- Green
operators as in Theorem ^.7, which is a subalgebra of the cone algebra on
R+.
Let us set
flf+(z) = (l-c-2^)-1l g-(z) = l-g+(z).
Moreover, fj(z) = 1 for j = 0, fj{z) = n{=1(k - z)~l for j > 1. Write the
asymptotic expansion of a(r) e S^(R) in the form
oo
a(r) ~ ^a^(*r)~J f°r r ~*i00)
i=o
i = x/3!. Then
a^(op+(a)(z) - {<</+(*) + a"^)}/^) for all j € N.
This expansion for j = 0 is due to Eskin [8]. For j G AT\{0} it was obtained
by Rempel, Schulze , cf. the monograph [20].
Let us finally note that there is an analogue of Theorem 4.7 to symbols
a(t,r) G Sj{R+ X i2), a(t,r) independent of t for large t. The analogue
of Corollary 4.8 is to be modified by subtracting finite-dimensional Green
operators on R+ with discrete asymptotics. Also the explicit formulcis for
the conormal symbols can be generalized to arbitrary a(t,r).
References
1. M.S. Agranovich and M.I. Vishik. Elliptic problems with parameter and parabolic
problems of general type. Uspekhi. Mat. Nauk, 19: 3, 53-161, 1964.
2. M.F. Atiyah and I.M. Singer. The index of elliptic operators. I. Ann. Math., 87:
484-530, 1968.
3. L. Boutet de Monvel. Boundary problems for pseudo-differential operators. Acta
Math., 126 (1-2): 11-51, 1971.
4. T. Buchholz and B.-W. Schulze. Anisotropic edge pseudo-differential operators with
discrete asymptotics (In preparation).
5. Ch. Dorschfeldt, U. Grieme, and B.-W. Schulze. Pseudo-differential calculus in
the Fourier-edge approach on non-compact manifolds. Preprint MP 1/96-79., Max-
Planck-Inst. fur Math., Bonn, 1996.
6. Ch. Dorschfeldt and B.-W. Schulze. Pseudo-differential operators with operator-
valued symbols in the Mellin-edge-approach. Ann. of Global Anal. Geometry, 12: 2,
135-171, 1994.
7. Ju. V. Egorov and B.-W. Schulze. Pseudo-Differential Operators, Singularities,
Applications. Birkhauser Verlag, Basel et al., 1997.
8. G. I. Eskin. Boundary Problems for Elliptic Pseudo-Differential Equations. Nauka,
Moscow, 1973. (Transl. of Math. Monographs 52, Amer. Math. Soc. Providence,
Rhode Island 1980)

225
9. B. V. Fedosov. Analytical index formulas for elliptic operators. Trudy Mosk. Mat.
Obshch., 30:159-241, 1974 (Russian).
10. B. V. Fedosov and B.-W. Schulze. On the index of elliptic operators on a cone. In:
Advances in Partial Differential Equations (Schrodinger Operators, Markov
Semigroups, Wavelet Analysis, Operator Algebras) Akademie Verlag, Berlin, 348-372,
1996.
11. B. V. Fedosov and B.-W. Schulze and N.N. Tarkhanov. On the index of elliptic
operators on a wedge (In preparation)
12. J. Gil and B.-W. Schulze and J. Seiler. Holomorphic operator-valued symbols for
edge-degenerate pseudo-differential operators, (to appear in Proceedings Conference
"Partial Differential Equations" Potsdam, 1996. Math. Research 100, Akademie
Verlag, 1997).
13. G. Grubb. Functional Calculus of Pseudo-Differential Boundary Problems. Birk-
hauser Verlag, Basel et al., 1986.
14. L. Hormander. The Analysis of Linear Partial Differential Operators. Vols. 1-4.
Springer-Verlag, New York, 1983/85.
15. V. A. Kondrat'ev. Boundary value problems for elliptic equations in domains with
conical points. Trudy Mosk. Mat. Obshch, 16, 209-292, 1967.
16. H. Kumano-go. Pseudo-Differential Operators. MIT Press, Cambridge, Mass., 1981.
17. G. Luke. Pseudo-differential operators on Hilbert bundles. J. of Diff. Equ., 12,
566-589, 1972.
18. F. Mantlik. Norm closure and extension of the symbolic calculus for the cone
algebra. Ann. of Global Anal. Geometry, 13: 4, 339-376, 1995.
19. R.B. Melrose and G.A. Mendoza. Elliptic operators of totally characteristic type.
MSRI, Preprint. 1983.
20. S. Rempel and B.-W. Schulze. Index Theory of Elliptic Boundary Problems.
Akademie-Verlag, Berlin, 1982.
21. S. Rempel and B.-W. Schulze. Parametrices and boundary symbolic calculus for
elliptic boundary problems without transmission property. Math. Nachr. 105: 45-
149, 1982.
22. S. Rempel and B.-W. Schulze. Complete Mellin symbols and the conormal asymp-
totics in boundary value problems. Proc. Journees Equ. aux Deriv. Part. Conf. No.
V, St.- Jean de Monts. 1984.
23. E. Schrohe and J. Seiler. An analytical index formula of Fedosov type for pseudo-
differential operators on non-compact wedges (In preparation).
24. E. Schrohe and B.-W. Schulze. Boundary value problems in Boutet de Monvel's
algebra for manifolds with conical singularities. I. In Advances in Partial Differential
Equations (Pseudo-Differential Calculus and Mathematical Physics), pages 97-209.
Akademie Verlag, Berlin, 1994.
25. E. Schrohe and B.-W. Schulze. Boundary value problems in Boutet de Monvel's.
algebra for manifolds with conical singularities. II. In Advances in Partial
Differential Equations (Boundary Value Problems, Schrodinger Operators, Deformation
Quantization). Akademie Verlag, Berlin.
26. E. Schrohe and B.-W. Schulze. Mellin operators in a pseudodifferential calculus
for boundary value problems on manifolds with edges. Preprint MPI/96-74., Max-
Planck-Inst. fur Math., Bonn, 1996.
27. E. Schrohe and B.-W. Schulze. Smoothing Mellin and Green symbols for boundary
value problems on manifolds with edges (To appear).
28. E. Schrohe and B.-W. Schulze. A symbol algebra for pseudo-differential boundary
value problems on manifolds with edges, (to appear in Proceedings Conference
"Partial Differential Equations" Potsdam, 1996. Math. Research 100, Akademie
Verlag, 1997).
29. B.-W. Schulze. Regularity with continuous and branching asymptotics for elliptic
operators on manifolds with edges. Integral Equ. and Operator Theory, 11: 557-602,
1988.

226
30. B.-W. Schulze. Pseudo-differential operators on manifolds with edges. In:
Symposium "Partial Differential Equation", Holzhau 1988. Teubner-Texte zur Mathe-
matik, Leipzig, 112: 259-287, 1989.
31. B.-W. Schulze. Pseudo-Differential Operators on Manifolds with Singularities.
North-Holland, Amsterdam, 1991.
32. B.-W. Schulze. The Mellin pseudo-differential calculus on manifolds with corners.
In: Symposium "Analysis in Domains and on Manifolds with Singularities", Breit-
enbrum 1990. Teubner-Texte zur Mathematik, Leipzig, 131: 208-289, 1992.
33. B.-W. Schulze. The variable discrete asymptotics of solutions of singular boundary
value problems. In: Symposium "Operator Calculus and Spectral Theory", Lam-
brecht 1991. Operator Theory: Advances and Applications Birkhauser Verlag, Basel,
271-289, 1992.
34. B.-W. Schulze. Pseudo-Differential Boundary Value Problems, Conical
Singularities, and Asymptotics. Akademie Verlag, 1994.
35. B.-W. Schulze. The variable discrete asymptotics in pseudo-differential boundary
value problems. I. In Advances in Partial Differential Equations (Pseudo-Differential
Calculus and Mathematical Physics), pages 9-96. Akademie Verlag, Berlin, 1994.
36. B.-W. Schulze. The variable discrete asymptotics in pseudo-differential
boundary value problems. II. In Advances in Partial Differential Equations (Boundary
Value Problems, Schrodinger Operators, Deformation Quantization), pages 9-96.
Akademie Verlag, Berlin, 1995.
37. B.-W. Schulze. Boundary value problems and singular pseudo-differential operators.
J. Wiley, Chichester. 1997 (To appear)
38. B.-W. Schulze and B. Sternin and V. Shatalov. Differential equations on manifolds
with singularities in classes of resurgent functions. Preprint MPI/95-88., Max-
Planck-Inst. fur Math., Bonn, 1995.
39. B.-W. Schulze and B. Sternin and V. Shatalov. On some global aspects of the
theory of partial differential equations on manifolds with singularities. Preprint
MPI/96-28., Max-Planck-Inst. fur Math., Bonn, 1996.
40. B.-W. Schulze and B. Sternin and V. Shatalov. An operator algebra on manifolds
with cups-type singularities. Preprint MPI/96-111., Max-Planck-Inst. fur Math.,
Bonn, 1996.
41. B.-W. Schulze and N.N. Tarkhanov. Green pseudodifferential operators on
manifolds with edges. Comm. Partial Differential Equations (To appear)
42. B.-W. Schulze and N.N. Tarkhanov. The index of elliptic operators on manifolds
with cusps (To appear in Proceedings Conference "Partial Differential Equations"
Potsdam, 1996. Math. Research, Akademie Verlag, 1997).
43. J. Seiler. Continuity of edge and corner pseudo-differential operators. Math. Nachr
(To appear).
44. B. Sternin and V. Shatalov. Borel-Laplace Transform and Asymptotic Theory.
CRC Press, Boca Raton, New York, London, Tokyo 1996.
45. F. Treves. Introduction to Pseudodifferential and Fourier Integral Operators. Vols.
1,2. New Jork, Plenum, 1985.
46. M.I. Vishik and G.I. Eskin. Convolution equations in a bounded region. Uspechi
Mat. Nauk, 20: 3, 89-152, 1965.

WODZICKI'S NONCOMMUTATIVE RESIDUE AND TRACES
FOR OPERATOR ALGEBRAS ON MANIFOLDS WITH CONICAI
SINGULARITIES
ELMAR SCHROHE
Max-Planck-A rbeitsgruppe
"Partielle Differentialgleichungen und Komplexe Analysis"
Universitat Potsdam
14415 Potsdam
Germany
Introduction
In 1984 M. Wodzicki found a trace on the algebra tyci(M) of all classical
pseudodifferential operators on a closed compact manifold M; he called it
the noncommutative residue. This trace vanishes on the ideal \P~°°(M) of
smoothing operators; it even is the unique trace (up to constant multiples)
on *C/(M)/*"°°(M), provided M is connected and dimM > 1.
Although it first seems a rather exotic object, this trace has found a wide
range of applications both in mathematics and in mathematical physics. In
appreciation of Wodzicki's accomplishment the name Wodzicki residue has
become generally accepted.
Also various extensions and analogs of the noncommutative residue
have been established, e.g. for certain algebras of Fourier integral operators
(Guillemin [11]), manifolds with boundary (Fedosov, Golse, Leichtnam, and
Schrohe [7, 8]), manifolds with conical singularities (Schrohe [26]), or cusp
pseudodifferential operators (Melrose and Nistor [21]).
In these four lectures I shall first give a short review of Wodzicki's
residue and some of its applications. Next I will explain the idea of B.-W.
Schulze's 'cone algebra5, a pseudodifferential calculus for manifolds with
conical singularities. For every conical singularity we shall obtain a trace
on this algebra. These traces vanish on operators supported in the interior
and are therefore different from Wodzicki's. On the other hand, there is a
natural ideal in the cone algebra having a trace which extends the classical
noncommutative residue. All these traces vanish on smoothing operators.
227
L. Rodino (ed.), Microfocal Analysis and Spectral Theory, 227-250.
(c) 1997 Kluwer Academic Publishers.

228
They are moreover seen to be the unique traces with this property on
a slightly extended version of the cone algebra. In view of the fact that
this ASI focuses on microlocal analysis and spectral theory, I shall finally
sketch Connes theorem linking Wodzicki's residue to Dixmier's trace. For
one thing this makes the noncommutative residue an important tool for
explicit computations in noncommutative geometry, see Connes [3]; it also
shows Weyl's law on the asymptotics of the eigenvalue of the Laplacian.
Lecture 1: Wodzicki's Noncommutative Residue for Pseudodiffer-
ential Operators
1.1 Definition. Let A be an algebra over C. A linear map r : A —> C is
called a trace if it vanishes on commutators, i.e., if
t(P, Q) = t{PQ -QP) = 0 for all P, Q e A.
Clearly, if r is a trace, then Xr is a trace for each A in C; moreover, the zero
map is always a trace. When we speak of a unique trace, we shall mean
that it is non-zero and the only one up to multiples.
1.2 Example. On Mr(C), the algebra of r xr matrices over C, there is a
unique trace, namely the standard one, Tr : A \-t YJj Ajj. Indeed, let Ejik
denote the matrix having a single 1 at position j, k (and zeros else). Then
the statement is immediate from the observation that [Ejik> Ekik] — Ejik for
j + k and [Ejtk, Ekij] = Ejtj - Ekfk.
In this lecture we shall be concerned with the following theorem, proven
by M. Wodzicki in 1984, as well as with several of its applications.
1.3 Theorem. Let M be closed, compact, connected, dimM > 1. Let
A — *C/(M)/*"°°(M) be the algebra of all classical pseudodifferential
operators on M modulo the ideal of the regularizing elements. Then there
is a unique trace on .4, the so-called noncommutative residue or Wodzicki
residue.
1.4 Applications, (a) As mentioned before, the noncommutative residue
plays an important role in Connes' noncommutative geometry due to Connes
observation that it coincides with Dixmier's trace on pseudodifferential
operators of order - dim M, cf. [2].
(b) As Wodzicki observed, it also is closely related to the residues of zeta
functions of elliptic pseudodifferential operators that were computed by
Seeley [30] as well as to the coefficients in heat kernel expansions.
(c) Wodzicki's trace is the multi-dimensional analog of the residue Manin
[19] and Adler [1] had found in 1978/79 in connection with their work on
algebraic aspects of Korteweg-de Vries equations in dimension one.

229
(d) Guillemin [10] had discovered the noncommutative residue
independently as an essential ingredient in his 'soft' proof of Weyl's formula on the
asymptotic distribution of eigenvalues. Under rather general axiomatic
conditions linking 'classical observables', i.e. functions p on a symplectic
manifold, with their 'quantum mechanical counterparts', namely self-adjoint
operators on a suitable Hilbert space, he showed that the counting function
Np(X) of the eigenvalues of P satisfies the relation Np(X) = cvol{p < A}
with a constant c independent of p or P.
(e) The noncommutative residue has been used in conformal field theory
in order to construct central extensions of the algebra of pseudodifFerential
symbols on the circle, cf. Khesin and Kravchenko [16].
(f) It has been applied to derive the Einstein-Hilbert action in the theory
of gravitation (Kalau and Walze [13], Kastler [15]).
We shall now go more into the details. We first recall a few facts about
pseudodifFerential operators:
1.5 Classical pseudodifFerential operators on manifolds. Let m e Z
and let a be as symbol in Hormander's class Sm = £™0(Rn x Kn). It defines
the linear operator A : S{Rn) -> S{Rn) by
Au(x) = /'eix*a(x,G)u(G)dt, u e S{Rn)
We say that A is a pseudodifFerential operator oF order m on Rn and reFer
to a as its symbol; it is uniquely determined by A, see [18]. We call a
classical iF it has an asymptotic expansion a ~ YfjLo am-j with aj e S3
homogeneous oF degree j in £ For large |£|, i.e., aj(#, A£) = \3aj(x,£) For
A > 1 and |£| > R. The ^ indicates that upon subtracting oF the first N
summands From a we obtain an element in Sm~N.
In the Following we let M be a compact maniFold oF dimension n, E a
vector bundle over M.
We say that a linear operator A : C°°(M, E) -» C°°(M, E) is classical
pseudodifFerential operator and write A e *c/(^) if? in each coordinate
neighborhood, the action oF A is given by a pseudodifFerential operator
with a classical symbol, modulo an operator with smooth integral kernel,
a so-called smoothing operator. We denote those by \P~°°(M). Note that
an operator will be smoothing whenever its symbol in in S~°° f]m Sm and
that *-°°(M) is an ideal in *C/(M). In the Following we let A = *c//*_0°.
Any smooth change oF the symbols aj on {|£| < R} modifies aj by an
element in S~°°. Over each coordinate neighborhood [/, the equivalence
clciss oF a pseudodifFerential operator oF order m in A can be thereFore be
identified with a Formal sum oF homogeneous Functions (taking values in
square matrices), £jl0am-j(z,£)> with ajix,€) € C°°(C/ x (Rn \ {0}))

230
homogeneous in £ of degree j. There are well-known rules for the behavior
of Yl aj under changes of coordinates.
1.6 Definition and Lemma. On Rn, n > 2, define the (n - l)-form
n
*(0 = £H)J+V6 A ... A d£j A ... A <*en.
i=i
The hat indicates that this differential is omitted. Let p be a smooth
function on Rn \ {0} which is homogeneous of degree -n. Euler's identity
Yl^jfyjP — ~nP implies that the form pa is closed:
d(pa) = (dp) A a + pda = -npd£i A ... d£n + pnd£i A ... d£n = 0.
The restriction of a to the unit sphere S71'1 is the surface measure.
We can now define the Wodzicki residue res A of an operator A:
1.7 Theorem. Let A G tyci{M), x e M. Suppose that in a
neighborhood U of x, the symbol A has the asymptotic expansion Ylaj Wl^ aj
homogeneous of degree j for |£| > 1. Denote by Tr the trace on C(E) and
define
resx A- ( / Tra_n(z,£)<T(£) jdxi A...Adxn.
This is a density on M. It therefore makes sense to set
res A = / resx A. (1)
JM
Then res is only depends on the equivalence class of A in A. It is a trace:
res [A, B] = 0 for all A, B e A. If M is connected, then any other trace on
A is a multiple of res.
Note. The local density a_n(z,£)<r(£) A dx\ A • • • A dxn can be patched to
a global density Qa with resA = Js*m^a: Denoting by u the canonical
symplectic form on T*M and by p the radial vector field one has
a_na A dxi A • • • A dxn = (-l)^"1)/2 —(ap J un)0,
Tit
where (.. .)o is the homogeneous component of degree 0 in an asymptotic
expansion of a p J un into homogeneous forms (J stands for the contraction
of forms with vector fields).
The proof relies on the following simple lemma. For a proof see e.g. [8].

231
1.8 Lemma, (a) Let the function p be a derivative of a smooth
homogeneous function q of degree — (n — 1) on W1 \ {0}, say p = gf-tf. Then
(b) Let p be a homogeneous function on Rn\ {0}. EacA of tie following
conditions is sufficient for p to be a sum of derivatives:
(i) degp ^ -n.
(ii) degp = -n and fspv = 0.
(iii) p = £adPq where q is a homogeneous function and \/3\ > \a\.
Proof of Theorem 1.7. Under a change of variables x tne symbol a
transforms to a symbol b with
b(y, V(y)fl ~ £ ^«(x(y),0v«(y,0, (i)
|a|>0
where the (pa{y,Q are polynomials in £ of degree < \a\/2 and </>o = 1
(see Hormander [12, (18.1.30)]. Changing the variable in the integral, and
applying first (1), then Lemma 1.8(b.iii) we get
/6-„(y,i/Mi7) = |detX,(y)|/6_B(y>Y(y)0^(0 (2)
Js Js
= |detx'(j/)|£ /(^a(x(y),0v«(y,0)-»^(0
l«l>o
= \^tx'(y)\Jsa-n(x(y)^HO-
Hence 1.7(1) is well-defined.
For the proof of the trace property we may employ the linearity of res
to confine ourselves to the case of two operators A, B with symbols a and
b supported in the same chart U. Also we may assume that we are in the
scalar case, since everything commutes under Tr. The symbol of [A, B] is
given by
£ t^(0taW-%b&°a). (3)
|or|>0 a*
We may rewrite this expression as Y^Jj-i ^ A? + dXjBj, where Aj and Bj
are bilinear expressions in a and b and their derivatives; they vanish for
x £ U. Thus, the integrals over S of (d^Aj)-na are zero by Lemma 1.8(a).
The same holds for the integrals of (dXjBj)-n over [/, since all Bj have
compact x-support in U.
To prove uniqueness, suppose r is another trace on .4, and consider an
operator A with symbol a ~ Ylaj supported in U. Let Xj and £j denote

232
any symbols with z-supports in U coinciding with xj and £j on the support
of a. The symbols of the commutators [A,opxj] and [A,op£j] then are
-D^a and DXja, respectively. Since the trace r vanishes on commutators,
it vanishes on all symbols that are derivatives with respect to either x or £.
Define a(x) = ^-5 fsa-n(x,£)a£. Applying Lemma 1.8(b) to clj for all
j z£ -n, there exist n functions bkj(x,£), k = 1,... ,n, homogeneous of
degree j + 1 in £ such that clj = ££=1 %^j- Let Mx,£) ~ Ej<m, j^-n &jy-
Then
a(x,0 - a(x)\t\-n = J2d*M*,t) + (a-n(x,t)-a(x)\tD .
Clearly, J5(a_n(z,£) - a(z)|£|~n)<r(£) = 0. So Lemma 1.8(b.ii) shows that
a_n(z,£) - a(a:)|£|~n is a finite sum of derivatives with respect to £. Hence
r(a) = r(a(x)|f |"n), soT : / ^ r(f{x)\^\-n) defines afunctional on C§°(C7)
which vanishes on derivatives. Now it is no restriction to assume that U is
diffeomorphic to an open ball. We easily deduce from Schwartz [29, II.4]
that Tf = c f f(x)dx for a suitable constant c. A priori, the constant might
depend on [/, but on the intersection of two coordinate neighborhoods the
constants must agree. If M is connected, then all are equal, and the proof
is complete. <
Note that no continuity condition is required for the uniqueness of the
noncommutative residue.
1.9 Examples and remarks, (a) Let A = (/- A)"n/2. Then a_n(z,£) =
|£|"n and res A = fMfSn-i \Z\-n<r(Qdx = volS71"1 • volM. So the volume
of M can be found as a noncommutative residue.
(b) If A is a differential operator, then res A = 0.
(c) If the order of A is < —n, then res A = 0, so res is not an extension of
the usual operator trace. In fact, as we shall see in Lecture 4, Wodzicki's
residue coincides with Dixmier's trace on pseudodifferential operators of
order -n and therefore vanishes on trace class operators.
1.10 Seeley's results on complex powers. We additionally assume A
to be invertible of order m > 0. In particular, a is elliptic, but we impose a
slightly stronger condition: There exists a ray Re = {z = re%e,r > 0} in C
with no eigenvalue of of am(x,£) on Rq for £ / 0. Then Seeley [30] showed
that the spectrum of A is discrete with only finitely many eigenvalues on
R$. Shifting 9 slightly, Re will not intersect the spectrum. Moreover:
(i) There exists a family of complex powers {^4S : s £ C}1 defined by
As = ^- [ X'iA-X^dX, Res<0;
27T Jc
As+k = AsAk Res<o,ifceN.

233
Here C is the path in C going from infinity along Re to a small circle
around 0, clockwise about the circle, and back along Rq.
(ii) As is a pseudodifferential operator of order m Re s, s *-+ As is analytic.
(iii) For Re s < -n/m, As is an integral operator with a continuous integral
kernel ks(x, y). For each xGM,mi;s(y) extends to a meromorphic
map with at most simple poles in sj = i2z:^, j = 0,1, There is no
pole in s = 0; the residue in Sj is given by an explicit formula. If A is
a differential operator, then also the residues at the positive integers
vanish.
1.11 The noncommutative residue and zeta functions. We use the
notation of 1.10. Since the spectrum {Xj} of A is discrete and As is trace
class for Res < -n/m we may define the zeta function
(a{$) = traceA~s = J^A~S, Res > n/m.
This is a holomorphic function. It coincides with JM k-s(x, x)dx hence has
a meromorphic extension to C with at most simple poles in the points Sj.
Wodzicki used Seeley's explicit formulas to show that
Resa=_iO = (27r)nres A/ord A; (1)
here ord A is the order of A; more generally
Ress=Sj (A = (27r)nres A-*/ord A. (2)
We can use this relation to define res via zeta functions: Let P be an
arbitrary pseudodifferential operator. Choose A satisfying the assumptions
of 1.10 with ord A > ord P. Then also A + wP, u G R, will meet the
requirements of 1.10, provided \u\ small, and (1) shows that
resP = —res (A + uP)\u=0 = (27r)"nord ARess=_i (a+uP-
au
1.12 Heat kernels. Starting from the assumptions in 1.10 we additionally
ask thatA is a positive operator and that the eigenvalues of the principal
symbol am lie in the right half-plane. Then one can define
e"M= I e-tX{A-X)~ldX,
where C is a suitable contour around the spectrum. The operator e~tA is
trace class, and trace e~tA = Y*%i e~Aj<- The identity
/ ts-le~xtdt = \~s / (Xty^e^diXt) = \-sT{s).
Jo Jo

234
shows that r(s)Ci(s) = f£° t8'1 trace (e~tA)dt is the Mellin transform of
trace e~tA. It is a well-known property of the Mellin transform that the
asymptotic behavior ~ t~si ln^ t near t = 0 produces a pole in Sj of order
k + 1 and vice versa. From the above results for the zeta function one
immediately deduces the asymptotic expansion near zero:
oo oo
tracee"M - J^a^A)^ + £/?*(>!)** In t
j=o k=i
Note that there is no term t° In t, since £4 regular in 0 while the Gamma
function has a simple pole; for the same reason there are no terms th In t if
A is differential
So we get res A = ord A • /?i(j4). Moreover, we can define the noncom-
mutative residue for a general pseudodifferential operator by choosing an
operator A with the above properties and ord A > ord P and letting
vesP=-ovdA—^(A + uP)\u^
Classically, A is the Laplace-Beltrami operator A associated with a Rie-
mannian metric on M, so that one really deals with the heat equation. It
is well-known that the coefficients clj(A) carry important geometric
information, see e.g. Gilkey [9].
1.13 Notes and Remarks. The original reference for Wodzicki's residue
is [32]; a much more elaborate presentation was given in [33]. Kassel's paper
[14] gives a good survey. The proof of Theorem 1.7 here follows [7].
In Theorem 1.7 we asked for simplicity that n > 2. For n — \ the
cosphere bundle has two components. A simpler version of the above
arguments then shows that one gets two residues when restricting to orientation
preserving changes of coordinates otherwise one residue as before.
Lecture 2: The Cone Algebra
In this lecture we shall review the cone calculus for manifolds with conical
singularities introduced by B.-W. Schulze. In the next lecture we shall deal
with noncommutative residues for these objects.
Following the general idea of noncommutative geometry, the
information about the underlying space is encoded in a suitable algebra of linear
operators. From the analysis of the classical case presented in Section 1,
we know that Wodzicki's residue recovers the geometric invariants detected
by the heat kernel expansion methods. One might therefore hope that a
similar result holds for the singular case.

235
In this context the choice of the operator algebra is very important.
Consider for example a manifold M with boundary. One possible operator
algebra is, of course, the algebra of classical pseudodifferential operators on
the open interior. Yet it is not difficult to see from the proof of Theorem
1.7 that there is no trace on this algebra.
On a manifold with boundary, it seems more natural to consider
boundary value problems. The canonical analog of the algebra of
pseudodifferential operators then is Boutet de Monvel's algebra. As it turns out we then
get the desired result [7, 8]:
2.1 Theorem. There is a trace on the algebra, Bc\ of classical elements in
Boutet de MonveVs calculus on M. It extends WodzickVs residue, va,nishes
on the ideal B~°°(M) of smoothing elements, and it is the unique trace
on the quotient algebra, Bci(M)/B~°°(M), provided M is connected and
dim M > 1.
We now introduce the basic elements of Schulze's cone calculus.
2.2 Manifolds with conical singularities. A manifold with conical
singularities, 5, is a second countable Hausdorff space which is, outside a finite
number of points v G 5, a smooth manifold.
In a neighborhood of each of the so-called singularities or singular points
u, the manifold is diffeomorphic to a cone X x [0, oo)/X x {0}, whose cross-
section, X, is a closed compact manifold.
In the following we shall confine ourselves to the case of one singularity v.
We blow up at v and obtain a manifold with boundary, and a neighborhood
of the boundary can be identified with the collar X x [0,1). We denote the
resulting object by B, while XA is the cylinder XA = X x K+.
2.3 Idea of the calculus. Apart from the technical complications the
basic concept is the following :
- On the smooth part of B use the pseudifferential calculus in its
standard form.
- Near singularities use Mellin calculus on X x R+ working with smooth
families of meromorphicMellin symbols taking values in the algebra of
pseudodifferential operators on X.
2.4 Mellin transform. For u G Co°(K+) we define the Mellin transform
Mu by
/•oo
(Mu)(z)= / tz-lu(t)dt, zeC
Jo
This furnishes an entire function which is rapidly decreasing along each line
Tp = {z G C : Re 2 = /?}. Plancherel's theorem for the Fourier transform
shows that M extends to an isomorphism L2(R+) -> L2(r1/2). The identity

236
(Mti)|r1/2_7(^) = Mt^z(t 1u){z + ^) motivates the following definition of
the weighted Mellin transform:
Mggu(z) = M(r1u){z + j).
The inverse of M7 is given by
For v = -tdtu one has Mv(z) = zMu(z) in particular -tdtu = M~lzMu.
2.5 Cut-off functions. Whenever we speak of a cut-off function or use the
notation a;,cD,a;i,a;2, • • • withour further specification we mean a function
w G Co°(R+) with a;(i) = 1 near £ = 0. We will also speak of cut-off
functions on IB, asking that they vanish on the part of IB not identified
with the collar.
2.6 Mellin Sobolev spaces. For s G N, 7 G R the Mellin Sobolev space
fts'-?(XA) is the set of all u G X>'(XA) for which tnl2-^{tdt)kDu{x,t) G
L2(XA) whenever k < s and D is a differential operator of order < s - k on
X. Interpolation and duality furnish %S'7(XA) for all 5,7 G R. Note that
duality is with respect to the pairing
2
and that U°>n/2{XA) = L2(XA).
These spaces make sense on IB, too: We pick a cut-off function u on IB
and let US^{B) = {u : uu e 7^(XA), (1 - u)u G tffoc(int IB)}.
2.7 Mellin Symbols and Mellin Operators. Let fi G Z,7 G R. By
LM(-X";R) denote the space of parameter-dependent pseudodifferential
operators of order /i on I with parameter space R. LM(-X", 1^/2-7) is the
corresponding space with 1^/2-7 identified with R.
Given / G C00(R+,L/i(X;r1/2_7)) define the Mellin operator with
(Mellin) symbol / and weight 7 by
W/]«(«) = ^ / rz/(t, *)[iM(z)<k
Z7H Jr1/2_7
for u G Qj°(XA) = C^(fi+,C°°(X)).Itiseasy to see that op^/ : Q°(XA) ->
C°°(XA) is continuous. Moreover,
"1 [°Pm/]^2 : fts'7+n/2 (XA) -> Us-™+n/2 {XA)
is bounded for all 5.

237
We shall now turn to the analysis of asymptotics.
2.8 Example. Let u be a cut-off function.
(a) Write M{u) = z-lM(-tdtu)(z). Since tdtu G Co°(R+)> we obtain
a meromorphic function with a single simple pole in z = 0; it is
rapidly decreasing along each Tp, uniformly for (3 in compact
intervals, including (3 = 0, provided we remove a neighborhood of z = 0
(by multiplication with a function which vanishes there and is 1 near
infinity).
(b) Let Rep < 1/2, ke N. Then M(t^lnktu(t))(z) = £c(Mu)(z-p).
This again is a meromorphic function with a single pole in z = p of
order k +1, it also is rapidly decreasing along each T^, uniformly for
(3 in compact intervals provided we remove a neighborhood of the
pole itself.
2.9 Asymptotic types and Mellin Sobolev spaces with
asymptotics. Fix 7 G R. Recall that a weight datum g is a triple g = (7 +
n/2,7 + n/2, (-1,0]) consisting of two reals and an interval.
(a) An asymptotic type associated with g is a finite set P = {(pj, raj, Cj) :
j = 1,... , J} with J G N (possibly J = 0, then P is the empty
set), pj G C with -1/2 - 7 < Repj < 1/2 - 7, raj G N, and Cj
finite-dimensional subspaces of C°°(X). We denote by ttqP the set
{Pj : j = l,... ,</}•
(b) A Mellin asymptotic type is a sequence P = {(pj,rrij,Lj) : j G Z}
with pj G C, Repj -> =foo as j —>• ±00, raj G N, and Lj finite
dimensional subspaces of finite rank operators in L~°°(X). As before
we write nc = {pj}.
(c) Given an asymptotic type P, and 5,7 G R we let %p7 (219) be
the space of all u G %s'7+n/2(2E?) for which there exist Cj^ G Cj,j =
1,... , J, A; = 0,... , raj, such that, for all e > 0,
J mj
t* - £ £ Cj*r^ lnktu(t) G ns^n/2+1~£{B)
2.10 Meromorphic Mellin symbols.
(a) Mq(X) is the space of all entire functions h : C -» L^(X) such that
/fclr^ € LM(X; T^) uniformly for /? in compact intervals.
(b) Let P be a Mellin type. Mp(X) is the space of all holomorphic h :
C \ 7rcP -> LM(X) with the following properties:
(i) In a neighborhood of pj we have h(z) = ^So "jfcO^Pj)"*"1 +
ho(z) with i/jjk G Lj and /iq analytic near pj]

238
(ii) for each interval [ci,C2] we find elements Vjk in Lj such that
rrij
h((3 + ir) - E E *j*Mt_»(r« In* tu(t)){P + ir) € L"(X, KT)
{i:Repi€[ci|C2]}A:=0
uniformly for (3 e [cb c2]. We set Mp°°(X) - fW£(X) •
2.11 Theorem. Af£(X) = Af£(X) + Mp°°(X) as a non-direct sum of
Frechet spaces.
With these notions at hand we are ready to define the full algebra. Fix
H and 7 and recall that g is the weight datum (7 + n/2,7 + n/2, (—1,0]).
2.12 The residual elements: Green operators. Cq{B ,g) is the space
of all operators G : Co°(intlB) -> X>'(int2E?) with continuous extensions
G:7Ts'7+n/2(2B) -> 7^Q;7+n/2(iB)and
G*:Us^-n/2(B) -> ?/J2'~7~n/2(lB)
for suitable asymptotic types Qi, Q2 and all 5. Here, G* is the adjoint with
respect to the pairing 7/s'7,7/~s'~7.
Note: Hg;7+n/2 <-> nN"+nl2 {B) is compact for each AT, hence CG(iB, g)
consists of compact operators.
2.13 An ideal: The algebra Cm+g(B ,g). Cm+g(B ,g) is the space
of all operators i? : Co°(int 2B) -» £>'(int2E?) that can be written
R = ui[oplIh]u2 + G,
where
(i) ho G Mp°°(X) for some Mellin asymptotic type Po,
(ii) ttcPo n r1/2_7 = 0,
(iii) o>i,o;2 cut-off functions, and
(iv) GeCG(B,g)
Note: These operators form an algebra which turns out to be an ideal in
the final algebra. The Green operators form an ideal in Cm+g{B , g). A
change in the choice of the cut-off functions results in a Green operator.
2.14 The full algebra. C^(B ,g) is the space of all operators
AM + A^ + R,
where Am = uiop1Mhu2, with h G C°°(K+,M£(X)), is a Mellin operator
supported close to the singularity, A^ is a pseudodifferential operator of
order \i supported in the interior and R G Cm+g{B,g).
Note: Cfi(B, g) is a Frechet space with the natural topology.

239
2.15 Theorem. The composition of operators yields a continuous map
C(B, g) x &{B, g) -+ C^\JB, g).
We Aave tie ideai structure:
CG(lBg)<CM+G(lB^)<C^lB,g).
2.16 Mellin quantization. For h € C°°(I+,M^(X)) there is a p €
C^R+.Z"^!*)) such that
oppsop^/imodi-00^).
Here C°°(R+, LM(X, R)) denotes the space of totally characteristic symbols
(also Fuchs type symbols), i.e. the elements of C^R+Z/^XjR)) that can
be written p(i,r) = q(t,tr) for some a € C00(R+L"(.X';R)). The symbol p
has the asymptotic expansion
OO -. jif. .A
^^)~E]M^W,-«"n*,0'-)-7rJ}l^
with T(t,t?) = ln*:[^,. Note that T^t') = t.
2.17 Symbols. To an operator in the cone algebra we can therefore
associate to important symbols, namely
(i) the interior pseudodifferential symbol which is in fact defined up to
the boundary with a totally characteristic degeneracy, and
(ii) the operator family {h(Q,z) + h0(z) : H*(X) -> HS-^{X) : z G
ri/2_7}, the so-called conormal symbol.
The conormal symbol plays an important role in the Fredholm theory on
manifolds with conical singularities. The Fredholm property for an operator
is equivalent to the invertibility of the interior principal symbol and the
invertibility of the conormal symbol on 1^/2-7•
2.18 Notes and Remarks. This is a simplified and comprehensive version
of the cone calculus. I used the material in the joint work [23, 24]. Other
good sources are Egorov and Schulze [6] and Schulze [28].
Lecture 3: Noncommutative Residues on Manifolds with Conical
Singularities
We start with a negative result:

240
3.1 Example. Wodzicki's residue does not extend to cone algebra. In
order to see this recall first that B is (n + l)-dimensional. Suppose h G
C°°(I+, Mjn_1(X)), and h vanishes for t > 1. For 7 = 1/2 we consider the
1/2
operator op^ h. According to 2.16 we can find a pseudodifferential symbol:
1/2
op^ h = opp modL-°°(XA) with p_n_i(x,£,£,r) = fc(t)_n_i(a?,£,-ttr).
In order to distinguish it from the densities we shall analyze below, we now
write W-res for the Wodzicki density introduced in Theorem 1.7. We then
have
W-res(a.|t)opp
= ( I p-n-i (x, t, £, r)a(£, r) J eta eft
= ( / fc(t)-n-i (&, f, -ttr)a(£, r) J eta dr
= ( / j / M*)-n-i(»ifi-^)^^(0)da:dr
= t"1 f / / fe(t)-n-i(&, f, -i$)d$ <r(£) J dx dr\
here <r(£, r) is the n form corresponding to the n - 1-form a used in Section
1. For the third equality we have used that the integrand is a closed form,
hence we can shift the contour.
In order to compute the noncommutative residue we would have to
integrate the density over the collar X x[0,1). This, however, is not possible,
unless /i(£)_n_i vanishes for t = 0.
We shall now define a different density:
3.2 Definition. Let A be as in 2.14. Near x € X let h(Q)(x, £, ir) be the
local symbol of/i(0,ir). The subscript -n-1 in the notation /i(0)_n-i(£,£, ir),
below, indicates the term of homogeneity -n-1 with respect to (£, r).
Define
res,
,A=( / Tr/i(0)_n_i(x,£,ir)dra(^)) dx\ A ...Adxn.
Since the operators may take values in a vector bundle £?, we also introduced
a trace Tr on C(E) in the integral above. For n — 1 replace integration over
Sn~l by h(0)-2(x, X>iT) + M°)-2(*, -1, ir).
3.3 Remark.
(a) The decomposition h + ho is not unique, but ho is of order -00 and
therefore gives no contribution.

241
(b) resxA = (jsn Tr/i(0)_n_i(£,£, ir)a(^r))dxiA.. ,Adxn in view of the
fact that h(0)-n-i(x,£,iT)a(£,r) is a closed form of degree -n - 1.
3.4 Lemma. res^A defines a density on X.
Proof. We fixed t as a global coordinate. So changes of coordinates are of
the form (x, t) \-t (x(x),t). Hence Lemma follows as in the standard case. 0
3.5 Definition. For A G C»(B,g) let
resA= / resxA = / / / Tr/i(0)_n_i(z,£, n-)dr<r(£)GtaiA.. .Adxn.
Jx Jx Js"-1 J-oo
May write
/oo
/i(0)_n_i(-, -,ir)dT
-oo
with Wodzicki's residue of the (-n)-homogeneous /^)O/i(0)_n_idr.
3.6 Example. Let A be the Laplacian for a two-dimensional manifold
with a conical singularity. Close to the singularity a computation shows
that t2A = c2dl + {tdt)2 has the Mellin symbol g(t,x,£,z) = -c2|£|2 +
t2. Her c is a suitable constant depending on the opening angle of the
cone. A parametrix A to t2A therefore has the Mellin symbol (-c2|£|2 +
z2)~l modulo lower order terms. This is the desired component of order -2,
'integration' over 5,*51 gives 2(-c2 + z2)'1. Thus
res
A = -s I r (c2 + T2)-1drdx = -47r2/c.
JS1 J-oo
We shall now produce an extension of Wodzicki's residue. On the collar we
consider the algebra of all Mellin operators with vanishing Mellin symbol
in t = 0.
3.7 Operators on the collar. Consider the operators in the cone algebra
that can be written in the form
A = u;i[op]f/i]u;2 + JS
with
(i) heC™(R+,MZ(X))MO) = o,
(ii) u>i, u>2 cut-off functions with uj\UJ2 = wi,
(iii) R€CM+G{lB,g).

242
For convenience cissume that suppu; C [0,1). What is important is that we
may choose h(0) = 0 and that U1U2 — u\. The latter condition normalizes
the representation in a certain sense. For (x,t) e X x (0,1) define
ves°x,tA = ( / n_i / a>i(t)M*)-n-i(&, £, ir)dra(m dxx A ... A dxn A y.
3.8 Lemma.
res° fA / Trwi(t)fc(*)-n-i(&,& ir)a(£,r)<ki A ... A —.
It is a density on X x K+.
Proo/. The asserted identity follows from the fact that h(t)-n-\ (x, £, ir)(j{^ r)
is closed. That it is a density can then be proven as before. <
3.9 Theorem. For A as above let
res°A = /'/ res^A.
Jo Jx
This makes sense in view of the condition h(0) — 0. Moreover, res0 is a
trace on operators of this form.
Proof. This can be shown just like in Theorem 1.7. <
3.10 Lemma. Let p be the totally characteristic pseudodifferential symbol
associated with A modL-°°(X x R+), cf 2.16.
p(t,r) ~ «!(*) f) ^,DkT{h(t, -iT{t,t')r)^ft}\t,=t
with T(t, t') = (t- t')/(ln t - In if). Then
res0 A = W - res p.
Proof.
I Trp(t).n.1{x,tr)dxAdt
JSZr
t'=t J
i,r VfcO "" / -n-1
= f Tr fc(t)-n-i (*, £, ~^>(£, r)dx A <
J5"

243
for the integrand can be rewritten as DlT .. .r7 with j < 1/2. Then
W-resp = / / / Trh(t)-n-i(x,£,-itT)a(Z,T)dxAdt
Jo JxJs^r
= 111 I Trh(t)-n-i(x,Z,-itT)dT<r(£)dxAdt
JO JX JS1}'1 J-oo
= res0 A
3.11 The extended cone algebra. On B choose a smooth function
t which coincides with the geodesic distance to the boundary near the
boundary of B and which is strictly positive in the interior. Let C(B, g)+
be the space of finite sums of operators of the form tmB with Rem > 0
and B e C(B,g). Similarly define CM+G{B,g)+.
C{B,%)t = (tmB:BeC(B,g),Rem>0)
CM+G(iB,g)J - (M:fieCM+G(iB,g),Rem>0)
Why is this an algebra? For the Green operators multiplication by t is
no problem, neither is it for the pseudodifferential operators in the interior.
For the Mellin operators we use the following computation. Note that we
may assume t = t, since we are close to the boundary.
opWB«)W = 2^jfri9 J0 (t/tTzHt,z)(t'mu(t'))vdz
fm /» /»oo sift
= 5s/r„,J «/<')-<M<,C-m)«M)^C
= r[opj,T-»*]«(t).
There is a minor problem if h is a meromorphic smoothing Mellin symbol
and T~mh has a singularity on 1^/2-7. Then we use the fact that we may
write
tmotfMh = tmov«h + G
with 7 - ji < 70 < 7 and a Green operator G.
With this notation we can state the theorem on uniqueness. A full proof is
given in [26].

244
3.12 Theorem.
(a) For each conical point we get precisely one continuous trace on the
quotient C{B, z)+/CM+G{IB, g)+.
(b) On C(J59,g)o*/CM+G(^9?g)J there is a unique trace, namely the
extension of Wodzicki's residue.
We understand (a) in the sense that near this conical point the operator
algebra is the cone algebra, i.e. there is not a fictitious conical point.
Lecture 4: The Noncommutative Residue and Dixmier's Trace
Dixmier's paper [4] settled a longstanding question: Is every completely
additive trace proportional to the standard operator trace on the set where
it is finite? Dixmier showed that the answer is 'no' by explicitly constructing
counter-examples. We start this section by reviewing his result, following
Connes [3] in presentation and terminology.
4.1 The spaces C^°°)(H) and C^90o)(H). Let H be an (infinite-
dimensional) Hilbert space, T e /C(ff), and \T\ = {T*T)ll2. Let /*0(T) >
Mi(r) > ... be the sequence of eigenvalues of |T|, repeated according to
their multiplicity. It is well-known that
Hj(T) = inf{||T - F|| : rankF = j} = min{||T|£x|| : dimE = j}. (1)
We define <rN(T) = EjLo/^CO and let C^'°°\H) = {T € K{H) :
<tn(T) = 0(logN)}, endowed with the norm
N>2 lOgiV
We have a natural subspace C^^H) = {T € K{H): aN{T) = o(logiV)}.
4.2 Lemma. Let aw be as in 4.1, and let T, T\, and T^ be compact.
(a) aN(T) = max{||TPE||i : dim E = N}, with the £x-norm || • || and PE
denoting the projection on E.
(b) an(T) = max{trace(Tfk) : dim E = N} for T>0.
(c) ajv(ri + T2) < ^(Ti) + aN{T2).
(d) <rN{T!) + aN{T2) < a2N{Ti + T2) ifTi,T2 > 0.
(e) £i1,oo), £(1-0°) are two-sided ideals in C(H).
Proof. For (a) use 4.1(1); the maximum is attained by choosing E the
eigenspace with respect to the first iV eigenvalues of \T\. (a) implies (b);
the maximum is attained for the same E. (c) is immediate from (b). Since

245
the dimension is subadditive one gets (d). Finally (e) is a consequence of
the estimate fij(TA) < fij(T)\\A\\ valid for bounded A. <
4.3 Cesaro Mean. We define the Cesaro mean Mf for / e L°°(R+) by
The function Mf is continuous and bounded; M : L°°(R+) -¥ Cb(l,oo) is
continuous. Moreover, Ml = 1 and M(/(A-)) - Mf £ Cj(0)(l,oo). Here,
A G C, and the subscript 0 indicates that the function vanishes at infinity.
4.4 The 'limit' limw. Let ThT2 € £(1,oo) be positive and
aN = Io^aT' ** = Io^aT' 1n = logiv •
Then {o^}, {Av}, and {jn} are bounded sequences. By 4.2 we have
-Yn<<*n + Pn< (log 2AT/log AT) 72iv, (1)
but in general no convergence. We embed £°° into L°°(K+) in the canonical
way by cissociating to the sequence {a^} the function f{aN} which has the
value a,j on the interval [j - 1, j[, j = 1,2, Next we choose a linear form
u on C&(1, oo) with (i) u > 0, (ii) w(l) = 1, and (iii) u(f) = 0 for / G C6(0).
Then we define lim^a^} = u;(M/{a}) with the help of Cesaro's mean.
Note that lim^ coincides with the usual limit on convergent sequences
by (ii) and (iii).
4.5 Dixmiers trace. For a positive operator T 6 £(1,0°) let
Tr-(T) = lmioiiVn?/"(r)-
As Proposition 4.6(a) shows, Trw is additive. We can therefore extend it
uniquely to a linear map on £(1'°°), also denoted Tr^.
4.6 Proposition. Let T,ThT2 € C^°°)(H),S € C{H)
(a) Tvw(Ti + T2) = Trw(Ti) + Tr^ (T2) for positive Ti, T2.
(b) Trw(r) >0 if T>0.
(c) If S is invertible, then Trw(5T5_1) = Trw(T). In particuiar, Tr^ is
independent of tie inner product in H.
(d) Trw(S:r)=Trw(rS).
(e) Trw = 0 on £j1,co), so it vanishes on trace class operators.

246
Proof, (a) follows from 4.4(1) together with property (iii) of u. We only
have to check (c) for positive T. Then use 4.2(b). Finally (c) implies (d). <i
4.7 Example. Consider the operator (1 - A)_n/2 : L2(Tn) -» L2(Tn),
where A is the Laplacian. The eigenvalues of A are known to be the lengths
\\k\\2 as k varies over Zn, so the eigenvalues of (1- A)"n/2 are (1- p||2)~n/2.
Let us show that (1 - A)"n/2 e C1^ and Tr„(l - A)"n/2 = fin/n,
independent of u. Here, Qn = vol(5'n"1): We let Nr denote the number
of lattice points in Br, the ball of radius R. Clearly, Nr ~ vol Br, hence
In Nr ~ n In R. Moreover,
£(l + |*|)n/2 ~ SlnjR{l + r2)-n'2rn-ldr
fR
Jo
\k\<R
We conclude that
(in^rix;(i+i*i)^~^ = v-
mn nlnR n
Recall that for (1 - A)"n/2 : L2(Tn) -» I2(Tn) we had computed in 1.9
that
res(l - A)"n/2 = vol5n"1volTn = ftn(27r)n.
In one special case we therefore have proven the following result:
4.8 Theorem (Connes 1988). Let M be closed compact, n-dimensional,
E a vector bundle over M; P : L2(M, E) -» L2(M, £?) a pseudodifferential
operator of order —n. Then
(a) PeC^°°\L2{M,E))
(b) resP = (27r)nnTrCt;P, independent of a;.
Proof We start with the observation that both res and Tr^ are local: If
{<pi,... , <pj} is a partition of unity on M and if {ifti,... , ipj} are smooth
functions with (pjtpj = <fj, then
res P = ]P res (fjPipj and Tr^P = ]P TTu(pjPipj,
since res<^jP(l - ^-) = res(l - iftj)(pjP) — 0, similarly for Tr^. Thus we
may assume that M — Tn.

