F.B.I. Transformation: Second Microlocalization and Semilinear Caustics

Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, Ziirich F. Takens, Groningen

1522

Jean-Marc Delort

E B. I. Transformation Second Microlocalization and Semilinear Caustics

Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo Hong Kong Barcelona Budapest

Author Jean-Marc Delort D6partement de Math6matiques Institut Galil6e Universit6 Paris-Nord Avenue J.-B. Cl6ment F-93430 Villetaneuse, France

Mathematics Subject Classification (1991): 35L70, 35S35, 58G17

ISBN 3-540-55764-4 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-55764-4 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1992 Printed in Germany Typesetting: Camera ready using Springer TEX macropackage 46/3140-543210 - Printed on acid-free paper

Foreword

This text grew up from lectures given at the University of Rennes I during the academic year 1988-1989. The main topics covered are second microlocalization along a lagrangian manifold, defined by Sjgstrand in [Sj], and its application to the study of conormal singularities for solutions of semilinear hyperbolic partial differential equations, developed by Lebeau [L4]. To give a quite self-contained treatment of these questions, we included some developments about FBI transformations and subanalytic geometry. The text is made of four chapters. In the first one, we define the Fourier-Bros-Iagolnitzer transformation and study its main properties. The second chapter deals with second microlocalization along a lagrangian submanifold, and with upper bounds for the wave front set of traces one may obtain using it. The third chapter is devoted to formulas giving geometric upper bounds for the analytic wave front set and for the second microsupport of boundary values of ramified functions. Lastly, the fourth chapter applies the preceding methods to the derivation of theorems about the location of microlocal singularities of solutions of semilinear wave equations with conormal data, in general geometrical situation. Every chapter begins with a short abstract of its contents, where are collected the bibliographicai references. Let me now thank all those who made this writing possible. First of all, Gilles Lebeau, from whom I learnt microlocal analysis, especially through lectures he gave with Yves Laurent at Ecole Normale Sup6rieure in 1982-1983. Some of the notes of these lectures have been used for the writing of parts of Chapter I. Moreover, he communicated to me the manuscripts of some of his works quoted in the bibliography before they reached their final form. Likewise, I had the possibility to consult a preliminary version of the paper of Patrick Gfirard [G], where is given the characterization of Sobolev spaces in terms of FBI transformations I reproduced in Chapter one. Moreover, this text owes much to those who attended the lectures, J. Camus, J. Chikhi, O. Gu6s, M. Tougeron and, especially, G. M~tivier whose pertinent criticism was at the origin of many improvements of the manuscript. Lastly, let me mention that Mrs Boschet typed the french version of the manuscript, with her well known efficiency. Let me also thank Springer Verlag, which supported the typing of the english version, and Mr. Khllner who did the job in a perfect way.

Main notations

T M = t a n g e n t b u n d l e to t h e m a n i f o l d M . TxM = fiber of T M at t h e p o i n t x of M . T*M = c o t a n g e n t b u n d l e to t h e m a n i f o l d M . T2M = fiber of T*M at t h e p o i n t z of M . TNM = n o r m a l b u n d l e to t h e s u b m a n i f o l d N of M . T~vM = c o n o r m a l b u n d l e to t h e submm~ifold N of M . F o r E a v e c t o r b u n d l e over M , E \ {0} or E \ 0 d e n o t e s E m i n u s its zero section. F o r E , F two fiber b u n d l e s over M , E XM F d e n o t e s t h e f i b e r e d p r o d u c t of E by F over M . O v e r a c o o r d i n a t e p a t c h of M , E XM F = { ( x , e , f ) ; e e Ex, f E F , }. If h : M1 --~ M2 is a d i f f e o m o r p h i s m b e t w e e n two manifolds, one d e n o t e s by h t h e m a p it induces ~ : T'M1 --+ T'M2. In local c o o r d i n a t e s h ( z , ~) = (h(x), tdh(x)-I • ~). If x0 E M1 a n d y0 C M2, one denotes b y h : ( M l , X 0 ) --* (M2,yo) a g e r m of m a p from t h e g e r m of M1 at x0 to t h e g e r m of M2 of y0. gr(~b) = g r a p h of a m a p ~b f r o m a m a n i f o l d to a manifold. d ( , ) = e u c l i d e a n (resp. h e r m i t i a n ) d i s t a n c e on t h e real e u c l i d e a n (resp. t h e c o m p l e x h e r m i t i a n ) space. d( , L ) = d i s t a n c e to a s u b s e t L. d = e x t e r i o r differential on a real manifold. 0 = h o l o m o r p h i c differential on a c o m p l e x a n a l y t i c m a n i f o l d . c~ = a n t i h o l o m o r p h i c differential on a c o m p l e x a n a l y t i c manifold. dL(:r) = L e b e s g u e m e a s u r e on C n. We will use t h e s t a n d a r d n o t a t i o n for the different spaces of d i s t r i b u t i o n s : C ~ (comp a c t l y s u p p o r t e d s m o o t h functions), S (Schwartz space), S ' ( t e m p e r e d d i s t r i b u t i o n s ) , H ~ ( S o b o l e v spaces), . . .

Contents

O. I n t r o d u c t i o n

...............

I. F o u r i e r - B r o s - I a g o l n i t z e r

. ................................................. transformation

and

first microlocallzation

1 ....

7

1. F B I t r a n s f o r m a t i o n w i t h q u a d r a t i c p h a s e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2. F o u r i e r - B r o s - I a g o l n i t z e r t r a n s f o r m a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

3. Q u a n t i z e d c a n o n i c a l t r a n s f o r m a t i o n s

17

..........................................

4. C h a n g e of F B I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

II. Second

28

microlocalization

.................................................

1. Second m i c r o l o c a l i z a t i o n along T{*0}IRn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

2. Second m i c r o l o c a l i z a t i o n along a l a g r a n g i a n s u b m a n i f o l d

31

3. T r a c e t h e o r e m s III. Geometric

......................

............................................................... upper

bounds

................................................

1. S u b a n a l y t i c sets a n d s u b a n a l y t i e m a p s

........................................

41 47 47

2. C r i t i c a l points a n d critical values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

3. U p p e r b o u n d s for m i c r o s u p p o r t s a n d second m i c r o s u p p o r t s . . . . . . . . . . . . . . . . . . . .

58

IV. Semilinear

72

Cauchy

problem

.............................................

1. S t a t e m e n t of the result a n d m e t h o d of p r o o f 2. S o b o l e v spaces a n d i n t e g r a t i o n s by p a r t 3. E n d of t h e p r o o f of T h e o r e m 1.3

..................................

.......................................

..............................................

4. T h e swallow-tail's t h e o r e m a n d various extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography

Index

..................................................................

.........................................................................

72 77 84 95 99 101

0.

Introduction

We will first recall some elementary results concerning the Cauchy problem for the linear wave equation. Then, we will indicate the new phenomenons appearing in the study of semilinear wave equations and we will describe the theorems obtained by Beals, Bony, Melrose-Ritter about semilinear Cauchy problems with conormal data. Lastly, we will state "swallow-tail's problem", which will be solved in the last chapter of this text, where we expose a method due to Lebeau. Let us consider on R l+d with coordinates (t, x) = (t, X l , . . . , Xd) the wave operator 02

~2

(1)

[] - Ot 2

A~ --

d

~2

Ot2 ;~1: Ox~ "

To solve the Cauchy problem is to find a solution u(t, x) to the problem Ou= f(t,x)

t > O

ul,=0 = u0(z) Ou

(2)

b 7 ,=0 = u l ( x )

where the functions f , u0, Ul are given in convenient spaces. Let us first consider the special case f - 0, u0 -- 0, Ul = 5, Dirac mass at the origin of R d. Using a Fourier transformation with respect to x, one sees that (2) adnlits a unique solution e + ( t , x ) in the space of continuous functions of t C ~ + with values in the space of tempered distributions on R d, whose Fourier transform with respect to x is given by

(a)

( 7 ~ + ) ( t , ~) - sintl~l I~l

l{t_>0l -

It follows from the preceding expression and from the Paley-Wiener theorem that e + ( t , x ) is supported inside the forward solid light cone/~ = { (t,x); Ixl _< t }. The elementary solution e+(t, x) allows us to solve in general problem (2):

T h e o r e m 1. Let f C L ~ ( R + , H ~ - I ( R d ) ) , uo e H~(Nd), u l e a unique solution u E C l ( ~ + , S ' ( N d ) ) . It is given by (4)

u(t, x) =

/o'/

H~-l(I~d). T h e n (2) has

!

c+(t - s, x - y ) f ( s , y) ds dy + e+ * [u0 ® St=0] + e+ * [ul ® &=0] •

2

0. Introduction

Proof. Let us remark first that because of the support properties of e+, the convolutions make sense. One then checks at once that the function u given by (4) is a solution of (2), and satisfies, because of (3), the regularity conditions given in the statement of the theorem. The assertion of uniqueness is trivial. One should remark that it follows from (4), and from expression (3), that if for every k E N D k I • L2(R+, H ~ - l - k ( R d ) ) , then D~u E L2(N+, H~-k(Nd)). This implies that ult>o is in the space Hlo¢(N+ ~ -l+d ) ff • f E H ~ I ( N I + d ) . In fact, one has just to write with k =

+ 1

[ f i ( r , ~ ) 2 ( 1 + ~2 + r2)~ d~dr < [ fi(r,~)2(1 + ~2)~ d~dr J I<_1~1 + f

a(r,~)2(1 + ~ ) ~ - k ( 1 + r~) k d ( d r

The formula (4) shows that the value of u at (t,x) depends just on the value of f at points belonging to (t,x) - /~ and on the value of u0, u~ at points of { y E IRa; (0, y) E (t, x) - / ~ } (finite propagation speed). If ~? is an open subset of N I + e one says that ~? is a determination domain of w = .(2 N {t = 0} if and only if for every (t, x) ~ 9 , the set { (s,y); (sgnt)(t - s) > Ix - y] and (sgnt)(sgn s) >_ 0 } is contained in ~2. Using convenient cut-off functions, one deduces from Theorem 1 and from the finite propagation speed property: T h e o r e m 2. Let ~ be a determination domain of w. Let uo E H~o¢(W), ul E Hlos -~- 1 (w) and let f be a distribution on 9 which is, locally in ~, in the space L~(IR, HS-I(Nd)). Then the problem [3u = f(t, x) (5)

ul,=o = Ilo

in Y2 on

O_~ t = 0 ~--- Ill

o n 03

has a unique solution u which is in C°(R, H ' - I ( N a ) ) locally in $2. Moreover u belongs 8--1 to H~oc(Y2) if f • Hio ~ (Y2). Let us now recall the theorem of propagation of C ~ microlocal singularities. We will use the notion of C ~ wave front set, whose definition is recalled in Section 1 of Chapter I. Let us denote by C a r d = {(t,x;T,~) C T*~?; ~2 = ~_2 } the characteristic variety of O. If A is a subset of T*Y2A {+t _> 0}, one will denote by P+(A) (resp. P _ ( A ) ) the union of A and of the forward (resp. backward) integral curves of the hamiltonian field of a([~) = ~2 _ ~_2 issued from the points of A N CarI~, and contained in/2: (6)

P+(A) = A kJ ({ (t, x; % ~); i t > 0, ~2 = r 2 and there is s E R < 0,

with

0. Introduction

3

Since ~ is a determination domain, as soon as there is (t, x; "r, ~) • ;O±(A) with ~2 = r2 and so • N such that ( t + so'r, x - s0~; T, ~ ) • 7:'± ( A ), then the points ( t + ST, x -- s( ; r, ~ ) belong to T'±(A) for every s • [0, so]. The theorem of propagation of microloeal singularities is then: T h e o r e m 3. Let u be a solution on (2 of the Cauchy problem (5). One has (7)

WF(u)l+t>0 C P ± [ ( W f ( f ) N { i t > 0 } ) U { ( 0 ,

x;%~); ~2 ='r2 and

(x,~) • WE(u0) U WE(u1) }1 . Proof. One knows (see [H], Section 8.2) that if vl and v2 are two compactly supported distributions (8)

WF(vl * v2) C { (z,(); 3(zl,z~) with (Zl,() • WF(vl), (z~,() • WF(v2) and z = zl + zz } .

Because of (4), we thus see that the inclusion (7) follows from the following lemma: L e m m a 4. One ha8

(9)

WF(e+) c T~*0/~'+d U { (t, x; "r, ~); t > 0, t ~ = x ~, (', 4) = ~ ( t , - x ) wi~h ~ • R}

Proof. To show (9) we will prove that e+ is conormal along the forward light cone. More precisely, let M be the C~(Na)-module of C ~ vector fields whose symbol vanishes on the right hand side of (9). We will show that if ( X 1 , . . . , Xm) is an m-tuple of elements of M one has X1 " " X m e - • HI~o¢(R l+d) for every a < !~A. One sees easily that .hd is generated by the fields 0

d z

0

tg+E J0xj 1

(10)

0 0 xj Ox---k - x k Oxj

1 < j 7~ k < d

0 0 xj-~+tOxj

1 <_j <_d.

The action of the first one on e+ gives - ( d - 1)e+ and the other ones cancel e+. For every compactly supported X and every m-tuple of vector fields X 1 , . . •, X m of the form (10) one has thus

(11)

Ix(Xl "'~'Xme+)[ _< Cmlx~+(-,~)l

_<

c1 + I'1 + 141

where the last inequality follows from (3). Let F = { (t,x); t = Ix[}. The inclusion (9) may now be deduced from (11) in 0 the following way: if (to,xo) • Supp(e+), t0z ~ x02, the fields X( X )b-77~j, J = 1 , . . . ,d, and

4

O. Introduction

X(X) o with X e C ~ ( l ~ l + d ) , Supp x n r = 0, are in .Ad and thus e+ is C ~ close to (to, x0). On the other hand, if (to, x0) satisfies to2 = x02 ¢ 0, there is, close to (to, x0), a system of 0 local coordinates (y0 ," . . , ya) such t h a t F is given by y0 = 0. Then, the fields X(Y)oy~, • .., X ( Y ) + are in Jet if S u p p x is small enough. It follows t h a t W F ( e + ) C T~-R a+a close to (to, x0). In the preceding proof, we used the u p p e r b o u n d (11) of IX~+[. In fact, there is a b e t t e r u p p e r bound, we will have to use in C h a p t e r IV: Lemma

(12)

5. For every X G C~X)(I~l+d) there is a constant C > 0 with

t2-~+(T, ~)1 < C(1 + I~l + I~-l)-x( 1 +

I1 1- I~-II) -1

•

Pro@ Because of the s u p p o r t p r o p e r t y enjoyed by e+, we m a y always assume t h a t X is a c o m p a c t l y s u p p o r t e d function of the single variable t. T h e n , by (3), X~+(v, {) = fo+oo ~~ - - i t r .; ([4"~sin ~ } ~ tl~l dt. Using that for any complex n u m b e r a one has

fo +~ x(t)e -it~

dt

<_ C ( l +

I~1) -1

the inequality (12) follows. Before beginning the description of the nonlinear p r o b l e m s we will be interested in, let us m e n t i o n that, of course, T h e o r e m 3 admits a m o r e precise s t a t e m e n t . In fact, as is well known (see [HI), W F ( u ) \ W F ( f ) is foliated by the integral curves of the h a m i l t o n i a n field of G(n). We will now s t u d y the p r o b l e m of control of microlocal singularities of the solution u, given in the space Hz'oc(~2 ) with s > I~A 2 , of a semilinear Cauehy p r o b l e m of the form []~ = f ( t , x, u)

(13)

~1,=0 = ~0

Ou ~ " t=0 ----ul

where f is a C ~ function over N l+d x ~ and u0, ul are given on w = D N {t = 0} in the space H~oc(W ) and H;~c](w) respectively. T h e new p h e n o m e n o n one has to cope with to solve such a problem, is the one of interaction of singularities. For instance, let us take two distributions with c o m p a c t s u p p o r t oi1 IR'~ vl, ve and assume that W F ( v j ) C { (0; A~J), A _> 0 }, where ~1 and ~2 are two non-zero elements of T0*R n such that there exist no negative real n u m b e r 0 with ~a __ ~ 2 . A s s u m e m o r e o v e r t h a t vl and v2 belong to H ~ ( R n) for some a > n / 2 . T h e n the p r o d u c t vlv2 exists, a n d defines an element of H~(R'~). Writing v---~, v~(~) = 01 *02((), one sees easily t h a t (14)

W F ( v ] v 2 ) C { (0, AI~ 1 + A2~2); A1 ~ 0, A2 ~ 0 } .

0. Introduction

5

In general, there is no better upper bound for WF(vlv2), i.e. there are, in this last set, directions belonging neither to W F ( v l ) nor to WE(v2). Moreover, if ~1 and ~2 belong to a same line and have opposite directions, the inclusion (14) is no longer true and any C T~N n m a y be inside WF(vlv2). A similar phenomenon happens when one computes f(v) with f a Coo function of v. This suggest that, in general, the solution of a semilinear problem like (13) will have much more singularities than the solution of the linear problem (5). As a matter of fact, it is reasonable to suppose that u will have at least the singularities of the solution to the linear problem, i.e. those given by the right hand side of (7) with f = 0. But then, in the nonlinear term f ( t , x , u ) of (12), these singularities will create new ones by interaction, that is W F ( f ( t , x, u)) will be bigger than WE(u). By (7) the upper bound for W F ( u ) will have to take into account the points obtained by propagation from W F ( f ( t , x, u)) fq Car [3. These new singularities will also, by interaction, contribute to increase W F ( / ( t , x, u)) and so on. In general, one cannot hope to obtain for nonlinear problems results like Theorem 3. In fact, Beals [Bell found an example of a solution of a semilinear Cauchy problem, with Cauchy data smooth outside 0, and whose singularities are dense inside the light cone {(t,x); Ixl _< t } . To get anyway results of control of singularities, one is thus lead to make specific assumptions on the nature of the singularity of the Cauchy data u0, ul. In particular, the notion of "conormal regularity" happened to be very well adapted to that. Let V be a submanifold of the hyperplane {t = 0}. One says that uj E H~/-j'+Oo if for every integer m and for every m-tuple of Coo vector fields X1, .. •, Xm tangent to V, one has X 1 - - - Xmu j ~_ H ~ j. In particular, W F ( u j ) is contained inside T~IR d. If one solves a linear problem like (5) with f = 0 and such initial data, it follows from Theorem 3 that

(15)

WF(

)I,>0 c {

t > 0,

= ,2 # 0, (x -

e

d }.

T

When V is a hypersurface, the projection of the right hand side of (15) on N l+d is close to t = 0 the union of two smooth hypersurfaces intersecting transversally along V. In the case of a semilinear Cauchy problem like (13), Bony [Boll, [Bo3] proved that the inclusion (15) remains valid for t close to 0. In fact, the solution u is conormal along the two outgoing hypersurfaces. This result thus shows that close to t = 0, the solution of the semilinear problem has the same singularities as the solution of the linear one. Anyway, on a longer interval of time, other singularities happen as a consequence of nonlinear interaction. Let us consider in 2 space dimension a solution u E Hio c, with s > ~ 2 , of the equation [3u = f(t, x, u), such that u]t to }, ~ , Z2, Z3 intersect transversMly at a single point 0. Then, it has been proved independently by Bony [Bo2] and Melrose-Ritter [M-R] that the solution u is C °o outside Z1 U Z2 U Z3 U P where F is the boundary of the forward light cone with vertex at 0, and that u is conormal along the smooth points of this intersection (see also Chemin [Ch] for an extension and Beals [Be2], [Be3] for a more elementary proof). In such a case, we thus see that interaction of singularities provokes the creation of new singularities along F.

6

O. Introduction

The fourth Chapter of this text will be devoted to the study of a phenomenon of interaction of singularities in the large. Consider in d = 2 space dimension a solution u of a semilinear wave equation, whose Cauchy data are conormal along a real analytic curve V of N 2, having at a single point a non-degenerate minimum of its curvature radius (for instance, a parabola). The projection on IR2 of the flow out of T~R a N Car O by the hamiltonial field is the union of two hypersuffaces of I{a, which are smooth close to t = 0, V+, and V_. One of them, say V_, remains smooth in the future, but the other one, V+, has a pinching point in t > 0 (V+ is a swallow tail). The aim of Chapter IV is to prove, following Lebeau [L4], that ult>0 is smooth outside the union of V_, V+ and of the two-dimensional forward light cone with vertex at the pinching point of V+.

I. F o u r i e r - B r o s - I a g o l n i t z e r t r a n s f o r m a t i o n a n d first microlocalization

This first chapter is devoted to the definition of Fourier-Bros-Iagolnitzer (FBI) transformation and to its application to the study of microlocal regularity of distributions. The first section studies FBI transformations with quadratic phases, as those introduced by Bros-Iagolnitzer [Br-I] and Sjhstrand [Sj]. In particular, we prove a characterization, due to P. G6rard [G], of H s microlocal regularity of distributions in terms of FBI transformations. We also give, following [HI, an inversion formula due to Lebeau [L1], expressing a distribution as an integral of its FBI transform. In the second section, we bring out the fundamental properties enjoyed by the quadratic phase i(~-¢)2 This enables us to define general FBI transformations, using 2 " phases satisfying these properties. We still follow the bibliographical reference [Sj]. The third section gives the definition of Sjhstrand's spaces and of transformations between these spaces given by convenient phase integrals. We introduce the notion of "good contour" and prove the "fundamental lemma" of [Sj]. The last section is intended for a proof of the theorem of change of FBI: following Sjgstrand, we show that one may pass from a FBI defined by a phase g to a FBI defined by a phase ~ using one of the transformations studied in the third section. This allows us to deduce from the results of Section 1 a characterization of microloeal H s regularity in terms of FBI transformations with general phases.

1.

FBI

transformation

with

quadratic

phase

Let u be a compactly supported distribution on R". The FBI transformation of u is the function on C '~ x [0, +oo[ defined by:

(I.I)

Tu(z,

[ :

i

J

e-}(~-t)~u ( t ) a

o

It is an entire function of the complex variable x, real analytic with respect to the p a r a m e t e r t . As u is of finite order, there exists an integer N and a constant C > 0 such that

(1.2)

ITu(x,y)l < C(1 + A + I l m x l ) N e -)(Im x)2

for z E C", ~ E [0, +co[. The transformation (1.1) is nothing else than a modified Fourier transform. As this one, it will allow us to characterize (microlocal) regularity of u through better estimates

8

I. Fourier-Bros-Iagolnitzer transformation and first microlocalization

than (1.2), the great parameter A playing now the same role than the norm of the frequency variable in usual Fourier transform. Let us begin by the study of Sobolev regularity. Recall the following:

-

Definition 1.1. Let u be a distribution on Rn. One says that u is HS microlocally at (to,ro)E T * R n- (0) (what will be denoted by u E Htto,+o)) or that (t0,ro) is not in 1 close to the H s - w a v e front set of u ( ( t o , r o$) WF,(u))if there is x E C F ( R n ) ,x t o , and r a conic neighborhood of ro in Rn - (0) such that

where (r)' = 1

+ r2.

Our first aim is to prove, following P. Gkrard [GI, that we may characterize the preceding Hs-wave front set using Tu. Assuming u compactly supported - which does not restrict the generality of the problem - we have:

Theorem 1.2 (P. Gkrard). The point ( t o , r o )E T * R n- (0) is not in W F s ( u )if and only if there exists W neighborhood of xo = to - ire in Cn such that

dL(x) standing for Lebesgue's measure on C7'. One should remark that, because of (1.2), one could replace in the first integral of (1.4) the lower bound 1 by any real positive number without changing the condition. The proof of the theorem relies on the following lemma.

Lemma 1.3. For a compactly supported distribution u E S ' ( R n ) ,let us put E,(u) = { ro E Rn \ (0); V r open conic neighborhood of ro

A point 7 0 E Rn - (0) is not in C s ( u )if and only if there exists a neighborhood V of in Rn such that

The Fourier transform of ?u(s - ir, A ) ( s ,T E R n ) with respect to s is

70

1. FBI transformation with quadratic phase

()-

n

27r ~ e__~(,_~)~fi(o)

(1.7) whence the equality

[Tu(x,A)[2e-)'(Im')2dL(x)=

(1.8)

""ff

"--iV

da

dTe-)'(r-~)2]ft(a)]2 .

'~

Since the contribution to the last integral coming from the d o m a i n {]a[ _< 1} is exponentially decreasing with respect to A, it is enough, to show the lemma, to prove that there is a relatively compact neighborhood V of 7o in l~ n - {0} such t h a t

,~~-1

(1.9)

K~(o,A)[fi(a)12 da dA < +co ]_>1

with

K~(a,

(1.10)

m

Let us show that if V C C V' are two relatively compact open subsets of ll~n - {0} and if F =- Ut>1 tv, F ' -- Ut>x tv', there exist C > 0 and ¢ > 0 such that for every a C ]R'~ with [a] _-1 one has: (1.11)

C-1)~-~ ] a [ 2 ~ l r ( a )

(1.12)

K~(GA ) G

.Iv e-(x*-*)~ d7 <_K~,(a, A)

CA~[o[2Slr,(o)/v,

e -()~'-~)~

d~"+ Ce -~(~+ ~lqlL) .

In fact, let V" be an open subset such that V C C V" C C V'. - If } ¢ Y", one has 1 7 - ~L > ~ ( 1 + ~ ) for every 7 e V. T h u s (1.12) is true because of the exponential term in its right h a n d side and (1.11) is trivial. - If ~ C V", one has l r , ( a ) = 1 and [0[ ~ cst .A. Then, if r denotes the distance between V " and

OV', r = d(V", OV') ,Iv

and on the other h a n d cst. ~2s-{

whence the inequality (1.12). In the same way, since ,~ ~ cst .1ol,

___ f

I<__~

& > cst.lol2

and

Iv e-(~r-~')2 d7 <_cst whence

(I.ii).

.A-"

-

10

I. Fourier-Bros-Iagolnitzer transformation and first microlocalization Modifying V if necessary, we deduce from (1.11) and (1.12) t h a t (1.9) is equivalent

to

(1.13)

dAfv e-('-o)2

<

with F = U t > l t V . One m a y always assume V of the form (1.14)

v = { T e 70; - < ITI < Z }

where 3'o is an open cone in R '~ - {0) and fl > c~ > 0. One has then An 1 dA

e -(x~'-~)2 d r = J l

A

v

_

+°° ~I{~V}(A)]

T h e last integral is uniformly b o u n d e d from above when ~ describes R ~, and uniformly b o u n d e d from below by a positive constant when ~r stays in F ' with F ~ C C / ' . It follows t h a t (1.13) (and thus (1.9)) is equivalent (after a modification of _r') to the condition (1.15)

f lal~[~(~)l ~ da < + ~ J a CF

which is equivalent to To ¢ Z~(u). T h e l e m m a is proved. P r o o f o f T h e o r e m 1.2: T h e distribution u is H s microlocally at (to, To) if and only if there exists X E C~¢(Rn), X = 1 close to to, such t h a t To ¢ Z ~ ( X u ) and thus, because of the lemma, such that there is a neighborhood V of To with

(1.16)

A"~+2s-1

Ir(xu)(x,

A)]2e -;~(I~ ~)2 d L ( x ) dA < + c ¢ .

n -iv

We just have to see that this condition is equivalent to (1.4). Assume first that (1.16) is true and let U be a neighborhood of to such t h a t X - 1 close to U. T h e integral defined as (1.16) with the integration d o m a i n R n - i V replaced by U - i V is finite. But if Re x E U, l(~Tg -- T ( X u ) ) ( x , A)I = IT((1 - X)u)(x, A)t _< Ce ~((Im z)2-e)

and so (1.4) is satisfied with W = U - i V . Suppose now that (1.4) is true. Let U be a n e i g h b o r h o o d of to, V be a neighborhood of To such that U - i V C C W and let X E C ~ ( U ) with X -z 1 close to to. Writing

1. FBI transformation with quadratic phase

11

we get, with the n o t a t i o n (1.6):

~(xu)(~ -

iv, a) = ( - A ) "

C u ( t - i~, ~ ) e ' A ' ( ~ - ~ ) 2 ( ~ ( ~ - ~)) ( 2 ~ ) "

Since 2 is rapidly decreasing, we obtain

£_, I~(~u)(x, A)I~ dL(~) < c £ I¢~(~, ~)t~ dL(~)+O(;,-~) whence (1.16). T h e o r e m 1.2 gives as a corollary a characterization of the C °° wave front set in terms of F B I transformation.Recall t h a t the point (to, r0) E T*N '~ - {0} is not in the C °° wave front set of the distribution u, W F ( u ) , if and only if there exist X E C~X~(Nn), X - 1 close to to and a conic neighborhood F of r0 in l~ ~ - {0} such t h a t for every integer N : (1.17)

sup(r)NI2~(-)l < +~. F

One has C o r o l l a r y 1.4. The point (to, To) E T*R n - {0} is not in W F ( u ) if and only if there exists a neighborhood W of to - i7o in C a such that for every N E N: (1.18)

ANITa(x, A)le-} (Ira x): < + c o . sup zEw, A>_I

Proof. T h e condition (to, TO) ¢ W F ( u ) is equivalent to the following assertion: There exists a conic n e i g h b o r h o o d F of r0 in N n - {0} and a n e i g h b o r h o o d U of to in N n such t h a t for every s E IR and every (t, r ) E g x _r, (t, 7-) ~ W F s ( u ) . On the other hand, condition (1.18) is equivalent to the existence of a n e i g h b o r h o o d W of to - i r o such that for every s E N

(1.19)

/ +o~ A~ +2s-1/w

ITu(x,/~)12 e -A(Im

Z)2 dL(x)

dA < +o0.

T h e result follows then from T h e o r e m 1.2: one has just to r e m a r k that, by inspection of its proof, one m a y choose in (1.19) a same neighborhood W for every s E IR as soon as one m a y take in (1.13) a same cone F for every s (and conversely). T h e t r a n s f o r m a t i o n u --* T u ( x , A) m a y also be used to characterize the analytic wave front set (sometimes called analytic singular s p e c t r u m or microsupport) and the Gevrey wave front set of a distribution u (in fact, it had been introduced for the first purpose in [SjD. Since we will just use this characterization, we choose to take it as a definition here. Its equivalence with the other possible definitions (using inequalities similar to (1.17) or t h r o u g h b o u n d a r y values of holomorphic functions, or t h r o u g h cohomological tools) m a y be found - in the case of the analytic singular s p e c t r u m - in [Sj], as well

12

I. Fourier-Bros-Iagolnitzer transformation and first microlocMization

as in the work of Bony [BOO] proving that there is at most one "reasonable" notion of singular spectrum. D e f i n i t i o n 1.5. i) One says that the point (to, 7"0) E T * ~ ~ - {0} is not in the analytic wave front set (or singular spectrum) of u, SS(u), if there exists a neighborhood W of to - ivo in C n and ~ > 0 such that

(1.20)

sup

Wx[1,+oo[

e-~[(Im~)~-~]lTu(x,A)l <

+co.

ii) One says that (t0,To) is not in the Cevrey-s wave front set of u, (s E ]1,+c~[), W F c , (u), if there exists a neighborhood W of to - i~'0 in C a and e > 0 such that (1.2"])

sup

e-~(Im ~)~+'"/' ITu(x, ;~)1 < + o o .

WX[1,4-oo[

We shall conclude this first section by an inversion formula, due to Lebeau, which gives an expression of a distribution u in terms of its FBI transform Tu. We follow HSrmander [H]. T h e o r e m 1.6. Let u be a compactly supported distribution on ~'~. For every t E R'* and r E ]0, 1[ set

(1.22)

u~(t)=½(27r)-"

e--~A'~-'dA

1-(w,

> Tu(t+irw,

A) dw

I=1 where D = (D1,.. • , D ~ ) and D j

--

1i

a

Oxj"

Then, for every r E ]0, 1], u~ is a real analytic function o f t , which converges in the sense of distributions towards u when r goes to 1 - . Proof. The analyticity of u~ follows from (1.2) and from the similar estimate for I ~ - T u ( x , A ) I (which is obtained applying Cauchy's formula on a polydisk with center at x, with radius of order 2)" Let ¢ E C ~ ( R n ) . By definition of T u ,

(1.23) for every r E ]0, 1[, the bracket in the right hand side standing for the duality between distributions and C ~ functions. Let

(1.24)

¢~(t) = ½(27r)-n f0+°~ e - ~ A'*-1 dA j(~l=l (1 + (w, D ) ) T ¢ ( t - i r w ,

A)dw.

