F.B.I. Transformation: 2nd 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

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F o r e w o r d

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 de- velopments 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 bibliograph- icai 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.

M a i n n o t a t i o n s

T M = t angen t bund le to the man i fo ld M. TxM = f iber of T M at the po in t x of M. T*M = co tangen t bund le to the man i fo ld M. T2M = fiber of T*M at the po in t z of M. TNM = n o r m a l bund le to the submani fo ld N of M . T~vM = conorma l bund le to the submm~ifold N of M. For E a vec tor bund le over M , E \ {0} or E \ 0 denotes E minus i ts zero section. For E , F two fiber bund les over M , E XM F denotes the f ibered p r o d u c t of E by F over M . Over a coo rd ina 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 feomorphism be tween two manifolds , one denotes by h the m a p it induces ~ : T'M1 --+ T'M2. In local coord ina tes h(z , ~) = (h(x), tdh(x)-I • ~). If x0 E M1 and y0 C M2, one denotes by h : (Ml ,X0) --* (M2,yo) a ge rm of m a p from the ge rm of M1 at x0 to the germ of M2 of y0. gr(~b) = g r a p h of a m a p ~b f rom a mani fo ld to a manifold . d ( , ) = euc l idean (resp. he rmi t i an ) d i s t ance on the real euc l idean (resp. the complex he rmi t i an ) space. d( , L) = d i s t ance to a subse t L. d = ex te r ior different ial on a real manifold . 0 = ho lomorph ic different ia l on a complex ana ly t i c manifo ld . c~ = an t iho lomorph ic different ial on a complex ana ly t i c manifold . dL(:r) = Lebesgue measure on C n. We will use the s t a n d a r d no t a t i on for the different spaces of d i s t r ibu t ions : C ~ (com- p a c t l y s u p p o r t e d smoo th funct ions) , S (Schwar tz space) , S ' ( t e m p e r e d d i s t r ibu t ions ) , H ~ (Sobolev spaces) , . . .

C o n t e n t s

O. I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

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 f i r s t m i c r o l o c a l l z a t i o n . . . . 7

1. FBI t r ans fo rma t ion wi th quadra t i c phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2. Four ier -Bros- Iagolni tzer t ransformat ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3. Quan t i zed canonical t ransformat ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4. Change of F B I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

I I . S e c o n d m i c r o l o c a l i z a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1. Second micro loca l iza t ion along T{*0}IR n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2. Second micro loca l iza t ion along a lagrangian submanifo ld . . . . . . . . . . . . . . . . . . . . . . 31

3. Trace theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

I I I . G e o m e t r i c u p p e r b o u n d s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

1. Subana ly t i c sets and subanaly t ie maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2. Cr i t ica l points and cri t ical values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3. Upper bounds for mic rosuppor t s and second microsuppor t s . . . . . . . . . . . . . . . . . . . . 58

I V . S e m i l i n e a r C a u c h y p r o b l e m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

1. S t a t emen t of the result and m e t h o d of proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

2. Sobolev spaces and in tegra t ions by par t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3. End of the p roof of T h e o r e m 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4. The swallow-tai l ' s t heorem and various extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

B i b l i o g r a p h y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

I n d e x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

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

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

~2

(1) [] - Ot 2 A~ --

02 d ~2

Ot2 ;~1: Ox~ "

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

O u = f ( t , x ) t > O

ul,=0 = u0(z) (2) Ou

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 t ransformation 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 t ransform 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 H~-l(I~d) . Then (2) has

a unique solution u E C l ( ~ + , S ' ( N d ) ) . It is given by

/o'/ ! (4) u(t, x) = 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 (Rd) ) , then D~u E L2(N+, H~-k(Nd)). This implies that ult>o is in the ~ - l+d • space Hlo¢(N+ ) ff f E H~I (NI+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:

s - -1 T h e o r e m 2. Let ~ be a determination domain of w. Let uo E H~o¢(W ), ul E Hlo ~ (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) in Y2

(5) u l ,=o = Ilo on

O_~ t= 0 ~--- I l l 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 Card = {(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 w i t h

< 0,

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(z l , 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

0 0 (10) xj Ox---k - x k Oxj 1 < j 7~ k < d

0 0 x j - ~ + t O x j 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

c - (11) Ix(Xl "'~'Xme+)[ _< Cmlx~+(- ,~)l _<

1 + 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

X 0 the following way: if ( to,xo) • Supp(e+), t0 z ~ x02, the fields X( )b-77~j, J = 1 , . . . ,d, and

4 O. Introduction

X(X) o with X e C ~ ( l ~ l + d ) , Supp xnr = 0, are in .Ad and thus e+ is C ~ close to (to, x0). On the o ther hand , if (to, x0) satisfies to 2 = x02 ¢ 0, there is, close to (to, x0), a sys tem of

," 0 local coordinates (y0 . . , ya) such tha 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. I t follows tha t W F ( e + ) C T~-R a+a

close to (to, x0).

In the preceding proof, we used the uppe r bound (11) of IX~+[. In fact , there is a be t t e r u p p e r bound , we will have to use in Chap te r IV:

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

(12) t2-~+(T, ~)1 < C(1 + I~l + I~-l)-x( 1 + I1 1- I~-II) -1 •

Pro@ Because of the suppor t p rope r ty enjoyed by e+, we m a y always assume tha t X is a compac t ly suppor t ed funct ion of the single var iable t. Then , by (3), X~+(v, {) =

+oo ~-- i t r . [4"~sin tl~l fo ~ ; ( ~ } ~ dt. Using tha t for any complex n u m b e r a one has

fo +~ x(t)e -it~ <_ C ( l + I~1) -1 dt

the inequal i ty (12) follows.

Before beginning the descript ion of the nonl inear p rob lems we will be in teres ted in, let us ment ion tha t , of course, Theo rem 3 admi t s a more precise s ta tement . In fact, as is well known (see [HI), W F ( u ) \ W F ( f ) is foliated by the integral curves of the hami l ton ian field of G(n).

We will now s tudy the p rob lem of control of microlocal singulari t ies of the solution I~A of a semil inear Cauehy p rob lem of the form u, given in the space Hz'oc(~2 ) with s > 2 ,

[]~ = f ( t , x , u)

(13) ~1,=0 = ~0

Ou

~ " t=0 ---- ul

where f is a C ~ funct ion 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 phenomenon one has to cope with to solve such a problem, is the one of in te rac t ion of singularit ies. For instance, let us take two dis t r ibut ions wi th compac t suppor t oi1 IR '~ vl , ve and assume tha t W F ( v j ) C { (0; A~J), A _> 0 }, where ~1 and ~2 are two non-zero e lements of T0*R n such tha t there exist no negat ive real n u m b e r 0 with ~a __ ~ 2 . Assume moreover tha t vl and v2 belong to H ~ ( R n) for some a > n/2 . Then the p roduc t vlv2 exists, and defines an e lement of H~(R'~) . Wri t ing v- - -~ , v~(~) = 01 *02((), one sees easily tha t

(14) WF(v ]v2 ) C { (0, AI~ 1 + A2~2); A1 ~ 0, A 2 ~ 0 } .

0. Introduction 5

In general, there is no bet ter upper bound for WF(vlv2), i.e. there are, in this last set, directions belonging neither to WF(vl ) 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 may 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 mat ter 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 WF(u) will have to take into account the points obtained by propagat ion 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, WF(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

~ of the equation consider in 2 space dimension a solution u E Hio c, with s > 2 , [3u = f( t , x, u), such that u]t<to<O is conormal along three characteristic hypersurfaces Z1, ~2, Z3 which, in t < to, intersect just two by two and transversally (conormal still meaning that uIt<t 0 keeps a fixed Sobolev regularity when one applies to it any number of C °O vector fields tangent to Z1 U ~2 U Z3). Assume moreover that in {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 IR 2 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. Four ier -Bros - Iago ln i t zer t r a n s f o r m a t i o n and first

micro loca l i za t ion

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. 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

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) [ e-}(~- t )~u t Tu(z, : i ( ) a o

J

It is an entire function of the complex variable x, real analytic with respect to the parameter 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 lmxl )Ne -)(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 bet ter 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*Rn - (0) (what will be denoted by u E Htto,+o)) or that (t0,ro) is not in the Hs-wave front set of u ( ( to , ro ) $ WF,(u)) if there is x E CF(Rn), x - 1 close to to , 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 ( to , ro) E T*Rn - (0) is not in WFs(u) if and only i f 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'(Rn), 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 Cs(u) if and only if there exists a neighborhood V of 7 0

in Rn such that

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

(1.7)

whence the equality

1. FBI transformation with quadratic phase

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

(1.8) [Tu(x,A)[2e-)'(Im')2dL(x)= ""ff da dTe-)'(r-~)2]ft(a)]2 . " - - i V '~

Since the contr ibut ion to the last integral coming from the domain {]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 tha t

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

(1.9)

with

(1.10) K~(a, m

Let us show that if V CC V' are two relatively compact open subsets of ll~ n - {0} and if F =- Ut>1 t v , 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 ) .Iv e-(x*-*)~ d7 <_ K~,(a, A)

(1.12) K ~ ( G A ) G CA~[o[2Slr,(o)/v, e -()~'-~)~ d~" + Ce -~(~+ ~lqlL) .

In fact, let V" be an open subset such that V CC V" CC V'.

- If } ¢ Y", one has 17 - ~L > ~ (1+ ~ ) for every 7 e V. Thus (1.12) is t rue because of the exponential te rm in its right hand 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 hand

c s t . ~ 2 s - {

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

___ f & > cst.lol2 - I<__~

and

whence (I.ii).

I v e-(~r-~')2 d7 <_ cst .A-"


Modifying V if necessary, we deduce from (1.11) and (1.12) tha t (1.9) is equivalent to

(1.13) dA fv e-('-o)2 <

with F = Ut> l t V .

One may 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 ) ]

The last integral is uniformly bounded from above when ~ describes R ~, and uniformly bounded f rom below by a positive constant when ~r stays in F ' with F ~ C C / ' .

It follows tha t (1.13) (and thus (1.9)) is equivalent (after a modificat ion of _r') to the condit ion

(1.15) f lal~[~(~)l ~ da < + ~ Ja CF

which is equivalent to To ¢ Z~(u). The lemma is proved.

P r o o f o f T h e o r e m 1.2: The distr ibution u is H s microlocally at (to, To) if and only if there exists X E C~¢(Rn), X = 1 close to to, such tha 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 I r ( x u ) ( x , A)]2e -;~(I~ ~)2 d L ( x ) dA < + c ¢ .

n - i v

We just have to see that this condition is equivalent to (1.4). Assume first tha t (1.16) is true and let U be a ne ighborhood of to such tha t X - 1

close to U. The integral defined as (1.16) with the integrat ion domain 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 ne ighborhood of to, V be a ne ighborhood of To such that U - i V C C W and let X E C ~ ( U ) with X -z 1 close to to. Wri t ing

1. FBI transformation with quadratic phase 11

we get, with the nota t ion (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).

Theorem 1.2 gives as a corollary a character izat ion of the C °° wave front set in terms of FBI t ransformation.Recal l tha t the point (to, r0) E T*N '~ - {0} is not in the C °° wave front set of the distr ibution u, WF(u) , if and only if there exist X E C~X~(Nn),

X - 1 close to to and a conic ne ighborhood F of r0 in l~ ~ - {0} such tha t for every integer N:

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

(1.17)

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 ) i f and only i f there

exists a neighborhood W of to - i7o in C a such that for every N E N:

(1.18) sup zEw, A>_I

ANITa(x, A)le-} (Ira x): < +co .

Proof. The condit ion (to, TO) ¢ W F ( u ) is equivalent to the following assertion: There exists a conic ne ighborhood F of r0 in N n - {0} and a ne ighborhood U of to in N n such tha t for every s E IR and every (t, r ) E g x _r, (t, 7-) ~ WFs(u ) . On the other hand, condit ion (1.18) is equivalent to the existence of a ne ighborhood 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.

The result follows then from Theorem 1.2: one has just to remark that , by inspection of its proof, one may choose in (1.19) a same neighborhood W for every s E IR as soon as one may take in (1.13) a same cone F for every s (and conversely).

The t ransformat ion u --* T u ( x , A) may also be used to characterize the analyt ic wave front set (sometimes called analytic singular spec t rum or microsuppor t ) and the Gevrey wave front set of a dis tr ibut ion u (in fact, it had been in t roduced 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 th rough b o u n d a r y values of holomorphic functions, or th rough cohomological tools) may be found - in the case of the analytic singular spec t rum - 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.