247
Part (a) now follows from writing P = (P(l - A)n/2)(1 - A)"n/2: The
first factor on the right hand side is bounded, (1 - A)"n/2 G £1,0°, and
£lj0° is an ideal.
In order to see (b) we first note that we proved it for (1 - A)~n/2, see
4.7. By linearity it is enough to consider T = P + A(l - A)_n/2 for P > 0
and large positive A. In that case, T : L2(M, E) -» Hn(M, E) is invertible,
and T = A'1 for a pseudodifferential operator A of order n satisfying the
assumptions of Seeley's theorem. By Wodzicki's formula 1.11(2),
resT res A"1 n n ^
,0 ,n = ,0 xn i , = Ress=iO = -Ress=_iCr = Ress=i ^ A ,
n(27r)n (27r)nordA ~ J
where Ao > Ai > ... are the eigenvalues of T, so the A"1 are the eigenvalues
of A.
Let Ao > ...Afc0_i > 1 > Xk0 > Aa:0+i, denote by 0 the characteristic
function of R+, and define fi(x) = Y^kLo @{x + l°g^HA:o)- This is a positive
measure. Its Laplace transform is
70 k=k0 A:=0
According to Seeley's result, this function is analytic for Res > 1 and
extends to {Re s < 1 - s} with a simple pole in s = 1. We can
therefore apply Ikehara's Tauberian theorem [5, Section 47] and conclude that
Ress:=iC4(s) = lime~xfj,(x) =: c.
NOW fl(x) = E{*:s>-logAfc+fc0} 1 = E{ib:c-*<Afc+fc0} 1 SO that KX) = J iff
Afco+j+i < e~x < \k0+j- From this we derive that jAfc0+j+i < fi(x)e~x <
jAfc0+j, hence Afc0+j ~ c/j with above c. We conclude that T G £(1,0°) and
Trw(T) = limjv-»oo &N/\ogN = c. Note that the limit exists and therefore
is independent of a;. <1
4.9 Corollary: WeyPs theorem. Let M be closed, compact, n-dimensional,
let A be the Laplace-Beltrami operator on M with respect to some Rie-
mannian metric. For the eigenvalues Xj of -A we then get the asymptotics
2/n / A \ 2/n
^(^rG9
Proof. We deduce this from the last part of the proof of the previous
theorem rather than from the assertion. Consider (1 - A)~n/2. Its inverse A
satisfies the assumptions of Seeley; (a is analytic on {Res > 1} and extends

248
to a larger half-plane with a simple pole. We know from 1.9 that res (/ -
A)"n/2 = ftnvolM, hence Tr„(/ - A)"n/2 = (27r)-nres(/- A)-n/2/n =
(27r)~nfinvolM/n =: cn. An application of Ikehara's Tauberian theorem
as above implies that the eigenvalues fij of (/ - A)~n/2 satisfy fij ~ cn/j.
— 2/n
The identity Xj = /j • ' then proves the result. <
4.10 Notes and Remarks. The idea of the proof of Theorem 4.8 was
adapted from Varilly and Gracia-Bondia [31].
In the article [2], Connes used the coincidence of the noncommutative
residue and Dixmier's trace in the following way:
For an algebra A, a p-summable Fredholm module (%, F) over A, and
a finite projective module E over A with an ^4-valued inner product, one
can introduce the notion of connections V and curvature 0.
He then considers the case of a 4-dimensional smooth compact Rieman-
nian Spinc manifold. The Fredholm module (%, F) consists of the Hilbert
space % of L2-spinors and F = D\D\~l, where D is the Dirac operator.
Under a compatibility assumption he can show that the (abstractly defined)
curvature 0 is an element of £2'°° so that the value of the Dixmier trace
Tr^fl2) = 1(0) defines a positive functional independent of u.
Moreover, given a classical connection A, the classical Yang-Mills action
YM(A) of A can be recovered by
YM(A) = 167r2inf/(0)
with the infimum taken over a suitable class of connections related to A.
References
1. M. Adler. On a trace functional for formal pseudo-differential operators and
the symplectic structure of Korteweg-de Vries type equations. Inventiones
Math., 50:219-248, 1979.
2. A. Connes. The action functional in non-commutative geometry. Comm.
Math. Physics, 117:673-683, 1988.
3. A. Connes. Noncommutative Geometry. Academic Press, New York, London,
Tokyo, 1994.
4. J. Dixmier. Existence de traces non normales. C.R. Acad. Sc. Paris, Serie A,
262:1107-1108, 1966.
5. W. Donoghue. Distributions and Fourier Transforms. Academic Press, New
York and London 1969.
6. Yu. Egorov and B.-W. Schulze. Pseudo-Differential Operators, Singularities,
Applications. Birkhauser, Boston, Basel, Berlin 1997.
7. B.V. Fedosov, F. Golse, E. Leichtnam, and E. Schrohe. Le residu non com-
mutatif pour les varietes a bord. C.R. Acad. Sc. Paris, Serie /, 320:669-674,
1995.
8. B.V. Fedosov, F. Golse, E. Leichtnam, and E. Schrohe. The noncommutative
residue for manifolds with boundary. J. Functional Analysis (to appear).

249
9. P. Gilkey. Invariance Theory, the Heat Equation and the Atiyah-Singer Index
Theorem. CRC Press, Boca Raton 1995
10. V. Guillemin. A new proof of WeyPs formula on the asymptotic distribution
of eigenvalues. Advances Math., 55:131 - 160, 1985.
11. V. Guillemin. Residue traces for cerain algebras of Fourier integral operators.
J. Functional Analysis, 115:391-417, 1993.
12. L. Hormander. The Analysis of Linear Partial Differential Operators
III. Grundlehren der mathematischen Wissenschaften 274, Springer Verlag,
Berlin, Heidelberg, 1985.
13. W. Kalau and M. Walze. Gravity, non-commutative geometry and the Wodz-
icki residue, J. Geometry Physics 16:327-344, 1995.
14. C. Kassel. Le residu non commutatif [d'apres M. Wodzicki]. Asterisque,
177-178:199-229, 1989. Seminaire Bourbaki, 41eme annee, Expose no. 708,
1988-89.
15. D. Kastler. The Dirac operator and gravitation. Comm. Math. Phys. 166:633-
643, 1995.
16. B.A. Khesin and O.S. Kravchenko. A central extension of the algebra of
pseudodifferential symbols. Functional Analysis and Appl., 25:152-154, 1991.
17. M. Kontsevich and V.I. Vishik. Determinants of elliptic pseudo-differential
operators. GAFA, 1995.
18. H. Kumano-go. Pseudo-Differential Operators. The MIT Press, Cambridge,
MA, and London 1981.
19. Yu.I. Manin. Algebraic aspects of nonlinear differential equations. J. Sov.
Math., 11:1-122, 1979.
20. R. Melrose. The Atiyah-Patodi-Singer Index Theorem. A K Peters, Wellesley,
MA 1993.
21. R. Melrose and V. Nistor. Homology of pseudodifferential operators I.
Manifolds with boundary. Preprint, MIT 1996.
22. B.A. Plamenevskij. Algebras of Pseudodifferential Operators (Russ.). Nauka,
Moscow 1986.
23. E. Schrohe and B.-W. Schulze. Boundary value problems in Boutet de Mon-
vePs algebra for manifolds with conical singularities I. In: Pseudodifferential
Operators and Mathematical Physics. Advances in Partial Differential
Equations 1. Akademie Verlag, Berlin, 1994, 97 - 209.
24. E. Schrohe and B.-W. Schulze. Boundary value problems in Boutet de Mon-
vel's algebra for manifolds with conical singularities II. Boundary Value
Problems, Deformation Quantization, Schrodinger Operators. Advances in Partial
Differential Equations 2. Akademie Verlag, Berlin 1995, 70 - 205.
25. E. Schrohe. Traces on the cone algebra with asymptotics. Actes des Journees
de Saint Jean de Monts, Journees Equations aux Derivees Partielles 1996.
Ecole Poly technique, Palaiseau 1996.
26. E. Schrohe. Noncommutative Residues and Manifolds with Conical
Singularities. J. Functional Analysis (to appear)
27. B.-W. Schulze. Pseudo-Differential Operators on Manifolds with Singularities.
North-Holland, Amsterdam 1991.
28. B.-W. Schulze. Pseudo-Differential Boundary Value Problems, Conical
Singularities and Asymptotics. Akademie Verlag, Berlin 1994.
29. L. Schwartz. Theorie des Distributions. Hermann, Paris 1966.
30. R. Seeley. Complex powers of an elliptic operator. Poc. Symp. Pure Math.
10:288-307, 1967.
31. J.C. Varilly and J.M. Gracia-Bondia. Connes' noncommutative geometry and
the Standard model. J. Geometry Physics 12:223-301, 1993.

250
32. M. Wodzicki. Spectral Asymmetry and Noncommutative Residue. Thesis,
Stekhlov Institute of Mathematics, Moscow, 1984.
33. M. Wodzicki. Noncommutative residue, Chapter I. Fundamentals. In Yu. I.
Manin, editor, K-theory, Arithmetic and Geometry, volume 1289 of Springer
LN Math., pages 320-399. Springer, Berlin, Heidelberg, New York, 1987.

LOWER BOUNDS FOR PSEUDODIFFERENTIAL
OPERATORS
CESARE PARENTT, ALBERTO PARMEGGIANI**
* Dipartimento di Informatica, **Dipartimento di Matematica
Universitd di Bologna
Piazza di Porta S.Donato 5, 40127 Bologna, ITALIA
1. Introduction and setting of the problem.
We start off by fixing some notation (see Sjostrand [6]). Let X be an open
subset of Rn (more generally, X can be a C°° n-dimensional manifold
without boundary) and let E C T*(X) \ 0 ~ X x (Rn \ {0}) be a C°° conic sub-
manifold. With \i G R and h G Z+ = {0,1,2,...}, we denote by N^h(X, E)
the set of all classical symbols of order //, p(x,£) ~ Ylj>oPv>-j(xiQi such
that for any j > 0 one has
where t+ := max{£, 0} and dists(£,£) denotes the distance of (#,£/|£|) to
{(y,7/) G E; |t/| = 1}. OPNM'*(X,E) will then denote the corresponding
class of (properly-supported) pseudodifferential operators.
Recall that the notation / < #, stands for: for any conic subset U ofT*(X)\
0 with compact base, there exists a constant Cjj > 0, for which f(x,£) <
Cug{x,Z),V(x,Z)6U.
We say that p (or the corresponding operator P) is transversally
elliptic (with respect to E) iff
\Wdistz(x,Qh<\Pll(x,Q\.
It is useful to bear in mind that
A G OPN"'*(X,E), B G OPNM''*'(X,E) => AB G OPN^'^'p^E),
A G OPN^(X,E) => A* G OPN^(X,E).
There are two basic objects related to an operator P G OPNM,/l(X, E).
251
L. Rodino (ed.), Microlocal Analysis and Spectral Theory, 251-262.
© 1997 Kluwer Academic Publishers.

252
1. The localized polynomial
Let P e OPN/i'/l(X,S) with principal symbol p^ let p e £ and v =
(va?j^) € TP(T*X). One defines the localized polynomial of P at p e E
by:
|a|+|0|=fc "^
Notice that for h = 0, pp(v) = Pn(p)- Furthermore, for h >1, pis transver-
sally elliptic iff for every p € S one has
p» ^ 0, Vu € AT,E = TP(T*X)/TP^ u ^ 0.
The polynomial pp has an invariant meaning (see Sjostrand [6]).
Examples.
For h = 1 :
pp(v) = (dp^(p), v) = a(v, HPll(p)v),
where a = £ • d^- A dxj is the canonical symplectic form of T*X and
Hp (p) = (d^p^p), —dxp^p)) is the Hamiltonian vector-field associated to
JV
For h = 2 :
^(u) = 2(Hess MpK u> = CTK F(P)U)>
where Hess pM(/o) is the Hessian-matrix of pM at p, and
F(p) = ijHess^(/0), </=(_°/n 7q ),
is the fundamental matrix of p^ at p.
2. The localized differential operator
The second main object is a differential operator with polynomial
coefficients attached at p 6 E, for every P € OPN^(X,E). More precisely,
define
Pp(v, Dy) := £ ^(^^V,)(P)2/^?? y € Rn
|a|+|0|+2j=/i tt^'
(D^ = dy/y/^1). It is well known that the operator Pp has an
"invariant meaning" (see Sjostrand [6], Boutet-Grigis-Helffer [1]). Notice that the
localized polynomial pp is the "principal symbol" of Pp.
We can now state the problem we are dealing with.

253
Let A = A* e OPN2m'2A:(X, £). Find necessary and/or sufficient conditions
on A (and E) in order to have the following lower bound: for every compact
K C X, there exists Ck > 0 such that
(Au,u) > -Ctf|<_Mi, V« € Co°°(X), (1)
where (, ) denotes the L2-scalar product and | • \\s (s e R) is the usual
Hs-Sobolev norm.
Some remarks are in order to justify the interest in studying the above
inequality.
1. The validity of inequality (1) depends only on the first k + 1 terms
«2m? «2m-i? • • • > ^2m-k of the symbol of A. In other words, inequality (1) is
stable under perturbations of A with operators in OPS2m"^+1^(X). In this
sense the exponent m - ^- is sharp.
2. We point out that possible applications are about the maximal hypoel-
lipticity, the well-posedness of the Cauchy problem for weakly hyperbolic
operators, just to mention a few of them (see, e.g., Helffer-Nourrigat [2],
Hormander [3], Parmeggiani [5]).
2. Necessary conditions.
We have the following necessary conditions
Proposition 2.1 Suppose that (1) holds. Then:
(i) a2m(x,Q>0,V{x,Z)eT*(X)\0;
when k > 1, for any p e £ :
(«) (AP(y, Dy)v, t;) > 0, Vt; € C0°°(Rn).
Proof. We will sketch only the proof of («'). Let E 3 p — (zo,£o), l£o| = 1,
and fix any compact neighborhood K of xq. Let K' CC K be such that
supp (Au) cK'ifue C$>(K). Take x € C$>{X) with x = 1 near K U K\
so that Au = xMxu) = a(x,D)t for u 6 C$>(K). For v e Cg°(Rn) and
t > 1, put
ut(z) = e,t2<^°>t;(i(z-a:o)).
For £ large, we have ut € Co°(.K) and
^='~V(*°'t%~^(f-^o).
Hence, as £ —>■ +00, we have
|t«*iL*±i = t4(m-'*l)-n(M8+ «(!))•

254
On the other hand,
Aut(x) = eit2^<f>Mx-xo)),
with
M*) = Wn j e^aixo + x/t, tri.+ t%)v(ri)dr,.
Taylor expanding then yields
0(^+7, tiH-^o)= '4(m_f)£ -^(flsrafo^^-j^xV+o^-**1)).
\a\+\f3\+2j=2k a-P-
Hence, by putting t(x - xq) = y, we get
(Aut, ut) = i4("H^n(Ap(y, A>,«) + 0(t^m-^-n) >
>-CKt*(m-^-n(\\v\l + o(l)).
Upon dividing by t4(m~k^2)~n, and letting £ —» +00, we obtain (ii). #
Remark 2.2 Conditions (i) and (ii) in the above proposition, being only
pointwise, are clearly holding true without supposing that £ is a C°°-
manifold.
Sufficiency is in general not true under the sole condition (tt). (From
now on we will take k > 1.)
In the double characteristics case (i.e. k = 1), the situation has been
greatly clarified by Hormander [3], who proved the following theorem:
Theorem 2.3 (Hormander [3]) Let A = A* e OPN2m'2(X,E), and
suppose:
(a) A is transversally elliptic,
(b) ranka is constant (i.e. dimfT^Sn (TpT,)a) is constant as p varies in
the connected components o/S).
Then the lower bound (1) (with k — 1) holds iff
(hi) a2m>0 onT*(X)\0;
(h2) sub(A)(p) + Tv+FA(p) > 0, Vp € E,
where sub(A) = a2m-i + i(dx, ^)a2m/2 is the subprincipal symbol of A,
FA is the fundamental matrix of a2m andTr+FA(p) = EM>0;t>€spec(FA(p)) ^
is the positive trace of FA(p).
Using Weyl-calculus, it can be easily shown that condition (h2) is equivalent
to the non-negativity of the localized operator Ap(y,Dy).
We now briefly recall the proof of sufficiency in Hormander's theorem, since
we shall need it later on (for the details, see [3]).

255
By a standard argument, we can suppose A e OPN2'2(X, S). At any fixed
p G E, we can find a symplectic basis in TP(T*X) so that, with v having
coordinates (z,£),
d d+l
ap{v) = a{v,FA(p)v) = Y,fi3{x2J+$)+ £ x\ (//, >0,j = 1,.. .,</).
The integers d, / are uniquely determined (because of the assumptions (a)
and (&)) by
2d + / = codim S, 2n - 2(d + /) = rank a\ .
Unfortunately, we cannot expect the above normal form to hold smoothly
in p varying in S. However, we have the following (smooth) fiber-bundle
decomposition of the normal bundle to S :
TVS = Im(FJ) 0 (Ker(FJ)/Ts).
We write (as smooth fiber bundles on S) :
C®(lm(FJ))=V0V,
with
V(p)= © Ker(F^(p)-^(p)), peS.
/i(p)>0
We can choose smooth vector-fields on £ :
r t>;0>)ef(p), i = i,...,rf
<
( t;^(p)eKer(F^(p)2)/TpS, * = 1,...,/,
so that for any p e E :
[ a(t;i(p),F>i(p)t;ib(p))=2*iib, j,*=l,...,d
<
( ^(^+i(p),F^(p)t;d+^(p)) =8ik, j,*=l,...,Z.
Now, using Morse Lemma, one can find, in a conic neighborhood of a point
p G E, 2d + / smooth real-valued functions /i,..., /bd+jj homogeneous of
degree 1 in £, with independent differentials, in such a way that
2d+l

256
Moreover, upon defining
xj - hj-i + V-1/2J, 3 - 1, • • •, d
Xj+d = /Wj, J = 1j • • • > f>
we have
#x» - FA{p)vj{p)i j = 1,..., d+ /.
As a consequence (microlocally),
^ " EXi(x> ^^i^. D) = Be OPS1.
i=i
Furthermore, the principal symbol of 5 is given by
h{p) = mh{A){p) - -±=Yja{HX}{p), HXj(p)), p € S.
To compute the Poisson brackets at a fixed p, we use the above normal
form of Fa (p). Define
Wj =
(ej = (0,..., 1,..., 0) G Rd). Since Fa{p)wj = y/^lfijWj, we get
Hence, fj(p) = ZjUi UjkWk, J — 1,.. .,d, for some unitary d xd matrix
t/ = (Ujk)j,k- Therefore,
d d
Yl*{Hx.ip),HXiip)) = ^(F^)^,^)^)) =
i=i i=i
d d
= E E UjkUjhHkP'h<T(wk,Wh) =
3=1 k,h=l
d d
= -2VC1E E UjkUjkfik = -2V=iTr+FA(p).
3=1k=l

257
In conclusion,
bl{p)=aub(A)(p) + Tr+FA(p),
which, by hypothesis, is non-negative on E. By a suitable Oth-order
modification of Xj(x,D), we can suppose &i > 0 in a conic neighborhood of
p e S. The Sharp Garding Estimate finally yields, (microlocally)
d+l
[An, u) = Y, \xAl + (Bu,u) > (Bu,u) > -C\u\l-
i=i
One can then patch together the above microlocal estimates . #
From now on, we will suppose A = A* G OPN2m'2*(X,E), transversally
elliptic with respect to S, and ranka ^constant. As far as we know, for
k > 2, no result as complete as the above Hormander's theorem is available.
In particular, we do not know how to get the lower bound (1), even from a
strengthened form of the necessary condition, namely
Vp € E : (A,(y, Dy)v, v) > 0, Vu e CS°(Rn), v / 0.
Note that the above condition implies (see Boutet-Grigis-Helffer [1]) the
microlocal hypoellipticity with loss of k derivatives for A. As the following
example shows, we need some extra-information on the "spectrum" of the
localized operator Ap in order to get the lower bound.
3. An example.
Suppose 2ra = 2k = 4, and that the localized polynomial ap(v) is the
product of two non-negative quadratic forms:
for which
and
ap(v) = a(v, Fi(p)v) a(v, F2{p)v),
KerFi(p) = KerF2(p) = TPE, Vp e E,
[F1(p),F2(p)] = 0, VpeS.
Hence, without loss of generality, we can suppose
A = PQ + R,
with P = P*,Q = Q* £ OPN2'2(X,E), p2,q2 > 0 on T*(X) \ 0 and
transversally elliptic with respect to E, Fi = Fp, F2 = Fq, and J? 6
OPN3'2(X, E). Since A = A*, we get
i?-i?* = [Q,P], and FR-R. = -^=[FP,FQ] = 0.

258
Therefore R - R* e OPN3'3(X,E) and, in particular, FR is real. We next
suppose
[FR(p),FP(p)} = [FR(p),FQ{p)] = 0, Vp € S.
Our last assumption will be
sub(P)| = sub(Q)| =0,
E
which is no retriction since it can be readily achieved by suitably modifying
JS, leaving all the commutation relations unchanged.
A standard argument shows that, for any fixed p G S, in suitable symplectic
coordinates, one can write
d d+l
a(v,FP(p)v) = Y,H(p)(x2j+ej)+ £ xl
j=l k=d+l
d d+l
°(v,FQ(p)v) = Y/\j(p)(x2j+ej)+ £ ak(p)zl,
j=l k=d+l
d d+l
o(v,FR(p)v)=j:uJj(p)(x2J+eJ)+ e fowl
j=l k=d+l
with //j, Aj, ak > 0, and o>j, /?* € R.
Using Weyl-calculus, it is not difficult to show that the necessary condition
(Ap(y,Dy)v,v) > 0, for any p 6 S, for any t; € Co°(Rn), is equivalent to
the following algebraic requirement:
F,
MCy) := ((//,C)+(yJ)+Tr+Fp(p))((A,C)+(a,y)+Tr+FQ(p))+(u;,0
(2)
+(/?, y) + E^j + Re sub(i?)(/>) > 0,
Vp € E, VC € (2Z+)d, Vy € R<+, where </i,C> = E"=i^0, etc., 1 =
(1,1,...,1)€R'+.
We now microlocally work near any fixed point of S. By using the
symbols Xj, j = 1,..., d + I constructed before, we can write
d+l
p = £xix; + 0PSl'
i=i
d+l
Q= £ XjAjkXh + OPS\
j,k=l

259
where Ajk = A\- G OPS0, and the matrix (ao(Ajk))j,k is positive. Let
{Cjk)j,k be a positive square root of (Ajk)j,k, and define
d+l
Yj = Y^ CjkXk, J = 1,..., d + /,
so that Q = £^+[ Y/%- + OPS1. We now define
d+l
J=l
It turns out that B G OPN2'2(X, S) is transversally elliptic and in some
coordinates as above
d+l
a{v, FB(p)v) = £ ^j(p)H(p)(*2j+t]) + £ V^)x*'
j=l k=d+l
so that 5 looks like a "square root" of PQ. More precisely, let us compute
d+l d+l
A:=(E*;*;)(En*n)-irfl.
It is not difficult to show that
d
i = L + £ {x;[Xi9YC]Yk - X*[Y3,YZ]Xk) + OPN3>3(X,S),
where L = L* G OPN4'4(X, E) and (Lu, u) > 0, for any u G C§°.
Thanks to the hypothesis [Fp, Fq] = 0, one can show that
d
£ (x*[X3,YZ]Yk- X*[Y3,YZ]Xk) GOPN3'3(X,S).
In conclusion,
A = PQ + fi = L + 5*5 + ]?,
with L > 0 as above, £ G OPN3'2(X, E) and
Fn = FR+ (Tr+FP)FQ + (Tr+ FQ)FP,
sub(i?) = sub(i2) - (Tr+Fp)(Tr+FQ),

260
on E. We now disregard L and work with B*B + R only. To get a lower
bound for B*B+R, we use a deformation argument analogous to Hormander's
one (see [3]). Namely, we look for C = C* G OPS1 such that, writing
5*5 + R = (B - C)*{B - C) + R + B*C + CB - C2,
and observing that (B-C)*(B-C) > 0, we can apply Hormander's theorem
to the operator
Ac := R + B*C + CB-C2 e OPN3'2(X, £).
Of course, we need only construct c = (J\(C) on S. To this purpose, we
need (on S) the following:
a(v, FAcv) = £ ("j + 2C)/A^ + (Tr+FP)Xj + (Tt+FQ)H) {x) + £?)+
(3)
+ J2 {Pk + 2<V«* + (Tr+Fp)afc + Tr+ FQ)4 > 0, Vt, € iV,E \ {0},
k=d+l
and
d /
Re sub(J4c)+Tr+Fic = Re sub(fl)-2c]T ^/Ai/xj-(Tr+FP)(Tr+FQ)-c2+
i=i
(4)
d
+ ^(Wi + 2c^A^ + (Tr+Fp)A,- + (Tt+Fq)^-) > 0.
3=1
It is now convenient to introduce the following quantity:
mar; ( h{p) + (Tr+Fp{p))ak{p) + Tr+F^ ()1
i<*<A ^VMF) '' -"'
Observe that the function S 3 p 1-4 7(p) > 0 is (in general) just continuous.
Moreover, define
d
J(p) = Re sub(R)(p) + £>,(,>) + (Tr+Fp(/>))(Tr+FQ(/,)) - j(p)2.
i=i

261
It can be readily seen that (with obvious notations)
Fp((,y) = J(p)+
+(«(p) + (Tr+FP(p))\(p) + (Tv+FQ(p))tx(p) + 2rr(p)y/\(p)n(p), C)+
+(P(P) + (Tr+FP(p))a(p) + (Tv+FQ(p))l + 2j(p)^[p), y)+
+(7(p) - (y/x(PHp),0 - (^U))2+
CV ,/ HP)\ /C
y
\ a(p) I \y I \ v<*(p) J \ v i
By virtue of the Cauchy-Schwarz inequality and the definition of 7, one has
If we have that J(p) > 0, Vp G E, then we can obviously construct a
smooth symbol c(p) > y(p) for which conditions (3) and (4) (in strict
form) are satisfied. To get J(p) > 0 we require:
(HI) F,(C,y)>0, VC€(2Z+)d, Vy€RV,
(necessary condition (2) in strict form) and
(H2) min F,(C,y)= min F,(C,y).
(C,2/)G(2Z+)^xR!f (C,y)xR$xRV
Let us clarify the meaning of condition (i?2) above. There are two cases in
which (if 2) is a trivial consequence of (if 1), namely when either j(p) = 0,
in which case J(p) = F/9(0,0), or
0 < T(p) = fiklp) + (Tr+Fp(/)))afc(p) + Tt+FqQo)^
Va*(p)
for some A; € {1,..., /}, in which case
J(p) = Fp(C = 0>y=-^=e*).
The troublesome case is when
0 < l[p) = "i(P) + (Tr+FpQ^A^) + (TY+Fqfr))^
2\/xj(p)»j{p)

262
for some j € {1,..., d}, and
w > mM ^M + ITr+FH^ + Tr+F^K
If this case occurs, the subset of R+ x{0} C R+ xR+ where Fp attains its
minimum J(p) may have empty intersection with (2Z+)d x{0}, as it can
be seen by trivial examples showing that J(p) < 0, while condition (if 1) is
still satisfied.
Hence condition (if 2) is highly non-trivial exactly in this case. Notice that
it can be spelt out as a "lattice" relation among the Uj, Xj and //j's.
In conclusion, we have the sought for lower bound for the operator A
considered above under conditions (if 1) and (if 2).
Remark 3.1 1) We are still unable to fill in the gap between the necessary
condition
FP(C,y)>0, (Pe^Xe(2Z+)d,yeRl+)
and the sufficient conditions (if 1) and (if 2).
2) There are at least two kinds of objections to the example we have
considered here.
First of all, the heavy requirements on the product form of the localized
polynomial and the commutativity of the involved fundamental matrices.
All this was somehow forced by the need of very precise informations on the
"spectrum" of Ap(y,Dy), required by our approach.
Secondly, in our proof of sufficiency, we completely threw away the
"possible" contributions of the J^th-order non-negative terms L and (B — C)*(B —
C). However, it is not clear (al least to us) how to take advantage of them.
References
1. L.Boutet de Monvel-A.Grigis-B.Helffer. Parametrixes D'Operateurs Pseudo-
Differentiels a Characteristiques Multiples. Asterisque 34-35, 1976.
2. B.Helffer and J.Nourrigat. Hypoellipticite Maximale pour des Operateurs Polynomes
de Champs de Vecteurs. Birkhauser, 1985.
3. L.Hormander. The Cauchy Problem for Differential Equations with Double
Characteristics. Journal D'Analyse Mathematique, Vol.32, 1977.
4. C.Parenti and A.Parmeggiani. A Necessary and Sufficient Condition fo a Lower
Bound for 4th-Order Pseudodifferential Operators. To appear in Journal D'Analyse
Mathematique.
5. A.Parmeggiani. An Application of the Almost-Positivity of a Class of 4th-Order
Pseudodifferential Operators. Preprint (1995).
6. J.Sjostrand. Parametrices for Pseudodifferential Operators with Multiple
Characteristics. Arkiv for Matematik 12, 1974.

WEYL FORMULA FOR GLOBALLY HYPOELLIPTIC
OPERATORS IN Rn
ERNESTO BUZANO
Dipartimento di Matematica, Universitd di Torino
Via Carlo Alberto 10, 10123 Torino, Italy
1. Introduction
It is well-known that the spectrum of the harmonic oscillator in Rn:
is given by a sequence of eigenvalues
n
Xa = £(2^ + 1), a€Nn,
3=1
(N = {0,1,...}), to each one of which there corresponds a single eigenfunc-
tion so that the eigenvalues are given by 2k + n, k € N. Each 2k + n is
semi-simple with multiplicity (n~i| ).
The asymptotic behavior of Xa as \a\ -> oo, i.e. of the eigenvalues 2k+n
repeated according to their multiplicities, can be easily deduced from the
one of the counting function:
N(X) = £ 1.
AQ<A
Because Xp < Xa whenever (3 < a (i.e. /3{ < oi{ for i = 1,..., n), we have
that N(X) i,s the number of points of Zn with odd positive co-ordinates
which belong to the n-simplex of side A. This means that N(X) is
asymptotically equivalent to the volume of the n-simplex of side A/2. Thus we
obtain
»W = MC + e(A., . _g_ j [A _ „,,,]<> „ + o(An(, (1)
263
L. Rodino (ed.), Microlocal Analysis and Spectral Theory, 263-306.
© 1997 Kluwer Academic Publishers.

264
as A -> oo, where an is the area of the unit sphere in Rn and [A - ||z||2] ,
is the positive part of A - \\x\\2.
In this paper all the asymptotic formulas concerning A are always for
A -» +00. From now on we shall omit sistematically the sentence "as A -»
+00". Moreover, in order to simplify the notation we employ the following
conventions. Given two functions /, g : Rn -» R, we say that
f{x) -< g{x)
if there exists a positive constant C such that
f(x) < Cg(x), for all x.
Furthermore, we write
R>0
to mean that J? is a positive, conveniently large constant. Thus by
f(x) <g{x), for||x||>fl>0,
we mean that we can choose R large enough so that there exists a positive
constant C such that
f(x)<Cg(x), for all ||z|| > it!.
Of course, one should pay attention to the order of the logical quantifiers.
For example, if we write that for all a G Nn we have
|0a/(aOH NrH, for ||*|| >fl»0,
we mean that we can choose R large enough so that for each a € Nn there
exists a positive constant Ca such that
\daf(x)\<Ca\\x\\-M, for ||a;|| > R.
Equation (1) is a special case of the Weyl formula, we now explain. Let
us consider the Schrodinger operator with real potential W(x) in Rn:
-A + W(x) (2)
and cissume that W(x) —> 00 as \\x\\ -> 00. Under these conditions the
spectrum of (2) consist of a sequence of real semi-simple eigenvalues Aj
diverging to +00. Let us cissume that the sequence {Xj} is arranged in

265
increcising order and that the eigenvalues are repeated according to their
multiplicity. Then we can define the counting function
N(X) = £ 1.
The Weyl formula for the counting function of the operator (2) is
N(\) = V(\)(l + 0(\-<)), (3)
where 6 is a suitable positive constant and
is called Weyl term.
The first result about (3) we want to mention is due to Tulovskii and
Shubin [14] in 1973, which proved the Weyl formula with e < 1/2 for
potentials W which are elliptic polynomials of second order, i.e. such that
||x||2 -< W(x) -< ||x||2, forW>fl>0.
This result has been improved by Hormander [8] in 1979, which obtained
6 < 2/3, and by Helffer and Robert [6] in 1981, which obtained the optimal
error estimate € = 1. The Weyl formula has been extended by Tamura [12]
to elliptic potentials of order ra > 1 and by Helffer and Robert [7] and
by Mohamed [10] to quasi-elliptic potentials, which satisfy some further
hypothesis. A smooth function W is quasi-elliptic if there exist n positive
constants mi,..., mn and 0 < p < mj1, j = 1,..., n such that for each
a G Nn we have
t i-pM
^(*H(i+£i^r
\ 3=1
and
w(x) yi + J2\xi\mj> for Nl ^ R> °-
Under the assumption that W has a principal part, i.e. there exists
W0(x)= lim t-1w(tmi1x1,...,tm"1xns),
such that for a suitable e > 0 we have
\\W(x)-Wo(x)\\^[l + J2\*j\m^

266
they obtained the Weyl formula
N(X) = V(A) (1 + C?(A-£)) = V0Xp (1 + C?(A-£)),
where
1 1 n
p = _ + ...+ _ +
In the next two sections we want to see how to extend the Weyl formula to
more general hypoelliptic potentials.
2. Newton polyhedra and hypoelliptic polynomials
Let us first consider a polynomial potential
W(x) = £ caxa
and let cissociate with W its Newton polyhedron, i.e. the convex hull Q of
{0} U A. The Newton polyhedron Q is contained in1 (Rq")71 and it is the
convex hull of a finite subset V(Q) C Nn of convex-linearly independent
points called the vertices of Q and univocally determined by Q.
Moreover there exists a finite set
AT(Q) = Ar0(Q)uiV1(Q)cRn,
such that
|M| - 1, for all v € N0{Q)
and
Q = {x e Rn : v • x > 0, Vi/ G A^o(Q)} n {x e Rn : i/ • x < 1, Vi/ G A^i(Q)} .
No(Q) and A^i(Q) are univocally determined by Q and the boundary of Q
is made up of faces Fu which are the convex hull of the vertices of Q lying
on the hyper plane orthogonal to v G N(Q) and of equation
i/-x = 0, if^eAT0(Q),
i/-a: = 1, if i/ G iVi(Q).
The following definition is due to Volevic and Gindikin:
xWe adopt the following notations:
R+ = {r € R : r > 0} , R+ = {r € R : r > 0} .

267
Definition 1 A convex polyhedron Q is complete if
1. V{Q) C Nn,
2. 0 € V(Q) ± {0},
3. N0(Q) = {e1,...,en}>
4- Nt(Q) c (R+)n,
where
e3; = (0,..., 0,1,0,..., 0), with 1 in j-position.
Given a complete polyhedron Q, for each a e Nn we have the estimate
(see [2], Ch. 1, Lemma 8.1)
\xa\-< AQ(x)k^a\ (4)
with
k(Q, a) = min (i£R+: t~la 6QU max v • a. (5)
V I J i/€Wi(C)
Newton polyhedra of quasi-elliptic polynomials are simplexes, in
particular they are complete. The following proposition shows that the class of
polynomials with complete Newton polyhedron is very wide:
Proposition 1 The Newton polyhedron of a hypoelliptic polynomial is
complete.
Proof: See [2], Ch. 1, Thm. 1.1 D
With a complete polyhedron Q we can associate a weight function
AQ(x) ■■
We have the estimates:
<*>"•
where
(X):
as standard, and
\aeV(Q) )
X Ae(x) -< (x)^ ,
= (i + IWIa)1/a,
llo = min lal, u,i = max lal.
aev(Q)\o] r aev(Q)
Moreover, for each a e Nn we have
#Ae(*) -< AeOc)1-'"!/^ (6)

268
where
/i = max{ — : j- 1,.. .,n, v 6 iVx(Q)
"i
is the formal order of Q.
To prove the estimate (6), we observe that if /3 < a € Q C\ Nn, we have
dpxa = fi\(a\xa-p
and
a - $ e Q.
Therefore
" •(<*-£)< 1 - —, for all v £ NUQ).
fj,
Hence (see (5))
*(e,«)<i-^,
thus from (4) we obtain for each a G Q fl Nn and j9 < a
|0V|^Afl(s)1-'0i//\
In general, we can prove the following
Proposition 2 IfW is a real-valued, bounded from below hypoelliptic
polynomial, with Newton polyhedron Q of weight Kq, then there exist
I < 1, -1/fi < r < 0,
such that for each a 6 Nn we have
\daW{x)\ -< W{x)AQ{x)T\a[ (7)
AQ(x)1 -< W(x) -< Ae(s), (8)
for ||s||>fl>0. D
If / = 1 we say that W is multi-quasi-elliptic or Q-elliptic . Multi-
quasi-elliptic polynomials have a long history in the theory of linear partial
differential equations. They have been introduced by Volevic and Gindikin
[15] and have been extensively studied by several authors among which
Friberg [5] and Cattabriga [4].
Of course it may well happen that I < 1: for example
-2A.2n
W(x, y) = [x2n~l - y2n) + x2n~2y

269
fails to be Q-elliptic along the curve y2n — x2n l. Indeed its Newton
polyhedron Q has vertices
(0,0), (0,4n), (4n-2,0),
the associated weight is
Ae(*,y) = (i + z8n-4 + 2/8n)1/2,
and we have
W{x, y) y kQ{x, y)S^, for \\x\\ + ||y|| > R > 0.
3. The Weyl formula for the Schrodinger operator with hypoel-
liptic potential
Definition 2 A complete polyhedron Q C (R^)n is non degenerate if the
intersection of its boundary with the diagonal o/(R+)n is an internal point
to a face Fv.
This means that there exists i/ 6 JVj(Q) such that
s - max < t e [0,1] : — e,- + -rre € F„, j = 1,..., n > > 0,
I "j M J
where
e = (l,...,l)GRn.
This inequality is equivalent to
1*1 ■■■Xnl^f- (\Xl\S^ +■■■+ \xn\Sl»») < AQ(x).
Now we can state the Weyl formula for non degenerate hypoelliptic
potentials. See [2] for the degenerate case in dimension 2 and [3] for fully
degenerate polynomial potentials in dimension n but without error
estimate.
Theorem 1 Consider a non degenerate, complete polyhedron Q C (R^)n
with weight Ag. Let W be a real-valued C°° potential satisfying (7) and (8),
with
/>0, —<r<-.
Assume that W = Wo + W, where

270
1. Wo is quasi-homogeneous:
W0{x){\Vlx!,...,X^Xn) = \Wo{x), forallx^O and\>0,
2. Wo is non degenerate:
\x1---xn\^t(\x1\s^ + ---+\xn\s^)^Wo(x),
for all \\x|| > #>0,
3. there exists s € [0,1) such that
Then
with
where
|W'(*)Hl + |xi|*M+---+|*„|sK
N{\) = {Vo + 0(h(X}) Al"l+n/2, (9)
v° = 7^I[1-Wo^/3^ (10)
A-£(logA)2n_2, if ex > e3 and e2> e3,
h(X) = { A-(logA)2"-1, */6i>£2 = £3) (11)
A e, otherwise,
€ < €i, € < €2l € < €3, (12)
M2 (i-s)«
€2 = 1 - S, €3 =
|i/| + n/2(l-s)S'
and
with
«i = g (^ - 5)'
--</><<$<-
/* /*
sucft f/jai /or eacft a, /? € Nn we Aave
|^(lKII2 + Wr(*))|
^(lie||2 + ^))(lH|2 + AS(,)2)MaWI)/2, (13)
/or ||*|| + |K||>fl»0.
Remark 1: Wq is called principal part ofW.

271
Remark 2: We can always take p = ^ and 5 = r+, but this choice can
be improved in several cases. For example, if W is a polynomial and
/ > 1/3 we have
1 AX1'21
p—- and o — —,
with 5 < 0 for / > 1/2.
We shall prove this theorem in Section 6.
From Theorem 1 one ecisily obtains the cisymptotic behavior of the
eigenvalues Xk of -A + W:
Corollary 1 Under the same assumption of Theorem 1 we have
A* =77
f—J (l + 0(h(k))), ask^oo.
EXAMPLES
I. Consider
W(x,y) = X>2V*J + f>^
\ v=1
+ ±sin
where /ij, fcj, /^ and k'j are non-negative integers such that
0 = hi < h2 < h3 < h4, fci > k2 > k3 > k4 = 0,
/&2 < &2> ^3 > Aft,
> t^—r- >
h4 — h3 hs — h2 h2
1 hj+i - hj , _ 1 fcj+1 - kj ,
2 fcj+i*j - hjkj+i 2 /ij+ifcj - hjkj+i
for j = 1,2,3 and 1 < / < J. Moreover
// = max
and
f_2_Mi_ 2 h4k3 \
\ k2-ki h4- h3)
-1-<!<*-.

272
\i is the formal order of the Newton polygon Q cissociated with W, i.e. the
polygon with vertices
(0,0), (0,2*0, (2/i2,2*2), (2/13,2*3), (2/14,0).
The face that intersect the diagonal has normal v with components
*3 - h h3- h2
v\ = -W7T1 rTT> "2 =
2(/i3^2 - ^2^3) ' 2(/l3*2 - fy^) '
the principal part is
W0(x} y) = x2/V*2 + x2h3y2k3.
Finally
5 = min {1 - 2h2\v\, ! ~ 2*3M) ,
5 = max J 21/2*1, 2z/i/i4, Vih'j + */2*j • J = 1> • • •></} •
Thus we have
AT(A)-(Vb + 0(A-e))A1+H?
where e satisfies (12) and
V0 = ^J[l-(x2h>y2k>+x2h*y2k*)]+dxdy
(B is the Euler Beta function).
II. The second example is a potential which is not multi-quasi-elliptic,
but still satisfies the hypotheses of the theorem.
W(x,y) =
with
^2»-l _ y2ny + a,2n-2J/2nl ^2* + y2m^ ^
, 2n ~ 1
n > 1, A; < m, m > 2n — 1.
_ 2n
The Newton polygon Q has vertices
(0,0), (0,4n + 2m), (4n-2,2m), {An - 2 + 2fc,0).
W fails to be multi-quasi-elliptic along the curve y2n = x2n_1, but it is
hypoelliptic because it is the product of two hypoelliptic polynomials.

273
On the other side Q is non degenerate because the diagonal of (R+)2
meets the boundary of Q in a point internal to the side of vertices (4n -
2,2m), (4n - 2 + 2fc, 0). Eventually we obtain
m+fc
jV(A) = v0As»*(2'»-i+»)+1 (1 + 0(\-e)),
with € > 0 satisfying (12) and
viv2B{vuu2)
V0 =
v\ =
V2 =
jr|i/|(M + l)
1
2(2m - 1 + *)'
2(2m-l + /s)m*
4. Symbolic calculus
In this section we develop a calculus for a class of symbols which contains
||£||2 + W(x), where W satisfies the estimates (7) and (8).
4.1. SYMBOL CLASSES
Given a complete polyhedron V C (Ro")2n with formal order \i and weight
i1/2
a*(*.0=| E ^
\(a,l3)eV(P)
we denote by Sft p * the class of symbols satisfying for each a, f5 € Nn the
following estimate:
with
rt(i,0UAP(x,0B-'|oWI,
-p < 5 < p < —, and m € R.
If, for each a, /? G Nn, a satisfies the estimates
|^#aOcfl| X |a(*,0|A?MrW+*l/?l
A^^.O' -< \a(x,t)\AP{x,t)m,
for ||z|| + ||£|| > i? >0 with
/ < m,
(14)
(15)

274
then a is called globally hypoelliptic. We denote by HS^'1 8 the class of
globally hypoelliptic symbols. If / = m, a is called globally multi-quasi-
elliptic or globally Q-elliptic.
4.2. PROPERTIES OF THE WEIGHT FUNCTION
Now we examinate some properties of the weight function.
1. For each m e R we have
Kv e svM» -i//i- (16)
Proof: We already know the result for m = 1 (see estimate (6)). For
m— — 1, (16) follows by induction and differentiation from the identity
A^A? = 1.
For general m (16) follows from the identity
^(Ac(»lOm) = AP(a!,Om-|fl,+/"amia+/,(aJ|0, (17)
with
a c(i-i/m)M
«m,7 t °7>,l//i,-l//i-
(17) is proven by induction. D
2. Ap is slowly varying in the sense that there exists e > 0 such that
Ap((&,0 -< A^(x + y,^ + 7/) -< Av(x,0,
for
||y||2 + ||i7||2 <«%>(*,0^.
Proof: Let Taylor expand A^(x + y, £ + i/)1^:
A^ + y^ + ^^A^O1/"
+ f\l-t) £ ify^#(Ar(* + *y,* + ti/)1>'*)<ft.
Because A^ € 5^: J*, _x, , the derivatives in the integral are bounded
and we have
lA^ + y^ + ^-ApM)1^
If we choose e > 0 small enough we obtain the result.

275
3. A-p is temperate in the following strong sense:
Av{x + y,Z+r)) <Av{x,£)Av{y,rj),
as one easily verifies.
4. We have
where
(a:,0=(l + N|2 + ||^||2)1/2,
as standard.
It is possible to define a metric in the sense of Hormander
G{Xt0(y,v) = \Ax,02iy\\2 + Mx,0-2ph\\\
so that the uncertainty principle is satisfied and our classes are the same
as Hormander's:
S?<p>s = S(g,A?).
However for didactic reasons we prefer to recall the basic properties of the
symbolic calculus in this particular setting.
4.3. ASYMPTOTIC EXPANSION
Let {rrij} be a sequence of real numbers tending to -oo and consider
a,j e S™3pS, for each j\
then for a G Sj>pS we write
if for each TV we have
3<N
The following two propositions are standard:
Proposition 3 Given a sequence a,j 6 S<p3 $ with nij -> —oo, there exists
a G Sp1 8 such that a ~ Ylaj- Moreover ifb£ S^ s is such that b ~ Ylaj>
then a-be <S(Rn). ' ' D
Proposition 4 Given a G Sj>tpt6 an^ a seQuence <*j £ ^v!p,8 w^ mj ~*
-oo assume that

276
1. for each a,/3 € Nn there exists katp such that
\d?dga(x,t)\*Av(x,t)k<>.(>,
2. there exists a sequence lj —> —oo, such that for each N we have
Then
a^J^ajU
<\vM)lN.
4.4. QUANTIZATION
The t-quantization associates with a symbol a € Sft s and a real number
t the pseudodifferential operator
Atu(x) = je^-y^a(tx + (1 - t)y,Qu(y)dyfa
where
fc=(2n)-ndS,
as standard. This procedure yields a family of continuous operators on the
Schwartz class:
At : <S(Rn) -> S(Rn),
which extend by double transposition to temperate distributions
At: <S'(Rn) -> S'(Rn).
Beside the Weyl quantization
Opw(a)u(x) = Al/2u{x) = J e^-vKa Q(x + y), ^ u(y) dy #,
we are also interested in the /e/t (or classical) quantization:
OpL(a)u(x) = A0u(x) = Jeil*-yKa(x,Z)u(y)dytf.
We say that a is the left symbol of OpL(a) and the Weyl symbol of Opw(a).
Given an operator A we denote by symbL(A) and by symb^(a) the left
and the Weyl symbol of A. We have
symbL(A) = e~ix<Aeix<.