Since T ¢ ( t - irw, A ) i s rapidly decreasing in A, uniformly with respect to t staying in a compact subset, w E S n - l , r E [0, 1[, ¢~(t) is locally uniformly convergent towards ¢1(r) when r --~ 1 - as well as all its derivatives. The theorem then follows from:

2. Fourier-Bros-Iagolnitzer transformations Lemma

13

1.7. For every function ¢ E C ~ ( N ' ~ ) , one has

(1.25)

= ½(2 )

L+~°

L --}(1+(w,D))T¢(t-iw,)Odw I=1

Proof. From Fourier inversion formula, we see (1.26)

¢ ( 0 ) = ~-~o+lim~ 1

JR- x~- eiS~-~lrl¢(s)dsdr"

We will deform the integration contour with respect to 7 in the complex domain. For ~r C C '~ staying in l i m a I < I R e a l , one has Recr 2 > 0 and so, one can set Is I = v / ~ where we choose the determination of the square root which is positive on the positive half-axis. Take a > 0 small enough so that alsl < 1 for every s c Supp(¢). If we p u t = ,+/asl~l, one has Re I~1 >- cst Id. Since dcq A - - . A d a n ----(l+ia(s, I@1})dT-1A...Advn, Stokes f o r m u l a applied to (1.26) allows one to replace the real integration contour in 7 by ~r = r + iasIrl, i.e.

There is a constant c > 0, independent of e, such t h a t O(isw-as21wl-clal)

>_cIw I.

By integrations by parts, and since ¢ is C a , we see t h a t in (1.27) the integral with respect to ds is rapidly decreasing in Ir[, uniformly in e. Passing to the limit when e --* 0 + we obtain

(1.28) ¢(O)----(2"rr)-nJ~"f x~" ei.....',rl(l

+ia(s,~))+(s)d$dT

"

This identity holds for every small enough real positive n u m b e r a. But since the right h a n d side of (1.28) is an holomorphic function of a in the half-plane Re a > 0, (1.28) is true for every such a. Take a = ½, apply (1.28) to ¢(t + .) and m a k e the change of variable r --= -,~w, ,~ E IR~_, w E S ~-1. One gets (1.29) ¢(t)

=(2?r)-njo+°°~"-IdAj~.l=idwi e'M'....)-('')'(i + and equality (1.25) follows.

~i (t - s,w)) ¢(s)ds

14

I. Fourier-Bros-Iagolnitzer transformation and first microlocalization

2. Fourier-Bros-Iagolnitzer transformations In the preceding section, we studied microlocal regularity of distributions in terms of a F B I transformation given by a quadratic phase.We wish now to define more general transformations, of the kind

Tgu(X, )Q -- / ei'Xg(z't)u(t) dt

(2.1) with phases

g(x, t),

(2.2)

looking like the phase of the first section

o(x, t) = i(x - t)2/2

We will first bring out the main properties g has to enjoy, so t h a t the associated transformation (2.1) shares the essential features of the one of Section 1. Let (xo,to) be a point of C" × It{" and let g : (x,t) ~ g(z,t) be an holomorphic function in a neighborhood of (xo,to) in C n x C ". We saw in the preceding section that, in the case of a quadratic phase, the microlocal regularity of u is reflected by the a s y m p t o t i c behaviour of e-X~(~)Tgu(x,/~) as A ~ +oo, where ~ is the "best weight function" such that there is N E N and C > 0 with

(2.3)

N

.

Thus, we will have

sup

(2.4)

tEIR n t c l o s e t o to

We want that, as in Section 1 where ~(x) = ½(Ira x) 2, ~2 be a s m o o t h function of x close to x0. It will be so particularly if the function t ~ - I m g ( x 0 , t), defined for t real close to to, has a non-degenerate m a x i m u m at t = to. This leads us to introduce the assumptions: (2.5)

Vt(V2( -

Img(xo,to)) = 0 Img(xo,to)) << 0

where Vt stands for the derivative with respect to real t, and Vt2 for the Hessian matrix. Under these conditions, multiplication by e i'~g(z't) localizes with respect to the space variables, i.e. (2.6)

[Tgu(x,

A)[ < Ce A(~(~)-~)

if u is a c o m p a c t l y s u p p o r t e d distribution, vanishing in a n e i g h b o r h o o d of to and x is close e n o u g h to x0. For x close to x0, we deduce from (2.5) t h a t the function (2.7)

t ---* - I m g ( x , t )

has a unique critical point in the real domain, close to to, analytic function of x and is a local m a x i m u m of (2.7).

t(x).

Moreover,

t(x)

is a real

2. Fourier-Bros-Iagolnitzer transformations

15

We shall now see t h a t e i~g(~,t) has also localizing properties with respect to the frequency variables. If u is c o m p a c t l y s u p p o r t e d in a n e i g h b o r h o o d of to on which g(xo, ") is defined, we have for x close to x0,

") dr dt Tgu(x,X)= (2~) n /J(]t-tol
(2.8)

B

Let us assume u regular enough so t h a t the integral in (2.8) is absolutely convergent. Let X E C ~ ( R n) be s u p p o r t e d in It - t01 < ~, X -- 1 on a neighborhood of ]t - to] _< ~. Let us consider the complex contour

Z = {t +iex(t)(T +

(2.9)

with ~ small enough to that has -

-

OImg .

~(x,t))(l+

g(xo, .)

]r]2)-1; I t - - t o [ < C, t r e a l }

be defined in a neighborhood of JT. If t" C Z , one

Im(g(x, [) + iT) =

O lm g O lm g ~m-t(z,t)¢x(t)(r + 0-Tm-~mt (X, t ) ) OImg .

- Img(x,t)

(2.10)

- v~x(t)(v

+

O---i-~-mt(x,t)) ( 1 + ITI2) -1

+

(1 + Ir12) -1

o(~2).

Because of Stokes formula, expression (2.8) is equal to

(2~)~ f~e~, / ~

(2.11)

On the piece of ~ where ] R e t because of (2.10) and (2.5), (2.12)

eiX(g(~'~)+~)~(AT)dtdT .

to] > ~, one has for Ix - x0] and ~ small enough,

-Im(g(x,t)

+ t r ) < ~(x) - c

for some positive constant c. On the piece of w where [ Re t - to ] < ~, we have for ~ small enough (2.13)

- Im(g(x, t) + tv) < - I m

g(x, t) - e (r +

Olmg x, t))2(1 + a--i-m-Trot(

-' +

2)

Ifl~- + aimt(Xo,to)l > C > 0 and if ~ and e are taken small enough, we see t h a t for x close to x0, (2.13) is less t h a n ~(x) - c for some c > 0. Using (2.11), we thus see that Tau(x, A) is equal to

(2.14)

(2~)"

~

+ oo~:{(zo,to)l
fl

Re[_E~,<~

e i~(~(~'t)+t~)u( AT ) dt dr

m o d u l o an exponentially decreasing remainder. It follows from (2.14) t h a t (2.1) describes u microlocally close to ( t o , - ~ ( x o , t o ) ) when the hypothesis (2.5) are fulfilled. It is then n a t u r a l to ask t h a t the m a p (2.15)

x ~ (t(x),

k

Olmg

16

I. Fourier-Bros-Iagolnitzer transformation and first microlocalization

realize an i s o m o r p h i s m from a neighborhood of x0 in C n to a n e i g h b o r h o o d of (to, To = _ O Im~ (x0,t0)) in T * R n To ensure that, it is enough to assume t h a t the differential 0Ira t of (2.15) at x0 be invertible. Taking (2.5) into account, this is equivalent by an easy calculation left to the reader to 02Im~

(2.16)

det

02Im~

/

, ~

OReO t RexIml7 (XOO t' )o2 ORetOlmx(XOb'O)lo~Im"g * ~1 # 0 . OImtc3Rex(Xo, tO) OImtOImx(XO,~O)]

We will impose hypothesis (2.5) and (2.16) to g. T h e y will allow us to show t h a t the F B I t r a n s f o r m a t i o n (2.1) enjoys similar properties to those of Section 1. First, let us give a new formulation of these hypothesis in t e r m s of holomorphic derivatives. We will denote by R 2~ the space C ° endowed with its underlying real s t r u c t u r e and we will identify T * C ~ to T * R 2~ using the i s o m o r p h i s m given on every fiber by T~• C n ~

(2.17)

T*(Re x,Im x) lt~2~

- Im

(u

for every u E T(Rex,Imx)• 2n -~ TxC n. In local coordinates, (2.17) is just (2.18)

(z; ~) ~ (Re x, Im x; - I m (, - Re ~) .

If f is a real valued C ~ function on C n, the section df of T*R 2~ is the image by (2.18) of the section 2_~Of of T*C ~ with Of = ~ Ox, o denoting the holomorphic derivative

0

(2.19)

, =

0 (bffez

i

0 ) almx

If g is an holomorphic function on C n, C a u c h y - R i e m a n n formulas show t h a t d ( - I m g ) is the image by (2.18) of the holomorphic differential Og. Formulating conditions (2.5) and (2.16) in terms of holomorphic derivatives, we set: D e f i n i t i o n 2.1. A phase of FBI in a neighborhood of a point (x0,t0) E C n x IRn is an holomorphic function (x, t) --* g(x, t) defined in a neighborhood of (x0, to) in C ~ x C" satisfying the following three conditions: i)

(2.20)

Og -5[(xo,to) = 02g

ii) I m ~ - ( x 0 ,

R~ -T0 c

t

iii) det ~ ( x 0 , t 0 ) UX(]~

- {0}

0) > 0 ~ 0.

As we saw before, conditions i) and ii) i m p l y t h a t for x close to x0 the restriction of t --* - I m g ( x , t ) to R n has a unique critical point t(x) close to to. This point is a local m a x i m u m of - I m g ( x , .) on the real domain. T h e m a p (2.15)

3. Quantized canonical transformations

17

is then, because of iii), a diffeomorphism from a neighborhood of x0 in C" onto a neighborhood of (to, r0) in T*R n. We put q0(x) = -- Im g(x, t(x)) .

(2.22)

This is a real analytic function. A direct computation, using (2.20), readily shows that is a strictly plurisubharmonic function in a neighborhood of x0 (i.e. the L6vi matrix of ~: \ ~ ] j , k The set

is positive definite for x near x0).

(2.23)

{

(t, Og

t); x, ~xx(X,t)); og (t,x)

close to (to,xo) }

is a complex analytic submanifold of T*C ~ x T*C ~ which is C-lagrangian (i.e. (2.23) is involutive and isotropic for the symplectic form ~ j d~-j A dtj + ~,j d~j A dxj). Moreover, (2.20) iii) implies that the natural projection of (2.23) on every factor of the product T*C ~ x T*C ~ is a local isomorphism. It follows that (2.23) is the graph of a complex canonical transformation (2.24)

X: (T*en,(to,ro)) ~ (T*C~,(xo, Og (xo,to)))

If one puts A~, = x(T*R"), one has (2.25)

A~, = { ( x , ~2 - ~ (x)); x E C" close to xo } .

Finally, one will remark that to, defined by (2.21), is a symplectic isomorphism when C n is endowed with the symplectic structure given by the two-form 2c50~.

3.

Quantized

canonical

transformations

In Section 1, we characterized Sobolev spaces using the FBI transformation associated to the phase i(x - t)2/2. In Section 4, we will obtain such a characterization using any FBI transformation. To do so, we will have to use a new class of transformations, whose study is the object of this section. Let us first define Sj5strand spaces, following [Sj] and

[c]:

D e f i n i t i o n 3.1. Let U be an open set in C n, ~ : U ~ N a continuous function, s a real number. One denotes by H~,(U) (resp. H~,(U), resp. N~o(U)) the space of functions

(3.1)

v : U x [1,+oo[--~ C (z, A) ~ v(z, A)

holomorphic in z, continuous in A, such that (3.2)

/ + ~ Iv e-2X~'(Z)lv(z'A)12A~-+2s-1 dL(z)dA < +oe

18

I. Fourier-Bros-Iagolnitzer transformation and first microlocalization

(resp. such t h a t there is N E 1N with

(3.3)

sup

<

zEU

X>l

resp. such t h a t there is ~ > 0 with sup ( e - ~ ( ~ " / - ~ ) l v ( z , A ) O < + ~ zEU )~>_1

If z0 E C " , one puts H"

).

= lira H$(U) (resp. H~,~ 0 = lim

H~(U), resp.

N , , , 0 --

lim N~(U)) where U ranges over the filter of all open neighborhoods of z0. z0E~

W h e n ~p - 0, the space Ho(U) or Ho,zo is called the space of symbols. One m a y define a subspace of classical symbols. D e f i n i t i o n 3.2. Let U be an open set in C ~ • T h e space of formal symbols of degree less or equal to d on U is the s p a c e Sd(v) of all formal series Ad ~+o ~ A-kak(z) whose coelCficients ak are holomorphic functions on U such t h a t there exists C > 0 (independent of k) with s u p v lakl < ck+lk! for every k. • T h e space of classical symbols of degree less or equal to d on U is the space Sd(u) of all a(z, )~) E Ho(U) such that there exists a formal symbol Ad ~ 0 + ~ A-kak(z) in Sd(U) fulfilling the following condition: 3 C > O, and VN E N, N

supA N + ] - d sup a(z, A) -- E

A>_I

zEU

ak(z)Ad--k <- cN+IN!

o

One should r e m a r k t h a t a symbol in No(U) defines a classical symbol associated with the zero formal symbol. Conversely, if a(z, A) is a classical symbol associated to the zero formal symbol, one has for every N E 1Nand every z E U, la(z, A)I < cN+IN!A -(N+O+d. Taking for N the integral part of A/C, one sees that a E No(U). Lastly, let us mention t h a t if Ad E +°° A-kak(z) is in Sd(U), there is always a classical symbol a(z, ~) in s d ( u ) associated with the given formal symbol. One has just to take 0

(3.5)

a(z,A)=Adao(z)+A d

~0e

oo

tk_ 1

e-AtEak(z)~_l)~.dt 1

with e0 small enough so that the serie converges when z E U. O u r aim is to define an operator acting on H~, with ~ real analytic, by a formula of the following kind:

(3.6)

/r

ei'~G(Y'~'e)a(y, x, ~9, A)v(x, A) dx d8

3. Quantized canonical transformations

19

where G is an holomorphic function verifying convenient assumptions, a is a classical symbol, v is an element of H~o and F is a contour to be chosen. Let us s t u d y first the case of pseudodifferential operators, i.e. the case when G(y, x, O) = (y - z ) . O. For every fixed y, the function (3.7)

(x, 0) --+ - Im[(y - x ) . 01 + ~o(x)

has a non-degenerate critical point at x = y, 0 = 72 ~ ( y). Moreover, an easy c o m p u t a tion shows t h a t this critical point is a saddle point, i.e. has signature 0. T h e quadratic p a r t of (3.7) at x = y, 0 = 72 ~_~,(y) is thus a quadratic form whose isotropic cone separates the set of points where it is positive definite from the set of points where it is negative definite:

>> 0

/

For fixed y, is chosen as non-tangent phase G, let

the u p p e r b o u n d of (3.7) when (x, 0) describes /~ will be the lowest i f / ~ on the figure, i.e. contained, except one point, in the open set << 0 and to the isotropic cone. To be able to do the same in the case of a general us set:

D e f i n i t i o n 3.3. Let ~ be a real analytic function in a n e i g h b o r h o o d of a point x0 in C n. A phase of quantized canonical transformation over H~,, o close to (yo,xo,Oo) E C" × C n × C N is an holomorphic function G ( y , x , O ) in a n e i g h b o r h o o d of (yo,xo,Oo) such t h a t the function

(3s)

(x, 0)

- Im a(u0, x, 0) + v(x)

has a non-degenerate critical point with signature 0 at (x0,00). Of course, the preceding condition implies t h a t for y close enough to y0, the function (x,O) --~ - I m G ( y , x , O ) + ~ ( x ) has a unique critical point ( x ( y ) , O ( y ) ) close to (xo,Oo). Moreover, this critical point is a saddle point and is a real analytic function of y. We will use contours of the following kind:

20

I. Fourier-Bros-Iagolnitzer transformation and first microlocalization

D e f i n i t i o n 3.4. Let w ~ f ( w ) be a Coo function in a n e i g h b o r h o o d of 0 in N q. Assume t h a t f has at w = 0 a non-degenerate critical point with signature (q+, q_). A good contour for f is a subset F of N q such that there exist a positive constant c, a ball B with center 0 in N q- and an injective immersion 7 of a n e i g h b o r h o o d of B into Nq such t h a t 3'(0) = 0, 7 ( B ) = F ( F will be endowed with the orientation coming from B t h r o u g h 7), and

(3.9)

f ( w ) - f(O) < - c l w l ~

Vm • r .

More generally, we will use the term "good contour" to designate the union of a good contour in the preceding sense and of a finite family of C °O immersed submanifolds of Nq of dimension q_, relatively compact in the open set { w; f ( w ) < f(O) }. Let us remark that the existence of good contours is clear. In fact, u n d e r the assumptions of the definition on f , Morse l e m m a shows that there is a system of local coordinates centered at 0 in which q+

f(w)=~wj 1

q 2

~

wj2 .

q++l

If c > O is small enough, F = { w • N q ; w j = O , j = l , . . . , q + , W q + + 2l + . . . + W q _ (with its n a t u r a l orientation) is a good contour for f.

2
P r o p o s i t i o n 3.5. Let (y, w) ~ f ( y , w) be a C °~ function in a neighborhood of (Yo, O) in ]Rp × N q such that w --~ f ( y o , w ) has at w = 0 a non-degenerate critical point with

signature (q+,q_). i~t r,o b~ a good contour b r f(Yo,') and let us denot~ by ~(y) the unique critical point of f ( y , .) close to O. Then there exists V a neighborhood of yo, c a positive constant and for every y • V, Zy a relatively compact immersed submanifold of N q, of dimension q_ + 1, and a good contour Fy for f ( y , .) such that aZy-(1"y o-Fy)

C {w; f ( y , w ) < f ( y , w ( y ) ) - c }

.

Proof. Let us choose a system of local coordinates w = (w', w") E Nq+ x Rq- close to 0 such t h a t f ( y o , w ) = w '2 - w ''2 (by Morse lemma). T h e tangent space of Fy 0 at 0 is then contained in ]w' I < Iw"l. So, if 6 > 0 is small enough, the Morse coordinates are defined on a neighborhood of { w; Iwl < 26 } and there exists a C °O function 7, with values in N q+, such that

(3.10)

r~0 n { w; Iw"l < ~ } = { (-r(t"),t"); t" • ~q-, It"L < c }

Over r~0 n { Iw"l > ~ }, one has f(y0,w) _< f(y0,0) - c d , whence, for y dose enough to Y0, f ( ~ , w ) _< f(y,w(~))- cd/2. On the other hand, there exists 5 > 0, independent of ~ such that (3.11)

f ( y , w(y) + (7(t"), t")) _< f ( y , w(y)) - cst 7t"i 2

if It"l < 5, lY - Y01 < 5. For e < 5, set

3. Quantized canonical transformations (3.12)

G=

e

21

[0,11, It"l _<

.

T h e conclusion of the proposition is fulfilled by Zv, Fu as soon as [y - Y0[ << ~One shows in the s a m e way that if/'10 and F~0 are two good contours for f(Yo, "), there exists Z~0 as in the proposition such that 0 Z y 0 - (Ful0 - Fu20) is contained in { w;

f ( y o , w ) < f(yo,O) - c }. We will now prove the " f u n d a m e n t a l l e m m a " of Sjhstrand [Sj]. Let us first recall t h a t a real quadratic form g on cq ~ N 2q is said of Levi type if g(iw) = g(w) for every w G C q and t h a t a real quadratic form h on C q is pluriharmonic (i.e. OOh = 0) if and only if h(iw) = - h ( w ) for every w E cq. An a r b i t r a r y real quadratic form on C q can be uniquely written as a s u m Q = g + h of a Levi form g and of a pluriharmonic form h (one has g( w ) = ½( Q( w ) + Q( iw ) ), h( w ) = ½( Q( w ) - Q( iw ) ) ). T h e form Q is plurisubharmonic (i.e. OOQ _> 0) if and only if g > 0 or, m o r e generally, if there is a pluriharmonie form h0 with g >_ h0. Let us prove then: 3.6. Let (y, w) ~ f ( y , w) be a plurisubharmonie C ~ function in a neighborhood of (Y0, w0) = (0,0) in C" x C.q. Assume that wo is a non-degenerate critical point of w --+ f(Yo, w), with signature O. For y close to Yo, call w(y) the unique critical point of w --+ f ( y , w) close to wo, and denote by @(y) the critical value ~(y) = f ( y , w(y)). Then @ is a plurisubharmonic function.

Lamina

Proof. We must see that the quadratic form \72y@(0) is plurisubharmonic. Since it depends just on the jet of f at order 2 at (0, 0), we m a y replace f by its 2-jet, i.e. assume I(Y, w) = f(0, 0) + ( V J ) ( 0 , 0)y + Q(v, w) with Q a quadratic form. T h e s u m of the first two terms is pluriharmonic. If w(y) is the critical point of w ~ Q(9, w), we must show t h a t Q(y, w(y)) is plurisubharmonic. Let us write Q(y, w) = Qo(W) + B(w, y) + QI(y) with Q0, Q1 quadratic forms and B bilinear form. By assumption, Q0 is plurisubh a r m o n i c with signature 0. Let F be a q-dimensional real subspace of G q such t h a t Q0 IF << 0. Since Q0 = g0 + h0 with go of Levi type, g0 > 0 and h0 pluriharmonic, one has Qo(iw) > ho(iw) = - h o ( w ) >_ - Q o ( w ) whence Qo[iF >> O. By a linear change of coordinates, we m a y assume F = Nq. Then, if y is close enough to y0, (3.13)

@(y) =

inf

sup Q(y, wl +iw2) •

w2ENq wl Ell~q

Since Q is plurisubharmonic, there exists a pluriharmonic form h with Q _> h. Since Q << 0 on {0} x Rq, h << 0 on {0} x R q and thus h >> 0 on {0} x iNq. This implies that w ~ h(0, w) has at w = 0 a non-degenerate critical point with signature 0. If one puts

(3.14)

@I(Y) =

inf

sup h(y, wi + iw2)

,

w2 G~q wl C~q

@I(Y) is,

for y close to 0, the critical value of w ~ h(y, w), and verifies @1 < 93, @(0) = @1(0) = 0. Since @ and @~ are quadratic forms, it follows from the results we recalled just before the s t a t e m e n t of the l e m m a that it is enough to see that @1 is pluriharmonic. But, since h is pluriharmonic, it is equal to the real part of an holomorphic function.

22

I. Fourier-Bros-Iagolnitzer transformation and first microlocalization

As a critical value of h(y, .), ~l(y) is thus also the real part of an holomorphic function, and so, is pluriharmonic. The l e m m a is proved. Let ~ be a real analytic function in a neighborhood of x0 E C '~ and let G(y,x, O) be a phase of quantized canonical transformation over H~,~ 0 in a neighborhood of (y0, x0,00) E C" x C" x C N, in the sense of Definition 3.3. Let us denote by (x(y), O(y)) (resp. ~(y)) the critical point (resp. the critical value) of (x, 0) ~ - Im G(y, x, O) + ~(x). If a is a classical symbol of order 0 at (y0, x0,00) and if F0 is a (germ of) good contour for (x, 0) --+ - Im g(yo, x, O) + ~(x) at (x0,00) in the sense of Definition 3.4, one has: T h e o r e m 3.7. Assume that y --~ x(y) is a real analytic diffeomorphism from (Cn,yo) to (C", x0). Then the operator A defined for v E H~,,xo by

Av(y,

(3.15)

a) =

fr0 ei;~c(v"'°)a(Y' x, O, ;~)v(x, A) dx dO

induces for every s E N a continuous operator from H~,,~o/N~,,xo to H~,yo/N(,,y o. Pro@ The function Av(y, ~) is holomorphic in a neighborhood of V0. If y is fixed close enough to Y0, there exists by Proposition 3.5 an n + N + 1-dimensional contour 57v, a positive constant c and an n + N-dimensional good contour Fy such that OE v -(1"o -Fv) is contained in (3.18)

{ (x,0); - I m a ( y , x , 0 ) + ~(x) _< ~(y) - c } .

Using Stokes formula we see that the integral in the right hand side of (3.15) is equal to fry + foE~-(ro-r,)" Because of (3.16), the modulus of the second term can be estimated by Ce A(~'(v)-c) in a neighborhood of y0. The relations (3.12) show that /'v may be assumed of the form (x(y), O(y)) + l"yo where Fyo is the intersection of F0 with a small neighborhood of (x0,00). If we choose a parametrization of Fv0 by a neighborhood of 0 in Nn+N t --+ (xt, Or), one has (3.17) e -)'e(v) [

eiXa(v"'°)a(y, x, O, A)v(x, A) dx dO

J Fy

Jlt I_
C(;l
C A - ( ' ~ + N ) / u e - 2 ~ ' ( ' ) l v ( x ' A)I2 dL(x)

where U is a convenient neighborhood of x0.

4. Change of FBI

23

T h e conclusion of the t h e o r e m follows from the expression (3.15) of Av and from (3.18).

4. Change of F B I In the first section, we o b t a i n e d characterizations of microlocal H ~, C ~° or analytic regularity using the t r a n s f o r m

(4.1)

Tu(x, ~) = / e-~(~-')%(t) dt.

We want now to obtain analogous characterizations using the m o r e general transformation (4.2)

T~u(x, ~) = / eixg(~'t)u(t) dt

where ~ is a phase verifying the conditions (2.20). Let us r e m a r k first t h a t if u is a c o m p a c t l y s u p p o r t e d distribution, one m a y write (4.1) as (4.3)

n

x~2

T U ( x , ~ ) = A- e - ~

~

t2

/~\

~(e-~u[-~))(ix)

where 9v denotes Fourier transform. Using Fourier inversion formula, one has (4.4)

u(t) = ~

ci~t. e~' - ' Tu(x,A)dx

where the integral has to be understood as Fourier transformation. We thus have formally: (4.5)

T~u(y,~) = ~

ei~. ei.~(y,t)+_~(t_~)2Tu(x,A)dxd t tER n

In fact, we will give a sense to the preceding integral by showing that ~(y, t) - ~(t - ~)2 is a phase of quantized canonical t r a n s f o r m a t i o n in the sense of Definition 3.3, and by c o m p u t i n g the integral over a good contour. More generally, if g is a n o t h e r phase of FBI, we will t r y to express T~u in terms of Tyu using a f o r m u l a of the form (4.6)

T~u(y,~) -= ~ " / i ei'x~(Y't)-i~g(~'t)a(y,x,t,~)Tgu(X,~)dxdt

where F is a good contour and a(y, x, t, ~) a convenient classical s y m b o l of order O. Let us prove first: 4.1. Let ~(y, t), g(x, t) be two phase~ o/ FBI, respectively defined in neighborhoods o[ (yo,to) and (xo,to) in C ~ x C" (with to real). Let y --* ~(y) = (t(y),-7(y)) and x ~ t~(x) -- (t(x),T(X)) be the isomorphisms associated to them by (2.21) and let us

Lemma

24

I. Fourier-Bros-Iagolnitzer transformation and first microlocalization

assume that to(x0) = k(y0). Let ~ and ~ be the weights associated by (2.22) to g, ~. Then, for y close to yo, the function (t, x) ~ - Im ~(y, t) + Im g(x, t) + ~(x)

(4.7)

has a unique critical poiut clo~e to (t0,x0) given by t = ~(y), x = x(y) ~f ~-l(~(y)). Moreover, this critical point is a saddle point and the critical value is ~(y). For (y,~) do~e to (y0,t0), th~ /unction (t, x) ~ - Im 9(y, t) + Im g(x, t) - I m g(x, s)

(4.8)

has a unique critical point close to (t0,x0) given by t = s, x = x(y,s) holomorphic function of (y, s). Moreover this critical point is a saddle point. Proof. Let us prove first the assertions about (4.7). T h e relations n(x(y)) = &(y), (2.12), (2.22) and an easy computation show that (t, x) = ( t ( y ) , x ( g ) ) is a critical point. T h e critical value is then 9~(y). T h e Hessian matrix of (4.7) at the critical point (t(y), x(y)) may be written (aij)l<_,,j<_4 with (4.9)

an --

02 Im(g - ~) O Re t 2

a21

tal2

02 Im(g - .~) =

a22

--

a31

=

a32

a33 a41

a42 a43

a44

--

ORetOImt

02 Im(g - g) 0Imt 2 02 I m g Ot(x) tal3 --_ 0 R e t 2 " ORex OT(x) O2 I m g

Ot(x) -ORex cgRetOImt a R e x t(Or(x) ~ ( 0 2 I m g ~ (Ot(x) = \ORex] \0Ret 2 ] \ORex] 02 I m g Or(x) = tal4 = ORet 2 " a I m x or(x) o 21rag at(x) = ta24 --- O I m x - O R e t O I m t O I m x = ta34 = t(Ot(x) ~ (02 Img'~ ( O t ( x ) \ O l m x ] k ORet 2 / k O R e x ] t(Or(x) ~ (02 I m g ~ ( O t ( x ) ---- \ O I m x / \ O R e t 2 ] \ 0 I m x / :

ta23

~--- - - - -

(using the relations obtained by differentiation of (2.20) i), (2.21), (2.22)). T h e value of the associated quadratic form over a vector of the form

(5(~ is equal to

Ot(x)

\

lOt(x)

Or(x). Imz),Rez, Imz)

. R e z + O I m ~ "Imz)'v~-d-Re-x" R e z -4- vqIm ~

4. Change of FBI 52 02 Im(g - .q) [ cgt(x)

(4.10)

Ot(x) z] 2 [ORex " R e z + O I ~ " I m Ot(x) . I m z ) [ °2 h n ( g - - g ) ] (Or(x) Or(x) . I m z) " R e z + 0Im---~ l O l m t O R e t J \ 0- R- e x - R e z + OIm-------x Or(x) . I m z ) [ 0-2 Im(g - 0).] ( O t ( x ) 0t(x) . I m z ) " R e z + 0Im-----x [cgRetOImtJ \ 0 R e x • R e z + 0 I m - ~ 0) ( O r ( x ) Or(x) 2

-ff-R-eet7

~tl" Ot(x) + 7° ~0R--eez

~tdOr(x) + 3'0 ~0--~-xex + 72 02 Im(g 0~--m-m~

\0Rex

( ~ -

7t

" R e z + I0m - - - ~ ' I m z )

Ot(x) . i m z ) [

02Img

](0r(x)

[0RT~-ez . R e z +

25(02Img~(Or(x)

Rez

\aRet 2 ] \cgRex " (0 2 Img'~ (Or(x) + \ 0R~-2-et2/ \ 0 R e x

&(x)

[OImtcgRetJ \ 0 R e x " R e z + Or(x) .Imz)[ O2Img ](Ot(x) Im~--~ 0 lORetOImtJ \O-ffe-ez . R e z +

• R e z + Im~----x 0

t/Or(x) -7

25

at(x)

~

,, fOr(x)

+ Olmx " Im~] 2 - zv~,0--R-~ex "

-OImx -

Ot(z ) -Olmx -

\

• Im

]z l

• Im z~

]

Or(x______)).i m z ) 2

Rez

+ aImx

]

Or(x) , i m z ) 2 ' R e z + Im------x 0

If one takes 5 = 1, "y > 0 small enough, one sees that (aij)lKi,j<4 is negative definite on a real linear suhspace of dimension 2n. On the other hand, taking 5 = 0, 7 < 0, ]3'1 small enough, one sees that it is positive definite on a real linear subspace of dimension 2n. T h e signature is thus zero and the other assertions of the l e m m a a b o u t (4.7) follow from that. T h e function (4.8) has a unique critical point close to (t0,x0) given by t = s, 0~ x = x(y, s), with by definition ~ ( y , s) = ~t (x(y, s), s) (see (2.20) iii)). T h e first two rows and the first two columns of the Hessian m a t r i x of (4.8) at y = Y0, x = z0, t = s = to are the same t h a n those of (4.7). T h e remaining block is equal to zero. T h e value of this quadratic form over a vector of the preceding form is given by (4.10) without its last term. If 5 = 1, 3' > 0 is small enough, one still gets a 2n-dimensional real subspace on which the Hessian m a t r i x is negative definite. If 5 < 0, V < 0, [V[ KK 6 2 KK 1, one has a 2n dimensional subspace on which it is positive definite. T h e signature is thus zero, and the second p a r t of the l e m m a follows from that. We will now prove: 4.2. Let O(y,t), g(x,t) be two FBI phases fulfilling the assumption~ of Lemma 4.1. There exists a classical symbol of order 0 a(y, x,t, A), defined in a neighborhood of (yo,xo,to), such that for every germ of good contour Fo at (xo,to) for

Theorem

(x, t) ~ - I m g(Y0, t) + I m g(x, t) + 9~(x)

(4.11)

and for every distribution u, with compact support close to to, one has: (4.12)

T~u(y,A) = A ~ / F ° ci;~(Y'O-i;~#(X'Oa(y,x,t,A)Tgu(x,A)dxdt

in H(o,yo/Ne,yo. To prove that t h e o r e m we will m a k e use of the s t a t i o n a r y phase formula we recall now. For a proof, see Sj6strand [Sj] - T h e o r e m 2.8.