De 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 e-~[(Im~)~-~]lTu(x,A)l < + c o . Wx[1,+oo[

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 < +oo . 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 , > T u ( t + i r w , A) dw I=1

where D = (D1, . . ,D~) and Dj 1 a • - - i O x j "

Then, for every r E ]0, 1], u~ is a real analytic funct ion 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~(Rn) . By definition of Tu,

(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 ) d w .

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 13

L e m m a 1.7. For every funct ion ¢ E C~(N '~) , one has

L+~° L --}(1 +(w,D))T¢(t- iw,)Odw (1.25) = ½(2 ) I=1

Proof. From Fourier inversion formula, we see

(1.26) ¢ ( 0 ) = ~-~o+lim ~ 1 JR- x~- e iS~-~lr l¢(s )dsdr"

We will deform the in tegra t ion contour with respect to 7 in the complex domain. For ~r C C '~ s taying 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 de te rmina t ion of the square root which is posit ive on the posit ive half-axis. Take a > 0 small enough so tha t alsl < 1 for every s c Supp(¢) . If we pu t

= ,+/asl~l, one has Re I~1 >- cst Id . Since dcq A-- .Adan ---- ( l+ ia ( s , I@1}) dT-1 A.. .Advn, Stokes fo rmula appl ied to (1.26) allows one to replace the real in tegra t ion contour in 7 by ~r = r + iasIrl, i.e.

There is a cons tant c > 0, independent of e, such tha t

O ( i s w - a s 2 1 w l - c l a l ) >_cIw I .

By in tegra t ions by par t s , and since ¢ is C a , we see tha t in (1.27) the integral wi th respect to ds is rap id ly decreasing in Ir[, uni formly in e. Pass ing to the limit when e --* 0+ we ob ta in

(1.28) ¢(O)----(2"rr)-nJ~"f x~" ei ..... ' ,r l( l +ia(s,~))+(s)d$dT "

This ident i ty holds for every small enough real posi t ive n u m b e r a. But since the right hand side of (1.28) is an holomorphic funct ion of a in the half-plane Re a > 0, (1.28) is t rue for every such a.

Take a = ½, apply (1.28) to ¢(t + .) and make the change of variable r --= -,~w, ,~ E IR~_, w E S ~-1. One gets

(1.29) ¢(t)

i (t - s ,w) ) ¢ (s )ds =(2?r)-njo+°°~"-IdAj~.l=idwi e'M' .... )- ('')' (i + ~

and equal i ty (1.25) follows.


2. Fourier-Bros-Iagolnitzer transformations

In the preceding section, we studied microlocal regulari ty of distr ibutions in terms of a FBI t ransformat ion given by a quadrat ic phase.We wish now to define more general t ransformations, of the kind

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

with phases g(x, t), looking like the phase of the first section

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

We will first br ing out the main propert ies g has to enjoy, so tha t the associated trans- format ion (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 funct ion in a ne ighborhood of (xo,to) in C n x C ". We saw in the preceding section that , in the case of a quadrat ic phase, the microlocal regulari ty of u is reflected by the asympto t ic 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)

Thus, we will have

(2.4)

N .

sup 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 smooth funct ion of x close to x0. It will be so part icular ly if the funct ion t ~ - Img(x0 , t), defined for t real close to to, has a non-degenerate m a x i m um at t = to. This leads us to in t roduce the assumptions:

(2.5) V t ( - Img(xo,to)) = 0 V2( - Img(xo,to)) << 0

where Vt stands for the derivative with respect to real t, and Vt 2 for the Hessian matrix. Under these conditions, mult ipl ication 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 compac t ly suppor ted distribution, vanishing in a ne ighborhood of to and x is close enough to x0.

For x close to x0, we deduce f rom (2.5) tha t the funct ion

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

has a unique critical point in the real domain, close to to, t(x). Moreover, t(x) is a real analytic funct ion of x and is a local max imum of (2.7).

2. Fourier-Bros-Iagolnitzer transformations 15

We shall now see tha t e i~g(~,t) has also localizing proper t ies wi th respect to the f requency variables. If u is compac t ly suppor t ed in a ne ighborhood of to on which g(xo, ") is defined, we have for x close to x0,

/J(]t ei)~(g(x't)+tr)tt(~7 ") dr dt (2.8) Tgu(x,X)= (2~) n -tol <C B

Let us assume u regular enough so tha t the integral in (2.8) is absolute ly convergent . Let X E C ~ ( R n) be suppor t ed in It - t01 < ~, X -- 1 on a ne ighborhood of ]t - to] _< ~. Let us consider the complex contour

OImg . (2.9) Z = { t +iex(t)(T + ~ ( x , t ) ) ( l + ]r]2)-1; I t - - t o [ < C, t r e a l }

with ~ small enough to tha t g(xo, .) be defined in a ne ighborhood of JT. If t" C Z , one has

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

O lm g O lm g (2.10) - Img(x,t) ~m- t ( z , t )¢x ( t ) ( r + 0-Tm-~m t (X, t ) ) (1 + Ir12) -1

OImg . - v ~ x ( t ) ( v + O---i-~-mt (x,t)) ( 1 + ITI2) -1 + o(~2).

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

(2.11) (2~)~ f~e~, / ~ eiX(g(~'~)+~)~(AT)dtdT .

On the piece of ~ where ] R e t - to] > ~, one has for Ix - x0] and ~ small enough, because of (2.10) and (2.5),

(2.12) - I m ( g ( x , t ) + t r ) < ~(x) - c

for some posi t ive cons tant c. On the piece of w where [ Re t - to ] < ~, we have for ~ small enough

Olmg x, t))2(1 + - ' + 2 ) (2.13) - Im(g(x , t) + tv) < - I m g(x, t) - e (r + a--i-m-Trot (

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

~ f l e i~(~( ~'t)+t~) u( AT ) dt dr (2.14) (2~) " + oo~:{ (zo,to)l<C Re[_E~,<~

modu lo an exponent ia l ly decreasing remainder . It follows f rom (2.14) tha t (2.1) describes u microlocal ly close to ( t o , - ~ ( x o , t o ) ) when the hypothes is (2.5) are fulfilled. It is then na tu ra l to ask tha t the m a p

O l m g (2.15) x ~ (t(x), k


realize an i somorph ism f rom a ne ighborhood of x0 in C n to a ne ighborhood of (to, To = _ O Im~ (x0, t0)) in T * R n To ensure tha t , it is enough to assume tha t the differential 0Ira t of (2.15) at x0 be invertible. Taking (2.5) into account, this is equivalent by an easy calculat ion left to the reader to

02Im~ 02 Im~ / , ~ (2.16) d e t ORetORex(XO'tO)o2 Iml7 ORetOlmx(XO'bO)lo~Img " * ~1 # 0 .

OImtc3Rex(Xo, tO) OImtOImx(XO,~O)]

We will impose hypothes is (2.5) and (2.16) to g. They will allow us to show tha t the F B I t r ans fo rma t ion (2.1) enjoys similar proper t ies to those of Section 1.

First , let us give a new formula t ion of these hypothes is in t e rms of ho lomorphic derivatives. We will denote by R 2~ the space C ° endowed with its under ly ing real s t ruc tu re and we will identify T*C ~ to T * R 2~ using the i somorph i sm given on every fiber by

• n (Re x,Im x) T~ C ~ T* lt~ 2~ (2.17)

( u - Im

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 ~ funct ion 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 denot ing the holomorphic derivat ive

(2.19)

If g is

0 , 0 i 0 ) = (bffez a l m x

an holomorphic funct ion on C n, Cauchy-Riemann formulas show tha t d ( - I m g ) is the image by (2.18) of the holomorphic differential Og.

Formula t ing condit ions (2.5) and (2.16) in te rms of ho lomorphic 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 IR n is an holomorphic funct ion (x, t) --* g(x, t) defined in a ne ighborhood of (x0, to) in C ~ x C" satisfying the following three conditions:

Og R~ i ) -5[(xo,to) = - T 0 c - { 0 }

02g t (2.20) ii) Im ~ - ( x 0 , 0) > 0

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

As we saw before, condit ions i) and ii) imply t ha t for x close to x0 the res t r ic t ion 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 . The 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

(2.22) q0(x) = -- Im g(x, t(x)) .

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 is positive definite for x near x0). The set

(2.23) { Og (t, t); x, og ~xx(X,t)); (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 C" . ~- (x)) ; E 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 . 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

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]:

Def 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

v : U x [1,+oo[--~ C (3.1) (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


(resp. such tha t there is N E 1N with

(3.3) sup < zEU X>l

resp. such tha t there is ~ > 0 with

sup ( e -~ (~" / -~ ) lv (z ,A)O < + ~ ). zEU )~>_1

If z0 E C" , one puts H" = lira H$(U) (resp. H~,~ 0 = l im 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 may 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 ~

• The 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 A d ~+o ~ A-kak(z) whose coelCficients ak are holomorphic functions on U such tha t there exists C > 0 ( independent of k) with supv lakl < ck+lk! for every k.

• The 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 A d ~ 0 + ~ A-kak(z) in

Sd(U) fulfilling the following condition: 3C > O, and VN E N,

N supA N + ] - d sup a(z, A) -- E ak(z)Ad--k <- cN+IN! A>_I zEU o

One should remark tha 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 1N and every z E U, la(z, A)I < cN+IN!A -(N+O+d. Taking for N the integral par t of A/C, one sees tha t a E No(U).

Lastly, let us ment ion tha t if A d 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

(3.5) ~0e 0 oo tk_ 1

a(z ,A)=Adao(z)+A d e - A t E a k ( z ) ~ _ l ) ~ . d t 1

with e0 small enough so that the serie converges when z E U. Our aim is to define an operator act ing 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


where G is an holomorphic funct ion 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 tudy first the case of pseudodifferential operators , i.e. the case when G(y, x, O) = (y - z ) . O. For every fixed y, the funct ion

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

has a non-degenerate critical point at x = y, 0 = 2 ~ ( 7 y). Moreover, an easy computa- t ion shows tha t this critical point is a saddle point, i.e. has s ignature 0. The quadrat ic par t of (3.7) at x = y, 0 = 72 ~_~, (y) is thus a quadrat ic form whose isotropic cone sep- arates the set of points where it is positive definite f rom the set of points where it is negative definite:

>> 0

/

For fixed y, the upper b o u n d of (3.7) when (x, 0) describes /~ will be the lowest i f /~ is chosen as on the figure, i.e. contained, except one point, in the open set << 0 and non- tangent to the isotropic cone. To be able to do the same in the case of a general phase G, let us set:

D e f i n i t i o n 3.3. Let ~ be a real analytic function in a ne ighborhood 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 funct ion G(y , x ,O) in a ne ighborhood of (yo,xo,Oo)

such tha t the funct ion

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

has a non-degenera te critical point with signature 0 at (x0,00).

Of course, the preceding condit ion implies tha t for y close enough to y0, the funct ion (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 analyt ic funct ion of y. We will use contours of the following kind:


D e f i n i t i o n 3 .4 . Let w ~ f ( w ) be a Coo function in a ne ighborhood of 0 in N q. Assume tha t f has at w = 0 a non-degenerate critical point with s ignature (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 ne ighborhood of B into Nq such tha t 3'(0) = 0, 7 (B) = F (F will be endowed with the or ientat ion coming from B th rough 7), and

(3.9) f ( w ) - f(O) < -c lwl ~ 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, under the assumptions of the definition on f , Morse lemma shows that there is a system of local coordinates centered at 0 in which

q+ q 2 2

f ( w ) = ~ w j ~ wj .

1 q + + l

2 2 < c 2 } If c > O is small enough, F = { w • N q ; w j = O , j = l , . . . , q + , W q + + l + . . . + W q _

(with its na tura l orientat ion) is a good contour for f .

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 ]R p × N q such that w --~ f ( yo ,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

a Z y - ( 1 " y o - F y ) 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 tha t f ( y o , w ) = w '2 - w ''2 (by Morse lemma). The 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 ne ighborhood of { w; Iwl < 26 } and there exists a C °O funct ion 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 ( y 0 , 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


(3.12) G = e [0,11, It"l _<

.