277
The Weyl symbol aw can be obtained by the left symbol ai in terms of
the oscillatory integral
<w(z,£) = J eT%y'J]aL (x + -y,£- 77J dyfa.
By Taylor expanding with respect to 7? and integrating by parts one easily
obtains
_ 0m—(p—8)
In particular
aw ~ Q>l
Let us denote by Lpp8 and HLV' s the classes of operators with Weyl
symbol belonging respectively to Sft $ and HSV1 s.
Also the Schwartz kernel Ka of A can be computed easily from its Weyl
symbol a in terms of inverse Fourier transform:
ffA(*,V)) = ^.-, {«(£(* + »)>*)}• (18)
We say that A is regularizing if it sends <S'(Rn) into <S(Rn). One can prove
the following
Proposition 5 The following statements are equivalent:
L A is regularizing,
2. KA G <S(R£ x RJ),
Ao€5(R»xR^) = nm6R%,«-
D
We set
l~°° = n lt,p,s-
The advantage of the Weyl quantization over the left quantization is
that it makes easy to compute the symbol a* of the formal adjoint A*. By
definition
(A*u, v)L2 = {u, Av)L2, for all u,v e <S(Rn),
but
Ju{x)Jei(*-y)<a(^(x + y),Av{y)dyfcdx
ei{y-x)<a(^{x + y),t\u^

278
so
a*(x,£) = a(x,£).
Thus A is formally self-adjoint if and only if a is real-valued.
4.5. COMPOSITION FORMULA
More complicated is the composition formula of two operators. If A =
Opw(a) and B = Opw(b), with a e SftpS and b e S\> p^ we denote by
6#a the Weyl symbol of BA. After some computations we obtain
6#o(x,0
= j ei(vn-»<)b ^ + ^y,£ + <) « (* + ^,£ + ^) dyfadwfc.
By Taylor expanding and integrating by parts we obtain:
= £ (-,,w
„ a!/J!2l«+Pl
In conclusion, by computing the integrals we obtain
b#a-Ej^J^D^D>. (19)
From this asymptotic expansion we obtain in particular that
and
->rra+/-(/9-<S)
6#a - ba e 5^j/9j5
4.6. FURTHER RESULTS
Using symbolic calculus one can prove the following standard results:
Proposition 6 If m < 0 every A e L?pp8 extends to a bounded operator
in L2(Rn), wich is compact ifm<0. D

279
Proposition 7 For each A G HLV' s, there exists a parametrix, i.e. an
7
operator B G if L^ ' ^ such that BA - I and AB - I are regularizing, D
Proposition 8 If A is globally hypoelliptic, then An G <S(Rn) =^ u G
5(Rn). D
The last proposition explains in which sense the operators are globally
hypoelliptic.
5. Friedrichs symmetrization and non negative operators
5.1. FRIEDRICHS SYMMETRIZATION
The Friedrichs symmetrization is a technical tool which allows to obtain
semi-bounds for operators with semi-bounded symbols. The idea is the
following. Given a G ^v,p,8i we define a new symbol ap G Sj>jPj$ such that
Opw(a)-OpL(aF)eL^-i:r5),
and Ap = OpL(ap) is non negative if a is non negative i.e.:
a(s,O>0, V(a?,0 => {AFu,u)L2>0, Vu G 5(Rn).
Let us show how to define the Friedrichs symmetrization ap of a. Choose
a real-valued q G C°°(Rn) such that
q(&) > 0, for all a,
q(a) = 0, for|H|>l,
g(<Ti,..., -(jj,.. .,<rn) = ?(<r), for all a and j = l,...,n,
and
Let
/*(*)
2d(7=l.
T = \(P+S)
and define
F(*ffcC) = Av(x,0~nT/2q (a?(*,0-t(C - 0) •
Then we set
&fo, *> 0 = / F(x> V,Oa(x,QF(x,Z,Q #,

280
and
aF(x,£) = J e-^-yW-tibfaytfdytr,.
Following Taylor [13], Ch. VII, §2 or Kumano-go [9] Ch. 3, §4, one can
prove the following
Theorem 2 We have
«f e SptPtS
and
aF(x,t)~a(x,t)+ £ 2/^.7^.0^(^,0, (20)
where each /a,/?j7 is independent from a and
fofi = 0, if |/J| = 1,
U € 5^% if H = l,
/^ € ^i;;:^;'^, *y m>i,
^- rtTtt —(p—£)
a-aFeSVtP*s \
and
7<Qf
Y.UfiA*1** e S3?"iT-')|o+/,|/2, /or|a + /3|>l.
7<a
5.2. NON NEGATIVE OPERATORS
Now we apply the Friedrichs symmetrization to the non negativity of
operators.
Theorem 3 If a is real-valued, then AF = OpL(aF) is formally
self-adjoint. If a > 0 then AF > 0, i.e.
(AFu, u)L2 > 0, for all u € «S(Rn).
Proof: We have
(AFu,v)L2 = J Jeix<aF(x,^)u(0^v^c)dx
= / eix< f e-^x-y>^-^b(n, y, 0«(0«R dV dV dx K

281
But
and b is real-valued if a is real-valued, thus we have that
{AFu, v)L2 = (w, AFv)L2.
Let now assume that a > 0 and let prove that
= J" jvit-n) || F(y> ^, C)0(y,C)F(y> £,C) #} fl(flfifo) #^ > 0,
for all y. By regularizing the integral we may assume that a(y,Q has
compact support in (. Then we can exchange the order of integration and
obtain
//
e*<F(y,t,Qmti
2
«(y,C)^C>o.
D
Now we can deal with lower bounds.
Theorem 4 Given A — Opw(a) € UppS there exists b 6 S™pS such that
0?w(a)-OpL(bF)£L-°°.
Moreover
a,f3 y<a
where each ga,pn is idependent from a and b and
9o,o,o — 1)
00,0,0 = 0, t/|/J| = l,
*»A<r € 5-^;:^', if\a\>l.
Proof: Let consider the sequence of symbols
b0 = a G Sv,p,8->
bs = symbw{0Pw(bj-1)-OpL((bj-1)F)}eS™Jp-S\ forj>l,
and define
6~5>

282
We have
Opw(bN+1) = Opw(bN) - OpL {(bN)F)
= Opw{bN-i) - OpL {{bN-i)F) - OpL {{bN)F) = •■■
= Opw(b0) - OpL ((60)f) OpL ((bN)F)
= Opw(a) - OpL ((60 + • • • + bN)F).
This implies that
0?w(a)-OpL(bF)£L-°°.
The asymptotic expansion of b follows quite easily from (20). D
Corollary 2 Let A = Opw(a) G HLV' s and assume that a is real-valued
and bounded from below, then also A is bounded from below, i.e.
(Au, u)L2 y |M|L2, for all u G S{Rn).
Proof: By Theorem 4 there exists b such that A - OpL (bp) is bounded,
because regularizing. So it suffices to show that b is bounded from below.
But from hypoellipticity
-< A^(a;,0"p|aM/?l, for \\x\\ + \\x\\ > R > 0,
thus from the asymptotic expansion of b we have
b(x,t) = 0(3,0 (1 + 0 (a^M)-1)) < M IMI + IKII ^ °°-
It follows that 6 is bounded from below. D
5.3. ANTI-WICK QUANTIZATION
We end this section by the construction of an isomorphism of given order
m G R.
Given a G Sj}p_p, with 0 < p < 1///, we define the anti-Wick
quantization of a as the operator
°?Aw(a) = Opw(a*a)
where
<r(*,0 = 7r-ne-INI2-lkll2.
The following properties are easily verified:
a * a is real-valued if and only if a is real-valued,
a(z,£)

283
a * a> c if a> c,
a * a € HS%'[ _„ if and only if a e HS% { _„.
J
Theorem 5 If a G HSv'p__p and a > 0, then A = Op^^(a) is an
isomorphism of S(Rn) with inverse belonging to HSp'~™.
Proof: A simple computation yields
|2
[Au, u)L2 = 2nW2 / / eix<e-Wx-yW2lAu{x) dx
a(y,t)dyK, (21)
thus, if An — 0, we obtain u = 0 and A is one-to-one.
To show that A is onto, we have to solve in <S(Rn) the equation
Au = /,
for / G <S(Rn). By hypoellipticity we can solve it in L2(Rn). Let b = a"1,
and 5 = Op^^(6), then
BA - / + R
with i? G Lp p compact, because of negative order, and B one-to-one,
because b > 0. Then also BA is one-to-one, and so it is onto by Fredholm
Theory. But then also A is onto because B is one-to-one.
Finally from the existence of the parametrix we obtain that A'1 G
HSA:-P- a
A simple, but useful example is given by
OPawW) € BL^_W
6. The Weyl formula for globally hypoelliptic operators
An operator A e LppS is always closable in L2(Rn). Let us denote by A
its closure. Then one easily proves that A is self-adjoint if and only if A
is formally self-adjoint i.e. its Weyl symbol is real-valued (see [2], Section
2.1).
Let us consider a hypoelliptic operator A e HLV' * with real Weyl
symbol a. If / > 0, then A has a parametrix of negative order -/, and
therefore A has compact resolvent. Thus we have that A has a real spectrum
which consists of an unbounded sequence of semi-simple eigenvalues Xj. By
hypoellipticity, the corresponding eigenfunctions (j>j belong to <S(Rn), thus
A and its closure A have the same spectrum. From hypoellipticity and
Theorem 4 one proves that a is either bounded from above or from below.
By a change of sign we can always assume that a is bounded from below,

284
so that the sequence of the eigenvalues diverges to +00. Then it make sense
to consider the counting function
N(X) = £ 1,
where the eigenvalues Xj are in increasing order and repeated according to
their multiplicity.
The following result has been proven by Boggiatto and Buzano in [1]
for S = —p, see also [2] for a more throughout exposition of this case.
Theorem 6 Let A = Opw(a) e HLV' s with a real-valued, semi-bounded
from below and
0 < / < ra, -p<S<p<-.
Assume that
1. V is non degenerate, so that there exist
0 < s < 1, and 1/ = (1/, 1/") G (R+)n X (R+)n
such that
l-s
\xl'~Xn'£l'~£n\ M '
• (laril-/^ H h 1*^1-/^ + l^ir7-^ H 1- |^»r/-^)
-< Ap (&,£),
2. a = ao + 5 wftere
^ ao w quasi-homogeneous:
do
(x^xu...,X<xn,X»"tu...,X^n)=Xma0(x,(i),
/or A>0 and (&,£)# (0,0),
f&j ao ^ ^on degenerate:
l-s
\xi--Xn'£l~-£n\li* '
. (|^|^ +...+ W*K + |6|*K +...+ l^l-K)
-< a0(s,0, /or||z|| + ||£||>#>0,
fcj fftere eziste 5 G [0,1) st/c/fc that
|fi(*,0| X 1 + l^l^i + • • • + |xn|S/"» + 1^1^" + • • • + KnP".

285
Then we have
N(\) = [V0 + O{h{\))]\Wm,
where
with
and
Vo = J x(Mx^)^)dx^
*«■*> = {!: Hit <22>
A_£(log A)2n_2, ifei > €3 and e2 > e3.
h(X) = { A-(logA)2"-1, «/€1>€2 = €3,
A e, otherwise
€<€i, €<62, £<*3,
2 (1 - 5)5 m
ao is the principal part of a (however, in general ao is not the principal
symbol of A).
We shall prove this theorem in Section 9.
We end this section by showing how to derive Theorem 1 from Theorem
6. Consider the convex polyhedron V C R£ X R? of vertices
{(0,2e1),...,(0,2en)}U{(/?,0):/?€y(Q)}.
Q is non degenerate, hence the diagonal of (R+)n intersects the boundary
of Q in a point interior to a face Fu and we have the estimate
|*i • • -*n|^f (l*!^1 + • • • + \xn\s^n) < AQ{x), for all x,
for a suitable
0<s< 1.
Then the diagonal of (R+)2n meets the face
F(*>H'
which is the convex hull of
F„x{0} and j(o, -eij ,..., (o, ^enj|,

286
in an interior point and we have the estimate
1*1 •••*»•& •••£»l11^-
(l^l'-M + ... + i^jWw. + ^2ts + ... + ^2^
with
/ \ 1/2
\(aj})ev(v) J
and
t-
H + «(l-«)/2'
Then we can apply Theorem 6 to -A + W(x) - Op^ (||£||2 + W(x)) with
/> and 5 satisfying (13) and st in place of s. We obtain (9), (10) and (11).
7. The method of approximate spectral projections
In order to evaluate the counting function N(\) we consider the spectral
projection:
E(X)= E (u,<t>j)L2<t>j,
where Xj and <pj are respectively the eigenvalues and the eigenfunctions of
A. Then
AT(A) = Tr£(A).
Recall that the trace of a self-adjoint compact operator T on a Hilbert space
H is given by the series of its eigenvalues p,j, repeated according to their
multiplicity:
If the series is absolutely convergent, we say that T is a trace-class operator
and we define its trace-class norm as
Pilaw = EM-
The space of trace-class operators is denoted be B\(H). We shall come back
on trace-class operators in the next section.
Now we compute the trace of the projection operator. Let
Kx{x,y)= Yl Mx)My)>
\3<\

287
be the Schwartz kernel of E(X). Since (f>j e <S(Rn) and the sum is finite,
K\ e <S(R2n), so E(X) is a regularizing operator with Weyl symbol:
ex(x,0 = T-\t |ka [x + -y+, a; - -yj |.
Therefore
tve(x) = x;wi»
A,-<A
= /lfc(.,.)«b
= e\(x,£)dxfc.
So we have to compute e\ in terms of the Weyl symbol a of A.
Let us first perform a formal computation we justify afterwards. Let
y(zX)-{l> ifRe*<A,
X^'Aj-\0, ifRe*>A,
and cissume that A is not an eigenvalue of A. Then we can express E(X) as
a Dunford-Taylor integral
E(X) = ^-JX(z,X)(A-z)-1dz (23)
where the integral is computed along a simple Jordan curve in Re z < X
and enclosing all Xj < A. We know that (A — z)~l is a pseudodifFerential
operator with Weyl symbol bz(x,£) such that
Let us substitute
(A - z)-lu{x) = j jl-*)<bt Q(* + y),{) u(y) dyfc
into (23). We obtain
E(X)u(x) = ^-Jx(z,X)je^-y^bz(^(x + y),^dy^dz
= Je^-y>^JX(z,X)bz(^(x + y),^j dzu(x)dy#.

288
By Cauchy Theorem
^/ri'1*g(.+i),e)t
Z7TI J a{X,£) - Z
= x(a(*,0^) + ----
Thus
and
eA(^0 = x(a(z,0>A) + ''->
N(X) = TvE(X)
What we want to do now is to make such an argument rigorous and to
estimate the remainder. Let us first remark that x{a{xiQity ls n°t smooth.
So the first step is to regularize x- Consider a positive real number e and a
smooth function
if): R -* R,
such that
and
i/>dt = l,
^{t) > 0, for all £,
tl>(t) = 0, for \t\ > 1.
Define
then
Xe(t, A) = A"*1"'* J x (s, A + Ax-72) j, (2X~^\t - sj) ds,
Xt[?'A)-\0, iff > A + A1-^,
and for each k € N we have
|d*Xe(*, A)| -< \-(1~t)k, for A > 0 and all t.
Now we set
ex,t(x,t) = Xc{a(x,t),\).

289
We have
CAfe^,^-|0j if <,(*,£) >A +A1-'. [ 4j
Thus eA,e has compact support and defines a regularizing operator
Ee(A) = 0Pwr(eAfe).
Of course
e\,e - e\ -» 0, as A"e -» 0,
so we can try to approximate Tr E(X) by TrI?e(A) for A large.
Observe that e\yC has compact support, but not uniformly in A. Actually
we have the following estimate
%dSexAx,Q\ ■< Xtla+0lAv(x,O-pH+m, (25)
for A > 1 and {x\\ + ||£|| > R > 0 This is a consequence
1. of the estimate of Xe(£, A),
2. of the fact that
A<a(a:,£) < A + A1_e,
on the support of the derivatives of e\^
3. of the following Faa di Bruno-type estimate for two smooth functions
g : R -» R and / : Rn -» R:
i«(#))i^ e |ffw(/w)|- e i^/w-^r/w
Estimate (25) is fundamental for what follows. Thus from here up to the
end of the paper we assume that A > 1.
Theorem 7 Assume there exist a function h : R+ —> R+ such that for a
suitable
p-S
0<e<
we have
where
(26)
2m '
v(\ + 0(\1-'j)=V(\)\l + 0(h(\))],
V(\) = JX(a(x,S),\)dxfi
is the Weyl term and x is defined in (22). Then, for each k 6 R we have
N(X) = V(X) [1 + 0(h(\))] + O (A"fc) . (27)

290
Proof: We need three lemmas:
Lemma 1 Ee(X) is a trace-class operator for which we have the trace
formula:
TTEe{\) = jeXt{x,Z)dxft.
Lemma 2 For each fcgRtoe have
\Et(X-Et(X)'
Lemma 3 We have
C[ = supA"^
A>1
Bi(L2)
= o(\-k(v{\ + \1-e)-v{\) + i)).
Et(X)(A-XI)Ee(X)
B(V)
< oo
C^ = infJA-^) inf ((I-Ee(\))(A-\I)(I-Ee(\))u,u) )
A>! [ IHIL2=i v JL J
> -oo.
We shall prove Lemmas 1 to 3 in the next section. Now we prove the
theorem. Let us denote by {/ij} the sequence of the eigenvalues of Ee(\),
repeated according to their multiplicity and let {^} be the sequence of
the corresponding eigenfuctions which belong to <S(Rn) because Ee(X) is
regularizing. Moreover we may assume that {iftj} is an orthonormal set.
Define
Ne(X) = £ 1.
/y €[1/2,3/2]
Let us show that
|iV£(A)-Tr££(A)|<4|E£(A)-E£(A)'
In fact we have
Bi(V)
Nt(X)-TvEt(X) = Y, 1-E^i
^€[1/2,3/2]
J2 0--H)- S Pi-
^€[1/2,3/2] ^£[1/2,3/2]
But
|l-/*il>2» if^'£
I 2
2' 2
thus
M> €[1/2,3/2] ^£[1/2,3/2]
= 4
£e(A)-££(A)'
Si(L2)

291
From Lemma 2 we obtain for any k € R:
Ne(X) - TtEt(X) = 0 (X~k (V(X + A1"') - V(X) + l)) .
From (24) and Lemma 1 we have
V(X)<TTEe{X)<V(X + X1-c).
Therefore we have for any k 6 R:
JVe(A) = V(X) + 0 (V(X + X1-') - V(X)) + 0(X~k).
Now we prove that there exists a positive constant C such that
N(X - CXX~C) < N€{X) < N(X + CA1_£).
(28)
(29)
Let M be the space generated by the eigenf unctions tpj corresponding to
fij e [1/2,3/2]. Then
dimM = N€(X).
Consider
and let
Then of course
Mi €[1/2,3/2]
Mi €[1/2,3/2]
/*j
Ee(X)v = w,
and we have from Lemma 3
(IW)U)l2 = (lE£(A)t;,£;e(A)t;)L2
= A II^CA)^!!^ + (^(AXA - A/)£?e(A)t;,«)
< MMh + C^'Ml
i?
because
^€[1/2,3/2]
Mi
<4 2 l«ii2 = 4|H&-
H3■ €[1/2,3/2]

292
Thus, if we choose
C > 4C'„
we have
Im (7 - E(X + CA1"6)) n M = 0,
rvppo 11 op
(Au,u)l2>(x + CX1-<)\\u\\12,
for u G Im (7 - E (A + CA1"*)). This implies that
Nt{\) = dim M < dim{ImE(X + CA1"6)} = N(X + CA1"6).
Let now consider the orthogonal complement Mx to M. Given
u = ^2 Ujipj 6 M1
/»> ^[1/2,3/2]
we have
with
u=(I-Et(X))v,
/»;*[l/2,3/2]
Pj
Computing as before and using Lemma 3, we have
(Au,u)l2 = ((I-Ee(\))A{I-Ee(\))v,v)v
= X ||(7 - Et(X))v\\2L2 + ((7 - Ee(X))(A - A7)(7 - E€(\))v, v)l2
> X\\u\\l2-C':X^\\v\\l2.
But
|2
Ml = £
^[1/2,3/2]
Uj
l-/*i
<4 J] K|2=4Hi2.
^[1/2,3/2]
Therefore
(AU)W)L2>(A-4C;'A1-)||M||i2.
This means that, if we choose
C > 4C't',
we have
Im£;(A-CAi_£)nM-L = 0,

293
because
for uelm E(X - CA1"6). Therefore
N{\ - CX1-6) = dim{Im£(A - CA1"6)} < dim M = N€(X).
From (29) we obtain that
Ne (A + ^(A1-6)) < N{\) < Ne (A + ^(A1-6)) . (30)
But from (28) and (26) we have
Ne (A + OiX1-')) = v(x + OiX1-6))
+ O {V (A + ^(A1-6)) - V (A + ^(A1-6)) } + O (A"*)
- V(A)(l + 0(/i(A))) + 0(A-*).
Thus from (30) we can conclude that
N{\) = v(\) (i + 0{h(\))) + o (x~k) a
8. Trace-class operators and proof of Lemmas 1 to 3
In order to prove Lemmcis 1 to 3 of the previous section, we need some
results on trace-class operators.
For simplicity we limit ourselves to self-adjoint compact operators on
a Hilbert space H (see [2] for general compact operators). Given such an
operator T we can consider the sequence of its eigenvalues fij repeated
according to their multiplicity and the corresponding eigenvectors ipj, we
may assume to be ortonormal. For each p > 1 consider the norm
imiw = (£Np)1/p-
^ ll^1l£P(#) < °°' we say ^at T is in the space BP(H). Operators in B\(H)
are called trace-class operators, while operators in 62(H) are called Hilbert-
Schmidt operators.
As we have already done, for operators in B\(H) we can define the trace
as
We have
\\T\\b2(H) < ||T||Bl(tf). (31)

294
Moreover 62(H) is a Hilbert space with scalar product given by
(5,r)ftW = Tr(r5). (32)
One can prove the following result
Proposition 9 IfT and S belong to B2{H), then ST € B\{H) and
\\ST\\Bl{H) < \\S\\B2(H)\\T\\B2(H).
D
Moreover, in the special case H = L2(Rn) we have
Proposition 10 T € #2(£2) if and only if it has a kernel K? £ L2(R2n).
We have
\\Th2(L*) = II#t||l2(R2").
In particular, ifT = Opw(t) € Up s, with m < -n/fiQ, where
Ho= min |a + /?|,
(«,/*)€ V(7>)\(0,0)
thenT £ B2{L2) and
\\T\\b2(L*) = II^t||l2(R2») = (27r)~nHillL2(R2«)-
D
Theorem 8 Let T = Opw(t) € L% s, with m < -2n//j,0> then T €
B\ (L2) and we have the trace formula
TiT = Jt{x,t)dxft£.
Moreover there exists M such that for each s < -2 we have
\\T\\Bl{L>) < £ /<M*i0'|^(*i0| dx fa (33)
\a+p\<MJ
for all t G Lp^j, where
A-m/2
wm = a * Av
and
a(x,0 = «-ne-M2-m*.

295
Proof: Let
From Theorem 5 we know that Wm is invertible. Thus we can write
Wm = Opw{wm) = OpAW(Avm/2).
T = W-1(WmT),
-ra/2
and W-1, WmT e L™'J8. From Propositions 9 and 10 we obtain T e 3X (L2)
and
= (27r)-"||^-1||B2(L2)||«;m#f||L2. (34)
Now it is easy to prove the trace formula. From (32) and Propositions 9
and 10 we have
TrT = {w„T,W-')eim
= / KwmT(z,y)Kw-i(y,x)dxdy.
But the kernel Kj is continuous because it is the Fourier transform of a
/^-function, so by Fubini Theorem we obtain
KT(x,x) = I KWmT(x,y)Kw-i(y,x)dy
and from (18) we have
KT{x,x) = Jt{x,t)#.
Eventually, we have
TrT = Jt{x,t)dxK.
Now we prove the estimate of IITHb^i^). This is done in several steps.
First we assume that t has support in the unit ball. Because wm G
Sp™s , from (19) we obtain that there exists M' such that
lK#*llL2x £ K#
\a+p\<M'
for all t G Sp s with support in the unit ball. Because t has compact
support
... / .../ dffitfaridyfa
-00 J — OO J — OO J — OO

296
where e = (1,..., 1) € Rn, so that, for example
d! = %•••%,.
It follows that
d^t
-<
J
de+ade+()t
dxfl£.
This shows that
IK#'IIl^ £ J\dPPj\dxfc
|a+/?|<M
with M = M' + 2n. Because £ has support in the unit ball and wm > 0,
because A^m/ > 1 (see subsection 5.3), we have
\a+0\<M
and (33) follows from (34).
Now we assume that t has support in a ball of radius 1 and center (x, £).
Let
i(x,€) = t(x-x,{-£),
and
Then
with
U is unitary, hence
Let
where
f = 0Pw(i).
f = U~lTU,
Uu(x) = e~tx*u(x + x).
Wm = Opw{wm),
Wm(x,Q = Wm(x - X,£ - f).
W"TO is the anti-Wick quantization of Ap(x - x,£- £)~m^2 and, as before
Thus
Wm = U-'WmU.
w-1 = irxw-lu

297
and
It follows that
mi*
\w
\r r i
ip)
-i
le2(L2)
\W
\ r r ,
-1
le2(L2)
-<
If
k(£2)
Iw-1
|w
m
k(L2)
B2(L2) 1
w«r|
^mf
|02(£2)
IIB2(L2) '
But £ has support in the unit ball. Thus we have already proven that
|a+/?|<M
■. s /
<£<#
<£#*
dxfe.
Now we consider a general £ with compact support. Let {0j} be a
partition of unity of R2n such that
1. each 0j has support in a ball or radius 1,
2. there exists / such that each suppflj intersects at most / suppfl*,
3. for each a/3 £ Nn there exists a positive constant Ca,p such that
\\^9j\\Loi><Ca^ for all j.
For each j let Tj = Op^(0j£). Because suppt intersects only a finite
number of supp 0j, say for j = j\,..., j#, we have that T is a finite sum of
It follows that
ll£i(L2)
L_1
-< E E /<|$f#(M|«fe#
k=l |a+)9|<M

298
X
E E /
Wl
k^t\
dx j£
suppf/^
< ' £ J<\o?dgt\dxft.
\a+p\<M
Thus we obtained the estimate (33) for each t with compact support.
Let now consider a general symbol t and let {tj} be a sequence of symbols
with compact support tending to t with all the derivatives, uniformly on
compact subsets. Choose s < -2. Then ws is integrable and we obtain
•lim I £ / w* \9!9^ Idx # } = £ / w* \dtdPAdx #•
Because each tj has compact support
|a+/?|<M
This implies in particular that {Tj} is a Cauchy sequence in B\ (L2(Rn)).
On the other side, Tj -> T in #2 (I2(Rn)), because tj -> t in L2(R2n).
Thus from (31) we obtain that Tj -> T in B\ (£2(Rn)), and the proof is
complete. D
Now we prove Lemmas 1 to 3 of Section 7. Lemma 1 is obvious because
e\i6 has compact support.
Let us prove Lemma 2. Let
w = w n i=cr*A
MO
From Theorem 8 we have that there exists M such that for any s < -2 we
have
E£(A)2 - Ee(X)\ ■< E [w* ka^(eA,e#eA,£ - eA,e)| dx#.
U ' \a+p\<MJ
We have the asymptotic expansion:
^A,e#eA,e " ^A,e =
O<|0+^|<7V
0ty!2l*+*l

From (24) and (25) we obtain
|^eA,e(x,fl| < hv{x,Z)-m+~m, for all (x,£) and A > 1,
where
p z= p - m€, and 8 = 8 + me.
Therefore we have that
uniformly with respect to A. Moreover
with
and
Thus
2 V 2m
p' = p-(e + J)m, 8' = 8 + (e + e')m.
|^rAfefJV(*,0| X A-e'(JV+l^l)A7,(«lO-(p'-'')JVVW+''W.
Finally we have
so (see (24))
4,0"'-(A V"0 + 2/ , for(x,£)esupp(ev).
Then, for any s < -2 and N > 0 we obtain the following estimate
*<A>a-*<A>U<i»,
= O (\~\^+ymS J{X(a(«,0,A+ A1-) - x(a(x,0,A)} <fe#
+o (a-'")
= O f A"(*+Om' (y (A + A1-) - V(X))) + O (A"'") .
This proves Lemma 2.

300
Let us prove Lemma 3. Consider Et(X)(A - XI)Ee(X). Its Weyl symbol
is
e\,S{a - A)#eA,e.
On the support of e\iC we have
a(x,0-A<A1_£
and, for \a + /3\ > 0
■< XAv{x,0~plal+m
for ||x|| + ||£|| > R > 0. p and 6 are defined in (36). Thus we have
\9?8l(a(x,t) - A)| ■< Xi-'AvM-M**™,
for (#,£) G supp(eA,e) and ||z|| + ||£|| > R > 0. Hence, from (19) it follows
that
e\
with
|a+/?|<n ^
= PA,e,AT + ^A,e,AT,
uniformly with respect to A. Thus we obtain that
A-(1-£)(eA,£#(a-A)#eA,£)e5°-)J-,
uniformly with respect to A.
This proves that
lA-(l-e)
Let now consider
sup,
AM
Ee(\)(A-\I)Ee(\)
B(I?)
< 00.
(/ - Ee{X)) (A - XI) (I - Ee(X))
with Weyl symbol
(l-ev)#(a-A)#(l-ev).

301
As before we have
1 - eA,e G SVjj,
and
A-(i-) {(i _ eA£) #(a _ A)# (1 - ev) - (1 - ev) (a - A) (1 - ev)}
uniformly with respect to A. Moreover
(1 - ex,e) (a - A) (1 - cA,e) > 0,
thus from Corollary 2 we have that
0Pw(A-(1-£)(l-ev)(a-A)(l-eA)£))
is uniformly bounded from below with respect to A and we obtain
M {A~"~" wfcf=1 (<'" £<<A» C* "A/) " - £«<A» "•")»} > -»•
D
9. Estimate of the Weyl term and proof of Theorem 6
Thanks to Theorem 7 in order to prove Theorem 6 we need only to show
how to choose the function h. Of course it is of interest to choose h such
that it goes to 0 as A —> oo. The following proposition shows how to do
this.
Recall that
V(\) = JX(a(x,t),\)dx#,
and
X(*'A)-\0, if t > A.
Proposition 11 Assume that for some constants Vb, p, <7, p', q' such that
Vb > 0 and
either p > p' or p = p' and q> q',
we have
v(\) = v^iog xy + (o(\p,{\og xy').

302
Then (27) is valid, with
h(X) = max{A-£, Ap'"p (log A)«'-«)}
and therefore also
N(X) = V(X)(l + 0(h(X)))
= V0X"(\ogXy(l + O(h(X))). (37)
Proof: We have
F(A+0(A1-£))-F(A)
= 0
x-( + xj>-p(\ogxy-q
V(X)
Then (37) follows from (27) with k < p. D
Now Theorem 6 is a consequence of this proposition and the following
Theorem 9 Assume that V is non degenerate, so that there exist
z, = (z/,z/') € (R+)n X (R+)n
and
0<s< 1,
such that
1-3
|3l---3n-fl"-fn| H '
. (l^./Kj + . . . + \Xn\s/< + |fl|.K + • • • + \Zn\*/<)
-< A?>(z,£),
for all (x,£).
Assume further that
a = a$ + a,
i. ao is quasi-homogeneous:
a0
(A^,...,A^n,A^
/or A> 0 and (x, £) 7^ (0,0),
5. ao is non degenerate:
. (|Xl|./^ +...+ \Xn\sl»'n + |6rK +...+ |&|'/*#)

3. there exists 0 < s < 1 such that
\&(x, oi1/m ■< i+1*1 ?K + • • •+i*»fK + i6r/i/JI+---+KnrK.
TAen
where
and
where
V(A)=(vb + 0(F(A)))Al"l/w>
A-3(logA)2-2, ?/63<£2)
T/(A)=<(A-^(logA)2-1, «/62 = £3,
A_£2, ife2>e3,
1 - (i-s)*M
€2 = 1-*, C3 =
(1 - s)s m
Outline of proof (see [2], Ch. 2, Thm. 6.1): We have to estimate
/ X(«(*)» *)dz- I x(a0(z), A) ck,
where
Let
i/j = j/,-, and i/' = j/j+n,
for; = l,...,n.
Of course we may limit ourselves to the first quadrant
z > 0, i.e. zj > 0, for j = 1,..., 2n.
Then we perform the following change of variables:
Zj = {\ujy*tm, forj = l,...,2n,
and let
b0(u) = a0(u?/m,...,uZn/m),
h(u) = A-1a((A«1r/"l,...,(Ati2n)^/TO).
\M/™ factors out and we reduce to prove that
2n „ 2n
... / 1 ~
as A —> +oo.
JkJk. J I -JLJL J
7 = 1 . .- 7 = 1
b0 + bA<l
» 2n « 2n
y n ^?/m"1 ^ - y n <j/m"1 ^=<wa))>
, + 6A<l i==1 *0<1 j=1
u>0 u>°

304
We can limit ourselves to the sector
U = {u e R2n : «i > u2 > ■ • • > u2n > 0} .
Now we make a second change of co-ordinates:
tu(0)
where
boH0)Y
0 = (9l,...,02n-l) ,
W(0) = (wi(«),...,W2w(«)),
and
U\ (0) = COS 6\,
Uk(0) = I Y[ sin0j I cos^, for 2 < jfe < 2n - 1,
2n-l
^2n(0) = ft Sin^'-
i=l
[/ is mapped over R+ x 0, where
© = {ee R2n_1:
0 < Oj < arctan(sec0j+i), for 1 < j < 2n - 1 and 0 < 02n-i < ^/^K
and the curve &o(^) = 1 is mapped over t = 1.
So we reduce to estimate
TZ{\) = / H(e)t\u\lm-ldtd6- I H(6)t\u\lm-ldtd6,
where
and
0<t<max{0,l
06©
H(0) =
'-'a}
fA(i,0) =
M
(6oMrM/m«r
0<t<l
06©
^oM»))J'
2n M^/m
/m TT i
** sin07-_i cos07_i
j=2 J J

305
From hypoellipticity
b0{u)l/m -< b0{u) + bx{u), for ||u|| > R > 0 and A > 1.
Therefore
ti/m ^ t + fx(t q for —1— > fl, and A > 1.
bo{u{0))
But on the domain of integration
* + fA(M)<l,
thus, either
t < Rma,xbo(u(0)) < oo,
0G0
or
*^(* + fA(M))m//<l.
Therefore, on the domain of integration t is bounded by a suitable constant
T.
Form the hypotheses we have
K)^,(«;/TO + - + 4TO)mxM«),
and
This yields
|fA(M)| -< A5"1 (^2+'--+^ -..^-i)"1^11,
for A > 1, -1 < t < T, and 0 e 0, and
_ii 2n ,/^/m
ff(tf) -< Ulm + ... + u;2s/m)"Hu;ri/m ft -r-^ 5— for 0 € 0.
w V * 2n / x Hsinfly-icosfl.-i
From these estimates we obtain the result. D
References
1. Boggiatto P., Buzano E. (1995) Spectral asymptotics for multi-quasi-elliptic
operators in Rn, submitted to Ann. Scuola Norm. Sup. Pisa.
2. Boggiatto, P., Buzano, E. and Rodino, L. (1996) Global Hypoellipticity and Spectral
Theory, Akademie Verlag, Berlin.

306
3. Boggiatto, P., Buzano, E. and Rodino, L. (1996) Spectral asymptotics for hypoel-
liptic operators, to be published in the Proceedings of the conference "Partial
Differential Equations", Potsdam, Germany, July 29-August 3.
4. Cattabriga, L. (1966-67) Su una classe di polinomi ipoellittici, Rend. Sem. Mat.
Univ. Padova, 36, 285-309; 37, 60-74.
5. Friberg J. (1967) Multi-quasielliptic polynomials, Ann. Scuola Norm. Sup. Pisa, CI.
Sc, 21, 239-260.
6. HelfFer B., Robert D. (1981) Comportement asymptotique precise du spectre d'opera-
teurs globalement elliptiques dans Rn, C.R. Acad. Sc. Paris, 292, 363-366.
7. HelfFer B., Robert D. (1982) Proprietes asymptotiques du spectre d'operateurs
pseudodifferentiels sur Rn, Comm. in P.D.E., 7, 795-882.
8. Hormander, L. (1979) On the asymptotic distribution of the eigenvalues of pseu-
dodifFerential operators ir Rn. Arkiv for Mat., 17, 297-313.
9. Kumano-go H. (1974) Pseudo-Differential Operators, The MIT Press, Cambridge,
MA.
10. Mohamed A. (1989) Comportement asymptotique, avec estimation du reste, des
valeurs propres d'une classe d'operateurs pseudo-difFerentiels sur Rn, Math. Nachr.,
140, 127-186.
11. Shubin M.A. (1987) Pseudodifferential operators and spectral theory, Springer-
Verlag, Berlin.
12. Tamura H. (1982) Asymptotic formulae with remainder estinates for eigenvalues of
Schrodinger operators, Comm. P.D.E., 7, 1-53.
13. Taylor M. E. (1981) Pseudodifferential operators, Princeton Univ. Press, Princeton.
14. Tulovskff V.N., Shubin M.A. (1973) On asymptotic distribution of eigenvalues of
pseudodifferential operators in Rn, Math. USSR Sbornik, 21, 565-583.
15. Volevic L.R., Gindikin S.G. (1968) On a class of hypoelliptic polynomials, Math.
USSR Sbornik, 4, 369-383.

SPLITTING IN LARGE DIMENSION AND INFRARED ESTIMATES
B. HELFFER
UA 760 du CNRS, Departement de mathematiques,
Bat 425,
F-91405 Orsay Cedex, FRANCE
Abstract. These notes for the NATO ASI conference in Microlocal analysis
and spectral theory consist in the analysis of the links between estimating
the splitting between the two first eigenvalues for the Schrodinger
operator and the proof of infrared estimates for quantities attached to
Gaussian type measures. They are mainly based on the "old" contributions of
Dyson, Frohlich, Glimm, Jaffe, Lieb, Simon, Spencer (in the seventie's) in
connection with more recent contributions of Pastur, Khoruzhenko, Bar-
bulyak, Kondratev which treat in general more sophisticated models. We
shall show how the recent semi-classical analysis permits sometimes to state
more precise results.
1. Introduction
Our aim is to understand in the large dimension limit m —> +00 the splitting
between the two lowest eigenvalues of the following Schrodinger operator
m q m
-h2A + Y,v(x3) + ±J2\xJ~*^\2 (1-1)
in the case when the so-called one-particle potential v defines a double
well (We take the convention that xm+\ = x{). As well known, the first
eigenvalue of the Schrodinger operator is simple, so the splitting is always
strictly positive. Its behavior with respect to m as m -> +00 depends
actually heavily on the nature of v and on the size on J which is always
assumed positive.
More generally we are interested in the similar problem attached to a d-
dimensional (periodic) lattice A = (2Z/mZ5)d, identified with [1 , m]d n
307
L. Rodino (ed.)> Microlocal Analysis and Spectral Theory, 307-347.
@ 1997 Kluwer Academic Publishers.

308
2Zd = {1, • • •, m}d. In the odd case m = 2L +1, we shall sometimes identify
A with [-1, ••-,+£]).
So we consider
-h2A + J2v(xi) + ^\x3-Xi\2, (1.2)
where i ~ j means that i and j are nearest neighbors in A considered as
living on a torus.
The parameter J which measures the size of the interaction satisfies the
condition
J > 0 . (1.3)
This model can also be written as
with
v(y) = v(y) + Jdy2,VyeIRn. (1.5)
Most of the time, we shall look at the case n = 1 but other results
considered for example by [40] will deal with the case n > 1. In this case we shall
usually assume that v is radial.
For a one dimensional lattice (d=l), we have analyzed with J. Sjostrand
[30] the case of the so called Kac model and it was shown in [21] how to
adapt this proof to treat other cases containing (1.1) in order to get that
6771
\2(m;h) - \i(m;h) < Cexp-(-) , (1.6)
for C, e independent of m and for h < fy), ho independent of m.
But we met an additional and unfortunate condition
m < Ch~No , (1.7)
without to know if it is a technical or deep condition. We have proposed in
[21] an example where the property was true without this condition,
-h*A+ f>J + | jr,Xj\ (1.8)
j=-L j=-L
but this example remains rather artificial, although it was motivated by
the study of the small temperature limit of a Kac type model.
C. Albanese mentioned to us the following model, called "Ising model with

309
transverse field", where the double well problem is replaced by the 2x2
matrix and the space <g)2L+1 L2 (Bn) by ®2L+1(P2.
This corresponds to the idea that we can replace, in the semi-classical
context, the one-particle Schrodinger operator -h2A + v by a two by two
matrix M (the interaction matrix) representing the restriction to the
spectral space attached to the two lowest eigenvalues.
This interaction can be expressed with help of the Pauli matrices
•"-(i-°.).-»-(Ji)
as
M = uld+\aw.
The parameter A corresponds to the measure of the tunneling effect for the
one particle problem and can, in the semiclassical limit and under suitable
conditions, be computed as having the order
A «exp-- , (1.9)
it
where S > 0 is the Agmon distance between the two minima (cf [27]).
Let us now describe the general hamiltonian. We denote by /H our Hilbert
space
+L
n= ® a?2, (1.10)
i = -L
where our line lattice is
A=[-L,+L]nZZ.
We shall also write it in the form
HA = S + AK, (1.12)
with
S = E^(1-43)43)):=E^, (1.13)
x^y x^y
and
K=E^i1)- a-14)

310
Here x ~ y means x ^ y and x nearest neighbour of y (that is, in 2Z/(2L +
1)2Z, x = y ± 1). Here the convention is that a\ ' acts only on the y
component:
°f\e-L ® • • • ® e„ ® • • • ® eL) = e_L ® • • • ® d(3)6y ® • • • ® eL .
The statement is
Theorem 1.1 :
If we denote by E± the two lowest levels, there exists C such that, for all
L,
|£+ -E.\ < (2L + l)C(2L+1)exp[(lnA)(2i: + l)] (1.15)
This theorem is due [37] (See [2] for another proof for an analogous more
complicate model [3], TY83).
What is quite important is of course to have a good control with respect to
the dimension. The standard perturbation theory gives of course a result
for A small because, according to symmetry arguments, we can consider
the two lowest eigenvalues as simple if we restrict Ha to suitable subspaces
%±. We observe also that the norm of Ylx #4 is a priori of order L and
this makes the usual argument of perturbation true only for |A| < C/L.
This means that we have to do something more tricky.
If we have in mind that, in this model, the parameter A satisfies (1.9),
the condition gives
exp~f-f' (L16)
and finally
L<Cexp|. (1.17)
This is in any case better than the condition (1.7) which we met with J.
Sjostrand in [30].
On the other hand, for connected results concerning Laplace integrals, we
have seen in [23] that this type of condition can be crucial and cannot be
considered a priori as simply technical.
But the main object of these notes is to analyze another approach based
on the so called infrared estimates which was developed by many authors
in the late 70's Frohlich-Simon-Spencer [12], Dyson-Lieb-Simon [9], and
Glimm-Jaffe [14]. More recently Pastur-Khozurenko [40] and Barbulyak-
Kondratev [4] look in the same spirit at other examples and it becomes
clear that this infrared approach gives also information about the splitting
for our initial questions. It will actually give a complete answer for all the
lattices of dimension strictly greater than 1. In this case, the result does

311
not seem to be related to tunneling properties. In the case when the lattice
is of dimension 1, we shall obtain only a partial result under this condition
(1.17). All these results are obtained through relatively easy extensions of
these contributions. Because, particularly in the paper by [4], semi-classical
analysis is involved, we have also tried to be more precise that in the original
paper, using our more precise knowledge of the tunneling [27].
But let us now present the main results obtained through the infrared
estimates.
Barbulyak and Kondrat'ev look1 in [4] at the d-dimensional extension of
the quantum model above which is denoted by
HT = -h2 E t£ + E «(**) -JZ*k*i. (Lis)
ke\ * JfceA k~j
Here A is a subset of 2Zd as defined before (with the periodicity convention).
The assumptions on v are the following:
- (Ha) veC°°{IR).
- (Hb) v(x) >ax2 + b,a,b£lR, a> Jd , x G M .
- (He) v(x) = v(—x).
- (Hd) For some qo > 0 the function v attains its strong global non
degenerate minimum at the points ±qo.
We could also consider potentials v on ZRn. In this case, the potentials
are assumed to be radial (invariance by SO(n)) and we take the natural
extension of the conditions (Ha)-(Hd).
(Ha) and (Hb) are much stronger than necessary. (Hb) gives however the
control of the interaction term J Ylk~j xk xj a* °° by the one particle term
^2v(xk). (Hd) will permit a detailed semi-classical analysis but a weaker
assumption can still work.
(He) is finally an important assumption in the analysis of the splitting but
seems to play only a technical role at the other steps.
Following [4], we now consider the operator exp -/3H^r and the associated
. Tv(BexP(-/3Hrr)) n 1Qx
{B)^ '= Tr (exp(-^r)) ' ( }
where B e ^a • Aa is a class of polynomials but we shall more specifically
analyze the case when
11 JfceA
lrThis is only a note without detailed proof.

312
In the case n > 1, this means that we consider with
1 ' A;GA
Physically the strictly positive parameter (5 corresponds to the inverse of
the temperature.
We now introduce the so called parameter of long-range order
P(/3) = lim PA(/J), (1.20)
|A|-*oo
with
^) = ((^E^) • (1-21)
The presence of the long range order, i.e. the strict positivity of P(/3), will
serve as a test for phase transition (cf [9] and [40]).
When the limit of (-)/3,a as |A| -4 +00 exists, we shall denote it by (• • )p.
Let
E(p) = £(1 - cos/)) , (1.22)
where A* is the dual lattice
A* = {p = (pW, • • .,PW) I p(0 = 2nk®/m , 0 < ib^) < ro-1 ; 1 < i < d} .
(1.23)
The method of infrared estimates will permit to get 2 the following lower
bound
pw > m, - ^ l,^^)' -* [M
dp .
. (L24)
This lower bound is deduced by a limiting argument (thermodynamic limit)
from the actually more useful inequality (for our questions relative to the
splitting), which is relative to the finite lattice case, and which is given by
the following theorem.
2Cf [9], (52), p. 368 (cf also Theorem 3.2 and Theorem 5.1 in this article).

313
Theorem 1.2 :
Under the assumption (Ha)-(Hc) on v, we have, for any k 6 A, the following
universal estimates
pa(/?)><4w-~ e (j^)icothf(^ft2jrE(p))
' ' p€A*\{0} ^' L
In
2
. (1.25)
Barbulyak and Kondrat'ev refer to [9] and [40]. The reference [40] treats
actually (1.24) in the particular case when v(x) = (x2 - l)2 and get in this
case an universal lower bound for (xfyp^. The specific part of [4] is probably
the semi-classical aspect which we shall develop further in these notes.
We emphasize that [40] works also for n = 1 but only for the particular
case. The nature at the semi-classical level of the splitting when n > 1 is
completely changed for the model v(x) = (\x\2 - l)2. The two first levels of
the one-particle hamiltonian are separated from each other by 0(h2). This
is indeed a Schrodinger operator with a uniformly degenerate well invariant
by SO(n). This theorem reduces the analysis of this long range order to the
analysis of (x\)p^ and this will be one of our goals to explicit how it can
be done in the semi-classical context. If we find indeed a lower bound for
(3 large independent of the dimension and if the second term in the right
hand side of (1.25) is small enough for h small enough, then we shall have
a proof of the existence of the long range order.
One part of the analysis consists in using monotonicity argument based
on Ginibre inequalities in order to reduce to an analysis of a one particle
problems. This will be recalled in section 2.
The second part is the analysis of the one particle problem (symmetric
double well problem) in the semi-classical limit. This will be presented in
section 3.
Section 4 will be devoted to the opposite situation (single well case) when
the potential is convex. The Brascamp-Lieb inequality then gives a rather
explicit way to control the situation.
Section 5 recalls the links between the splitting and the study of the trace of
(#)/?,A- Taking the limit /3 —> +oo before taking the thermodynamic limit
leads to some improvement of the results concerning the splitting. We get
for example the following theorem
Theorem 1.3 :
Let us consider the family of Hamiltonians Hv^r defined in (LI8) where
A C 2Zd with d>2. If the potential v and J satisfy (Ha)-(Hd), then there
exists ho independent of A such that we have) for 0 < h < ho,
lim (A*-Af) = 0, (1.26)
|A|-*+oo
where \± and A£ are the two lowest eigenvalues of Hv^r.

314
The direct application of the result by Barbulyak-Kondrat'ev would have
given, in the limit f3 -» +00, the condition d > 3.
Section 6 is devoted to a more precise analysis of the case /3 large and
presents essentially the results obtained by Barbulyak-Kondratiev and Pastur-
Khozurenko with some improvements.
The two last sections will be devoted to a short presentation of the infrared
estimates obtained by Frohlich-Spencer-Simon in the classical case ([12])
and by Dyson-Lieb-Simon in the quantum case ([9]).
From recent discussions with J. Frohlich in Ascona (June 1996) where we
presented some results contained in these notes, we learn that the infrared
estimates are not the optimal approach when n = 1 and discrete symmetry
is involved. J. Frohlich indicates that, using the techniques developed by
Glimm-Jaffe-Spencer [15], it is possible to prove that the splitting tends to 0
when n = 1 (Theorem 1.3) without any restriction on the dimension of the
lattice. The alternative proof giving in principle better results when d = 1
and n = 1 is based on the so-called Peierls argument. But the proof in [15]
is written in the framework of the Field theory and is difficult to understand
for non-specialists in this field. A nice but non self contained presentation of
the subject is also given in [10], particularly in Subsection 8 which presents
many other models. Nethertheless the results of Dyson-Lieb-Simon and the
results concerning the splitting are not explicitely analyzed in Fro#olich's
lectures.
We heard also at this conference in Ascona of more recent contributions
by Kondratev and Rebenko [41] using also the Peierls argument.
In any case, what seems still open is the complete analysis of the
tunneling with control of the size of A.
2. Ginibre type inequalities
Ginibre type inequalities3 make it possible to estimate (#!)/?,a from
below by the average (xl)* £ taken over the measure corresponding to the
"formal" Hamiltonian
H = ~h2 E |W E «(**)= E *> (2-1)
ke%d k ke%d ke%d
3The authors refer to [43]. It is more explicitely written in [44]. This result of Ginibre
generalizes previous inequalities of Griffiths extended by Kelly-Sherman.