26

I. Fourier-Bros-Iagolnitzer transformation and first microlocalization

T h e o r e m 4.3. Let U be an open neighborhood of 0 in C ~ and let h be an holomorphic function in U. Assume that 0 is the only critical point of h in U and that this critical point i~ non-degenerate with zero signature. Denote by i" a good contour for h. Let /~ be a differential operator in a neighbourhood of O, which, in an hoIomorphie system of i o coordinates 2. such that h(z) = h(O) + g(z 1 +..- + is equal to/~ = ~o + . . . + o~." Denote by Y the jacobian determinant of % with respect to z. Then there exist C > O, ~ > 0 such that for every bounded holomorphic function v on U, one has (4.13)

e -~h(°) f eiXh(Z)v(z) dz Jr

1

Z

o_
~

a)

I

V

+

with (4.14)

IR(A)[ < -le-~A sup Iv(z)] . U

Proof of Theorem ~.2. We are looking for a classical symbol of order O, a(y, x, t,/~) such that f

(4.15)

= ] K(y, s,

ds

with

(4.16)

K (y, 8,/~) = /~n fifo eiAg(Y't)--iAg(x't)+iAg(x'S) a(Y' x, t, /~) clx dt

Since we may always assume u compactly supported in a small neighborhood of t0, it is enough to study K ( y , s, A) for (y, s) close to (Y0, so). If F(y,~) is a good contour for (4.8), Lemma 4.1 and Proposition 3.5 imply that K(y, s, A) is equal to (4.17)

)n /_

ei'Xg(Y't)-i)~g(z't)+i'xg(z'S)a(y, x, t, A) dx dt

Jc( modulo a remainder bounded by ½e ~(Im~(y,~)+~) (¢ > 0 independent of (y,s) close to (yo,to)). Because of the proof of Proposition 3.5, we may assume that this contour depends holomorphically on (y, s) (since the critical point is an holomorphic function of (y, s)), and then, the remainder is also holomorphic in (y, s). So its derivatives are also bounded by Le-~(Im ~(Y'~)+~). By Definition 4.2, there exist holomorphic functions ak(y, x, t), C > 0 large enough, ~ > 0 small enough such that (4.18)

a(y,x,t,)~)-

E )~-kak(y'x't) <- 1-e-~ C k<_x/c

for (y, s) close to (Y0, to) and dist((x, t), F(y,~)) small enough. Modulo a remainder bounded by ~e -:~(Im ~(Y'~)+~) and holomorphic in (y, s), we may replace in (4.17) a(y,x,t, A) by the preceding development. Because of Theorem 4.3, (4.17) is equal to

4. Change of FBI (4.19)

ei~g(Y'~) E

(27r)"(l!A/)-I

o
27

A-k[l~(Y,s)]l(ak/J(y,s))(x(y's)'s)

E O
modulo an holomorphic remainder bounded by le-~(Im~(Y'S)+e). In (4.19) z~(y,8) is a differential operator in (x, t) whose coefficients depend holomorphically on (y, s), J(y,8) is an holomorphie function of (y,x,t, s) and (x(y, s), s) is the critical point of (4.8). One m a y then choose successively a0, al, a2 ... so that (4.19) be equal to e i~](y'e), modulo a remainder of the same kind than above (using that (y, s) --+ (x(y, s), s) is an holomorphie diffeomorphism). To conclude the proof of the theorem, one has just to verify that ak satisfy the estimates of Definition 3.2 for every k. This is done by an easy induction left to the reader. C o r o l l a r y 4.4. Let g(x, t) be a FBI phase at (xo, to), ~(x) the strictly plurisubharmonic weight associated to it and

Og

x0)

the isomorphism (2.21). Let u be a distribution with compact support close to to. Then (4.20)

Og

g(x0) = (to, - ~ ( x o , t o ) ]

f[ WEe(u) <===V Tgu • H;,~o .

Proof. The corollary follows from Theorem 1.2 of characterization of He-wave front set, Theorem 4.2 and Theorem 3.7, one may apply since its hypothesis is verified by the operator (4.12). The reader will easily state the analogous results for C °o or analytic wave front sets. To conclude this section, let us remark that one may use Corollary 4.4 to obtain another proof of the conical structure of the wave front set (which does not rely on its original Definition 1.1 but on the characterization (4.20)). If r > 0 is given, put

(4.21)

i g(x,t)

=

i -

t) 2 ,

O(x,t)

=

-

t)

.

The associated identifications are given respectively by g : x ~ (Re x, - I m x), k : x --* (Rex,-lImx). Then, if (t0,T0) ¢ WYe(u), one has by (4.20) Tgu(x,A) e g (Ira 8 x)2/2,zo with x0 = to - iT0. Since Ton(x , A) Tgu(x, A/r), one has T~u(x, A) e g(~imz)2/2~,~o whence by (4.20), k - l ( x 0 ) = (t0,rr0) ¢ WEe(u). One argues in the same way for W F ( u ) and SS(u).

II. S e c o n d m i c r o l o c a l i z a t i o n

This second chapter deals with the definition and the study of second microlocalization along a lagrangian submanifold of the cotangent bundle to a real analytic manifold, as it has been defined by Sj6strand [Sj] and Lebeau [L2]. T h e first section is an introduction to second microlocalization along the conormal to 0 in N '~. Starting from the notion of conormal regularity, we guess what should be, in this peculiar case, the good notion of FBI transformation of second kind allowing one to define a second wave front set resembling the one studied by Bony in [Bo2]. T h e second section is devoted to the definition of the second wave front set and of the second m i c r o s u p p o r t along a general lagrangian submanifold, following ISj] and [L2]. One first defines the notion of FBI phase of second kind and proves the existence of such objects, following [L2]. Then, one introduces the good contours naturally associated to such phases. This allows one to define FBI transformations of second kind and second wave front set along a lagrangian submanifold, still following [L2]. One should remark t h a t we present here the definition of second wave front set with growth taken from [L3], which is essentially a uniform version with respect to the small p a r a m e t e r # of the second wave front set of [Sjl , [L21. The last section gives a proof of a trace formula due to Lebeau [L3]. Given a submanifold N of R n, this formula gives an upper b o u n d for the wave front set of the restriction to N of a s m o o t h enough distribution u on R '~, in terms of the wave front set of u and of its second wave front set along the conormal bundle to N in R n.

1.

Second

microlocalization along

T{*o}R n

In this first section, we will give an heuristic introduction to the notion of second microloealization along the lagrangian submanifold A = T~0iR n of T*R n. T h e precise and rigorous definition of that notion will be given in the next section. First microlocalization (i.e. the F B I transformation presented in the first chapter) allowed us to give a m e a n i n g to assertions like: "the distribution u is s m o o t h close to to E Ii~n, in the direction TO E N n -- {0}". We would like to have a quite analogous notion, which would say "how" a distribution is singular. We reminded in the introduction the i m p o r t a n t notion of conormal regularity along a submanifold V. Let us recall its definition when V = {0}: one says that a distribution if for every integer k and for every k-tuple of s m o o t h vector u is in the space rr,,+oo ~{o} fields X1, • • •, Xk vanishing at 0, one has X1 - • • X k u E H~oc. This is equivalent to the fact that for every multiindex a = ( a l , . . . , a,,)

1. Second microlocalization along T(*0}l~~

29

+l~l ' t"U ---~t71 ' ' ' t n ~ " t t E H Sloc

(1.1)

O n e says t h a t t h e d i s t r i b u t i o n u is c o n o r m a l a l o n g {0} if t h e r e is a n s E R such t h a t A e o n o r m a l d i s t r i b u t i o n at 0 is t h u s s m o o t h over l~ n - {0} a n d h a s at 0 a u E ,/#-s,+ , {0} oo .,. s i n g u l a r i t y of a special kind. T h e space of c o n o r m a l d i s t r i b u t i o n s will p l a y w i t h r e s p e c t to second m i c r o l o c a l i z a t i o n t h e s a m e role t h a n t h e space of C °° f u n c t i o n s w i t h r e s p e c t to t h e first one: a d i s t r i b u t i o n will b e 2 - m i c r o l o c a l l y r e g u l a r if a n d o n l y if it is a c o n o r m a l distribution. As we gave a m e a n i n g to t h e n o t i o n of m i c r o l o c a l r e g u l a r i t y of a d i s t r i b u t i o n u in a cone of t h e p h a s e space, one m a y give a m e a n i n g to t h e n o t i o n of c o n o r m a l r e g u l a r i t y of u in a d o m a i n of the f o r m F A V where V is an o p e n n e i g h b o r h o o d of 0 a n d F an o p e n cone in N '~ w i t h v e r t e x at 0: if s E N, we will say t h a t u is in r_p,+oo ( 2 - m i c r o l o c a l l y ) in "'{o} F n V if for every m u l t i i n d e x 7 w i t h [71 -< s a n d for every f a m i l y X1, • . . , Xk of s m o o t h v e c t o r fields v a n i s h i n g at 0, (1.2)

X l .. . X~(O~u) ~ L 2 ( F n V ) .

O n e s h o u l d notice t h a t t h e cone _P a b o v e is a cone of the b a s e s p a c e N n a n d not of the p h a s e space. It is t h e n p o s s i b l e to define t h e n o t i o n of c o n o r m a l r e g u l a r i t y close to a d i r e c t i o n 6t o E N ~ - {0}: t h e d i s t r i b u t i o n u will be said c o n o r m a l in t h e d i r e c t i o n 5t o if t h e r e exist an o p e n conic n e i g h b o r h o o d F of 5t ° in I~'~ - {0} a n d an o p e n n e i g h b o r h o o d V of 0 such t h a t (1.2) is satisfied for every f a m i l y X1, . . . , Xk. We s h o u l d like to give an equivalent definition of t h a t n o t i o n using t r a n s f o r m a t i o n s looking like t h e F B I t r a n s f o r m a t i o n of C h a p t e r 1. If f o r y E C n w i t h R e y # 0 a n d f o r M C R * + we set T u ( y , A ' ) = j e[

(1.3)

- v~' (y - ' t ,2 u(t) dt

we saw t h a t t h e a s y m p t o t i c b e h a v i o u r of T u ( y , M) w h e n M --+ + o c allows one to s t u d y the d i s t r i b u t i o n u in a n e i g h b o r h o o d It - Re Yl < e of Re y. If # is a real p o s i t i v e n u m b e r a n d if we p u t (1.4)

T~,u(y,A') =

f

e_~_(y_ , 2~(t)dt, ~' 7)

T ~ u ( y , A') enables us to s t u d y u in an o p e n set of the form ] R e y - ~ ] < c. If # varies in a n i n t e r v a l ]0, a[, a > 0 fixed, a n d y is close to a p o i n t y0 w i t h Rey0 ~ 0, t h e fmnily i n d e x e d b y # (T~,(y, M)) u controls the r e g u l a r i t y of u in a d o m a i n (1.5)

A, =

t; 3 # C ] 0 , a [ a n d

~-Rey0

<¢

w i t h e > 0 fixed, i.e. in a d o m a i n of the form F , N V, w i t h _P, conic n e i g h b o r h o o d of Y0 in R ~ - {0} a n d V, n e i g h b o r h o o d of 0. More precisely, if for i n s t a n c e , t h e r e is ¢ > 0 such t h a t S u p p ( u ) Cl A , = 0, t h e i n t e g r a l (1.4) has a n u p p e r b o u n d :

(1.6)

IT.u(y,

-< C . - m e ~ ( I r n Y):~--)~'~

30

II. Second microlocalization

for y close to Y0, # E ]0,a[, M >_ 1, with C positive constant and rn real n u m b e r depending just on the order of the distribution u. We will show t h a t under the a s s u m p t i o n (1.2), the function (1.4) is rapidly decreasing with respect to M for y close to a convenient point y0. First of all, let us express the t r a n s f o r m a t i o n (1.4) using an usual F B I t r a n s f o r m a t i o n (1.7)

~

Tu(x,A) =

2

e - ~ (~-t) u ( t ) d t .

Let us put (1.8)

T2u(y,A,#)=

c 2(1-~,)

~

ru(ix, A)dx.

P u t t i n g (1.7) into t h a t f o r m u l a and c o m p u t i n g the i n t e r m e d i a t e integral, we get (1.9)

T 2 u ( y , A, #) =

e /1

-

#2x~/2

-~'2

-~-b~Ayt

[Jx

u(~) dt

..

_~_eAy~

.

.

Let us show now: P r o p o s i t i o n 1.1. Let 5t ° C l~ ~ - {0} and let u be a compactly supported f u n c t i o n of L2(R '~) such that there exist a neighborhood of O, V and an open conic neighborhood of 5t ° in R '~ - {0}, F such that f o r every f a m i l y of C a vector fields X 1 , . . . , X k vanishing at O, one has

(1.10)

XI "" Xku E Z2(V M V) .

T h e n there is c > 0, a E ]0, 1[, ~ E R and f o r every N E N, CN > 0 such that

IT2u(y, A, ~)1 <

(1.11) for

every

(~, ~, ~)

satisfying

~1 _< [ R e y

CNA~(A,~)

I _< 2 , [ h n y

-Ne

~ (Ira y)

-- St°liSt°If

< E , # E ] 0 , a[, A# 2 > 1.

P r o o f . Let X be a C °° function on the unit sphere S n-1 of R n, with s u p p o r t contained in F M S " - ] , such t h a t X --- 1 on a neighborhood of St°/[St°[. Let us write

(1.12)

T 2 u ( y , A , # ) = T 2 [x(t/[t])u](y, A,#) + T ~ [(1 - x ( t / [ t [ ) ) u ] ( y , A , , ) .

For ~ small enough, I m y is far from Supp(1 - X) and so the second t e r m of (1.12) m a y be e s t i m a t e d by the right h a n d side of (1.6). To s t u d y the first t e r m , p u t

(1.13)

w(y, A, ;') = / e-~t'+i~'~((t/]t])u(t)

dt .

Let us r e m a r k first t h a t if v is a c o m p a c t l y s u p p o r t e d distribution belonging to H s (s non-negative integer), one has (1.14)

I T v ( x , A)I <_ C A - } - % }

(I~)~

2. Second microlocalization along a lagrangian submanifold

31

when ] I m xJ stays between two positive constants. In fact (1.14) is trivial if s = 0. Let us prove it for general s by induction. If we assume (1.14) for s - 1, let us r e m a r k (1.15)

0 n 1 ~ 0 (Ov~ -o~Tv(x,A) = - ~ - ~ T v ( x , A ) - ~ oxjTkotj](x,A)

.

j=l

Using C a u c h y ' s formula on a polydisk of radius equivalent to $, 1 we see t h a t the induction hypothesis implies J ° T v ( x ,

(1.16)

A) J _< CA-½-~e-}0m ~)2 if v E H ~. Let us write

Tv(x,

: Tv(x, 1) +

Tv( ,

By w h a t we just saw, the p r o d u c t of the second t e r m of the right h a n d side by e--} (ira x)2 has m o d u l u s b o u n d e d from above by A e r-@(Im

(1.17)

2

1

1 s

~) r - ~ -~ dr < C A - ~ -

.

Moreover ITv(z, 1)J < Ce½ (~m x)2 _< C , A-~1 _ 2 e } ( i m z)~ for A >-- 1 if C ' is large enough. We thus get (1.14). To obtain (1.11) we argue in a similar way. Using the a s s u m p t i o n (1.10) and combining (1.8) and (1.14) applied to t~u for any multiindex a , we see t h a t (1.18)

JT2(x(t/ltJ)t~u)(u,/~,~#)1

~ C / \ - ½ - J a ' ] e ~2-'~(Imy)2 •

Using this inequality for a = 0 and (1.9), we get

(1.19)

iw(y '/~,/~,r)j <. c/vn/2~_½e@(lmy)2

(setting A' = A#2). We m u s t show that we m a y replace in the right h a n d side of (1.19) M n/2 by A' - g for every integer N (with a constant C depending on N ) . This follows from an easy induction using that, because of (1.13) and (1.18), for every k E N (1.20)

0k -o-~w(y,/~,/~') < ek.~'n/2-k~ - ½e ~ ( I m

y)2

This concludes the p r o o f of the proposition.

2. S e c o n d

microlocalization

Let N be a submanifold such that N is given by N in N", A = { (t', 0; 0, C n which is the inverse z --~ ( R e x ; - I m x): (2.1)

along a lagrangian

submanifold

of N n and let us choose local coordinates on IR~ t - (t',t") N = {t" = 0}. Let us denote by A the c o n o r m a l bundle to T " ) } . We will denote by L the real analytic submanifotd of image of A by the isomorphism from C" to T * N " given by

L = { x E C"; I m x ' -- 0, R e x " = 0 } .

32

II. Second microlocalization

If u is a c o m p a c t l y s u p p o r t e d distribution, let us denote by Tu(x, A) = f e- ~-(~-t)2u(t) dt. We will associate to A a F B I t r a n s f o r m a t i o n of second kind, generMizing the one defined in Section 1 when N = {0}, setting

(2.2)

~I

T2tt(Y'/~'#)

=

Xt~2

e - 2(1-~'2)

mx'=0

t

t 2

It

• tt 2

A

[(u-= ) +(~ +,= ) ]+~-=

tl2

Tu(x,A)dx

Re x " = 0

Let us denote the phase of (2.2) by

(2.3)

a ( ~ , x, ~) -

i'~ [(y' - z')~ + (y" + ix") ~-] - i ~''~ 2(1 - #2) 2

and put (2.4)

f(y, x, #) = - I m G(y, x, #) + ½(Ira z) 2 .

T h e n f(y,x,O) = ½ ( I m x ' ) 2 + ½(Rex") 2 vanishes at order 1 oil L and its transverse hessian along L is positive definite. Moreover, the term in #2 in (2.3) appears to be a cut-off, b o t h with respect to space and phase variables, as a phase of F B I transformation. In particular, the restriction to L of the coefficient of #2 in the a s y m p t o t i c development of (2.4) at # = 0, has a non-degenerate m a x i m u m at a unique point of L. To define F B I transformations of second kind, like (2.2), but associated to lagrangian submanifolds which are no longer necessarily a conormal, we will have to introduce a class of phases, more general than (2.3), but still enjoying the features we displayed above for (2.3)-(2.4). Let (to,To) be a point in T*R '~ - {0} and let g(x,t) be a F B I phase defined in a n e i g h b o r h o o d of the point (x0, t0) in C '~ × C '~, such that the image of x0 by the associated diffeomorphism n be equal to (to, T0). Let ~(x) be the strictly plurisubharmonic function in a n e i g h b o r h o o d of x0 which is the critical value of t -~ - I m g ( x , t). We saw t h a t is a symplectic diffeomorphism when we endow C" with the symplectic structure given by the symplectic form ~c50~. Let A be a germ at (to, To) of real analytic lagrangian submasfifold of T * R '* and set (2.5)

L = ~;-'(A) .

It is a germ at x0 of real analytic lagrangian submanifold of C n. Let Y0 be a point in C n and

(2.6)

(y, x, ~) ~ a(y, x,

#)

an holomorphic function of (y, x) in a n e i g h b o r h o o d of (y0, xo) in C n × C n, real analytic in # C [0,#0[. Let us put +oo

(2.7)

f(x, y, #) d¢2 -- Im G(y, x, #) + ~(x) d¢=fE #kfk(Y' x) . k=0

We shall assume the following conditions:

2. Second microlocalization along a lagrangian submanifold (2.8)

33

i) f01c- ×L = 0, (df0)[c-×L = 0, x ---* f0(Y0, x) has a positive definite transverse Hessian along L. ii) fl (Y, x) ~ O. iii) T h e function f2(Yo,') restricted to L has at x0 a non-degenerate critical point with a negative definite Hessian.

We will have then: 2.1. Under the preceding assumptions, for (y, #) close enough to (Yo, 0), t~ ~ O, the function defined in a neighborhood of xo in C n

Lemma

x --~ f(y, x, #)

(2.9)

has a unique critical point x(y, #) satisfying the following conditions: •

(~, ,) --~ ~(y,,)

• For e~ery y , ,

is a real analytic map,

--* ~ ( y , , )

is a ~ur~e issued from a point :~(y, O) ~ L and ~(yo, O) = xo,

• d(x(y, #), L) = 0(#2), # --* O+ (where d(., .) is the hermitian distance on Cn). Moreover this critical point is a saddle point. Let us prove first: 2.2. Let L be a real analytic submanifold of C'*, tagrangian for the form ~ 0 ~ . Then L is a totally real submanifold of C ~, i.e. T L • (iTL) = T C ~.

Lemma

Proof. One has (2.10)

2c50~z = i-2 E

02kOz------~02~dSk A dzj .

j,k Let us consider the i s o m o r p h i s m from the tangent space to R n to the complex tangent space of C '~ given by "

(2.11)

0

0

~ ~(aj+

0 ibj) Oxj

- + bj 0 I m xj j~: l aj 0 Re xj

~7

U

T h e action of (2.10) over a couple (U, 1~) E T R ~'~ x T ~ 2~ is equal to (2.12)

2. ((AV, U) - (AU, 12)) = - 4 Im(AU, V) $

where (.,-) denotes scalar p r o d u c t on C n x C ' , {., .} h e r m i t i a n p r o d u c t on C '~ x C • and A is the positive definite m a t r i x k ~ J " Because of (2.11) we m a y consider T L as a subspace of T C '~. Let U be a vector of T L such t h a t iU C TL. Since L is lagrangian, the action of (2.10) over the couple (V, iU) must then give 0. So using (2.12) we m u s t have (AU, U) = 0, whence U = 0

34

II. Second microlocMization

since A is p o s i t i v e definite. T h e n TL N (iTL) = 0 which implies t h e r e s u l t since L is of d i m e n s i o n n. If L is a real a n a l y t i c s u b m a n i f o l d of C n which is t o t a l l y real a n d if x0 • L, t h e r e is a h o l o m o r p h i c c h a n g e of c o o r d i n a t e s in a n e i g h b o r h o o d of x0 such t h a t L is t r a n s f o r m e d into t h e s u b m a n i f o l d I m x = 0. To see t h a t , let us first r e m a r k t h a t w h e n L is linear a n d x0 = 0, one j u s t has to t a k e the d i f f e o m o r p h i s m Re x + i I m x ~ M Re x + i M I m x w h e r e M is a linear i s o m o r p h i s m from IRn over L. In general, we m a y t h u s a s s u m e t h a t ToL = N n a n d so t h a t t h e r e is a real a n a l y t i c f u n c t i o n h in a n e i g h b o r h o o d of 0 in N", satisfying h(0) = 0, h'(0) = 0 a n d such t h a t L = { t + ih(t); t • R ~, t close to 0 } close to 0. T h e n , close to x = O, x ---+x + ih(x) is a h o l o m o r p h i c d i f f e o m o r p h i s m whose inverse fulfills o u r r e q u i r e m e n t s . Let us give now the p r o o f of L e m m a 2.1:

Proof of Lemma 2.1. B e c a u s e of L e m m a 2.2 a n d of t h e p r e c e d i n g r e m a r k , we m a y p e r f o r m a h o l o m o r p h i e change of c o o r d i n a t e s such t h a t x0 -- 0, L = { I m x = 0 }. Let us set (2.13)

f(y,z,u) = ~f(y, Rez + i#hnz,#) 1 = ~ f o ( y , Re z + i p Im z) + f2(Y, Re z + i # I m z) + 0 ( # ) .

T h e H e s s i a n m a t r i x of f(Yo,', 0) at z = 0 is equal to (2.14)

~(yo,

0)

0

0 02 o

(y0,0)

(see (2.S)i)) and so, because of (2.S)i) anti iii), is non-degenerate with signature 0. For (y, #) close e n o u g h to (0, 0), z ---+ f ( y , z, #) has thus a u n i q u e critical p o i n t close to 0, z(y, t.t). T h i s critical p o i n t is a s a d d l e p o i n t a n d (y, #) ----,z(y, #) is a real a n a l y t i c m a p . F o r # = 0, we have (2.15)

1 02fo f(y, z, 0) -- $ 0 I m x 2 (y' Re z ) ( I m z) 2 + f2(Y, Re z) .

By u n i q u e n e s s of t h e critical p o i n t , we t h u s see t h a t t h e critical p o i n t z(y, 0) of (2.13) w h e n # = 0 is j u s t t h e u n i q u e p o i n t in IR'~ such t h a t 0 R°h (y,z(y,O)) = 0. If we p u t e:c x(y,#) = R e z ( y , # ) + i # I m z ( y , # ) , we have I m x ( y , # ) = 0 ( , 2 ) . T h e l e m m a is proved. In t h e s y s t e m of c o o r d i n a t e s used above, in which L is given b y I m x = 0, it is very e a s y to w r i t e an a s y m p t o t i c e x p a n s i o n of x(g, p) at # = 0. O n e has in fact (2.16)

o/

1 Ofo

Oi--~mx(Y,Z,#) - # O~mx(Y, R e z + i # I m z )

+ #~(y,

Re z + i# Im z) + 0(# 2)

olmx

whence I m z ( y , # ) . . . .

{~-a

t*\Olmx27

oi2 2 : \(,~ ' z(y,O)) + 0 ( # 2 ) . W e t h u s get:

OIm

2. Second microlocalization along a lagrangian submanifold

Imx(y,#)=--#2(

(2.17)

02f°

~-'

35

Of 2

0 mx (v'x(v'°)) +

O/2 0Re

(y,x(u,0)) = 0.

We will set (2.18)

e ( y , #)

=

~ f ( y , x(y, #), #)

$

Because of (2.13), ~ ( y , # ) is real analytic in ( y , # ) for ( y , # ) close to (y0,0), # > 0. Moreover, applying L e m m a 3.6 of C h a p t e r I for fixed positive #, we see t h a t y ~ k~(y, #) is p l u r i s u b h a r m o n i c for every # > 0. Let us define (2.19)

¢ ( y ) = k0(y, 0) .

It is a p l u r i s u b h a r m o n i c real analytic function (as the limit of the p l u r i s u b h a r m o n i c functions ~(y, #) when # goes to 0). R e m i n d t h a t we endowed C n with the symplectic s t r u c t u r e coming from the sympleetic form w = ~c~0q0. Since L is lagrangian, we m a y use the h a m i l t o n i a n i s o m o r p h i s m to identify TLX and T'L: (2.20)

TLX --* T*L V

~

ivco

where ivw is the linear form on the fibres of TL associating to u E T L the scalar co(v, u). If Y is a n o t h e r copy of C '~ in a neighborhood of Y0, we shall define a m a p (2.21)

A : y --. T*L

in the following way. If y C Y, we write (2.22)

x(y, #) = ao(y) + #el(y) + #2az(y) -q- O(# a)

# ---+0 ,

we associate to y the class of the vector (x(y,0); a2(y)) in TLX ~-- T X I L / T L and we take the image of this last object by (2.20) to get A(y) E T*L. Such a m a p is well defined since, if H is a holomorphic diffeomorphism in a n e i g h b o r h o o d of x0, one has (2.23) H(x(y, #)) = H(ao(y)) + #H'(ao(y)). a,(y) 1 tt q- #2[H'(ao(y)) " a2(y) -l- gH (ao(y)) " (al(y),a,(y))] q-O(# 3 )

and g"(ao(y))(aa(y)),al(y)) e T H ( L ) because (2.17) shows t h a t al(y) e TL. This m a p A will play for second microlocalization the s a m e role t h a t the i s o m o r p h i s m ~ defined in I-(2.21) with respect to first microlocalization. Of course, such a thing is possible only if A is an isomorphism. T h e next l e m m a gives a necessary and sufficient condition to ensure that. Lemma (2.24)

2.3. Let & be the natural 3ymplectic form on T*L. We have: A'c5 = 2.0a¢ .

Thus, A is a (local) isomorphism if and only if ¢ is strictly pluri3ubharmonic.

36

II. Second microlocalization

Proof. Let us choose a holomorphic system of coordinates on C n such that L is given a ~j, ~ = ~ vjo I ma x j by I m x = 0. Let us denote by w the 2-form 2 a 0 ~ . If fi = ~ u j O--~-~-e and if we set u = ~

uj-~7~s, v = ~ ivj

, we have, because of (2.12)

w(O, ~) : - 4 Im(Av, u)

(2.25)

where A = (\ a-A3-~-h O~:kOzj J" Since fo(x)- ~(x)= -ImG(y,x,O)is pluriharmolriC, }0O~ = {Oafo and (2.25) shows that the isomorphism (2.20) is given by

( a2fo "~,

(2.26)

~-\O~mx2]

"

Using (2.17), we see that

a f2 ,

= (xIv,0), 0 mx Y, xIy, 0)))

(2.27)

Of 2 a i~e~(y, x(y, o)) : o . Then (2.28)

A*& = ~k,l

(~ j

02f2 Oxj(g,O)) dReyk A dReyl OlmxjOReyk OReyl

O2f2 . Oxj(y,O_)~ +~(~-~cOImxjOlmyk Ohnyi ]dlmykAdlmyl k,l

02f2 + ~(~-~g3lmxjaReyk k,l

j

Oxj(y, O)

02f2

a lmyt

0hnxj0Imyt

02~b = ~ OImykOReyl

(dReyk A dReyl

j

Oxj(y, O) 0Reyk )dReykAdlmyt.

On the other hand (2.29)

200~b*

+ dlmyk

A dlmyl)

k,l

02¢

+

-~k3(OReykOReyt Using that ¢(y) = f2(Y, x(y, 0)) and that two preceding expressions are equal.

02¢

OlmykOImy,) dlmykAdRey'" f2(y, x)

is pluriharmonic, one sees that the

Let us now give the following definition. D e f i n i t i o n 2.4. Let ~ be a real analytic strictly plurisubharmonic function in a neighborhood of x0 C X = C n. Let L be a germ at a0 of real analytic submanifold of X, which is lagrangian for the symplectic form 2c30~v. A phase of FBI of second kind along L over H~ at (yo,xo) E C n × C n will be a function G(y, x, #), holomorphic with respect to (y, x) close to (y0, x0), real analytic in # varying in an interval [0, #0[, such that f(y, x, #) -- - I m G(y, x, #) + ~(x) satisfies

2. Second microlocalization along a lagrangian submanifold

37

the conditions (2.8)i), ii), iii) and such that the critical value ¢ ( y ) defined by (2.19) is strictly plurisubharmonic in a neighborhood of y0. It is easy to see t h a t for any couple (~, L) fulfilling the preceding conditions, one m a y find a phase of F B I of second kind along L over H~. Let us denote by x the generic point of X = C n and by x ---+:~ = M(x) an holomorphic change of coordinates defined in a n e i g h b o r h o o d of x0, such that M(xo) = 0 and M(L) is given by I m ~ = 0. Let us show t h a t thcre is a holomorphic function gL dcfined close to 0 such t h a t qOL = - - Im gL satisfies

(2.30)

0~

~ ( ~ ) = ~(~) (~)_ o~

0Im~

0IMP"

"

for every point ~ • [Rn. In fact, since L is isotropic for 230qo, (2.12) shows t h a t we must have

Iml( °=~° "~U V} = 0 for every couple of tangent \ \ O~kO~'j ] ' aj~-f~,V=~a b~ao~swith aj, bj real numbers. By (2.31)

ORe.~kOIm~j

= 0Im~kcqReh:j

vectors

(U,V)

of the form U =

an easy c o m p u t a t i o n , this means

if ~ C IR'~, for every

j, k.