The conclusion of the proposi t ion is fulfilled by Zv, Fu as soon as [y - Y0[ << ~-

One shows in the same way tha t if/ '10 and F~0 are two good contours for f(Yo, "),

there exists Z~0 as in the proposi t ion such tha t 0Zy 0 - (Ful0 - Fu20) is conta ined in { w;

f (yo ,w) < f(yo,O) - c }. We will now prove the " fundamenta l l emma" of Sjhst rand [Sj]. Let us first recall t ha t a real quadra t ic form g on cq ~ N 2q is said of Levi type if

g(iw) = g(w) for every w G C q and tha t a real quadra t ic fo rm h on C q is p lu r iharmonic (i.e. OOh = 0) if and only if h(iw) = - h ( w ) for every w E cq. An a rb i t r a ry real quadra t ic fo rm on C q can be uniquely wri t ten as a sum Q = g + h of a Levi form g and of a p lur iharmonic form h (one has g( w ) = ½ ( Q( w ) + Q( iw ) ), h( w ) = ½ ( Q( w ) - Q( iw ) ) ).

The form Q is p lur i subharmonic (i.e. OOQ _> 0) if and only if g > 0 or, more generally, if there is a p lur iharmonie form h0 with g >_ h0. Let us prove then:

L a m i n a 3 .6 . Let (y, w) ~ f (y , w) be a plurisubharmonie C ~ function in a neighbor-

hood 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.

Proof. We mus t see tha t the quadra t ic form \72y@(0) is p lur i subharmonic . 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 quadra t ic form. The sum of the first two te rms is p lur iharmonic . If w(y) is the critical point of w ~ Q(9, w), we mus t show tha t Q(y, w(y)) is p lur isubharmonic . Let us write Q(y, w) = Qo(W) + B(w, y) + QI(y)

with Q0, Q1 quadra t ic forms and B bil inear form. By assumpt ion , Q0 is plurisub- ha rmonic with s ignature 0. Let F be a q-dimensional real subspace of G q such tha t Q0 IF << 0. Since Q0 = g0 + h0 with go of Levi type, g0 > 0 and h0 plur iharmonic , one has Qo(iw) > ho(iw) = -ho(w) >_ -Qo(w) whence Qo[iF >> O. By a l inear 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 p lur i subharmonic , there exists a p lur iharmonic fo rm 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 tha t w ~ h(0, w) has at w = 0 a non-degenera te critical point wi th s ignature 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 quadra t ic forms, it follows f rom the results we recalled just before the s t a t ement of the l e m m a tha t it is enough to see tha t @1 is p lur iharmonic . But , since h is p lur iharmonic , it is equal to the real pa r t of an holomorphic function.


As a critical value of h(y, .), ~l(y) is thus also the real part of an holomorphic function, and so, is pluriharmonic. The lemma 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

(3.15) Av(y, 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"y o where Fy o is the intersection of F0 with a small neighborhood of (x0,00). If we choose a parametrizat ion 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_<cst

The square of the L 2 norm of the right hand side of (3.17) over a small neighborhood V of Y0 is bounded from above by

C(;l<cst e-;~clt'2 dr)(fv/tl<_cst e-)~c'tl2-2~°(~(y)+~t)'v(x(y) + xt' )~ )'2 dt dL(y)) .

Since y ---+ x(y) is by hypothesis a local diffeomorphism, this expression can be estimated by

(3.18) CA-( '~+N)/u e - 2 ~ ' ( ' ) l v ( x ' A)I2 dL(x)

where U is a convenient neighborhood of x0.

4. Change of FBI 23

The conclusion of the theorem follows f rom the expression (3.15) of Av and f rom (3.18).

4. Change of F B I

In the first section, we ob ta ined character izat ions of microlocal H ~, C ~° or analyt ic regular i ty using the t r ans fo rm

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

We want now to obta in analogous character izat ions using the more general t r ans forma- t ion

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

where ~ is a phase verifying the condit ions (2.20). Let us r emark first t ha t if u is a compac t ly suppor ted dis tr ibut ion, one m a y write

(4.1) a s

n x ~ 2 ~ t 2 / ~ \ (4.3) TU(x ,~ ) = A- e - ~ ~(e -~u[ -~ ) ) ( i x )

where 9 v denotes Fourier t ransform. Using Fourier inversion formula, one has

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

where the integral has to be unders tood as Fourier t rans format ion . We thus have for- mally:

ei.~(y,t)+_~(t_~)2Tu(x,A)dxd t (4.5) T~u(y,~) = ~ ei~. t E R n

In fact, we will give a sense to the preceding integral by showing that ~(y, t) - ~(t - ~)2

is a phase of quant ized canonical t r ans fo rmat ion in the sense of Definit ion 3.3, and by comput ing the integral over a good contour. More generally, if g is ano ther phase of FBI, we will t ry to express T~u in te rms of Tyu using a fo rmula of the fo rm

(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 symbol of order O. Let us prove first:

L e m m a 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


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

(4.7) (t, x) ~ - Im ~(y, t) + Im g(x, t) + ~(x)

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

(4.8) (t, x) ~ - Im 9(y, t) + Im g(x, t) - I m g(x, s)

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). The relations n(x(y)) = &(y), (2.12), (2.22) and an easy computa t ion show that (t, x) = ( t (y ) ,x (g) ) is a critical point. The critical value is then 9~(y). The Hessian mat r ix of (4.7) at the critical point (t(y), x(y)) may be wri t ten (aij)l<_,,j<_4 with

02 Im(g - ~) (4.9) an -- O Re t 2

02 Im(g - .~) a 2 1 = t a l 2 - -

O R e t O I m t 02 Im(g - g)

a 2 2 - - 0 I m t 2 02 Img Ot(x)

a 3 1 = t a l 3 --_ 0 R e t 2 " ORex OT(x) O 2 Img Ot(x)

a 3 2 : t a 2 3 ~--- - - - - - - ORex cgRetOImt a R e x

t(Or(x) ~ (02 Img~ (Ot(x)

a33 = \ O R e x ] \ 0 R e t 2 ] \ O R e x ]

02 Img Or(x) a 4 1 = t a l 4 =

ORet 2 " a I m x or(x) o 21rag at(x)

a42 = ta24 --- OImx - O R e t O I m t OImx

a43 = 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 Img~ ( O t ( x ) a44 ---- \ O I m x / \ O R e t 2 ] \ 0 I m x /

(using the relations obta ined by differentiation of (2.20) i), (2.21), (2.22)). The value of the associated quadrat ic form over a vector of the form

( 5 ( ~ Ot(x) \ lO t ( x ) Or (x ) . I m z ) , R e z , I m z ) . R e z + O I m ~ "Imz)'v~-d-Re-x" R e z -4- vqIm ~

is equal to

4. Change of FBI 25

(4.10) 52 02 Im(g - .q) [ cgt(x) Ot(x) z] 2 -ff-R-eet 7 [ORex " R e z + O I ~ " I m

~tl" Ot(x) Ot(x) . I m z ) [ °2 h n ( g - - g ) ] (Or(x) + 7° ~0R--eez " R e z + 0Im---~ l OlmtORet J \ 0 R e x

~tdOr(x) Or(x) . I m z ) [ 0-2 Im(g - 0).] ( O t ( x ) + 3'0 ~0--~-xex " R e z + 0Im-----x [cgRetOImtJ \ 0 R e x

+ 72 02 Im(g - 0) ( O r ( x ) Or(x) 2 0~--m-m ~ \ 0 R e x " R e z + 0 I m - - - ~ ' I m z )

( ~ Ot(x) . imz ) [ 0 2 I m g ] ( 0 r ( x ) - 7 t • R e z + 0 Im~----x [OImtcgRetJ \ 0 R e x " R e z + - -

t /Or(x) Or(x) . Imz)[ O2Img ](Ot(x) - 7 [0RT~-ez . R e z + 0 Im~--~ lORetOImtJ \O-ffe-ez . R e z + - -

Or(x) . Im z) - - - R e z + OIm-------x

0t (x) . I m z ) • R e z + 0 I m - ~

&(x) \

• I m z l OImx ]

Ot(z ) • I m z ~

Olmx ]

25(02Img~(Or(x) R e z at(x) ~ ,, fOr(x) R e z Or(x______)). i m z ) 2 \ a R e t 2 ] \ c g R e x " + Olmx " Im~] 2 - zv~,0--R-~ex " + a I m x ]

( 0 2 Img '~ ( O r ( x ) Or(x) , i m z ) 2 + \ 0R~-2-et2 / \ 0 R e x ' R e z + 0 Im------x

If one takes 5 = 1, "y > 0 small enough, one sees tha t (aij)lKi,j<4 is negat ive definite on a real l inear suhspace of dimension 2n. On the other hand, taking 5 = 0, 7 < 0, ]3'1 small enough, one sees tha t it is posi t ive definite on a real l inear subspace of dimension 2n. The s ignature is thus zero and the other assert ions of the l e m m a abou t (4.7) follow f rom that .

The funct ion (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)). The first two rows and the first two columns of the Hessian ma t r ix of (4.8) at y = Y0, x = z0, t = s = to are the same than those of (4.7). The remaining block is equal to zero. The value of this quadra t ic form over a vector of the preceding form is given by (4.10) wi thout its last term. If 5 = 1, 3' > 0 is small enough, one still gets a 2n-dimensional real subspace on which the Hessian ma t r ix is negat ive definite. If 5 < 0, V < 0, [V[ KK 6 2 KK 1, one has a 2n dimensional subspace on which it is posi t ive definite. The s ignature is thus zero, and the second pa r t of the l e m m a follows f rom that .

We will now prove:

T h e o r e m 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

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

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 tha t t heorem we will make use of the s t a t ionary phase formula we recall now. For a proof, see Sj6st rand [Sj] - Theo rem 2.8.


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 o coordinates 2. such that h(z) = h(O) + g(z 1 + . . - + is equal to/~ = ~ + . . . + 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 J r

with

1 ~ I V

Z a) + o_<t<~

(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 27

(4.19) ei~g(Y'~) E (27r)"(l!A/)-I E A-k[l~(Y,s)] l (ak/J(y ,s))(x(y 's) ' s ) o<I<A/C O<k<A/C

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 may 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) g(x0) = (to, Og

- ~ ( 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

i i (4.21) g ( x , t ) = - t ) 2 , O ( x , t ) = - t ) .

The associated identifications are given respectively by g : x ~ (Re x, - I m x), k : x --* ( R e x , - l I m x ) . Then, if (t0,T0) ¢ WYe(u), one has by (4.20) Tgu(x,A) e g 8 (Ira 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 ( x0 ) = (t0,rr0) ¢ WEe(u). One argues in the same way for WF(u) and SS(u).

II. Second micro loca l i zat ion

This second chapter deals with the definition and the s tudy 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].

The first section is an introduct ion to second microlocalization along the conormal to 0 in N '~. Star t ing from the notion of conormal regularity, we guess what should be, in this peculiar case, the good notion of FBI t ransformat ion of second kind allowing one to define a second wave front set resembling the one studied by Bony in [Bo2].

The second section is devoted to the definition of the second wave front set and of the second microsuppor t along a general lagrangian submanifold, following IS j] and [L2]. One first defines the not ion of FBI phase of second kind and proves the existence of such objects, following [L2]. Then, one introduces the good contours natural ly associated to such phases. This allows one to define FBI t ransformations of second kind and second wave front set along a lagrangian submanifold, still following [L2]. One should remark tha 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 parameter # 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 bound for the wave front set of the restriction to N of a smooth enough distr ibution 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 . S e c o n d microlocalization along T{*o}R n

In this first section, we will give an heuristic int roduct ion to the not ion of second mi- croloealization along the lagrangian submanifold A = T~0iR n of T*R n. The precise and rigorous definition of tha t notion will be given in the next section. First microlocalization (i.e. the FBI t ransformat ion presented in the first chapter) allowed us to give a meaning to assertions like: "the distr ibution u is smooth 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 distr ibution is singular.