315
with separate variables.
Less formally, we introduce
HA = J2Hk (2.2)
Jb€A
and we shall get the
Proposition 2.1 ;
Under the assumptions (Ha)-(He) on the potential v and J, the following
inequality is true) for any fceA,
(4W > (4W,#a • (2.3)
Let us describe briefly the version of the Ginibre inequalities (we just treat
for simplification the case d = 1) which is needed here. This is related to the
control of the sign of the correlations attached to the more general measure:
m
Z~l exp]T JijXi xj 11 dvj(xj) . (2.4)
ij i=i
We look at the partial derivative of ((#jfe)2) with respect to J{j. Here (•) is
taken with respect to the measure (2.4).
This gives
djij ((xk)2) = (xi xj (xk)2) - ((xk)2) • (xi xj ) .
The right hand side appears as the pair correlation of the two functions
/ = X{ Xj and g = x\.
We shall deduce from the Ginibre's inequalities that this expression is
positive.
dj,3((xk)2)>0. (2.5)
Let us briefly recall some elements of this theory (cf [13], [43], p. 271-279
and [44], p. 119-124), for a nice exposition). We recall the
Theorem 2.2 :
Let T\ be the set of functions on IR which are nonnegative and monotone
increasing on [0, +oo) and either even or odd. Let Tm be the functions
on IRm of the form f\(xi) • • - fm(xm) with f{ G T\. Let d\i be a
probability measure of the form (2.4) where J{j > 0 and each dvj has the form
exp(fj(x))d\j(x) with fj G T\ and d\j even. Then
(GKSl) (/)>0
(GKS2) (fg)>(f)-(g), (2'bj
for all f,g € Tm.

316
The reader can find the proof of the theorem in [44].
For (GKS1), this is essentially obtained by expanding the exponentials
exp/j and exp£-_^ JijX{Xj and controlling the sign of each term of the
expansion.
The inequalities (GKS2) are obtained by a more sophisticated duplication
method.
The measure we are actually considering, attached to the restriction to the
diagonal of the distribution kernel of exp-/?Jff^er, where J becomes a
parameter in [0, JbL has not exactly the structure which is introduced above.
It is consequently useful to use the Trotter product formula which describes
the kernel of exp -/3H^r as the limit in a weak sense of (exp - ^ rfx' exp ^)N
and this kernel satisfies the assumptions of the theorem for J > 0.
We have just to control the limit procedure in order to get the result.
This argument is sketched in [44] p. 120 and p. 122. Let us recall the
argument.
What we need to prove is some GKS-inequality for a measure which appears
as a limit in a weak sense of a family of measures satisfying (GKS2). The
measure d\i is the measure whose density with respect to the Lebesgue
measure is the restriction to the diagonal of the kernel distribution of
exp-f3H^r. The measures dfiw are the measures whose density with
respect to the Lebesgue measure is the restriction to the diagonal of the kernel
distribution of (exp -§Vexp ^A)N.
The starting point is the Trotter-Kato product formula saying simply that
^nmJ(exp-^exp^A)"/ | g)L2 = (exp-/?^/ I g)v ■ (2-7)
We have to verify two points.
- Verify that dfipj satisfies the assumption of the Ginibre's Theorem.
- Go from a convergence property in the weak sense for the kernel to a
convergence property for the trace.
The two points will be actually more intricate. We first observe that for the
very specific kernels K (or Kn) which are involved we have
JjRn f{x)9{x)K{xi x)dx
= lime_+0 (<T2 fRnxRnf{x)g(y)K(x,y)exp-\\x - y\2dx-dy) ,
We have also for more general slowly increasing / or g
/ f(x)g(x)K(x,x)dx = lim / f{x)g(x) exp-2r]x2 K(x,x) dx .
JRn V-+Q JRn

317
What is important here is that we can stay, in all the limiting procedures,
inside the assumptions of the Ginibre's Theorem. This is indeed the case
by our choice of regularization. (GKS2) has the following structure
(fRnK(x,x)dx) X (fRn f(x)g(x)K(x,x)dx)
> (fRn f(x)K(x,x)dx) x (fRn g(x)K(x,x)dx)
with f(x) = X{ • Xj and g(x) = (#jt)2. We observe that / and g are in ^2-
Application:
The Hamiltonian HA describes a system of non interacting particles.
Consequently, we get immediately, for fee A,
(xl)pM = (xl)p,Hk = (x2o)p,H0 , (2.8)
for all k e Zd.
Here we recall that H^ is the one-particle Hamiltonian at k G 2Z .
We consequently have obtained
Proposition 2.3 ;
Under the assumption (Ha)-(He) on v and J, we have
(4W > (xl)pji0 ■ (2.9)
Remark 2.4 ;
When n > 1, we can no more apply this technique directly. The case when
we have a rotational symmetry can probably be treated by taking polar
coordinates. The results in this case is mentioned in [1\]. Pastur and Kho-
ruzenko proceed differently in the case of the model v(x) = (1 - \x\2)2.
3. Semiclassical analysis of the one particle problem
We have seen in Section 2 how one can replace with help of the Ginibre
inequalities the study of the quantity (^^)/?,a by the study of the simpler
quantity
2 Trfogexp-ffffp)
<Xo>WJ" = Tr(exp-^o) (3b1)
attached to the one particle Schrodinger operator Ho = -h2j^ + v(x)
where v satisfies the condition (Ha)-(Hd). When considering the one particle
problem, we sometimes write simply x instead of x$.
The main topics of this section is the semi-classical analysis of (£q)^//0 as
h -» 0 and /3 -» +oo.
Two conditions on v could be relaxed. The assumption n = 1 in (Ha) is not

318
important in this section although we shall write the results in this case.
It is actually sufficient, instead of the strong (Hb), that exp-(3Ho is trace
class in order to perform the analysis of this section.
We shall be rather sketchy in this part and refer for example to [17] for
the presentation of the semi-classical theory involved, which is mainly due
to Simon [45], [46] and Helffer-Sjostrand [27]. Actually, we need sometimes
weaker results which are probably much older (particularly in the case when
n = l).
Let (j)\ (x; h) be the ground state of Ho. Observing that the first eigenvalue
is simple, the first remark is that
Urn (xl)p,Ho= I x2M^h)2dx. (3.2)
/?-*+oo J ft
The right hand side of (3.2) is clearly related to the localization of the first
eigenfunction <j>\. In the case of the double well problem, one has indeed
the
Proposition 3.1 :
Under the assumptions (Ha)-(Hd) on v,
lim / x2(j)i(x\h)2dx = q% . (3.3)
h-*o Jr
We can actually prove by standard semi-classical analysis that there exists
a constant C and ho such that
/ x2^ (a; h)2dx > ql - Ch , V/i < h0 . (3.4)
Jr
More precisely, this analysis based on the harmonic approximation gives
the existence of a complete expansion of the type
/,
x2(j>i (x\ h)2dx rsj q% + ^^ hJjj . (3.5)
R i>i
An immediate consequence is
Proposition 3.2 :
For any e > 0, there exists ho such that, for 0 < h < ho, one can then find
/?i such that
(xl)p>Ho>(l-e)q2Q, (3.6)
for all /3 greater than (3\.
This statement was used by Barbulyak and Kondratev. It seems actually
interesting to relate /3\ and h.

319
The formal expression for {x2)p,H0 *s
2 Ej exP -PXJ (I*2<j>j(x; h)2 dx)
{X h'H° = E^exp-^ ' (3J)
where \j is the sequence of the eigenvalues of Ho arranged in increasing
order and 4>j is the corresponding orthonormal basis of eigenvectors.
The semi-classical analysis says that near the first level of Ho which is given
modulo 0(h2) by the first level of the harmonic approximation at qo
-h2$ + r"My2' (3-8)
they are two eigenvalues Ai and A2 which are exponentially close. There
exists indeed S called the Agmon distance between qo and -qo and given
(in the case n = 1) by
S = f ° yjv{x) - v(q0) dx , (3.9)
J-q0
such that
A2 - Ai - hh • exp-- • a(h) , (3.10)
h
with a(h) ^ 0 and admitting a complete expansion in powers ofh. The other
point is that the third eigenvalues A3 is given modulo 0(hi) by the second
eigenvalue of the harmonic approximation leading for A3 to a splitting of
order h
A3(h) - X2(h) - hy/2v»(qo) . (3.11)
So we formulate the following natural question:
Do we have to assume
- /?i ~ exp jr corresponding to the inverse of the splitting between the
two first eigenvalues,
- or the weaker P\ ~ \ corresponding to the inverse of the splitting
between the packet of the two first eigenvalues and the third one ?
We shall see that we are actually, under the assumption (Hd) (symmetric
double well problem), in the second case of the alternative.
Let us indeed prove the following result.
Proposition 3.3 :
There exists constantsC, ho and 70 > 0 such that, when/3h > 70, h e]0, ho],
we have
(x2o)p,Ho > (jx2<f>l dx) ■ (l - Cexp(-^) - Cexp(-^fe)) . (3.12)

320
This leads immediately to
Corollary 3.4 :
For any e > 0, there exist constants C\(e) and ho(e) such that, for any
h e]0, ho(e)] and P, such that
Ph>Ci(e), (3.13)
then we have
(x2o)0,ho> flg(l - «) - (3-14)
Proof of Proposition 3.3:
One observes indeed that
exp ~p\\ • (J x24>\ dx) + exp -/?A2 • (/ x24>\ dx)
= exp-pX, • (Jx^ldx) [l + exp-/J(A2 - \x) (|^)]
= exp-/JAi.(/»2^d»)[l + exp-/J(A2-Ai)(l + 0(exp-|))],
for any S < S, where 5 is introduced in (3.9). Here we have used the
more precise information coming from the semi-classical analysis that, in
the double well problem, one can find normalized (j)\ and </>2, such that
<f>x + fa := 2<f>(lefi) is exponentially localized 4 in the left well and such that
<f>x - (f>2 := 2<f>(r%9hi) is exponentially localized in the right well. The second
eigenvector fa being odd, we have also the property
^i)(x) = ^righi\-x).
We consequently obtain
[x2<fidx = 2 fx2{^left\x))2dx + 2 f x2^left\x)^ri9ht\x)dx ,
and
jx24>\dx = 2[x2(<t>(left\x))2dx -2Jx2^t\x)^ht\x)dx .
But according to the exponential decay of <f>(le^) and c/)(rt9ht\ we get
[ x2^left\x)^ri9ht\x)dx = [ x2^left\x)^leJt\-x)dx = 0{exp-j-) ,
4The decay is, for any e > 0, like G(exp - ^=1), with
d(x) = inf{| / y/v{t)-${-*>) dt\,S]
J—an

321
for any S strictly less than the Agmon distance S between the two wells.
We summarize what we have obtained in the following lemma
Lemma 3.5 :
For any S < S, we have
[ x2(j)\dx= lx2(j)22dx + 0(ex?-^) . (3.15)
We have now to control the other forgotten terms in the computation of
(x2)p,Ho- Because we are mainly interested by a lower bound of this
quantity, we shall have to find an upper bound for Tr (exp -/3Hq) = £]j>i exp -/?Aj,
and more precisely the quotient, as a function of (/?, /i),
0M)»-> (exp-^Ax^+exp-^^))-1 l^exp-jSA,-J , (3.16)
which we want to be small in a suitable domain.
Here we recall from (3.11) that, for some strictly positive Co, we have the
estimate
\j{h)>\i{h)+C0h, (3.17)
for j > 3.
It is sufficient to prove the existence of C > 0 such that
exp/3Ai (^exp-^Aj) < Cexp--f3h , (3.18)
i>3 C
for C large enough.
In order to control this expression, we can divide the sum in two parts
exp/3Ai (E^sexp-^Aj) = exp/3Ai({£j>3, ^.^exp-^)
+ exp/3A1(£Aj>aexp-/3Ai) (3.19)
for some a > 0 (possibly /i-dependent) to be determined.
We may assume without loss of generality that
%o) - 0 . (3.20)
The first part I\ of the sum can be estimated by
#{j| Xj <«}exp-/3(A3-A1) KCah^exp-^h. (3.21)

322
Here we have used a very weak version of the Weyl Formula (compare
for example with the harmonic oscillator) and the fact that, for a small,
f / £\<adxd£ ~ a, where p(x,£) = £2 + v(x). (See for example [26].)
Note here that, in order to look for an optimal result (by playing with a),
we need an estimate which is uniform with respect to a 6 [0, ao].
We have obtained
h <Cah~lexp-^h. (3.22)
Let us now analyze the second part I2 of the sum. In order to get this
estimate, we first use the Golden Thompson inequality [38]) saying that,
for any t > 0,
Tr exp-tHo < ( [ exp-th2£2dt)( f exp-tv(x)dx) . (3.23)
This leads to
Tr exp-tHo < C—([exp-tv(x)dx) . (3.24)
Uh J
If we use that v has non degenerate minima (Hd) and (3.20), we obtain,
for t > 1, the existence of a constant C such that, for all h e]0, feo],
Tr exp -tH0 < ^ . (3.25)
th
Coming back to the definition of I2 we write the inequality
h < exp —^ • (exp -(4AX - a) • ( ]£ exp -- Xj
When a > 4Ai, we obtain
^Aj>a
h < exp -^ • Tr [exp-^Ho] . (3.26)
Taking a = Dh, with D large enough, we get, from (3.26) and (3.25) with
h < const. /T1 /T1 exp--/?/i . (3.27)
Combining (3.27) and (3.22) with a = Dh, we have obtained (3.12)undef
the condition f3h > 70 > 0. This completes the proof of the proposition.
Remark 3.6 :
These arguments can be extended to any dimension. The semi-classical
analysis involved in the argument is presented for example in [27].

323
Remark 3.7 :
In the case when the potential is v(x) = (\x\2 - l)2 then the Bogolyubov
inequality permits to have a universal lower bound for (x2)p^. This is men-
tioned in [40] which refers also to [6]. Again this argument is true for any
n. We observe that when the dimension n is strictly larger than 1, we are
no more in a double well situation. We have indeed a uniformly degenerate
well given by \x\2 = 1 (cf [28]) and no tunneling is involved. Note that the
splitting A2 — Ai is of order 0(h2) as can be seen for example by taking
polar coordinates.
4. The strictly convex case
This section will be devoted to the case when the involved potential or
phase is convex.
4.1. STRICTLY CONVEX CASE, CLASSICAL CASE
Let us first consider the "classical" case. The result is analyzed (for one
dimensional lattice) for example in [31], where under suitable assumptions
of strict convexity of the family of potential $(m) on iRm, it was shown
that
lim Jn-(iE.-«»)2«p-^H(«)<fa = 0> {41)
m-H-00 fexp-/3$(m)(x)dx
This is indeed a consequence of the following control of the correlations
y^ 1 Ir™ xlXi exp -/?$(")(s)cfa, ^ ^
V fexp-0*W(x)dx l~ P [ ]
which leads to a convergence in O(^).
Let us discuss these results more precisely. The starting point is the Brascamp-
Lieb inequality.
Theorem 4.1 :
Let F(x) = exp(-$(#)),£ G Rm, with $ in C2 and strictly convex. We
assume that $ has a minimum and consequently F decays exponentially in
all directions. Let f e C1(IRm)f and let us assume that var / < 00. Then
var/^V/,^)-^/) (4.3)
where V/ is the gradient of f.
Here all the mean values (•) are with respect to the measure exp -$(x)dx.
An immediate consequence of (4.3) is

324
var f < \ ll|V/l2|lL2(i?";exp-<E(E)d*) _ ^
~ miX£Rm Amin(Hess$(a;)) fexp-$(x)dx
When $ is even and when f(x) = ^ Yl^Li xii we obtain
J (jS^'exp-*(«)& < ^ 1
fexp-$(x)dx ~ m inf Amtn(Hess$(a;))
In particular, we obtain
Proposition 4.2 :
Let $ = $(m) (m G Wj a family of even strictly convex potentials on IRm
such that there exists a > 0 such that, for all m G IN',
Hess$(m)><7, (4.6)
then we have, for any /3 > 0,
lim ~ r n^( W x i =°- 4J
m-^+oo /exp-/?$(m)(£)<to
As typical example, we can apply this result for
m st m
*<■»>(*) = £ ^(^)+^ x; I*,- - ^+ii2, (4.8)
where vj is even and satisfies t>"(#) > a > 0.
We observe that, when v = Vj, we have also the property (4.2) in the non-
convex case (using the approach of the transfer matrix [20]).
4.2. THE STRICTLY CONVEX CASE, QUANTUM CASE
Similarly to the classical case, it is interesting to observe the following
proposition
Proposition 4.3 :
Let $ = $(m) (m G JN)be a family of strictly convex potentials on IRm such
that (4.6) is satisfied for some a > 0. Then, for H^ = -h2A + $(m\ we
have
Tr
(^zti^y^p-H^
m-H-oo Tr exp-if(m)
*- = 0 . (4.9)

325
Corollary 4.4 :
Let v satisfying (Ha)-(He) and, instead of (Hd),
v"(x) > 2J2 > o, \/x e m,
for some u > 0, then,
P(P):= lim PM=0 (4.10)
|A|-*+oo
About the proof of the proposition:
We shall again apply the Brascamp-Lieb inequality [7] and techniques
developed in their article.
We just give the proof for the case considered in the corollary. We know that
the restriction to the diagonal of the distribution kernel of exp -fiH^' is
logconcave as a limit of logconcave densities associated to the distribution
kernel of fexp-^exp^^J . What has to be verified is a quantitative
control of the strict logconcavity in the procedure. The techniques
developed by Brascamp-Lieb [7] as recalled in Simon [44] and also in [20] are
actually relevant. Theorem 4.3 in Brascamp-Lieb (and the arguments
presented in Section 6 of this paper for the proof of Theorem 6.1) say indeed
that it is sufficient, for the study of the case when
with v(x) = u2x2 + R(x) and R positive, convex, to analyze the quadratic
case and that the distribution kernel Ka{P) of exp -f3H^er' can be written
as the product
- of the kernel obtained for exp-/? (Z^ga(~~^2A^ +^2|^|2)) .
- and of a logconcave kernel.
But the kernel of exp-/? [£^GA(-/i2A^+u;2|a^|2)] is explicitely known
using the Mehler's formula. We recall indeed that the Mehler's Formula,
which is obtained by explicit computations (see the book of Simon [44] p.
36-38), expresses the kernel of exp -t(-^d2/dx2 + \x2 - \) by
Ht(x,y) = tt-^I - exp -2t)~2
exp[-(l - exp-2t)-1(l(l + exp-2t)(x2+y2) - 2exp-txy)] .
(4.11)

326
or
Ht(X,J/)= i (4 12)
^-i(l_eXp-2t)-5exp[-(27iie)(^ + 2/2) + idhi^]> ^ '
We consequently get the existence of a > 0 and independent of |A| such
that
Hess[-ln*A(j8)](aO >^>0
where
h((3)(x) = Kii([])(x,x).
a can be chosen as equal to
u cosh t - 1
2h sinh t
or
a = ^-tanh(|), (4.13)
with
t = /3hu. (4.14)
This gives finally the stronger version of the corollary
Proposition 4.5 :
IfV is strictly convex with (Hess V% > 2u2 , Vz G Rm, then
0<FA(/3)<^f, (4.15)
with
Remark 4.6 :
Let us consider two limiting cases.
When (5 —> +oo, we find that
and consequently
When h —> 0, we get
lim a = —- ,
/?-++oo 2h
lim PA(/?) < ttt —
lim a = 3u2 ,
/i-K)
15™ P. ( R. h\ S
77ms corresponds to the classical result for the measure exp-f3V(x)dx
(4.17)
(4.18)
(4.19)
(4.20)

327
4.3. THE QUANTUM CASE, LIMIT AS /? -»■ +00
Let us further analyze (4.17) and (4.18). If we perform the limit /3 -4 +00,
we get, with B = (mE;^)2,
(^Bexp-^r, = /(lr,W|l). *. (4.21)
0-H-ooV Tr exp-f3HAPer ; J V|A| ;
We have used here that the lowest eigenvalue Ai = A^ of H^r is simple
and (j)± is the associated normalized strictly positive eigenfunction.
When V is strictly convex, we can use Brascamp-Lieb and get
/(jXjE^)2^*)2^ * j^j(inf A^^Hesst-ln^^x)]))"1 . (4.22)
We recall indeed a lower bound for Amtn(Hess[-ln<^](#))) through the
maximum principle ([48]) by the square root of infa,Gi?|A| \min(H.essV(x))).
As expected, we recover (4.18).
5. Connection with the Splitting.
We analyze in this section different standard relations connecting the
splitting between the two first eigenvalues of the Schrodinger operator and the
different quantities introduced in the preceding sections. Our initial remark
is that we can also write (without using Brascamp-Lieb or the condition of
strict convexity but keeping V even) the general inequality
(]X| Z^)2^! (*A)2*faA < ]Xj(^ - Af )"1. (5.1)
/
This is indeed just the minimax principle5 saying that
(AA-AA)= inf ^§^, (5.2)
ff{xA)(tf{xA)2dx/i = 0
where (•) is computed for the measure (f>i(xA)2dxA :
(•> = OocA , (5.3)
(remembering that it corresponds to the case (3 = +oo).
This is then applied to the function
/(*A) = T7T I> • (5-4)
11 t'GA
5 One can extend C£° to slowly increasing C°° functions in our case.

328
The condition of symmetry on V (which is a consequence of (He) for our
specific example) gives the orthogonality of /$^ to $^.
In the case when X{ e IRn with n > 1, we have to introduce the functions
/(•?)(#A) = 4t YlieA xi (f°r 3 — 1»""" » n)- This change the discussions by
unimportant constants. We assume n = 1 in all this section.
Taking account of the lower bound for the splitting given by J. Sjostrand
[48] in the case of the Schrodinger operator with strictly convex potential,
we recover (4.18).
Conversely, we can interpret this inequality as
aa - a? < jxjt/Cjii?^2^^)2^"1 • (5-5)
This means that if we can prove by other means the property that
[/(jX[Ea!i)a^(«A)2«faA]>P>0, (5.6)
with p independent of A, then we get that
Jim |AA-AA| = 0. (5.7)
|A|-*+oo
A lower bound for Pa(+oo) gives an upper bound for the splitting but
Theorem 1.2 gives the starting point for finding this lower bound and this
will then permit to prove Theorem 1.3. We shall mainly follow the proof of
Barbulyak-Kondrat'ev but with a small change.
In [4], the authors take indeed first the limit |A| -> +oo and then the limit
/3 -> +oo. We shall proceed for the application to the splitting in the
inverse order.
Proof of Theorem 1.3:
We start from (1.25)
h
> (4)pa ~ 2fkf EP€A*\{o}( jfey)5 coth
Taking the limit /3 -> +oo, we first obtain
(h*pJE(p))
(5.8)
<- \xk)+°°A ~ 2|A| Lp£A*\{0}\ JE(p))2 •

329
Now, as |A| -» +00 and if d > 2, the right hand side is estimated from
below by
F(/?)>lim inf (x2k)+00iA - \j'*h , (5.10)
where we recall that
'< = W<Lrm'Up- (5-u)
We now observe that Id is finite for d > 2.
We have seen in Section 3 that
<4W > (xlh,H0 • (5.12)
We can then take the limit /? —>• +00 in this inequality and obtain
(Zfc)oo,A > (*o)
00, Ho •
(5.13)
The right hand side is consequently independent of A and we get, for any
A,
^n, > (/,w*)- mjjjk/*' (5-14)
where (j)\ is here the first normalized eigenfunction of the one particle hamil-
tonian Ho = -h2-^ + #(#).
In the thermodynamical limit, we get first
^IM^oo/^S^2^^)2 dxk * (J4M*)2dx) - \{j)*h ■
(5.15)
Using the semi-classical analysis of Section 3, we obtain, observing also that
the condition
ql-yJ-"h>U (5.16)
is satisfied for sufficiently small /i, the following
Proposition 5.1 :
Let v and J satisfying (Ha)-(Hd) and d > 2. Then there exists ho and
po > 0 such that, for h e]0, ho], we have
limJni [(^-.Y:^M(^)2dxA>po. (5.17)
|A|-*+oo J \A\ *-r

330
In particular we have obtained the proof of Theorem 1.3 through (5.5).
We recall that qo corresponds to the minimum of the potential v(x) =
v(x) + Jdx2. This potential may become convex as J increases without the
same property for v. In the case when v defines a symmetric double-well,
the inequality v"(0) < 0 is satisfied in the most simple generic case and for
J large enough, more precisely when t/'(0) + 2Jd > 0, we get cases when
v has a unique minimum at 0. The mean value /x2(j)i{x)2dx satisfies then
semiclassically (using the harmonic approximation)
/x2<j>1(x)2dx ~ k . (5.18)
J y/\v»(0)+dj
The eigenfunction is indeed localized near the minimum of 0, that is at 0.
This changes of course the discussion but a part may remain true if the
following inequality is satisfied
i 1 - hJ~* > 0. (5.19)
This is clearly satisfied when t/'(0) + 2d J is a strictly positive sufficiently
small number. This case is treated by the following proposition.
Proposition 5.2 :
Let v and J satisfying (Ha)-(He) and d>2. Let us assume that v has a
unique non degenerate minimum at 0. Let be satisfied the condition
(\s"m-'-\-J-'h>0. (5.20)
Then there exists h\ and p\ > 0 such that, for h e]0, hi], we have
^IaS+oo/^?*1'^^2 ^ ~ Pl • (5'21)
When v"(0) > 0, we are happy to verify that the method does not work. We
have indeed the inequality Id > -4= (as a consequence of Cauchy-Schwarz)
showing that (5.20) can not be true, (cf Theorem 7.1 in [9]).
Remark 5.3 :
Another case where /(mZ^i)2^^)2 dxA is controlled is the case con-
sidered by Pastur and Kozurenko [40], The potential is (1 - \x\2)2 and the
proof is also valid for n > 1. We emphasize that no tunneling is involved
as n > 1.

331
Remark 5.4 :
As will be clear in Section 6, we note here that, by taking the limit /3 -» +oo
before to take the thermodynamic limit, we have eliminated a singularity at
the origin which makes the argument valid for d > 2 instead of d > 3 in
the proof of Barbulyak-Kondrat 'ev. The direct study of the limit /3 —> +oo
could also lead to weaker estimate on v but we shall loose the control of the
limit (3 —> +oo.
The case: d = 1.
Another interesting point is to analyze the case d = 1. The sum
is divergent as |A| —> +oo, but this divergence is controlled in ln(|A|). This
leads to the statement that the splitting remains in 0(-^) for h<h0 and
|A| < exp ^, where T is explicitely computable.
This condition is much weaker than the condition (1.7) given in [30].
This may be connected to phenomena discussed in [23] and to the Ising
model with transverse field discussed in the introduction. As h -» 0, the
condition for the limit case is when
ln|A|~^[2J]2 . (5.22)
We note also that the phenomenon is effectively related to J ^ 0. To
summarize, we prove in this case that
Theorem 5.5 :
Let d = 1 and v satisfying (Ha)-(Hd), then there exists a constant C such
that
C 1
A"-A^JAJ(l-CMog|A|)+; (5-23)
As mentioned in the introduction and as was communicated to us by J.
Frohlich, this condition on |A| is due to the method (when n = 1). An
approach using the Peierls trick could be more effective in this case [10].
6. On the heat kernel for (5 large
This section describes mainly the results obtained by Barbulyak and Kon-
drat'ev but take account of the stronger results obtained by semi-classical

332
dp .
analysis in Section 3 concerning (#2)/?,//0.
We assume that d > 3 and n = 1.
Let us start from (1.24) in the form
W > <4>«. - 2^1,^^^ [(W*«)*
(6.1)
We want just to precise under which conditions on /3 and h we can get the
strict positivity of P(/?).
We observe as in (5.16) that
ql ~ \hh J'* > 0 , (6.2)
for sufficiently small h.
A natural critical value of f3 is the solution f3 = f3\ (h) of
q2° = \ (2^F /_ „[d{JE{p))~*coth [wje<p))*] dp ■ (6-3)
Clearly f3\ satisfies fi\{h)h < Co.
Combining with (3.14) which analyzes the convergence of (x2)^^ to q% as
h —> 0, we obtain the following theorem (due essentially to [4])
Theorem 6.1 :
If d > 3, then, for v satisfying (Ha)-(Hd) and for any e > 0, there exists
ho(e) > 0 and Co(e) such that, for all h < ho and (3 such that /3h > Co(e),
we get
P((3)>q2(l-e). (6.4)
Remark 6.2 :
It is interesting to look at the limit J —> 0 . We find
lim [ d(JE(p))"2 (coth [/3h(JE(p))$]) dp = +oo . (6.5)
On the other hand, we know that the situation (say for the splitting) is quite
different between the two cases:
For J = 0, the splitting is in 0(exp -f) for any S < S, while, for J > 0,
one hopes a splitting in 0(exp—*-%-).
In the same direction, let us observe that this strict positivity of P(f3) is no
more true in the case J = 0 as can be seen by direct computation. We have
indeed, for A = {1, • • •, m},
1 Tr(x2exp-/3ff0)
Fa(/?) " m TY(exp-/?ff0) ' ( }

333
which tends to 0 as m -» +00.
We have used here that, by symmetry of v,
Tr(sexp-j3ffo) = 0.
About the Pastur and Khoruzenko results:
These authors discussed two different cases of operators in the case when
v(x) = -%\x\2 + f |z|4 with a > 0, b > 0.
- The first case with a > 2Jd is called ferroelectric model of the disorder
type. The corresponding v describes a double well (if n = 1)
6W=<^)^*<. (6.7)
- The second case corresponds to 0 < a < 2Jd and is called ferroelectric
model of the displacement type. The potential v is now convex.
They prove actually in this context and without the distinction between
the two cases the following theorem for the above model,
Theorem 6.3 :
If d>3 and if
J>h2(n + 2)2b2I2d/4a2 , (6.8)
then there exists a temperature Pq1 such that, for (5 > /?o, the corresponding
P(fi) is strictly positive.
The control of (#|)/?,a is obtained by using the Bogolyubov's inequality. We
recall that this inequality (See [42], Lemma 5.5.1) gives in particular that
<[[C*,tf],C])M>0. (6.9)
In our context, this inequality is applied with
c = 75X-- (6-10)
This leads, by considering the limit (5 —> +00 before to take the
thermodynamic limit, to the following result for the splitting between the two first
eigenvalues as |A| tends to 00.
Theorem 6.4 :
1
If d > 2, then, for v defined by (6.7) and ho = /*+2)bi > we have> for
he]0,ho[,
Urn |A£-Aft = 0. (6.11)
|A|-*+oo

334
Let us just detail a variant of the argument given by Pastur-Khozurenko
for the case when (5 = +00.
In the limit /3 -» +00 and in the case when H = -A + V is a Schrodinger
operator, the inequality (6.9) becomes simply
/([ [C*, H],C\$)(x)$(x)dx > 0 , (6.12)
where $ is the first normalized positive eigenfunction of H. This is easily
and directly obtained by the minimax principle. In particular, if C is given
by (6.10), we get
J2 [{d2V/dxjdxk){x)${x)2dx > 0 . (6.13)
jk J
When V(x) has the form (we take for simplification n = 1)
V(x) = J2 v(xj) + JJ2 \x* ~ xj\2 '
j i~j
we simply get
J2 [v^x^ix^dxyO.
j
By invariance of $ in this case, we get that, for any j,
[v"(xj)<$>{x)2dx>0. (6.15)
In the more specific case when v(x) = \x* - \x2 we obtain
[x2^{x)2dx>^-. (6.16)
The proof through the infrared estimates is then easy.
Remark 6.5 :
Jf we compare with the semi-classical lower bound obtained in Proposition
3.2, we note that when n = 1 and a > 2Jd, we have ql = a~^rf. This
suggests that the semi-classical result is far to be optimal and that we should
be able to get the results with assumptions on v instead of v.
An easy extension of the proof by Pastur- Khozurenko gives that Theorem
1.3 and some weak form of Theorem 6.1 is true if the pair (v, v) satisfies
(Ha), (Hb), (He) and (H'd) with
(6.14)
{H'd) v"{x) < -70 + Jix2 ,
(6.17)

335
for real 70, 71 with 70 > 0. In many cases, we can take 70 = -V'(0). The
point 0 corresponds for example to the top between the two wells.
We note also that this approach does not make use of the Ginibre
inequalities.
Extensions to n > 1 could in this spirit be also interesting, because avoiding
some assumption occuring in the validity of these inequalities.
7. Infrared estimates: the classical case.
The basic reference is the paper by Frohlich, Simon, and Spencer [12]. This
is also presented in detail in the book by Glimm-Jaffe [14]. We treat actually
here a rather simple example.
We change a little the notations in order to follow this last reference. The
proof is more general in the reference. The interaction Hamiltonian J (A)
corresponds to the interaction term
m = -j £ xt-xj. (7.1)
i rsj j
i,je A
In this section, we take J — 1 because J can be included in (3. Here X{
belongs to Mn and X{-Xj denotes the scalar product in ZRn. The other terms
are put in the one-particle measure and we are considering the measure
<W,A = Z~l exP -/^(A) II dw(*0 , (7.2)
i'ga
with d/j>i(xi) = exp -v(x{)dxi, where v satisfies some natural condition (for
example (Hb)) at 00 permitting to control the interaction.
Moreover, v(x) = v(—x) (condition (He)) when n = 1.
When n > 1, Glimm-Jaffe assume that v is invariant by SO(n).
This assumption of symmetry permits to have the property
(^)/?,A := / xi dfifsA = 0 . (7.3)
We introduce
9A{p,x) = -7F=iY,exP-iP'i x*' (7-4)
VIAI^A
This is a function defined on A*, the dual lattice introduced in (1.23) of
A, with value in (Cn. For j G {l,-,n}, we denote by gl (p,x) the j-th
component. We now introduce
SfiM = (9(p, ■)■§(-?, -)W = it(9(j)(p, -yg^i-P, -)W = (\h(P, -)l2> •
(7.5)

336
We then have
5/?,a(p) = 5ZexP ~i£P (x° ' *')/*.a • (7-6)
This is simply the discrete Fourier transform of the correlation function
A3£^\A\-2(x0'X£)pA.
The main result in the finite lattice version is
Theorem 7.1 :
For all p6A*\ {0}, we have
Corollary 7.2 :
Lei d > 3, and /e£ /? 6e sufficiently large, so that
Km |A|inf <*^,a • 0 > (2jt)-' / ■ ". dp, (7.8)
F^:=mlim. <(T7iE^)V>0. (7-9)
|A|-H-oo |A| ££
Proof of the corollary:
We have:
<(iTT E *<) V = (TTt)2 E <** • **>AA = 777^(0) • (7-10)
By the inverse discrete Fourier transform, we have also (Plancherel formula)
£ TtAaW = <*oW - (7.H)
pGA* |iV|
We then write this identity in the form
^AA(0) = (x20)PA - Y, \T&M ■ (7-12)
11 P€A*\{0} ' '
The theorem gives
^(,W4'-Hi101«5^)' (7-13)

337
Taking the thermodynamic limit we then obtain
P(/3)> Mm inf (xg)«A - (27r)_d / . n „ dp. (7.14)
V- |A|-+ooV 0/ftA V ' i]-^4/?£i=isin2(^) ' * '
In order to complete the proof, we need some control of (#o)/?,A in the limit
/3 -> +00 and |A| -> +00. This may lead to conditions permitting to apply
the Ginibre inequalities as analyzed in Section 2.
The proof of Theorem 7.1 is based on the following lemma
Lemma 7.3 :
Let d be the forward finite difference quotient. Let fa 6 £2(h\IRn) (a =
V",<0, / = (/a)a=i,.,de^2(Ax{l,...,d});iRn). Then
(exp L • (£ Oafa))) < exp ((2/?)-1||/||22) , (7.15)
where
11/16 = £/aW2 = £l/£W- (7-16)
This inequality is sometimes called the "Gaussian domination estimate".
We recall that, for / <E £2{2Zd; Bn),
(daf)(£) = f(£ + ea)-f(£), (7.17)
where ea is the unit vector in the a-th coordinate direction. We recall also
that
x-g= £ xi-g(i), (7.18)
for ge£2{A).
Proof of Theorem 7.1, assuming Lemma 7.3:
We substract 1 from both sides of (7.15), substitute ef for /, multiply by
€"2 and let e -> 0.
Thus we obtain6
{(*'Y,d*f°) )<r1\\f\\l. (7.i9)
6 We shall need the natural extension of this inequality for functions with values in

338
Note here that we have used the invariance by translation of (•) in order to
obtain (x • dafa) = 0. We note also that this inequality can be extended to
complex valued functions.
With d* the negative of the backward lattice difference quotient, we define,
for a given p£A^\ {0}, for j = 1, • • •, n, fj e £2(A X {1, • • •, d}; IRn) by
/aj = «(-AH"Wii (7.20)
where Sj is the vector in IRn
We recall that the periodic Lattice Laplace operator Aper on A has as
eigenvalues
with
The corresponding eigenvectors being defined as
A9^H>Xp(^) = |A|~2 expip-i.
This gives the bound (7.7), by easy computations (cf also [14]).
Remark 7.4 :
If we consider, say in the case n = 1, for g constant and orthogonal to
KerAper, the vector fa = 3£(-A*cr)-10, the inequality (7.19) gives
((..ffl^f^t-AH-1?)'?). (7.21)
This is interesting to compare with some forms of the Brascamp-Lieb
inequality (See [7], [20], [51] and [39]).
About the proof of Lemma 7.3:
The argument for this lemma is called a "multiple reflection bound". We
first observe that
x-dag = J29(t)(~xt + xi-eQ) •
Taking g = fa and summing over a, we get
/ =(eMZax-dafa)eM-rp\\f\\2p))
/e*p(-E<,q f ((-»f+«<-ea+/?-1/a(<))a) 11^
/exp(-E*,a f (-*H-X,_ea)2J n^<

339
The desired inequality is then
This is obtained by reflection arguments. The first step is an easy but basic
inequality:
Lemma 7.5 :
Under weak assumptions on the measures n and v defined on Mn, we have
for any a in Rn
(/exp[-±(x - y - a)2]dn{x) ■ du{y))
< (leM~Ux - y)2W(x) • Mv)) (leM-lix - y)2¥v{x) • rfj/(y))
(7.22)
The proof is simple. We just use the Plancherel formula
(/exp[-±(x -y- a)2]dn(x) • du(yj)
= const, f/exp-^2 expect z>(£) • jj.(£)dH .
We then Cauchy-Schwarz the right-hand side and get
(/ exp - ±£2 exp i£a v(£) • £(£)#)
< (/exp-±£2 |j>(£)|2^) • (jexp-ie HOI2 d() .
(7.23)
(7.24)
The inequality (7.22) is then clear, using again the Plancherel formula.
A quantum variant of this argument is proposed in Lemma 4.1 in [9]. It
involves the Trotter product formula. We shall come back to this point in
Section 8.
The reflection argument :
For any h G £2{A x {1, • • •, d}; 2Rn), we introduce
^({Mm)= /«p-f ss^-x^+zj-xw)2 n^- (7-25)
The proof is then reduced to the proof of the inequality:
Z({ha(£)}) < Z0 , (7.26)
where Z0 is by definition Z({ha(£)}) for ha(£) = 0:
Zo = Z({0}).

340
We assume that A = [0, m - l]d with m even. We first observe that
s\iphZ({ha(£)}) is attained (Z(h) tends to 0 as \\h\\ -> +oo). Inside the
class of the maximizers, we can choose h° such that Z(h°) = s\xph Z({ha(£)})
and which has a maximal number of components equal to 0 inside this class.
Let us show that this leads to a contradiction if h° is not identically 0.
After easy manipulations we can assume that h\(m - 1,0, • • •, 0) ^ 0.
We now rewrite the measure
exp-f EE(*<-*<+«. + P~lh*W)2) IId^
ieAa=l ) t£A
as the tensor product of two measures multiplied by an interaction density.
The first measure is
fi' = (exp -f [£P(a* - xe+ea + /T1/^))2])
xri£€A,A€[0,f-l]^ )
with
T,._f (^,a)€Ax{l,..-,d}
~\ti
€[0,f-l],^ + ea€[0,f-1] J
and the second one is
M" = (exp -f [Ep,^ - x,+ea + /HM4)2])
xn^eA,Ae[f ,m-i]^ )
with
r//._/ (<,a)ax[l,-,<|| 1
' '"I £i € [f,m-!],£! +ea€[f,m-l]. j
The interaction density is then given by
I(x) =exp-f (DJ(^f-M' -*»^ + /J-1fci(f - M'))2])
xexp-§ (E^[(^m-i^-^' + /?"1^i(m~l,f))2])
If we change the name of the variables by posing yt = x^ where a denotes
the symmetry around the hyperplane: l\ = mf^. The interaction takes the
form of some product of exp — § (a?/ — y/ — /i^)2, corresponding to some I e A
such that t\ — 0 or t\ = y - 1. We now apply the inequality given by the
lemma and obtain
Z0 = Z({h°a(£)})2 < Z{{h\(£)})Z({hl{l)}).
Here h\{£) and h2a(£) satisfy
hi(e) = hi(e)(v(t)),

341
and are equal to 0 when ai = 1 , i\ — y -1 and when ol\ — 1 , £i = m — l.
We observe also that necessarily
Z0 = Z({hi(£)}),j = l,2.
But at least h1 or h2 has a larger number of vanishing components than h°
and has the same property as h°.
This gives the contradiction.
8. Infrared estimates: the quantum case.
8.1. INTRODUCTION
We mainly describe the results given by Dyson-Lieb-Simon [9] with some
additional remarks given by Pastur, Khozurenko, Barbulyak, Kondrat'ev.
We shall consider unbounded operators on ®ie\L2(IRn;IR) of the form
H = HpAer with
#Tr - T,£eA Hi + 2 J Et~j \xi - xj\2 /g j\
The operator Hi works only in the variable xf.
Ht = -h2AXt + v(xt). (8.2)
and Hi has the same form with t; replaced by v.
In all this section, v satisfies (Ha)- (He), but n is not necessarily equal to
1. When n > 1, we assume that v is invariant by SO(n) which implies
Tr^exp-^n^O. (8.3)
8.2. THE DUHAMEL TWO-POINT FUNCTION
The first point is the introduction of the Duhamel two-point function. For
quantum systems in finite volume (attached with some selfadjoint hamilto-
nian H on L2(iRn'A')) with partition function Z = Tr (exp -/?i?), we define
the Duhamel two-point function (DTF) by
(A, B) = Z~l [ Tr [exp -x/3H A exp -(1 - x)/3H B] dx . (8.4)
Jo
A and B are non-bounded operators and we assume that (8.4) has a sense.
This will be clear in our applications.
This (DTF) has the following properties.
(A,B) = (B,A), (8.5)

342
by cyclicity of the trace and change of variable in the integral x »->■ (1 - x).
{A\A)>0. (8.6)
We can indeed, with Cx = exp - ((^)l3Hj A exp -(§/?#), write (A*, A)
as
Tr [exp -X0H A" exp -(1 - x)/3H A)] = Tr [C*CX] .
Let us also observe that one recovers the mean value of A through
(A) = (A, 1). (8.7)
We shall use also the following property which actually explains the
introduction of the (DTF) function. If
(B)^ = {Tr [exp -(3H + M)]}"1 Tr [B exp(-(3H + pA)] (8.8)
then
= (A,B)-(A)(B). (8.9)
d(B),
thet
in a perturbation expansion of \l \-> Tr [exp(-/?Jff + fiA)]
In particular when A = S, the term ^/j,2(A,A)Z is the second order term
8.3. A TRICKY FUNCTION
The second point is a tricky lemma
Lemma 8.1 :
There exists a function f from [0, +oo] to [0,1] defined implicitely by
f(x tanh x) = x~l tanh x .
This function is convex, monotone, decreasing and satisfies
lim fix) = 1 , lim fix) = 0 .
x-+q v ' :r-++oo V '
We now introduce three thermodynamical quantities.
g(A) = \{A*A + AA*) = ^-Z-1Tv[(AkA + AAk)exp-fiH] . (8.10)
Zd Zd
b(A) = (A*,A). (8.11)

343
c(A) = ([A\[H,A}}). (8.12)
From essentially the Jensen's inequality, one gets under suitable
assumptions the following theorem [9]
Theorem 8.2 :
b{A)-9{A)f{^)h (8'13)
This is extended by convexity to a finite sum A{ of operators in the form
£6(4) > (Zg(Ai)) ■ /(4^,C^)) • (8-14)
We can now deduce from an estimate on b and c an estimate on g through
the following Theorem
Theorem 8.3 :
Suppose b > gf(z), b,g,c > 0 and b < bo, c < c$. Then we have
g<go (8.15)
where
I So = ^(c0b0)2 cothxo (8 16)
8.4. GAUSSIAN DOMINATION IN THE QUANTUM CASE
We recall a small extension of Lemma 7.5 which was used in the classical
case.
Lemma 8.4 :
Let %\ be a finite-dimensional vector space and let H = H\ ®H\. If A, S, • • •
are operators on %\) we use the same symbol for A® Id, B®Id,- • -, and the
symbols A, B, • • • for Id® A, Id®B, • • •. Then for any selfadjoint operator
A, S, C{- - with real matrix representations and real numbers h\, • • •, hk
(Tr {exp[A + B-EL(C.-a-M2]})2
< Tr {exp [A + A- £?=i(Ct- -Q- h{)2]} (8.17)
x Tr {exp [B + B- £-=i (d -Q- h{)2]} .
We now apply the Gaussian domination argument. Lemma 7.3 is now
replaced by

344
Lemma 8.5 :
Let H be a Hamiltonian of the form (8.1). Let {ha(£)} , £ £ A, a = 1, • • •, d
be d\A\ vectors in Mn. Let A be [1, • • •, m]d with m even. Let (j>(h) = J2e W)'
X£.
Trex?[-(3H + <f>(j:adaha)] '»>»2
Trfo-m ^XP^' (8J8)
where \\h\\2 = Zt,a\ha(e)\2.
Similarly to the way we get (7.19) from (7.15) and taking account of (8.9),
we deduce from (8.18) the inequality
d / d d \
b(x-^dafa):= U^U,^EU 1 <0_1||/||*. (8.19)
a=l \ a=l a=l /
We now introduce
b^ = b(Aj) = (g^\g^), (8.20)
with
Aj = 9JP •■= KXpSi ■ (8-21)
The analog of Theorem 7.1 is then true when one uses the (DTF).
Theorem 8.6 :
For hamiltonians of the form (8.1) in square boxes A of size m ( m even
integer) and under the same conditions as in Lemma 8.5 holds,
bW<(2pjE(p))-11j = lr~1n. (8.22)
8.5. END OF THE PROOF OF THE INFRARED ESTIMATE
We can now finish our sketch of the proof of Theorem 1.2.
In order to use Theorem 8.2, we have also to control the bracket: [g^ , [/?if, #_p]].
We immediately get
\9? , W . 9^]] = Wh2 ■ (8-23)
We can then apply Theorem 8.3 with
b"=wjm' <8-24>
and
c0 = 2n/3h2 . (8.25)

345
The corresponding xo is given by
x0 = y/f32h2JE{p) . (8.26)
Coming back to the definition of g(A) = Ylj9{Aj), we apply Theorem 8.3
and get the equivalent of Theorem 7.1.
Theorem 8.7 :
For all p6A*\ {0}, we have
g(A) = SPA(p)<^(j^)l2coth
Uh2p2JE(p)y\ . (8.27)
The end of the proof is then similar to the proof of (7.13).
Acknowledgements :
I would like to thank F. Klopp for communicating the paper of V.S.
Barbulyak and Y. Kondrat'ev, J. Frohlich for his informations and J.P.
Solovej for motivating discussions. The redaction of these notes was also
stimulated by the preparation of the proposal for the european contract
"Postdoctoral training program in Partial Differential Equations and
applications in Quantum Mechanics" and by L. Rodino who initiated this
NATO ASI conference.
References
1. Agmon, S. (1982) Lecture on exponential decay of solutions of second order elliptic
equations. Math. Notes, t.29, Princeton University Press .
2. Albanese, C. (1989) On the spectrum of the Heisenberg Hamiltonian, Journal of
statistical Physics 55 (1-2), 297-309.
3. Albanese, C. (1990) Unitary dressing transformations and exponential decay below
Threshold for Quantum spin systems, Comm. in Math. Physics 134, 1-7.
4. Barbulyak, V.S. and Kondrat'ev, Y. (1992) The quasiclassical limit for the
Schrodinger operator and phase transitions in Quantum statistical physics,
Functional Analysis and applications, 26(2), 61-64.
5. Barbulyak, V.S. and Kondrat'ev, Y. (1990) Methods of functional analysis in
Problems of mathematical Physics (in Russian), Institut Mat. AN USSR, Kiev, 30-41.
6. Bouziane, M. and Martin, Ph. A. (1976) Bogolyubov inequality for unbounded
operators and the Bose gas, J. Math. Phys. 17(10), 1848-1851.
7. Brascamp, H.J. and Lieb, E. (1976) On extensions of the Brunn-Minkovski and
Prekopa-Leindler Theorems, including inequalities for Logconcave functions, and
with an application to diffusion equation, Journal of Functional Analysis 22 366-
389.
8. Brunaud, M. and Helffer, B. (1991) Un probleme de double puits provenant de la
theorie statistico-mecanique des changements de phase (ou relecture d'un cours de
M.Kac), Preprint LMENS.
9. Dyson, F.J., Lieb,E.H. and Simon, B. (1978) Phase transitions in quantum spins
systems with isotropic and nonisotropic interactions, J. Statistic. Phys. 18(4), 335-
383.