One m a y find a real analytic function gl defined in a n e i g h b o r h o o d of 0 in ~'~ such that (2.32)

Ogl

It is then enough to take for Let us now set

(2.33)

099

--(2)0 Re &j

oqIm ~j

gL a

(a~) ;

j=l,...,n,

~ • R n.

holomorphic extension of gl - iqD.

i# 2 c(~,x,.)

= 7-~y

- M(x)) 2 - ~L(z)

We have

f(y,x,p) = fo(x) + #2fu(y,x) with (2.34)

f0(x) = ~(x) - ~L(x) f2(y, *) : _ 1 Re(y - M(x)) 2

Because of (2.30), the conditions (2.35)

fOIL, (dfo)lL =

02f0

1

0 are fulfilled. One has

02fo

02kO3sj = a(Olmh;kOIm~j) and since f0 is strictly plurisubharmonic, 02fo(~)/OIm~ 2 >> 0 if ~ E R n. Condition (2.8)i) is thus realized. Moreover, condition (2.8)iii) is trivial, and the critical value ¢ ( y ) is here equal to } ( I m y ) 2 and so is strictly plurisubharmonic. We conclude that the phase (2.33) is a phase of F B I of second kind along L over H~o in the sense of Definition 2.4.

38

II. Second microlocalization

Of course, any phase which differs from (2.33) by a t e r m which is O(# 3) when # --* 0 + is also a F B I phase of second kind along L over H~,. In particular, we will make use in C h a p t e r 3 of the phase (2.33)'

(y a(y,

f,) -

2(1

M(x))

-

•

-

Since x ~ f(y, x,#) has a saddle point at x(y,#), one m a y define a notion of "good contour". D e f i n i t i o n 2.5. A good contour for f(y,., .) is a real analytic m a p (t, #) -* 7(t, #) defined for ( t , # ) E B x [0, #1], where B is a bali of center 0 in N '~ and #1 a positive number, with values in C n, such that: • V~ C 1 0 , ~ l , t --, 7(t, ~) is an injective immersion such that 7 ( 0 , # ) = x(y,#). • 3C>0andV(t,#)EBx[0,#l[ (2.36)

f(y,7(t,#),#) <_ f(y,x(y,#), #) - Clt[2 # 2

• T h e r e exist a ball B ~ C B, #2 E ]0,#1], a holomorphic diffeomorphism H defined on a neighborhood of [-Jvel0,v~] 7 ( B ' , #) sending L onto H(L) = R n and a m a p

(2.37)

--.

B ' x [0,~2[--'

C~

which, for every # C [0, #2], is an injective immersion with respect to t, such that 7(t, #) = H -1 (Re ~(t, #) + i t Ira'S(t, #)). T h e fact t h a t good contours exist m a y be proved as in the case of good contours for F B I phases (see Section 3 of C h a p t e r I): if we choose coordinates such t h a t L = { I m x = 0 }, we m a y define the function f(v, z, #) given by (2.13). Using Morse lemma, it is then possible to find a real analytic change of coordinates z ~ w, depending on the p a r a m e t e r s (y, #), such t h a t f(y, z, #) = f(y, z(y, #), #) + Q(w) where Q is a quadratic form. An n dimensional plane of the w-space on which Q is negative definite gives then a good contour for f(y,., .). One m a y also prove a result similar to Proposition 3.5 of C h a p t e r I: let /'vo : ( t , # ) ~ 7y0(t,#) be a good contour for f(yo,','). Then, there exist a neighborhood V of y0, a positive constant c and for every y C V a good contour Fy for f(y,., .) such t h a t the following holds: • for every y E Y, there is a real analytic m a p (2.38)

~ v : B x [0, 1] x [0, #1]

-~

cn

(t, which is, for every fixed # E ]0,#1], the restriction to B x [0, 1] of an injective immersion, and which is such that, for every # E ]0, #1 ],/'~0 - F ~ - O Z ~ ' is contained in

2. Second microlocMization along a lagrangian submanifold

39

{ x; f(y, x, #) <_I(Y, x(y, #), #) - c# 2 }

(2.39)

( F 2 ' -/'~0' Zy~ being the contours at fixed #). Let us give the following definition: D e f i n i t i o n 2.6. Let U be an open neighborhood of a point y0 in C '~ and (2.40)

(y, #) -* k~(y, #)

be a continuous function defined for y E U, # close to 0 in ~ + . We will denote by H$(U) (resp. N$(U)) the space of functions (2.41)

(y, ~ , , ) -~ ~(y, A, ~)

holomorphic in y E U, continuous in A >_ 1, # close to 0 in R+, such that there is e > 0, M E R with (2.42)

sup

e-~"2e(Y,U)lw(y,A,#)])~ -M < +oo

yEU

~e]0,~], )~t,2>_1 (resp. (2.43)

sup yEU

t,c]0,e], )~t,2_>1

One defines as in the case of first microlocalization the spaces of germs H~,,~ 0 2 and 2 " One should notice that the space N~, depends only on ¢(y) = k~(y, 0). Nc~,yo To define the second microsupport, we will use the notion of FBI transformation of second kind:

D e f i n i t i o n 2.7. Let c~ be a germ at x0 E C'* of strictly plurisubharmonic real analytic function, L be a germ at x0 of real analytic submanifold of C '~, lagrangian for ~(~0~, and let G(y, x, #) be a FBI phase of second kind along L over H~, close to (yo,xo,O) e C '~ × C n × N+. Let k~(y, #) be the critical value given by (2.18). The transformation of FBI of second kind associated to G is the operator from H~,,xo/N~,,xo into 2 2 H~,yo/N~,y ° given by (2.44)

T~v(y, ~, #) = / r

eiAa(Y'~'~)v(x' ;~) dx ~o

where Fy 0 is a good contour for - Im G(y0,., .) + ~ in the sense of Definition 2.5. The fact that the preceding definition does not depend on the choice of the good contour Fv0 and that T 2 acts on the indicated spaces is a consequence of the remarks following Definition 2.5. We are now in position to give the definition of second wave front set and of second microsupport.

40

II. Second microlocalization

Let A be a germ at P0 = (to, r0) of lagrangian real analytic submanifold of T*[R n. Let us choose a F B I phase g(x,t) in a neighborhood of (xo,to) and let ~p be the associated strictly plurisubharmonic weight. Let ~ be the isomorphism associated to these d a t a by formula (2.21) of C h a p t e r I. Then, L = x - l ( A ) is a germ at x0 E (7~ of submanifold, lagrangian for 200~0. We will denote by ~ : T*L ~ T*A

(2.45)

the isomorphism induced by ~. Let u be a distribution defined on a n e i g h b o r h o o d of to. We have: D e f i n i t i o n 2.8. One says that the point (P0,P~) C T*A is not in the second wave front set (resp. the second microsupport) with growth of u along A if there exists: - a F B I phase of second kind along L over H~ G(y, x, #), defined close to a point (y0,x0) E C n x C" with y0 = / 1 - 1 0 ~--l(p0,p~), -- a neighborhood V of y0 in C n, - two constants M C JR, s > 0, such that for every N E N, one has (2.46)

sup

e -xÈ2~(u'u) [T~Tgu(y, )~, # ) [ A - M ( ~ # 2 ) N < + o o

yEV

uE]0,~], ~u=_>l 2 ). We will denote this set by (resp. if there exists G as before such that T~Tgu E N¢,v0

WV~'a(u)

(resp. SS~a(u)).

It is a closed subset of T*A.

In fact, as for the wave front set, condition (2.46) is satisfied by any transformation associated to a phase G along L over H~ as soon as it is satisfied by one of them. In the same way, the definition is independent of the choice of the phase g of the first microlocalization. The proof is quite similar to the one of T h e o r e m 4.2 of C h a p t e r I, but is m u c h more technical and will be a d m i t t e d (cf. [L2]). We will deduce from these properties the following result about the conical structure of the second wave front set. 2.9. i) The sets W F I ' 2 ( u ) and 582'1(72) are conic subsets of T*A (i.e. are invariant under the dilatations (p,p*) ---+(p, rp*), r > 0).

Proposition

ii) Assume that A is conic in T*R n and for every r > 0 let mr : A --+ A be the restriction to A of the map of multiplication by r on the fibers of T*R n. Denote by ~ : T*A --+ T*A the map it induces on the cotangent bundle. Then WF~'a(u) (resp. SS~'2(u)) is invariant under the action of mr, r > O.

Proof. Let us choose a F B I phase g(x, t) allowing one to characterize the wave front set microlocally d o s e to P0 = (to; w0) E A, and let x be the isomorphism associated to it by formula (2.21) of C h a p t e r I. Let us choose a F B I phase of second kind G(y, x, #), characterizing the second wave front set close to (Po,P~) C T*A and let A be the isomorphism (2.21). If (po,p~) • W F ~ ' I ( u ) , then T~Tou satisfies (2.46) close to the point y0 = /1-1 o g-a(po,p~ ).

3. Trace theorems

41

Put a , ( y , x , # ) = a(y,x,#v/-~ ). Then Ta.Tgu satisfies (2.46) close to Y0 and since the associated isomorphism k o A N is the composition of ~ o A and of the m a p (p,p*) --* (p, rp*) on T ' A , we see that (P0, rp~)) ¢ WF2'l(u) whence i). When A is conic, set g~(x,t) = rg(x,t). The associated identification x~ is the composition of n = nl and of rn~. On the other hand, put G~(y, x, #) = rG(y, x, #/v/~). The associated identification ,5 is the same than for r = 1: in fact, with the notations (2.20)-(2.21), it is the composition of two arrows Y --* T L X --* T ' L , the first one being modified by a factor 1_ on the fibers and the second by a factor r, since the sympleetic 7" form one has to consider on X is rO0p, if ~ is the strictly plurisubharmonie weight determined by g. Then

T~ Tg u(y, A, #) = T~Tgu(y, At, #/~/7)

(2.47)

is rapidly decreasing with respect to A# 2 for y close to y0, and thus rh,(p0,p;) ¢

3.

Trace

theorems

Let N be a submanifold of N" of dimension n ~, to a point of N and u a distribution defined on a neighborhood of to in R n. Let us choose a system of local coordinates t = (tl,t ") centered at 0, such that N is given by t" = 0. We will denote by A the conormal bundle to N in Rn: (3.1)

A = ~Nn

= {(t', 0; 0 , - " ) }

and by e the canonical projection from T*NnlN to T * N deduced from the injection of N into R", and given in local coordinates by (3.2)

e(t', 0; , ' , T") = (t'; T') .

On the cotangent bundle T ' A , we have local coordinates (t', T'; t'*, ~-"*) and the projection from A to N induces an injection j : A ×N T * N ~ T ' A , given in the preceding coordinates by (3.3)

j((t', T"),(t',t'*)) = (t', T";t'*,O) .

We will denote by ~ : j ( A XN T ' N ) --~ T * N the composition o f j -1 and of the natural projection A x g T * N --~ T*N. In local coordinates, we have

(3.4)

0((t',

t'*, 0)) = (t'; t'*).

We will assume that u is compactly supported and fulfills the following smoothness property: (3.5)

3 M E N, 5 > 0, C > 0 such that for every r / E R ~'

f(1 +

C ) I d,/' _< C(1 + I,/I) M .

Such a condition implies in particular that U]Nis well defined. It is verified for instance when u belongs to H s ( R n) with s > ~-@. The main result of this section is the following theorem:

42

II. Second microlocMization

T h e o r e m 3.1. Under the assumption (3.5), (3.6)

WF(uIN) C 0(WF(u) Cl T*N"IN ) U 0(WF24'l(u) VIj( A XN T ' N ) f3 T*AIA_N ) SS(uIN) C ~(SS(u) n T*IR~IN) U ~(SS~'I(u) ~ j ( A x y T ' N ) r? T*AIA-N )

where A - N stands for A minus its zero section. If f is a compactly supported distribution fulfilling assumption (3.5), we will set (3.7)

1

S2f(y,s,;~')=-~?

f

~, ii II 2 #2 e-Z-(" +.~r~) - ~ + , , ,

•

I #

yll . 7711

(

,]

The proof of the theorem relies on the following lemma: L e m m a 3.2. Let us put

(3.s)

T ( f l g ) ( y ' , ~') = f e - ~ - ( " - ¢ ) ~ f ( t ', O) dr' .

We have (3.9)

/0 "l- °°

IS

n II -- 1

dy"S2f(u,

,a ') =

~

.

X'

n - n "

H(a')T(flN)(y',a')

where n" = n - n' and H()d) is a continuous function, equivalent to (~, )n"/2 when M goes to +oo. Proof. One has just to set yll

(3.10)

g(a') =

nil _

e - T (' + ~ )

- -

.

r]tt-

) dy"

.

1

Let us consider a point (t~, r~) which is not in the right hand side of the first inclusion (3.6). It means

(3.11)

(t0,0; ' T0,T I " ) ¢ WF(u) for every v" E N n'' (to, 7"; T~, 0) ~( WF24'1(u) for every r" E

with IT"[

1.

(One should remark that (3.11) is really equivalent to the fact that (t~; r~) is not in the right hand side of the first inclusion (3.6), since because of Proposition 2.9, WF2'I(u) is preserved by the maps

(3.12)

T* A

-~

T* A

(t',r";t'*,7"*) ~ ( t ' , r l r " ; r f l ' * , r ; l r 2 r ''*)

for every rl > 0, r2 > 0). We will apply (3.9) to the distribution (3.13)

f(t) = e

Atttl2

2 u(t) .

3. Trace theorems

43

One has uiN = f i N . Moreover, if u satisfies (3.5), f also with the constant C replaced by CA ~/2. We will then prove t h a t if a point ' "",t~* ,~-o (to,~0 "* ) ¢

(3.14)

WF~'I(u)

there is so > 0, B > 0, a neighborhood W of y0 = (t~ - zt " '0 * , - T " + ZT~ " I*) and for every N , CN > 0 such that

IS2 f(y,

(3.15)

At, 8)[

_~. CN,sBAt-Ne3~ [(Imy')2q-I-i'~* (Imy")2]

for y E W, A' _> 1, s C [So,+Cx~[. On the other hand, we will also show that if for a given s ~ ]0, s0[, the point (Rey~, ' - I m y ' o , s R e y ' o ' ) is not in W E ( u ) , there is a neighborhood W of y0, A~ > 0 and for every N C N, CN > 0 such t h a t for every y E W, ), > ),~, xi CN.~t_Ne~_[(Im

IS2f(y, A',s)[

(3.16)

~2 y , ) 2 + l_TT,.r(i m y11)21

To prove the theorem, we will see first that the second a s s u m p t i o n (3.11) implies the estimate (3.15) for every Y0 satisfying yto = t'o --iTS, ytj real, iY~'I = 1. Then, we will show t h a t the first a s s u m p t i o n (3.11) implies a version of (3.16) uniform with respect to s. These two inequalities, together with (3.9), will then allow one to conclude the proof. L e m m a 3.2. Under the second a~sumption (3.11), there exist so > O, B > O, a neighborhood V of Y~o = t~o - i7~ and for every N E N, CN > 0 such that

IS2/(y,A',s)l _< C N ~ B A

(3.17)

'

-N

M

t

e-~(~mY)

2

for every y' E V , y" real lY"I _-- 1, s > so, A t >_ l . Proof. We have

(3.1s)

s~f(y,a',~)=

t n'2

(2~),

s

n"

~,t ,iz ytt

e-~

o e~-~'. 2 s)~ f (_ y,;,s Oy"

A Is

with (3.19)

~;2f(y,A,,s ) =

)l

i

e--~(,-t)

t

2

x I

2

-z-st

It2

,

t

+,~t

11

It

~ f(t)dt.

Because of (3.13), we know moreover:

(3.20)

~2 f ( y , A , , s ) = S 2 u ( y '

s

. ,

Let us consider the following F B I phases of first and second kind given respectively by g(x,t)

(3.2t) G(y,x,#)=

-

i(x - t) ~

2 ' i , ~ ( y ' - x')2 ~ ( y " + ix") 2 2 ( 1 _ # 2 ) + i 2 ( 1 _ #2 + # 4 )

.X

it2

2

44

II. Second microlocalization

The FBI transformation of second kind T2Tg associates to u an element of the space H~, where (3.22)

(Im y") 2

#(y, #) = 1(i m y,)2 +

2(1 + It4) "

The critical point x(y, It) of the function - Im G(y, x, It) + ½(Im x) 2 with respect to x, is given here by

x'(y,#) = Rey' + iitImy' (3.23)

,,

It2

x (y, i t ) -

l+it4Imy"+iRey".

By Definition 2.7, T~(Tgu) is given by the integral of ei)'c(Y'~'#)Tgu(x, A) over a good c o n t o u r / ' passing through the critical point (3.23). Modulo a remainder in N~ we may replace this contour by (3.24)

{ (x',x"); x' e N"', x" E iN"" } .

One may then compute explicitly the integral with respect to x in T~Tgu and gets ~

(3.25)

T~%~(y,~,it)

=

~-

i

i

:t

.X 1

4

~ ( ~ - ' ) - ~ ~.-~'

,

J,2

II I~

+'~ ' ~(~)~t.

The identification between T*A and Y = C" associated to (G,g) is given by (t', T"; t'*, T"*) ~ (t' - i t ' * , - - c ' +

iv"*) .

The second assumption (3.11) implies that there exist It0 > 0, B > 0, and for every NcN CN > 0 s u c h t h a t (3.26)

IT~T.u(y,l, it)l

<_ CNIt-2B(,\It2)--N,s

2

J2

(Imy)

for y' close to y~ in C n', y" real with ly"l = 1 and Ait2 > 1. If we set s = 1/it 2, A' = Ait2, we deduce from (3.19), (3.20), (3.25) that

~2 f(y, A', s) = T~Tgu(y, A, tt) .

(3.27)

Using that (3.26) is also fulfilled by t~u for j = n' + 1 , . . . , n = n' + n", we deduce from these inequalities, (3.27) and (3.18) that (3.17) holds, after maybe a modification of the constant B. Let us now fix a real positive number 7 such that (3.28)

7(B + 1) <

1

and let us cut the integral in s in the left hand side of (3.9) into

_,

....

where N is a fixed integer. Because of (3.17), the second term is less or equal than

3. Trace theorems

45

CN)~tT(B-t-1)N--Ne@(Im y')~ ~_~CN)~ t- -~ e ~(Im y,)2 .

(3.30)

To estimate the third one, let us use that the modulus of the integrand of (3.7) is bounded from above by e-- x-~(~Z~'+Im Y')2--~@ (1- ~L~t )2 [TIHI ]Yttl e-g-(Ira ~'' y ,) 2 A's 2

(3.31) and that

(3.32)

,~N

e -~-(1- .:,''

[CI ds < A'-(~N+X)~lr/'l~ [ +O~e_T ,x'(1 - ;1)2 ds

52)tt

--

J0

82--5

_< CA'-('fN+l)elr/'le . Because of (3.5), we have thus (3.33)

L,+: ds/s.,,_ dy"lS~f(y,.X',s), < CA 15/2 / e- 2Y~'--Im Y"*/'(1 Jr- It/I) M d~'/~t--(~,N+I)5

<_ CA'e/2+M+~'-(~N+a)6e@O

TM

y')~

which shows that the third term of (3.29) is also rapidly decreasing. To conclude the proof of Theorem 3.1, we thus have to show that the same is true for the first term in this decomposition, using the first assumption (3.11). Towards this end, let us remark that because of (3.19) ~2f(y, A', s) = Tg, u(y', iy", A') where g~ is the FBI phase depending on the parameter s given by (3.34)

g (x,t) = i

' - t')2 + i -(1- 2 - -s2) ''2 - ist" • x "

The identification x~ naturally associated is (3.35)

(x',x")--+

Re x" Rex',-s l+s~;-Imx',sImx''

)

.

The result on characterization of WF(u), SS(u) in terms of FBI transformations given at the end of Section 4 of Chapter I (Corollary 4.4) and the assumption (3.11) imply that for every s > 0, ~2f(y,A,,s) is rapidly decreasing in A' when (y',iy") stays in the inverse image by (3.35) of a conic neighborhood of { (t~, 0; r~, r"); r " E N '~'' }. To conclude, we must obtain such estimates uniformly with respect to s G [0, so]. To do so, we have to prove a version "with parameters" of Theorem 4.2 of Chapter I. This is the aim of the following lemma: L e m m a 3.3. Let v(z", A') be a holomorphic function of z", continuously depending on

A', such that there exists D E I{ with (3.36)

sup

z" EK A~>I

(A'-DIv(z",A')le -~-(Im='')=) < +c~

46

II. Second microlocMization

for every compact K . For y" E ~ (3.37)

n It

, I v " l = 1, let us set

A ' ~ " " fj s e - - r ~~' ~,"~-i)¢'~" A~v(y i, , ~ 1 ) = (\~_~] * t ~ - ~"'~" , v(z",A)dx"dz"

where Z i~ the contour X 11 =

0 "ll

(3.38)

l+s 2

z 11 = s y

11

2

~rI ' c C

, la"l
Im~"-iRecr"

Then, there exist~ a continuous function s --~ C(s), defined on R+ with values in R~_ such that one has the equality (3.39)

A~ [S2u(yl,., t I, 1)] (y") = C(s)S2 f ( y , A', s)

modulo a remainder exponentially decreasing in )~', uniformly with respect to s. Proof. Let us consider the contour deformation x II = OztII + crtl (3.40)

z" = sy II

1 + s2 - I m a " -- i r e a" + ic~s2t" 2

la"l -< c, a e [0,1].

At c~ = 0, one gets the contour (3.38). On the other hand, if one expresses the left h a n d side of (3.39) in terms of u using the contour (3.40) at c~ = 1, and c o m p u t e s the integral in a ' , one obtains the right h a n d side of (3.39) m o d u l o a remainder with uniform exponential decay. Since an easy calculation shows that the contribution to (3.37) of the b o u n d a r y of the deformation (3.40) gives also such a remainder, the l e m m a is proved. We thus deduce from (3.39) that under the first assumption (3.11), the first term in (3.20) is rapidly decreasing with respect to AI for yl close enough to y~. Using also (3.30) and (3.33), we get, taking (3.9) into account (3.41)

]T(flg)(y',/~t)l < C ~ 'cst -Ninf(½"~6)e~-~-(Im Y')2

for yl close to y~. This proves the first inclusion of T h e o r e m 3.1. T h e proof of the second inclusion (3.6) is identical: one just has to break the integral f : o o S 2 f d s at s = e ~ ' instead of s = 1,~N.

III.

Geometric

upper

bounds

In this third chapter, we will prove results giving a geometric upper bound for the singular spectrum, and for the second microsupport along a lagrangian submanifold, of distributions defined as boundary values of convenient ramified functions. The estimates we will obtain will depend just on the geometric data of the problem, that is on the (singular) hypersurface around which the distribution under consideration is ramified. The method relies on technics of deformation of the integration contours in the integral expressions of FBI transformations of first or second kind. It uses on an essential way the theory of subanalytic sets and functions. The first section thus recalls without proofs the main basic definitions and results about these notions. Proofs m a y be found in [Hi] or [Bi-M], except in the case of Theorem 1.13, for which one should consult [Hall or [T]. We also give the definition of Whitney's normal cone, following the lines of [K-S1], [K-S2], which will play an essential role in Section 3. Section 2 studies critical points and critical values of locMly lipschitzian subanalytic functions. We show that a function obtained from an analytic function by a minimax formula is a critical value of the latter. We prove also that its derivative, at points where it exists, m a y be computed in terms of the derivatives of the starting function at critical points. All these results come from [D-L] and [L4]. Finally, the third section is devoted to the proof of geometric upper bound formulas. They rely heavily on the use of the results of Section 2. The references used are still [D-L] and [L4]. To conclude this presentation, let us mention that the results of Section 3 we will apply in Chapter IV could be replaced by similar upper bounds obtained using the theory of holonomic D-modules (see [La]).

1. Subanalytic sets and subanalytic maps In this first section, we will recall the definitions and the main properties of subanalytic sets and subanalytic maps we will have to use subsequently. Let us begin by the definition of subanalytic sets:

D e f i n i t i o n 1.1. Let M be a real analytic manifold and A a subset of M. One says that A is subanalytic at x E M if and only if there exist a neighborhood V of x in M and a finite family (Xi,j)~,j (resp. (fi,j)i,j), 1 < i < p, j = 1,2, of real analytic manifolds (resp. of proper real analytic maps fi,j : X i , j ~ V ) such that

48

III. Geometric upper bounds P

(1.1)

A n V = U ( f i , , ( X i , , ) - fi,2(Xi,2)) . i=1

One says that A is subanalytic in M if A is subanalytic at every point of M. Subanalyticity is thus a local property in the ambient space M. It follows also from the definition that the class of subanalytic sets is stable under finite intersection, finite union, difference. One m a y prove: T h e o r e m 1.2. Let M and N be two real analytic manifolds, f : M --* N a proper real analytic map. i) If A is a subanaIytic subset of M , f~ is subanalytic in M . it) I f A is a subanalytic subset of M , f ( A ) is subanalytic in N . One m a y prove that the class of subanalytic sets is the smallest class of sets containing all closed real analytic manifolds, and stable under finite intersections, finite unions, differences and proper projections. We will use the following result, known as "curve selection l e m m a ' . T h e o r e m 1.3. Let A be a subanaIytic subset of M and a a point of A. There exists a curve 3' : [0, 1] ~ M , real analytic on a neighborhood of [0, 1] such that ~/(0) = a and 7(t) E A for every t E ]0, 1]. Another important result of the theory of subanalytic sets is the "lemma of regular separation", known also as "Lojaciewiecz inequalities": T h e o r e m 1.4. Let A and B be two compact subanalytic subsets of N n. There exist C > 0 and N E N such that for every x E A, d ( x , B ) >_ Cd(x, A N B) N (d(.,-) denoting usual euclidean distance on R'~). D e f i n i t i o n 1.5. A continuous mapping from a real analytic manifold M to a real analytic manifold N is said subanalytic if its graph is subanalytic in M x N. We will now recall the notion of subanalytic stratification. D e f i n i t i o n 1.6. A stratification of a real analytic manifold M is a partition S of M, satisfying the three following conditions:

=

(Si)ieI

i) For every i E I , Si is a connected real analytic and subanalytic submanifold of M. it) The family

(Si)iEI

is locally finite in M.

iii) If Si and Sj are two elements of S such that Si M Sj 7£ O, then Sj C St. One should remark that in condition i) above, in spite of the fact that Si is a submanifold, one cannot drop the condition of subanalyticity: in fact, subanalyticity is

1. Subanalytic sets and subanalytic maps

49

a local p r o p e r t y in M (not in St) and so, since Si is not assumed to be closed, it might be an analytic submanifold without being subanalytic in M .

Example. Let us take M = R 2 with coordinates (x, y). T h e family So = ({x = 0), {y = 0, + x > 0}, {=Ex > 0, + y > 0}) is not a stratification since condition iii) is not satisfied. On the other hand, the family

s = ({x = u = 0}, {z = 0, + y > 0}, { i x > 0, y = 0}, { + x > 0, + y > 0}) does verify all the conditions of Definition 1.6. One says that a stratification S of M is compatible with a locally finite family

C = ( C j ) j e j of subanalytic subsets of M if for every i • I, j • J , one has either S i n Cj = 0 or

S i c Ci.

One m a y show that for every subanalytic subset A of M , there exists always a stratification S of M , compatible with the family (A, M - A). One then defines the dimension of A as the m a x i m u m of the dimensions of the s t r a t a of S contained in A (it is of course independent of the choice of the stratification). D e f i n i t i o n 1.7. Let M and N be two real analytic manifolds and f : M ~ N a real analytic map. A stratification of f is a couple (S,.T) where 8 (resp. 5t') is a stratification of M (resp. N ) such t h a t for every s t r a t a Si of S, one has f ( S i ) C .T" a n d r k ( f l s , ) -- dim f(Si). T h e m a i n result a b o u t stratification we will use in the sequel, is the following theorem. 1.8. Let M and N be two real analytic manifolds, f : M -~ N a real analytic map, C (resp. 79) a locally finite family of subanalytic subsets of M (resp. N), 1"2 a subanalytic open subset of M such that flD is proper. Then, there exists a stratification (S,9 ~) of f l ~ with S compatible with C and 27: compatible with 79.

Theorem

Let h : R ~ R be a real subanalytic function, and denote by gr(h) C R x It~ its graph. Applying the preceding theorem to M = R x N, N = R, f = first projection on R x If{, C = { gr(h), M - gr(h) } we see that there is a discrete subset of N, (ti)iez, ti < ti+l for every i such t h a t hl]t~,t~+l[ is real analytic o n ]ti,ti+l[. In fact, one m a y show t h a t for every i E Z, there are integers r + r~- and real analytic functions on a n e i g h b o r h o o d of 0, g+ and g7 such that h(ti + t) = gi~(tl/r~) for t > 0 close to 0. We will deduce from T h e o r e m 1.3 the following result; P r o p o s i t i o n 1.9. Let S = (Si)iei be a stratification of M = R n, xo a point of M and for every Q > 0, denote by B e the closed ball of center xo, of radius Q, and by OBq its boundary. There exists 6o > 0 such that for every 6 E ]0, 60[

(1.1)

T~B M n (U TL* M ) c T ~*M . jEI

50

III. Geometric upper bounds T h e p r o o f will make use of the following lemma:

L e m m a 1.10. If S = (Si)iEI is a subanalytic stratification of a real analytic manifold M , U i e ! T},M is subanalytic in T*M.

Proof. Since (Si)ieI is locally finite, (T~,M)ieI also. So, we just have to prove t h a t if L is a real analytic and subanalytic submanifold of M, T~M is subanalytic in T*M. Let us choose an analytic coordinate system on a neighborhood of a point of L in M. T h e set A = {(x,~,X,t); (x,~) e T ' M , ( x , X ) e TL, ]X I = 1, t = [(X,~)]}

(1.2)

is subanalytic as soon as TL is subanalytic. But since the projection (1.3)

q~:

A ~ T*M ( x , ~ , x , t ) -~ (x,~)

is proper, we see that qS(A M {t > 0}) is subanalytic, and so, that the same is true for [C~(A M {t > 0})] ]L

= { (x,~) E T*MIL; VX with IX] = 1 and ( x , X ) C TL, one has (~,X) = 0 } de~

T; M

.

We are thus reduced to prove that T L is subanalytic in TM. For every open subset U of M over which exists a system of local coordinates, let us set (1.4)

C(L)] U = { (x, X ) E TU; there exist sequences xn, y,, in L M U, cn in ]R+ with xn ~ x, Yn ~ x, cn ~ + c o such that c,~(xn - yn) ---*X }.

One sees easily t h a t the preceding set is independent of the choice of the coordinates on U and that (1.4) intrinsically defines a closed conic subset C(L) of TM. Moreover C(L)IL = TL, and so it is enough to prove that C(L) is subanalytic. Of course, we need verify t h a t just for C(L)Iu where U is any subanalytic open set endowed with a system of analytic local coordinates. Let us then set E= (1.5)

{((z,X),x,y,u)•TV×LxL×[0,1];

E0 = E -

y-x=uX}

{u = 0}

7r : TU × L × L × [0, 1] ~ TU the n a t u r a l projection. One has C(L)iu = 7r(E00 N {u --- 0, x = y = z}) and the right h a n d side is subanalytic by T h e o r e m 1.2.

Proof of Proposition 1.9. We m a y assume x0 = 0. If the proposition is false, there exists ~0 • R'* with ]~01 = 1 such that the point (0, ~0,0) is in the closure of the set (1.6)

(x,~,o) e T * M × ] O , 1]; ( x , ~ ) E U T ~ , M , ~ = ~ - ~ , t ~ = i x I

.

1. SubanMytic sets and subanMytic maps

51

Because of the lemma, (1.6) is subanalytic in T * M x [0,1]. By T h e o r e m 1.3, there is thus an analytic curve t --~ 7(t) = (x(t), ~(t), o(t)) with 7(0) = (0, G0, 0) and 7(t) belonging to (1.6) for every t E ]0, 1]. Since U i e i T ~ M is subanalytic in T ' M , T h e o r e m 1.8 applied to t ~ (x(t), ~(t)) shows t h a t there exist e > 0 and i0 E I such t h a t (x(t), ((t)) E T}io M for every t E ]0,@ Using t h a t the one form ~dx vanishes identically on any conic isotropic submanifotd o f T * M , we see t h a t ( ( t ) . d x ( t ) - 0 on ]0, e[. Since 7(t) is in (1.6), this implies x ( t ) . ~?(t) -= 0 whence Ix(t)l 2 -- cst on [0,~]. This contradicts the fact that e(t) = Ix(t)l > 0 for t ¢ 0. T h e set C(L) we used in the proof of L e m m a 1.10 is a special case of " W h i t n e y ' s n o r m a l cone" we will now define. Let M be a real analytic manifold, S and V two subsets of M . If x is a point in M , and if we assume chosen a system of local coordinates on a n e i g h b o r h o o d of x, we set (1.7)

C,(S, V) = { 0 E T,M; there exist sequences sn E S, vn E V, cn E N~_ with

s.,x,

x, c . ( s . - v . ) - - , 0 } .