We reminded in the introduct ion the impor tan t notion of conormal regulari ty along a submanifold V. Let us recall its definition when V = {0}: one says tha t a distr ibution

rr, ,+oo if for every integer k and for every k-tuple of smooth vector u is in the space ~{o}

fields X1, • • •, Xk vanishing at 0, one has X1 - • • Xku E H~o c. This is equivalent to the fact tha t for every mult i index a = ( a l , . . . , a,,)

1. Second microlocalization along T(*0}l~ ~ 29

HS+l~ l (1.1) t"U ---~ t71 ' ' ' t n ~ " t t E loc '

One says t h a t the d i s t r i bu t ion u is conormal a long {0} if the re is an s E R such tha t /#-s,+ oo A u E , , {0} . , . eono rma l d i s t r i bu t ion at 0 is thus s m o o t h over l~ n - {0} a n d has at 0 a

s ingu la r i ty of a special k ind. The space of conormal d i s t r i bu t ions will p l ay wi th respect to second mic ro loca l i za t ion the same role t han the space of C °° funct ions wi th respect to the first one: a d i s t r i bu t ion will be 2-micro loca l ly regu la r if and only if it is a conormal d i s t r ibu t ion .

As we gave a mean ing to the no t ion of micro loca l r egu la r i t y of a d i s t r i bu t i on u in a cone of the phase space, one m a y give a mean ing to the no t ion of conormal r egu la r i ty of u in a d o m a i n of the fo rm F A V where V is an open n e i g h b o r h o o d of 0 and F an open cone in N '~ wi th ver tex at 0: if s E N, we will say tha t u is in r_p,+oo (2-micro loca l ly) in "'{o} F n V if for every mul t i i ndex 7 wi th [71 -< s and for every fami ly X1, • . . , Xk of smoo th vec tor fields vanish ing at 0,

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

One should not ice t ha t the cone _P above is a cone of the base space N n and not of the phase space.

It is then poss ib le to define the no t ion of conormal r egu la r i t y close to a d i rec t ion 6t o E N ~ - {0}: the d i s t r i bu t ion u will be sa id conorma l in the d i rec t ion 5t o if the re exist an open conic ne ighbo rhood F of 5t ° in I~ '~ - {0} and an open n e i g h b o r h o o d V of 0 such t ha t (1.2) is sat isf ied for every fami ly X1, . . . , Xk. We should like to give an equivalent def ini t ion of tha t no t ion using t r ans fo rma t ions looking like the 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 wi th R e y # 0 a n d f o r M C R * + we set

[ ~' (y t ,2 (1.3) T u ( y , A ' ) = j e - v - ' u( t ) dt

we saw tha t the a s y m p t o t i c behav iou r of T u ( y , M) when M --+ + o c allows one to s tudy the d i s t r i bu t i on u in a ne ighbo rhood I t - Re Yl < e of Re y. If # is a real pos i t ive number and if we pu t

f ~' 7) ~(t) dt, (1.4) T~,u(y,A') = e_~_(y_ , 2

T~u(y , A') enables us to s t u d y u in an open set of the form ] R e y - ~ ] < c. If # varies

in an in terva l ]0, a[, a > 0 fixed, and y is close to a po in t y0 wi th Rey0 ~ 0, the fmnily indexed by # (T~,(y, M)) u controls the regu la r i ty of u in a d o m a i n

(1.5) A , = t; 3 # C ] 0 , a [ a n d ~ - R e y 0 < ¢

wi th e > 0 fixed, i.e. in a d o m a i n of the form F , N V, wi th _P, conic n e i g h b o r h o o d of Y0 in R ~ - {0} and V, ne ighbo rhood of 0. More precisely, if for ins tance , there is ¢ > 0 such t ha t Supp(u ) Cl A , = 0, the in tegra l (1.4) has an 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, wi th C posi t ive cons tan t and rn real n u m b e r depending jus t on the order of the d is t r ibut ion u.

We will show tha t under the a s sumpt ion (1.2), the funct ion (1.4) is rap id ly decreasing with respect to M for y close to a convenient point y0. Firs t of all, let us express the t r ans fo rma t ion (1.4) using an usual F B I t r ans fo rma t ion

~ 2

(1.7) T u ( x , A ) = e - ~ (~-t) u ( t ) d t .

Let us put

(1.8) T 2 u ( y , A , # ) = c 2(1-~,) ~ r u ( i x , A ) d x .

P u t t i ng (1.7) into tha t fo rmula and comput ing the in te rmedia te integral , we get

- ~ - b ~ A y t [Jx (1.9) T2u(y , A, #) = e . . u(~) dt

/ 1 - # 2 x ~ / 2 - ~ ' 2 _ ~ _ e A y ~ . .

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 func t ion 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 for every fami ly of C a vector fields X 1 , . . . , X k vanishing

at O, one has

(1.10) X I " " X k u E Z 2 ( V M V) .

Then there is c > 0, a E ]0, 1[, ~ E R and for every N E N, CN > 0 such that

(1.11) IT2u(y, A, ~)1 < C N A ~ ( A , ~ ) - N e ~ (Ira y)

1 < [ R e y I < 2 , [ h n y St°l iSt°I f < E , # E ] 0 , a[, A# 2 > 1. f o r e v e r y (~, ~, ~) s a t i s f y i n g ~ _ _ - -

P r o o f . Let X be a C °° funct ion on the unit sphere S n-1 of R n, wi th suppor t conta ined in F M S " - ] , such tha t X --- 1 on a ne ighborhood of St°/[St°[. Let us wri te

(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 f rom Supp(1 - X) and so the second t e r m of (1.12) m a y be e s t ima ted by the r ight h a n d side of (1.6). To s tudy the first t e rm, pu t

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

Let us r emark first t ha t if v is a compac t ly suppor t ed d is t r ibut ion belonging to H s (s non-negat ive integer), one has

(1.14) ITv(x , A)I <_ C A - } - % } ( I ~ ) ~

2. Second microlocalization along a lagrangian submanifold 31

when ] Im xJ s tays be tween two posi t ive constants . In fact (1.14) is t r ivial if s = 0. Let us prove it for general s by induct ion. If we assume (1.14) for s - 1, let us r emark

0 n 1 ~ 0 ( O v ~ (1.15) -o~Tv(x ,A) = - ~ - ~ T v ( x , A ) - ~ o x j T k o t j ] ( x , A ) .

j = l

Using Cauchy ' s fo rmula on a polydisk of radius equivalent to 1 $, we see t ha t the induct ion

hypothes is implies J ° T v ( x , A) J _< CA-½-~e-}0m ~)2 if v E H ~. Let us wri te

(1.16) Tv(x, : Tv(x, 1) + Tv( ,

By wha t we jus t saw, the p roduc t of the second t e rm of the right hand side by e--} (ira x)2 has modu lus bounded f rom above by

A e r -@(Im 2 1 1 s ~) r - ~ -~ C A - ~ - (1.17) dr < .

, 1 _ 2 e } ( i m z)~ -- C ' Moreover ITv(z , 1)J < Ce½ (~m x)2 _< C A-~ for A > 1 if is large enough. We thus get (1.14).

To ob ta in (1.11) we argue in a similar way. Using the a s sumpt ion (1.10) and com- bining (1.8) and (1.14) appl ied to t~u for any mul t i index a , we see tha t

(1.18) JT2(x(t/ltJ)t~u)(u,/~,~#)1 ~ C / \ - ½ - J a ' ] e ~2-'~(Imy)2 •

Using this inequal i ty for a = 0 and (1.9), we get

(1.19) i w ( y '/~,/~,r)j <. c/vn/2~_½e@(lmy)2

(set t ing A' = A#2). We mus t show tha t we m a y replace in the right hand side of (1.19)

M n/2 by A ' - g for every integer N (with a constant C depending on N) . This follows f rom an easy induct ion using tha t , because of (1.13) and (1.18), for every k E N

0 k ( 1 . 2 0 ) -o-~w(y,/~,/~') < ek.~'n/2-k~ - ½ e ~ ( I m y)2

This concludes the p roof of the proposi t ion.

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

Let N be a submani fo ld of N n and let us choose local coordinates on IR ~ t - ( t ' , t " )

such tha t N is given by N = {t" = 0}. Let us denote by A the conormal bundle to N in N", A = { ( t ' , 0; 0, T " ) } . We will denote by L the real analyt ic submani fo td of C n which is the inverse image of A by the i somorphism f rom C" to T * N " given by z --~ (Rex ; - I m x):

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


If u is a compac t ly suppor ted distr ibution, let us denote by Tu(x, A) = f e- ~-(~-t)2u(t) dt. We will associate to A a FBI t ransformat ion of second kind, generMizing the one defined in Section 1 when N = {0}, sett ing

(2.2) ~ I Xt~2 t t 2 It • tt 2 A t l 2

- [(u-= ) +(~ +,= ) ]+~-= Tu(x,A)dx T 2 t t ( Y ' / ~ ' # ) = m x ' = 0 e 2(1-~'2)

Re x " = 0

Let us denote the phase of (2.2) by

(2 .3) a ( ~ , x , ~) - i ' ~ [(y' - z')~ + (y" + i x") ~-] - i ~''~ 2(1 - #2) 2

and put

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

Then f(y,x,O) = ½(Imx ' ) 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, bo th with respect to space and phase variables, as a phase of FBI t ransformat ion. In part icular , the restriction to L of the coefficient of #2 in the asympto t ic development of (2.4) at # = 0, has a non-degenerate max imum at a unique point of L.

To define FBI t ransformat ions of second kind, like (2.2), but associated to lagrangian submanifolds which are no longer necessarily a conormal, we will have to in t roduce 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 FBI phase defined in a ne ighborhood 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 p lur isubharmonic funct ion in a ne ighborhood of x0 which is the critical value of t -~ - I m g ( x , t). We saw tha t is a symplect ic diffeomorphism when we endow C" with the symplect ic s t ruc ture given by the symplect ic form ~c50~. Let A be a germ at (to, To) of real analyt ic 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 funct ion of (y, x) in a ne ighborhood of (y0, xo) in C n × C n, real analyt ic in # C [0,#0[. Let us put

+oo

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

We shall assume the following conditions:


(2.8) i) f01c- ×L = 0, (df0)[c-×L = 0, x ---* f0(Y0, x) has a posi t ive definite t ransverse Hessian along L.

ii) f l (Y, x) ~ O.

iii) T h e funct ion f2(Yo,') res t r ic ted to L has at x0 a non-degenera te critical point with a negat ive definite Hessian.

We will have then:

L e m m a 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

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

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

• (~, , ) --~ ~(y , , ) is a real analyt ic map,

• For e~ery y , , --* ~ ( 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:

L e m m a 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) = TC ~.

Proof. One has

(2.10) 2c50~z = i-2 E 02kOz------~02~ dSk A dzj . j,k

Let us consider the i somorph ism from the tangent space to R n to the complex tangent space of C '~ given by

" 0

~ aj 0 Re xj (2 .11) j : l

~7

0 ~ ~ ( a j + 0 - - + bj 0 Im xj ibj) Oxj

U

The act ion 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 roduc t on C n x C ' , {., .} he rmi t i an p roduc t on C '~ x C • and

A is the posi t ive definite ma 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 tha t iU C TL. Since L is lagrangian, the act ion of (2.10) over the couple (V, iU) must then give 0. So using (2.12) we mus t have (AU, U) = 0, whence U = 0

34 II. Second microlocMization

since A is pos i t ive definite. Then TL N (iTL) = 0 which impl ies the resul t s ince L is of d imens ion n.

If L is a real ana ly t i c submani fo ld of C n which is t o t a l l y real and if x0 • L, there is a ho lomorph ic change of coord ina tes in a ne ighbo rhood of x0 such t h a t L is t r ans fo rmed into the subman i fo ld I m x = 0. To see tha t , let us first r e m a r k t ha t when L is l inear and x0 = 0, one jus t has to take the d i f feomorphism Re x + i Im x ~ M Re x + iM Im x where M is a l inear i somorph i sm from IR n over L. In general , we m a y thus a s sume tha t ToL = N n a n d so tha t there is a real ana ly t i c funct ion h in a ne ighbo rhood of 0 in N", sa t i s fy ing h(0) = 0, h ' (0) = 0 and such tha t L = { t + ih(t); t • R ~, t close to 0 } close to 0. Then , close to x = O, x ---+ x + ih(x) is a ho lomorph ic d i f feomorphism whose inverse fulfills our requi rements . Let us give now the proof of L e m m a 2.1:

Proof of Lemma 2.1. Because of L e m m a 2.2 and of the p reced ing r emark , we m a y pe r fo rm a ho lomorph ie change of coord ina tes such t ha t x0 -- 0, L = { Im x = 0 }. Let us set

(2.13) f ( y , z , u ) = ~ f ( y , Rez + i # h n z , # )

1 = ~ f o ( y , Re z + ip Im z) + f2(Y, Re z + i# Im z) + 0 ( # ) .