346
10. Frohlich, J. (1976) Phase transitions, Goldstone bosons and topological superselec-
tion rules, Acta Phys. Austriaca Suppl. XV, p. 133-269.
11. Frohlich, J., Israel, R., Lieb, E.H. and Simon, B. (1978) Phase transition and
reflection positivity, I General theory and long range lattice models, Comm. in
Math. Physics 62 (1), 1-34.
12. Frohlich, J., Simon, B. and Spencer, T. (1976) Infrared bounds, phase transitions,
and continuous symmetry breaking, Comm. Math. Phys. 50, 79-85.
13. Ginibre, J. (1970) General formulation of Griffiths inequalities, Comm. in Math.
Phys. 16, 310-328.
14. Glimm, J. and Jaffe, A. (1987) Quantum physics (a functional integral point of
view), Springer Verlag, Second edition.
15. Glimm, J., Jaffe, A. and Spencer, T. (1975) Phase transitions for <j>\ quantum fields,
Comm. in Math. Phys. 45, 203-216.
16. Globa, S.A. and Kondrat'ev, Yu. G. (1987) Application of functional analysis in
Problems of mathematical Physics (in Russian), Institut Mat. AN USSR, Kiev,
4-16.
17. HelfFer, B. (1988) Introduction to the semiclassical analysis for the Schrodinger
operator and applications, Springer lecture Notes in Math., n° 1336.
18. HelfFer, B. (1992) Problemes de double puits provenant de la theorie statistico-
mecanique des changements de phase, II Modeles de Kac avec champ magnetique,
etude de modeles pres de la temperature critique. Preprint LMENS.
19. HelfFer, B. (1992) Decroissance exponentielle des fonctions propres pour l'operateur
de Kac: le cas de la dimension > 1, in Operator calculus and spectral theory,
Symposium on operator calculus and spectral theory, Lambrecht (Germany), December
1991, edited by Demuth, M., Gramsch, B. and Schulze, B.-W., Operator Theory:
Advances and Applications 57; Birkhauser Verlag Basel, 99-116.
20. HelfFer, B. (1993) Spectral properties of the Kac operator in large dimension.
Proceedings on Mathematical Quantum Theory II: Schrodinger Operators, edited by J.
Feldman, R. Froese, and L.M. Rosen. CRM Proceedings and Lecture Notes.
21. HelfFer, B. (1996) Recent results and open problems on Schrodinger operators,
Laplace integrals and transfer operators in large dimension, to appear in Advances
in PDE, Akademie Verlag.
22. HelfFer, B. (1995) Semiclassical analysis for Schrodinger operators, Laplace integrals
and transfer operators in large dimension: an introduction, Paris 11- Edition.
23. HelfFer, B. (1996) On Laplace integrals and transfer operators in large dimension:
examples in the non convex case, to appear in Letters in Math. Physics (1996).
24. HelfFer, B. (1996) Semi-classical analysis for the transfer operator: WKB
constructions in dimension 1, in Partial differential equations and mathematical physics,
The danish-swedish analysis seminar (1995), Birkhauser (1996).
25. HelfFer, B. (1996) Semi-classical analysis for the transfer operator: formal WKB
constructions in large dimension, Preprint Universite Paris 11.
26. HelfFer, B. and Robert, D. (1984) Puits de potentiel generalises et asymptotique
semi-classique, Annales de VIHP (section Physique theorique) 41 (3), 291-331.
27. HelfFer, B. and Sjostrand, J. (1984) Multiple wells in the semiclassical limit, Comm.
in PDE 9 (4), 337-408.
28. HelfFer, B. and Sjostrand, J. (1987) Puits multiples en limite semi-classique
Vile cas des puits varietes-, Annales de VIHP (Section Physique theorique) 46 (4),
353-373.
29. HelfFer, B. and Sjostrand, J. (1992) Semiclassical expansions of the thermodynamic
limit for a Schrodinger equation, Asterisque 210, 135-181.
30. HelfFer, B. and Sjostrand, J. (1992) Semiclassical expansions of the thermodynamic
limit for a Schrodinger equation II, Helvetica Physica Acta 65, 748-765 and Erratum
67 (1994), 1-3.
31. HelfFer, B. and Sjostrand, J. (1994) On the correlation for Kac like models in the
convex case, J. Statist. Physics, 74 (1-2), 349-369.

347
32. Kac, M. (1962) Statistical mechanics of some one-dimensional systems, Studies
in mathematical analysis and related topics in Essays in honor of Georges Polya,
Stanford University Press, Stanford, California, 165-169.
33. Kac, M. (1966) Mathematical mechanisms of phase transitions, Brandeis lectures,
Gordon and Breach.
34. Kac, M. and Helfand, E. (1963) Study of several lattice systems with long range
forces, J. Math. Phys. 4, 1078-1088.
35. Kac, M. and Thompson, C. (1966) On the mathematical mechanism of phase
transition, Proc. N. A. S. 55, 676-683 and Erratum 56, 1625.
36. Kac, M. and Thompson, C. (1969) Phase transition and eigenvalue degeneracy of a
one dimensional anharmonic oscillator, Studies in Applied Mathematics 48, 257-264.
37. Kirkwood, J. and Thomas, L. (1983) Expansions and phase transitions for the
ground state in quantum spin systems, Comm. in Math. Physics 88, 569-580.
38. Lenard, A. (1971) Generalization of the Golden Thompson inequality, Indiana
Univ. J. Math. 21, 457-468.
39. Naddaf, A. and Spencer, T. (1995) On homogeneization and scaling limit of some
gradient perturbations of a massless free field, Preprint Princeton.
40. Pastur, L.A. and Khoruzhenko, B. A. (1987) Phase transitions in quantum models
of rotators and ferroelectries, Teor. Mat. Fiz., 73 (1), 111-124 .
41. Rebenko, A.L. (1996) Peierls argument and long range behavior of quantum lattice
systems with unbounded spins. Lecture in Ascona (June 1996).
42. Ruelle, D. (1969) Statistical mechanics, Math. Physics monograph series,
W.A.Benjamin, Inc..
43. Simon, B. (1974) The P(<t>)2 Euclidean quantum field theory, Princeton Series in
Physics.
44. Simon, B. (1979) Functional integration and quantum physics. Pure and Applied
mathematics, n°86, Academic press, New york.
45. Simon, B. (1983) Instantons, double wells and large deviations. Bull. A MS. 8,
323-326.
46. Simon, B. (1983) Semi-classical Analysis of low lying eigenvalues.
I Ann. Inst H. Poincare 38 (1983), 295-307.
II Annals of Mathematics 120 (1984), 89-118.
47. Sjostrand, J. (1993) Potential wells in high dimensions I, Ann. Inst. Poincare,
Section Physique theorique 58 (1), 1-41.
48. Sjostrand, J. (1993) Potential wells in high dimensions II, more about the one well
case. Ann. Inst. H. Poincare, Section Physique theorique 58 (1), 42-53.
49. Sjostrand, J. (1994) Evolution equations in a large number of variables, Mathema-
tische Nachrichten 166, 17-53.
50. Sjostrand, J. (1994) Ferromagnetic integrals, correlations and maximum principles,
Ann. Inst. Fourier, 44 (2), 601-638.
51. Sjostrand, J. (1996) Correlation asymptotics and Witten laplacians, to appear in
St Petersburg Mathematical Journal.
52. Sokal, A.D. (1982) Mean-field bounds and correlation inequalities. Journal of
statistical Physics, 28 (3), 431-439.
53. Thomas, L. and Yin, Z. (1983) Quantum Ising lattice systems, Comm. in Math.
Phys. 91, 405-417.
54. Thompson, C. (1972) Mathematical statistical mechanics, The Macmillan company
New York.

MICROLOCAL EXPONENTIAL ESTIMATES
AND APPLICATIONS TO TUNNELING
A. MARTINEZ
Universite Paris-Nord
Institut Galilee - Departement de Mathematiques
Av. Jean-Baptiste Clement
F-93430 Villetaneuse (FRANCE)
1. Introduction
The purpose of this lecture is to present a technique related to the study of
the behavior as h -> 0+ of the solutions u G L2(Rn) of partial differential
equations of the type:
P(x,hDx;h)u = 0 (1.1)
1 d N
where Dx = :— and the operator P(x,hDx;h) = Y^ h pk{x,hDx) is
1 dx to
assumed to have analytic coefficients. In particular, the eigenfunctions of
semiclassical operators (such as the Schrodinger operator: -h2A + V(x)
with V analytic) can be investigated in this way.
It is a well known fact in microlocal analysis that the behavior of u is
strongly related to the geometric properties of the principal symbol Po(#>£)
of P{x, hDx\ /&), where (#,£) G R2n. Actually, if P(x, hDx\ h) has analytic
coefficients, then some phenomena occurring for (#,£) complex can also
give rise to particular properties of the solution u. As a simple example,
if P is elliptic at some point xq (in the sense that po(x,£) never vanishes
for x close to x0 and f real with some uniformity as |£| -> 00), and u is
normalized by \\u\\L2 = 1, then there exists a positive constant S such that
u = 0(e~8lh) uniformly near x0. Not much is known about this £, but
one expects that it is a reflect of the distance between {xq} x Rn and the
complex characteristic set of P:
Ch<ir(P) = {(x,Z)eC2n;p0(x,Z) = 0}.
349
L. Rodino (ed.), Microlocal Analysis and Spectral Theory, 349-376.
© 1997 Kluwer Academic Publishers.

350
This is the case e.g. when P is the Schrodinger operator (see [2]):
outside the classically allowed region U = {V(x) < 0}, the solution is known
to decay like exp(-d(U,x)/ti) where d is the so-called "Agmon distance"
(or "Lithner-Agmon distance"), that is the pseudodistance associated to
the degenerate metric Max(V(x),0)dx2. Here this distance lives on the
"position-space" {£ = 0}, but as we shall see, different situations can be
considered in which the decay is described by an (#, ^-dependent function.
This is precisely for studying such situations that the tool we present
here is made, but we shall start by showing how it can also be applied to
recover in a particularly easy way (but in a somehow simplified context)
several well-known results of microlocal analytic singularities, in the spirit
of the book of J.Sjostrand [16].
Actually, we shall remain very close to the considerations of [16] [17], in
particular by working with a so-called "Fourier-Bros-Iagolnitzer" (in short:
FBI) transform, which has been intensively studied in [16]. From this point
of view, the way in which we recover some results of [16] does not really
contain new ideas, but constitute to our opinion a quite simpler
presentation. Our main originality lies on the fact that we derive all the main
features of analytic microlocal analysis from a single a priori estimate, the
proof of which, moreover, turns out to be elementary.
Anyway, this a priori estimate also permits to work with essentially
arbitrarily large exponential weights, and therefore gives access to
phenomena occurring far in the complex domain. As applications, we use it to
investigate several spectral problems involving (microlocal) tunneling.
At first, we apply our technique to adiabatic theory, that is to evolution
equations of the type:
te-jfip = H(t)(p
where H (t) is a selfadjoint operator depending analytically on the time t,
and where we investigate the behavior of the solutions as e —> 0+. Assuming
that the spectrum of H(t) admits a gap which depends continuously on t,
we show that the transition probability from -oo to +oo between the two
separated parts of the spectrum, can be upper-bounded by 0(e""£/e) where
E > 0 is a geometrical constant explicitly related to H(t). Therefore this
result (the details of which can be found in [8]) permits to specify the
previous upper bounds given in [5], [15].
Next, we consider a problem where two semiclassical Schrodinger
operators interact to create resonances. The two potentials are assumed to have
no crossing on the real, so that at a classical level the interaction can take
place only in the complex. Applying our a priori estimate, we get that the
width of the resonances can be upper bounded by 0(e~slh) where, here
again, S > 0 is related to geometrical quantities associated with the sym-

351
bols of the two operators. Such a problem has been studied in [9], and then
the one dimensional case has been specified in [13], [1].
Finally, we show how our a priori estimate can be used to justify WKB
constructions, in the case of a semiclassical operator P whose (real-analytic)
symbol admits a non degenerate minimum at some point (#o,£o) of R2n.
After a convenient linear change of symplectic variables, we prove that the
FBI-transform Tu of the first eigenfunction of P admits a WKB expansion
in a neighborhood Q of (#o,£o)- Moreover, Q can be described in terms of
deformations of Lagrangian manifolds, and one can show that it contains
at least a ball centered at (a?o,£o) w^h radius explicitly given in terms of
some constants attached to the symbol of P.
2. Microlocalization
For h > 0 small, (#,£) e R2n, and u (possibly ^-dependent) in L2(Rn), we
define:
Tu(x, £; h) = c(n, h) je^x-y^lh-^^l2hu{y)dy (2.1)
where c(n, h) = 2~"n/2(7r/i)"~3n/4 is chosen in such a way that:
\\Tu\y{K2n) = |M|L2(Rn). (2.2)
T is called the Fourier-Bros-Iagolnitzer transform, and has been studied
by J.Sjostrand in [16] and [17]. In some sense, Tu(x,£;h) describes the
behavior of u both in the space variable x and in the momentum (or Fourier)
variable £. The behavior of Tu as h tends to zero is called the microlocal
behavior of u. In particular, one can consider the microsupport of u which
is the closed subset MS(u) of R2n defined by:
Definition 2.1
There exists 8 > 0 such that
Tu = 0(e~8/h) uniformly for
(x,£) G R2n close enough to
(#o, £o) o.nd h > 0 small enough.
Noticing that Tu satisfies the equation:
(hDx - £ - ihD^Tu = 0 (2.3)
we see is an holomorphic function of z = x - i£. As
a consequence, we get for (x,£), (t, r) 6 R2n:
Tu{x + it,£ + iT;h) = e(t2+T2-2ttt+iTM2hTu(x + r^-t;h). (2.4)
(so.fo) $MS{u) <»

352
Therefore we see that in the definition of MS(u), one can equivalently
take (#,£) in a complex neighborhood of (#o,£o)> and also, using Cauchy
formulas, replace the uniformity with respect to (#,£) by a local L2 (or
even Lp, p > 1) -norm.
In the case when u does not depend itself on /i, this microlocal
behavior is closely related to the analytic singularities of u: in fact, denoting
WFa(u) C T*Rn\0 the analytic wave front set of u (see [16]), one can
prove:
MS(u) = WFa{u) U Suppux{0}.
Moreover, in many instances one can recover from MS(u) the points x
where u is (or is not) exponentially small. Actually, if one has an a priori
estimate of the type
\\Tu\\LHKxm>c}) = 0{e-s'h) (2.5)
for any compact K C Rn and for some positive constants C = Ck, & =
&K, then one can show that the ^-projection of MS(u) is precisely the
complementary of the points near which u is uniformly exponentially small
as h tends to 0 (see [7]). Note that an estimate such as (2.5) is automatically
satisfied when u is solution of a partial differential equation which is elliptic
in the classical sense (that is with symbol p(x,£) polynomial of degree
m with respect to £ such that, locally with respect to x, the quantity
l^|mp(^?0_1 *s uniformly bounded as |£| tends to infinity).
One of the interests of working with T is that one can write easily
and explicitly how it transforms the pseudodifferential operators acting on
L2{Rn). More precisely, denote
Sn(l) = {pe C°°(R2n) ; V a e N2n, dap = 0{l) uniformly} (2.6)
and for p G <Sn(l) and t G [0,1], consider the t-semiclassical quantization of
p defined by:
0?htt(p)u(x,h)= j^je^-y^hp(tx + (l-t)y,0u(y)dyd4. (2.7)
Then, by the Calderon-Vaillancourt theorem, Opht(p) defines a bounded
operator on L2(Rn), and we have:
Proposition 2.1 For all p G Sn(l) and t G [0,1], one has
ToOPhit(p) = 0Phit(p)oT
where p G e>2n(l) Z5 defined by (denoting (x*, £*) the dual variables of (x, £)):
P(x,t,x*,e)=p(x-e,x*)-

353
Proof- For u 6 C£°(Rn), we have:
OpfcXar-r.OJru^.O (2-8)
= ^^ /R5n J*'hp(tx + (1 - t)x' - f, x*)u(y)dydx'dt'dx*de
with
* = (x - *')** + (£ - ?)? + (*' - y)£ + i(x' - yf/2.
Then, integrating first with respect to £' and using the fact that
j jV-v-W/htf = (2irh)nS^{x,_y)
we get from (2.8):
0pw(p(3-r,0)r«(*,o
= ^^ / e*1^** - te' + y, x*)u(y)dydx'dx* (2.9)
with
*i = (*' " y)f + {x - x')x* + i{x' - j/)2/2.
Finally, making the change of variables x' i-* z = a; - x1 + j/ in (2.9), the
result follows easily. o
3. Exponential weighted estimates
In this section we state and proof the basic a priori estimate from which
we shall derive all the results of this lecture.
Let a, b > 0 and p e Sn(l) such that p extends holomorphically to
the complex strip S(a,b) = {(#,£) G C2n ; |Imx| < a , |Im£| < &}, and
satisfies:
Va G N2n , 0ap = 0{1) uniformly in S(a, 6). (3.1)
Let also ^ = ^(#, £) G <5n(l) be a real-valued function on R2n satisfying:
Sup |V^| < b and Sup |V^| < a. (3.2)
R2n R2"
We denote
dz = \(Vx + iV() (3.3)
which corresponds to the usual holomorphic derivation with respect to z =
x - i£.

354
Theorem 3.1 Assume (3.1) and (3.2) and denote P = Ophj(p) where
t G [0,1] is fixed. Let also a G <Sn(l). Then for all u, v G L2(Rn), one has:
(ae^TPu.e^Tv)^ = ((q(x,bh) + R(h))e^hTu,e^hTv)L2
where q(x,£;h) (which depends also on t) has an asymptotic expansion of
the form:
q{x,£;h)~J2h3<lj{x,0
with
qo(x,£) = a(x,£)p(x - 2dzip(x,Z),Z + 2idzip(x,Z))
and all the qj are smooth bounded functions on R2n. Moreover, R(h) is a
bounded operator on L2(R2n) satisfying:
\\R(h)\\cm = 0(h~)
as h tends to zero.
Sketch of the proof- Using proposition 2.1, we have ae^lhTP = Qe+/hT
with Q = ae+'hOpht(p(x -£*,x* ))e ^lh. Then by standard arguments, we
see that Q is a classical /i-pseudodifferential operator with principal symbol
q(x,£,x*,£*) = a(x,£)p(x - £* - id^^x* + idxt/)). Now, denoting
q(x,t, **, f) = q{x,Z, »*,O ~ ?(»,t,£ ~ fyfa dx*l>)
and using standard /i-pseudodifferential calculus, we get in particular the
existence of two pseudodifferential operators Q\ and Q2 satisfying:
Ophit(q) = \(QiA + AQ1) + ^(Q2B + BQ2) + R1
with
A = hDx-t + d^ ; B = hDi-dxi> ; \\Ri\\c(V) = 0{h).
Moreover as a consequence of (2.3), we have
A o e^/hT = %B o e^/hT.
Using the fact that both A and B are symmetric on L2(R2n), we then
obtain
{{QXA + AQje+^Tu, e^hTv) = i([Qu B]e^hTu, e^hTv)
and
((Q2B + BQ2)e^hTu, e^hTv) = i([A, Q2]e^hTu, e*'hTv).

355
Since [Qi, B] and [A,^] are both pseudodifferential operator with symbol
uniformly 0(h), and q{x,£,£- d^,dxf/)) = a(x,£)p(x - 2dztl>(x,£),Z +
2idzt()(x,£)), we have proved the result up to a remaining operator of order
0(h) instead of 0(h°°). However, an iteration argument plus a resummation
procedure permits to get a 0(h°°) remainder term, and therefore to finish
the proof. We refer to [7] or [14] for more details. o
Let us immediately state two corollaries of theorem 3.1, that will be
useful in the sequels.
Corollary 3.1 Under the assumptions of theorem 3.1 one has
\\ae*lhTPu\\2 = ||p(o: - 2^^, € + 2i^^)ae^/,lr^||2 + 0(/i)He^^r^lj2
uniformly for h > 0 small enough and u 6 L2(Rn).
Corollary 3.2 If moreover p is real on the real, then there exists a constant
C > 0 such that
\\e€^hTPu\\2 > e2\\(Hv^)e*lhTu\\2 - C(h + e3)\\e€^hTu\\2
uniformly for e,h > 0 small enough and u 6 L2(Rn).
Here, Hp = d^pdx - dxpd( is the Hamiltonian field associated to p.
Corollary 3.1 is an easy consequence of the proof of theorem 3.1, and
corollary 3.2 is obtained from corollary 3.1 by taking the imaginary part of the
first order Taylor expansion of p(x - 2edzifi,£ + 2iedzip) with respect to e.
In view of the applications to tunneling, let us also mention that corollary
3.1 has the following immediate consequence:
Corollary 3.3 Assume in addition that Pu = 0 and, for S > 0, denote
A8 = {(*,0 e R2n ; \p(x - 2dztl>,t + 2idztl>)\ > 6}.
Then, for all S > 0 one has:
lle^rtill^A,) = 0(Vh\\eVhTu\\L2{R2n\Af))
uniformly for h > 0 small enough. In particular, if Ag D Supp ij) for some
positive S, then \\e*/hTu\\ = 0(Vh\\Tu\\).

356
As we shall see, it is sometimes useful to introduce an extra parameter
H > 0 into the definition of T, by setting:
Ttu(x,bh) = p-nt2Tu(x&-). (3.4)
Of course, all the previous estimates on T have analog for T^, obtained by
a change of variable in £ and a modification of the parameter h.
We conclude this section with a few remarks.
First, by restricting the space where u and v are taken, we can allow
some polynomial growth at infinity for p(x,£). This permits to consider
directly partial differential operators also. However, one can always reduce
to the case where p is bounded by composing P with a suitable regularizing
elliptic operator. Also, adding to ^ a function of the type h ln((x)~Sl (£)~S2),
one can replace the space L2(Rn) by any polynomial weighted Sobolev
space.
Next, concerning the assumption of analyticity, one can show that it
can be locally relaxed in the regions which are sufficiently far (depending
on the smallness of the quantities one wants to study) from SuppVV? (see
e.g. [9]). It is also clear from our proof that if ij) = ij)(x) depends only on
x, the analyticity of p is required in the £-variables only, and vice versa.
In particular, the result applies without any analyticity assumption when
ij) = 0, and permits e.g. to recover the semiclassical version of the Sharp
Garding inequality, or even the unicity part of a theorem of Levy-Mizohata
(see [7] for more details).
Finally, let us notice the existence of a Gevrey version of this estimate,
which permits to work with weights of the type e^x'Wh ° (with s > 1): see
[6]. Actually, the method (which relies on the almost analytic extensions
introduced by Melin and Sjostrand in [12]) can also probably be adapted
in the C°° case, with weights of the type h'^x^\
4. Microsupport of solutions of P.D.E.
As a first application of theorem 3.1, we are going to recover many results
concerning the microlocal behavior of the solutions of partial differential
equations of the type
P(x,hDx;h)u = 0. (4.1)
As we have already mentioned, one can always compose P(x,hDx\h) to
the left by an elliptic regularizing pseudodifferential operator, so that we

357
can actually reduce us to the case where
P(x,hDx;h) = P = Oph{p) (4.2)
N
with p = ^2 hkpk{^-,Oi Pk £ £n(l) (here we do not specify which quanti-
k=o
zation we use, since this will have no relevance in the sequels). Assuming
that the initial partial differential operator has its coefficients holomorphic
in a complex strip, we also get that p satisfies (3.1) for convenient a, b > 0.
We first have:
Proposition 4.1 Assume (4*2) and let u 6 L2(Rn) be a solution of (4-1)
normalized by \\u\\j^ = 1. Then:
(i) MS(u) C Char(P);
(ii)(Hanges theorem) For any real integral curve 7 of Hpo, either 7 C
MS(u), orjnMS(u) = ®.
Proof- For part (i), we fix (x0, £0) outside Char(P), and we apply corollary
3.3 with ij) non negative, ij) supported near (#o,£o), ^(#o?£o) > 0, and ifr
flat enough so that p0(x - 2d^,£ + 2idzip) ^ 0 on Supp ^. We then get
\\eHhTu\\LHSupf4) = 0(1)
from which the result follows.
Concerning (ii), let us first prove it when po is real on the real. In
this case, we are going to apply corollary 3.2 with a function i/> adapted
to 7, which, in some sense, will permit to 'slide' along 7. More precisely,
assume there is a point (#o,£o) of 7 which is not in MS(u). Without loss
of generality, we can also assume that po |7 = 0 and Hpo ^ 0 on 7. Then,
denote (j/i, y') a system of coordinates centered at (#o,£o) such that Hpo =
d/dyi near 7 (and therefore 7 is given by y' = 0), and define ip(x,l;) =
/(yi)xfly'l) with X € C0°°(R+) supported near 0, X(0) = 1, x' < 0, and
where / G Cq° satisfies for some y® > 0 arbitrary:
/(0) = 8\ > 0 small enough;
/' > 0 on (-00,0] and /' < 0 on [0, +00);
|/'(2/i)| > $2 > 0 for |2/i| G [*3,y?] , $3 > 0 small enough;
f(±y°1) = 8^2.
Then, we choose e > 0 small enough so that e2S\ > ACe3 where C is the
constant appearing in corollary 3.2, and we apply corollary 3.2. Since by

358
construction \HPofa > 82 on 7fl{|j/i| G [£3,2/?]}, and (having chosen 81 and
83 in a convenient way) ee^lhTu = 0(1) on 7fl{|j/i| < £3}, we get for some
8 > 0 small enough, denoting Vs = {|j/i| < j/i ; \yf\ < 8}:
\\e^hTu\\Vs = 0(l+ ||^/fcm||RanXV|) (4.3)
Looking more carefully, we see that we can also arrange in such a way that
X(W\) < 1/4 on \y'\ > 8, and then we can deduce from (4.3) that
ee8il*hTu = 0^ near 7f]{yi= ±yoja
Since y® has been taken arbitrarily, the result follows in this case.
In the general case where po is not necessarily real on the real, the proof
is more subtle. One has to consider the solution fa = fa(x^\e) of the
system:
edtfa = -x{x,Z)lmp0(x - 2edzfa,£ + 2iedzfa)
fa |t=o = ^0
where x G Co° is a cut-off function supported around a fixed segment of
7 containing (#o,£o)? and ^0 is supported near (#o,£o) and is chosen in
such a way that Vd(*o,6>) > 0 and \\ee^lhTu\\ = 0{1). Then one can see
that for small enough values of e and t (but independently of the choice
of (#o?£o))> and for (#,£) on 7, fa behaves like ipo(exp(-tHpo(x,£))). In
particular fa(exp(tHpo(xo,£q))) > 0, and if we denote
f(t) = \\e€^/hTu\\2
we obtain by applying corollary 3.1:
hf'(t) = -2Im(xe^t^rP^,e^/^) + C?(/i)||e^//lr^||2
= 0(h)f(t).
As a consequence, we get for some positive constant C
f(t) = O(e°W)/(0) = 0(1)
from which the result follows. o
As another application of theorem 3.1 to the study of the microsupport,
we have the following celebrated theorem of Kawai and Kashiwara. To the
previous assumptions, we add that there exists a real C°° function 4>(x,£)

359
defined near some point (a?o,fo) £ Char(P) n R2n such that <f>{x0,tio) = 0
and for all (x,£) in a neighborhood of (a?o,fo) and all £ > 0 small enough:
p0{x + iedt<l>(xo,Zo),Z- ied^xo^o)) ^ 0. (4.4)
When (4.4) is satisfied, P is said to be microhyperbolic at (a?o,£o) m *^e
direction H^. Then we have
Theorem 4.1 (Kawai-Kashiwara) Assume (4-2) and (4-4)> an^ tet u £
L2(Rn) be a solution of (4*1) normalized by \\u\\u2 = 1. Assume also that
there exists a neighborhood V of (#o,£o) such that
M5(u)n{(a:,OeV;0(a:,O<O} = 0-
Then, (z0,£o) <£ MS{u).
Proof- Apart from the fact that we use theorem 3.1 (which makes the
proof simpler), the idea is taken from [16]: we fix S > 0 small enough and
X G C£°(R2n) (0 < x < 1), such that x = 1 on the ball B5 centered at
(#o,£o) of radius £, and x is supported in a small enough neighborhood of
(#o?£o)- Then we set
so that V<fo(z0,£o) = V<£(z0,£o), to(xo,Zo) < 0, and 4>s > (<£ + <*3/2)x
outside B$. Applying corollary 3.3 with the weight e~€<t>s^h (e > 0 small
enough), and using the microhyperbolicity (as well as a standard "Bochner'
tubes" theorem), one finds:
\\e-^hTu\\LHBs) = O (l + ||e-*''fcTtt||L8(Ra.XB#)) . (4.5)
Now, since (R2n\Bs) D {<j>s < 0} C (R2n\Bs) n{<f>< -63/2}, we see that
the assumption on MS(u) implies that
He-^rtil^d^B,) = 0(1)
so that the results follows from (4.5) and the fact that <fo(#o,£o) < 0. o
As the last application of this section, let us show how theorem 3.1 can
also be applied to boundary problems, and permits for instance to recover
in an easy way the so-called 'microfocal Holmgren theorem' (see [16]).
Let P = P(x,hDx) be a partial differential operator on Rn, with
bounded analytic coefficients (in a complex strip as before), such that the

360
hyperplane {xn = 0} is non characteristic for P, which means that P can
be put under the form:
m—1
P = (hDXn)m + £ Ak(x,hDx,)(hDXn)k (4.6)
k=Q
where for any fc, Ak
is a Xfi~ dependent partial differential operator in the
variables xf = (a?i, ...,#n_i) of order at most m - k. Here we shall work
locally near 0 with respect to a?n, but for simplicity we remain global with
respect to x'. We also extend the definition of MS(u) in an obvious way
for those u which are defined only for small xn (e.g. by inserting a cut-off
function with respect to yn in the definition of Tu). Then the theorem is:
Theorem 4.2 Let P given as in (4-6), and let u be a solution in L2(Rn_1 X
Jo) (where Jo is a small interval around 0) of the equation Pu = 0, nor-
malized by |M|l2(r„-ixj0) = 1. Let also (x'q,£q) € R2(n-1) such that:
(*Uo) i MS{u \Xn=0) U MS(dXnu \Xn=0) U ... U MS^u \Xn=0).
Then, there exists 6 > 0 such that
K,xn,Zo,Zn) iMS{u)
for all xn G [—<5, 8] and all £n £ R.
Proof- By setting v = (w, hDXnu,..., (/lD^)™""1!*), the equation becomes
ftDa.nt; = i4(a:,feDa.0v (4.7)
where A is a m x m matrix of partial differential operators of order at
most m. Let T" denotes the partial PBJ-transform defined as in (2.1) but
acting in the variables x1 only. In particular, T'v is a (vectorial) function
of (#',£n,£';/i), and it is enough to prove that T'v is exponentially small
near (&o,0,£d) uniformly as h tends to 0.
Let $ = il>(x',t') e C^R2^"1)) real such that ^K,Q > 0 and
\^/hrv\Xn=o\\LHm^)) = 0(l). (4.8)
For any A > 0, denote ip\(x,£f) = if{x'^') - \\xn\ and set
h(*») = IKO-m/a«*A(*,c')/fcr'«iea(Ra(-i))
Then, using theorem 3.1, we get on Iq fl {xn > 0}:

361
+0{h)\\e^x'WhT'v\\2 (4.9)
with i(x,0 = iA(x'-2dz,tl>,xn,Z'+2idz,ip) = 0{(?)m) (here z' = x'-i£).
In particular, taking A large enough so that (in the sense ofmxm self-
adjoint matrices) ReA(x, £') - A < -So < 0 on Supp^> X 7o> we get for h > 0
small enough:
^(*n)<C||r'»||ia(R2(.-.))
where C is a positive constant. Integrating from 0 to xn > 0 and using the
fact that /a(0) = 0(1) uniformly as h tends to 0, this gives
fx(xn) = 0{h~l).
Using a similar argument for xn < 0, we therefore get for any xn 6 1$:
||<0-m/V<*''*')/fcr't;||£2(R2<»-i>) = 0{h-le2X^\lh)
and since ^(#o,£o) > 0? ^e result follows for \xn\ < S by taking 8 > 0
sufficiently small. o
5. Adiabatic transition probabilities
As our first application to tunneling, let us look at some evolution equation
of the type:
ie-y = H(t)y (5.1)
where for all t 6 R, H(t) is a selfadjoint operator (uniformly semibounded
from below) on a Hilbert space %, and where we investigate the behavior
of the solutions as e —> 0+ (the so-called adiabatic limit).
We see that equation (5.1) involves the operator P = eDt + H(t) which
can be interpreted as a semiclassical operator (e playing the role of h) with
(operator-valued) symbol r + H(t).
In this context, the characteristic set becomes
Char(P) = {(/, r) e C2 ; r + H(t) is not invertible }
= {((,r)GC2; -rea(H(t))}
where a(H(t)) denotes the spectrum of H(t).

362
Now, assume a(H(t)) admits a gap, that is there exists two bounded
continuous functions €j(t) (j =1,2) such that
r0:=lnf(e2(t)-e1(0)>0 (5.2)
and
a(ff(O)fl[ci(O,c2(O] = 0. (5.3)
Denote IIi(Z) the spectral projection of H(t) associated to a(H(t)) n
(-oo,ei(/)], and 112(f) = 1 - III(*)- Then for s, £ e R one can define the
so-called transition probability between Ranlli(s) and RanII2(£) by:
^i,2(*,*):=lin2(0^(*,*)ni(*)||2
where [/(/, s) is the unitary evolution operator defined by:
ie-U(t,s) = H(t)U(t,s) ; U(s,s) = ln
and the norms are those of the bounded operators on %. Physically, the
quantity 7\2(s, £) represents the probability for a particle with energy in
a(H(s))n(-oo, ei(s)] at time s, to have its energy in cr(//"(/))PI[e2(/), +00)
at time t.
Now, denoting E(t) = \(ei + e2)(t), assume in addition that H(t)-E(t)
depends analytically on £ in a complex strip Sa = {|Im i| < a}, and that
on each side of Sa it tends sufficiently rapidly towards a limit, in the sense
that there exist two operators H± such that for some p > 1:
Sup (1 + \t\)p\\H(t) - E{t) - F±|| < +00. (5.4)
tesa
±Re t>0
Here the norm of the operators are those of the bounded operators from the
domain H\ of H(t) (which is assumed not to depend on /) to %. Possibly by
taking a smaller, we also assume that for t € Sa the spectrum of H(t) ~E(t)
remains separated into two disjoint parts, which deform continuously into
a(//(0-^W)n(-oo,i(ei-e2)(0]anda(//(0~^(0)n^(e2-ei)(/),+oo)
as / becomes real (see [8] for a more precise statement of this property).
Under theses assumptions, one can show that that 7^i,2(5,t) has a limit
^i,2(—oO)+oo) as s —> -00 and t —> +00 (see e.g. [15]), and the problem
is to know its behavior as e becomes small. This problem has been studied
by many authors, and we send the reader to the bibliography of [8] for
references about it.
Now, for r e (-^-, ^-), we denote
/c(r) = Sup {k e (0, a) ; r + H(t) - E(t) is invertible for all t G SK} (5.5)

363
and we set
So = Jl^ *{T)dT. (5.6)
Then S0 > 0 and we have:
Theorem 5.1 Under the preceding assumptions, for any 8 > 0 arbitrarily
small there exists a constant C$ > 0 such that:
Vh2(-oo,+oo)<C8e-2V°-5Ve
uniformly as e —> 0.
Idea of the proof- First of all, one can reduce to E(t) = 0 and show that
Pil2(-oo,+oo)= Sup \(<p-{t)t<p+{t))n\2 (5.7)
lk±(o)||H=i
where <p~(t) and <p+(t) are %1-valued functions solutions of:
U2(t)(p-(t) -»■ 0 as t -> -oo ; (5.8)
IIi(i)y>+(t) -»■ 0 as t -> +00.
Setting ¥>*(<) = IIj^)^* (7° = 1,2) and denoting
D-£d
Dt~~idt
the system (5.8) becomes:
(A + H(t) + fepftx, IIi])^ = teil2{t)<fi ;
(A + H(t) + fepfti,ni])^= = felli(t)vf ; (5.9)
[ V^i" (*) ~^ 0 as t -> +oo ; (p2 (t) -> 0 as t -> -oo.
(where IIj denotes the derivative of II j with respect to /). When e tends to 0,
one can consider (5.9) as two semiclassical systems with (operator-valued)
principal symbol given by:
p(t, r) = (r + ff (t))I2.
Instead of working with the FBI transform of the previous section, here we
prefer to work with T^ given in (3.4), which in this case becomes:
Z>(i,r;*) = 2-1/2(7rs)-3/y/4 / c*-t>/e-rtt-*r/* ^t')dt'. (5.10)

364
The extra parameter fi > 0 will be fixed small enough later on. Setting
On = ~^: + 1^-
fiot or
we have the following analog of corollary 3.1:
Proposition 5.1 Let g = g(t,r) e C£°(R2;R) such that Sup \dTg\ < a.
Then for all (p G £2(R; %\) one has:
||^TM(A + ff(0)vllLa(W;70
= ||e*/£(r + ifid^g + H(t - d»g))Ttf\\l?{W<H)
+0{y/i)\\<*l'T,tp\\l?V?m
uniformly with respect to e > 0 small enough and <p G L2(R;7i\).
Then, in the same spirit as for corollary 3.3, this proposition permits to
show that the T^^'s are exponentially small in the elliptic region {\r\ <
To/2} with a rate of decay given by
9i(r) = 1/ Eo. r^OOMinj / K(s)ds ; / /c(s)ds}
where the precise meaning is the following: for all 6 > 0, k,l 6 N, and
H > 0 sufficiently small, one has
Denoting t\ the point where g\ reaches its maximum (the value of which
is So/2), the estimate (5.9) permits to study T^<pf separately in the two
regions {r < T\} and {r > ri}, in the following way : Let x be a C°° function
on R such that x = 1 on {r < ri}, and x = 0 on {r > t\ + v} where */ > 0
is arbitrarily small. Then one can deduce from (5.9) the following result:
Lemma 5.1 For all 8 > 0 and //, v both positive and small enough, one
has
\\(tyl\H{t - £>T),x(r)]7>±|| = O (e-(s°-W2£)
uniformly with respect to e > 0 small enough.
This lemma permits to insert the function x into the system (5.9) (after
having transformed this one with TM), and to control the error terms that it
makes appear. Forgetting these error terms, we are reduced to a new system
that acts on functions supported in {r < t\ + v}. Since in this region the

365
operator Dt + H(t)Ui(t) is elliptic, the first equation of this system permits
to get the estimate:
IK0-'/2x(r)2>r II = O {e\\{t)-^x{r)T^-2\\ + e"^)/2*) . (5.12)
Inserting (5.12) into the second equation of the system, one finds:
\\(t)^\Dt + HA(t-DT))x(r)T^\\
= O {e2\\{t)^l\{r)T^-2\\ + e-(so-*)/2*) (5.13)
where Ha = H + ie[fli,Ui\. Then, evaluating dt\\x{r)T^(f2\\2L2m.n\ and
using the fact that </?2 vanishes at -oo, one can deduce from this an estimate
on x{T)T^2 which finally gives by (5.12):
\\(t)-»l\{T)T^-\\ = 0(e-(so-*)/2*). (5>14)
A similar estimate holds for xT^"1-, where this time x cuts off in {r > ri}.
Now, coming back to the elliptic region {\r\ < To/2}, we see that (5.14)
permits to apply again proposition 5.1 to <p~ with a new weight #2 which
is constant on {r < To/2} with value So/2. As a consequence, one obtains
an improvement of (5.9) in the region {r < ri}, that one can again
propagate along the negative values of r as before. Iterating this procedure and
working in a similar way for </?+, one finds that for all S > 0, there exists
\i = /j,(5) > 0 arbitrarily small such that:
\\(t)-^e^£T^-\\ = 0(e8f£)
\\{t)-pl2e9+l£T^+\\ = 0{e8'£) (5.15)
where (extending the function k by zero outside (-To/2, To/2)):
rr0/2
5_(r) = Max(j), / n(r)drj
5+(r) = Max(0, / K(r)dr).
v J-r0/2 J
-r0/2
Finally, noticing that (<f>-(t),<f>+(t))n = (T^-(/, .),T^+(/, .))L2(Rr^),
theorem 5.1 is a consequence of (5.15) and of the fact that #_ +#+ = So. o
6. Widths of resonances
Now, we investigate another problem of tunneling, related to the resonances
of molecules. Let P(x,hDx) be the matricial system:
f-tfA + Viix) hR(x,hDx) \
V hR*(x,hDx) -h2A + V2(x)J {°'l)

366
acting on L2(Rn) © L2(Rn), where R(x, hDx) is a (pseudo)differential
operator of order less than 2, and Vi, V2 are real-analytic functions on Rn.
Such kind of systems occurs for the study of molecules when one
considers the so-called Born-Oppenheimer approximation, in which the nuclei
are supposed to be much heavier than the electrons. In this situation, the
parameter h represents the inverse of the square root of the mean mass of
the nuclei.
Here, we assume that the symbol of P extends holomorphically in a
complex domain of the type
D = {(s,0 G C2n ; |Im x\ < a + S0\Re x\ , |Im£| < 6 + £0|Re£|}
(where a, 6, S0 > 0) and that V2 admits a compact well at some fixed energy
#o G R, that is:
Uo := {xeRn ', V2{x) < Eo} is compact. (6.2)
We also assume
U0C{xeRn; Vx(x) < Eo} (6.3)
(so that the two characteristic sets {£2+Vi(x) = #0} and {£2+V2(x) = Eo}
do not intersect on the real) and that there is no trapped trajectories of
energy close to #0 for pi(x,£) = £2 + V\(x), that is:
V E close to Eo, V (#,£) G Pi1(E), \exptHPl(x,£)\ -> 00 as \t\ -> 00.
(6.4)
In this situation, there exists near #0 a discrete subset of C (whose elements
are called resonances of P), which can be characterized by the following
property: p is a resonance of P if there exists u G [C°°(Rn)]2 such that:
Pu = pu
Tu G [L2((l + it)Rn x (1 - it)Rn)]2 for some t G (0, So). (6.5)
Then, the quantity |Imp| is called the width of the resonance />, and its
inverse can be physically interpreted as the life-time of an unstable state
associated to u. As a consequence, any upperbound on \lmp\ permits to
predict how long such a "molecule" will exist at least. This is precisely the
purpose of this section.
Denote also P2{x,£) = £2 + ^2(2), Sj = p]l{Eo) (j = 1,2), and for any
S > 0, Ej(<5) = p~l(\Eo - 8,Eo + S\). For p > 0, we consider the set £M of
functions ^ G C°°(R2n) such that V^ G Cg°({£2 + Vx(x) < E0}, ip\u0 = 0,
Sup|^^| < 6, Supjd^j < a, and:
V S > 0, 3CS > 0 such that \p2(x - d^,£ + ipd^)\ > -J- on Rn\E2(<$);

367
3C > 0 such that \pi (x - d^tfi, £ + ipd^tfi) \ > — on Supp V^.
Then the number:
S0 = Sup Sup Inf ^(s,0 (6.6)
is positive, and the result is:
Theorem 6.1 Assume (6.2)-(6.4), and let p = p(h) be a resonance of P
tending to Eq as h —> 0. Then for any 6 > 0 there exists a constant C(S)
such that:
|Im p\ < C(8)e-2^-5^h
uniformly for h > 0 small enough.
In this result, the constant So can seem rather abstract since it rests on a
good choice (not necessarily easy to make) of a function ip in £M. However,
there are situations where it can be specified, for instance if Uq is a non
degenerate point-well (that is Uo = {xo} with Hess 1^(0) > 0): see [9]. In
the particular case V^z) = #2 and V\(x) = -xn - 1, the best choice for ij)
is of the form (see [10]):
with p = 1+4 ? So = 3 4 5 and where x is a cut-off supported in {pi(#,£) <
£0}.
Notice that when n = 1, this estimate can be improved by making
a convenient symplectic change of variables (see [13]), and, using exact
WKB expansions, it is even possible to understand the precise nature of
the tunneling (see [1]).
Here we do not sketch the proof of theorem 6.1, but it is based on the
same idea as e.g. corollary 3.3: if u is a resonant state (suitably normalized)
associated to />, the definition of £M permits to estimate ||e^Tu||t for every
^ £ £/i> where T is a FBI transform introduced by Helffer and Sjostrand
in [3] which (for technical reasons attached to the fact that we deal with
resonances) is more complicated to write down than our previous TM, but
whose properties are essentially similar (see [9] for more details), and where
|| . ||t is a norm similar to the L2-norm outside a neighborhood of Si, but
slightly modified near Si. Then, writing
Im(TPtt,Ttt)L2(a)
Am P = ,, ,,2
HTwIIl2(o)

368
for any Q CC {pi{x,£) < #o}, and using the fact that P is formally
selfadjoint on L2, the result can be deduced by choosing Q in a proper way.
7. Microfocal WKB expansions
As our last application of theorem 3.1, we present a joint work in
preparation with V. Sordoni, where we investigate the existence of microlo-
cal WKB expansions for the eigenfunctions of pseudodifferential operators
whose symbol admits a non degenerate minimum at some point (#o,£o) of
R2n. This is e.g. the case for the electromagnetic Schrodinger operator
PA(x, hDx) = J2 {hD*, ~ M*))* + V{x)
i=i
when V admits a non degenerate minimum at some point xq. In the case
where the Aj'& can be taken small enough (that is when the magnetic field
is small enough) and everything is analytic, it has been shown by HelfFer
and Sjostrand [4] that the first eigenfunction u of Pa admits near xq a
WKB expansion of the form
u(x,h)~e-+AWhYthk"k(x)
k>0
where <f>^ and the a^s are smooth functions. Moreover, the set of a?'s where
such an expansion is valid can be estimated geometrically by means of
the minimal geodesies starting from #o, relatively to the so-called Agmon
distance (i.e. the distance associated to the degenerate metric (V(x) -
V(xo))dx2). But the problem remains entirely open for greater magnetic
fields.
Here, we are going to show that in any case (but still under assumptions
of analyticity), a similar WKB expansion exists near (a?o,£o) f°r the FBI-
transform Tu of w, at least if one choose convenient symplectic coordinates
in R2n. Moreover, the set of (a?,£)'s where the expansion is valid can be
estimated by means of simple constants attached to the symbol of P, and
a more general notion of "admissible open set" will be given, in terms of
deformation properties.
Now, let us specify our assumptions.
For the sake of simplicity we take p in «Sn(l) (although everything could
be generalized to symbols with polynomial growth at infinity), and we
assume that p satisfies (3.1) for some positive a, b. We also assume that p(x, £)
is real non negative for real (#,£), p-1(0) = (0,0), Hess p(0,0) is positive

369
definite, and there exists Si > 0 such that p(x,£) > 8\ outside some
neighborhood of 0. It is standard to show that there exists a linear symplectic
change of variables such that, in the new coordinates, p satisfies:
P(*iO = Ew(^+^) + 0(|x^|3) (7.1)
where 0 < ji\ < ... < /J,n. Moreover, since p can be written under the
form p = p + k with Inf p > 0 and k G Co°(R2n), the Weyl' theorem of
perturbation implies that the spectrum of P :=Oph i(p) is discrete near
0. Also, because of (7.1) one can use the same arguments as Helffer and
Sjostrand in [2], which show that the first eigenvalue E of P is simple and
has an asymptotic expansion of the form
E~hJ2ekhk
k>0
as h tends to 0. Denote u the first eigenfunction of P, normalized by \\u\\i2 =
1. We are going to show that, in suitable neighborhoods of (0,0), Tu admits
a WKB asymptotics of the form:
Tu(x,&h) ~ fc-^c^^-^-^^Va,-^ - iO (7.2)
j>0
where (p and the a/s are holomorphic near 0 G Cn. Notice that in the
particular case where p(x,£) in (7.1) is even with respect to £, then the
method of [2] can be directly generalized and gives a WKB expansion for
u(x;h). However, if it is not the case (as for the general electromagnetic
Schrodinger operator), then the usual Agmon estimates fail to give the
expected expansion.
Since TPu = PTu with P = Opfc|i(p(a? - £*,&*)), if we look for a
solution Tu of the form (7.2) we are lead to solve the eiconal equation:
p(z-dz<p,idz<p) = 0 (7.3)
where z = x - i£. To solve it, we proceed in an analogous way as in [2]:
Define
?(*, 0 = -p(z - C, O
so that (7.3) becomes q{z,dz(p) = 0. Near (0,0) e C2n we have:
q(z,0 = J2t*j(2zj(j-z2j) + O(\zX\3)

370
and therefore the fundamental matrix of q at (0,0) is
where fi = diag(//x, ...,//n). The spectrum of Fq is {±2//j ; j = l,...,n},
and the direct sum of the eigenspaces associated to {+2//j ; j = 1, ...,n}
(resp. to {-2//j ; j = l,...,n}) is the Lagrangian space £+ = {( = z/2}
(resp. £_ = {z = 0}). Then by adapting the analytic version of the 'stable-
unstable manifold theorem' which is in the appendix of [18], one can show
that there exist two holomorphic complex Lagrangian manifolds A±
containing (0,0), stable under the action of Hq, and such that T(0)o)A± = £±.
In particular, A+ projects bijectively on the base {( = 0}, and therefore
there exists a holomorphic function <p such that in a complex neighborhood
of 0, A+ is given by:
A+ = {C = /(*)}.
Since g(0,0) = 0 and q is constant on A+, we see in particular that <p solves
(7.3). Notice that if we normalize (p by setting </?(0) = 0, we also have:
V{z) = \* + 0{\z\*)
and therefore
|+Rev>{x _ i0 = f!±i! + o(\x,s\*).
Now, first working with z real, one can construct as in [2] an analytic
symbol a(z, h) ~ ^ hkdk{z) defined near 0, such that formally
Opfcfi(p(z+ iC.C) - E)(a(z,h)e-«Wh) ~ 0
and after resummation, this means that there exists e > 0 such that, for
(#,£) small enough:
(P - E) (a(x - i£, h)e-^'2h-^x-^lh) = O (e-*2/2fc-Re*(*-*)/W>) .
(7.5)
where the action of the pseudodifFerential operator P on the function a(x —
it,h)e~Z2/u-v(x-iO/h (which is defined only near (0,0)) is defined via a
formal stationary phase expansion. Moreover, the estimate (7.5) is valid
locally uniformly in the maximal connected open set fio where both <p and
the ajt's extend holomorphically and \lm(z - dz<p)\ < a, |Re dz<p\ < b .