T h e definition is independent of the choice of the coordinate system and one puts: D e f i n i t i o n 1.1. One calls Whitney's normal cone to S along V the set (1.8)

C(S,V)=

U C~(S,V) c T M . xEM

W h e n V is a submanifold of M, C(S, V)Iv is TV-invariant and one denotes by

Cv(S) C T v M

(1.9)

the image of C(S, V)lv m o d u l o TV. W h e n M is a symplectic manifold and when V is a lagrangian submanifold of M , T v M m a y be identified with T*V using the hamiltonian isomorphism, and one then considers C v ( S ) as a subset of T*V. W h e n L and S are subanalytic in M , one shows, as in the proof of L e m m a 1.10, t h a t C v ( S ) is subanalytic in T*V. Let us consider the case when M = T * X with X real analytic manifold, V is a lagrangian submanifold A of T * X and S is a conic subset of T*X. Let X be a canonical t r a n s f o r m a t i o n defined in a n e i g h b o r h o o d of a point q0 E A, sending A onto the zero section X of T*X. Let us denote by )~ : T*A --~ T * X the isomorphism it induces on cotangent bundles. If X is endowed with a local coordinate system close to x(qo), one sees easily, coming back to the definition, t h a t on a neighborhood of x(qo)

(1.10)

)((CA(S)) = { (:/:, ~); there exist sequences (ym,~]m) E S, Urn E N*+, Um --* 0 snch t h a t i f

=

x,.

x,

}.

To conclude this section, we will recall two topological results a b o u t subanalytic sets. 1.12. Let A be a closed subanalytic subset of a real analytic manifold M. There exists a subanalytic open neighborhood of A in M and a subanalytic retraction by deformation

Theorem

52

III. Geometric upper bounds

r : U x[O,1]--,U

(1.11)

of U over A. 1.13. Let M and N be two real analytic manifolda, K a subanalytic compact aubaet of N , p: M x K -+ M the first projection, ~ a finite family of subanalytic aubseta of M x K . There exists:

Theorem

- a finite stratification (Mi)ie, of M , compatible with the family (p(F))FeF, - for every i E I, a point ei E Mi and a subanalytic horneomorphism gi from p - l ( M i ) to M~ x p-l(ei), such that for every E C ~-, g , ( p - ' ( M , ) n F) = M~ x (p-l(e~)n V), and making the following diagram commutative: p-l(Mi)

g, -----+

pN'. N

Mi

X

p-1 (el)

~117.¢- 1

Mi

2. C r i t i c a l

points

and critical values

T h e next section will be devoted to the study of u p p e r b o u n d s for the microsupport or the second microsupport of distributions which can be continued holomorphically in the complex domain. To do so, we will first get estimates for F B I or second F B I transformations in terms of weight functions ~b given by minimax formulas like formula (3.14) of C h a p t e r I. The new feature here, is that the functions to which these m i n i m a x will be applied will no longer have a single critical point, depending s m o o t h l y on the p a r a m eters. T h e "critical value" ~b(y) will thus no longer be a real analytic function. In fact, ~b will just be a subanalytic function. The aim of this section is to prove this result, and to obtain informations about the derivative of ~b (at points where it exists) in term of the geometric d a t a of the problem. T h e assumptions we do below m a y seem quite complicated and technical. We make t h e m because they will a p p e a r to be the natural hypothesis in the applications we will treat in C h a p t e r IV. Let us first introduce the geometric data. We assume given: - Z a closed real analytic submanifold of 1R", - ~Tj, j = 1 , . . . , k, k real analytic hypersurfaces in Z x ]0, 1], which are subanalytic in Z x [0, 1] and transverse to the fibres of the second projection Z x ]0, 1] -+ ]0, 1], -

~7 a real analytic submanifold of Z x ]0, 1], subanalytic in Z x [0, 1], transverse to the fibres of the second projection a x ]0, 1] --+ ]0, 1].

We shall assume the following condition:

(T)

For every z E Z x ]0, 1], at most three manifolds a m o n g the ~Tj and X' pass t h r o u g h z. Moreover, these submanifolds are m u t u a l l y transverse, and when three of t h e m pass t h r o u g h z, each of t h e m cuts transversally the intersection of the other two.

2. Critical points and critical values Lastly, we will suppose that every 2 j is the b o u n d a r y of an open half-space ~ ?

53 of

Z x l 0,1] such that ~2 = N k 22+ is bounded. We will denote by D~ for e e 10, l] the section of ~2 at ¢. We will use an analogous notation for the sections of G. On the other hand, let U be a b o u n d e d open subset of N p x ]0, 1], which can be realized also as the intersection of a finite n u m b e r of open half-spaces, whose boundaries are real analytic submanifolds of N p x ]0, 1], subanalytic in R p x [0, 1], satisfying the evident analogue of (T). Let /" be a real analytic m a p defined on a big enough open subset of NP, with values in N N, such that the m a p

(2.1)

~t, x 10,1] -~ ~t N x ]0,1] (t,e) ~ (/'(t),s)

is an e m b e d d i n g of U into $2 \ 2 sending OU n ( R , x ]0, 1]) into (0~? U 2 ) Cl ( Z x ]0, 1]). For every e • ]0,1] we will denote by Fo,~ the slice at e of the i m a g e of U by (2.1). For e • ]0, 1], let ~ be the set of all C °~ homotopies H : r0,~ x [0,1] --*

(2.2)

(z,~)

a~.

~ H(z,~,)

satisfying the three conditions: H ( z , O) = z for every z e F0,~ (2.3)

H ( z , ~r) = z for every z C OFo,~ M 0 ~ H ( z , a ) E 2~ if and only if z E 2 , .

Let Y be an open neighborhood of y0 = 0 in N n and V a neighborhood of the set {z•Z;3e•[0,1]with(z,e)•~} inR N.Let #:YxV--*

(2.4)

R

(y, z) -~ ~(v, z)

be a real analytic function and set (2.5)

¢(y,e) =

inf

sup

Hc'H,~ zEFl,¢ =H(F0,t ,1)

@(y,z) .

We will denote by $ the stratification of ~ M (R N x ]0, 1]) whose s t r a t a are the connected c o m p o n e n t s of the following manifolds: •

Z/~\(U

ZjUZ j#jl

k

•

(U

)]

/]

N~,

jlE{1,...,k}

j=l

2j,rlXj~\(uj=~.2,jU2)] /~.~#j, N~-,

j,¢j2C{1,...,k}

III. Geometric upper bounds

54

[

~.,;1 r"l .~, \ ( U .~j) ] I~ ~ ,

jl • {1,..., ]¢}

j~t jl

[Z A NXj~ ;qZj~] N - ~ ,

j l , j 2 , j3 all different in { 1 , . . . , k }

nr] nn,

ji,

di erent in { 1 , . . , k}

n\2. F o r ~ • ]0, 1], S, will be the stratification of ~2, which is the slice at ~ of S. In the same way, one defines a natural stratification Or of the image of U by (2.1) intersected with R N x ]0, 1]. For g • ]0, 1], Or, will be the stratification of F0,, given by the slice of Or at ~. We will denote by OFo,, = 0I'o,~ N OY2---~.Let us prove the following:

2.1. Let (y, c) G Y × ]0, 1] be such that ¢(y, ~) > sup~ebr0,. ~(y, z). There e~ists (z, ¢) • Us,~s, T*s, R N such that

Proposition

¢ = d~(y, z),

(2.6)

¢(y, ~) = ~(y, z)

T h e proof will use the following lemma. Lemma

2.2. Let (y,c) • Y×]0,1] be such that ¢ ( y , c ) > supbr0,. ~(y,z). Suppose that

- with ~(y, z) = ¢(y, ~), one has (z, d ~ ( ~ , z)) ¢ Us. es. for every z • -n~ there exists a C ~ homotopy

T *s. ~ g

• Then,

L

(2.7)

h : Y2~ × [0,1] ~

(z,~)

~2~

-~ h(z,~)

such that the following conditions hold: •

h(.,0)

=

Id,

• for every a • [0, 1], h(., a) is a diffeomorphism from Y2¢ to itself, • for everya • [0,1], h(-,cr)l$r0, = Id and h ( Z e , a ) C Z~, • there is 5 > 0 such that ~5(y, z) < ¢(y, c) + 5 implies q~(y, h(z, 1)) < ¢(y, ¢) - 5.

Proof. It is enough to construct a C + vector field on a neighborhood of $2~ in Z, tangent to Zj,, for every j and to Z, vanishing close to (3_F0,,, transverse to the smooth hypersurface { z; 4~(y, z) = ¢(y, e) } at each of its points, and pointing towards the open set { z; # ( y , z) < ¢(y, ~) }. In fact, the flow h(z, a) of such a vector field, with the time p a r a m e t e r suitably chosen, gives a map fulfilling all the conditions of the proposition.

2. Critical points and critical values

~

55

Z~'~C(y,z)= V(y,e)

Using a partition of unity, it is enough to build such a vector field locally. Let z0 be a point of Z through which are passing three submanifolds of the family (rj)j=l ..... k U Z. Because of assumption (T), we may find a system of local coordinates on Z, centered at z0, z = (z 1, z 2, z 3, z 4) such that these three submanifolds are given respectively by the equations z I = 0, z 2 = 0, z 3 = 0. If we write d~qh(y, z0) = ((1, ~2, (3, (4), the assumption oe " 0 then suits us in a neighborhood of the lemma means (4 # 0. T h e vector field - y;Zz4 of

Z0 .

One argues in a similar way close to any point of Y2~ (one should remark that, by hypothesis, 0P0,, does not meet { z; # ( y , z ) = ¢ ( y , e ) }).

Proof of Proposition 2.1. Assume, by contradiction, that for every z 6 9~ with

= one has ¢ Us. s. T*s. R/v For any 6 > 0, there is, because of Definition (2.5) of ¢, an homotopy H E ~ such that F~,~ = g(Fo,~, 1) C {z; ~ ( y , z ) < ¢ ( y , e ) + 5 }. If ~ is small enough, L e m m a 2.2 shows that the hom o t o p y (z, or) ~ At(z, cr) = h(H(z, o), a) is in 7"~ and satisfies ]~r(fo,~, 1) C { z; • (y, z) < ¢(y, e) - 6 }. This last inclusion contradicts the definition of ¢(y, e). •

D e f i n i t i o n 2.3. A real number c is said to be an St-critical value of qh(y,.) if c > supzesr0., ~(y,z) and if there exists ( z , ( ) 6 Us~es~ T*s, It~N with ( = dz~(y,z) and c = 4~(y, z). Proposition 2.1 thus says that for every fixed (y, e), either ¢(y, e) = supzebr0., ~(y, z) or ¢(y, e) is an St-critical value of ~(y,-). Let us show now 2.4. Let (y, s) be a point in Y x ]0, 1]. The function z -~ ~(y, z) has a finite number of Se-critical values.

Theorem

Proof. T h e set

(2s)

{z c

(z,

z)) c

(_J T's. R N } S~ 6S~

56

III. Geometric upper bounds

is subanalytic (see Theorem 1.2). Since it is compact, it has a finite number of connected components C1, . . . , Cl, each of them being subanalytic. The set of critical values is equal to Ull ~(y, Cj) minus possibly sup~csr0,, ~(y, z). Let cl and c2 be two points in • (y, C~). There exist z~ and z2 in C~ with ~(y, zj) = cj, j = 1, 2. By the connectedness of C1, we may choose a subanalytic curve 3' contained in C1, joining zl to z2. By Theorem 1.8, we may choose a stratification of 3' compatible with the family (S~)s, es,. If-~ is a one dimensional stratum of that stratification, there exists S~ E S~ with ~ C S~. Since moreover, ~ C C1, we have for every z E ~/, (z, d~qb(y, z)) E T~, R N. This implies that ~i(y,.) is constant along -~, whence cl = c2. The number of critical values is thus finite. We will deduce from that result the subanalyticity of ¢(y, e). C o r o l l a r y 2.5. The function (y,e) ~ ¢(y,e) i~ locally lipschitzian in y uniformly in e E]0, 1], and it~ graph is subanalytic in Y x [0, 1] x N.

Proof. Since ~2 is compact, for every compact K of Y, there is C > 0 such that ] # ( y l , Z ) - ~5(y2,z)] < Ciyl -y21 for yl, y2 in K and z belonging to {w; 3~ E 10,11 with w E $2~ }. The assertion of the corollary about lipschitz regularity follows from this inequality. Let us consider the set (2.9) A = { (y,e,c) E Y x ]0,1] x R; either c = sup ~(y,.)

Oro, or

3z E $ ? ~ w i t h c = ¢ ( y , z )

and (z,d.qb(y,z)) E U

"r* .L Se NN }

S~ E N~

Let us first remark that for every z such that ~(g,z) = sup~r~,, 4~(y,-), the point (z, d,~(y, z)) is in the union of the conormal bundles to the strata of the natural stratification .T'~ of 0_F0,~. To see that, one argues as in the proof of Lemma 2.2: since U - and thus 0F0,~ satisfy by assumption condition (T), if this property is not true, one may build a C °° vector field tangent to the strata of 0F0,~, transverse to { z E ~ ; ~5(y, z) sups/.0,. ~5(y, .) } and pointing towards qb(y, z) > sup$r0,, ~i(y, .) whence a contradiction. The set .4 is then contained into ----

A ={(y,s,c)

EYx]0,1]xR;?zE~-~withc=@(y,z)

and

either (z, dzqS(y, z)) E U T*F~RN F, EY, or(z,d~¢(y,z)) E U

T's,RN ~1 •

S, 6S~

This is a subanalytic subset of Y x [0, 1] x N. In fact, one deduces from Lemma 1.10 and Theorem 1.2ii) that for every stratum S of S (resp. F of f ) , the set { (¢,z,() E ]0,1] × T ' a N ; (z,() E T*S, R N (resp. (z, ¢) E T*F. N N) } is subanalytic in [0, 1] x T*N N. The subanalyticity of A then follows by the elementary properties of subanalytie sets.

2. Critical points and critical values

57

Let us denote by 7r : A ---* Y x [0, 1] the proper projection (y, ~, c) ~ (y, ~). By Theor e m 2.4, 71"]Ahas finite fibers and because of the inclusion A C A and of Proposition 2.1, these fibers are non empty. This implies dim A = dim Y. Using T h e o r e m 1.8, we find a stratification of 7r c o m p a t i b l e to the partition Y x {0)tJ Y x ]0, 1] of Y × [0, 1]. Since ~r has finite fibers, its restriction to every s t r a t u m of A is a local diffeomorphism onto its image. Let W be a s t r a t u m of Y × ]0, 1]. Because of Proposition 2.1, gr(~b)N ~r-1 ( W ) is contained in, and thus equal to a s t r a t u m of A whose image by 7r is W. So g r ( ¢ ) is the union of a locally finite family of subanalytic strata, and is thus subanalytic. This proves the corollary.

A

Y x ]0,1] H

W For every fixed y E Y, the function ¢ ~ ¢(y, 6) has a g r a p h which is subanalytic in [0, 1]. By the description of subanalytic functions of a single real variable we gave after T h e o r e m 1.8, we then see that l i m e - 0 + ~b(y, ~) exists. Let us denote it by ¢(y, 0). Since y ---* ~b(y, ¢) is lipschitzian, uniformly with respect to 6, we see that ¢(y, 0) is also lipschitzian. Moreover, if we extend ¢ to Y x [0, 1] by its limit ¢(y, 0) at ~ = 0, we get a subanalytic function on Y x [0, 1]: in fact, the g r a p h of this function is just the closure of the g r a p h of ¢[Y×]0,1]" Let us show now: T h e o r e m 2.6. Assume that ¢ i~ real analytic in a neighborhood of ( y l , s l ) C Y × ]0, 1] with ¢(Y1,61) > supsr0,c 1 ~(yl,-). Then there exists z E f2~ 1 such that ¢(Y1,~1) -~(Yl, z) and (2.10)

d y ¢ ( y l , 6 1 ) ----dy~5(yl,z)

and

(z, dzqh(yl,z)) E

U T*S . 1 •N S, l ES, 1

58

III. Geometric upper bounds

Proof. T h e set

B = {(V,z,c; d~(y,z)); y c Y , c ~10,1], z ~ , (2.11)

(z, d ~ ( y , z))

6 U

T* R N

S, 6S~

¢(~, c) = e(y, z) } is subanalytic. Let us consider a stratification of the projection ~- : B ~ Y x ]0, 1]. Because of Proposition 2.1, ~ is onto on a neighborhood of (yl, cl). Let (Y0, ~0) be in an open s t r a t u m of Y x ]0, 1] such that ¢(Y0, c0) > sups/%.,0 ~(yo, "). Let us choose a point b0 6 B with ~'(b0) = (y0,c0). Since ~" is a submersion from the s t r a t u m of B containing b0 onto an open subanalytic neighborhood of (y0,c0) in Y × ]0, 1], there exists a real analytic m a p (y,e) ~ b(y,c) defined close to (y0,e0), such t h a t b(y, e) 6 B and #(b(y, e)) = (y, c). We thus have

b(y, c) = (y, z(y, ~), c; d ~ ( v , z(v, c))) (2.12)

(z(y, c), (~z~)(y, z(y, c))) e

U

T*~ R N

S~ ES~

¢(~, c) = ~(y, z(y, ~)) Then, dye(y, e) = (dyq~)(y, z(y, c)) + (dz#)(y, z(y, c))(dyz)(y, c). Since the 1-form (dz vanishes identically on Us, es, T~, ~N, the second relation (2.12) implies that the last t e r m in the preceding s u m vanishes identically. Since the point (yl,Cl) is in the closure of the set of points (y0, c0) fulfilling the preceding condition, the conclusion of the t h e o r e m follows letting such a sequence of points (y0, co) converge to (Yl, Cl).

3. Upper bounds for microsupports and second microsupports Let Z be a real analytic submanifold of R m passing through z0 = 0 and Z c the complexification of Z in C N in a neighborhood of 0. If r is a positive real n u m b e r , we will

pat Br = { z e ZC; Izl < r } and By = Br n Z. Let h be a real analytic function on a neighborhood of 0 in Z satisfying h(0) = 0 and such that there exist r > 0 and a connected c o m p o n e n t A of 23~ - h - l ( 0 ) with h]A > 0 and 0 6 A. We will also denote by h the holomorphic continuation of this function to Z c and we will assume that r has been chosen small enough so t h a t h be defined in a n e i g h b o r h o o d of Br- We will set g2~ = B r - h - l ( 0 ) and will denote by ~r : Q~ ---, ~ r the quotient of the universal covering of S2r by the equivalence relation identifying to 0 every loop of ~2~ h o m o t o p i c to a loop of A. Then, 7r-l(A) is the disjoint union of connected components, each of t h e m being isomorphic to A by lr. Let a : A ~ C be an analytic function satisfying the following conditions:

3. Upper bounds for microsupports and second mierosupports (3.1)

i) 35 > 0, 3 C > 0 such that for every small enough c, f{,; C e c,

lh(z)l_<~}la(z)l

59 dz <_

ii) T h e r e exist an holomorphic function g : g2"~--* C, a connected c o m p o n e n t of rr-a(A), a real n u m b e r g such t h a t gl~ = (a°Tr)]/i and t h a t for every open subset V of J2"-'~with s u p p e r Card[Y M 7r-a(Tr({))] < +oo, there is C v > 0 with

la({)l _< CvIh( ( ))l -K for every t" E V. Let 6z be the integration eourant on Z, associated to the euclidean metric on R N. Denote by l = c o d i m ~ Z and chose l independent vector fields with real coefficients transverse to Z. I f / 9 = (/3,,.-.,/31) 6 N', let us set 6(zz) = X ~ t . . . X t ~ ' 6 z and let us consider the distribution (3.2)

u : a l A ' 6 (~)

(where 1 is the characteristic function). T h e aim of this section is to obtain u p p e r bounds for the m i c r o s u p p o r t and the second m i c r o s u p p o r t of u in terms of the geometric d a t a of the p r o b l e m i.e. Z c and the hypersurface of Z c given by the equation h = 0. Distributions like u m a y be considered as a special case of "conormal distributions" along the singular hypersurface of Z given by the equation h = 0. To state the result we are looking for, we must first define some geometric objects. D e f i n i t i o n 3.1. One denotes by T~_~(0)Z c the closed subset of T * Z c given over every holomorphic chart of Z c by: (3.3)

{ (z, ~); 3(Zn)n (resp. (cr,~),,) sequence of Z c (resp. C) such t h a t an" h ( z , ) ~ 0 and a . O h ( z n ) ~ ¢ } .

One checks at once that the preceding definition is intrinsic. Moreover, one m a y show (cf. [K]) that T~_I(0)Z c is a C-isotropic complex analytic subset of T * Z c (that is the holomorphic symplectic form on T * Z c vanishes on the open set of its s m o o t h points). Let •z : T * C N I z c --+ T * Z ¢ be the natural projection. W i t h the preceding notations, we want to prove: T h e o r e m 3.2. Let A be a real analytic lagrangian submanifold of T*]~ N and A c its complexification in T * C N. One has:

(3.4)

SS(u) C ez-1 (T;-x(o)Zc) M

T*NN

SS~t'I(u) C CAc(@zI(TTt-I(o)ZC)) f') T * A .

Let us r e m a r k first that it is enough to prove the t h e o r e m when /3 = O. In fact, assume t h a t we have proved (3.4) in such a case and let us consider an u given by (3.2)

60

III. Geometric upper bounds

with a non-zero ft. Since b o t h sides of inclusions (3.4) are intrinsically defined, it is enough to prove (3.4) locally. But on any coordinate system defined on a small enough chart, one m a y write u on the form

u= ~

O~u7

bl_ 0 small enough, let us put

= {z e 9T; Ih(z)l _> (3.5)

g

}

= A~ = A n { z ;

[h(z)l > c } .

Let (t0,v0) be a point in T*R g - {0} with to close to 0. If x E C y is in a small enough n e i g h b o r h o o d of to - iTo, let us write the F B I transform with quadratic phase of u, Tu(x, A) = f e-~ (~-t)2 u(t) dt as a sum: (3.6)

f

Tu(z, A) = J ¢--~(x-t)21 A_Aa(t)~Z(t)

at -~-/ e-~(x-t)21A a(t)SZ(t ) dr.

Because of (3.1), the m o d u l u s of the first term is b o u n d e d from above by (3.7)

Ce" e -}(Irax)2

T h e sccond one m a y be written fA, e--~(~-Z)2a(z)dz where z denotes the variable on the manifold Z and dz the r i e m a n m a n volume on Z. Let us consider the variety (3.8) We have

E = { (z,~) C h~ × 10,e0]; h(z) = ~ } .

3. Upper bounds for microsupports and second microsupports

61

L e m m a a . 3 . There exists ro > 0 such that for r • ]0, r0[, ~3B ZCfqT~-,(o)Z c is contained into the zero section of T * Z c. Particularly, for e small enough, the subvarieties of B~ given by the equations h(z) = e or Ih(z)l = e are smooth and transverse to OBj.

Proof. T h e first assertion of the l e m m a follows from Proposition 1.9. In fact, one knows ([K-S1], [K-S2]) that the isotropy of T~_~(0)Z c implies t h a t there exists a stratification of Z c such t h a t T*h_,(0)Z c is contained in the union of the c o n o r m a l bundles to the strata. One m a y then a p p l y Proposition 1.9 to that stratification. T h e subvarieties of B~ given by h(z) = e or Ih(z)l = e are s m o o t h for r, e small enough since, if not, there would be a connected c o m p o n e n t of the complex analytic set { z; h'(z) = 0 } meeting { z; h(z) ---- 0} and not contained in the latter. But this is impossible, as one sees at once, using for instance the curve selection l e m m a (Theor e m 1.3). T h e fact t h a t the manifolds h(z) = e or Ih(z)l = ~ cut OB~ transversally, if r, e are small enough, is now clear since T~_~(0)Z¢ contains the limits of the conormal directions to these submanifolds when e ---* 0. T h e l e m m a shows t h a t if r, e0 are small enough, the open set

~ =

(z,e) • Z c × ]0,~0[; ~ < Ih(z)l and Iz[ < r

a n d the submanifold 22 defined by (3.8) fulfill condition (T) of Section 2. T h e s a m e is true for the initial contour F0,~ = A,. Let H be a C a h o m o t o p y of the form (2.2), with values in D .... satisfying conditions (2.3). T h e r e is a unique h o m o t o p y

~,~ -~/~(z,~)

H : Fo,~ x [0,1] ~

(3.9)

(z,o)

such t h a t /~l~=0 = (Tr]~i) -1 : F0,, --~ F0"-~_~ (Trl.~)-l(F0:~) and t h a t 7r o H = H . Let FI,'-"-~-- H(T'0,e, 1). Then, using Stokes formula, one sees that the second t e r m of (3.6) is equal to =

(3.10)

f~

e-

~(~-~(~))~(~)&(~)

I

Actually, the fact t h a t the b o u n d a r y of the integration chain is not kept fixed by the h o m o t o p y / ~ r does not m a t t e r . T h e p a r t of the b o u n d a r y which could move during the d e f o r m a t i o n must stay on the complex submanifold Se, over which the complex form of degree equal to d i m e Z ¢ dz vanishes identically. Let us consider the function (see (2.5)) (3.11)

~(x,e) =

inf

sup

- 7 1 Re(x - z) 2 .

HET"I. zEFI,.=H(I,Fo,.)

Because of Corollary 2.5, q0 m a y be extended to e = 0 and this extension is subanalytic. T h e m a i n point, in the p r o o f of the inclusions (3.4), is to show that the lower b o u n d in (3.11) is reached by a contour/~l,e whose volume is b o u n d e d uniformly with respect to e close to 0 and x close to a given point x0. Such a result will be proved precisely

62

III. Geometric upper bounds

in the d e m o n s t r a t i o n of the second inclusion (3.4). For the time being, we a d m i t t h a t property, and go on with the proof of the first formula (3.4). Because of (3.1), the possibility of choosing a contour realizing the infimum in (3.11) and whose volume is uniformly b o u n d e d with respect to the p a r a m e t e r s implies that the m o d u l u s of (3.10) m a y be e s t i m a t e d by (3.12)

c c - g e ~(~'~) .

One has ~(x, c) < ½(Ira x) 2. Let us show: P r o p o s i t i o n 3.4. Let xo be a point with ~ ( x 0 , 0 ) = ½(Imx0) 2. If x --* ~(x,O) is not differentiable at xo, there exist~ c > 0 such that for x in a neighborhood of xo (3.13)

[Tu(x, A)[ < c-Xe -}[(Im~)~-~] •

Proof. Because of (3.7) and (3.12), one has ,

-~loglTu(z,A)l < s u p

L

A

+ ½ ( I m x ) 2,

~

+~(x,~)]

+O

.

Since T is subanalytic, it follows from Lojaciewicz inequalities ( T h e o r e m 1.4) t h a t there exist fl > 0, C > 0 with (3.15)

~ ( x , ~ ) ___ ~ ( x , 0 ) + C~ ~

for x close to x0. Consider the function (3.16)

g(x) -- @(x,0) - ½(Ira x0) 2 - ( I m x 0 ) ( I m x - I m x 0 ) .

One has g(z) < ½ ( I m x - I m x 0 ) 2 and by a s s u m p t i o n f(x0) -= 0. In particular, if ~ is differentiable at x0, its derivative at t h a t point must be 0. So, if f is not differentiable at x0, there exist 5 > 0 and a sequence x , converging to x0 such t h a t for every n (3.17)

g(xn) < - 5 [ z n - n0[ .

By the curve selection l e m m a ( T h e o r e m 1.3) there is a reM analytic curve issued from x0 and contained (except its origin) inside the subanalytic set { x; g(x) < - 5 I x , - x0 ] }. Using t h a t g is lipschitzian, we deduce from that the existence of a cone F with vertex at x0 in C g and of a neighborhood V of x0 such that for every x C F N V , g(x) < - ~ [ x - x o [. T h e function 1 f ( x ) -= ~ log }Tu(x, A)] - l ( I m x 0 ) 2 - ( I m x 0 ) ( I m x - I m x 0 ) is plurisubharmonic. Its value at xl, close to x0, is b o u n d e d from above by its average over the sphere centered at xl with radius g. At a point x of this sphere, f ( x ) m a y be e s t i m a t e d in general by +

-

+ o

If m o r e o v e r x E F VI V, using (3.14), (3.15) with c = e - x ~ (V > 0 to be chosen) and the choice of F, we get for f ( x ) the u p p e r b o u n d

3. Upper bounds for microsupports and second microsupports (3.19)

6+ sup [ - 3 ' a + ½ ( ~ + Ix0 - x , I ) 2 , K 7 - ~

6 ~]x0 - x,[ + C e _ ~ ]

63

+ 0(1).

T h e r e is e0 > 0, c > 0 such t h a t for Q < ~0 and xl • B(xo, ~2), the quotient of the volume of { x; ] x - x 1 [ = ~ and x • F } by the volume of the sphere S(xa, ~) = { x; I x - x 1 [ = ~ } remains b o u n d e d from below by the uniform constant c. Let us write then 1 f ( x , ) = ~ log [Tu(x,, A)[ - ½( I m x0)2 _ ( I m x 0 ) ( I m xl - I m x0)

-<

1

Is

f ( x ) dx

1

f ( x ) dx +

f ( x ) dx .

Using (3.19) to e s t i m a t e the first integral in the preceding s u m and (3.18) to e s t i m a t e the second one, one sees t h a t if Q and 7 are fixed with Q2 << "7 << Q << I there is d > 0 with

(3.20)

1

log

ITu(x,, A)I _< ½(Ira x,) - c'

for xa in the ball B(xo, ~2) and A large enough. T h e proposition follows from (3.20). T h e proposition shows th.at if (to; T0) • SS(u) and if xo = to - iTO, then necessarily T ( x 0 , 0 ) = ½(Ira x0) ~ and x ~ T(x, 0) is differentiable at x0. We must now deduce f r o m these properties t h a t (to, T0) must be in the right h a n d side of the first inclusion (3.4). Towards this end, let us prove: 3.5. Let ~2(x,e) be a (continuous) subanalytic function on IRk × [0, 1] such that x ---* ~ ( x , 0 ) is differentiable at a point xo. Then there is a sequence (xm,¢m) in IRk × ]0, 1] converging to (x0,0) such that the following holds: • For every rn E N, ~ is real analytic in a neighborhood of(Xrn,¢m). Lemma

•

W h e n m g o e s to i n a n i t y ,

c o n v e r g e s to

Proof. We may assume that the derivative o f T ( . , 0) at x0 is zero. Since T is subanalytic, there exists a stratification of IRk × [0, 1], compatible to IRk X {0}, with the p r o p e r t y t h a t t0 is analytic on the open s t r a t a and t h a t !p[,=0 is analytic on the s t r a t a of IRk × {0} opened in IRk x {0}. T h e c o m p l e m e n t F of the union of these ones in IRk × {0} is a closed subanalytic subset whose dimension is less or equal to k - 1. T h e r e exists then a n e i g h b o r h o o d V of x0 in IRk and an open cone F of IRk with vertex at x0 such that F N V does not meet F: to see that, one has just to show t h a t C,o(F, {x0}) # T~oiR k, and this is true since C,o(F , {x0}) is subanalytie with dimension less or equal to k - 1 (this last p r o p e r t y is a consequence of wing's lemma: cf. [Th]). Let W be an open s t r a t u m of IRk x ]0, 1] such that F N V C W. Assume t h a t the conclusion of the l e m m a is false. Then, possibly after replacing W by its intersection with a n e i g h b o r h o o d of (x0,0) in IRk × [0, 1], there is c > 0 with (3.21)

inf

0~

> c.