The Hess ian m a t r i x of f(Yo,' , 0) at z = 0 is equal to

(2.14) ~ ( y o , 0) 0 0 2 o

0 (y0,0)

(see (2.S)i)) and so, because of (2.S)i) anti iii), is non-degenerate with signature 0. For (y, #) close enough to (0, 0), z ---+ f ( y , z, #) has thus a unique cr i t ical po in t close to 0, z(y, t.t). This cr i t ica l po in t is a saddle po in t and (y, #) ----, z(y, #) is a real ana ly t i c map . For # = 0, we have

1 02fo (2.15) f (y , z, 0) -- $ 0 I m x 2 (y ' Re z ) ( Im z) 2 + f2(Y, Re z) .

By uniqueness of the cr i t ica l po in t , we thus see t ha t the cr i t ica l po in t z(y, 0) of (2.13) when # = 0 is jus t the un ique po in t in IR '~ such t ha t °h (y,z(y,O)) = 0. If we pu t

0 R e : c

x(y,#) = R e z ( y , # ) + i # Imz (y ,# ) , we have I m x ( y , # ) = 0 ( , 2 ) . The l e m m a is proved.

In the sys t em of coord ina tes used above, in which L is given by Im x = 0, i t is very easy to wr i te an a s y m p t o t i c expans ion of x(g, p) at # = 0. One 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) o l m x

whence I m z ( y , # ) . . . . { ~ - a oi2 ( , z(y,O)) + 0 ( # 2 ) . We thus get: t * \ O l m x 2 7 O I m 2 : \ ~ '


(2.17) I m x ( y , # ) = - - # 2 ( 02f° ~-' Of 2 0 mx (v'x(v'°)) +

O/2 0 R e ( y , x ( u , 0 ) ) = 0 .

We will set

(2.18) e ( y , #) = ~ f ( y , x(y, #), #) $

Because of (2.13), ~ ( y , # ) is real analyt ic in (y ,# ) for ( y ,# ) close to (y0,0), # > 0. Moreover, apply ing L e m m a 3.6 of Chap te r I for fixed posi t ive #, we see tha t y ~ k~(y, #) is p lu r i subharmonic for every # > 0. Let us define

(2.19) ¢ (y ) = k0(y, 0) .

It is a p lu r i subharmonic real analyt ic funct ion (as the l imit of the p lu r i subharmonic funct ions ~(y , #) when # goes to 0).

Remind tha t we endowed C n with the symplect ic s t ruc tu re coming f rom the sympleetic fo rm w = ~c~0q0. Since L is lagrangian, we m a y use the hami l ton ian i somorph ism to identify TLX and T'L:

(2.20) TLX --* T*L V ~ i v c o

where ivw is the linear fo rm on the fibres of TL associat ing to u E TL the scalar co(v, u). If Y is ano the r copy of C '~ in a ne ighborhood 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 d i f feomorphism in a ne ighborhood 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 TH(L) because (2.17) shows tha t al(y) e TL. This m a p A will p lay for second microlocal izat ion the same role tha t the i somorph i sm ~ defined in I-(2.21) wi th respect to first microlocal izat ion. Of course, such a th ing is possible only if A is an i somorphism. The next l e m m a gives a necessary and sufficient condi t ion to ensure tha t .

L e m m a 2.3 . Let & be the natural 3ymplectic form on T*L. We have:

(2.24) A'c5 = 2.0a¢ .

Thus, A is a (local) isomorphism if and only if ¢ is str ict ly pluri3ubharmonic.


Proof. by Im

and if

(2.25)

where

{Oafo

(2.26)

Using

(2.27)

Then

(2.28)

Let us choose a holomorphic system of coordinates on C n such that L is given u a x = 0. Let us denote by w the 2-form 2 a0~ . If fi = ~ j O--~-~-e ~j, ~ = ~ vjo a Im x j

we set u = ~ uj-~7~s, v = ~ ivj , we have, because of (2.12)

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

A = ( a-A3-~-h Since fo(x)- ~(x)= -ImG(y,x,O)is pluriharmolriC, }0O~ = \ O~:kOzj J"

and (2.25) shows that the isomorphism (2.20) is given by

( a2fo "~, ~- \O~mx2] "

(2.17), we see that

a f2 , = (xIv,0), 0 mx Y, xIy, 0)))

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

( ~ 02f2 Oxj(g,O)) dReyk A dReyl A*& = ~ OlmxjOReyk OReyl

k,l j

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

k,l j

02f2 Oxj(y, O) 02f2 Oxj(y, O) + ~ ( ~ - ~ g 3 l m x j a R e y k a lmyt 0 h n x j 0 I m y t 0 R e y k ) d R e y k A d l m y t .

k,l j

On the other hand

(2.29) 02~b (dReyk A dReyl + dlmyk A d l m y l ) 200~b* = ~ OImykOReyl

k,l

02¢ + 02¢

-~k3(OReykOReyt OlmykOImy,) dlmykAdRey'"

Using that ¢(y) = f2(Y, x(y, 0)) and that f2(y, x) is pluriharmonic, one sees that the two preceding expressions are equal.

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


the conditions (2.8)i), ii), iii) and such that the critical value ¢ (y ) defined by (2.19) is strictly plur isubharmonic in a neighborhood of y0.

It is easy to see tha t for any couple (~, L) fulfilling the preceding conditions, one may find a phase of FBI 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 ne ighborhood of x0, such that M(xo) = 0 and M(L) is given by I m ~ = 0. Let us show tha t thcre is a holomorphic funct ion gL dcfined close to 0 such tha t qOL = - - Im gL satisfies

~ ( ~ ) = ~(~)

(2.30) 0 ~ ( ~ ) _ o ~

0 I m ~ 0 I M P " "

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

Iml( °=~° "~U V} = 0 for every couple of tangent vectors (U,V) of the form U = \ \ O~kO~'j ] '

aj~-f~,V=~a b~ao~s with aj, bj real numbers. By an easy computa t ion , this means

(2.31) ORe.~kOIm~j = 0Im~kcqReh: j if ~ C IR '~, for every j, k.

One may find a real analytic function gl defined in a ne ighborhood of 0 in ~'~ such that

O g l 099 (2.32) - - ( 2 ) - (a~) ; j = l , . . . , n , ~ • R n.

0 Re &j oq Im ~j

It is then enough to take for gL a holomorphic extension of gl - iqD. Let us now set

(2.33)

We have

with

(2.34)

i# 2 c ( ~ , x , . ) = 7 - ~ y - M ( x ) ) 2 - ~ L ( z )

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

f0(x) = ~(x) - ~L(x)

f2(y, *) : _ 1 Re(y - M(x)) 2

Because of (2.30), the conditions fOIL, (dfo)lL = 0 are fulfilled. One has

(2.35) 02f0 1 0 2 f o 02kO3sj = a(Olmh;kOIm~j)

and since f0 is strictly plurisubharmonic, 02fo(~)/OIm~ 2 >> 0 if ~ E R n. Condit ion (2.8)i) is thus realized. Moreover, condit ion (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 FBI of second kind along L over H~o in the sense of Definition 2.4.


Of course, any phase which differs f rom (2.33) by a t e rm which is O(# 3) when # --* 0 + is also a FBI phase of second kind along L over H~,. In par t icular , we will make use in Chap te r 3 of the phase

(2.33)' ( y - M ( x ) ) - • a ( y , f , ) - 2 ( 1 -

Since x ~ f (y , x ,#) has a saddle point at x(y,#), one m a y define a not ion of "good contour" .

D e f i n i t i o n 2 .5 . A good contour for f (y , . , .) is a real analyt ic 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 posit ive number , wi th values in C n, such that:

• V~ C 1 0 , ~ l , t --, 7(t , ~) is an injective immers ion such tha t 7 (0 ,# ) = x(y,#).

• 3 C > 0 a n d V ( t , # ) E B x [ 0 , # l [

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

• There exist a ball B ~ C B, #2 E ]0,#1], a holomorphic d i f feomorphism H defined on a ne ighborhood 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 immers ion with respect to t, such tha t 7(t , #) = H -1 (Re ~(t , #) + i t Ira 'S(t , #)).

The fact tha 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 Chap te r I): if we choose coordinates such tha t L = { I m x = 0 }, we may define the funct ion f(v, z, #) given by (2.13). Using Morse l emma, it is then possible to find a real analyt ic change of coordinates z ~ w, depending on the p a r a m e t e r s (y, #), such tha t f (y , z, #) = f (y , z(y, #), #) + Q(w) where Q is a quadra t ic form. An n dimensional p lane of the w-space on which Q is negat ive definite gives then a good contour for f (y , . , .).

One m a y also prove a result similar to Propos i t ion 3.5 of Chap t e r I: let /'vo : ( t , # ) ~ 7y0(t ,#) be a good contour for f (yo, ' , ' ) . Then, there exist a ne ighborhood V of y0, a posit ive constant c and for every y C V a good contour Fy for f (y , . , .) such tha t the following holds:

• for every y E Y, there is a real analyt ic m a p

(2.38) ~ v : B x [0, 1] x [0, #1] - ~ c n

(t,

which is, for every fixed # E ]0,#1], the restr ic t ion to B x [0, 1] of an injective immers ion, and which is such tha t , for every # E ]0, #1 ],/'~0 - F ~ - O Z ~ ' is contained in

2. Second microlocMization along a lagrangian submanifold 39

(2.39) { x; f(y, x, #) <_ I(Y, x(y, #), #) - c# 2 }

(F2 ' -/'~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 y E U

~e]0,~], )~t,2>_1

(resp.

(2.43) sup y E U

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:

Def in i t ion 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.


Let A be a germ at P0 = (to, r0) of lagrangian real analytic submanifold of T*[R n. Let us choose a FBI phase g(x , t ) in a neighborhood of (xo,to) and let ~p be the associated str ict ly p lur isubharmonic weight. Let ~ be the isomorphism associated to these da ta by formula (2.21) of Chapter I. Then, L = x - l ( A ) is a germ at x0 E (7 ~ of submanifold, lagrangian for 200~0. We will denote by

(2.45) ~ : T*L ~ T*A

the isomorphism induced by ~. Let u be a distr ibution defined on a ne ighborhood of to. We have:

D e f i n i t i o n 2.8. One says tha t the point (P0,P~) C T*A is not in the second wave front set (resp. the second microsuppor t ) with growth of u along A if there exists:

- a FBI 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 ne ighborhood 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 < +oo yEV

uE]0,~], ~u=_>l

2 (resp. if there exists G as before such that T~Tgu E N¢,v0 ). We will denote this set by

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 t ransformat ion 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 Theorem 4.2 of Chapter I, but is much more technical and will be admi t ted (cf. [L2]). We will deduce f rom these propert ies the following result about the conical s t ructure of the second wave front set.

P r o p o s i t i o n 2.9. i) The sets WFI '2 (u ) and 582'1(72) a re conic subsets of T*A (i.e. are

invariant under the dilatations (p,p*) ---+ (p, rp*), r > 0).

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 FBI phase g(x, t) allowing one to characterize the wave front set microlocally dose to P0 = (to; w0) E A, and let x be the isomorphism associated to it by formula (2.21) of Chapter I. Let us choose a FBI 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~) • WF~' 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 map (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 --* TLX --* 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

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

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

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

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 = ( t l , 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 = ~ N n = {(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 M E N, 5 > 0, C > 0 such that for every r / E R ~' (3.5)

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

Such a condition implies in particular that U]N is 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:


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) VI j ( 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

1 f ~, i i II 2 #2 • I # (3.7) S2f(y , s , ;~ ' )=-~? e-Z-(" +.~r~ ) - ~ + , , , (

, ]

The proof of the theorem relies on the following lemma:

L e m m a 3.2. Let us put

(3.s)

We have

(3.9)

y l l . 7711

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

/ 0 "l- ° ° I S ~ . n - n " dy"S2f(u, ,a ') = X' H(a ' )T( f lN)(y ' ,a ' ) n II -- 1

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

(3.10) g ( a ' ) = e - T (' + ~ ) - - nil _ 1

y l l . r]tt- ) dy" .

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

(t0,0; I " every E ' T 0,T ) ¢ WF(u) for v" N n'' (3.11)

(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

T* A -~ T* A (3.12)

( 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

At t t l2 (3.13) f ( t ) = e 2 u(t) .