371
By an abstract spectral argument (still as in [2]), one can deduce from
(7.5) that for convenient constants mo and c*o, one has near 0:
Tu - a0h-m°a(x - t£, h)e'^l2h'^x^^lh = 0(e~e'lh)
with some e1 > 0. In particular, denoting
v(x,£,h) = a0h-mQa(x - t£,fe)c-*2/2*-v(*-«'0/*
we have
eei2h+n*<p(x-ii)lhpu _v^ = 0(e-*'/2h} (7>6j
for | x, £| sufficiently small compared with e'. Since not much is known about
this e1, the problem is now to extend (7.6) in a neighborhood of (0,0) that
one can control in a more geometrical way.
Coming back to the variables (*,£), we set:
Ao = {( = ilmz}
(so that (z, Q G Ao if and only if (z - £, i() is real), and for t > 0:
At = exptHq (A0).
Since Ao is R-Lagrangian (that is Lagrangian for the real symplectic form
Re(d( A dz) on C2n w R4n), and the map exptHq is a complex canonical
transformation, we have that At is R-Lagrangian for all t. Moreover,
approximating Hq by its linearization at (0,0), one can see that At is
transversal to {z = 0} at (0,0). As a consequence, At admits near (0,0) an equation
of the form: _
d(j>t{z^)
Kt ' C " ~~dT~
where <f>t is a real C°° function defined in a neighborhood of 0 and vanishing
at 0. Looking carefully at the proof of [18]Appendix, one can also see that
there exists a fix neighborhood Q, of 0 in Cn such that for all t > 0, (/>t is
smooth in £2, and
<j)t{z,~z) -+ 2Re(p(z) as t -+ +oo (7.7)
in C°°(fi).
Now, for (z, C) = (^, ilmz) G A0, we have -q(z, () = p(Rez, -lmz) and
is therefore real non negative, with a non degenerate minimum at z = 0.
Since q is constant along {exptHq(z) ; t > 0}, we deduce from this that for

372
any t > 0, —g |At is real non negative and there exists a constant C\ > 0
such that: _
v,en,-,(,,*£2)>iW>. (7.8)
One can also prove that on £2, <f>t satisfies the evolution equation:
^ = -2q(z,dzj>t)
and therefore, (j>t{z) is an increasing function of t for z ^ 0. As a
consequence, we have for all t > 0 and z ^ 0:
Mz,*) < Mz,z) < 2Re</>(2:) (7.9)
where </>o = ~(lmz)2.
Then we introduce the following notion of "admissible open set":
Definition 7.1 Let Qi be an open subset of Q, containing 0. Then Q\ is
said to be "admissible" if for any compact subset K ofQ\ there exist sk > 0
and a neighborhood Vk of dQ with the following property:
For every t > 0 large enough, there exists tpt G C°°(Cn) real such that
tpt = <f>t on K ;
ij)t - 4>q is constant outside Q ;
ij)t < 2Re</? everywhere;
tfit < 2Re</? - sk °n Vk ;
Sup |Im(z - dzipt)\ < <*> and Sup |Re dzil>t\ < b ;
3C't>0 such that \q(z,dzipt)\ > -pr,\z? °n fi.
In terms of deformation of R-Lagrangian manifolds, this means that one can
deform (in a somehow non increasing sense) A* into Ao within an arbitrarily
small neighborhood of fl\fli, in such a way that q remains elliptic along
the deformation. Moreover, the deformed weight is required to be smaller
than 2Re(f near #£2, since the WKB constructions may cease to exist there.
Before proving that the estimate (7.6) remains valid locally uniformly
in any admissible open set, let us state a result of existence of such an
admissible open set, which actually permits to exhibit such a set in terms
of some constants attached to q and easy to compute.
First of all, working with the quadratic approximation of q it is easy to
see that for all t > 0, one has:
M*rf = \ E ((1 - e-^KRez^2 - (1 + e-4^)(Im^)2) + 0(\zf)
3=1

373
where the 0(|2|3) is uniform with respect to t.
Then, noticing that 2Re<p(z) - <f>o{z,z) = \z\2/2 + 0{\z\3), we set
\z\v = {2Re<p(z) - <fo{z,z))> (7.10)
and we define the five constants 70, jj > 0 (j = 0,1,2,3) in the following
way:
70 = SuP]^
zen \z\v
70 = Sup -rf
zm \z\
71 = Sup r-j
t>o |Z|(^
2 en
\d,M*tf -(z-e-^z)/2\
T2 = SUP U|2
t>0 \Z\*
zen
73- ^Up — .
lfl<7ll*lV ^
Then we have:
Proposition 7.1 J/r > 0 satisfies:
7o(272Mn + ^73)v^ + 7o7273r < Mi (7.11)
and
To,
7iV^<& ! (72 + f)v^<« (7.12)
then the set
Br = {zeCn; {Imz)2 + 2ReV(*) < r} = {|*|J < r}
is an admissible open set in the sense of definition 7J.
We refer to [11] for the proof of this proposition, and now we state the main
result of this section:
Theorem 7.1 Assume (7.1) and let Q\ be any admissible open set in the
sense of definition 7.1. Then for any compact set K C fii, there exists
e > 0 such that

374
uniformly for x - i£ £ K and h > 0 small enough. Here u is the first
normalized eigenfunction of P = Op^ \(p), and
v(x,& h) = a0h-moa{x - if,fc)c-«2/2MM)A
is the WKB solution constructed at the beginning of this section.
Proof- Using (7.6), let £0 > 0 and V0 be a neighborhood of 0 e R2n such
that
^€?/2h+B*<p(x-it)/hpu _ v)\\12{Vq) = 0(e-€Q/h). (7.13)
Then, fix K CC fti, and let sr > 0 and ^ {t > 0 large enough) be given
by definition 7.1. By (7.7), we can fix to sufficiently large so that:
|^0-2Re^|<^Min(^0,^) on K. (7.14)
Let also x G ^^(fi) be such that x = 1 on K and SuppVx is included
in the interior of the neighborhood Vk of dQ where ipto < 2Re<^ - sk, and
define
k; = X^ - Tu.
Then by construction, we have
(hDs-Z-ihDt)w = /i(DlX-%)« + 0(c"(t2/2+Rev+e)/fc)
= 0(c-«2+4*o+«*)/2*) (7>15j
since D^x - ^£X *s supported in Vk- Moreover
(P-E)w = {P-E)Xv
= [P, x]v + 0{e~e/2h-Reip{x-it)/h-ei/h) (7.16)
with S\ > 0. We set
^ = 2 V + ^° + 2Min^0' ^)
(which is constant outside £2), and we plan to apply theorem 3.1 with this
^, but with Tu replaced by w. Actually, since w does not satisfy (2.3) but
only (7.15), following the proof of theorem 3.1 we see than an extra error
term appears, namely:
\\e+'h(P-E)w\\2 = ||^(o: — 2^^0, ^ + i^^)e^/^xt7||2 + C7(^) ||e^/^xt7||2
+0 (h\\e+'hv\\»{Vx) + e~^h) \\e^hw\\. (7.17)

375
However, since ij) < — + Rey —— on Vk, we have
\\e*lhv\\u(vK) = 0(e~^h)
and therefore, using also the fact that p(x-2dzip,£+2idzip) = -g(2,0^to)
is elliptic outside 0, we get from (7.17) and (7.16):
\\e^hwf = O (l + II^Nl Vo) + 11^V. *HI2) ■ (7-18)
Using (7.14), we also have
|^_L_Re^|<lMin(60,^) (7.19)
and thus, by (7.13):
¥*lh™\\h<y0) = O{e-^h). (7.20)
In view of (7.18), it remains to study the term Ue^/^fjP, x]^||2- We write:
= J^Je%(X-Y)XVhP(^X*)MY) ~ x(X))v(Y)dYdX*
and, denoting X = (#,£), Y = (y,r)) and X* = (#*,£*), we make in (7.21)
the change of contour of integration:
R2n 3 (**,f) H- (** + ^j^r + ia'|5^) (7.21)
with Sup \lm(z-dz<p)\ < a' < a and Sup |Re dz(p\ <b' < b. Then, denoting
e'K = Min^o,^), we get:
\\e*,h[P, XM = 01+ Sup e-^-Y\/h+s'K/h (7#22)
\ X{X)*X{Y) )
with some 8 > 0 depending only on K. Now we have
Inf \X-Y\>0
y&k
X(X)*X(Y)
and we see that sk can possibly be taken smaller without modifying the set
{Y i VK , x(X) ^ x(Y)}. Then we deduce from (7.22) that \\e+'h[P,x]v\\
is exponentially small, and in view of (7.18) and (7.20), we finally get
||e*/*ti;|| = 0(1).

376
Since ^(a, £) > -£2 + Re<p(x - if) + -Min(£0, £k) on K, this completes the
2 8
proof of theorem 7.1. o
References
1. [Ba] H. BAKLOUTI, Asymptotique de largeurs de resonances pour un modele d'effet
tunnel microfocal, These Universite Paris-Nord (1995)
2. [HeSjl] B. HELFFER, J. SJOSTRAND, Multiple Wells in the Semiclassical Limit I,
Comm. P.D.E., vol. 9, (4), 1984, p. 337-408
3. [HeSj2] B. HELFFER, J. SJOSTRAND, Resonances en limite semi-classique, Bull. Soc.
Math. France, Memoire n. 24/25, tome 114, (1986)
4. [HeSj3] B. HELFFER, J. SJOSTRAND, Effet tunnel pour Vequation de Schrodinger
avec champs magnetique, Ann. Sc. Norm. Sup. di Pisa, Ser. IV, 14(4), 625-657 (1987)
5. [JoPf] A. JoYE, C.-E. PFISTER, Exponentially small adiabatic invariant for the
Schrodinger equation, Commun. Math. Phys. 140, p. 15-41 (1991)
~Ju] K. Jung, Phase space tunneling in Gevrey class regularity, preprint (1995)
Mai] A. MARTINEZ, An introduction to semiclassical analysis, book in preparation
Ma2] A. MARTINEZ, Precise exponential estimates in adiabatic theory, J. Math.
Phys. 35 (8), (1994)
9. [Ma3] A. MARTINEZ, Estimates on complex interactions in phase space, Math. Nachr.
167 (1994)
10. [Ma4] A. MARTINEZ, Estimations sur Ifeffet tunnel microfocal, Seminaire E.D.P. de
l'Ecole Polytechnique 1991-92
11. [MaSo] A. Martinez, V.Sordoni, paper in preparation
12. [MeSj] A. MELIN, J. SJOSTRAND, Fourier integral operators with complex valued
phase functions, Springer Lecture Notes in Math., No.459, 120-223 (1976)
13. [Nal] S. Nakamura, On an example of phase-space tunneling, Annales Inst. H.
Poincare, Vol. 63, 211-229 (1995)
14. [Na2] S. NAKAMURA, On Martinez' method on phase space tunneling, Rev. math.
Phys. vol 7, p.431-441 (1995)
15. [Ne] G. NENCIU, Linear Adiabatic Theory. Exponential Estimates, Commun. Math.
Phys. 152, 479-496, (1993)
16.
17.
Sjl] J. SJOSTRAND, Singularites analytiques microlocales, Asterisque 95 (1982)
Sj2; "
J. SJOSTRAND, Function spaces associated to global I-Lagrangian manifolds,
Preprint Ecole Polytechnique de Palaiseau No. 1111, (1995)
18. [Sj3] J. SJOSTRAND, Analytic wavefront sets and operators with multiple
characteristics, Hokkaido Mathematical Journal, Vol. XII No.3, 392-433 (1983)

A TRACE FORMULA AND REVIEW OF SOME ESTIMATES
FOR RESONANCES
J. SJOSTRAND
Centre de Mathematiques
Ecole Polytechnique
F-91128 Palaiseau, France
and UA 169 CNRS
Abstract. The main part of theses notes from the NATO ASI on
microfocal analysis and spectral theory at II Ciocco, Sept.-Oct. 1996, is devoted to
a new trace formula for resonances, which is valid for long range
perturbations of the Laplacian in all dimensions. We work in the frame work of
complex scaling and have a natural opportunity to review that method. We
also review some lower bounds and some upper bounds on the density of
resonances near the real axis, mainly following joint works with M.Zworski.
The lower bounds however, are new in the case of even dimensions and
form a first application of the new trace formula.
1. Introduction
The original plan for these notes was to explain some estimates for the
density of resonances for compactly supported perturbations of the
Laplacian, largely obtained in collaboration with M.Zworski, and closely related
to work of G.Vodev as far as the upper bounds are concerned. For the
lower bounds a well-known Poisson type trace formula, valid in odd
dimensions, plays an important role. This formula has been elaborated in
the frame work of the Lax-Phillips theory successively by Lax-Phillips,
Bardos-Guillot-Ralston, Melrose, Sjostrand-Zworski. During the
preparation of some lectures at Ecole Polytechnique in Spring 1996, I tried to
obtain "my own" approach to this formula. Instead however, I obtained a
new trace formula with a remainder, valid in many situations where the
resonances can only be defined in some neighborhood of the real axis. This
formula seems to have new applications for instance to even-dimensional
377
L. Rodino (ed.), Microfocal Analysis and Spectral Theory, 311-431.
© 1997 Kluwer Academic Publishers.

378
cases and to long range scattering for the Schrodinger equation, and
despite the fact that there has not yet been enough time to develop most of
these applications, I found it natural to give a detailed account here. The
sections 2-8 are devoted to the statement and the proof of the new result.
In those sections we also have the occasion to review the method of
complex scaling, (though our proof is of such a general nature that it should
be easily adaptable to some other frameworks for the study of resonances,
such as the one developed with B.Helffer in [14]). It would no doubt be
useful to understand better the links with a recent approach by Guillope and
Zworski to a Poisson type formula on hyperbolic surfaces, which is based
on general scattering theory and especially on the Birman-Krein formula
for the scattering phase. Our approach uses no scattering theory.
In section 9 we compare our trace formula with the Poisson type formula
of Lax-Phillips theory in the more restrictive situations, where the latter
one applies.
In section 10, we review some lower bounds on the density of resonances
(which are new in the case of even dimensions), and in section 11, we review
some upper bounds.
2. The main result
Our trace formula will concern a pair of self-adjoint operators Po, Pi, but
much of the work will concern each of these two operators individually, so in
order to ease the notation, we will often suppress the subscript j = 0,1 and
simply write P. or P, and similarly for the various quantities attached to P..
We shall use essentially the abstract setting introduced with M.Zworski in
[34], but our assumptions will be weaker in the sense that we do not assume
that P. be equal to -A near infinity, and we do not assume the dimension to
be odd. It will also be convenient to adopt a semiclassical framework from
the very beginning, so that P. and the corresponding quantities, depend on
a Planck's constant h e]0, /io], where h0 > 0.
Let %. be a complex separable Hilbert space with an orthogonal
decomposition:
n.=H,Ro®L2(Rn\B(0,Ro)), (2.1)
where R0 > 0 is some fixed constant and B(x,R) = {ye Rn; \x - y\ < R}.
The corresponding orthogonal projections will be denoted by u^ u\b(o,r0)i
and u *-+ ^|Rn\^(o,JR0) or simply by the characteristic function (1^) of the
corresponding set (L). We consider an unbounded self-adjoint operator
P. : ft. -> n. with domain V. = V(R). (2.2)
Assume that
iR-Wo.jyP = #2(Rn \ 5(0, Ro)), (2.3)

379
uniformly with respect to h in the following topological sense: Equip H2(Rn\
B(0, i?0)), with the norm \\(hD)2u\\L2, where (hD) = (l+(hD)2)2, (hD)2 =
Ei(fe£,a?i)2» and e<luiP V- with the norm ll(* + P)u\\u.- Then in (2.3), we
require that the restriction map from V to H2(R2 \ B(0, i?o)) is bounded
uniformly with respect to h and has a uniformly bounded right inverse (that
we may call an extension map).
Assume
1b(o,Ro)(p- + i)~1 is comPact- (2.4)
We also assume that,
lR"\B(0,R<))P-V< = Q.tt, Q.U = T,\a\<2a'AX^h){hDx)aU,
a.,a(x\ h) = a.,a(#) is independent of h for |a| = 2,
and a.,a G C£°(Rn) are uniformly bounded w.r.t. h. (2.5)
Here C£°(Rn) denotes the space of C°° functions on Rn which are bounded
together with all derivatives. Observe that if ^ G C£°(Rn) is constant
near £(0,i?o), then there is a natural way of defining the multiplication:
%. 9 u \-t ij)u G %., and we have ^m G P. if u G P.
It is further assumed that Q is formally self-adjoint on Rn with:
E|a|=2fl-,a(x)e>iK|2,
uniformly with respect to h. (2.6)
It is quite likely that the second part of this assumption can be
weakened, so that we could allow operators of the form -h2A + V(x), where
V(x) may be unbounded. In some cases like that, the final trace formula is
much easier to obtain and results very quickly from Lidskii's theorem. This
fact has been used by L.Nedelec [25] for Schrodinger operators with linear
matrix-valued potentials.
Assume
K,a(z; h) - a0,a(x; h)\ < C(x)~n, n > n. (2.7)
This assumption will guarantee that /(Pi) - f{Po) is "of trace class near
infinity" , when /eCg°(R).
Let R > R0 and M = (R/RZ)n, where R > 2R. Then we can view
B(0, R) as a subset of M, and as in [34], we can define an unbounded self-
adjoint operator P* in lit = HRo 0 L2(M \ ff(0, flp)), which coincides
with P. (in the natural sense) near B(0, R) and which outside B(0, i?o) is
of the form Qr and has the same properties (except for the behaviour at
infinity) as in (2.5),(2.6). As in [34], we see that Pt has discrete spectrum.

380
Let N(P.', I) denote the number of eigenvalues of P. in the interval J. We
assume:
N(P*; [-A, A]) = 0((\/h2r/2), A > 1, (2.8)
for some number n. > n. As in [34], this assumption does not depend on the
choice of i?, i?, or QT. We briefly explain why. Let Ai < A2 < A3 < .. be the
eigenvalues of P* = P. , so that, if fij(K) denotes the j:th characteristic
(or singular) value of the compact operator/^, ^((P*-^)"1) = | i — A ^ |"~x =
(Aj)"1. (See [9].) Then it is easy to see that the property (2.8) is equivalent
to the property:
Recall that if K is a compact operator on some separable Hilbert space,
then the characteristic values, fJ>i(K) > l^2(K) > •• are defined to be the
eigenvalues of yjK*K.
If M is a second torus and P. a corresponding operator analogous to
P. , we can identify M and M by means of a diffeomorphism which is
the identity map near B(0, i?o), and achieve that the two operators act in
the same Hilbert space and coincide near £(0,i?o). The invariance of the
assumption (2.8) then follows from the resolvent identity
(i - P*)-1 = (i- p*)-1 + (i - P*)-\P* - P*)(i - P*)-1
and the general identities for characteristic values:
fij+k_1(A + B)<fij(A) + fik(B),
tij+k-i(AB) <Hj(A)nk(B).
(See [9], where both identities are easily derived from the Ky Fan identity
f*M) = inf. \\A-R\\.
rank(K)<j-l
Here || • || denotes standard operator norm.)
Let Sm denote the standard symbol space of functions a G C°°(R),
satisfying a^(t) = 0((t)m~k) for every k e N. In the next section, we
shall prove:
Proposition 2.1 Let f e S~m^ be independent of h, where ™U)'^t^ >
1> ^max =def max(no?fti)- Let x £ Co°(Rn) be equal to 1 near S(0,i?o).
ThenxW), W)X, (l-x)/(Pi)(l-x)-(l-x)/(W-x) are of trace
class and
wtr(/(Pi)-/(Po)r =
= [tvxf(Pj)x+ tr (1 - X)f(Pi)x + trx/(P;)U - x)]}=o
+tr[(l-x)/(^)(l-x)]}=0 (2-9)

381
is independent of the choice ofx and is 0(h nma*). Here we write [aj]j=o =
ai - a0.
This proposition will be proved in section 3. Notice that /(Pi) - f{Po)
is not a well defined operator in general and this is the reason for the use of
quotation marks. Also notice that [(1 - x)f(Pj)(l - x)]}=o *s a we" defined
operator in L2(Rn), %o, and H\ and has the same trace in all these spaces
(as soon as it is of trace class as an operator in one of the spaces).
Our trace formula will involve resonances of P0? Pi and for simplicity,
we will use the frame work of complex scaling, or complex distorsion ([1],
[18]), and we will follow the presentation in [34]. For that, we shall use the
following assumption:
There exist 0O £ [0, tt[, e > 0, and R > i?0, such that
the coefficients a.i0((x;h) of Q. extend holomorphically in x
to {ru; u e Cn, dist (u, S71"1) < 6, r G C, \r\ > R, arg r G [-€, 0O + e[}
and (2.7) and the second half of (2.6) remain valid
in this larger set. (2.10)
It is quite likely the main result below remains valid in other frame works,
such as the one in [14]. We can now define the resonances X.j of P. in the
sector So0 = {z € C\ {0}; 0 < -arg z < 20o} as the eigenvalues of P. on a
suitable contour in Cn (see [34], and section 5) and it follows from section 8
and also from the methods of [30], [34]) that if ft CC 5^0 is independent of
ft, then the number of resonances of P. in ft is 0(h~n ). The same estimate
holds for the number of eigenvalues in any fixed compact subinterval of
]-oo,0[.
Let W CC ft be open relatively compact subsets of e^~2^0'eo']0,+oo[,
where €o > 0 may be arbitrarily small, such that the intersections J, J
with the open positive half axis are intervals, and denote by ft_, W-, the
intersections with e]~2^°'°]]0,+oo[. Also assume for simplicity, that ft is
simply connected.
Theorem 2.2 We make the assumptions above and assume that W, ft are
independent of h. Let f = f(z\ h) be holomorphic in z on ft, and satisfy
\f(z; h)\ < 1 for z G ft \ W. Let x £ Co°(^) be h-independent and satisfy
X — 1 near I. Then
"tr((x/)(Pi;fc)-(x/)(Po;fc))" =
[ E /(A.j;fc)- £ /(A*;fc)]S + 0(fc-n~) (2.H)

382
The proof of the above theorem will occupy the sections 3-8. We end
this section by showing that the theorem is still valid under slightly different
assumptions about /. This will be useful in section 9.
Proposition 2.3 Let f = f{z',h) satisfy the assumptions of Threorem 2.2
with the following modification: Instead of assuming that \f(z\h)\ < 1 in
Q \ W, we assume that for all a:
\daf(z; h)\ < Caec^z^lh, z e Q \ W,
for some constant C > 0. Let W C £2 be (relatively) open with W CC
W CC £2. Then there exist 0 < eb < €o, and a holomorjrfiic function f in
£2fl{argz < ?o} such that f(z\h) — 0(1), when z 6 Q\W, argz < €q9 and
such that f - /= (9(1), z G ft-, da(f'- f) = Oa(l), zeJ.
Proof. Let x £ Cq>(W) be equal to 1 near W. Treating separately the
equations
^i=%/)l[o,r0](arg^),
^2=5(x/)l]-20o,o](arg*)i
we get a function g = g\ + g2 in Q n {argz < ?()}> such that dg = d(xf)i
with 5 = 0(l)expC(argz)+//i, such that d£ej2<7 = ^it(l) in J, and such
that g(z) = 0(1) in (fi\ W) narg-1([0,60]), if €0 is small enough.
Indeed, we can treat the ^-equations in "polar" coordinates ( given by
z — e^ and solve the equation for g2 using the convolution with the standard
kernel l/?r£. For </i, we use the convolutor (7r()~1eq^/h, where q(() is a
suitable quadratic polynomial with q(0) = 1. To understand the choice of 9,
consider for instance the case ?o = 1? so we have an equation J=# = fc, where
|fc| < eclm<, suppA; C {0 < Im( < 1}. Take q(() = -c<2 + t'-1(C+ 1)C,
where a > 0 will be small. The convolution at a point £ is then bounded
by 0(l)erK)M, where
r(C)=_sup Re(-a((-()2-i(C+l)((-0)+Clml
CGsuppA;
If K is the projection to the real axis of suppfc, we get
r(C) < (C + l)ImC-adist(ReC,K)2 + a(ImC)2+max(0,-2aImC + a-l).
Taking into account that Im£ is bounded for z G ft, we choose a > 0 small
enough and get,
r(C)< (C+l)ImC + a(ImC)2-adist(ReC,K)2.
We then get 51 with the required properties.

383
We finally take f = xf"9 an^ check that f has the required properties.
a
If / is as in the last proposition, then we can still apply Theorem 2.2.
In fact, we first check that "tr[(x/)(ip.)]o" only changes by £?(/T"nmax), if
we replace / by /. In section 8 we shall see that the number of eigenvalues
and resonances that appear in the sums to the right in (2.11) is (9(/i"nmax),
so the sums also only change by this amount, if we replace / by /.
3. Trace class estimates before complex scaling
We start by estimating the characteristic values of certain resolvents and
truncated resolvents. In doing so, we let the spectral parameter belong to
the open set in C, defined by:
|Im*|>^(|IM-C)+. (3.1)
From (2.8), we get:
N{{P*-i)-l)<C{l + h2j2lnrl. (3.2)
Using the resolvent identity, we get for a more general z in the set (3.1):
«P* \-i\ <r J C(M + *2i2/w-)"1. when \lmz\ * Const. > 0,
HW. z) ) S j _£_(! + h2p/n)-\ when |Imz| < Const..
(3-3)
Let r C Rn be a sufficiently widely spaced lattice and let 0 < ^„ €
Co°(Rn), v € T, be a translation invariant partition of unity (ifiv(x) =
ifto(x - v)) with ^o = l near 5(0, Rq).
Lemma 3.1 Let % varV in some bounded subset of C£°(Rn) with % —
Const, near B(0, Rq) and let v vary in I\ Then for z satisfying (3.1):
/i;(X(P. - z)-10,), HiMP'-^X) <
c(z) , r~^g^(dist(suppytsupp^)~l)4. (o a\
\Imz\{(z)+h*pln-)e ' ^'^>
where (z) = y/1 + \z\2.
Proof. Assume first that v — 0. If /, g are real functions on Rn, we write
/ -< g, if supp / is contained in the interior of the region where </ = 1. Let
$>o G Co°(Rn) with tpo -< ^o- If we choose P* and the corresponding torus
suitably, we have:
(F. - z)~lifo = MP* - z)-1^ - (P - z)-l[P,M{P# - z)"Vo. (3.5)

384
MP- -*)_1 = MP* -fT1^ + MP* -z)-l[P,M{P -z)~l- (3-6)
Here (P. - 2)_1[F.,^o], [P.,M(P ~ z)~l are bounded operators of norm
<9(fi~r). (We use the ellipticity of P. near supp [P., M-) Assume first that
\lmz\ ^ c^it:- Then (3-3) implies that
N((R - ,)"Vo), N(MP- - z)-1) < {{z)+Ch2p,n.y (3-7)
and we get (3.4) without the exponential factor. If dist (supp x, supp ^o) >
1, choose iJ)q with support sufficiently close to that of fa and notice that
by a Combes-Thomas argument,
X{P. -z)-l[RMl [RM]{R-z)-lx =0(fc)c-A(di8t(»uPPx.»upp*o)-i)+j
in operator norm. (A typical Combes-Thomas argument is employed in the
proof of (7.8) below.) When multiplying (3.5) to the left or (3.6) to the
right by x, only the last terms survive in the right hand sides, and we get
(3.4).
Still in the case v = 0 it remains to treat the case when |Im z| is bounded
(so that |z| is also bounded). Using the resolvent identities,
(P. - 2)~Vo = (P - 0~Vo + (P - z)~\z - i)(P - 0~Vo, (3.8)
MP - z)"1 = MP - *)_1 + MP - 0_1(* - i){P - z)~\ (3-9)
and (3.7) with z = i, we get,
K((P. - z)"Vo), H(MP ~ -)"1) < {lmz\{llh2pin.y (3-10)
which gives (3.4) without the exponential factor.
Assume that d = dist (supp x, supp ^o) > 1- Let x £ C£° be equal to 1
on
{x e Rn; dist (a, suppx) < d/3},
and equal to 0 on {x G Rn;dist (#,supp^0) < d/3}. From (3.8) we get
X(P - *)~ Vo = X(P ~ i)-^o +(z- i)X(P ~ z)-'x{P ~ i)~ Vo
+(z - i)X(P - z)-1^ ~ X)(P ~ i)~XM (3.H)
and from (3.9) we get a similar relation:
MP - z)~xx = MP - 0_1x + (z- i)MP - i)~lx{P - z)~lx
+(z - i)MP - 0_1(1 " X)(P ~ z)~lX. (3.12)

385
We can estimate the characteristic values of the first two terms of the RHS
of (3.11) and (3.12) by using (3.4) with z — i (already established). A
Combes-Thomas argument shows that the norm of x(^ - j?)~1(l ~ x) is
OWlmzl-1 exv-d/(Ch\Imz\).
Combining this with (3.7) for z = i, we can estimate the characteristic
values of the last terms of the RHS of (3.11) and (3.12) and we get (3.4)
for v — 0.
Consider finally the case v ^ 0. The proof is the same except that in
the formulas (3.5), (3.6), we replace "0" by 'V and now let P* be an
operator on a torus containing the support of ^„ and be equal to P. in a
neighborhood of the latter support. D
As a special case of Lemma 3.1, we have
M^-»)-^<^|(W + ww. (3.13)
When */, fi G r \ {0}, then \^)y{P. - ^"Vjo *s a we" defined operator in
any of the spaces %0, %i, L2(Rn) and we have the following result, where
the assumption (2.7) is used for the first time:
Lemma 3.2 For v, ji G T \ {0}, we have
Mi([^(P.-«)-VjS)<
~ |Imz| , ,,
\lmz\2({z)+h2j2/n™*x)
C(«)2(ma*(M,H)-"+e-^ ,3 U)
Proof. It suffices to treat the case when |j/|, |//| >> 1, since we otherwise
get (3.14) from (3.4). For the same reason, we may restrict the attention
to the case when \v - ji\ « \fj,\.
We need a resolvent identity with cut-offs. Let x G Co°(Rn) be equal to
1 near B(0, i?o) and let v G L2(Rn \ 5(0, Rq)) so that t; can be considered
as an element of %j, j = 0,1. Let Uj G Wj be the solution of (Pj - z)^j = u.
Then
(P3-z){l-X)u3 = {l-x)v-[P3,x]ur
Rewrite the equation for j = 1 as
(P„ - *)(1 - x)«i = (1 - X)v ~ [Pi, x]«i - (Pi - Po)(l - X)«i.
Then,
(Po-z)((l-x)tti-(l-x)«o) = -([Pi,xK-[Po,x]«o+(Pi-Po)(l-x)«i),

386
which we can write
(1 - X)(Pi - z)~xv - (1 - x)(Po - z)-*v =
-(Po-*)-1([fi.X](fi-*)-1»-
[Po,x](Po ~ zy'v+iP, - P0)(l -X)(Pi - ^"M-
Let xi» X2 have the same properties as x- Then, if (l-xi)(l~x) — l"Xi»
[(i-xi)(P.-^)-1(i-X2)]S =
-[(i - xi)(Po - ^[P.xKP - ^U - X2)]S -
(1 - Xi)(Po - z)-^ - Po)(l - x)(Pi - z)~Hl - X2). (3.15)
Multiplying this identity from the left by ij)u and from the right by ij)^ we
get the same relation with 1 - Xi replaced by ij)u and 1 - X2 by ^M. Choose
X = Sjler;|/I|<|i/|/2^JI" ^n estimating the characteristic values of the first
term of the RHS of (3.15) (with the substitutions: 1 - Xi ^ ^, 1 — X2 "->■
^/x) we use (3.4) to estimate the characteristic values of ^(Pq - z)~l and
estimate the norm of [P.,x](.P. - ^)~V/i by
\lmz\
It follows that the characteristic values of the first term of the RHS of (3.15)
satisfy (3.14). In estimating the characteristic values of the second term of
the RHS of (3.15), we observe that the coefficients of Pi - Po are 0((fi)~n)
on the support of 1 - x, and it follows that the characteristic values of this
term satisfy (3.14) without the exponential factor. In order to get also the
exponential factor, when dv^ =def dist (supp^„, supp^M) > 1, we split the
term into
£>(P0 - *r7*(Pi - Po)(i - x)(Pi - *)-V„,
0
where fx + f2 = 1, fk G C6°°, fi(x) = 1 when dist (a, supp^„) < 4,^/3,
/2(#) = 1 when dist (#,supp^) < d^/3. Let /^ 6 C£° be equal to 1 on
supp /jfe and have its support in supp (fk) + P(0,1) and write the term for
k = 1 as
iMPo " ^MPi ~ Po)(l ~ x)fi(Pi ~ z)-1^. (3-16)
Here the norm of ^(P0 - zYlh{Px - P0)(l - x) is 0{fyn\-») while
the characteristic values of f\(P\ - z)~ltj)^ can be estimated by (3.4). The
characteristic values of (3.16) then obey (3.14). It remains to treat ^(Pq -

387
z)~1f2{Pi ~ Pq)(1 ~ X){Pi ~ z)"1^^ and we do essentially the same thing,
estimating now the norm of (Pi - P0)(l ~x)(Pi - ^)~V/i as before and the
characteristic values of ^(Po - z)~1f2 by means of (3.4). Putting all the
estimates together, we get (3.14) for the second term of the RHS of (3.15).
a
Lemma 3.3 Let m 6 N with (1 + m)zr— > 1 and let zn G C \ R be some
v ' "max *
fixed point. Then for z as in (3A), andv,fi 6 r, /ipl/(P-zo)~m(P-z)~1,ip^
is of trace class and
\\MP--zo)-m(P--z)-l1>,*\\to<
Ch-n. (z)-min(l,m+l-»./2)(1 + 8mn/2 log(z))
J«Lc-min(l,|Im*|)(|i/-/«|-C)+/(Cfc) (o 17)
\Imz\ ' V ' /
Here we use the standard notation for the Kronecker delta. For v, fi 6 Y \
{0}, we have
\\[MP--Z0)-m{P--z)-l1>Mtr<
^_nmlX&(^)"min(1,,n+1"nmax/2)(l + ^,nmax/2l0g(2))X
(|„|-S + e-1^1^^^^^-1^1^^^-^-^ (3.18)
Proof. Write
MP--*o)-m(P •-*)-% =
Y, MP - zoTHai{P - 2o)_V«a..(P. - ^o)_V«m(F. - z)-Vm,(3.i9)
aGrm
with a = (ai,..,am). Using (3.4) we can estimate the j:th characteristic
value of the general term in the sum by
^(i+/lv/nrm((^)+^2/n)-1 x
\\mz\
e-cM(K«il-a)++(l«i-«2|-C)^^
and hence the trace class norm of the term is bounded by the sum over j
of the values (3.20). We estimate
oo
5 = £(1 + h2j2'n)-m{{z) + h2^)'1
1
< / {l + h2x2ln)-m{(z) + h2x2ln)-ldx. (3.21)
Jo

388
After a change of variables, we get
S< {l + t)-m{(z) + t)-hn'2-Utx^h-n-. (3.22)
Jo 2
The integral is convergent, since m + 1 > n./2 and if we treat separately
the integrals over the intervals [0, (z)] and [(z), oof, we get
S < 0(l)h'n (z)-min(l,m+l-n./2)(1 + ^ ^ ^^ (333)
The trace class norm of the general term in (3.19) is bounded by
0{l) jlmTj (^)"min(1'm+1"n /2)(1 + <W l0g(^»^n X
e-^((k-a1|-C,)++(|a1-a2|-C)++..+(|am_1-am|-C)++|Im2|(|am-M|-C)+) ^Mj
Summing over a, we get (3.17).
Turning to the proof of (3.18), we write first for v, n £ T \ {0}:
[MP- ~ zo)-m(P. - *)-V„]o =
aerm\(r\{o})m
^^(P.-^o)-1^^-^)-1^
+ E [^(^•-^)"Va1.^am_1(P.-^)"Vam(P.-2r)-VX (3-25)
aG(r\{0})^
In each term in one of the two sums constituting the first term of the RHS
of (3.25) at least one of the components aj is equal to 0, and the proof of
(3.17) shows that the trace class norm of the first term of the RHS of (3.25)
is bounded by
e}(1)l£l(2)-min(1',n+1-nmax/2)(l + *TO,W2log<*» x
e-min(lChZ|,1)(maX(l^l^l)-Cr)+fe-nmaxt (3.26)
The general term of the last sum in (3.25) is a sum of m expressions of the
form
(P0 - ^b)-1K+2.^am(Po - z)-1^ (3.27)
for k = 0,.., m - 1 with the convention that o>o = v, and a term
MPi - zo)-^ai..(Pi - zQ)-l[^am{P. - z)-lMl- (3-28)

389
Combining (3.4) and (3.14), we can estimate the j:th characteristic value
of the operator (3.27) by
0(l)(*)(max(|afc|,|afc+i|)-»
|Im z|(l + h2pln™)m{{z) + h2p/n™*)
e-^h((l'/-ai|-<:7)++(lai-a2|-C')++..+(|am_1-am|-C')+ + |Im2|(|am-M|-C)+)^3i29^
and of the operator (3.28) by
C?(l)(2)2(max(|«m|,|Ml)~" + e-J^1(max(|am|,W)-C)+)
|Im z\2(l + h2pln™*)m{(z) + h2pln™*)
x A, (3.30)
where A denotes the same exponential factor as in (3.29).
As in the proof of (3.17) it follows that the trace class norm of (3.27) is
/2\/2.\-min(l,m+l-nmax/2)
C(l)]7^| (l + <Ww/2)log<*>)
max(|afc|, K+il)~^~nmax * A, (3.31)
and that the trace class norm of (3.28) is
(max(|am|, |M|)-^+ e-^(max(la-l'W)-a)+)/i-nm- X A. (3.32)
Possibly after modifying the constant C in the exponential factor A, we
may replace max(|ajt|, |ajfe+i|) in (3.31) by \v\. In (3.32), we may similarly
replace \am\ (at two places) by \u\ (noticing that the term e-JW1(max(|am|,|/i|)-c)+
can be ignored all together when \lmz\ > 1). It then follows that the trace
class norm of the last term in (3.25) is
^-nmaxe-^ min(l,|Im2|)(|j/-/i|-C)+ ^ (3.33)
Combining this with (3.26), we get (3.18). □
Proof of Proposition 2.1. The first part of the proof will be to treat the
case when / G Co°(Rn). We shall use the operator version of the Cauchy-
Green-Riemann-Stokes formula ([15]):
f(P) = lj%(P--*)-lL(dz), (3.34)

390
where L(dz) is the standard Lebesgue measure on C = R2 and / G Cq°(C)
is an almost analytic extension of/, so that /|r = /, and || vanishes to
infinite order on R. Almost analytic extensions were introduced by Hormander
[16]. To get trace class operators under the sign of integration, we fix some
zo e C \ R and write
f(z) = (z-z0)-mg(z), 5eC0°°, (3.35)
/(P.) = (P. -zo)-mg(P.) = \jdj^(p'~*o)-mGP. "z^Hdz). (3.36)
Here we choose m as in Lemma 3.3.
Let x € C£°(Rn) be equal to 1 near B(0,i?o). It follows from (3.17)
that for Imz ^ 0:
\\x(P. - z0)-m(P. - zy'xiUr <
Ch-n. {z)- mm,m+l-»./2)(1 + ^ log(z))^, (3.37)
||(1 - X)(P. - Z0)-m(P. - zyhWtr, \\X(P ~ Z0Ym{P ~ z)~\\ ~ X)||tr
< Ch~»- (z)-ni-(l.m+l-»./2)(1 + 8mn/2 hg{z))
\lmz\
From (3.18) we get (since n > n):
\m-X)(P-Z0)-m(P-z)-l(l-x)}h\\tr<
Ch~n™\h^{z)~ min(1'm+1-nmax/2)(l + ^,„mix/2 log(z»
(1+ ■ * ,J2n- (3-39)
min(l, |Imz|)
From these three estimates and (3.36), we get,
\\xf(P.)x\\tr, ||(l-X)/(P.)x||tr, ||x/(P.)(l-X)||tr,
||[(1 - X)/(P.)(1 - X)]J||tr < 0(h~—), (3.40)
and hence also the corresponding estimates for the traces. The RHS of (2.9)
is therefore 0(h~nm*x) and it is straight forward to see that it is independent
of the choice of x-

391
We now turn to the general case and let / € S~mW as in the
proposition. Let x € Co°(R\ {0}) and consider for A > 1:
/(P.)X(A"2F.) = /(A2(A-2F.))X(A"2F.)-
The function t y-> f(X2t)x(t) has its support in a bounded A-independent
interval and
l#(/(A20x(0)l<C*A~2m(/)- (3.41)
Consider the operator P. = \~2P. It is then straight forward to check
that P. satisfies the assumptions for P. in the proposition, provided that we
replace h by h = h/X.
In view of (3.41), the uniform control of the support of the function t \->
f(X2t)x(t) and the fact that we already have established the proposition in
the case of Cq° functions, we get
"tr[/(P.;/i)x(A-2P.)]o" = 0(\-2m^h-n^) = 0(A-2m(/)+n»«)fe-n»«.
Here -2ra(/) + nmax < 0 so we can decompose f(P.) into terms of the
tyPe f(P')x(^~2P-)i with a sequence of A's which grow exponentially, and
the corresponding estimates above can then be summed and we obtain the
proposition in the general case. □
4. Review of functional calculus for PT
In this section we essentially only use material from [34] in a straight
forward way. Let M be a torus containing B(0, R) for some R > i?o, and
define P. : %t -> %t as an unbounded operator with domain Vf as
in section 2 and in [34]. Then if x G C^(M \ B(0,i?o)), we get as in
Proposition 4.1 in [34], that x ls a uniformly bounded operator with
respect to h: V*M -> H2k(M), H2k{M) -> V*M, for every fc G R, if we
equip H2k(M) with the norm \\(hD)2ku\\L2 and let V*M denote the
domain of (P.)k, for k > 0, and the dual of V*'~k for k < 0. An operator
A = A{z\ h) :U* ->Ut, defined for 0 < h < h0 and for z in some subset
of C \ R will be called negligible if for every JVgN, there exists M > 0,
such that
A(z\ h) = 0{hN\lm z\~M) : V*'~N -> V*^. (4.1)
Lemma 4.1 Let ^i?^2 G C°°(M), be constant near B(0,i?o) o,nd have
disjoint supports. If we restrict z to a bounded subset of C\R, then ^i(F. -
z)~l^2 is negligible.
Proof. We follow the proof of the corresponding statements (Propositions
5.1 and 6.1) in [34]: Let ^2 -< ^3 -< ^4 -< •• -< i>N < 1 - i>\ and consider the

392
identity:
MP* - z)-1^ = (-i)NMP* - z)-\p*An]{p* - z)~\.
{P* - z)~l[P*,i/>3](P* - *)"V2.
Notice that {P* - z)"1 = Odlmz]-1) : V*'k -> vt'k+1 (since z varies
in a bounded set), and that [P.#,V>j] = 0{h) : V*'k -»• X>#'fc_2. Then the
lemma follows. D
Lemma 4.2 Let Q. = Yl\a\<2h-Ax'^h){hDx)a, b.,a € C°°{M) satisfy the
same assumptions on M as Pr outside B(0, Rq) . Also assume that Q. is
self-adjoint (necessarily with domain H2). Let Q C M\B(0, R0) be an open
set where Q. and Pr coincide. Then for every x € Cq^(Q), the operator
x(Q- — z)~lX ~ x(P- ~ z)~lX JS negligible when z is restricted to some
bounded set.
Proof. Let x X V € Cg°(fi). Write
(J* - z)-\ = 0(0. - ^X - (P* ~ z)-l[P*,i>}{P# - zy'x,
so that
X(P* - z)-'X ~ X(Q- - z)~'x = ~X(P* - z)-l[P*^){P* - z)-lX.
Here the RHS is negligible since [FT,^](P. - z)~lx is, as we see from
Lemma 4.1, if we notice that [P?,i>] = 0(h) : 2?#»* -> £>#'*-2. D
Notice that if / G Cq°(R) is independent of /i or varies with ft in a
bounded subset of Cq°, then under the assumptions of Lemma 4.2, xf(Q-)x~~
Xf(P-)x ls negligible in the sense of operators independent of z: We say
that A = A(h) is negligible if A : 0{hN) : V*>~N -> £#»* for every JVgN.
Further it follows from the results of Helffer-Robert, see [28] (and also
[30], [5] for a presentation based on the operator Cauchy-Green-Riemann-
Stokes formula,) that f(Q.) is a an h-pseudodifferential operator (,from now
on h-pseudor for short,) with leading symbol /(</o).
5. Review of complex scaling in the semi-classical case
Complex scaling or analytic distorsion is a standard technique in resonance
theory since the work of Aguilar-Combes [1]. Among the numerous later
works, we can mention the work of Hunziker [18]. Here we shall follow [34]
since we also need large angle distorsion. More precisely, we give a quick
review of section 3 of [34] with some minor modifications, due to the fact
that our operators are slightly more general. We refer to [34] for more
details.

393
A smooth submanifold T C Cn is said to be totally real if TXY n iTxY =
{0} for every x e I\ where TXT is viewed as a real linear subspace of
TxCn ~ Cn, and i denotes (multiplication by) the imaginary unit. We say
that T is maximally totally real (m.t.r.) if V is totally real and of maximal
(real) dimension n. The standard example of such a manifold is V = Rn.
Let T C Cn be a locally closed m.t.r. manifold and indentify T*T with
a submanifold of Cn X Cn, via the map T*T 9 (x,du(x)) h->> (x,dxu(x)),
where u is an almost analytic extension of the real valued smooth function
wonT, and du is the holomorphic part of the differential of 2, here
identified with the corresponding n-vector of holomorphic partial derivatives.
By almost analytic extension we mean a smooth extension such that du
vanishes to infinte order on T. (Here du denotes the antiholomorphic part
of the differential of 5, so that du — du + du. In [34], we reviewed the
existence and quasi uniqueness of almost analytic extensions of functions
on m.t.r. manifolds, due to Hormander [16] and Hormander-Wermer [17].)
Let Q C Cn be an open neighborhood of V such that V is closed in £2,
and let
P(x,Dx)= £ *a{x)Da (5.1)
\a\<m
be a differential operator on 0, with holomorphic coefficients. Define P? :
C00(r)-»CC0(r) by
Pru=(Pu)]r, (5.2)
where u is an almost analytic extension of u as above. Pp is a then a
differential operator on V with smooth coefficients and for the principal
symbols, we have the relation:
Pv = P|T*r- (5.3)
It is well known, that if Pp is elliptic and Ppu = u, u e ^'(r), where t; has
a holomorphic extension to a neighborhood of T, then the same holds for u.
Lemma 3.1 in [34] gives a deformation version of this and says roughly that
if I\, / G [0,1] is a smooth family of m.t.r. manifolds which are independent
of t outside a compact in £2, and with the property that Prt is elliptic, then
if Pr0^ — v ai*d t; has a holomorphic extension to a neighborhood of the
union of all the I\, then the same holds for u.
Let
P(x,hDx;h)= £ aa(x;h)(hDx)a, (5.4)
\a\<m
where aa are holomorphic on Q, and uniformly bounded with respect to h for
x in any fixed compact subset of Q. Then Pp has an analogous form with C°°
coefficients, for every choice of local coordinates on T, and the coefficients

394
are locally uniformly bounded. The semi-classical principal symbol is then
denned modulo 0{h(§m) by
pM]h)= J2 M*;W, (5-5)
\a\<m
and similarly for Pp. The relation (5.3), also holds (modulo 0(h(£)m)) for
the semiclassical symbols. If aa(x; h) = a%(x) + 0(h), then we can make p
and py ^-independent by choosing p(x,£) — Yl\a\<m a^(x)^a an^ similarly
for jF^, and (5.3) holds without any remainder term.
For given €o > 0 and R\ > i?o, we can construct a smooth function
[0,7r] x [0,oo[9 (0, t) ^ fe(t) G C, injective for every 0, with the following
properties,
(i) /*(*) = tforO<t<fli,
(ii) 0 < arg/*(*)< 0,0*/*^0,
(iii) Mgfe{t) < Mgdtfe{t) < Mgfe{t)+e0,
(iv) fe(t) = e%et for t > To, where To only depends on €o and R\.
For later use we shall give an explicit construction and derive a fifth
property, which will be convenient though probably not essential. We look
for /(/) = fe^{t) of the form f(t) = tei@(s\ where s = logt and 0 = 0^,eo
depends smoothly on 0, €q. Then f(t) = (l+i& (s))et@(s\ so it is enough to
take 0 smooth in all variables with 0 < &(s) < €o, 0(s) = 0 for s < log i?i,
0(5) = 0 for s large enough. This is easy, but we make an explicit choice
of 0: Let 0 < <j) G Cg°(]0, l[) with f <j){s)ds = 1, &(*) = e^^s/e). Let 0
be the solution which vanishes far to the left, of the equation,
e'(s) = €0(</>eo * l[o,0/eo]H5 ~ logiZi).
Then (i)-(iv) hold and moreover:
(v) arg/#(£) is an increasing function of 0 and of t and if 0\ < 02 and
foAt) ¥" fe2(t), then (log*-log Pi) > 0i/eo and wehave0i-€§ < arg/^(t).
Consider the map
kq : Rn 9 a: = tu H> /^(t)u; G Cn, t = |s|.
The image is a m.t.r. manifold which coincides with Rn along B(0,Pi).
Let U.fi = n.tRo © L2(Te \ B(0, P0)). If X G C§°(fl(0, Pi)) is equal to 1
near B(0,Po), let V.fi = {u G ^;x« G ©(P.), (1 - x)« G P2^)}, where
P2(r#) is equipped with the natural semi-classical norm. Let P.,q be the
unbounded operator W.j —> W.j with domain P.^, defined by
P.f*tt = P.(xtO + P.fr(l-x)ti.