64

III. Geometric upper bounds

Let us show t h a t this implies I ~ ( ~ , 0 ) l > c on r n I°o-~(xl,0)[ < c. Let ~ > 0 be such t h a t (3.22)

Ix -- Xl[ ~<~5, X # X 1 ~

v . If not, there is x~ in F l q V with

[ ~ ( x , O ) -- ~t~(Xl,O)[ < C I X - - Xl]

a = { (x, ¢); I~ - x, I -< ,~, ¢ • ]o,,~] } c w. Let us denote by ~((x, c), s) the flow of the vector field iv.~(,,¢)l v~(~,~) over G. On has:

~2(~((x,,¢),s),¢)

(3.23)

- ¢p(x,,~) :>

cs

I¢((~,,~), ~) - ~,1 -< Id because of (3.21). Fixing s > 0 and letting then ~ go to zero (along a subsequence) we get a contradiction with the first relation (3.22).

We thus have I ~ ( ~ , 0) J > c over r n y On the other hand, since ~(~, 0)is assumed to be differentiable at x = x0, with a vanishing derivative, then is for every a > 0 a n e i g h b o r h o o d V~ C V of x0 such that

(3.24)

17:'(x, O) - ~(x0,0) I _< al x - xol

v=~(~,0) if x • V~. Let us now consider the flow ~ 0 ( x , s ) of W,~(,,0)] over F A V. Restricting V and F if necessary, we see that there is 5 > 0 such that s --~ ~50(x,s) is defined on [-~lx - xol,5lx - x01] for every x • P a V. If x • r f~ V~ with a small enough, we will thus have

(3.25)

d~l < ~;(¢o(X, ~), O) - W(~, O) < Od¢o(x, s) - Xo[ + o,1~ - xol I~o(x, ~) - xol < I~1 + I:~ - xol

for ~ • [ - ~ l x - x 0 h 6 1 ~ - x01]. Taking Id = ~1~ - x01 and choosing a small enough, the two inequalities (3.25) become contradictory. T h e result is proved.

Proof of the first assertion of Theorem 3.2. Let (to, To) E SS(u), with to = O. Because of Proposition 3.4 and of L e m m a 3.5, there is a sequence (Xm, em) converging to (to --iro, O) such that, for every m, q~ is real analytic close to (Xm,em) a n d t h a t 2_ioo_~(xm,¢m) converges to 2i ~ ( x 0 , 0 ) when m goes to infinity. Since qp(x,0) _< ½(Imx) 2 with equality at x0 (for (to, To) • SS(u)), we have 2i °o-~z(x0,0) = - I m x0. Since for m large enough, we have T(Xm,¢m) > supbr0." --y1 Re(x - z) 2 , where c~F0,~ still denotes 01"o, , N c9f2.... it follows from T h e o r e m 2.6 t h a t there exist

(3.26) (3.27)

20w

2

0

z,,, • f2~,~

(Zm--Zm)

such t h a t

2

'T* ( z , ~ , i ( x m - z m ) ) E T * C N [{l~l=,.}U.,ilhl=~/2}

Since for ]z] ---- r and R e x close to zero, - Re ( ~

C N

* UTih=~m}CNuT~NCN.

< ½(Imx) 2 - c and since, on the

other hand, 2_i°o-~(x0,0) ~ 0, we just have to keep, in the right h a n d side of (3.27), the two terms T*{Ihl=em/2} a~N ~ and T*{h=~m} C N " C o m i n g back to the definition of T~_I(o)Z*c, the first inclusion (3.4) now follows.

3. Upper bounds for microsupports and second microsupports

65

E s t i m a t e s for t h e s e c o n d m i c r o s u p p o r t We will now prove the second inclusion (3.4). We still study u microlocally in a neighborhood of (to, T0) E T * R N using the FBI transformation (3.28)

T u ( x , A) = /

e--~(~-tPu(t) dt

for x close to Xo = to - ivo. We will denote by g the associated isomorphism from C N to T * R N. Its inverse is given by (t,T) ~ t - i v . Let L = g - l ( A ) C C N and let us choose a holomorphic change of coordinates in a neighborhood of x0, centered at xo, x --+ M ( x ) = 2 such that L = {2; I m 2 = 0} (for 2 close to 0). Let gL be the holomorphic function on a neighborhood of x0 such that TL = - - I m g L fulfills condition (2.30) of Chapter II. If Y is a neighborhood of a point Y0 in C N with Re Y0 = 0 and if y E Y, set (3.29)

~e2 ( Y - ~ ) 2 - i ~ g L ( M - I ( ~ ) ) T u ( M -1 ( 2) , A) d2 e - 2(,-,~)

T 2 u ( y , A, #) =

EF0 where Fy o is a good contour for the phase at Y0. Because of formula (2.33)' of Chapter II, (3.29) is a FBI transformation of second kind along A. Because of formula (2.27) of Chapter II, the associated identification A : y --+ T * L is given, when T * L is endowed with the local coordinates coming from 2 on L, by y --* (Re y , - Im y). Let us introduce the following canonical transformation: XL : T * c N

-~

T*C N

(,-iT,,-

(3.30)

Because of the construction of gL, A is the submanifold of T * N g given by the equations T = O--~'(t Ox~ -- iT), (t, 7-) C T * R N. The complexification Ac of A in T * C N has thus for equations (3.31)

T = ~x

( t - - i'c) ,

(t,T) • T*C N .

Its image by XL is the zero section C N of T * C N and X L ( A ) is the submanifold L of C N. Modulo an exponentially decreasing remainder, one may replace in (3.29) Fy o by a ball { 2 • RN; ]21 < R } and thus write, using (3.2) (3.32) T 2 u ( y , X , # ) f~

=

). 2

~-Rn e - ~ ( y - x )

- 2

•

--,XgL(M

--i -

X

--I -

(x))---f(M

2

(x)--t) 1 A a ( t ) S z ( t ) d t d 2 .

" I~[
II~1 Z

C~°~e~(Imy)=

-

66

III. Geometric upper bounds

We will use the following notations: (3.34)

$2~,, -- { (~,z) e C N × B~; I~1 ___R, Ih(z)] > ~ } ~2,-,"~ = (Id

XTi')--l("~rr,e)

s~ = { (~,z,~) • c N × z c × ]0,~0[; I~l < R, ~ < ]h(z)l, lzl < ~ } E = { (~,z,~) • C N × Z c × ]0,~0[; I~l < R, Ih(z)l = ~, Izl < ~ } r0,~ = {~ • RN; I ~ I < R }

×d,.

(remind that the notations B~, ~- have been defined at the beginning of this section). We take r and e0 small enough so that L e m m a 3.3 implies that condition (T) of Section 2 is satisfied by ($2, 57) and by _r'0,,. If H is an h o m o t o p y of the form (2.2), with values in $2.... fulfilling conditions (2.3), we still denote by H : F0,, x [0, 1] ~ g2~,, the unique h o m o t o p y which lifts H and is such that ~rl.=o = Id x ( r r [ ~ ) - I :/'o,~ ~ r0--7~ - - ( I d x ( r r l A ) - ' ) ( r 0 , ~ ) . If/~,~ = ~r(/~o,., 1), Stokes formula shows that I~ is equal to /~1*

(3.35)

e - ~ ( yA - 2z )

- 2

-,),g,(M-~(~))-~(M-~(z)-,r(z))2a(5)dTr(~,)d ~ .

Let (3.36)

#2 1 ~(y, ~, z, #) -- 2(1 - #2) Re(y - ~)2 _ ¢pL(M-1 (~)) _ ~ R e ( M - 1 (~) _ z)2

and let us define, as in (2.5), the function (3.37)

e l (y, #, e) --- inf sup ~(y, ~, z, #) . He'He (.~,z)Erl,~=H(1,ro.e)

Let us put (3.38)

e2(y, #, e) =

sup q~(y,~,z,#) (~,~)eOro,,

with O~Fo,~ = OFo,~ f) 00,.,~ and

(3.39)

e(y, ,, ~) = sup(e, (~, ,, ~), e(y, ,, ~)).

One should remark that the function (y,#) --~ e(y,#,c) is lipschitzian uniformly in c E ]0, ~0[. Moreover, there is a constant C > 0 such that y --~ k~(y, #,~) is lipschitzian with a lipschitz constant b o u n d e d from above by C # 2 for # > 0 close to 0 and c E ]0, e0] (this follows from the fact that in (3.37), (3.38) the only t e r m of • depending on y is multiplied by #2). Moreover, by Corollary 2.5, the graph of e is subanalytic in Y x [0, 1/2] x [0, e0]. Lastly, since there is c > 0 with 2

(3.40)

#2

-~[(Im y) 2 - c] ___~2(y, ~,~) < ~ ( I m Y) 2

for # > 0 close to 0, c E ]0,~0], a similar inequality is satisfied by e . Let us put

3. Upper bounds for microsupports and second microsupports (3.41)

67

k~(y, #, O) = lim k~(y,#, c) . e--*0+

Then ~(y, #, 0) verifies (3.40) and it follows from the preceding remarks that (3.42)

(y, #) -* ~ ( y ,

#, 0)

is bounded, locally lipschizian in y uniformly with respect to # • ]0, ½], continuous on Y x ]0, ½] and its graph is subanalytic in Y x f0, ½] (because it is the image of the subanalytic subset of Y × [0, ½] given by the graph of (y, #) ---+ ~(y, #, 0) by the map (y, #, k~) --* (y, #, q~/#2), which is proper when restricted to this set). We will set (3.43)

¢(y) = lira l e ( y , # , O ) . /~---*0+

It is a lipschitzian subanalytic function. L e m m a 3 . 6 . There exist C > O such for y • Y, # • [O, ½], c • ]O, eo], A > 0 the second term I~ of the decomposition (3.32) satisfies

(3.44)

1I~1 ___ c S %

~+(~'"'~)

Proof. Let Y0 be a relatively compact subanalytic neighborhood of V0 in Y. Let ~- be the following family of relatively compact subanalytic subsets of C N × R x C N x C N × I~ × •: (3.45) { (V, #, ~, z, c, t) EY0 x ]0,½] x H x ] 0 , 1 ] ;

c_<e0 and

+(y,~,z,~)

< ~(v,~,c) +t }

{ (y,~,~,z,c,t) e Yo × 10, ½1 × ~ × 10,1]; c e ]0,c0] } { (y,F,,~,z,c,t) e r0 × ]0, ½] × o H × ]0,1]; c • ]0,c0] } { (y, ~, ~, z, c, t) • Y0 × ]0, ½] × c N × c u × ]0, c0] × ]0,1]; (~, z) • r0,~ } Let K be a large enough compact subanalytic subset of C N × C N and let us apply Theorem 1.13 to the projection: (3.46)

p: Y × ]0, ½1 × K × ]0, ~o] × 10, 11 --+ M = Y × ]0, ½1×]0, c0] × ]0, 11 (y,~,~,z,c,t) -+ (y,~,~,t)

and to the preceding family Y. We thus get a stratification (Mi)i6I of M and for every i • I, a point ei = (yi, #i, ei, ti) in Mi and a subanalytic homeomoFphism gi such that for every subset F of the family (3.45), gi]F : P - I ( M i ) n F ---+Mi × ( p - l ( e i ) VI F) is a trivialization. This trivialization gives for every e • Mi a subanalytic homeomorphism

9~,~:/'(e)

-+ p - l ( ~ )

such that for every F in the family (3.45), gi,e[Fop-a(e) is an homeomorphism from F N p-I(e) onto F VIp-a(e~). By the definition (3.39) of k~, for every i • I there is an homotopy Hi fulfilling conditions (2.3) such that if FI,,, = Hi(l, Fo,~,) one has

68

III. Geometric upper bounds sup ~(yi,5:,z,#i) <_ k~(yi,#i,ei) + lti . (~,z)EFl,,i

Let /ti be the lift of Hi and denote by FI,~--~= H'-~(1, _r'0,~, ) the lift of the contour/"l,e,. In general, Fl,~i and/"1,~ are not subanalytic contours. Anyway, it is easy to see that there exists a subanalytic chain/~/, which is homotopic to FI,~ through an h o m o t o p y preserving OFo,~ and compatible to Z ~ , such that sup ~(yi,~c, Tr(z.),l~i) ~ kO(yi,#i,ci) -}-ti (~,~)Er~ (in fact, one has essentially to approximate the C ~ mapping H i ( l , .) by a real analytic map preserving the different constraints). For every i E I and every e = (y,/~, e, t) E Mi, gi,~ is an h o m e o m o r p h i s m from J2~,~i onto ~2.... Let us denote by gi,~e the h o m e o m o r p h i s m deduced from gi,e on the coverings gi,"~: ~,~-'~ --~ J2~--/~and put ~ = g~,e(Fi). By construction, there is an h o m o t o p y from Fi onto F0,~. Conjugating this homotopy by gi~'~, we thus get a continuous h o m o t o p y from Fe onto Fo,~. Moreover, by the compatibility of gi with the sets (3.45), the composition of this h o m o t o p y with 7r satisfies conditions (2.3). Using the definition of the first set (3.45), we see also that (3.47)

sup ~(y, 2,7r(5), if) < kV(y, #, e) + t . (~,~,)Er~

Stokes formula shows that if e = (y, if, e, t) E Mi, the integral I~(y, if) equals the integral (3.35) computed at (y, if,c) over the chain F¢. From (3.1) and inequality (3.47) we deduce that there exists C > 0 such that for every (y, #, e, t) E Mi one has (3.48)

IZ;(y, ~)1 < C m e a s ( 5 ) c -/'J'~'(y'"'')+~''

-

Since the chain Fe is obtained using a trivialization over M,, one knows (el. [Ha2] or [T]) that meas(Fe) is uniformly b o u n d e d for e E Mi. Since the strata Mi are in finite number, the inequality (3.48) implies that there is C > 0 such that for every (y, #, e, t) E ]7o x ]0, 1] x ]0, e0] x ]0, 1] one has

I-r:~(y, ~)1 < C~ - K e ~ ( ~ ' " ' ~ ) + : ' ~ Letting t go to zero we get (3.44). T h e second inclusion (3.4) will be a consequence of: P r o p o s i t i o n 3.7. Assume that the function y --~ ¢(y) defined by (3.43) is not differentiable at a point yo such that ¢(y0) = l ( h n y 0 ) 2. Then, there exist I E R, c > O, #0 E ]0,½] such that for every y in a neighborhood of yo, every # E ]0,#0], every ;~ >_ 1/# 2

(3.49)

iT2u(Y, ~, ~)l

~ X--)~le~2-2~[(ImY)2-c] • C

3. Upper bounds for microsupports and second microsupports

69

Proof. Because of (3.33), (3.44) we have 1 A/22 log [T2u(y, A, v)[ / a l o g ~ + ½(Im v)~ + O ( ~ ) , _<supt,-T7

K log ¢ - A/22

+

+

Since k~(y, #, e) and #-2kr,(y, #, 0) are subanalytie, it follows from Lojaeiewiez inequalities ( T h e o r e m 1.4) that there is a positive/3 with (3.50)

kV(y,/2,e) _< kV(y,/2, 0) + cste z _
If one takes ~ =/22/~e - ~ 7 Proposition 3.4.

the result follows by the same argument as in the proof of

Proof of the second assertion of Theorem 3.2. Let (qo,q~) be a point of SS~'l(u) such that q0 = (to,to). T h e point Y0 = A - l ( ~ ( q 0 , q ~ ) ) corresponding through the identifications satisfies Rey0 = 0. Because of (3.33), (3.44), (3.50) we must have ¢(y0) = ½(Imy0) 2. Proposition 3.7 then implies that ¢ is differentiable at y0 and then 2i 0~ Oy = _ Imy0. By L e m m a 3.5, there exist a sequence (Ym,/2m)m with/2m > 0 for every m, converging to (Y0,0) such that for every m ~P(y,/2, 0) is real analytic in a neighborhood of (ym,/2m) and that ,m ~ 2~ og, 0,k (Y0) when m goes to b~y(Y",/2,,, 0 ) converges to 72 -g~y infinity. Applying once again L e m m a 3.5, we see that for every m, there is a sequence (ym,k,/2re,k, em,k)k with e,~,k > 0, converging to (gin,/2,~, 0) such that kV(y,/2, e) is real analytic in a neighborhood of (Ym,k,/2m,k,Sm,k) and that #z) k 2_ ) i Og'(y,~,k,/2,,,k,em,k Oy OqJ

converges to ~tt~ 2_ ~v(ym,/2m, 0) when k goes to infinity. i

For m large enough, k __ k0(m) large enough and (y,/2, ~) close to (ym,k,/2re,k, era,k), !P(y,/2,e) is close to ½(Imy0) 2. Because of (3.38), (3.39), we thus have ~(y,/2,e) = k~(y,/2, e). T h e o r e m 2.6 then implies that there is a point (~m,k, zm,k) C JQ~,~.~,k with ~(Y.~,k,/2m,k,e,.,k ) = ~(y.~,k,X.~,k,Zm,k,/2,,~,k ) and

(3.51)

20¢'(y~,k,/2~,k,em,k ) = -i2 N0¢ (ym'k'xm'k'zm'k'/2m'k) i Oy (~m,~, zm,k; T ~ (

)' T ~("

S*m,~ 6S*m,~

s.m~

where ,.q~ is the stratification of ~2r,~ defined before Proposition 2.1. -- 2m,k) T h e right hand side of the first formula in (3.51) is equal to z/2,.,k(ym,k " 2 and since i -2i -~-~v(ym,k,/2m,k,em,k) o~ converges to - - I m y 0 , one sees that Xm,k goes to #m,k

Rey0 = 0. So, for m large enough and k >_ ko(m) large enough, I~m,kl < n and we deduce from the second relation (3.51) that ~ ( Y m , k , Xm,k, Zm,k, ~2re,k) = 0 i . e . (3.52)

i(vm,k -

~,~ )

-- k[- ~@m~kglL(M--l(~m,k)) + "'7~'--(M-i 1 (Xm,k) -- Zm,k)] TOM-1 (X'ra,k) /2m,k

•

70

III. Geometric upper bounds

On the other hand, still using (3.51), we see that the point

(Zrn,k, 2 -~z (Yrn,k,fGrn,k,Zm,k,#rn,k)) = (Zm,k,-- i~' -'(M-l(Xm,k)#m,k

(3.53)

--Zm,k))

must be in T*{}hl. . . . ,/2} C N U T*{ h = e m . ~ } C N u T ~ C N U { ( z , ( ) E T * C N ;

zeZ

C, I z l - = r }

When k goes to infinity, em,k goes to 0 for every fixed m, so (3.3) implies (3.54)

(Zm,

i

1

\

G 0-I(T•-I(0)Z C) U { (z,~) G T * c N ; z G g c, Izl =

} •

At every point k where I m k = 0, by construction of gL, g ' L ( M - I ( k ) ) = - - I m M - ] ( k ) • Multiplying (3.52) by #2 and letting k and m successively go to infinity, we see that Zm,k converges to z0 = R e M - l ( R e y 0 ) = to. For m large enough, (3.54) thus implies that (Zm, era)= (Z,~,---~-~(M-l(~:m)- zm)) belongs to 0-1(T~_,(0)ZC ). Using the definition (3.30) of XL and (3.52) we have

m,k

(3.55)

(Re Y0, - Im y0) = l i r n

2 1 . t d M _ 1 o XL(Zm,--~m¢rn) #~,

_ _

where the dot - stands for multiplication on the fibers. Using (1.10), this means that (3.56)

( R e y 0 , - Imy0) C M o XL C A¢(O-I(T~_I(o)ZC) )

where M o XL is the m a p from T * A to T * L deduced from M o XL. Since this m a p is nothing else than the identification in the definition of SS~ 1 , the result follows. In (3.4) we obtained upper bounds for the microsupport or the second microsupport by quantities involving points of the complexifications of the varieties naturally associated with the problem. To conclude this chapter, we will give an example (coming from [L6]) showing that there is no hope to obtain similar inclusions using only the real part of the geometry. More precisely, we will prove that in the right hand side of the second inclusion (3.4), one cannot replace AC and T~_~(o)Z * c respectively by A and T~_~(o)Z. * To see that, let us take on Z = R 3 the real analytic function (3.57)

h ( x , y , t ) = t + x 6 + x4y 2 .

The distribution (3.58)

u(x, y, t) = [sup(0, h(x, y, t))]

is of the form (3.2) and thus, the inclusions (3.4) are valid for it. Let N be the submanifold of IR 3 with equation t -- 0. Denote by (x, y, T) the coordinates on A = T/~R 3 and by (x, y,~-; x*, y*, T*) the coordinates on T*A. The restriction of u to N is the function

3. Upper bounds for microsupports and second microsupports

71

uo(x, y) = x 2 ( x 2 + y2)½ .

(3.59)

A straightforward c o m p u t a t i o n shows t h a t +

=

and thus T~[~ 3 is contained in SS(u0). Let us consider in particular the point (x = 0, y -- 0; ~ = 0, r] = 1). Because of the second inclusion (3.6) of C h a p t e r II, either there is T0 • R with (3.61)

(x = 0, y = 0, t = 0;~ = 0, r / = 1, r0) E SS(u)

or there is TO E l~ with IT01 ---- 1 a n d (3.62)

(X = 0, y :

0, T0;X* = 0, y* ~--- 1 , T * = 0) • S S 2 ' l ( t t )

.

Since

(3.63)

T;_I(o)C 3 = { (x, y , - x 6 - x4y2;T(6x5 + 4x3~2), 2T~4~, T) }

the first inclusion (3.4) excludes (3.61). Because of the second inclusion (3.4) and of (3.62), we have necessarily (3.64)

(z = 0, y = 0, T0; x* = 0, y* = 1, T* = 0) C CAc(Th-~(o)C 3) •

Using the characterization (1.10) of W h i t n e y ' s n o r m a l cone (with the canonical transformation X( X, y, t; ~, rh T) = ( x, y, T; ~, r], - t ) we see that there are sequences (z,~, y,~, tn; • 3 and un ~ 0 with ~,,~,,,T,~) C T~_~(0)C x~ -~ 0, y~ --* 0, 7~ --* To, -~" ---*x * =0, /t n

-7/~ ---* y. =1,

-t~ - - - * ~_. = 0 .

Un

Un

By (3.63), this means 3 2 --* 0 1 ( 6 ~ + 4~.yn)

(3.6~)

Un

1(2x4~)

-~ ±

It n

TO

--* 0 . ~ ( x ~ + x~u,~) ~ It n

Writing the first relation (3.65) as !(2x~y~)(3 ~ Un

Yn

+ 2 y" ) --* 0 Xn

and taking the second one into account, we see that a sequence of points of T~_~(0)C 3 allowing one to get the point (3.64) m u s t verify 3Xn + 2Yn ___,0 . Yn

Xn

But the function s ---+3s + ~ never vanishes on ll~, and so the quotients *~Y~ must be imaginary numbers. T h u s we cannot replace the right h a n d side of (3.64) by CA(T~_~(o)It~3).

IV.

Semilinear

Cauchy

problem

In this last chapter, we will state and prove a theorem of Lebeau [L4] giving a geometric upper bound for the wave front set of the solution of a semilinear wave equation with Cauchy data conormal along an analytic submanifold of the Cauchy hyperplane t = 0. The interest of this result is that it is valid in large time, in particular after the formation of caustics. The method of proof relies on the theory developped in Chapter II and Chapter III. In Section 1, after stating the theorem, we display on an example the main ideas of the demonstration. Sections 2 and 3 are devoted to the detailed proof of the theorem. It is divided into two steps. In the first one, given in Section 2, we show that the wave front set estimates we are looking for follow from upper bounds for the wave front set of a family of explicit distributions. These distributions may be expressed as products of elementary solutions of the wave equation and of distributions built from the Cauchy data. In the second step, which forms the matter of Section 3, we deduce from the results of Chapters II and III geometric estimates for such distributions. To do so, we first write the products involved in the expressions under study as restrictions to the diagonal of tensor products, and we use the trace formula of Section 3 of Chapter II. Thus, we have just to get upper bounds for the wave front set and the second wave front set of distributions like those studied in Section 3 of Chapter III. Using the geometric estimates obtained there, we are able to conclude the proof of the theorem. In Section 4, lastly, we state the "swallow-tail theorem" and give some indications about various extensions of the results of this chapter.

1. Statement of the result and method of proof On N l+a = R x N d, let z = (t,x) denote the coordinates, with x = ( X l , . . . , X d ) and let

02

(1.1)

C-- at 2

02

A - - at 2

~ 02

~'Ox~ ~

be the wave operator. Let 12 be an open subset of N T M which is a domain of determination of w = 12 (q {t = 0}. Let u be a real valued continuous function on f2, locally belonging to the space C°(1Rt, H~(Na)) of continuous functions of t with values ill the Sobolev space H~(Nd), with a > d given. Let P ( t , x , u ) = ~ om p j ( t , x ) u j be a polynomial in u with real coefficients smoothly depending on (t, x) E 12. Assume that u solves the following semilinear Cauchy problem

1. Statement of the result and method of proof

73

Ou = P(t, x, u)

(1.2)

Ult=0 ~ U0 0,~t{t=0 = It 1

where u0, ul are elements of H}~c(~ ) and Hi~o-~i(w) respectively. Let V be a real analytic submanifold of w. We will assume: (1.3)

u0 a n d ul are C ~ classicM conormM distributions along V.

Let us recall the m e a n i n g of the words "classical conormal". R e m i n d first t h a t if U is an open subset of V on which exists a system of local coordinates x' = ( x ~ , . . . , x~), a C ~ symbol of degree r on V x N is a C ~ function (x', A) --~ a(x', A) such that for every a E N d-1 a n d f l E N , (1.4)

sup

z' EU, .kEN

[ ( 1 + I~l)-r+lZllD:,D{a(x',~)l]

< +oc .

Such a symbol is said classical if for every k E N, there is a function ak(x', X) s m o o t h on g x (N - {0}), positively homogeneous of degree r - k in X, such t h a t N

(1.5)

VN E N,

sup a(x',a) - E ak(x', a) XN+I < + c ~ [~1-~1 o

z'EU

T h e n (1.3) means that uo, ul are C °o outside V and t h a t every point of V has a neighborhood W, endowed with a system of local coordinates x = (z', x") E I~d-1 × R in which V M W = {x" = 0}, such t h a t u0[w and ul]w m a y be written on the form (1.6)

f e iXtt.~IIaj(x', ~") d~"

j = O, 1

for convenient C ~ classical symbols a0, al on ( W M V) × N. We want to estimate the Coo wave front set of the solution of (1.2) by an object built from V and [] in a geometric way. Let us first define some sets of sequences. Let (z,~;(,n) = (t,~,x,~;rm,~m) be a sequence indexed by m E N of points of T*C ~+d. Consider the following conditions: i) (Zm) m converges to a point of Y2,

ii) there exists a converging sequence (~/m)m Of C ~+d with t~?ml = 1 for every m and a sequence ($,~)m of C* with (,~ = $mqm, 2 for every m. iii) (zm, ~m) E Car [] i.e. ~2m = r m D e f i n i t i o n 1.1. We shall say t h a t a set g of sequences fulfilling conditions i), ii), iii) above is admissible if it contains every subsequence of any of its elements a n d if it satisfies the four following axioms: A.I: E contains every sequence (zm ;(m)m fulfilling i), ii), iii) and such t h a t (,~ --+ 0. Zlm) -----0 A.2: If (zm; ~m)m E g and if z m is a sequence of C l+d such t h a t lim(zm and lim Izm - z ~ l . I~-~l = 0, there is a subsequence of (Z'm, ~m)m which is in E.

74

IV. Semilinear Cauchy problem

A.3: If (Zrn;~m)m • ~ and the b o u n d a r y of the open half t = 0, and such that for every bicharacteristic of 0, there is a

if (Z'm)m is a sequence of C l + d , s u c h t h a t limz~m is on light cone with vertex at lim z,, which does not meet m, (Zm; (m) and (z~; (,n) belong to the s a m e complex t • (m)m belonging to g. subsequence of ( Z m,

A.4: If (Zm; (Jm)m, j = 1,... , g are g sequences in g with a same base point for every m, and if (Zr,; ~-*),n is a sequence fulfilling conditions i), ii), iii) and such t h a t lira(f,,, - ( ~ . . . . . ( N ) = 0, there is a subsequence of (zm; (m)m which is in g. Let us set now: D e f i n i t i o n 1.2. For every admissible set of sequences g, we denote by Z(E) the closure of the set of points (z, ff) • T * C I + d I o such that there is an integer N and N sequences (Zm, ~Jm)m in g, j = 1 , . . . , N , with same base point, such t h a t z = limzm, = lim(4~ + . . - + 4N). We will denote by w c a small enough neighborhood of ~o in C d and by V c the complexification of V in w c. We put (1.7)

My = { (zm, £,~)meN; (zm, ~,~)m satisfies i), ii), iii) Zm = (0, Xm),

(Zm,s¢,,,)

=

'T'* a)C

• -vc

(Tm, m) 2

and Tm

=

2 ~rn }

T h e aim of this chapter is to prove: Theorem

(1.8)

1.3. If u is a solution of (1.2) with Cauchy data satisfying (1.3), we have

WF(u[,>0) c Z(C) n T* 2

for every admissible se~ of sequences g containing .Av. We will give now on an example the principle of the proof. A detailed a n d complete d e m o n s t r a t i o n for the general case will be done in the next sections. ~r Let us take the space dimension d be equal to 3 and let u E Hlo¢(~ ) with 2 < a < 5 be solution in $2 of (1.9)

Du = u 2

u],= 0 = u0 • HL( a,,.,I,=o =

)

•

Let us denote by v the solution of the homogeneous p r o b l e m o b t a i n e d when one replaces u 2 by 0 in the right h a n d side of the first equation (1.9) and let f = u - v. Let e+ be the e l e m e n t a r y solution of the wave o p e r a t o r s u p p o r t e d in the forward light cone and denote by E + the o p e r a t o r of convolution by e+. We will use the following fact: if g • H1:c(12), then E+(llt>__0}g ) • Hl~oc(Y2) and is s u p p o r t e d in {t >_ 0}. To see that, write l{t>0Ig = a(D)l{t>_o}g + (1 - a ( D ) ) l { t > o } g where a(7-,~) is a symbol of order 0 s u p p o r t e d close to ( = 0, equal to 1 on a conic neighborhood of ~ = 0. Since l{t>0}g • L2(Nt, g a ( N d ) ) (locally), (1 - a(D))l{t>o}g is in g ' , and thus the same

1. Statement of the result and method of proof

75

5 is true for its image by E+. On the other hand, because a < ~, one sees easily that l{t>0}g E H a-2 and thus a(D)(l{t>_o}g) E H "-~. If a is conveniently chosen, this last function is supported microlocally in the domain where [] is elliptic, and thus its image by E+ is in H ~. We will use also the fact that the first two traces of E+(l{t>_0lg) on t = 0 are identically zero. If we set f + = fl{t>0}, it follows from (1.9) that

(1.10)

f+ -- E+((vl{t___0} + f+)~) •

We will study flt>o using the relation (1.10). Developping the square in (1.10), we get (1.11)

WF~+2(f+)lt>0

C [Wf~,+2(E+(v21lt>o})) U WF,+2(E+(vf+)) U WF~+2(E+(f~_))] t>0 (where we use the notation introduced in Definition 1.1 of Chapter I). The first t e r m in the right hand side may be considered to be known, since it just depends on the solution of the linear problem v. On the other hand, we will see that the action of E+ improves the regularity by 1, that is (1.12)

WF,~+2(E+(v f+ )) C 79+(WF~+l(v f+ ))

where 79+ is the operator of propagation along forward null bicharacteristics defined in r4~+1 , and since this the introduction (see Theorem 3 of Chapter 0). Since f+lt>0 E *qoc space is a n algebra by the assumption (r > 2, E+(f~_)lt>o is in ~loc~r~+2and thus the last term in (1.11) is empty. We just have to study the right hand side of the inclusion (1.12). To do so, let us use again (1.10) and write (1.13)

v f+ = vE+(l{t>o}V 2) + 2vE+(vf+) + vE+(f~ ) .