One has uiN = f iN . Moreover, if u satisfies (3.5), f also with the constant C replaced

by CA ~/2. We will then prove tha t if a point

(3.14) ' "" WF~ ' I (u ) (to,~0 ,t~* "* ,~-o ) ¢

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

(3.15) IS2 f (y , At, 8)[ _~. CN,sBAt-Ne3~ [(Imy')2q-I-i'~* (Imy")2]

for y E W, A' _> 1, s C [So,+Cx~[.

given s ~ ]0, s0[, the point (Rey~, ne ighborhood W of y0, A~ > 0 and ), > ),~,

(3.16) IS2f(y , A',s)[

On the other hand, we will also show tha t if for a

' - I m y ' o , s R e y ' o ' ) is not in WE(u) , there is a for every N C N, CN > 0 such tha t for every y E W,

xi ~2 CN.~t_Ne~_[(Im y , )2+ l_TT,.r (i m y11)21

To prove the theorem, we will see first tha t the second a s sumpt ion (3.11) implies the es t imate (3.15) for every Y0 satisfying yt o = t' o --iTS, ytj real, iY~'I = 1. Then , we will show tha t the first a s sumpt ion (3.11) implies a version of (3.16) uni form 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 neigh-

borhood V of Y~o = t~o - i7~ and for every N E N, CN > 0 such that

(3.17) - N M t 2

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

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

Proof. We have

t n ' 2 n" ~,t ,iz ytt (3.1s) s~f(y,a',~) = ( 2 ~ ) , s e - ~ o e~-~' . 2 s ~ f ( y , ; , s ) _

A I s Oy"

with

) l i t 2 x I 2 I t 2 , t 11 I t

(3.19) ~;2f(y ,A, , s ) = e - - ~ ( , - t ) - z - s t + , ~ t ~ f ( t ) d t .

Because of (3.13), we know moreover:

(3.20) ~2 f ( y , A , , s ) = S2u (y ' s . ,

Let us consider the following FBI phases of first and second kind given respect ively by

(3.2t)

g ( x , t ) - i ( x - t ) ~

2 '

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

.X i t 2

2


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

(Im y") 2 (3.22) #(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 ' + i i t Imy'

(3 .23) ,, It2 x (y, i t ) - l + i t 4 I m y " + i R e y " .

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

~ i i :t .X 1 4 J ,2 , I I I~

(3 .25) T ~ % ~ ( y , ~ , i t ) = ~ - ~ ( ~ - ' ) - ~ ~.-~' + ' ~ ' ~ ( ~ ) ~ 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 N c N CN > 0 s u c h t h a t

J 2 ( 3 . 2 6 ) IT~T.u(y,l, it)l <_ CNI t -2B( , \ I t2 ) - -N,s 2 ( I m y )

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

(3.27) ~2 f(y, A', s) = T~Tgu(y, A, t t) .

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

1 (3.28) 7(B + 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


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

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

,~N 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) i ' - t')2 - - 2 - - - ist" • x g (x,t) = + i (1 s2) ''2 "

The identification x~ naturally associated is

Re x" ) (3.35) (x',x")--+ R e x ' , - s l + s ~ ; - I m x ' , s I m x ' ' .

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) s u p (A'-DIv(z",A')le -~-(Im='')=) < +c~ z" EK A~>I


n I t for every compact K . For y" E ~ , I v " l= 1, let us set

i, ( A ' ~ " " f ~' ~ ,"~- i )¢ '~" ~"'~" (3.37) A~v(y , ~ 1 ) = \~_~] j s e - - r ~ * t ~ - , v ( z " , A ) d x " d z "

where Z i~ the contour

X 11 = 0 " l l

(3.38) l + s 2 ~r I ' c C , l a " l < c . 11 11 z = s y I m ~ " - i R e c r "

2

Then, there exist~ a continuous funct ion 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 = Ozt II + cr tl

(3.40) 1 + s 2 la"l -< c, a e [0,1]. z" = sy II - - I m a " -- i r e a" + ic~s2t"

2

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 computes the integral in a ' , one obtains the right hand side of (3.39) modulo a remainder with uniform exponential decay. Since an easy calculation shows that the contr ibut ion to (3.37) of the boundary of the deformation (3.40) gives also such a remainder, the l emma is proved.

We thus deduce from (3.39) tha t under the first assumpt ion (3.11), the first te rm in (3.20) is rapidly decreasing with respect to A I for yl close enough to y~. Using also (3.30) and (3.33), we get, taking (3.9) into account

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

for yl close to y~. This proves the first inclusion of Theorem 3.1. The 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.

I I I . G e o m e t r i c u p p e r b o u n d s

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 may 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, may be computed in terms of the derivatives of the start ing 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:

Defini t ion 1.1. Let M be a real analytic manifold and A a subset of M. One says that A is subanaly t ic 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 f i , 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 may 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) I f 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 may 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 part i t ion S = (Si)ieI of M, satisfying the three following conditions:

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 roper ty in M (not in St) and so, since Si is not assumed to be closed, it might be an analyt ic submanifold without being subanalyt ic in M.

Example. Let us take M = R 2 with coordinates (x, y). The family

So = ({x = 0) , {y = 0, + x > 0}, {=Ex > 0, + y > 0})

is not a stratif ication since condit ion 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 tha t a stratif ication S of M is compatible with a locally finite family C = (C j ) j e j of subanalyt ic subsets of M if for every i • I , j • J , one has either S i n Cj = 0 or S ic Ci.

One may show that for every subanalyt ic subset A of M, there exists always a strat if icat ion S of M, compatible with the family (A, M - A). One then defines the dimension of A as the m a x i m um of the dimensions of the s t ra ta 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 analyt ic manifolds and f : M ~ N a real analyt ic map. A stratification of f is a couple (S, .T) where 8 (resp. 5 t ') is a stratification of M (resp. N ) such tha t for every s t ra ta Si of S, one has f ( S i ) C .T" and

rk( f l s , ) -- dim f (Si) .

The main result about stratif ication we will use in the sequel, is the following theorem.

T h e o r e m 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.

Let h : R ~ R be a real subanalyt ic 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 project ion 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 tha t hl]t~,t~+l[ is real analyt ic o n ]ti , t i+l[. In fact, one may

show tha t for every i E Z, there are integers r + r~- and real analyt ic functions on a

ne ighborhood of 0, g+ and g7 such that h(ti + t) = gi~(tl/r~) for t > 0 close to 0. We will deduce f rom Theorem 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[

* M * (1.1) T~B M n (U TL ) c T ~ M . jEI


The proof 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, Uie ! T},M is subanalytic in T*M.

Proof. Since (Si)ieI is locally finite, (T~,M)ieI also. So, we just have to prove tha t if L is a real analyt ic and subanalyt ic submanifold of M, T~M is subanalyt ic in T*M. Let us choose an analyt ic coordinate system on a ne ighborhood of a point of L in M. The set

(1.2) A = {(x,~,X,t); (x,~) e T'M, (x ,X) e TL, ]X I = 1, t = [(X,~)]}

is subanalyt ic as soon as TL is subanalytic. But since the project ion

(1.3) q~: A ~ T*M (x ,~ ,x , t ) -~ (x,~)

is proper, we see that qS(A M {t > 0}) is subanalytic, and so, tha t the same is t rue 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 subanalyt ic 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 tha 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 tha t just for C(L)Iu where U is any subanalyt ic open set endowed with a system of analyt ic local coordinates. Let us then set

E = { ( ( z , X ) , x , y , u ) • T V × L x L × [ 0 , 1 ] ; y - x = u X }

(1.5) E0 = E - {u = 0}

7r : TU × L × L × [0, 1] ~ TU the na tura l project ion.

One has C(L)iu = 7r(E00 N {u --- 0, x = y = z}) and the right h a n d side is subanalyt ic by Theorem 1.2.

Proof of Proposition 1.9. We may assume x0 = 0. If the proposi t ion 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) eT*M×]O, 1]; ( x , ~ ) E U T ~ , M , ~ = ~ - ~ , t~= ixI .

1. SubanMytic sets and subanMytic maps 51

Because of the lemma, (1.6) is subanalyt ic in T*M x [0,1]. By Theorem 1.3, there is thus an analyt ic 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 Uie i T ~ M is subanalyt ic in T ' M , Theorem 1.8 applied to t ~ (x(t), ~(t)) shows tha t there exist e > 0 and i0 E I such tha t (x(t), ((t)) E T}io M

for every t E ]0 ,@ Using tha t the one form ~dx vanishes identically on any conic isotropic submanifotd ofT*M, we see tha t ( ( t ) . dx ( 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 tha t e( t ) = Ix(t)l > 0 for t ¢ 0.

The set C(L) we used in the proof of Lemma 1.10 is a special case of "Whi tney ' s normal cone" we will now define.

Let M be a real analyt ic 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 ne ighborhood of x, we set

C,(S, V) = { 0 E T,M; there exist sequences sn E S, vn E V, cn E N~_ with (1.7)

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

The definition is independent of the choice of the coordinate system and one puts:

Def in i t i on 1.1.

(1.8)

One calls Whitney's normal cone to S along V the set

C ( S , V ) = U C~(S,V) c T M . x E M

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

(1.9) Cv(S) C T v M

the image of C(S, V)lv modulo TV. W h e n M is a symplect ic manifold and when V is a lagrangian submanifold of M,

T v M may be identified with T*V using the hamil tonian isomorphism, and one then considers Cv(S) as a subset of T*V. When L and S are subanaly t ic in M, one shows, as in the proof of L e m m a 1.10, tha t Cv(S) is subanalyt ic in T*V. Let us consider the case when M = T*X with X real analyt ic 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 ransformat ion defined in a ne ighborhood 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, tha t on a neighborhood of x(qo)

)((CA(S)) = { (:/:, ~); there exist sequences (ym,~]m) E S, Urn E N*+, Um --* 0 (1.10)

snch tha t i f = x , . x , } .

To conclude this section, we will recall two topological results about subanalyt ic sets.

T h e o r e m 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


(1.11)

of U over A.

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

T h e o r e m 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:

- 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 ( M i ) g, p-1 -----+ Mi X (el)

pN' . N ~117.¢- 1

Mi

2. C r i t i c a l p o i n t s a n d c r i t i c a l v a l u e s

The next section will be devoted to the s tudy of upper bounds for the microsuppor t or the second microsuppor t of distr ibutions which can be continued holomorphical ly in the complex domain. To do so, we will first get est imates for FBI or second FBI transfor- mat ions in terms of weight functions ~b given by minimax formulas like formula (3.14) of Chap te r I. The new feature here, is tha t the functions to which these minimax will be applied will no longer have a single critical point, depending smooth ly on the parameters. The "critical value" ~b(y) will thus no longer be a real analyt ic function. In fact, ~b will just be a subanalyt ic 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 geometr ic da ta of the problem.

The assumptions we do below may seem quite complicated and technical. We make them because they will appear to be the natura l hypothesis in the applications we will t reat in Chapter 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 subanalyt ic in Z x [0, 1] and transverse to the fibres of the second project ion Z x ]0, 1] -+ ]0, 1],

- ~7 a real analyt ic submanifold of Z x ]0, 1], subanalyt ic in Z x [0, 1], t ransverse to the fibres of the second project ion a x ]0, 1] --+ ]0, 1].

We shall assume the following condition:

For every z E Z x ]0, 1], at most three manifolds among the ~Tj and X' pass (T) th rough z. Moreover, these submanifolds are mutual ly transverse, and when

three of them pass th rough z, each of them cuts transversal ly the intersection of the other two.

2. Critical points and critical values 53

Lastly, we will suppose tha t every 2 j is the bounda ry of an open half-space ~ ? of

Z x l 0,1] such tha t ~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 no ta t ion for the sections of G.