395
These definitions do not depend on the choice of (the /^-independent) x«
Parametrizing Yq by means of kq, we get outside the origin:
-Are = (fit)-1 A)2 - (f(t)f'(t))-\n - \)iDt + (f(t))-2Dl (5.6)
where -D^ is the Laplacian on Sn_1. If u>*2 denotes the principal symbol
of D%j and we let r be the dual variable of t, then the principal symbol of
-Are is
p0l* = (r//'(0)2 + ("7/W)2, / = /*, (5.7)
so pointwise on Yq, - Are is elliptic and the principal symbol takes its values
in an angle of size < 2€o, while globally, po,0 takes its values in the sector,
-2(0+ €o) <arg*<0. (5.8)
In the following we shall always take 0 < 0$ (< 7r) so when €o is small
enough, the angle 2(0 + £o) of the sector (5.8) is < 7r.
Choosing R\ large enough, we get the following facts in view of the
assumptions (2.6), (2.10):
In Rn \ B(0, i?o), h~2P.j is an elliptic differential operator whose
principal symbol (in the classical sense) over each fixed point in
Yq takes its values in an angle of size < 3€o, and globally
in a sector - 20 - 3€o < arg z < €o. (5.9)
In Rn \ S(0, i?i), the difference between the semiclassical
principal symbol of Rf and the principal symbol of h~2P.^
is o(l)(£}2, when Rx -> oo. (5.10)
The coefficients of Rj - e~2te(-h2A) and all their derivatives
tend to zero uniformly with respect to h when r# 9 x —> oo,
and we identify Ye and Rn, by means of kq. (5-11)
Here we write the operators semiclassically as in (5.4)
Lemma 5.1 If z e C\ {0}, arg z ^ -20, then Rj - z : V.j -> %.$ is a
Fredholm operator of index 0.
This is essentially a consequence of a certain ellipticity near infinity and
the proof is the same as the one of Lemma 3.2 in [34]. The only difference
is that to the operators K(z), L(z) there, we have to add operators with
arbitrarily small norm (depending in the choice of the partition of unity).
It follows from Lemma 5.1 that if arg z ^ -20, z ^ 0, then z belongs to
the spectrum of Rq iff Ker (P q - z) ^ 0.

396
Lemma 5.2 Assume that 0 < 0\ < 02 < O0 and let z0 G C\e-2il6l>ed[0, oo[.
Then dim Ker (P^ - z0) — dim Ker (Rft2 - z0).
This is practically identical to Lemma 3.4 of [34] and the proof is the
same as there, using (the extension to our present situation of) Lemma 3.1
of [34] evoked after (5.3).
The lemma above and analytic Fredholm theory (as developed for
instance in the appendix of [14]) show that the spectrum of P.j in C \
e~2^[0, oof is discrete and in particular (when 0 = 0) that the spectrum of
P. in ] - oo, 0[ is discrete. If 0 < |, this discrete set consists of the negative
eigenvalues of P. plus a discrete set in the sector e""2tfo,^]0, oof. If 0 > |,
then the spectrum of P^ in C \ e~2^[0,oo[ is contained in e~2^°'^]0, oo[.
Lemma 5.2 tels us that the spectrum in e~2^°^°t]0, oof is independent of 0
in the following sense: We say that z G e~2^°^°t]0,oo[ is a resonance for
P. if and only if z G &{R,e) for some (and hence for all) 0 G]O,0o] with
Cee-2W[]o,oo[.
By analytic Fredholm theory (see for instance the appendix of [14]), we
know that if z0 G e-2i&el]0, oof is a resonance, then the spectral projection
-. f{z-R,e)-ldz, (5.12)
^ 2,rt,7
with 7 : [0,27r] 9 s y-> zq + €ets, and e > 0 small enough, is of finite
rank. The image F.j^ is contained in the domain of any power of Rj
and is invariant under Rj. Moreover the restriction of P.^ - z$ to F.jlZo
is nilpotent, so F.,^0 = Ker(P,0 - z0)k° for some k0 G N. If 0 G [O,0O]
is a second number with zq G e"~2*[°'^]0, oof, then since Lemma 5.2 can be
extended to "dim Ker {Rfii ~ zo)k = dim Ker (Rte2 ~ zo)k f°r a^ ^"? 7r-,^,«o
and 7r ~ have the same rank, which by definition is the multiplicity of the
resonance z$. Further, in Lemma 5.2 and in the above mentioned extension,
we have invariance not only under changes of 0, but also under changes of
the family /#.
One can also define the resonances as the poles in e~2^0^°t]0,oo[ of
the meromorphic continuation from the upper half plane across ]0, oof of
(P. - z)'1 : ft.,comP -> P.,ioc- (See [34] for the definition of H.iComp/\oc,
^•,comp/loc-) We shall not use that point of view in the following, so we do
not give details.
In the following result we use the special family /#, given after (i)-(iv)
with €0 > 0 small enough.
Proposition 5.3 Let K C eQ0>mW*>2*-2e°ft]Q,oo[ be compact, and let m G
N with (1 + m) ^—^ > 1. Then for 0 < 0 < 0q and z0, z G K, we can define
"tr[(P,0 - £o)~m(P,0 - z)"1]ov as in the case 0 = 0 and this quantity is
independent of 0.

397
Proof. Identifying Ye with Rn by means of kq, we can use the same cutoffs
and partitions of unity on Ye as on Rn. We also choose €0 > 0 in the
construction of Ye sufficiently small depending on K. Then (P.^ - z)~l :
H.j -> V.j is uniformly bounded for z G if, h e]0,/io], 0 < 0 < 0O and
we can go through the first part of the proof of Lemma 3.1 and replace P.
everywhere by P.,0. We get the same estimates and obtain with x and ^„
as in Lemma 3.1:
/i;(x(P.,0 - *)-V,), /ii(0v(P.,tf - «)~1X)
< C p-cM<iist(suppx,supp</v)-l)+ /k iq\
^ (l+/i2j2/n.)e 1 l0-1^
W(^(P,, - ,)-V.) < (1 + ^-2/ra.)^^(k"Ml"C)+. (5-14)
uniformly for 0 < /i < feo, 0 < 0 < 0o> ^ M £ T.
Similarly, we can replace Po, Pi, by Po^, Pi^ in the proof of Lemma 3.2
(avoiding the more delicate discussion there of the case when Im z —> 0,)
and obtain:
C
(l + ft2j2/nmax)
max(M, |//|)-V^diy-'il-a)+, (5.15)
for i/, /* e r \ {o}, o < fe < h0, z e k, o < 0 < e0.
The proof of Lemma 3.3 gives:
WMP-.o-zo^iP.j-zr^WtT < Ch-ne-m(\"-»\-c)+, z/,M € I\ (5.16)
\\[MP;9-zo)-m(Re-z)-^]l\\tr<
CTt-rw maxd^j, |m|)-»c-ot(I"-/«I-C)+ , „, M € r \ {0}, (5.17)
with h, z, 9 as in (5.15).
By applying some of the arguments (see in particular (3.36), (3.37),
(3.38)) of the proof of Proposition 2.1, we see that "tr [(P.tg - z0)~m(P.,e -
2)_1]J" can be defined and is (9(/i-nmax).
To prove the independence of 0 of "tr [(P.t9 - z0)-m (R>e - z)_1]J", it
suffices to show that we get the same value for 0 = 0\ and for 0 — 02 provided
that 0!,02 € [O,0O] and |0i - 02\ is small enough. Let x € Cg°([0,1[;[0,1])
be equal to 1 on [0, ^], and consider for R > 1, the intermediate contours
Teue2,R = neltg2iR(Rn), where
KBuei,R : Rn 9 x = tu H- /*lA,fl(t)w £Cn,t= |x|, (5.18)

398
/«„«„«(«) = /*(0 + x(|)(/*2W ~ feM- (5-19)
For R < Ri (given in (i) after (5.5),) we have feue2,R = felt and r^,^
converges to Tq2 pointwise when R tends to oo. We can define P,eue2,R =
P.\rg e R in the obvious way. We claim that
"tr[(F.A^,fl-^o)-m(i5.A^1fi-^_1]i"->"tr[(i5,fl2-^o)-m(F,fl2-^-1]S"
(5.20)
when R ->■ oo, provided that |02 - #i| is small enough. In fact, this follows
from the following two statements, which are easily verified:
(A) (5.15)-(5.17) are uniformly valid if we replace P.j by P,0i,e2,R with
|0i-02| and R> 1.
(B) MP-fiuhrR ~ *o)-m(P;elt<hJi ~ z)-1^ -> MP;<h ~ zo)~m{P,e2 -
z) 1^/i in trace norm when R —> oo.
It now only remains to prove:
"tr [(^,01,02,* " ^rm(fyiA,fl " ^)"1]5" is independent of J2. (5.21)
Let i? vary in some compact interval I contained in [i?i,+oo[. Then
/fliifciM*) ls independent of R except for t in some compact interval J C
]i?o, +00 [ and on this interval we have
arg/ft.fc.Mt), arg/^(£) = 0X + O(e0) + O{\0x - 02\).
By abuse of notation, we write P.^r for P.fluo2,R and similarly for the
contours. Let x = x(0 G Co^I^r) have support disjoint from B(0, i?o) and
be equal to 1 near t 6 J. (Strictly speaking, x wiH depend on i?, since we
let x live on T^, but we arranged so that x is identically equal to one on
the part of Yr which varies when R varies in J.)
The operators Xi(z~Pfi)~1X2 are independent of i?, if Xi?X2 G C^I^r)
are constant on £(0,i?o) and supp(xj) are disjoint from the set where
X = 1. In fact, this follows from the same principle of non-characteristic
deformation as the one which is behind Lemma 5.2. It is now clear that
(5.21) will follow from
tr (x o {RiR - z0)~m{P.lR - z)-1 o x) is independent of R G /. (5.22)
Let x G Q°(rfi) be = 1 near suppx- Put Gt{x) = Cne-n/2e-^e~ie^\
0 — 9\ with Cn chosen so that
/ eieGe(x)dx = l.

399
Denote by Ge* the operator with kernel Ge(x - y)dy on any one of Tr.
x{Gt*)x, x(Ge*)x tend strongly to (multiplication by) x m the space of L2
bounded operators over each of these contours, so
X(P,R-Z0)-m(P.,R-z)-1X =
lim£_,0 x(Ge*)x(R,R ~ zo)-m(P.,R ~ z)-xx{G^)x (5.23)
in the space of trace class operators.
ForueC?(rR):
X{G€*)X{P.,r ~ z0)-m(RR - z)-lX{Gt*)xu{x)
= IrR X{x)l<e{x, y)x{y)<y)dy, x e TR, (5.24)
where Kt(x,y) is an entire function which is independent of R. The trace
of the operator (5.24) is therefore
/ x{x)2I<e{x,x)dx, (5.25)
JrR
which is independent of R. From this and (5.23), we get (5.22) and the
proof is complete. □
Simpler proofs are likely to exist. For instance, one might try to prove
that the ^-derivative of the "trace" in the proposition is the trace of a
commutator, hence 0.
6. Grushin problem for the scaled operator
In the following, we write 0 instead of #o- Let F be a smooth mapping from
a neighborhood of e~^~eo'2^+eo^[0,oo[ into itself, such that
F(z) = z for \z\ sufficiently large, (6.1)
F(z) = z for z in a neighborhood of e~2 [0, oof, (6.2)
Q, is disjoint from the range of F. (6.3)
Here Q is the same set as in Theorem 2.2
Let / = F|r. Choosing Yq conveniently as in section 5, let p.j be the
semi-classical principal symbol of P.^, defined on T*(Tq \B(0,i?o))- Then
F o p.Q is a well-defined smooth function with values away from Q such
that {F op.j)(x,£) = p.,0(#,£)? when \x\ > i?2, if i?2 > #o is large enough.
(Here we identify Yq with Rn by means of k$.)
Using functional calculus, we shall first construct a finite rank
perturbation of Pq for which Q belongs to the resolvent set, when h > 0 is small

400
enough. Then using this perturbation, we construct a Grushin problem for
P.,6 ~ z which is well posed for z near Q.
Let R\ G]i?o> #2[ have the property that Yq coincides with Rn near
S(0, i?i). Let P. be the same operator as in the sections 3, 4, realized on
a torus which contains 5(0, R\). Then
f(F*) = P* + K*, (6.4)
where K* is a uniformly bounded operator (w.r.t. h) with
T*nkK* = 0(h-n-). (6.5)
Moreover, Q belongs to the resolvent set of /(P ) and
(*- /(P*))"1 = 0(1) for z in an /^-independent
neighborhood of Q. (6.6)
Let Q. be an h-differential operator on the same torus M where p* is
defined, which is elliptic self-adjoint and which coincides with P. outside
S(0,i?o). If Xi?X2 G C°°(M) are constant on S(0,i?o) and have disjoint
supports, then in [34] we showed that Xi(z - P )~1X2 is negligible for z
in a fixed compact set, in the sense that for every JVfN, there exists
M = M(N) such that
||Xi(^-P#)-1X2||^#,P(P#)) = C?Ar(^|Im^rM). (6.7)
The same holds for Q. with V(P*) replaced by H2(M). As in [34] we
also obtain that x(z ~ P*)~lX ~ *(* - Q)~lX is negligible, if x G Q°(M\
5(0, i?o))« Using some standard integral formula, we obtain that xf{P-)x~
Xf{Q*)x is negligible, i.e. of norm Ow(hN) in C(L2,H2) for every N.
Now according to the functional calculus for /i-pseudors, due to Helffer-
Robert (see [28] and also [31], [5]), we know that xf{Q> )x is an h-pseudor
on M of the natural class with leading symbol x(x)2f(P- O^O)? where pf
is the (semi-classical) leading symbol of P. .
Let 1 = Xo + Xi + X2> where Xj > 0 are smooth, Xo is equal to 1 on
S(0, i?o) and has its support close to that set, Xi £ C* has support disjoint
from 5(0, R0) and xo + Xi = 1 near 5(0, R2), so suppx2 is disjoint from
S(0,i?2) and X2 = 1 near infinity. Let Xj -< Xj? where suppXj is close to
supp Xj and consider
Pfi = Xo/(P#)Xo + Xi£,FXi + X2^,^X2, (6.8)

401
where R.tp is an fo-pseudor with leading symbol F(p_i$) and such that the
total symbol of R.tp — Pj = S.tF has compact support in £.
It is easy to see that
(z - P.)g)-1is well defined and
0(1) : %.fi -¥ V.fi for z <E £2, (6.9)
Write,
P.,e = P.te + Xo{f(P*) ~ P*)Xo + XiS.,FXi-
In view of the properties of S. f we can find T. f of finite rank 0(h~n) such
that xi (S.,f - T.,F)xi = 0{h"). Put
P.,9 = P.,9 + Xo(f(P*) - ^#)Xo + XiT.jXi- (6-10)
Then
(z - F.^)_1is well defined and
0(1) : n.fi -¥ V.<e for z£0, (6.11)
and
P.,9 = P.,9 + K., where
K. = 0(1) in £(H.,o,H.,o) and rank A". = 0{h~n •). (6.12)
In the following we identify To with Rn by means of no, and consequently
we shall not always write the subscript 9 for the spaces %. and V. On Z>.,
we use the scalar product,
(u\v)v. = ((P2 + l)u\v)u., (6.13)
where P. is the original (unsealed) operator. Let e.,i, e.,2i •• be an O.N. basis
in V. such that
e.,i,..,e.,jv. span ImK®, N. = 0(h~n ), (6.14)
where K® \U.->V. is the adjoint of K. : P. -»• U.. Notice that
If.® = (P.)~2K*, (6.15)
where /£* is the adjoint of K. : %. —>• %.. It follows from (6.14) that
c.1jv.+i,...e (ImX®)1 = Ker/s:..
Put
i2.1+u(j) = (u|c.j)d. , 1 < j < AT., (6.16)

402
N.
R.,-(z)u. = J2u-(j)f,3, f.j = (P.,o + K.- z)e.j, z € Q, (6.17)
1
so that jR.i+ : V. -*• CN-, R.t_ : CN- -»• ?{.. Consider the Grushin problem:
/ (P.to-z)u + R.t-u-=v,u£V.,u-€CN- lR 1Q.
\ J2,+u = »+,«€ 7*., i;+eC". (fU8j
This problem is of index 0, so in order to show that it is well posed, it
suffices to show the injectivity. Suppose that
f (P.,-,)„ + « «_=0,
[ R.,+u = 0. v '
If ^ — Y^ UjC.j, we first get u\ — .. = un = 0, and the first equation in
(6.19) becomes:
oo N.
£ u3{P,e ~ z)e.j + £*-(*)(*,* + K. - *)e,, = 0. (6.20)
AT.+l 1
Since K.je.j — 0 for j = 1,.., N., we can write this (dropping temporarily
the subscript "•"):
N oo
(Pe + K - z)(£u-ti)ei + E *i*i) = ° (6-21)
1 AT+1
and the bijectivity of Pq + K -z implies w_ (j) = 0, Uk = 0, fc > N +1, and
we have shown the injectivity of (6.19) and the wellposedness of (6.18).
We also need a priori estimates for (6.18): Writing u = u' + Ya v+(j)€ji
v! — Y?N+\ eji the first equation in (6.18) becomes:
N
{Pe - z)v! + R-u- = v - J2 v+(J)(pe - z)ev
i
and as in the proof of the injectivity, we get:
AT oo N
(Pe + K- z)(Y,u-(j)e3 + £ u3e3) =v-(Pe- z)(5>+(j)e;),
1 JV+l 1
which gives
IH|d + ||i»-||c»<C(|M|w + ||i;+||cW). (6-22)

403
6i,.., epj do not necessarily have compact support, but (6.15) shows that
(P.)2ej have support in some fixed compact subset of B(0, R2) and by a
Combes-Thomas argument, we infer that
MmiT^Biojh)) = 0{e-&), l<j<N.
Let x £ C^(Te) be equal to 1 near £(0,i?2). We still have a well-posed
Grushin problem, if we replace ej by xej and fj by (P# + K - z)(xej), and
the preceding estimates remain valid. In the following, we shall refer to this
modified problem. After increasing R2 slightly, we may then assume that
(for the new modified quantities):
suppej, supp/j C S(0,i?2), 1 < j < N. (6.23)
7. Trace class estimates for the inverse of the Grushin problem
For z e Q, let
/ E.{z) E.,+(z) \
k-W-\E.,.(z) E.,-+(z)) [7A>
denote the inverse of
*M = (FCo^)- (7'2)
Here 0 = 0o and we identify Rn with Ye by means of «$. Our estimates will
be essentially the same as those of section 3 for (P. - w;)""1, when w is in a
compact set disjoint from R.
Let M be a large torus containing P(0, R2) and define P* = P. as in
the preceding section, so that P# = P in B(0, #2). Since ei,.., e#, /1,.., /at
have their support in B(0, #2), according to the last modifications in section
6, we can define Rf:V*-> CN, R* : C^ -> ft# as before and get
it
with a uniformly bounded inverse
\ il. 0 /
"■(5J)-
(We sometime drop the subscript •.)
Let w e fi with Imw > Co^st . Then
£#(w) = (P# - w)-1 - (P# - m)"1 £*(«;)£*(«;),

404
where the last term is uniformly bounded and of rank 0(h~n). Then
where R is uniformly bounded of rank 0(h~n ). It follows that,
fij(S*(w))<C(l + h2j2,n-)-1-
If z G £2, we use the relation
£*(*) - £#(w) = -£#(2)(7>#(*) - V*{w))£*{w),
which together with the previous estimate gives:
Mi(f *(*)) < C(l + h2j2'n')-\ z e fl. (7.3)
Let x G C£°(Rn) be = 1 near B(0,i?2) and choose x G Cg°(Rn) with
X •< X? and assume that we have taken M large enough, so that x also lives
on M and P* = Pq on suppx. Then for z eQ:
<w(s°/)=(r,K>(s!)
-^)([p,oa !!)«*w(j J). p-4)
since i?+x — ^+> X#- — ^-j so that,
N(r,)
We also used that
in the obvious sense.
Similarly,
(50/)^ = (5?K)(S!) +
(o °/)^)([Pf] !!)'<*>■ <™)
0
(7.5)

405
From this and (7.3), we obtain
M*(*)(J J)w(j 0j)£(z))<c(i + h2j2/n)-\zen. (7.7)
Actually, when using (7.4) to estimate one half of the characteristic
values above, we need to estimate 8(z)hA, where A — f ft n ), A =
^-1[^)X]- We have A = £|a|<i aa(z;^)(/i-D)a, with aa(-;h) bounded in
C6°° and suppaa(-;/i) bounded away from 5(0, R2). Then [7>,.4] = 0(/i) :
V -4 H and hence, [£,.4] = -S[P,A]£ = 0{h) :U-*V. (For the special
.4 appearing in (7.4) we will also be able to gain exponential decay at large
distances.) It follows that €A = AS + [£, A] = 0(1) : U -4 Vll2.
Next we derive exponentially weighted estimates for E(z) in the usual
way. Let <f> e C°°(Rn) with <f> = Const, near B(0, R2) and V<£, V2<£ = 0{e)
in sup norm, and with V<£ € C6°°. Then in £(V xCN;7lx CN):
' eHh o \
0 e^lh )
Pe-z + 0{e) R. \
R+ o J
has a uniformly bounded inverse if e > 0 is sufficiently small. Approximating
<f> with functions that are constant near infinity and passing to the limit,
we see that:
('T %p,,»)'m(*T %,/»)=W)=*xC"-H)xC»,
(7.8)
or more explicitly:
e^lhE(z)e^h = 0(1) in C{U\V),
e^hE^(z)e^°^h = 0(1) in C(CN;V),
e-m/hE_(z)€Hh = 0(1) in L(U\ CN). (7.9)
Let xi belong to a bounded set in C£°(Rn), such that dist(suppxi,
SUPPX) > 1/Const.. Combining (7.3), (7.4), (7.6), (7.8) with <f> properly
chosen, we get
< c(l + /t2j2/**-)-ic-Uh<1wt(8uppxi,8uppx)# (7.10)

406
Let now x belong to some subset of Cq° with uniform bounds on dax for
every a, on the diameter of the support of x and with dist (suppx, 5(0, R2))
> 1/Const.. Then near suppx we approximate Pq by a (new) operator P#
defined on a torus M et c, so that (P# - z)"1 is 0(1) for 2 e ft. If
X -< X £ Co°> where x has only slightly larger support, then
^)(1P;aS)((F#o2)^)(^). <-)
(^)((P#r)_1 S)(lPox1S)^ <-)
As before, we get
Mi(*(*)(j S)),W((J ^(^^Ca + fcV^)-1, *€ft, (7.13)
and if xi is chosen as before (with respect to the new x) i and constant near
5(0, R2), we get
«<(xo !)'«UXM51!)*>(?!)>*
(7(1 _|_ /t2j2/n.\-lc-^dist(suppxi,8uppx)? (7.14)
where * = 1 when suppxi H S(0, #2) ^ 0 and 0 otherwise.
Let 0 < tyv g Cq°, // 6 T be a partition of unity as in section 2, with
^o = 1 near S(0,i?2). Put
*-(U,o/)' <"5>
Then (7.10), (7.14) imply that
Hj{VvS.{z)*t) < C(l + /i2i2/n)-1e-^^-"l-c)+)I/)^ r. (7.16)
We also wish to estimate the characteristic values of 9l/(Si(z)-Eo(z))9li
when v,fj,^0. Let V>o € Co°(Rn) with 1B(0R \ ■< V'o and put

407
Consider,
V = {1- *0)£l*M - (1 " *0)^0*m-
Here
_ / (1 - i„)E% 0 "\
-{ o o)'
and since i?.,+ (l - ^o) = 0:
pJ(i-*.)&«,=(<^-*><i0-*,)&*' j).
Consequently
A = (PO,0 - z)(l - &)£iV>M - (ft,* - *)(1 - ^0)^0^
= (PM -z)(l- faErfu - {P0,e -z){l- ^oWm
-(Pi,e-Po,e)(l-^o)Eii>li. (7.18)
Here
(P.,* - z)(l - &>)E. = -[P,o, ME. + (1 - $o){P.,o ~ z)E.
= -{P,eME + {l-M,
since (P.j - z)E. + #.,_£.,_ = / and (1 - rp0)R.- = 0. It follows that
A = [-[P.,,, $o]E]l% - (Phe - Po,«)(l - faErtv (7.19)
and applying E$ to the left in (7.17), we get,
"-*($!)■
Vv^oA 0
E0A 0
£0,-A 0 J '
*"V=V 0 0
We have shown that
/ -[^^)[P.,fl,^)]£?.0M]S - ^^o(Pi,e - Po,«)(l - ^oWm 0 \ (720)

408
This is analogous to (3.15) with (1 - xi), (1 - X2), X replaced by ^„, ^M,
^0 and the same discussion as after (3.15) leads to
Cmax(|/*|, li/D-V^t^"^"^ (1 + tfj2'"™*)-1. (7.21)
Actually, to obtain this is, there is a minor technical difficulty, similar to
the one we encountered in order to obtain (7.7). We do that in the
following way: For j = 1,2, let Aj = £|a|<i ajia(x;h)(hD)a, a.j%a(-;h) be
uniformly bounded in C%° with support in x uniformly at a distance > 0 from
B(0,R2). Put Aj = (AQJ JV Using the identity [£,Aj] = -£[P,Aj]£,
we get:
£A\A2 =
^M2f - .4i£[P, ^]£ - A2£[V, Ai]£ + £[V, Ai]£[P, A2]£
+£[V, A2]£[V, Ax}£ - £[[V, Ai], A2]£,
and we get exponentially weighted estimates under suitable assumptions
on <j>:
e-+lh£AiA2e+lh = 0{1) :HxCN-->HxCN\
We can then replace A\A2 by a more general operator B =
B = 5D|or|<2 ba{%\ h)(hD)a, where ba have the same properties as a^a above,
and we then have the additional control over weighted norms that is needed
to obtain (7.21).
The proof of Lemma 3.3 gives
Lemma 7.1 Let m, zq be as in Lemma 3.3, z £ Q, and i/,/i G T, Then
ij)v(P.fl — zo)~mE.(z)tl)tJb is of trace class and
\\MP-,e ~ *o)-m£.WJtr < Ch-ne-m(\x-»\-ch. (7.22)
For v,/j,^0, we have
\\[MP-,8 ~ ^o)-m^.(^)^]olltr < Ch~n™ max(|i/|, |M|)-V^(I"-"I-C)+.
(7.23)
In particular we get,
Proposition 7.2 "tr [(P.,e - zQ)-mE.{z)]ln is well defined and = 0{h~n).
B 0
0 0

409
8. End of the proof of the trace formula
Let fi_|_ be the intersection of Q with the sector 0 < arg z < e > 0 and let
f2_ be the intersection of f2 with the sector —26 < argz < 0, where 6 = 0q.
We define W± similarly. Write f(z) = [z — zo)~mg(z), where
(1 + ra)-^- > 1 (8.1)
^max
as in Lemma 3.3. Then
(Xf)(P-) = (P:-zo)-m(X9)(P.)
= (P. - z0ym± J9(z)^(P. - z)~lL{dz)t (8.2)
where x £ Co° (fi) is an almost analytic extension of x with support in a
small neighborhood of J. We first look at
/.- = (P. - zQ)-m- I g(z)^(P - z)-lL{dz). (8.3)
7T JQt_ OZ
Let x £ Co°(^) be equal to 1 near W-, X — X near J and be almost
analytic also near R_ in case ft reaches that set. If 17 n R_ = 0 we can
replace x by X in (8-3) but in general we have
/.' = -(xlK-f)(P-) + - f 9(z)^-(P. - z0)-m(P - z)-lL{dz). (8.4)
7T JQ_ OZ
As in the proof of Proposition 2.1, we get:
"tr (/f - /0-)» = -[ £ W]l + 0(h-n™). (8.5)
iiea(P.)nK-nW
Next look at
/.+ = I f 9(z)^(P - z0)-m(P - z)-H{dz).
TV J ft, OZ
Green's formula gives for every 5 > 0
1+ = I I S(Z)^(P - zQ)-m(P - z)-H{dz)
7T Jn+n{lmz<8} OZ
+ ^ / f , 9{z)x{z){R - z0)-m(z - P)~ldz, (8.6)
27TZ Jn+n{Im^=5}
where the integration contour in the last integral is oriented in the direction
of decreasing Re z. If S > 0 is small enough (independently of /i), the

410
integrand in the first integral in (8.6) has support in the region where
g = 0{1) and as in the proof of Proposition 2.1 we get
"tr (/+ - /+)" = "tr Ji - Jo" + 0(/i"nmax), (8.7)
where J. denotes the last integral in (8.6). Committing another error 0(h~nrn!ix)
for the evaluation of "tr Ji — Jo", we may replace J. by
J-= idl9{z){R - *)-m(z - p)~ldz> (8,8)
where 7 is a segment with end points a, b with Im& = Ima = <$>0 small,
belonging to Q \ W so that a is close to the left end point of J and b is close
to the right end point. Moreover 7 is oriented in the direction of decreasing
Rez. According to Proposition 5.3 we then have
»tr Ji - Jo" = " tr Ki - K0n, (8.9)
where
K=^-J 9(Z)(P;0 ~ zo)-m(z - R^-'dz, (8.10)
and where we choose Tq as after (i)-(iv) in section 5.
Using the Grushin problem of sections 6, 7 we write
(z - P,,)"1 = -E.(z) + E..+W£?.,_+(^)-1f7,_(z), (8.11)
K. = -L. + M., (8.12)
where
L=hil 9{z){R'e ~ z°)~mRwdz> (8-13)
M = ^TiJ 9{z){R'e ~ *rm^*K-+(*rl£s-(*)<fe- (8-14)
In the expression for L., we can replace 7 by a curve 7, contained in Q\W
and with the same end points as 7. Then g = 0(1) along 7, so Proposition
7.2 implies that
"trLi - L0" = 0(/Tnmax). (8.15)
M. is of trace class and in order to study its trace, we shall first simplify
the integral formula (modulo errors). We want to replace (F.^ — zo)~m by
(z — zo) ~m. The difference of these two quantities is a finite sum of terms
(P.,b - zo)~k(z - z0ye(P.,e - z) (8.16)
with k +1 = m + 1, k,£ > 1. Since (P0 - z)E.t+ = -R.-E.-+, we get

411
£l!,9(z)(P,e - z0)-k(z-z0)-<(P,e - z)K+(z)E.,-+(z)-lE.t.(z)dz
= ~is J,9(z)(Re - z0)-k{z - zo)-<R.t4z)R,-(z)dz. (8.17)
In the last integral we can replace 7 by 7 and we get an operator which
is 0(/i~nmax) in trace norm. Consequently,
M. = N. + 0(/Tnmax) in trace norm, (8.18)
where
N' = hl f(z)E-Az)E,-+{z)-lE,-{z)dz. (8.19)
When taking the trace of N. we can pass the trace inside the integral and
use the cyclicity of the trace,
tr£1+(z)£I_+(*)-1£,-(z) = tTE,-+{z)-lE,-{z)E.Az)- (8-20)
From
j-£{z) = -S(z)^(z)£(z), (8.21)
we get
£L(*)£7+(*) = EL+{z)+E-{z)&4z)E-+{z), (8.22)
so iV has the same trace as Q + i?, where
Q = ~tilf{z)E-+{z)~'E'-+{z)dz' (8-23)
R=^-. [ f(z)E-(z)R'_{z)dz. (8.24)
In (8.24) we can replace 7 by 7 and see that
R = <9(/T'w) in trace norm. (8.25)
We sum up the discussion so far in:
"tr [*/(P.)]o" = [tvQ.]l - [ Y, /OOlo + 0(h-*™). (8.26)
a(P.)r\R-r\W
It remains to study trQ.. In £2, £L+ is a matrix of size 0(h~~n ) and of
norm 0(1), so
D(z;h) =dcf detE_+(z) = 0(eC'hn). (8.27)

412
For #o > 0 small but independent of h, let fl+f$0 = {z G Q+;lmz > 50}.
Choosing Ye conveniently, we have
{z-pB)-l = o{i)> zen+jo, (8.28)
and using that E-+(z)~l = -R+(Pq - z)~~lR-, we get
E-+{z)-l = 0{\), *efi+A. (8.29)
Consequently,
\D(z;h)\>e-c'hn\ *efl+A. (8.30)
It is easy to see as in [14] (or as in [33], where the Dirichlet to Neumann
operator plays the same role as £L+ here,) that the resonances in £2, i.e.
the eigenvalues of Pq in £2, coincide with the zeros of D in Q, and that the
multiplicities agree. Let N(P,Q;h) be the number of resonances in Q.
(8.27) remains valid for z in a slightly larger domain and combining this
with (8.30) and Jensen's inequality, we get
N(P,tyh)<Ch-n\ (8.31)
(Cf. [30], [34], [23], [44], [41].)
Let Zj be the resonances in Q repeated according to their multiplicity
and put
Dw{z;h) = Uj(z-zj). (8.32)
Then thanks to (8.31), we have
\Dw(z;h)\ <ec/hn', ze&. (8.33)
Since the zj are confined to £2_, we also have,
\Dw(z;h)\>e-c'hn\ *€fi+A. (8.34)
In Q, \ fl+,$0, we can get the same lower bound, if we avoid to go too
close to the resonances. To see that, we first establish the simple
Lemma 8.1 Let x\,.., xn G R and lef/cR be an interval of length \I\ G
]0,oo[. Then there exists x G /, such that U^\x-Xj\ > exp[-Af(l+logjjr)].
Proof. Consider F{x) = Eflogi^-j. Notice that
f 1 fW2 1 2
/ log- -dx < 2 / \og-dt = |/|(l + log—),
JI \X - Xj\ J0 t \1\

413
so that
/,
We can therefore find x € I such that F(x) < N(l + log A-X i.e.
a
Now make 7 = ya depend smoothly on a parameter a in some bounded
interval J but with fixed end points 6, a, so that 7 moves transversally in
ft \ ft+,50/2> in such a way that if Zj 6 jaj:, then dist (^j, 7a) > |<* - »j| for
all a € J, and if ^ belongs to no 7a, then dist(zj,7a) > dist (a, dJ). It
follows from Lemma 8.1 , that if / is a non-trivial subinterval of J, then we
can find a 6 I such that
\Dw(z\h)\ >e-Ch~n', zeya, (8.35)
where C = C\i\. We factorize D:
D(z] h) = G{z\ h)Dw(z; h), * G ft, (8.36)
where G and \jG are holomorphic in ft. Combining (8.27), (8.34), (8.35),
we get (with a new constant)
\G(z;h)\<ec/hn\ zett+,80U%- (8.37)
The maximum principle gives
\G(z;h)\<ech~n\ z e ft, (8.38)
where ft CC ft is any simply connected relatively open ^-independent set
with W in its interior, provided that we choose the family % such that
7a n ft n ft_ = 0.
(8.30), (8.33) imply that
\G(z]h)\>e-c'hn\ *efi+A. (8.39)
Consider the harmonic function on ft:
0 < i(z; h) = ChTn- - log \G(z; h) |. (8.40)
Harnack's inequality for non-negative harmonic functions tells us that for
every /{ CC fi, we have
sup^<Ctfinf^, (8.41)
K K

414
i.e. that the function t{z\ h) is uniformly of constant order of magnitude
on K. After an arbitrarily small decrease of fl, we have (8.41) with K = Q
and if we use (8.39), we get l[z\ h) < Chrn- on fl+i$0 and hence by (8.41)
with K = &:
£(z;h) <Ch~n-, on fi. (8.42)
We conclude that log \G(z; h)\ > -Chrn- on & and with (8.38), we get
| log |G(*;/i)|| <C/Tn', *efi. (8.43)
Since log \G(z\ h)\ = Re \ogG(z\ h) is harmonic we get after an arbitrarily
small decrease of ft:
VRelogG = 0(l)/rn-. (8.44)
The Cauchy-Riemann equations then give the same estimate for Vim log G
and consequently
-^\ogG = 0(l)h-n\ (8.45)
dz
Choose a family 7a, a £ J in ft \ W with the same properties as after
Lemma 8.1. If zj e 7a,, we see that
J,
3"
1
|dz| = 0(l)log: -. (8.46)
'7«l*-*jl |a~aj
Consequently we can apply Lemma 8.1 or rather its proof, to see that there
is a 7 = ya, such that:
jj± log Dw(z)\\dz\ = 0(l)h~n-. (8.47)
Since fz log D = fz log G + fz log Dw, we get from (8.47), (8.45) that
£\±logD\\dz\ = 0(l)h-»: (8.48)
From this and (8.23), we get since ^ \ogDw = £ ^-:
E_ fM + '^llm-fzhgD(z)dz, (8.49)
J2j between 7 and 7
where the last integral is 0(l)h~~n-. Combining this and (8.26), we get the
theorem. Q

415
9. Comparison with the Poisson formula for resonances in Lax-
Phillips theory.
Let P0, Pi be independent of h and satisfy the assumptions of Theorem 2.2
with h = l. Assume in addition that P0, Pi are semi-bounded in the sense
of self-adjoint operators: P. > -C. We can then define
u(t) = 2"tr[cos*x/P]J" G V'(R) (9.1)
by the following formal computation: For <f> e Cq°(R), we should have
(u,0) = 2 /</>(t)"tr[costx/P]S"dt = "tr[2 /<£(£) costy/P.dt]J". (9.2)
Here 2 J>(t) costy/F.dt = f (ftt)^*^ + e^^dt = 0(P), where 0(z) =
0(v^)+^(-\/^) (wither) = f e~~ltr(l)(t)dt denoting the Fourier transform)
is an entire function, independent of the choice of sign for y/z, and of
Schwartz class S on [-C, +oo[ for every C > 0. According to Proposition
2.1, and the semiboundedness of P., we can then define u e V(R), by
<«,<£) = "tr[^(P.)]J", ^GC0°°(R). (9.3)
In the case when n is odd > 3, Po = -A and P\ coincides with -A
outside a compact set, it is known that resonances are naturally defined as
a discrete set of numbers fij (repeated at most finitely many times
according to multiplicity) in the closed upper halfplane or alternatively as the the
conjugate numbers JIj in the lower half-plane. The relation with the
resonances defined by complex scaling as in section 5 is given by: ~pj = y/\j.
(See [34] and [13] and further references, given there.) It is then known that
we have the following Poisson type formula,
u(t) = Y, e^3t ^ £>'(]0, +oo[), (9.4)
where the sum converges in the sense of distributions, in view of known
polynomial bounds on the number of resonances in large discs. For more
concrete classes of operators Pi and on an interval of the type ]i2, oo[, this
relation was established by Lax-Phillips [20], Bardos-Guillot-Ralston [4].
Melrose [21] extended the validity of (9.4) to the full interval ]0, +oo[ and
Sjostrand-Zworski [36] further extended it to the case of operators Pi like
the ones in Theorem 2.2, but still with the restriction that Pi = -A outside
a compact set and that the dimension is odd > 3. The proofs in these works
are based on the Lax-Phillips theory (cf [20]).
In the case of certain hyperbolic surfaces a formula of the type (9.4) was
recently established by Guillope-Zworski [11] and their proof uses more
general scattering theory including the Birman-Krein formula for the scattering

416
phase. According to private communications from the authors it is quite
possible that their proof can be adapted to give (9.4) under the assumptions
described above.
When the Poisson formula is valid, we have for every x G Co°(]0, +oo[),
AeR:
xfc(A) = £*(A-w)i (9.5)
where we notice that the Paley-Wiener theorem assures the convergence of
the sum. This version of the Poisson type formula was used in [35] together
with asymptotic informations about the LHS when A tends to infinity, to
get lower bounds on the density of resonances in certain neighborhoods of
the real axis.
Under the more general assumptions of the beginning of this section,
we shall now see how to get an asymptotic version of (9.5) which is strong
enough to recover the lower bounds in [35] and extend them to the case
of even dimensions. (Lack of time in preparing these notes has prevented
us from developing other applications of Theorem 2.2, and we have the
intention to return to these questions in some future work(s).)
For the remainder of this section, we let the self-adjoint operators Pj,
j = 0,1 be semi-bounded from below, independent of h and satisfy the
assumptions of Theorem 2.2 with h = 1. Define u(t) 6 V'(R) as in the
beginning of this section. Let x £ Co°Qa?&D> where 0 < a < b. For AeR
with |A| large, we consider
^(A) = "tr[x(A-x/R) + x(A + x/R)]i".
Write A = £, fi = ±1, where h > 0 is small, and write P. = jph2P.. Then,
X*(£) = "tr [x{\{li - y/VE)) + x({(A* + V^))]J".
As already noticed, the function
h(z) = x(£(/i - V~z)) + X(^(M + V^)) (9-6)
is entire since the odd powers of the square root of z dissappear from the
series expansions.
For z < 0, write z = -t. Then fh(z) = x(J(M - itf)) + x(J(m + iy/i)),
so even though x G £(R), we may have exponential growth when t » h2.
Fortunately, h2P. > -h2C, so we are only concerned with the region 0 <
t < Ch2, where fh(z) and all its derivatives are 0(h°°).
We next look at the case of positive z. For 0 < z < 1 - ^tjy, we have
dkzfh(z) = ok(h™),
(9.7)

417
also in this region.
1
0(1
dkzh(z) = 0N,k(hN(l + |*|)-"), V*,N € N. (9.8)
For z > 1 + ^fjT, we have \/jl - yfz\, \p + yfz\ ~ 1 + y^, so
Noticing now that h2P. satisfy all the assumptions of Theorem 2.2 with
variable h, we can combine the preceding remarks about fh and Proposition
2.1 to conclude that the contribution from the region \z - 1| > ^tjt to
"tr[//l(/i2P.)]J" is 0(h°°) (in the sense that we change the "trace" only by
0(h°°), if we replace fh(z) by ip(z)fh(z), where tp e Co°(R) is equal to 1
near z = 1).
It remains to consider the contributions from the inteval a < z < /?,
where 0 < a < 1 < /?, and we shall study A(^) for complex * satisfying,
a<Rez<(3, |Imz| < 7. (9.9)
For such a z we have Re yfz = ^*l+2Re*, Im y^ = (sgn Im ^)^^
\-Rez
lmV~z = -1=^==, (9.10)
V2(p| + Re z)
so
I*™1, <|ImVil<i^d. (9.11)
Since x £ Co°(]a?&D> we ^ave according to the Paley-Wiener theorem:
|x(r)|<0*(l)(l + M)-Nxj ^y^lll] (9-12)
If fi = 1, we see from this, that the second term in the RHS of (9.6) is
0(h°°) with all its ^-derivatives for a < z < /3, while the first term of the
RHS extends holomorphically to the region (9.9) and is
0N{1)(^^)-Ne~VW+n\+e)lmz for im* > 0
it
and
0N{l)^—^)-Ne-^lmz for Imz < 0.
lb
Similarly for /x = -1, the first term of the RHS of (9.5) is 0(h°°) with
all its z-derivatives on the interval [a, (3] while the second term extends
holomorphically to the region (9.9) and is 0N(l)(B±fi)-Ne'7^W+^+^ mz
for Imz < 0 and 0N(l)(li±^-)-Ne^Imz for Imz > 0.

418
We can then apply the variant of Theorem 2.2 given after Theorem 2.2
up to Proposition 2.3, and obtain the following: Let h2\u(P.) denote the
resonances oih2P. in the intersection fi_ of the region (9.9) with the closed
lower half-plane. Then when A —> -oo, we get:
&W = i £ X(^(-1 + y/h2K(P-))) + Of*00)]*
or equivalently:
&W = [ £ x(A + yA^))]J + 0(|A|-°°) (9.13)
a„(p.)ga2o_
When A -> +oo, we change the sign of the angle of complex scaling
in section 5, and define the resonances h2\u{P) in fl+, where Q+ is the
intersection of the region (9.9) with the closed upper halfplane. Theorem
2.2 remains valid with the obvious modifications, and we get
W) = [ £ x(A-V^))]o + 0(|A|-°°) (9.14)
A„(P.)GA20+
10. Lower bounds near the real axis
In this section we review the work [35] and extend the results to even
dimensions. Let u(t) be the distribution defined in the preceding section.
To start with, we make the general assumptions of the preceding section,
but we also assume that Pq has no resonances in a set of the form {ji e
C; \z\ > C, 0 < argz < 1/C}. Here and in the remainder of the section,
we work with the convention that the resonances are defined in the upper
half-plane (by switching the sign of the angle of scaling in section 5). In
many cases, it turns out that u(t) has conormal singularities on the positive
half- axis and we can then apply a Tauberian argument to get lower bounds
on the density of resonances in certain logarithmic neighborhoods of the
positive half-axis. We start by discussing the Tauberian argument (of [35])
before discussing some applications.
Let A = {/ij} be a discrete subset of {// G C; \z\ > C, 0 < argz < 1/C}
for some fixed C > 1, where the fij may be repeated with some multiplicity,
and assume that for some n\ > 0, we have:
N{r) =def #{H € A; |^| < r} = 0(rni). (10.1)
For p > 0, we define
Ap = {fij € A;Irafij < plog\hj\}.