The first t e r m in the right hand side depends only on v and so is essentially known. The two other terms have a regularity which is not better than H~¢ and moreover the unknown function f+ is involved in their expression. The solution v of the linear problem m a y be written as a linear combination of integrals of the form (1.14)

f

+(z ° - z l ) w ( z l ) d z

'

j =0,1

where

(1.15)

w0(z) = u0(x) ®

wl(z) = ul(x) ®

(see Theorem 1 of Chapter 0). Then, vE+(vf+)(z °) is a linear combination of integrals of the form

(1.16)

/ e+(Z 0 -- z'l)e.t_(z 0 --

z'tl)~_t_(z'¢l

--

z2)wi(z'l)f+(z'n)wj(z2)dz '1 dz"' dz 2

for i,j E {0, 1}. One should remark that because of the support properties of e+(.), the integration in (1.16) is done on a bounded domain for every z ° fixed. The main point of the proof is to show that if (z °, (o°) e T*R 4 is such that for every (z",z"l,z 2) ~ tt 4 x R 4 x R 4

76

IV. Semilinear Cauchy problem (zO,ztl,zttl,z2;~g,O,O,O)

(1.17)

¢ WF(e+(z ° - z" )e+(z ° - z 'n)¢+(z ''1 - z2)wi(z'l)wj(z2)) for i,j E {0,1}, then (z°,¢ °) ~WF,,+](vE+(vf+)). This property will follow from the improvement of Sobolev regularity provided by the operator E+. In fact, if X is a compactly supported Coo function, one has (1.18)

]~--~+(¢)] _< C(1 + I(]) -1 .

Let 8 E C ~ ( R 4) supported close to zoo and let X E C~(IR 4) be such that X - 1 on a neighborhood of { (x,t) - (y, s); (x,t) e SuppS, t -- s ~ 0, Ix - Yl --< It -- s I }. Let us set (1.19)

U(z°,z'l,z

m , z 2) = 0 ( z ° ) ( X e + ) ( z

0 - ztl)()~e+)(z 0 - zttl)(xe+)(zttl

- z 2) .

The Fourier transform of the product of (1.16) by 8(z °) is equal to

(1.2o)

(2~)3(l+d)f o(¢o,c',,¢,,,,¢2)<(-c,,)/+(-c,,l)~j(-¢2)< ,1de it1 de 2

and because of (1.18), we have (1.21)

[~(¢0 ¢,1, cttl, (2)] <

CN(1 + [¢o + (,1 + (-1,1 Jr ¢2D-N (1 + I¢,11)(1 + 1¢-1 + ¢21)(1 + 1¢21)

Moreover, it follows from assumption (1.17) that (1.22)

/ 0 ( ¢ 0 , ¢,1, ¢,,1 ,¢ 2 )wi(-( ^ tl )wj(-( ^ 2 )de ,1 d ( 2

< CN(I+[¢°I+[¢"II)-N

if ]¢'n t _< ¢]¢°1, Supp8 is small enough and ¢0 stays in a small conic neighborhood 7 of ¢o. The contribution to (1.20) of the integration over a domain ]¢,a] < e]¢0[ is thus rapidly decreasing in ¢0 for ¢0 6 7 i.e. gives a microlocally C °O contribution to (1.16). To see that vE+(vf+) is in H "+1 microlocally close to the points (z °, ¢0) satisfying (1.17), we just have to see that (1.23)

CN(1 -~- ]¢ 0 "~ ¢t'1 _~_ ¢,,1

(1 + [¢o[)~,+1 ~

..{_

¢21)--N

,,,l>,lel (1 + I¢"1)(1 + I¢"' + ¢21)(1 + 1¢21)

x ]t~i(-¢'l)l I]+(-("1)l I~j(-¢2)1

de '1 de ''1 de 2

is in L2(d¢ °) for every i,j • {0, 1}. Let us treat the case i = j = 1. In (1.23), write

(1.24) 1{1¢,,~1_>~t¢o5 = l{]¢,,~]_>~]¢olandl¢,,~+¢~l_>~]¢~l} + l{i¢,,,]>ei¢Oland]¢,,~+¢~l<~[¢ot} and decompose (1.23) into a sum/1 + / 2 . In the expression giving I1, u s e the inequality (1 + l¢''1 + ¢2l)-1 _< cst(1 + 1¢°1)-1. Using the inequality

]l+lr°+r"+r"l+r21)-~&,,dr2 < cst / (1 + I~-,11)(1 + 1~-21)

-

(1+ It° + r"l + r2l)-' dr~ 1 + >21

< c l o g ( p -° + ~-"11 + 1) 1 + [r ° + r"l[

2. Sobolev spaces and integrations by part

77

where C is a constant independent of (r °, r 'n), we obtain for I12 the upper bound

(1.25)

(1 + IClY" [i(1+ 1~° +p +~"' + ¢21)-N+4(1+ I('1) -''+' x (1 + 1~2I)-2"+~(1 + IC''~ I) -2" d4 '1 d~'2 dC "1]

x

IS(1+!~°+{"+{"1+{

l)

+l.o +."' ~, x+b_o+~_.,i

i)).

X t~1(~11)2/+ (( H1 ) 2 W2(~-2 ) 2 df ,1 dr2 dCt,l ] where (,1 = ((,1, ~_n), (2 = ( ~ m2) and t51, ]+, ~2 are L2-functions of their argument, depending only on Wl, f+, w2 respectively. If N is large enough, the first factor in (1.25) is uniformly bounded in (0 and the second one integrable in (0. To treat the integral /2, one argues in the same way, using the inequality (1 + I¢~1) -~ < cst(1 + I¢°1) -~ on the support of the last term in (1.24). By a similar method, one shows that if (z °, C0°) is such that for every (z '1, z 'n ) (1.26)

(~°,z'~, z"l; C°,o,o) ¢

WF(~+(z ° -

z')~+(z ° - z"l)~,(z"))

for i = 0, 1, the point (z0°,¢°) 9~ WF~+](vE+(f{)). We thus proved that the H ~+2 wave front set of f+lt>0 is contained in the set of points (z °, (°) satisfying (1.17) or (1.26). The second task one has to cope with is to show that this set is included into the set Z(E) of the statement of the theorem. To do so, we will have to use the results of Chapter III. This is why we are obliged to do analyticity assumptions on the geometry.

2.

Sobolev

spaces

and

integrations

by part

In this scction, we will give the first step in the proof of Theorem 1.3. We follow closely the reference [L4]. First of all, to get rid of the limitation a < 75 we encountered in Section 1, we will measure the regularity of functions in spaces looking like L°°(•, H~(Rd)) where ~ is a d fixed real number with ~ > ~. D e f i n i t i o n 2.1. One says that a distribution u E D'(R l+d) belongs to the space A close to (to, x0) E Rl+d if there exists ~o ff C~°(R l+d) supported in a neighborhood of (t0,x0), W -= 1 close to (to,xo) such that

(2.1)

II~ull~, clef

i

(1 + I~'1)="

(i

12de < +oo

I ~ 0 - , ~)1 d~"

If ~7 is an open subset of •l+d, A(~7) will denote the space of distributions on $2 which are in A close to every point of $2. We have the following lemma: L e m m a 2.2. i) The space A(~) is a C~(~2)-module and a subalgebra of the space of

continuous functions on Q.

78

IV. Semilinear Cauchy problem

ii) Let vo E H~(Rd), V l e H~-~(Rd). Then the solution v of the Cauchy problem [3v = O, vlt=o = Vo, Otvlt=o = v~ is in A(NI+4). iii) Ira E L~(N,H~(Nd))

then E+(l{t___0}a) is in A(R~+d).

Proof. i) Let u E A(~) and ¢ E C ~ ( ~ ) . Then if ~ E C~(J?) is such that ~u satisfies (2.1), let us write

Since the first factor in the right hand side is rapidly decreasing in I~ - 771 it follows at once that II¢~U[]A < cst 119zuIIA whence the C ~ ( ~ ) - m o d u l e property. Since n > d, it follows from (2.1) and from Cauchy-Schwarz inequality that A(~) is contained in the space of continuous functions: in fact, if u is a compactly supported element of d(f2), II~IIL1 < cst [IuilA. Now, if v is another compactly supported element of A(f2), the inequalities f I~(-, ~)] dv <_ ff la(a, ~ - 7)1 da f 15(r, v)l dr dn and ( l + l ~ [ ) ~ _< c s t ( ( l + l ~ - r ~ l ) ~ + ( l + l q [ ) ~) imply IluvllA <_ cst(lli~llL, llvllA + llu[lAll5l]L, ) <_ cat IlulIAII~IIA. ii) By finite propagation speed, it is enough to prove ii) when v0, vl are compactly supported. Then, if ¢ E Cg~(~:), it follows from formula (3) of the introduction that A f f -i, . . . . . sintlgl ^ ¢ v ( r , ~ ) - - a e-"~O(t)c°s(tl~[)~°(~)dt + j e w l ~ ) - - ~ v'(~)dt

1

^

-- ½[¢(~ - I~1) + ¢ ( ~ + I~1)100(~) + 2T/~[~(~ - I~1) - ¢ ( ~ + I~1)1~(~) The result follows since the integral in r of the first (resp. the second) bracket is bounded (resp. less than cst(1 + I~1)-1). iii) We may also assume a compactly supported. If b = E+(l{t>_0}a) and if ¢ E C ~ ( N ) one has (denoting by a 2 Fourier transform with respect to x):

f

0'

sin(t~t')l~la2(t',~)l{t,>_o} dt')dt

= 2gill 1 fo +~ 82(t,, ~)e -it'~ [o(t , ,~ - I~1) - o ( t ' , , + I~1)] dr' where O(t',r) = e i''~ ft +~ e-it~¢(t) dt. Using that when iv I ---* cc O(t', r) = ~ ¢ ( t ' ) + o(I,I -~) and that fR I(r - )~ + i) -1 - (7 + A + i)-11 dr = O(log A) when a ~ + ~ , one obtains

/ l~(r,

~)1 d , _< cst

//"

la2( t', ~)l dt'(1 + I~l) -x log(2 + I~1)

whence the result. We will now define a class of Sobolev spaces which will contain distributions like the integrand of (1.16). The regularity we will require with respect to the integration variables of (1.16) will be dual to the kind of regularity enjoyed by the functions of the algebra A.

2. Sobolev spaces and integrations by part

79

Let still 17 be an open subset of R l+a which is a d o m a i n of d e t e r m i n a t i o n of w = 17 VI {t = 0}. If k E N*, we will denote by ( z , w ) = ( z l , . . . , z k , w ) the generic point of 12k x I7 with for every j = 1 , . . . , k , zj = ( t j , x j ) a n d w -- ( s , y ) . T h e dual variables will be denoted ({, w) = ( { 1 , . . . , ek,W) with {j = (rj, ~j), w = (a, 7/). Let F be the solid forward light cone of [] and let us fix a point qo = (wo,wo) E T*~2 \ 0 and a real n u m b e r v. If v is a c o m p a c t l y s u p p o r t e d distribution on g2k x f2, let us set k

(2.2)

I l v l l ~ ( , . , ) - -j /- ]- - [ ( l + l ~ j l ) -~" j=1

sup I~((rj,¢j)j=l ..... k~W)12 d~ . (rl ..... rk)

D e f i n i t i o n 2.3. One says that a distribution u E D ' ( O k × f2) is in the space M~(qo) if and only if: i) S u p p ( u ) C { ( z , w ) E ~k x ~; Vj = 1 , . . . , k ,

w - zj E -P },

it) T h e r e is 9 E C~'~(~), T - 1 close to w0 and for every q~ E C ~ ( ~ k) an integer N E N such t h a t (1 + IwI)-NN~i @ ~PUIIM(W) E L 2 ( ~ l + d ) , iii) T h e r e is T E C ~ ( J 2 ) , ~ -= 1 close to w0 and 3' open conic neighborhood of w0 in R T M - {0} such that for every • E C~(J2k), (1 + Iw[)~]l~ • ~uliM(w ) E L2(7). We will m a k e use of the following properties of the space we just defined: L e m m a 2.4. i) The space M[(qo) is a C~(f2k+l)-module. If u E M[(qo) is compactly supported in z, then v = f~k u ( z , w ) d z is in the space H qo ~ (i.e. v is microlocalty g ~ at qo ). it) / f u E MV ( qo ) and f i e ( z , ) is in A( f2 ), then the product a( zl )u( z, w ) is well defined and is in M~(qo).

Proof. i) Let u E M[(qo) and O E C~(~2k+1). We leave the verification of properties i) and it) of Definition 2.3 for 0u to the reader. To check iii) let us write 0~((,~) = f~(¢

- ¢',~ - J)~(C

J)d('

d~'

with ~2 = q5 ® Tu. Writing ( = (~-, ~), {' = (~-', ( ' ) we thus have k

1,

sup,. Io~(O-, ~), ~)1 _< cN J(1 + I~ - ~'1) -N 1-IO + I~j - ~} I) -N sup~.,I~((~', ~'), O)l)l d~' d J 1

for every large integer N. Squaring this inequality, multiplying by 11j(1 + [~j[)-2~ (1 + ]w[) 2v, integrating in ~j E R d, j -- 1 , . . . , k, and in w E "y small conic neighborhood of wo, we get by Cauchy-Schwarz inequality

L

~.(1

+

I~,l)"llO,~ll~,(~) d~ k

JJJ ' + I<~-<"'l)-'+l"i]-I(1

+1~'-~)1)-"+'(1

+l<JIT"

k

× [[(1 + I~} 17 '< sup 1,7(0-', ~"), ~')1 ~ d~' d<~'d~ <~. 1

,,r.#

80

IV. Semilinear Catchy problem

for N large enough. Let 7' be a conic neighborhood of ~ - {0} such that (1 + I~'1)" II~IIM(~') • L~(~')- Then write the preceding integral as a s u m / 1 + h where I1 (resp. h ) is given by the integration over the domain co' • 7' (resp. co' ~ 7'). If we estimate I, by f~,(1 + Ico'l)vItfill~(w')dw' and h using condition ii) of Definition 2.3, we see that

f.,(1 + Icol)"ll0~li~u(co)dco < +oo. To prove the second assertion of i), we m a y assume that u is compactly supported. If we choose 0 e C~(£2 k) such that Ou = u, one has ,3(co) = f~(4,co)O(-{)d{. The result follows from this equality. ii) We m a y assume a, u compactly supported and l = 1. If we set v = al (Zl)U(Z, w) we see that

sup 1'3((ri, 4j)j=, ..... k,~)l T

S(s~Pi~((TI'41 -- ~#')'(TJ'4J)J=2..... k'W)li Since ~ >

IgII(T:'4II)IdT~)d4tl "

d/2, the inequality I1(1 + 1~1 I)-'V

g(6)llL~
*

holds. It follows at once that (1 + I~l)'llVllM(~) • L2(7) • We will denote by e_(z) = e+(-z) the backward fundamental solution of the wave operator (e_ is supported in - F ) . If u is in M[(qo), we set P

k

E-~%)(z,w) = / 1-I e_(zj - z})~(z~,... ,4,~)dz',." .dz'~.

(2.3)

d

j=l

Because of condition i) of Definition 2.3, the integrand in (2.3) is compactly supported in z t • ~2k. We saw in Section 1 that the fundamental point, in the proof of T h e o r e m 1.3, is the use of the improvement of regularity coming from the action of E+, under the assumption (1.17). This fact appears in the second part of the following lemma: L e m m a 2.5.

Let u E M~(qo). Then

i) E~_k(u) • M;(qo). ii)

Assume that for every z • ~2k, (z, ~ = 0, q0) • WE(u). Then E~_k(u) • M; +1(qo).

Proo]. We m a y assume u compactly supported. If • C C~(~? k) there is ¢(t) E C ~ ( N ) such that, close to w0, one has @Ee_k(u) = ~Ee_k,¢(u) where (2.4)

E~_k,¢(u)

P

k

J II(¢e-)(zj

--

z))u(z',,..., 4 , w) dz~.., dz'~.

j=l

It follows from formula (3) of Chapter 0 that ICe~-0-, 4)1 < C(1 + Irl + 141)-'. Since

2. Sobolev spaces and integrations by part

81

k ~ H ¢'~''~ ¢Jdef = E~_k,¢(u)=e _ ( rj,~j).fi j=l

the first claim of the l e m m a follows. T h e a s s u m p t i o n of ii) implies t h a t there is e > 0 such t h a t fi((, w) is rapidly decreasing for I(~l--< elwl a n d w • 7. If one writes ~(~,w) = ~(~,w)l{t¢l<_~l~,i} + ~3(~,w)liNl>,i~,i} , the first t e r m is rapidly decreasing and the s u p r e m u m of the second in ~- is less t h a n (1 + ]w I) -~ s u p , 1~((, w)l. T h e second assertion of the l e m m a follows. In the sequel, we will have to m a k e use of the following kind of operators. If ( / 1 , . . . , lk) is an element of (N*) k, we denote by k

~ -~ I I ~'~ (~)

(2.5)

j=l

the o p e r a t o r sending the distribution u E ~D'(~ k+l) on the distribution on $2h+'''+l~+l (with coordinates ((Zl)j=l 3 J ..... l l , ' ' ' , (Zk)j=l ..... l,, w )) given by: (2.6)

u(z~,...,Zk,1 ZO) @ [~'(Z~ -- Zl2) @ ' ' " @ ~(Z/, t - 1 -- Z[1)] @ ' ' " . ® [ ~ ( d - z~) ® . . . ® ~(z~ ~-~ - z ~ ) l .

We have Lemma

2.6.

The operator (2.5) maps M[(qo) into M~+...+tk(qo ).

Proof. By induction on (ll,...,Ik), it is enough to prove the l e m m a when 11 = 2, 12 . . . . . lk = 1. If u is a c o m p a c t l y s u p p o r t e d element of M~(qo) and if v is its image by (2.5) we have ^

I

~(~, ¢1, ¢~,---, Ck, ~,) = ~.(¢1 + ¢I, ¢~,--., ~ , w) where we denote by (~ the dual variable of z 1i = z12. Using that, since n > 7d sup [(1 + 1~11)2~ ~

(1 + I ~ 1 - ~'1])--2~(] + I~'11)--2~ d~;] < -}-00

we deduce from this equality that v E Mk~+1(qo).

An e x a m p l e of an element belonging to one of the spaces M~(qo) we just and studied, is the distribution 5(z - w), which is in M~°(qo) for every u0 < every q0 E T * Q - 0. We will now describe the m e t h o d of integration by p a r t s which is one of steps in the proof of T h e o r e m 1.3. Let v be the solution in Q of the linear problem

(2.7)

Dv = 0 t~ I t=0 ~" u0

OtVlt=O ~ It 1

defined - 1 and the key Cauchy

82

IV. Semilinear Cauchy problem

where u0, Ul are the Cauchy data of the semilinear problem (1.2). By Lemma 2.2, ii) v belongs to A(~2). We decompose the solution u of (1.2) in the following way: (2.8)

u=v+f,

f=f++f_

,

f+=fll+t>0

} .

As in Section 1, we write

f+ = E+[P(t,x,vl{t>_o} + f+)]

(2.9)

and we will substitute in the right hand side f+ by the expression (2.9) itself and iterate. More precisely, let us set (2.10)

ao=O,

so = f + ,

f+ = a o + s o .

Assuming that for some integer l we obtained a decomposition f+ = at + st, we write (2.9) on the form f+ = E+[P(t, x, vl{t>0} + at + st)]

(2.11)

We then develop the right hand side in powers of at and of sl and we write f+ = al+l + sz+l where st+l (resp. at+l) is the sum of the monomials involving a positive power of st (resp. involving no power of st). We get the following expressions (2.12)

a/+l = E + [ E p j ( t , x ) ( J j,k

s,+, :

~ v j - k .l{t>o}al]k]

F, ,,>1 j,k

~k)v

at

s,ml],

pj(t, x) still being the coefficients of P. Using Lemma 2.2, one sees that for every l, at and st belong to A(~2). We will now obtain an expression for sl in terms of so for every I. First we must +oo define a family of vector subspaces of M = (~k=l [NqeT* n-0 M ; ° (q)] where v0 is a fixed real number with v0 < - ! 2" We define Vl° to be the C-vector subspace of M~'° = RqeT* n--{O} MkVo(q) generated by the distribution 5(z - w).We then define Vt' , 1 < i < l, by induction: Vt' is the C-vector subspace of M generated by all the distributions of the form k

k

j=l

j=l

at_ i

(zj)l{t,>_o}V"'

(zi)p,j(zj)E2k(b(zl,...

,zk, w))

where b ( z l , . . . , z k , w ) d e s c r i b e s V/i-1 and (mj, k j , n j ) satisfies 0 _< rnj < kj <_ nj < d e g P . It follows from the algebra property of A(~2) and from Lemma 2.4ii), Lemma 2.5i) and Lemma 2.6 that 17/i is contained in M if the same is true for Vzi-1. Moreover, for every i, Vz' is finite dimensional. We may now state: L e m m a 2.7. For every i C {O,...,l}, st(w) is a finite linear combination of functions of the form

2. Sobolev spaces and integrations by part

83

/ ~,_i( zl ) . . . ~,-i( zk )b( Zl , . . • ,zk,w)dzl ... dzk

(2.14)

with b C V~'. Proof. For i ---- 0, we just write st(w) ----f 6(z --w)st(z)dz. Assume t h a t 8t has been expressed in t e r m s of s t - i + l using expressions like (2.14). By (2.12), st-i+1 = E + ( ~ t - i + I ) where st-i+1 is a s u m of m o n o m i a l s of the form pjvJ-ka~[--ims'~_i. Moreover we have

/ H E+ (~t-i+l)(zJ)b(z1"" 'zk'w) dZl . . . dzk J

: / II

zk,

ez

3

where we can p e r f o r m the integration by part since, on one hand, it follows from (2.13) t h a t for every fixed w • ~, z ---* b(zl,...,zk,w) is c o m p a c t l y s u p p o r t e d in D k and on the other h a n d ~t-i+l • A(D). If we write every monomial vJ-kakl_~im.s~n_i(Zj) of gt--i+l(Zj) under the form

VJ-kak--irn f 6rn [f i st_i(zh)] dz2 . ..dz~ h=l

we obtain the expression we were looking for. T h e expression (2.14) with i = t will play the s a m e role t h a t the expressions (1.16) in Section 1. For 0 < i < l let us define (2.15)

Z~= {qeT*Ye;

3 b ( z , , . . . , z k , w ) • V t i, 3 ( z a , . . . , z , ) G Y e ' such that ( z l , . . . , z k , ¢ l

= 0 , . . . , ¢ k -- 0;q) e W F ( b ) } .

We have Proposition

2.8. For every 1 e N, WF~0+t(u)[,> 0 C WF~0+/(at) t_JZ~ U... t2 Z[ -1.

Proof. We have WF~0+l(u)[t>0 C WF(az) U W F ( s t ) and it is enough to see that if q ~ Z ° U . . - U Z[ -1, then st is microlocally in H ~°+l at q. Because of (2.14) with i = l and of L e m m a 2.4i), it is enough to see that if q ~ Z ° U - - - U Z[ -1 every element b(za,..., zk, w) of Vtt is in M~,°+t(q). Let us show that every element b(zl,..., zk, w) of V~i is in M~°+l(q) by increasing induction on i. If i = 0, this follows from the inclusion Vt° C M~~0 (q) for every q when u0 < - 3 1" Assume that the result has been proved at order i - 1. If b ( z l , . . . , Zk, w) E Vti-1 and q ~/ Z~ -1 we see from L e m m a 2.5 ii) that E~_k(b) E M~°+i-1)+l(q). It then follows from (2.13) and L e m m a 2.4 t h a t every element of Vti is in ""k az~0+l (q) (under our assumption: q ¢ Z~ U . . . . . . Z[ -1). T h e second step in the proof of T h e o r e m 1.3 is to obtain a geometric u p p e r b o u n d for W F , o + l ( a l ) U Z ° U---U Z [ - x . To do so we will prove t h a t for every element b(zl,..., Zk, w) of Vti, E°_k(b) m a y be written in t e r m s of e+ and the Cauchy d a t a t h r o u g h an expression

84

IV. Semilinear Cauchy problem

generalizing (1.17). Applying then the results of Chapters II and III we will get a geometric estimate which will provide the conclusion of Theorem 1.3.

3.

End

of

the proof

of Theorem

1.3

Let us first recall that a tree is a finite set I with art order relation such that the set of strict minorants of any element of I has only one maximal element (when it is nonempty). All the trees we will consider will be assumed connected i.e. they have just one minimal element, denoted by 0.

C: We will denote by f : I - {0} ~ I the map sending j E I - {0} onto the unique maximal element of the set of strict minorants of j. We will denote by I °° the set of maximal elements of I: I °~ = I - f ( Z - {0}). Let us set: D e f i n i t i o n 3.1. A diagram is a 4-tuple D = (I, J ' , J " , ¢ ) where i) I is a tree (connected, with minimal element denoted by 0), ii) J ' and J " are two disjoint subsets of I ~ , iii) e : J = J ' U J " --* {0,1} is a map. Let us still denote by e+ the forward elementary solution of [] and by u0, ul the Cauchy data of problem (1.2). If D is a diagram, we associate to it the following two distributions:

(3.1)

=

1-I

- zj)

jeI-{0} {e}((zJ)Jel) = H jEJ t

"da-~(')) ® u~(j)(xj) H "{tj=0}

.~(e(j)) ~{ti=0} ® 1 "

jEJ"

The products in the preceding expressions are all tensor products. The two distributions (3.1) are thus well defined. We have: L e m m a 3.2. Let ¢ E C ~ ( N ) and let [D]¢ be the distribution obtained when one replaces in the first formula (3.1) e + ( t , x ) by ¢ ( t ) e + ( t , x ) . A s s u m e that Uo E H~oc(Rd),

3. End of the proof of Theorem 1.3 Ul

E

85

with e; > ~. There exists ~ > 0 such that for any g E C~(12),

loc ~,

(3.2) is a tempered ]unction of ~ (we denoted by ¢ = ((¢j)jeI) and by ~' = ( ( ~ ; ) j E I ) and g i.~ a function of Zo alone). Proof. It is enough to prove that (3.2) may be estimated by cst(l+I¢I) M for some integer M when # is of the form e(xj) where ~ E C~(Y2), e E C~(ov). If we set &3 = (1 + I~ji)~-J~j(~j) E L2(R d) for j E J', we have

1-I~e~_j~(z~)1-IieJ,,

jeI--J

j~J'

j~g"

By a slight change of notation we may thus assume J " = 0 without affecting the generality of the result. If we do the change of coordinates

zjI t = - z j + z~(j)

j E I-

{O},

z 0I t = z 0

the dual variables are related by the formulas

E

j e I - {0),

k~f-~(j)

E

j E I-

{0}, jEI

ker(j)

where l ( j ) = { k E I; 31 E N with f(O(k) = j }. Using that in the z"-coordinates [D]¢ is a tensor product, we deduce from the preceding formulas

jex

jei-{0)

k~1(j)

Let us use the expressions we just obtained for qD{D} and g[D]¢ to compute in (3.2) the integral with respect to d ~ , j E I - J. Since the convolution of two rapidly decreasing functions is a rapidly decreasing function, we see that the modulus of (3.2) may be estimated by

jEI

jEJ

jCJ

where/9 is a rapidly decreasing function. This integral is less or equal than the product

/ ( l + [ j e ~ J rj ) - 2 l _ i ( l + l ¢ } l ) 8 ( l + i r j l + ] f } ] ) _ l ( l + i i r ; l _ i ~ } l l ) _ l irjil_~(j)(l+K}l)~(j ) dr' jEJ

86

IV. Semilinear Cauchy problem

integrated with respect to d ~ , j E J (where we used formula (12) of the introduction to A

e s t i m a t e ICe+ ].) To conclude the proof it is enough to show t h a t it is less or equal t h a n cat Ilj~j(1 + I~1) ~' log(2 + I~1) with 6' small relatively to ~; - d. To do so, decompose the domain of integration into I~1 >> (1 + I~1) and I~'1 < (1 + I~1) for every j and integrate first with respect to the indices j for which lr~[ >> (1 + I~1), using that if 6 1 > 0 , ~ 2 > 0, 6 1 + 62 < l one has

/

+°°(1 -4- ]a' - T'l)-1+~1(1

-4- I~'1) - x + ~ d~' < est(1 + la'l) -~+~1+~ .

T h e n integrate with respect to the other indices using the following inequality, left as an exercise for the reader:

+~(1 + I~'1 + IC'1)-1(1 + ]1~'1 - It'll) -1 d~' ~ cst(1 + IC'I) -1 log(2 + IC'I) • One thus obtains an estimation by cst I-[jej(1 + l~t) ~' log(2 + t~l) where 6' is a constant multiple of 6. If 6 is taken small enough, we get the conclusion. L e m m a 3.2 implies that the product (3.3)

]D] = [D]. {D}

is well defined. T h e explicit distributions whose wave front set will allow one to get an u p p e r b o u n d for the quantity WFv0+~(at) U Z ° U . . . U Z~ -1 we defined in Section 2, are given by the following definition: D e f i n i t i o n 3.3. Let k E N. One denotes by Y~/k the vector Space of distributions over Y2 x Y2k generated by all distributions of the form

(3.4)

a(z0, z') = f [Dl(z0, z', z")~(z0, z', z") dz"

for all d i a g r a m D = ( I , J ' , J " , e ) , all ~ E C~(/21II), such t h a t if I ' = I °° - J , [I' I = k (we denoted by z' = ( z j ) j e I, and by z" = ( z j ) j e I , , with I " = ( I - I °°) U J). One should r e m a r k t h a t if (z0, z ~) stays in a c o m p a c t subset of ~2 x f2 k, then z" --~ is c o m p a c t l y s u p p o r t e d in 12]/"1: in fact, for every j E I - {0} we have zj E zo -- 1-' by definition of [D]. Moreover if j E I " - {0} there is j E I °° = I ' U J and an integer l with f ( 0 ( j ) = j . One has then z j E z l + .F. But if j E I ' , zj stays in a c o m p a c t by assumption, and if 3 E J, zj = (tj, xj) with tj = 0. This implies that zj stays inside a c o m p a c t subset of Y2 (see the figure).

iD[(zo,z',z")

3. End of the proof of Theorem 1.3

87

Zo

We have: 3.4. i) The space M o is an algebra, all of whose elements are supported inside { z z (t, z); t > o }, stable under the action of E+.

Lemma

ii) If a(zo,z') E Adk, aj(zj) E Ado, lj E N for j E I', we have

(3.5)

E-*r'~ [H ~'~ H jEI ~

~,(zk)a(zo,(zj)j~,,)] e AdZ,~

kEI'

Proof. i) An element a E .A40 is a distribution of the form a(zo ) = / [Dl(z0, z")~(Zo, z") dz" .

(3.6)

It is an evidence that the product of two expressions like (3.6) gives an expression of the same form. On the other h a n d since I ' = 0, it follows from the discussion before the last figure t h a t a is s u p p o r t e d inside {t _> 0}. Lastly, since E+(a)(zo) = f e+(zo-Z'o)a(4) dz'o we see that E+(a) E .£40. ii) T h e assertion (3.5) is an i m m e d i a t e consequence of the definition of the operators E_~ Z~ b and 1-IjeI, ~b" We will use also the following lemma: Lemma

3.5. For every integer l, the distribution al defined by (2.12) is in Ado. More-

over f o r 1 < i < l, for every b(zl,... ,zk,w) C VL E°-k(b) ~ Adk

Proof. Write if w = (s, y) 1{~>0} = [ e+(w - z)~,=01 ® 1 ~z

?3(w)l {s>0/ - f e + ( w - z)[~,=0/® u0 + ~.=0/® u,] d z This shows t h a t these two distributions belong to Ado. Using (2.12) and L e m m a 3.4. i) we see t h a t at E .Ado.

88

IV. Semilinear Cauchy problem

To prove the second assertion, remark first that E_(6(z - w)) = e+(w - z) E J~l whence E_(Vt °) C 3J~. In general, if b E Vti, we deduce from (2.13) and Lemma 3.4 that E~_k(b) C 34k. Remind that, by Proposition 2.8, we know that WFvo+t(u) C WF,o+t(at)U Z~ U..-U Z[ -~. By Lemma 3.5, at E .M0. Moreover if b E Vli, E~_k(b) E 34k. Thus, since we may write b = (1-I~ [J~¢)E°-k(b), we see using the dcfinition of A//k that WE(b) C WF(E_~kb) is contained inside a finite union

UWF(/

(3.7)

indexed by diagrams D, and smooth functions ~ (were we denoted z 0 = w a n d z ' = ( z ~ , . . . , z k ) ) . By (2.13) every element b C Viz is supported in the domain { t j > 0; j E I ' }. Because of the remark following Definition 3.3 and of (3.7) we see that

(3.8)

WF(b) c { (z0, (zs be1'; C0, ((s)Jex'); 3D = (I, J', J", ~) a diagram with I °° - J = I I

3(zj)je(i_l~_{o})ug E J2Ill-I/'l-1 such that (zo, (zj)j~i,, (zj)je(1_l~_{o})uj; (0, ((j)jeI', 0) E WF(]D D }

n {

zs

=

> 0}

Coming back to the definition of Z], we thus get: P r o p o s i t i o n 3.6. Let u be the solution of problem (1.2) on X? and let u E R and q0 = (zo;(o) = (t0,x0;v0,~0) be a point of WF~(u) with to > O. There exist a diagram D and a point z = (Zj)jei_{O } E J~II[-1 with zj = ( t j , x j ) , t j ~ 0 for e v e r y j E I - - {0} such that (3.9)

(zo, (zj)je1-{o}; ¢o, O) ~ WF(ID D .