On the other hand, let U be a bounded open subset of N p x ]0, 1], which can be realized also as the intersect ion of a finite number of open half-spaces, whose boundar ies are real analyt ic submanifolds of N p x ]0, 1], subana ly t ic in R p x [0, 1], sat isfying the evident analogue of (T). Let /" be a real analyt ic m a p defined on a big enough open subset of NP, wi th values in N N, such tha t the m a p

(2.1) ~t, x 10,1] -~ ~t N x ]0,1]

( t ,e) ~ ( / ' ( t ) , s )

is an embedd ing 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 a t e of the image of U by (2.1). For e • ]0, 1], let ~ be the set of all C °~ homotopies

H : r0,~ x [0,1] --* a~. (2.2) (z ,~) ~ 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 ne ighborhood of y0 = 0 in N n and V a ne ighborhood of the set { z • Z ; 3 e • [ 0 , 1 ] w i t h ( z , e ) • ~ } i n R N . L e t

# : Y x V - - * R

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

be a real analyt ic funct ion and set

(2.5) ¢ ( y , e ) = inf sup @(y,z) . Hc'H,~ zEFl,¢ =H(F0,t ,1)

We will denote by $ the s trat i f icat ion of ~ M (R N x ]0, 1]) whose s t r a t a are the connected componen t s of the following manifolds:

• /] Z / ~ \ ( U Z j U Z N ~ , j l E { 1 , . . . , k }

j # j l

• )] k

(U j = l

2j, rlXj~\( uj=~./~.~#j, 2,jU2)] N~-, j,¢j2C{1,...,k}


[ ~.,;1 r"l .~, \ ( U .~j) ] I~ ~ , j l • {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 stratif ication of ~2, which is the slice at ~ of S. In the same way, one defines a na tura l stratif ication O r of the image of U by (2.1) intersected with R N x ]0, 1]. For g • ]0, 1], Or, will be the stratif ication of F0,, given by the slice of Or at ~. We will denote by OFo,, = 0I'o,~ N OY2---~. Let us prove the following:

P r o p o s i t i o n 2.1. Let (y, c) G Y × ]0, 1] be such that ¢(y, ~) > sup~ebr0,. ~(y, z). There

T* R N such that e~ists (z, ¢) • Us,~s, s,

(2.6) ¢ = d ~ ( y , z ) , ¢(y, ~) = ~(y, z)

The proof will use the following lemma.

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

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

L

(2.7) h : Y2~ × [0,1] ~ ~2~ (z,~) -~ 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 ne ighborhood of $2~ in Z, tangent to Zj , , for every j and to Z , vanishing close to (3_F0,,, t ransverse to the smooth hypersurface { z; 4~(y, z) = ¢(y , e) } at each of its points, and point ing towards the open set { z; # (y , z) < ¢(y , ~) }. In fact, the flow h(z, a) of such a vector field, with the t ime pa ramete r sui tably chosen, gives a map fulfilling all the conditions of the proposit ion.


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

Using a par t i t ion 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 ( r j ) 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 ne ighborhood of the lemma means (4 # 0. The vector field - y;Zz4 " o f Z 0 .

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 contradict ion, tha t for every z 6 9~ with T* R/v For any 6 > 0, there is, be- • = one has ¢ Us. s. s.

cause 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, Lemma 2.2 shows that the ho- mo topy (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 T* It~ N with ( = dz~(y,z) c > supzesr0., ~(y,z) and if there exists ( z , ( ) 6 Us~es~ s,

and c = 4~(y, z).

Proposi t ion 2.1 thus says that for every fixed (y, e), ei ther ¢(y , e) = supzebr0., ~(y, z) or ¢(y , e) is an St-cri t ical value of ~(y,-) . Let us show now

T h e o r e m 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.

Proof. The set

( 2 s ) { z c (z, z)) c (_J T's. R N } S~ 6S~


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) = c j, 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,

o r 3z E $ ? ~ w i t h c = ¢ ( y , z ) and (z,d.qb(y,z)) E U "r* NN } .L Se 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

= { ( y , s , c ) E Y x ] 0 , 1 ] x R ; ? z E ~ - ~ w i t h c = @ ( y , z ) A and

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

o r ( 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

T* R N (resp. (z, ¢) E T* N N) } is subanalytic in [0, 1] x T*N N. ]0,1] × T ' a N ; (z,() E S, F.

The subanalyticity of A then follows by the elementary properties of subanalytie sets.


Let us denote by 7r : A ---* Y x [0, 1] the p roper projec t ion (y, ~, c) ~ (y, ~). By Theo- rem 2.4, 71"]A has finite fibers and because of the inclusion A C A and of Propos i t ion 2.1, these fibers are non empty. This implies d im A = d im Y.

Using T h e o r e m 1.8, we find a s trat i f icat ion of 7r compat ib le to the par t i t ion Y x {0)tJ Y x ]0, 1] of Y × [0, 1]. Since ~r has finite fibers, its restr ic t ion to every s t r a t u m of A is a local d i f feomorphism onto its image. Let W be a s t r a t u m of Y × ]0, 1]. Because of Propos i t ion 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 subana ly t ic s t ra ta , and is thus subanalyt ic . This proves the corollary.

H

W

A

Y x ]0,1]

For every fixed y E Y, the funct ion ¢ ~ ¢(y , 6) has a g raph which is subanaly t ic in [0, 1]. By the descript ion of subana ly t ic funct ions of a single real var iable we gave after T h e o r e m 1.8, we then see tha t l ime -0+ ~b(y, ~) exists. Let us denote it by ¢(y , 0). Since y ---* ~b(y, ¢) is lipschitzian, uni formly with respect to 6, we see tha t ¢ (y , 0) is also lipschitzian. Moreover, if we extend ¢ to Y x [0, 1] by its l imit ¢ (y , 0) at ~ = 0, we get a subanaly t ic funct ion on Y x [0, 1]: in fact, the g raph of this funct ion is jus t the closure of the g raph 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) dy¢(y l ,61 ) ---- dy~5(yl,z) and (z, dzqh(yl,z)) E U T* •N S . 1

S, l ES, 1


Proof. The set

(2.11)

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

(z, d ~ ( y , z)) 6 U T* R N S, 6S~

¢(~, c) = e(y, z) }

is subanalyt ic . Let us consider a s t rat i f icat ion of the project ion ~- : B ~ Y x ]0, 1]. Because of Propos i t ion 2.1, ~ is onto on a ne ighborhood of (yl , c l) .

Let (Y0, ~0) be in an open s t r a t u m of Y x ]0, 1] such tha t ¢(Y0, c0) > sups/%.,0 ~(yo, ").

Let us choose a point b0 6 B with ~'(b0) = (y0,c0). Since ~" is a submers ion f rom the s t r a t u m of B containing b0 onto an open subana ly t ic ne ighborhood of (y0,c0) in Y × ]0, 1], there exists a real analyt ic m a p (y ,e) ~ b(y,c) defined close to (y0,e0), such tha t b(y, e) 6 B and #(b(y, e)) = (y, c). We thus have

(2.12)

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

(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 relat ion (2.12) implies tha t the last t e rm in the preceding sum vanishes identically.

Since the point (y l ,Cl ) is in the closure of the set of points (y0, c0) fulfilling the preceding condition, the conclusion of the theorem follows let t ing such a sequence of points (y0, co) converge to (Yl, Cl).

3. Upper bounds for microsupports and second microsupports

Let Z be a real analyt ic submanifold of R m passing th rough z0 = 0 and Z c the complexification of Z in C N in a ne ighborhood of 0. If r is a posit ive real number , we will pat Br = { z e ZC; Izl < r } and By = Br n Z.

Let h be a real analyt ic funct ion on a ne ighborhood of 0 in Z sat isfying h(0) = 0 and such tha t there exist r > 0 and a connected componen t A of 23~ - h - l ( 0 ) wi th h]A > 0 and 0 6 A. We will also denote by h the holomorphic cont inuat ion of this funct ion to Z c and we will assume tha t r has been chosen small enough so tha t h be defined in a ne ighborhood of Br- We will set g2~ = Br - h - l ( 0 ) and will denote by ~r : Q~ ---, ~ r the quotient of the universal covering of S2r by the equivalence relat ion identifying to 0 every loop of ~2~ homotop ic to a loop of A. Then, 7r- l (A) is the disjoint union of connected components , each of t hem being isomorphic to A by lr.

Let a : A ~ C be an analyt ic funct ion satisfying the following conditions:

3. Upper bounds for microsupports and second mierosupports 59

(3.1) i) 35 > 0, 3C > 0 such tha t for every small enough c, f{,; lh(z)l_<~} la(z)l dz <_

Ce c,

ii) There exist an holomorphic funct ion g : g2"~ --* C, a connected componen t of r r -a(A), a real number g such tha t gl~ = (a°Tr)]/i and tha t for every open

subset V of J2"-'~ with suppe 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 in tegra t ion eourant on Z, associa ted to the euclidean met r ic on R N. Denote by l = c o d i m ~ Z and chose l independent vector fields wi th real coefficients

t ransverse to Z. I f /9 = (/3, , . - . , /31) 6 N', let us set 6(z z) = X ~ t . . . X t ~ ' 6 z and let us consider the dis t r ibut ion

(3.2) u : a l A ' 6 (~)

(where 1 is the character is t ic function). The a im of this section is to ob ta in upper bounds for the mic rosuppor t and the

second mic rosuppor t of u in te rms of the geometr ic da t a of the p rob lem i.e. Z c and the hypersur face of Z c given by the equat ion h = 0.

Dis t r ibut ions like u m a y be considered as a special case of "conormal dis t r ibut ions" along the singular hypersurface of Z given by the equat ion h = 0.

To s ta te the result we are looking for, we must first define some geometr ic 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 char t of Z c by:

(3.3) { (z, ~); 3(Zn)n (resp. (cr,~),,) sequence of Z c (resp. C) such tha t

an" h ( z , ) ~ 0 and a . O h ( z n ) ~ ¢ } .

One checks at once tha t the preceding definition is intrinsic. Moreover, one m a y show (cf. [K]) tha t T~_I(0)Z c is a C-isotropic complex analyt ic subset of T * Z c ( tha t

is the holomorphic symplect ic form on T * Z c vanishes on the open set of its smoo th points) .

Let •z : T * C N I z c --+ T * Z ¢ be the na tura l project ion. Wi th the preceding notat ions , 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) -1 T * N N SS(u) C ez (T;-x(o) Zc) M

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

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


with a non-zero ft. Since both 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_<lZl

with u 7 of the form (3.2) with/3 -- 0 (and another function a). To see that , one has just to decompose locally every vector field Xj as Xj = Xj,T + Xj,N with Xj, T (resp. Xj,N) tangential (resp. normal) to Z. Then Xj,TbZ is a multiple of 5z and Xj,N commutes to a 1A, whence the result. We are thus reduced to prove that for any dis tr ibut ion v, SS(Ov) C SS(v), SS~I(0v) C SS~'a(v). Such inclusions are t rue when one replaces 0 by an analyt ic pseudodifferential operator . Anyway, for a single derivative, and for the microsuppor t , the result is easy: use formula (1. t ) of Chapter I and write T(au)(x, A) = A -10Tu(x, A). The inclusion then follows from an applicat ion of Cauchy ' s formula over a polydisk with center at x and radius ,~ 1/A. For the second microsuppor t along the conormal to a submanifold (which is the only case we will use in the sequel) one may apply a similar me thod after a convenient choice of coordinates, using a t ransformat ion like (3.25) of Chapter II. For a general A, it seems that such tricks are no longer possible, and one is obliged to use the general result we alluded to before (see [Sj] and [L2] for details).

We will first indicate the main lines of the me thod of proof of the theorem in the case of the microsupport , and will give a more detailed demons t ra t ion for the second microsupport . If e is a real number in ]0, co], with e0 > 0 small enough, let us put

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

(3.5) =

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 ne ighborhood of to - iTo, let us write the FBI t ransform with quadrat ic phase of u, Tu(x, A) = f e-~ (~-t)2 u(t) dt as a sum:

f ¢--~(x-t)21 A_Aa(t)~Z(t) at -~- / e-~(x-t)21A a(t)SZ(t ) dr. (3.6) Tu(z , A) = J

Because of (3.1), the modulus of the first term is bounded f rom above by

(3.7) Ce" e -}(Ira x)2

The sccond one ma y be wri t ten fA, e--~(~-Z)2a(z)dz where z denotes the variable on the manifold Z and dz the r i emanman volume on Z.

Let us consider the variety

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

We have

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 con-

tained 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 asser t ion of the l e m m a follows f rom Propos i t ion 1.9. In fact , one knows ([K-S1], [K-S2]) tha t the isotropy of T~_~(0)Z c implies tha t there exists a s t ra t i f icat ion

of Z c such tha t T*h_,(0)Z c is conta ined in the union of the conormal bundles to the s t ra ta . One m a y then app ly Propos i t ion 1.9 to tha t s trat i f icat ion.