419
AUr) = #{Mj<EA,;ReMj<r}.
Let u 6 ^'(R), and assume that for every x € C£°(]0, oof), we have:
&W= £ x(A-W) + 0(A-°°),A->+oo, (10.2)
where £2+ was defined in the preceding section. We then have the following
minor modification of a result of [35]:
Theorem 10.1 Letk G R, d, b > 0 and suppose that for all <f> G Co°(]0,oo[)
with support in a sufficiently small neighborhood of d and with 4>(d) = 1,
we have:
|<MA)I> (b-o(l))Xk, A->+oo.
Then for every e > 0 and p > ^fpS we have
a) Ifk> 0, then Np{r) > {B - o(l))rk+\ B = ^^ and
J- e-(d-e)lm»3 >(B-0€(l))rk+1.
/ijGAp,|Re/ij|<r
b) Ifk < 0, then for all 6 > 0, there exists r(S) > 0, such that Np(r) > rl~8,
for r > r(8).
Proof. We will here only prove the part a). Let <f> G Cq°(] - 1,1[) with
0(0) = i, $ > 0. For 7 > 0, we put 07|d(t) = 0(±(i - d)), so that
07ld(r) = 70(7r)e~^T. If 7 is small enough, then according to (10.2) and
the assumption in the theorem:
I £ <Q(A - W)l > (^ " o7(l))Afc, A -► +00. (10.3)
/ijGA
Here we also used that
£ Si(A-w) = 0(A-°°)f
which is an easy consequence of the Paley-Wiener theorem:
\M<)\ < TCMe(d±7)ImC(l + |tCI)"M, ±ImC > 0, M e N.
Using this estimate, we shall also estimate away the contribution to the
sum in (10.3) from the ///s in A\ Ap. Using the bound on N(r) we get for

420
A > 2 with constants depending on M, 7, d - 7 > 0:
E \m*-k)\<c e e-^'^u+iA-^ir"
/0€A\Ap /tj€A\Ap
/CO
r(^"(i + |A-*|)-A/diV(t)
< C(-(l + |A - 1|)-MW(1) - y°° ^(r(d-^p(l + |A - t\)-x')N(t)dt)
/*co /V
< <? / l3-(r(d"7)p(1 + lA " ^l)"M)kni^ = ©(A111"^"7^) = o(Afc),
where we chose M large enough and used that that n\ - (d - y)p < k if 7
is small enough.
Combining this estimate with (10.3), we get
I E <QA - N)\ > (A - o7(l))Afc, A -> +00. (10.4)
MjGAp
For € > 7 > 0, we introduce
^(r) = sup|e-(d-e)a(Q(r + ur)|, r G R.
We claim that
2
tf(r) = 7^) + -X-C?M((7r)-M). (10.5)
€-7
First, it is immediate that ^(r) > 7^(77-). In order to have the upper
bound, we write for a < 0:
et<T|0(7r + 17a)I < eta\$(jt + iya) - ^(yr)\ + $(jt),
where the first term of the RHS is
f<ewi\<r\ I \4>\ir + isy<r)\ds <CMet(Tl\<j\ I e5|7lW(l + |7r| + \sja\)-Mds
Jo Jo
< C'Mle-^°\\o\{\ + |7r|)-M < CM^(V)-M,
which gives the required upper bound in (10.5).
We next show that for 7 < €2,
f i>(r)dT =(1 + 0(j2))H(r) + 0M{{ir)-M), (10.6)
J—00

421
where H(r) = l[o,+oo[(r) is the Heaviside function: Since
/+oo ^
<t>(T)dT = 27T<£(0) = 1,
-co
we have
f It \a (ir 2( \a i Om((h)-m), r < 0,
This shows that the integral of the first term of the RHS of (10.5) gives the
expected contribution to (10.6). It remains to treat the contribution from
the remainder in (10.5). Since e > y/y, we have
j2 -Om((it)-m) = 71/20m(7(tt)-M)
€-7
so the integral of this can be treated the same way, and we have verified
(10.6).
Introduce the positive measure
H(dr)= J2 e-^-^^r-Re/^dr.
MjGAp
Using (10.4) and the subsequent definition of ^, we get:
'A
k27T
(A _ 07(i))A* < £ |^,d(A-Mj)|<
Mj GAP
Y, VKA-Re/^e-^1"1^ = [^(X-t)h((It) = V>*m(A), A > 1(10.7)
Notice that M(r) =def M([0,r]) < Np(r) and that f ip(\ - r)^(dr) <
^7,m(|A|-m) for A < -1. Then integrating (10.7) from 1 to r > 1, we
get:
<£(FfI)-"'<1»,*H* />*">{x)dX
= [i>(t)([r l*(d\))dt= fip{t)M(r-t)dt
= I\p(r-T)M{T)dp. (10.8)
Since M(r) = 0(rni), we deduce from (10.5) that for S e]0,1[, when r > 1:
/ i)(r - T)M(r)dT = 07,M(rni_<5M), VM. (10.9)
JR\[r-rs ,r+r*]

422
Then
rr+r*
r+ri * r* x
< ( / ^(r - T)dr)M(r + r5)= / i>(t)drM(r + r5)
Jr-rs J-rs
= M(r + r8)(l + 0(71/2) + 07,M(r"5M)).
Since we can choose 7 arbitrarily small, part a) of the theorem follows easily
by a substitution in r. D
Let O CC Rn, n > 2 be an open set with C°° boundary. Let Pi be the
self-adjoint realization of -A on Kn\0 with Dirichlet, Neumann or Robin
conditions on the boundary. Let P0 be -A on Rn. Then we can define u
as in the preceding section and (9.14) holds for \ £ Q^QOi+ooDi where
the only contributions come from the resonances Xj of Pi (here with the
convention that ImAj > 0). With fij = y/Xj, we define Np(r) as above.
In Rn \ O or rather in a certain topological space sitting over Rn \ 0, we
can define generalized bicharacteristics, also called C°°-rays, in the sense of
[24]. For a more complete discussion of this and for some other points below,
we refer to the book [27] by Petkov-Stoyanov, and further references, given
there. Roughly, a generalized bicharacteristics is a curve / 9 t ^ 7(f) 6
Rn \ 0, where J is some interval, such that jf(t) exists and is constant of
norm 1 on every open subinterval on which j(t) £Rn\0. If y(t) G Rn\O
for t0 - e < t < t0 and 7(^0) G dO and Y(t0 - 0) £ TdO (the tangent
space of dft), then it is required that 7 reflects at 7(^0) according to the
rules of geometrical optics, so that j(t) G Rn \ O for small t - to > 0, and
7'(£o + 0) ^ Y(to-0) and Y(to + 0)-Y{to-0) is a multiple of the normal of
dO at 7(^0)- Such a point t0 is called a transversal reflection point. When 7
hits the boundary tangentially, the description is a little more complicated,
and we refer to [24], [27] for a more complete decription of C°°-rays near
such diffractive points.
A C°°-ray: R9t^ j(t) e Rn \ O is called periodic with period T > 0
(or T-periodic) if j(t + T) = j(t) for all t e R. If 7 is T-periodic, we let
T* denote the primitive period, i.e. the smallest period > 0 of 7. Clearly,
T = kTf, for some 1 < k e N. We say that 7 is transversally reflected if
all boundary points of 7 are points of transversal reflection.
If 7 is a T-periodic transversally reflected C°°-ray then one can define
a corresponding linearized Poincare map P1 (also depending on the choice
of the period T), which can be viewed, up to symplectic conjugation, as a
symplectic map P7 : R2(n-1) -» R2^"1). We say that 7 is non-degenerate
if det(/-P7)^0.

423
Let 7 be a C°°-ray and assume:
7 is a transversally reflected periodic non-degenerate C°°-ray
of period T7 > 0 and of primitive period J*. (10.10)
There are no other T7-periodic C°°-rays,
up to time translations and time reversals. (10.11)
Then in a neighborhood of t = T7, we have
T*
u(t) = -^Re(e*2^(i - r7 - iO)"1)|det(JRy - I^'^modL^. (10.12)
Here /?7 is a real number which depends not only on 7, but also on the
type of boundary condition, and whose actual value is unimportant in the
following. The corresponding result for tr cos ty/Pn, when P = -A on Q
with Dirichlet, Neumann or Robin condition and Q, is bounded with smooth
boundary was obtained in [10] (see also [27]) very much as a consequence of
the general result on propagation of C°°-singularities in [24]. Then (10.12)
follows, if we notice that the property of finite propagation speed for
supports of solutions to the wave equation implies that for every T > 0, we
have
u(t) = 2(tr costJpn\G - tr cos£\/^), for -T <t<T,
if Q = S(0,i?) and R > 0 is large enough depending on T. Here Pq\oi
Pq denote the realizations of -A on Q and Q \ O respectively, with the
appropriate boundary conditions.
If 0 G C'o0(]0,oo[) has its support close to T7 and 0(T7) = 1, then it
follows from (10.12) that
'*<*>'=idrt(p;.-op/*+aW'^*'0' (10-13)
so the assumptions of the previous theorem are satisfied with d = T7,
b = T*\ det(P7 - I)\~1/2, k = 0. Consequently for p > n/Ty, we have
Np(r) > (^ - o(l))r, (10.14)
where Np(r) is previously defined with fij = y/Xj, and \j being the
resonances in a an angle attached from above to the real axis. For odd dime-
nions > 3, this result is due to [35], (even if not stated there in the same
generality).

424
The next application is also due to [35] in the case of odd dimensions >
3. Let S(R) = {x e Rn+1; \x\ = R} be the sphere of radius R > 1, equipped
with the induced metric gs(R)- We will assume that R is sufficiently large.
Consider
XR = (S(R) \ Bs{R)(x0,1)) U (S"-1 x]0,1[) U (R» \ 5(0,1)). (10.15)
Here x0 is some fixed point on S(R) and Ss^j^o? 1)) denotes the open ball
in S(R) of center x0 and radius 1. We give Xr the structure of a smooth
manifold by introducing a parametrization of Xr near Sn~1x]0, 1[ of the
form (£,u;), -1 < t < 2, u G S71'1 with the following rules of assigning a
corresponding point in Xr: First we identify the tangent space of S(R) at
xq with Rn in a way to get a linear isometry between the two spaces. Then
by using geodesic coordinates, we can identify points in BS(r){xq, 2) \ {x0}
with corresponding points (r,u) with u G Sn~11 0 < r < 2. If 1 < t < 2,
then we let (£,u;) correspond to a point in BS(r)(xo, 2) \BS(r)(xq, 1) in the
way just described. If 0 < t < 1, then (t,u) designs a point in ]0, llxS71'1
in the obvious way. If -1 < t < 0, then the corresponding point should be
(l - t)u e Rn.
Using this parametrization, we can easily construct a metric qr on Xr
which coincides with that of S(R) in S(R) \ Bs(r)(xq,3/2) and with that
of, Rn in Rn \ £(0,3/2) and which has a perfectly uniform behaviour in
the "coordinates" (£,u;), when R -> oo.
Let -Pi be the corresponding Laplace operator A9R on Xr. As before
we choose Pq = -Arm.
The geodesies on S(R) are all periodic of period 27ri? and most of these
geodesies are also geodesies on Xr, we therefore expect u(t) to have strong
singularities at the points 2trRk, k G Z. In [35] it was indeed proved that
with <f>dn as in the proof of Theorem 10.1, we have
Ifa^kMW > (2^n + 0(i?))A^1 + CWA"-2), (10.16)
for € > 0 small enough. We conclude that for every p > 0, we have Np(r) >
^rn, for r>r(p).
This estimate was obtained in [35] in the case of odd dimensions and is
new in the case of even dimensions.
In the 3-dimensional case, Fahry and Tsanov [6] have recently obtained
the same lower bounds in much thinner neighborhoods of the real axis.
I was unable to see how they obtained from [35] some crucial uniformity
w.r.t. fc, in their estimate (3.2).
It has not been possible to include here any discussion of the Lax-
Phillips conjecture (see [19], [26]) or of the consequences that can been
drawn from the singular behaviour of u(t) in the Poisson formula when t —y

425
0 (see [36], [40], [22], [29]). One might expect that many questions around
the Lax-Phillips conjecture can now be studied also in even dimensions and
perhaps in long-range situations.
11. Upper bounds near the real axis
In this setion, we review some upper bounds on the density of resonances
near the real axis.
We start with a result from [30]. Let
P = -h2A + V(x) (11.1)
be a semi-classical Schrodinger operator on Rn, n > 2, where V is a real-
valued analytic potential with a holomorphic extenstion V to a set of the
form {x G Cn; |Ims| < (Rex)/C} with V{x) -> 0, x -> oo. Let E0 > 0.
Then by the method of complex scaling, we can define the resonances Xj
of P in a /^-independent neighborhood of E$, and we have Im \3 < 0.
Let p(x,£) = £2 + V(x) be the semi-classical symbol of P, and let
Hp = 2£ • §j - W(x) • -g? be the corresponding Hamilton field. Following
[8], we introduce V± = {(z,£) G p-1([^o - £o,E0 + €0]); \exp(tHp)(x,£)\ -fr
oo, t -> =foo}, K = T+ PI T_. Here €0 > 0 is small. Then V± are closed and
K is compact. A basic result implicit in [14] says that if K = 0, then there
are no resonances in some /^-independent neighborhood of [£^o~ £o, £?o+€o]?
when h is small enough.
We assume that Hp generates a hyperbolic dynamical system near K in
the following sense: Define K, T± as above with €o replaced by 2€0. Assume:
In a neighborhood QPo of every point po G K, we can represent
V± as a union of closed disjoint C1 manifolds of dimension n + 1
such that if p G fiPo n f+ and if Ej = Tp(f+lP)
(tangent space of f+iP at /)), where f+iP is the corresponding leaf,
then E+ depends continuously on p G fiPo PI T+ and
contains Hp(p). Same assumption with "+" replaced by "-". (11.2)
E+ and E~ intersect transversally for every p G K. (11.3)
We also assume that there exists a constant C > 0, such that
||d(exptffp)(t;)|| < Ce-*/c||t;||, v G Tp{R2n)/E}, p G K, t > 0, (11.4)
||d(exp(-tffp))(t;)|| < Cc-*/c||t;||, t; G r,(R2-)/£;;, p G K, t > 0. (11.5)

426
Here d(exptHp) is considered as a map T(R?n)/Ef -> T(R2n)/E±xptH , },
and we equip the various spaces with their natural (induced) Euclidean
norms.
We say that d > 0 is a Minkowski codimension of K if
limsup€-dVol{(z,£) e R2n; dist((z,£),tf) < *} < +00. (11.6)
e-K)
We then have
Theorem 11.1 Under the above assumptions, let d be a Minkowski
codimension of K. Then there is a constant Co > 0 such that for 0 < h <
l/Co, Coh < S < 1/Cq, the number of resonances in the rectangle ] -
60/2, €0/2[-t[0, S[ is < C06dh-n.
The proof of this result is based on the theory in [14], with additional
work in finding and using escape functions of limited regularity. The use
of the Weyl-inequality without any considerations of determinants is also
introduced here, and is also used in obtaining the other results that we
review below. Unfortunately it would lead us too far to outline the proof
of Theorem 11.1, but we wanted to recall the result, because it goes quite
far in linking resonances to properties of dynamical systems, the analogous
results should be obtained in other settings, and also, the trace formula in
Theorem 2.2, is perhaps an encouragement to pursue the efforts towards
even finer results, involving also lower bounds.
We next review an upper bound from [37] in a general abstract
setting. Let n > 2 and let % be a complex separable Hilbert space with an
orthogonal decomposition,
n = nK®L2{Rn\K),
where K C Rn is a bounded convex set. As in section 2, we use the notion
of restrictions or characteristic functions to denote the corresponding
projection operators. Let P : % -> % be an unbounded self-adjoint operator
with V C U. Assume that
V\Rn\KcH2(Rn\K),
and conversely that if u 6 H2(R2\K) vanishes near K, then u £ V. Assume
(Pu)\Rn\K = -A(u\Rn\K), UeV,
1k{P + *)_1 is compact.
Let Kt = {x £ Rn; dist (x,K) < e} for small e > 0. Put Uk, = Uk ©
L2(Ke \ K). Define Pf on this space with domain
V* = {u£ nKe; xu € V, (1 - X)u <E H2{IQ n ^(int (Kt))},

427
where x € Co°(int (Kt)), x = 1 near K. For u € Pf, we put
P*u = P(Xu) + (-&)((l-x)u)£U*e.
Then P* is self-adjoint with discrete spectrum. Introduce the counting
function for the positive eigenvalues:
$£(r) = #HP#)n]0,r2[).
Assume that in the limit r -> +00:
$e(r) = (l + oe(l))*f(r), $f(r)= f <MO<*C,
Jo
where <£e satisfies:
1/C(C) < MnVMr*) < C(C), for 1/C < r^ < C, r^ > C(£),
with C, C(C), C(e) positive constants.
Let fij = \/Aj, where Aj are the resonances of P in a small sector
attached to the positive half-axis from below, as in section 5 and define
Ne(r) = #{Mi! 1 < \N\ <r,~0< argN < 0}.
The main result in [37] is:
Theorem 11.2 Under the assumptions above, we have for small 0 > 0:
Ne(r) < (1 + Ce(0))<f>e{e)(r) + Ce(6)rn, for r > r(6),
where e(0) = 02/7 in general, and e{9) = 02/5, when K is strictly convex
with smooth boundary.
Using the Weyl asymptotics for the eigenvalues of second order
operators in bounded domains, we get the following consequence ([37]):
Theorem 11.3 Let 0 CC Rn f>e open with smooth boundary such that
Rn \ O is connected. Let P be an elliptic second order operator with smooth
coefficients on Rn\0, equal to -A near infinity, and equipped with Dirichlet
boundary conditions. Letp(x,£) > 0 be the principal symbol. If'di" denotes
"convex hull of" and Q = ch (supp (P + A) U O) \ O, where supp (P + A)
denotes the closure of the set of points in Rn \ O, where P ^ -A, then
< (1 + y7))rn / dxd£ + 0(62'7)rn, r > r{0).

428
We shall explain some ideas in the proof of Theorem 11.2 and
concentrate on the case of a general convex and compact K. One uses complex
scaling which is adapted to K and constructs m.t.r. submanifolds of Cn of
the form
re = {z = x + if'(x)-xeR"},
where e £]0, €o], €o > 0, so that the following holds:
i)Kc re>
ii) x + ife(x) — (1 + it)x for \x\ > C,
iii) Pe = -A|p€ is uniformly elliptic also w.r.t. €.
iv) For x G K: pe(x,£) = £2. Here pe denotes the principal symbol of Pe.
v) For 0 < dist(x,K) < C<it, we have -0o < argPe(#,0 < 0, where
tan 0O = 2>/5.
vi) For Ctf < dist (x, K) < C, we have -0O < argpe(z,£) < -e2/C.
vii) For dist (x,K) > C, we have argpe(z,£) < -0i, tan0i = 2e/(l - €2).
In order to obtain this, one uses geometric considerations, to find a
function (j) = (j)t e C°°(Rn; R), such that
4>\K < 0, (H.7)
4>{x) > ^dist (x, K) - Ce, x e Rn \ K, (11.8)
^ < 4>"(x) < Ce, (11.9)
<j>{x) = x2/2 - a, |x|>C, (11.10)
|^(x)|<C,|x|<C. (11.11)
One then takes
f(x) = C2eg(<t>(x)/Cle),
with C\ » C2 » C, where g 6 C°°(R) is a convex function with g (t) = 0,
for i < 0, </(*) = i — Const., for i > 1. We do not go into the details of
this and simply give the calculation of pe: With z = x + if(x), f = /e,
we get the relation between tangent vectors, Sz = (1 + if"(x))8x, and the
corresponding relation for cotangent vectors, £ = (1 + ^/"(a;))"1^. Hence
pe(x,0= ((H-tH*))"1*)2- Using that (l + i/"^))"1 = (1-*7"(*))(1 +
/"(x)2)"1, we get
pe(«,0 = ((1 - i/"(x))02 = ((1 - f"(x)2)U) - Hf"(x)U),
with £ = (1 + /"(x)2)_1£. The construction of / also gives ||/"|| < 1/V^,
so |(| ~ |£j.

429
Let Pe be the realization of P on Tc, defined as Pq is section 5. The
resonances of P in {z 6 C\\z\ > 1, 0 < -argz < e/C} can then be
identified with the square roots of eigenvalues of Pt in the same set.
Introduce the auxiliary function
/(*) = * +a/*2-1,
defined for all complex z which do not belong to the vertical half-lines
±l + t[0, +oo[, and with the branch of the square root chosen so that /(0) =
-i. Then /(±1) = ±1 and / is bijective: {z e C; Re 2: < 0} -> {z e
C; Re2: < 0, \z\ > 1}. The inverse is given by f~x(w) = \{w +w"1).
For flf(€) > 0 small, put F€(z) = F(z - ig(e)), F(z) = f(2z - 3). Using
some pseudodiffererential and functional calculus from [34], we can then
define Fe(/i2Pe), and verify that the eigenvalues of this operator in 1 <
\w\ < l + 8,lmw < 0, for 8 < e/C are precisely the images in this set under
Fe of the resonances of h2Pe. One can also show:
For 0 < h< h(e) > 0, we have Bt > 1, where Bt = Fe(Pe)*Fe(Pe).
If fii < ii2 < .. are the eigenvalues of \[Bt and S < e2/C, and M$e(h) =
#{W;Mi < 1 + *}, then Me,5 < ^e(h-1(2 + 0(82))) -<f>e(h-i(l-0(82))) +
0{e)h-n.
If ^1,^2,.., are the eigenvalues of Ft(h2Pt), with |Ai| < |A2| < .. < 1 + ^,
then we have the Weyl inequalities,
Mi ••••/** < |Ai|-..-|Ajt|.
Let N€i$(h) be the the number of \j with |Aj| < 1 + S. For #i < ^2/2, we
then get with M = AfC|*2, A^ = N^, if N > M:
1M(1 + 82)N-M<(1 + 81)N,
so
logfjg S2
The remainder of the proof is then book-keeping. We choose 81/82 << 1.
Applying Fe-1, we get a bound on the number of resonances inside an ellipse
with focal points l+ig(e), 2+ig(e) of diameter 1+Si + 1/(1+Si) = 2+0(8\)
and width 1 + £1 - 1/(1 + ^1) = 28\ + 0(82). The required smallness of h
depends on e and on the choice of g(e) but not on the other constants
in the estimates above. To finish the proof one finally takes a geometric
progression of fe's and tries to cover the required sector as economically
possible with dilated ellipses.
The result in Theorem 11.3 is far from perfect. It is quite possible that
the method of scaling can be further improved even in the general convex

430
case, so that we can get smaller powers of e in the errors. Also, if we
restrict the attention to the case O = 0 (in order to fix the ideas) it is quite
possible (and indicated by preliminary results by M.Zerzeri [42]) that we
can replace volchsupp(P+A) by the generally smaller phase space volume
of the union of certain trapped trajectories. Further one may sometimes ask
for estimates in certain parabolic neighboroods of the real axis (in analogy
with those of Theorem 11.1.
In [38] we looked at the exterior Dirichlet problem in Rn \ O when
O is strictly convex with smooth boundary and showed, by using scaling
up to the boundary, that Ne(r) = 0(63/2)rn, r > r(0). The example of
the ball, shows that the exponent 3/2 is sharp. Later Harge-Lebeau [12]
obtained results about the Gevrey regularity in the time direction for the
wave-equation with an obstacle as above. They used the same type of
complex scaling up to the boundary as in [38] and made the additional and
important observation that the angle of scaling 7r/3 is very convenient. As
a consequence they showed that there are only finitely many resonances in
an inverse cubic neighborhood of the positive real axis, a result wich was
previously known in dimensions 2 and 3 ([2], [7]) and in general dimension
for analytic boundaries ([3]). The work [12] promted us to improve the
estimates on the resonance density for C°° boundaries in [39] and in [32],[43]
we continued with the analytic case. In the following we shall mainly
concentrate on [32], since the estimates in that case depend on dynamical
properties, but the proof relies a lot on [39], and the latter work contains
some fine estimates that (so far) were not possible to carry over to the case
of analytic boundaries.
To be more precise, we know from the works [2], [7], [12], [3], that
if O CC is open strictly convex with C°° boundary, then there exists a
constant Co > 0 such that for every e > 0, there are at most finitely many
resonances (jij = V^i> w^h ^i as defined in section 5) in
Rez>l, Im >-(C0-€)(Re*)1/3. (11.12)
An explicit value of Co can apparently be obtained from the works cited
above and was also obtained in [39], [32]. In the general C°°-case, we can
take Co = Coo, where
c. = r'/*a»fc,jjf0)OM,/* (n.18)
In the case of analytic boundary dO, we can take Co = Ca, where
Ca = 2-1/3cos^Cisup inf -/ Qirr'(t))2,3dt. (11.14)
0 T>0 7 boundarygeodesic 1 JO

431
Here S(dO) denotes the tangent sphere bundle, i.e. the bundle of
normalized tangent vectors, (for the induced Riemannian metric on the
boundary)- Q{v) denotes the curvature of the boundary in the direction v, or in
other words the second fundamental form defined by Q{y) — {(f>"(x)v,v),
v G Sx(dO), where 0 is a convex smooth function such that </> = 0,
||V0|| = 1 on dO. In (11.14) we only consider boundary geodesies 7 such
that the derivative 7' is normalized.
In order to state the result, we shall work on S(dO). Put
CiM = (2Q(*))2/3Ci, (11-15)
where 0 > —Ci > —C2 > •• are the zeros of the Airy function, which we
define up to a non-vanishing factor to be the solution of (D2 + t)Ai(t) = 0
for t e R, which is rapidly decaying when t -» 00. Let $t • S(dO) -> S(dO)
be the geodesic flow, and put
<f(i>) = ^j\i(*t(i>))dt, (H.16)
CiTmi„ = infCiT- (11-17)
We also introduce the almost everywhere limit whose existence is
assured by the Birkhoff ergodic theorem:
Ci°° = Km CiT (11.18)
1 —>-00
General arguments give:
supCimin = Km #,„*. < essinfCT- (H-W)
2* 1 —>-oo
Put
W0
,(M)=/ (/i-cos^cni72^, (n-20)
where dS is the natural Euclidean volume element on S(dO). The main
result in [32] is:
Theorem 11.4 There exists a constant C > 0 depending on the obstacle
O, such that if k > 0, ji < cos § limx_^oo Cfmin + 1 A? an^ N(k, fi) denotes
the number of resonances (fij = y/Xj) situated in the closed lower half
plane, above the parabola Imz = a(Rez - k)2 - jik1^, which crosses the
real axis at the two points k ± J^r^k2^3, with r$ > 0 some fixed constant
> 0, is
(11.21)
when k —> +00.

432
As one can see from the the proof, the Woo-term is a disguised phase
space volume. If \i < cosf limx-+oo (£min, then the leading term in (11.21)
is zero, in agreement with the pole-free region result. What may be more
remarkable is that if limx_hX) (fmin < essinf (i,min then we have the same
conclusion if 0 < //-cos f lim Ci"min *s small. (Here we run into delicate
questions about the boundary dynamics, and we have no example showing that
this interesting phenomenon may occur.) Zworski [43] has obtained more
precise results for surfaces of revolution. When the boundary is only C°°,
the theorem remains valid provided that we refrain from taking time
averages, or in other words that we have to replace limx_HX) C^mim by m^ Ci {v)
and tf° by (i. See [39] and appendix b (jointly with Zworski) in [32].
One would get a much more natural estimate if one were able to replace
the factor 3 in the estimate by 1. The loss appears in the application of the
Weyl inequalities, and we have not seen any trick similar to the one in the
proof of Theorem 11.2, which would help us.
In order to give some ideas about the proof, we start by discussing the
corresponding result for C°° boundaries, where there can be no averaging
involved. We shall do scaling up to the boundary, and it turns out that
the interesting things happen near the boundary, so we will only discuss
that region. Near the boundary, we choose geodesic coordinates (x',xn) 6
dO X [0, +oo[, so that xn becomes the distance from the point described by
x to the boundary, and x' is the corresponding boundary point. In these
coordinates, we get the well known description of -h2A:
-h2A = (hDXn)2 - 2xnQ(x', hDx,) + R(x', hDx.)
+0(x2n(hDx,)2) + 0(h2Dx) + 0(h2). (11.22)
Here we take h = k~l and Q = h2Q(x',Dxi), R = h2R(xf,Dx>) are
positive elliptic operators. The principal symbol of Q can be identified
with the second fundamental form and to leading order, -R = A#e>,
the Laplace-Beltrami operator on 80. We now use complex scaling which
near the boundary and in geodesic coordinates takes the form: (x1\xn) ^
(x*, et7T/3xn). Let V be the image contour. Then we get the scaled operator,
Pr = e-2^\(hDxJ2+2xnQ(x',hDx,))+R(x>\hDx,)+0(..) + .. . (11.23)
This is a degenerate elliptic operator, so we have a degenerate
elliptic boundary value problem, and following the general philosophy for such
problems, we view Fp as a vector valued /^-differential operator in the
tangential variables with operator valued symbol:
= e-2*i'3{(hDXn)2 + 2xnQ(x',Z')) + R(x',t') + ... (11.24)

433
The eigenvalues for the Dirichlet problem on the positive half-axis for
the operator (hDXn)2 + 2xnQ(xl',£') are of the form,
h2/%(x',S') = hW(2Q(x',Z'))y%, (11.25)
so the eigenvalues of Pr{x', £') become, if we ignore the (9-terms: R(xf, £') +
e-2*i/3h2/3Qt
From this we can conclude with some work, that if u>q = $uq + &Vo,
belongs to the open first quadrant, then Pr(x,hDx;h) has no eigenvalues
in the disc of center u>0 and radius
ro + cosV/3 inf Ci(*',0-©(*),
0 R(x'4')=Reujo
which is essentially the result on absence of resonances near the real axis
in the case of C°° boundary.
Also with quite a lot of work, it is possible to estimate the
accumulation of small eigenvalues of \/(Pr ~ ^o)*{Pr ~ ^o) in terms of W(fJ,) =
Js(d<9)(M"~C0S (K0+ ^ and combining this with the Weyl inequalities in a
rather precise way, we can get the C°° analogue of Theorem 11.4 essentially
due to [39].
In the case of analytic boundaries, it is possible to use exponential
weights microlocally in the bounary variables, and global FBI-transforms
provide a convenient frame-work. Let X be a compact analytic Rieman-
nian manifold. Let <f>(a,y) be an analytic function on {(a, y) £ T*X X
-X";dist(ar,y) < 1/C}, such that:
(A) (j) is holomorphic and = 0(|(a^)|) on {(a,y); llma^l, |Imj/| < 1/C,
|Ima€|<^|<ac>|}.
(B) <^(a,ar) = 0, {dr)<l>)(a,ax) = -a{, lm(0^)(a,ar) ~ |(af)| •/. Here we
expressed the conditions in terms of canonical coordinates induced by some
system of local coordinate charts, and used the notation: a = (axia^). By
Taylor expansion, we get </>(a, y) = af • (ax - y) + 0(l)(af)|ar - y|2.
If a is a suitable elliptic analytic symbol and x G C°°(X x X) is equal
to one near the diagonal and has its support in a small neighborhood of
the diagonal, then we introduce the global FBI-transform:
Tu(a; h) = J eWa>4a(a, y; h)X(ax, y)u(y)dy, (11.26)
taking distributions on X into holomorphic functions m_a defined on some
neighborhood of T*X in a suitable complexification T*X of this manifold.
It is possible to construct an approximate left inverse of T which works up
to certain exponentially small errors, but we will not go into the details of

434
that essentially well-known fact here, and simply notice that the theory is
very close to the one developed on Rn in [14].
Let A C T*X be an J-Lagrangian manifold, i.e. a manifold which is
Lagrangian for the real symplectic form which is equal to the imaginary
part; Imcr, of the complexification a of the standard symplectic form on
T*X. We assume that A is close to T*X in the C°°-sense and coincides
with T*X outside a bounded set (in the fiber directions). Locally we can
then find a smooth function H on A, such that
dH = -Im (a£ • dax)\A. (11.27)
We now assume that
(C) (11.27) has a global solution H e C°°(A; R).
Notice that we can normalize the choice of H by requiring H (a) to be 0
for large a%.
An^example of such a manifold can be produced from a function G G
Co°(T*X;R), that we can view as a weight. For t G R, \t\ small, put
At = exp(tH^a)(T*X). Then the assumption (C) is fulfilled.
Definition. Let A, H be as above so that (C) holds. For m 6 R and for \t\
small, we put
tf (A; (ai)m) = {«£ V(X); Tku € L2(A; e-MJ/fc|<ac>|2TOA»)}> (11.28)
equipped with the natural norm. We observe that the norm also depends
on the choice of ff, and actually these spaces coincide with the Sobolev
spaces, because of the condition that A should coincide with T*X far away.
It is rather the exponentially weighted norms introduced here which are
interesting.
If P(x,hDx;h) = Yl\a\<mak{z\h)(hDx)k is an /^-differential operator
with analytic coefficients, uniformly bounded in a complex neighborhood
of X, then for \t\ small enough, P is 0(1) : H(At; (a^)m) -> H(At\ 1), and
can be viewed as an h-pseudor with leading symbol P|A + 0(h(a^)m).
In the proof of Theorem 11.4 we apply the above theory with X = dO,
t = 0(h2/3) depending on h. Let G(#',£') be a smooth realvalued function
on T*dO with compact support (in £). We then consider the scaled operator
Pp on H(Kh2iz)®L2Xn (with suitable modifications when xn becomes large).
According to the previous remark we can then view Pp as a vector valued
/i-pseudor operator with leading symbol
p(*',o =
= rw/3((fcj)JJ + 2xnQ(x',e)) + J2(*',0 - ih*/3(HRG)(x',?) + O...
= e-2^3((hDXn)2 + 2x„Q(x',0 + ££ffflG(x',0) + R + °~ (n-29)

435
and this means that the discussion we gave in the C°° case can be applied
with Cj(x\£) replaced by
S(*'.0 = G(*'.0 + Z^HRG(x\Z'). (11.30)
COS ft
It is natural to try to choose G so that that the infimum of £i over the
cosphere bundle S: #(#',£') = 1 becomes as large as possible, and the
natural way of doing this is to average.
The vectorfield v — Hr conserves the Liouville measure on S. Let k be
a smooth (at least Lipschitz) functions on R except for a jump +1 at 0
and assume also that k has compact support. Put fcx(t) = k(t/T). Then if
v G L°°(R), the convolution u — kj * v satisfies ^ = v - ^£t * v, where
lT = t(t/T), and -i is the derivative of kR\{0} e I°°(R), so that fidt = l.
Let q be a real C°°-function on E. Put Gt = ~ J kT(~s)q o exp(sv)ds.
Then,
q + v{Gt) = j; ^T{~s)q o exp(sv)ds.
With a suitable fc, we have 1= l[-i,o]- Then
1 fT
q + v{GT) = 7p I 9 ° exp(w)d5 =def qT-
Applying the discussion to d, we may find G, so that on E:
_ 1 P
Ci = CiT = y J Ci ° exp(stffi)ds
Applying the arguments outlined for the C°°-case, we get the estimates
in the theorem with Woo replaced by a function Wt, which is defined the
same way, but with (™ replaced by C^, and with limx-+oo (fmin replaced
by Cfmin- The last step in the proof is then to use the ergodic theorem to
check that we can pass to the limit T —> oo.
An extension of the results concerning absence of resonances in inverese
cubic neighborhoods of the reals, for stricly convex obstacles with Gevrey
boundary, has recently been obtained by B. and R. Lascar.
References
1. J.Aguilar, J.M.Combes, A class of analytic perturbations for one-body Schrodinger
Hamiltonians, Comm. Math. Phys. 22(1971), 269-279.
2. V.M.Babich, N.S.Grigoreva, The analytic continuation of the resolvent of the
exterior three dimensional problem for the Laplace operator to second sheet, Funcional
Analysis i Prilozen. 8(1)(1974), 71-74.

436
3. C.Bardos, G.Lebeau, J.Rauch, Scattering frequencies and Gevrey 3 singularities,
Inv. Mat. 90(1987), 77-114.
4. C.Bardos, J.C.Guillot, J.Ralston, La relation de Poisson pour Vequation des ondes
dans un ouvert non borne, Comm. PDE 7(1982), 905-958.
5. M.Dimassi, J.Sjostrand, Lecture notes in preparation.
6. L.S.Fahry, V.V.Tsanov, Scattering poles for connected sums of Euclidean space and
Zoll manifolds, Ann. I.H.P. (phys. th.) 65(2)(1996), 163-174.
7. V.B.Filippov, A.B.Zayev, Rigorous justification of the asymptotic solutions of
sliding wave type, Journal of Soviet Math. 30(2)(1985), 2395-2406.
8. C.Gerard, J.Sjostrand, Semiclassical resonances generated by a closed trajectory of
hyperbolic type, Comm. Math.Phys., 108(1987), 391-421.
9. I.Gohberg, M.Krein, Introduction to the theory of linear non-self-adjoint operators,
Amer. Math. Soc, Providence, RI, (1969).
10. V.Guillemin, R.Melrose, The Poisson summation formula for manifolds with
boundary, Adv. Math. 32(1979), 128-148.
11. L.Guillope, M.Zworski, Scattering asymptotics for Riemannian surfaces,
Preprint (1996).
12. T.Harge, G.Lebeau, Diffraction par un convexe, Inv. Mat. 118(1)(1994).
13. B.Helffer, A.Martinez, Comparaison entre les diverses notions de resonances, Helv.
Phys. Acta 60(1987), 992-1003.
14. B.Helffer, J.Sjostrand, Resonances en limite semiclassique. Bull, de la SMF 114(3),
Memoire 24/25(1986).
15. B.Helffer, J.Sjostrand, Equation de Schrodinger avec champ magnetique et equation
de Harper, Springer Lecture Notes in Physics 345(1989), 118-197.
16. L.Hormander, On the singularities of solutions of partial differential equations, in
"Proc. internat. conf. on functional analysis and related topics, (Tokyo, 1969)",
Univ. of Tokyo Press, Tokyo (1970).
17. L.Hormander, J.Wermer, Uniform approximation on compact sets in Cn, Math.
Scand. 23(1968), 5-21.
18. W.Hunziker, Distorsion analyticity and molecular resonance curves, Ann. Inst.
Poincare, Phys. Theor. 45(1986), 339-358.
19. M.Ikawa, Singular perturbation of symbolic flows and poles of the zeta functions,
Osaka J. Math. 27(1990), 281-300 and 29(1992), 161-174.
20. P.Lax, R.Phillips, Scattering theory, Academic Press, New York(1967).
21. R.Melrose, Scattering theory and the trace of the wave group, J. Funct. Anal.
45(1982), 29-40.
22. R.Melrose, Geometric scattering theory, Cambridge University Press (1995).
23. R.Melrose, Polynomial bounds on the distribution of poles in scattering by an
obstacle, Journees Equations aux derivees partielles, St Jean de Monts (1984).
24. R.Melrose, J.Sjostrand, Singularities in boundary value problems, I and II, Comm.
Pure Appl. Math. 31(1978), 593-617, and 35(1982), 129-168.
25. L.Nedelec, Work in preparation.
26. V.Petkov, Sur la conjecture de Lax et Phillips pour un nombre fini d'obstacles
strictement convexes, Seminaire equations aux derivees partielles, expose no 11
(1995-96).
27. V.Petkov, L.Stoyanov, Geometry of reflecting rays and inverse spectral problems,
J.Wiley & Sons, (1992).
28. D.Robert, Autour de 1'approximation semi-classique, Progress in Mathematics, 68,
Birkhauser (1987).
29. A.Sa Bareto, M.Zworski, Existence of poles in potential scattering, preprint (1995).
30. J.Sjostrand, Geometric bounds on the density of resonances for semiclassical
problems, Duke Mathematical Journal, 60(1) (1990), 1-57.
31. J.Sjostrand, Microlocal analysis for the periodic magnetic Schrodinger equation
and related questions, CIME-lectures, Montecatini, (Juillet 1989), Springer Lecture
Notes in Math. 1495 (1991), 237-332.

437
32. J.Sjostrand, Density of resonances for strictly convex analytic obstacles, Can. J.
Math., 48(2)(1996), 397-447
33. J.Sjostrand, G.Vodev, Asymptotics of the number ofRayleigh resonances, preprint,
(Dec. 1995, revised Sept. 1996),
34. J.Sjostrand, M.Zworski, Complex scaling and the distribution of scattering poles,
Journal of the AMS, 4(4)(1991), 729-769.
35. J.Sjostrand, M.Zworski, Lower bounds on the number of scattering poles, Comm.
in P.D.E., 18(5-6)(1993), 847-857.
36. J.Sjostrand, M.Zworski, Lower bounds on the number of scattering poles, II, J.
Funct. An., 123(2)(1994), 336-367.
37. J.Sjostrand, M.Zworski, Distribution of scattering poles near the real axis,
Comm.P.D.E., 17(5-6)(1992), 1021-1035.
38. J.Sjostrand, M.Zworski, Estimates on the number of scattering poles near the real
axis for strictly convex obstacles, Ann. Inst. Fourier, 43(3)(1993), 769-790.
39. J.Sjostrand, M.Zworski, The complex scaling method for scattering by strictly
convex obstacles, Ark. f. Matematik, 33(1) (1995), 135-172.
40. G.Vodev, Asymptotics on the number of scattering poles for degenerate
perturbations of the Laplacian, J. Funct. An. 138(2)(1996), 295-310.
41. G.Vodev, Sharp bounds on the number of scattering poles for perturbations of the
Laplacian, Comm. Math. Phys., 146(1992), 205-216.
42. M.Zerzeri, Thesis, in preparation
43. M.Zworski, Appendix C in [32].
44. M.Zworski, Sharp polynomial bounds on the number of scattering poles, Duke Math.
J. 59(1989), 311-323.

Index
Adiabatic limit, 361
Admissible open set, 372
Agmon distance, 319, 368
Almost analytic extension, 393
Analytic
bilagrangian distribution, 108
distorsion, see Complex scaling
hypoelliptic problem, 39
Baouendi and Goulaouic example, 39
Analyticity, partial, 76
Anti-Wick quantization, 282
Approximation
Born-Oppenheimer, 366
harmonic, 318
APS
boundary problem, 118
index problem, 118
operator, 118
problem, 117
Argument
Combes-Thomas, 384
Gaussian domination, 343
Peierls, 314
Asymptotic
expansion of a symbol, 275
type, 237
Mellin, 237
Weyl, 427
Atiyah-Patodi-Singer problem, see APS, problem
Bicharacteristic
generalized, 422
strip, 33
Bilagrangian
conic submanifold, 93
distribution, 97
analytic, 108
Bineighborhood, 13
Birkhoff ergodic theorem, 431
Bogolyubov inequality, 323
Born-Oppenheimer approximation, 366
Brascamp-Lieb inequality, 323
C°°-ray, see Bicharacteristic, generalized
Canonical
2-formonT*(Cn, 101
transformation, 5
Cauchy-Green-Riemann-Stokes formula, 389
Cesaro mean, 245
Characteristic
manifold, 2
value, 380
Combes-Thomas argument, 384
Complex scaling, 392
Convex potential,phase, 323
Counting function, 265, 427
Diffractive region, 91
Distribution
bilagrangian, 97
analytic, 108
Dixmiers trace, 245
Double well, 307
DTF, see Duhamel two-point function
Duhamel two-point function, 341
Edge-degenerate
operator, 201
symbol, 202
Ellipticity
of partial differential equations, 41
Fuchs type, 191
of operators on a manifold with edges, 213
transverse, 251
439

440
Estimate
Gaussian domination, 337
infrared, 310
classical case, 335
quantum case, 341
trace-class, 383, 403
Eta function, 122
Exponentially localized, 320
FBI transform, 75, 351
global, 433
Ferroelectric model
disorder type, 333
displacement type, 333
Formal order, 268
Formula
Cauchy-Green-Riemann-Stokes, 389
Mehler, 325
Poisson type, 415
trace, 294, 378
Trotter-Kato product, 316
Weyl, 265, 322
Fourier-Bros-Iagolnitzer transform, see FBI
transform
Friedrichs symmetrization, 279
Fuchs
ellipticity type, 191
operator type, 189
symbol type, 239
Function
2-phase
equivalence of, 97
local, 94
bihomogeneous, 93
counting, 265, 427
Duhamel two-point, 341
eta, 122
holomorphic operator, 194
local phase, 92
localizing, 42
microhyperbolic, 81
microlocal behavior of, 351
microsupport of, 351
partially analytic, 76
representation, 65
strongly poly homogeneous, 134
Function
sublinear, 61
weight, 267
with infraexponential growth at infinity, 61
zeta, 122, 233
modified, 122
Functional calculus, 391
Fundamental matrix, 252
Gaussian domination
argument, 343
estimate, 337
Generalized bicharacteristics, 422
Geodesies, 11
Gevrey
class,space, 2, 40
hypoellipticity, 40
Ginibre
inequality, 315
theorem, 315
GKS inequality, 316
Golden-Thompson inequality, 322
Green edge symbol, 207
Green operator, 183, 238
on a manifold with conical singularities, 199
on a manifold with edges, 199
singular, 139
Grushin problem, 402
Hanges theorem, 357
Harmonic approximation, 318
Heat
kernel, 233, 331
operator, 121
Hilbert-Schmidt operator, 293
Holmgren microlocal theorem, 359
Holomorphic operator function, 194
Hyperbolic dynamical system, 425
Hyperfunction, 63
Hyperfunctional boundary value, 64
Hypoelliptic symbol, see Symbol, globally hypoel-
liptic
i-Lagrangian manifold, 434
Inequality
Bogolyubov, 323

441
Inequality
Brascamp-Lieb, 323
Ginibre, 315
GKS, 316
Golden-Thompson, 322
Weyl, 429
Infraexponential growth, 61
Infrared estimates, 310
classical case, 335
quantum case, 341
Interaction, size of, 308
Involutive
homogeneous submanifold, 8
submanifold, 103
Kawai-Kashiwara theorem, 358
Ky-Fan identity, 380
Large dimension limit, 307
Lax-Phillips theory, 415
Left
quantization, 276
symbol, 276
Levi form, 44, 102
Limit
adiabatic, 361
large dimension, 307
thermodynamic, 312
Localized
differential operator, 252
exponentially, 320
polynomial, 252
Lower bound, 253
Manifold
characteristic, 2
i-Lagrangian, 434
stretched, 167
with conical singularities, 166, 235
with edges, 166
Mehler formula, 325
Mellin
asymptotic type, 237
discrete asymptotic type of symbols, 195
operator, 236
pseudo-differential, 190
Mellin
operator-valued symbol with continuous
asymptotic data, 212
smoothing edge symbol, 209
Sobolev space, 236
transform, 235
weighted, 236
Microfunction, 5, 70
higher order, 78
Microlocal behavior, 351
Microsupport, 351
Minimax principle, 327
Minkowski codimension, 426
Multi-quasi-elliptic
polynomial, 268
symbol, see Symbol, globally multi-quasi-
elliptic
Multineighborhood, 75
Multiple reflection bound, 338
Newton polyhedron, 266
Normal kernel of an operator, 141
Normal trace of an operator, 141
One-particle
potential, 307
Schrodinger operator, 317
Operator
APS, 118
edge-degenerate, 201
elliptic, 2
transversally, 251
Fourier integral, 5
Fuchs type, 189
Green, 183, 238
on a manifold with conical singularities, 199
on a manifold with edges, 199
singular, 139
heat, 121
Hilbert-Schmidt, 293
localized
differential, 252
polynomial of, 252
Mellin, 236
micro-, 5
micro-hypoelliptic, 10

442
Operator
normal kernel of, 141
normal trace of, 141
Poisson, 139
potential, 185
power of, 121
principal type, 2
pseudo-differential, 3, 171
analytic, 4
classical, 229
in higher microlocalization, 71
Mellin, 190
with operator-valued symbol, 180
regularizing, 277
resolvent of, 121
Schrodinger
one-particle, 317
semi-classical, 425
special parameter-dependent, 157
sum of squares of real vector fields, 44
trace, 185
trace-class, 286, 293
near infinity, 379
with multiple characteristics, 2
Pair of Lagrangian submanifolds, 96
Parameter of long-range order, 312
Parameter-dependent
ellipticity, 187
parametrix, 187
pseudo-differential calculus, 185
special symbol, 157
Partial differential equation
elliptic, 41
subelliptic, 43
Peierls argument, 314
Phase function, 92
equivalence of 2-, 97
local, 92
local 2-, 94
Phragmen-Lindelof principle, 88
Poisson
formula type, 415
operator, 139
Polyhedron
complete, 267
Polyhedron
complete
formal order of, 268
non degenerate, 269
weight function of, 267
Newton, 266
Polynomial, multi-quasi-elliptic, 268
Positive trace, 254
Potential
one-particle, 307
operator, 185
quasi-elliptic, 265
strictly convex
classical case, 323
quantum case, 324
Power operator, 121
Primitive period, 422
Problem
analytic hypoelliptic, 39
APS, 117
Grushin, 402
Profile, 93
Pseudo-convex domain, 39
Q-elliptic
polynomial, see polynomial, multi-quasi-
elliptic
symbol, see Symbol, globally multi-quasi-
elliptic
Quantization
anti-Wick, 282
left, 276
t-semi-classical, 352
Weyl, 276
Quasi-elliptic potential, 265
Quasi-homogeneous subset, 13
H-Lagrangian submanifold, 371
Regularizing operator, 277
Residue, Wodzicki, 230
extension of, 244
Resolvent, 121
Resonance, 366, 381
width of, 366
Second microlocalization, 14

443
Semi-classical
analysis, 317
principal symbol, 394
Schrodinger operator, 425
theory, 318
Size of interaction, 308
Sobolev space
abstract wedge, 175
Mellin, 236
weighted, 191
Spectral
boundary condition, 118
projection, 286
Spectrum, analytic singular, 69
Splitting between eigenvalues, 307
Submanifold
conic, 92
bilagrangian, 93
involutive, 103
Lagrangian, 92
pair of, 96
m.t.r., see Submanifold,totally real,
maximally
H-Lagrangian, 371
totally real, 393
maximally, 393
Sum of squares, 44
Symbol, 2, 170
asymptotic expansion of, 275
classical, 170
conormal, 239
discrete asymptotic type of Mellin, 195
edge-degenerate, 202
Fuchs type, 239
fundamental matrix of, 252
globally hypoelliptic, 274
globally multi-quasi-elliptic, 274
Green edge, 207
homogeneous
strongly, 132
weakly, 132
homogeneous principal part of, 170
hypoelliptic, see symbol, globally hypoellptic
left, 276
operator-valued, 178
classical, 178
Symbol
operator-valued
Mellin type with continuous asymptotic
data, 212
principal, 2
semi-classical, 394
principal part of, 285
smoothing Mellin edge, 209
special parameter-dependent, 157
sub-principal, 7, 254
totally characteristic, 239
transversally elliptic, 251
weakly polyhomogeneous, 134
Weyl, 276
Symmetric double well problem, 319
Symplectic
manifold,variety, 39
two-form, 5
Theorem
Birkhoff ergodic, 431
Connes, 228
Ginibre, 315
Hanges, 357
Holmgren microlocal, 359
Kawai-Kashiwara, 358
Weierstrass preparation, 21
Weyl, 247
Thermodynamic limit, 312
Trace, 228
Dixmiers, 245
formula, 294, 378
of a self-adjoint operator, 286
of an operator, 293
operator, 185
positive, 254
Trace-class
estimate, 383, 403
near infinity, 379
norm, 286
operator, 286, 293
Transform
FBI, 75, 101, 351
global, 433
Mellin, 235
weighted, 236

444
Transition probability, 362
Transversal reflection point, 422
Transversally
elliptic symbol, 251
reflected ray, 422
Trotter-Kato product formula, 316
Tunneling effect, 309
Ultradistribution, 3
Uniformly degenerate well, 323
Wave front set, 4
analytic k-, 74
higher order, 74
second, 93
second analytic, 75
Wedge, 63, 166
Weight function, 267
Well
double, 307
symmetric problem, 319
uniformly degenerate, 323
Weyl
asymptotic, 427
formula, 265, 322
inequality, 429
quantization, 276
symbol, 276
term, 265
Width of resonances, 366
Wodzicki residue, 230
extension of, 244
Yang-Mills action, 248
Zeta function, 122, 233
modified, 122