To prove Theorem 1.3, we must now obtain an upper bound for WF(ID[) with the help of the results of Chapters II and III. We will associate to every diagram D the two following complex lagrangians:

(3.10)

f

A{D} = / ( ( z j ) j ~ I , ( ~ j ) j e l ) E T*(cI+d)III; J, zj = (O, xj) if j c J, * Cd ifjcJ', ( j = ( T i , ~ j ) a n d ( x j , ~ j ) E T*V~C d UT~d (j=0ifjff

(j = (ri,~j) and (xj,~j) < T ~ C d i f j E J " } where V c is the complexification in wc of V C ~v, and

3. End of the proof of Theorem 1.3

AID] =

(3.11)

89

{ ((zj)jEx, (~j)jEI) C T*(Cl+d)JII; there exist :~-j E C TM, j E I - {0} such t h a t

(zj - zi(j) , ~ j ) E Ao

¢0=

j E I - {O}

}--~ ~= jEY- ~(0)

z,, kef-l(j)

where we denoted by AD the set (3.12)

Acl = { (t,x;At,-Ax);

(t,x) E C l+d, t 2 = Z 2, /~ e C }

U T~o}CTM U T~I+~CTM • Let us now introduce a notation. If F1 and F2 are two conic subsets of a cotangent bundle T*N M, with coordinates (z, ~), let us set, following [K-S1], [K-S2] (3.13)

F1 + F2 = { (z, ~) E T*RM; there exist sequences (z j , ¢ ~ ) , , E Fj, j = 1, 2, with z m j ~z,

j =

1,2,

C1 + C~m-~ C,

-

z ll¢ll

One should remark that, in spite of the fact that we gave the preceding definition in a local coordinate system, the object we defined is intrinsic in T*R M. We want now to prove:

Proposition

3.7.

One has

(3.14)

W F ( I D I ) C (A[D] ~ AID}) N T * R 0+d)lli .

Proof. Let us first reduce to the case when uo, ul are analytic conormal along V. In fact, the right h a n d side of (3.14) being closed, we just have to see t h a t it contains W F , ( I D I ) for every integer v. Close to a point of V, let us choose a local coordinate system (x ~, xd) flattening V to Xd = 0. Since uj, j = 0,1, is classical conormal, its associated symbol aJ(x', ~a) (see (1.5)) has an a s y m p t o t i c development which m a y be written as

(3.15)

E

j

,

ak(x' 1)('~a)+J

-k

kEN

•

+ E a~(x',--1)(--{d)+ j -k. kEN

If X E C°~(R), X ~ 0 close to 0, X - 1 outside a neighborhood of 0, the functions

(3.16)

ei~d'¢aX(~d)(~d)--Id~d

ei~d'¢dX(~d)(--~a)-td~a

and oo

with l > 0 are ramified over C - {0}. If we decompose the restrictions of these functions to N as sums of functions supported in 4-x~ > O, we deduce from (3.15) t h a t there is for every integer v a decomposition

90

IV. Semilinear Cnuchy problem

uj(x) =

(3.17)

~ ~(x)g~(x) aEA(v)

+ gJ(x)

j=0,1

where:

• A(u) is a finite set of indices, • (~)~eA(~) is a family of C ~ functions, j = 0, 1,

• gJ is an element of C"(Nd), j = 0, 1, • (g,)aeA(v) is a family of functions, supported in one of the half-spaces determined locally by V, and equal on this half space to the boundary value of a ramified function on C d - V C. Using (3.17) we estimate WF,(IDI) by the g " wave front set of the family of distributions obtained by replacing in the expression of ]DI, u0, ul by the g~'s. Changing notations we see that it is enough to estimate WF(IDI) when u0, ul satisfy the same properties than the g~. Let us now set (3.18)

IDI = [D] @ {D}[ N

where N is the diagonal of N m ~ f ~ (1-t-d)lll X ~ (l+d)lll. By Lemma 3.2, the distribution [D] ® {D} satisfies locally the assumption (3.5) of Chapter II. We may thus apply Theorem 3.1 of Chapter II and conclude that (3.19)

WF(IDI) C ~[WF([D] @ {D}) N T*R M] U ~[WF~'I ([D] ® {D}) R j ( A XN T ' N ) n T*AIA_N ]

where A is the conormal bundle to N in ]I~M and where the maps j, ~, ~ are defined by the relations (3.2), (3.3), (3.4) of Chapter II. We will now use the results of Chapter III to get geometric upper bounds for WE([D] ® {D}) and w r ~ ' l ( [ D ] ® {D}). The elementary solution e+(t,x) of [] satisfies (3.20)

e+(t,x) = cst E]klll~l
if d = 2k + 1 if d = 2k.

Since for any distribution U, W f ( [ ] k V ) C WE(U), WF2'I([3kU) C WFA4(U), it is enough to estimate WF([])] N {D}) and WF~'~([/)] ® {D}) where (3.21)

[bl((zj)keI) =

II

l{l*,u)-*sl-
if d is odd

jGI-{0}

[b]((zj)kei) =

H l{l~m)-~Jl
Let us denote by ( ( z j ) j e I , ( ~ j ) j e r ) the variable on N M = NO+d)III x ]RO+d)lq with zj = ( t j , x j ) , £j = ([j,kj). Let Z be the submanifold of N M

(3.22)

z = {

b = 0 vj • J }

and Z c be its complexification in C M. Let us consider the holomorphic function on Z c given by

3. End of the proof of Theorem 1.3

=

(3.23)

1-I [(tj(s)-ts)2-(xs(j)-xj) jeJ-{0}

91

II jeJ'

where for j E J', 0j is an equation of V C such that Oj(Y:j) E N+ if :~j C Supp(u~(/)) (remind that we reduced ourselves to the case when uj, j = O, 1, are supported in a half-space with boundary V). Let A be the connected component of Z - h - l ( 0 ) given by tl(j) --tj > Ixl(j] --xjl if j E I - {0), Oj(~j) > 0 i f j E J'. One has h]A > 0 and the function a(z, ~) = [D] ® {D} satisfies condition (3.1) of Chapter III. Since gz ~(T~_~(0)zC) is nothing but A[DI x A{D), it follows from Theorem 3.2 of Chapter III that (3.24)

WF([/)] ® {D}) C SS([/)] ® {D}) C (AID] × A{D}) N T*R M WF~I([/)] ® {D}) C SS~'I([/)] ® {D}) C CA~:(A[D] x A{D}) 71T*A.

Using formula (1.10) of Chapter III and putting (3.24) into (3.19), one gets by a direct computation the inclusion (3.14). This concludes the proof of proposition 3.7. We will now begin the last part in the proof of Theorem 1.3. Let $1, . . . , Sp be holomorphic submanifolds of C l+d and denote by 2 ( S 1 , . . . , Sp) (resp. $ ( S 1 , . . . , Sp)) the set of sequences (z,~, (m),~ in T*C l+a satisfying conditions i) and ii) (resp. i), ii) and iii)) of Section 1, and such that for every holomorphic vector field with lipschitz coefficients X, tangent to $1, . . . , Sp, with principal symbol a ( X ) , one has

(3.25)

Cm )

0

along a subsequence (Zmk, ~mk)k of (zm, ~m)m. When p = 1, one may without changing g(Sl) or E ( S l ) a s s u m e (3.25) only for vector fields with C ~ coefficients tangent to $1. Moreover, if $1 is a characteristic hypersurface for [3, one sees using a change of coordinates flattening $1, that the module of C °~ vector fields tangent to $1 is generated by d + 1 vector fields Xo, . . . , Xd, such that for every j, there are differential operators of degree 1, Aj,k, Bj, 0 <_ j, k <_ d, with d

(3.26)

[D, xj] = Z

+ Bj

i = 0,...,

k=O

One has then L e m m a 3.8. If $1 is a characteristic hypersurface for [3, the set g( S1) satisfies axioms A.1, A.2, A.3, A.4 of Section 1.

Pro@ The first two axioms A.1 and A.2 are readily verified. Axiom A.4 follows from the fact that ~ --* a(X)(z, ~) is linear for any vector field X. The verification of A.3 will make use of (3.26). Let (t, x; T, ~) E r*~2 c be a characteristic point and let

be the complex bicharacteristic starting at that point (with ( = ( r , ( ) ) . By a direct computation

92

IV. Semilinear Cauchy problem

=

o(xj)}

for j = 0 , . . . , d. Denote by aj,k(z, ~) = 2-~la(Aj,k)(z, ~). It follows from (3.26) that

d4xj)(

(s)) = }2 k

If ~4 is the m a t r i x ..4 = (aj,k(O(s)))j,k and ~(s) = t(~r(X0)(O(s)),..., (r(Xa)(O(s))), • is a solution of the differential equation d~qli(s) = A(s)~(s) and so there is a continuous function C(s) with values in N*+ with I~(s)l < C(s)l~(0)l. Axiom A.3 follows from that. Let us denote by V+ and V_ the two characteristic hypersurfaces of ~Q issued from the submanifold V of {t = 0}. On a small interval of time, V+ and V_ are smooth. We will denote by V c, V+c, V_c the complexifications of V, V+, V_ in ~QC. We will use the following lemma: L e m m a 3.9. Let E be a set of sequences satisfying condition~ i), ii), iii) and axioms A.1 to A.4. Assume that $ contains the set .47 given by (1.7). On a neighborhood of {t = 0} small enough ~o that V+ and V_ stay smooth on it, we have: i) $(V+c) U g(V_c) C $ and $(V+c) U g(V_c) verifies axioms A.1 to A.4. ii) g(V+c, V_c, {t = 0)) = g(V+c, V c) U g(V_c, V c) U $({t = 0}, V c) iii) £(V+c, V_c, {t = 0}) = g(V+c) U ~a(v_C).

Proof. Let us choose on a neighborhood of a point of V in X2 a system of real analytic coordinates (yo,gl,.-.,yd) centered at 0, such that V_ = {Y0 = 0}, V+ = {Yl = 0} and t = Y0 - Yl- Let us prove ii): the inclusion of the right h a n d side in the left hand one is obvious. Let us consider a sequence (zm, ~m)m E g(V+c, V_c, {t = 0}) with zm = ( y g , . . . , Y2)- For rn large enough one has, after extracting a subsequenee, ]Y~I -< 1 X plY?[ or [y•[ _< lly~,[ or 7]Y~[ < [Ylm [ < 2[Y~I" Let us treat the second case. If o is a vector field with lipschitz coefficients tangent to V+c, V c, one has X = x-,d z_,0 bJb--~vj bl(yo,O, y2,...,yd) -- 0 and bo(O,O, y2,...,yd) -- O. Then, if 0 E C ~ ( N ) is such that

O(s)=_lif]d< ~,O(s)_Oiflsl>

},thevectorfieldO(l~l)(b

~° + b l ~ j +,E 2 b j d

has lipschitz coefficients and is tangent to V+c, V_¢, {t = 0}. Thus, its symbol computed at (Zm, ~,,~) goes to zero, and so (Zm, ~m)m e g(V+c, Vc). T h e two other cases are similm'. To prove iii), it is enough, because of ii), to show that (3.28)

$ ( V f , V c) c

g(v ) u g(v c)

= 0}, v c) c g(v+c)

u

Since V~ is characteristic and since {t = 0} is non-characteristic, the principal symbol of [] in the chosen system of coordinates may be written (3.29)

ayo~7g -~ b7]o7]1 ~- CylT] 2 + 7logo(y; 7]t) -~ Ill/1 (y; 7]t) -k q(y; rl' )

where a, b, c are real analytic in y, b(O, O, Y2,..., Yd) • O, lj(y, .), j = 0, 1, (resp. q(y, ")) is a linear form (resp. a quadratic form) in r/' = (r/2,..., rid), real analytic in y. Let then

3. End of the proof of Theorem 1.3

93

(zm, ~m)m be a sequence of g(V+c, V c) with zm converging to a point of V c. One has then YF " ~ ---+0, YF " 77{" --* 0, y~ • r/~ ---+0, r/m ~ 0 and to prove the first inclusion (3.28) we must show that either y~' • YF --* 0 or T/~ --~ 0, at least for a subsequence. Assume that ]y~ • ~?F] -> c > 0. Since (Zm,4m) is characteristic, we see, multiplying (3.29) by y p than cst Ib(ym),~ '~ + o(1)1 < l u ~ , F ( b ( y ' ~ ) , ~ + e ( y m ) y ? . , / ( + l l ( y m , r / m ) ) I < o(1) whence r/~'~ ~ 0 (since b(0, 0, y ~ , . . . , Yd) 7~ 0). To prove the second inclusion (3.28), let us consider a sequence (Zm,4m)m of g({t = 0}, V c) with Zm converging to a point of V c. One has then y~(~;n + ~F) ~ 0, yF(r/~ + r/F ) ~ O, (y~ - yF)(r/~ - r/F ) ~ 0, 7/'m --~ 0. Using these relations and (3.29) we see that Ib(ym)l I~211~?l -- (1 + I~"l + I~71 + I~g~?l) x o(1) which implies, after extracting a subsequence, that r/~n --* 0 or ~/F --~ 0, whence the result. To show the first inclusion in i), one has just to remark that every sequence of g(V+c) U E(V_c) may be obtained by propagation from a sequence of ,Av and so, from a sequence of g. The fact that g(V+c) tO S(V_c) satisfies A.1, A.2, A.3 follows from Lemma 3.8. The fact that it satisfies A.4 is a consequence of the inclusion ~(Vc , V_c) C $(V+c) U g(V_c) which follows from iii). E n d o f t h e p r o o f of T h e o r e m 3.1. It follows from Proposition 3.6 and from Proposition 3.7 that if q0 = (z0, 40) E W F , ( u ) for some integer v there is a diagram D and points zj = ( t j , x j ) E Jh, tj >_ 0 for j E I - {0} such that (3.30)

(zo, (zj)jEi_{o }, 40,0) E (A[D] + A{D}) • Supp(IDI) .

By definition there are thus sequences (3.31)

(zj(1, k), 4j(1, k))jEI E A[D] ,

(zj(2, k), Cj(2, k))jEI E A{D}

such that 40(t, k) + 40(2, k) -~ 40 when k ~ +co and (3.32)

i) zj(1, k) --* zj, zj(2, k) --* zj, k -+ +c~, j E I, ii) 4j(1, k) + Cj(2, k) ~ 0, k ~ +co, j E I -

{0},

iii) 14j(1, k)l Izj(1, k) - zj(2, k)l ~ 0, k ~ + ~ , j E I. By (3.11), there are sequences ~j(1, k), j E I -- {0} in C l+d such that (3.33)

(zj(1, k) - zl(j)(1 , k), ~j(1, k)) E AD 4o(1, k) =

E

Zj(1, k)

jEf-l(O)

4j(1, k ) = - Z j ( 1 ,

k)+

E

~t(1, k),

j E[-{0}.

94

IV. Semilinear Cauchy problem

Using the first of the preceding conditions, we see that if zj(1, k) - zl(j)(1 ,k) does not belong to the complexification F c of F, one has ~1(1, k) = 0. Then, such a term contributes for zero to ~0(1, k), and so we may suppress all the vertices l such that there is p with f(P)(l) = j without affecting the final result. Thus we may assume that zj(1, k) - zs(j)(1 , k) E F C for every j E I - {0). Let us consider the set (3.34)

-? = { j E I;

zj = (t j, z j) and tj

= 0 } .

Because of (3.10), J C ]. If j E I ~ -- J, one has, by (3.10), ~j(2, k) = 0 and thus by (3.32) ii) ~j(1, k) --~ 0 i.e. by (3.33) ~j(1, k) --* 0. Such a vertex may be suppressed from the diagram and we may assume

j=F~c/ Let us prove the following assertions: • I f j E I ~ , then

(zj(1, k),~j(1, k))k E $(V+c, VC_,{t = 0}).

Since I ~ = J, it follows from (3.10) that (zj(2, k), ¢j(2, k)) e g(Y+c, V_c, {t = 0}) by definition of this set. By (3.32)ii) and iii) and by the fact that g(V+c, V_c, {t = 0}) verifies axiom A.2, it follows that (zj(1, k ) , - ~ j ( 1 , k) = ~j(1, k))k E $(V+c, V_c, {t = 0}). • I f j E I and

f(j) ¢ i, then (zj(1,k),~j(1, k))k E $.

If j is in i let us show that (zj(1, k), ~j(1, k))k E g(V+C, V_c, {t = 0}). When j E I ~ , we saw it just above. If j E I - I ~ , it follows for the definition of ] and from the fact that (3.30) is in Supp(iD]) that f - l ( j ) C -T. Assume by induction that for every l E / - l ( j ) , (zl(1,k),Zl(1, k))k E g(V+C,V_C,{t = 0}). For indices l such that z~(1, k) zj(1, k), we have (zj, (1, k), ~ ( 1 , k))k E 2(V+c, V_c, {t = 0}). For the other indices, the fact that (zt(1, k ) - zj(1, k),F~l(1, k))k E Aa implies that ~l(1, k) E C h a r d and thus (z,(1,k),.~,(1, k)) E = 0}) = ~(Vc) u 2(v_c) by Lemma 3.9iii). Since this set satisfies axiom A.3, it follows that (zj(1, k),3t(1, k)) E g(v+C,v_C,{t = 0}). Since 2(V+c, V_c, {t = 0}) verifies axiom A.4 and since ~,(1, k) --~ 0, it follows from the last equality (3.33) that (zj(1, k),F.j(1,k)) E $(V+c,V_C,{t = 0}) for j E -r. If, moreover, f(J) f~ ], then necessarily ~j(1, k) E Charm and so (zj(1, k), ~j(1, k))k E $(V+c,V_C,{t = 0}) C g by Lemma 3.9i).

2(v+C,v_C,{t

• If j E I -

{0}, then (zj(1, It), ~j(1, k)) k E $.

We reduced ourselves to the case when I °~ = J C/~. Taking into account the assertion we have just proved, it is enough to see that if j E I - {0} is such that Vl E f-l(j), (zl(1, k), ~1(1, k))k E $, then (zj(1, k), ~j(1, k))k E g. This follows from the fact that g satisfies axioms A.3 and A.4 by a similar reasoning than above. The theorem follows from that last assertion and from the definition of the set Z($).

4. The swMlow-tail's theorem and

various

extensions

95

4. T h e s w a l l o w - t a i l ' s t h e o r e m a n d v a r i o u s e x t e n s i o n s T h e o r e m 1.3, we finished to prove in the preceding section, gives an u p p e r b o u n d for the C °~ wave-front set of the solution of (1.2) with C a u c h y d a t a classical c o n o r m a l along a real analytic submanifold V, in terms of any admissible set of sequences g, satisfying axioms A.1 to A.4 a n d related to V by the condition A v C g (with the n o t a t i o n (1.7)). For any given geometric d a t a V, if one wishes to get an "explicit" geometric u p p e r b o u n d for W F ( u ) , one is thus reduced to the construction of a set g, satisfying the different conditions recalled above, a n d such that Z ( g ) can be estimated explicitly. This has been done by Lebeau in [L4] when V is a curve close to p a r a b o l a in two space dimension (d =- 2). In this last section, we will describe this result, w i t h o u t proof, and mention extensions of t h a t theorem. Let us consider problem (1.2) with d = 2 and assume that the C a u c h y d a t a u0, ul are classical conormal along a real analytic curve V of ll{2 which has at a unique point a non-degenerate m i n i m u m of its curvature radius (for instance, V m a y be a parabola). Let A be the union of all null bicharacteristics of [] issued from T ~ N 3 N C h a r []. T h e n A is a s m o o t h lagrangian submanifold of T * R a. If 7r : T * R 3 ~ N 3 is the projection, 7r(A) is a singular analytic hypersurface of R 3 which is the union of two irreducible c o m p o n e n t s 17+ and V_. One of them, for instance V_, is s m o o t h in t > 0. T h e other one V+ is s m o o t h close to t = 0 but develops, in t > 0, a singularity: it is a swallow-tM1, whose behaviour is shown on figure 1. This variety admits the following n a t u r a l stratification: - the singular point O, - the curve of cusp points C, - the curve of transverse self-intersection points T, - the set of s m o o t h points, (since we imposed to the s t r a t a of a stratification to be connected, one should in fact take the components of the previous subsets). T h e singular point O is of course the image by lr of the unique point of A at which rrlA has rank 0. W h e n one studies the solution of a linear C a u c h y problem, with d a t a (classical) conormal along V, one knows that u is (classical) lagrangian along A (see [H] for the definition of that last notion and for a proof of this assertion). In particular, it follows that u is C ~ on R 3 - 7r(A) = N 3 - (V_ U V+) and even t h a t u is (classical) conormal along the s m o o t h points of V_ U 17+. W h e n one studies the solution to a semilinear problem, because of the p h e n o m e n o n of interaction of singularities we recalled in the introduction, one expects new singularities p r o d u c e d by the singular point O of V+. More precisely, if F is the b o u n d a r y of the forward light cone with vertex at O, the best one can hope is t h a t the solution will be s m o o t h on lt~a__- (V- U V+ U F ) . Let (Si)iel be the following stratification of V+ U / " (see figure 2): - S0 = {O} singular point of V+, -

5'1 = C curve of cusp points of V+,

- $2 = T curve of transverse self intersection of V+,

- Sa = I curve of transverse intersection of V+ a n d / '

96

IV. Semilinear Cauchy problem - $4 = L ray of tangency of 17+ w i t h / " -

$5 = smooth points of 17+ U -P.

We have: T h e o r e m 4.1. Let u E C°(N+, Hl%¢(R~)) with a > 1 be aolution of (1.2) with classical conormal Cauchy data along V. Then 5

(4.1)

s~

•

j--~0

As we mentioned above, this theorem follows from Theorem 1.3 as soon as one is able to build a set of sequences g, satisfying the different requirements, and explicit enough so that one may prove that Z ( g ) is contained in the right hand side of (4.1). This set is built first over the smooth points of V_ U V+ U F: if we denote by S the regular part of V_ U V+ U/" (i.e. S = (V_ - V) U Ss), g is defined in a neighborhood of every point of S' by $ = d ( S c) (with the notation used in Section 3). Moreover, close to a point of $4 = L, one defines g by the equality £ = g(V+c, L c) tO $(V_c, Lc). The definition of $ at the other singular points is given by propagation fl'om the points where this set yet has been defined. It is then easy to prove that $ satisfies axioms A.1, A.2, A.3. The difficult point is to show that A.4 is also valid. The proof of this last property requires lengthy computations involving a parametrization of the swallow-tail. We refer the rash reader to the appendix of [L4] for the details. Since by construction Z(E) is contained in the right hand side of (4.1), one gets Theorem 4.1. Let us now describe briefly some extensions of Theorem 1.3 and Theorem 4.1. First of all, one can prove both results for Cauchy data which are conormal along an analytic submmlifold V, but not necessarily cIasaical conormal, i.e. one may just assume (with ,r4-o',q-oo (see [91]). O11 the other the notations of the introduction) u0 E ~ v , ul E ~ra--1,-I-oo **v hand, both theorems may be proved when the right hand side of (1.2) is more general than a polynomial in u with Coo coefficients. In fact Lebeau proved in [L5] that the same results remain true when one assumes that u is a solution of Flu = f ( t , x, u, Vu) where f is a C °o function of its arguments. Another point of interest is the eonormality of the solution u along V_ U V+ U F in the future. It has been proved in [D2] that ult>o is conormal along the smooth points of V_ U V+ U F, and also along the points of transverse intersection of 17+ U F. To conclude, let us mention that Theorem 4.1 has been proved very recently for Cauchy data conormal along a Coo submanifold V of {t = 0} by S£ Barreto [SgB] (see also [M-S£B]). The method is completely different from the one we explained above and relies on an explicit blowing-up of the singularities of V+ U/~. Of course, such an approach cannot give general results for arbitrary geometries like Theorem 1.3, but has the advantage that it needs no assumption of analyticity and provides informations about the conormality of the solution, including at singular points. Similar technics have been applied to the study of diffraction of conormal waves by Melrose-Sg Barreto-Zworski [M-Sb,B-Z].

~j

/

©

0

0

P

c~ 0

Z

t~ b~

O"Q

II 0

e~

Bibliography

[Bel]

[Be2] [Be3] [Bi-M] [BOO] [Boll

[no2] [Bo3]

[Br-I] [Ch] [D1] [D2] [D-L] [G] [Hal] [Ha2] [Hi] In] [K] [K-S1]

Beals, M.: Self spreading and strength of singularities for solutions of semi-linear wave equations. Ann. of math. 118 (1983), 187-214. Beals, M.: Vector fields associated to the non linear interaction of progressing waves. Ind. Univ. Math. J., vol 37, n ° 3, (1988), 637-666. Beals, M.: Propagation and interaction of singularities in nonlinear hyperbolic problems. Progress in Nonlinear Differential Equations and Their Applications, Birkhguser (1989). Bierstone, E.; Milman, P.D.: Semi-analytic and subanalytic sets. Inst. Htes Etudes Sci. Publ. Math., n ° 67 (1988), 5-42. Bony, J.M.: Equivalence des diverses notions de spectre singulier analytique. S~minaire Goula~uic-Schwartz, exp. n°3 (1976-77). Bony, J.M.: Interaction des singularit~s pour les ~quations aux d~riv~es partielles nomlin~aires. S~mlnaire Goulaouic-Meyer-Schwartz, exp. n°2 (1981-82). Interaction des singularit~s pour les ~quations de Klein-Gordon non lln~aires. S~minaire Goulaouic-Meyer-Schwartz, exp. n ° 10 (1983-84). Bony, J.M.: Second microlocalization and propagation of singularities for semilinear hyperbolic equations. Proceedings of the International Taniguchi Symposium HERT, Katata and Tokyo 1984, Academic Press, 11-49. Bony, J.M.: Singularit~s des solutions de probl~mes de Cauchy hyperboliques nonlin~aires. Pr~publications de l'Universit~ Paris-Sud (1985). Bros, J.; Iagolnitzer, D.: Support essentiel et structure analytique des distributions. S~minaire Goulaouic-Lions-Schwartz, exp. n ° 18 (1975-76). Chemin, J.Y.: Interaction de trois ondes dans les ~quations semi-lin~aires strictement hyperboliques d'ordre 2. Comm. in P.D.E., 12 (1), (1987), 1203-1225. Delort, J.M.: Deuxi~me microlocalisation simultan~e et front d'onde de produits. Ann. scient. Ec. Norm. Sup. 4~me s~rie, t. 23, (1990), 257-310. Delort, J.M.: Conormalit~ des ondes semi-lin~aires le long des caustiques, Amer. J. Math., 113 (1991), 593-651. Delort, J.M.; Lebeau, G.: Microfonctions I-langrangiennes, J. Math. Pures et Appl. 6 7 (1988), 39-84. G4rard, P.: Moyennisation et r~gularit4 deux-microlocale. Ann. scient. Ec. Norm. Sup. 4~me s4rie, t. 23 (1990), 89-121. Hardt, R.: Semi-algebraic local triviality in semi-algebraic mappings. Amer. J. Math. 102 (1980), 291-302. Hardt, R.: Some analytic bounds for subanalytic sets, in Geometric control theory. Birkhguser (1983), 259 267. Hironaka, H.: Introduction to real analytic sets and real analytic maps. Quaderni dei gruppi ... Inst. L. Tonelli, Pisa, 1973. HSrmander, L.: The analysis of linear partial differential operators. Grundlehren der Mathematischen Wissenschaften, Springer-Verlag (1983-85). Kashiwara, M.: B-functions and holonomic system. Invent. math. 38 (1), (1976), 33-53. Kashiwara, M.; Schapira, P.: Microlocal study of sheaves. AstSrisque 128 (1985).

100

Bibliography

[K-S2] [La] [L1] [L2] [L3] [L4] [L5]

[L6] [MR] [M-SkB] [M-SkB-Z] [s~B] [sj] [W] [Th]

Kashiwara, M.; Schapira, P.: Sheaves on manifolds. Grundlehren der Mathematischen Wissenschaften, Springer-Verlag (1990). Laurent, Y.: Probl~me de Cauchy 2-microdiff~rentiel et cycles ~vanescents. Pr@publication de l'Universit~ Paris-Sud (1988). Lebeau, G.: Fonctions harmoniques et spectre singulier. Ann. scient. Ec. Norm. Sup. (4), 13 (1980), n ° 2, 269 291. Lebeau, G.: Deuxi~me microlocalisation sur les sous-vari~t~s isotropes. Ann. Inst. Fourier, Grenoble 85, 2 (1985), 145-216. Lebeau, G.: Deuxi~me microlocalisation £ croissance. S@minaire Goulaouic-MeyerSchwartz, exp. n ° 15 (1982-83). Lebeau, G.: Equations des ondes semi-lin~aires II. Contr61e des singularit~s et caustiques non-lin~aires. Invent. math 9S (1989), 277-323. Lebeau, G.: Front d'onde des fonctions non-lin~aires et polyn6mes. S~minaire EDP, Ecole Polytechnique, exp. n ° 10 (1988-89) and Singularit~s des solutions d'~quations d'ondes semi-lin~aires, Pr@publications de l'Universitg Paris-Sud (1990). Lebeau, G.: PersonnM communication. Melrose, R.; Ritter, N.: Interaction of nonlinear progressive waves. Annals of Math. 121 (1985), 187 213. Melrose, R.; S~ Barreto, A.: Non linear interaction of a cusp and a plane. To appear. Melrose, R.; S£ Barreto, A.; Zworski, M.: Semilinear diffraction of conormal waves. To appear. S£ Barreto, A.: Evolution of semilinear waves with swallow tail singularities. Preprint, Purdue University. SjSstrand, J.: Singularit~s analytiques microlocales. Ast~risque 95 (1982). Tessier, B.: Sur la triangulation des morphismes sous-analytiques. Inst. Htes Etudes Sci., Publ. Math., n ° 70 (1989), 169-189. Thorn, R.: Ensembles et morphismes stratifies. Bull. Amer. Math. Soc, vol 75 (1969), 240-284.

Index

A,

77 74 [D], {D}, ID[, 86 E_~k, 80

Good contour,

Av,

84

20, 38

HS-wave front set,

8

Inversion formula,

12

E_%, s0

Lojaciewiecz inequalities,

A{D},

Phase of FBI transform, 16 -of second kind, 37 Phase of quantized canonical transformation, 19

A[D], M~,

88 88 86

M;~qo), 79

II-IIM(¢'-'), 79 &~, 81 +~ 89 V~', 82 Z(g), 74 Z~, 83

S~-criticM value, 55 Second microsupport (SS2A't(.)), 40 Second wave front set (WF2A'1(.)), 40 Semilinear wave equation, 73 Singular spectrum (SS(-)), 12 Sj5strand spaces ( H ; , H~, N~,), 17 Sobolev microlocal regularity, 8 Stationary phase formula, 26 Stratification --ofamap, 49 --ofaset, 48 Subanalytic - - map, 48 -set, 47 Symbol (formal ~,d, classical cod), 18

l~

Admissible set of sequences, 73 Analytic wave front set (SS(.)), 12 Characterization of WF~(-), 27 Classical conormal distribution, 73 Conormal distribution, 29 Curve selection lemrna, 48 C~-wave front set, 11 Diagram,

84

FBI transformation -of second kind, 40 -with general phase, 14 -with quadratic phase, 7 FundamentM lemma, 21 Gevrey-s wave front set W F a . (-),

48

Totally real submanifold, Trace theorem, 42 Tree, 84

33

Upper bounds for microsupports, 12

Printing: Druckhaus Beltz, Hemsbach Binding: Buchbinderei Schfiffer, Grfinstadt

Whitney's normal cone,

51

59