The subvariet ies of B~ given by h(z) = e or Ih(z)l = e are smoo th for r, e small enough since, if not , there would be a connected componen t of the complex analyt ic set { z; h'(z) = 0 } meet ing { z; h(z) ---- 0} and not conta ined in the la t ter . But this is impossible , as one sees at once, using for ins tance the curve selection l e m m a (Theo- r em 1.3). T h e fact tha t the manifolds h(z) = e or Ih(z)l = ~ cut OB~ t ransversal ly, if r, e are small enough, is now clear since T~_~(0)Z¢ contains the l imits of the conormal directions to these submanifolds when e ---* 0.

The l e m m a shows tha t if r, e0 are small enough, the open set

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

and the submani fo ld 22 defined by (3.8) fulfill condit ion (T) of Section 2. T h e same is t rue 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), wi th values in D .. . . sat isfying condi- t ions (2.3). There is a unique h o m o t o p y

H : Fo,~ x [0,1] ~ ~,~ (3.9)

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

such tha t /~l~=0 = (Tr]~i) -1 : F0,, --~ F0"-~_~ = (Trl.~)-l(F0:~) and tha t 7r o H = H . Let

FI,'-"-~ -- H(T'0,e, 1). Then, using Stokes formula, one sees tha t the second t e r m of (3.6) is equal to

(3.10) f ~ e- ~(~-~(~))~(~) &(~) I

Actually, the fact tha t the b o u n d a r y of the in tegra t ion chain is not kept fixed by the homotopy /~ r does not mat te r . The pa r t of the b o u n d a r y which could move dur ing the de format ion mus t s tay on the complex submanifo ld Se, over which the complex form of degree equal to d ime Z ¢ dz vanishes identically.

Let us consider the funct ion (see (2.5))

1 Re(x - z) 2 (3.11) ~ ( x , e ) = inf sup - 7 . HET"I. zEFI,.=H(I,Fo,.)

Because of Corol lary 2.5, q0 m a y be extended to e = 0 and this extension is subanalyt ic . The ma in point , in the p roof of the inclusions (3.4), is to show tha t the lower bound in (3.11) is reached by a contour/~l ,e whose volume is bounded uni formly wi th respect to e close to 0 and x close to a given point x0. Such a result will be proved precisely


in the demons t r a t i on of the second inclusion (3.4). For the t ime being, we admi t t ha t proper ty , and go on with the proof of the first fo rmula (3.4).

Because of (3.1), the possibil i ty of choosing a contour realizing the in f imum in (3.11) and whose volume is un i formly bounded with respect to the p a r a m e t e r s implies tha t the modu lus of (3.10) m a y be es t ima ted 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 ~(x0 ,0 ) = ½(Imx0) 2. I f 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 subanalyt ic , it follows f rom Lojaciewicz inequalit ies (Theo rem 1.4) tha t there exist fl > 0, C > 0 wi th

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

for x close to x0. Consider the funct ion

(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( Imx - I m x 0 ) 2 and by a s sumpt ion f(x0) -= 0. In par t icu lar , if ~ is differentiable at x0, its derivat ive at tha 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 tha t for every n

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

By the curve selection l e m m a (Theo rem 1.3) there is a reM analy t ic curve issued f rom x0 and conta ined (except its origin) inside the subana ly t ic set { x; g(x) < - 5 I x , - x0 ] }. Using tha t g is l ipschitzian, we deduce f rom tha t the existence of a cone F with ver tex at x0 in C g and of a ne ighborhood V of x0 such tha t for every x C F N V , g(x) < - ~ [ x - x o [.

The funct ion

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 p lur i subharmonic . I ts value at x l , close to x0, is bounded f rom 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 es t ima ted in general by

+ - + o

If moreover x E F VI V, using (3.14), (3.15) wi th c = e -x~ (V > 0 to be chosen) and the choice of F , we get for f ( x ) the uppe r bound


6 6 + C e _ ~ ] 0 ( 1 ) . (3.19) sup [ - 3 ' a + ½ ( ~ + Ix0 - x , I )2 ,K7 - ~ + ~]x0 - x,[ +

There is e0 > 0, c > 0 such tha t for Q < ~0 and xl • B(xo, ~2), the quot ient of the volume of { x; ] x - x 1 [ = ~ and x • F } by the volume of the sphere S(xa, ~) = { x; Ix -x1 [ = ~ } remains bounded f rom below by the uni form cons tant c. Let us wri te then

1 f ( x , ) = ~ log [Tu(x,, A)[ - ½ ( Im x0)2 _ ( Im x0) ( Im xl - I m x0)

1 I s f ( x ) dx -<

1 f ( x ) dx + f ( x ) dx .

Using (3.19) to es t imate the first integral in the preceding sum and (3.18) to e s t ima te the second one, one sees tha t if Q and 7 are fixed with Q2 << "7 << Q << I there is d > 0 wi th

1 (3.20) l og ITu(x,, A)I _< ½(Ira x , ) - c'

for xa in the ball B(xo, ~2) and A large enough. T h e propos i t ion follows f rom (3.20).

The propos i t ion shows th.at if (to; T0) • SS(u) and if xo = to - iTO, then necessari ly T(x0 ,0 ) = ½(Ira x0) ~ and x ~ T(x, 0) is differentiable a t x0. We mus t now deduce f rom these proper t ies tha t (to, T0) mus t be in the right hand side of the first inclusion (3.4). Towards this end, let us prove:

L e m m a 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).

• W h e n m goes to i n a n i t y , converges to

Proof. We m a y assume that the derivat ive o fT( . , 0) at x0 is zero. Since T is subanalyt ic , there exists a s t ra t i f icat ion of IRk × [0, 1], compat ib le to IRk X {0}, wi th the p rope r ty tha t t0 is analyt ic on the open s t r a t a and tha t !p[,=0 is analyt ic on the s t r a t a of IRk × {0} opened in IRk x {0}. The complemen t F of the union of these ones in IRk × {0} is a closed subana ly t ic subset whose dimension is less or equal to k - 1. There exists then a ne ighborhood V of x0 in IRk and an open cone F of IRk with ver tex at x0 such tha t F N V does not meet F : to see tha t , one has jus t to show tha t C,o(F, {x0}) # T~oiR k, and this is t rue since C,o(F , {x0}) is subana ly t ie wi th d imension less or equal to k - 1 (this last p rope r ty 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 tha t F N V C W. Assume tha t the conclusion of the l e m m a is false. Then , possibly after replacing W by its intersect ion with a ne ighborhood of (x0,0) in IRk × [0, 1], there is c > 0 wi th

(3.21) inf 0~ > c .


Let us show tha t this implies I ~ ( ~ , 0 ) l > c on r n v . If not , there is x~ in F lq V with

I°o-~ (x l ,0) [ < c. Let ~ > 0 be such tha t

(3.22) Ix -- Xl[ ~<~ 5, X # X 1 ~ [ ~ ( 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 v~(~,~) iv.~(,,¢)l over G. On has:

(3.23) ~2(~((x, ,¢) ,s) ,¢) - ¢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 contradict ion with the first relation (3.22).

We thus have I ~ ( ~ , 0) J > c over r n y On the other hand, since ~(~, 0 ) i s assumed

to be differentiable at x = x0, with a vanishing derivative, then is for every a > 0 a ne ighborhood 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. Restr ict ing

V and F if necessary, we see that there is 5 > 0 such that s --~ ~50(x,s) is defined on [ - ~ l x - x o l , 5 l x - 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. The result is proved.

Proof of the first assertion of Theorem 3.2. Let (to, To) E SS(u), with to = O. Because of Propos i t ion 3.4 and of Lemma 3.5, there is a sequence (Xm, em) converging to (to --iro, O) such that , for every m, q~ is real analyt ic close to (Xm,em) and tha t 2_i oo_~(xm,¢m ) converges to 2i ~ ( x 0 , 0 ) when m goes to infinity. Since qp(x,0) _< ½(Imx) 2 with equali ty

at x0 (for (to, To) • SS(u)), we have 2i °o-~z (x0,0) = - I m x0. Since for m large enough,

1 Re(x - z) 2 , where c~F0 ~ still denotes 01"o, N c9f2 .... it we have T(Xm,¢m) > supbr0." - -y , ,

follows f rom Theorem 2.6 tha t there exist z,,, • f2~,~ such tha t

(3.26)

(3.27)

2 0 w 2 0 ( Z m - - Z m ) 2

* N 'T* C N * ( z ,~ , i ( xm-zm) )ET C [ { l~ l=, . }U. , i lh l=~/2} UTih=~m}CNuT~NCN.

Since for ]z] ---- r and R e x close to zero, - Re ( ~ < ½(Imx) 2 - c and since, on the

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


E s t i m a t e s for t h e s econd m i c r o s u p p o r t

We will now prove the second inclusion (3.4). We still s tudy 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; Im2 = 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

- ~e2 (Y-~)2- i~gL(M-I (~) )Tu M -1 2 (3.29) T 2 u ( y , A, #) = e 2(,-,~) ( ( ) , A) d2 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

(3.30) (,-iT,,- Because of the construction of gL, A is the submanifold of T * N g given by the equations

O--~'(t -- iT), (t, 7-) C T * R N. The complexification A c of A in T * C N has thus for T = Ox~ 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 - 2 • - - i - X --I - 2 = ~-R n e - ~ ( y - x ) --,XgL(M (x))---f(M (x)--t) 1 A a ( t ) S z ( t ) d t d 2 .

" I~[<R

Using the definition (3.5) of A~, we break (3.32) into a sum I~ + I~ writing 1A = 1 A - A , + 1A~. Because of (3.7), we have

( 3 . 3 3 ) I I ~ 1 Z C ~ ° ~ e ~ ( I m y ) = -


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 tha t the nota t ions B~, ~- have been defined at the beginning of this section). We take r and e0 small enough so tha t Lemma 3.3 implies tha t condit ion (T) of Section 2 is satisfied by ($2, 57) and by _r'0,,.

If H is an homotopy of the form (2.2), with values in $2 .. . . fulfilling condit ions (2.3),

we still denote by H : F0,, x [0, 1] ~ g2~,, the unique homotopy 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 , ~ ) .

I f /~ ,~ = ~r(/~o,., 1), Stokes formula shows that I~ is equal to

/ ~ 1 * A 2 - 2 - - - e - ~ ( y - z ) -,),g,(M-~(~))-~(M-~(z)-,r(z))2a(5)dTr(~,)d ~ . (3.35)

Let

#2 1 (3.36) ~(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 funct ion

(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, ,, ~)).

O n e 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 tha t y --~ k~(y, # ,~) is lipschitzian with a lipschitz constant bounded from above by C# 2 for # > 0 close to 0 and c E ]0, e0] (this follows f rom the fact tha t in (3.37), (3.38) the only t e rm of • depending on y is mult ipl ied by #2). Moreover, by Corollary 2.5, the graph of e is subanalyt ic in Y x [0, 1/2] x [0, e0]. Lastly, since there is c > 0 with

2 #2 (3.40) -~ [ ( Im y) 2 - c] ___ ~2(y, ~ ,~) < ~ ( I m Y) 2

for # > 0 close to 0, c E ]0,~0], a similar inequali ty is satisfied by e . Let us pu t


(3.41) 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:

p: Y × ]0, ½1 × K × ]0, ~o] × 10, 11 --+ M = Y × ]0, ½1 ×]0, c0] × ]0, 11 (3.46)

( 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 (Mi) 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 VI p-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


sup ~(y i , 5 : , z , # i ) <_ k~(yi ,#i ,ei) + l t i . (~,z)EFl,,i

Let / t i 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 subanalyt ic contours.

Anyway, it is easy to see tha t there exists a subanalyt ic chain/~/, which is homotopic to FI ,~ th rough an homotopy preserving OFo,~ and compatible to Z ~ , such tha t

sup ~(yi,~c, Tr(z.),l~i) ~ kO(yi,#i,ci) -}-ti (~,~)Er~

(in fact, one has essentially to approximate the C ~ mapping Hi( l , .) by a real analyt ic map preserving the different constraints) .

For every i E I and every e = (y,/~, e, t) E Mi, gi,~ is an homeomorph i sm from J2~,~ i onto ~2 .... Let us denote by gi,~e the homeomorphism deduced from gi,e on the coverings

gi,"~: ~,~-'~ --~ J2~--/~ and put ~ = g~,e(Fi). By construct ion, there is an homotopy from Fi

onto F0,~. Conjugat ing this homotopy by gi~'~, we thus get a continuous homotopy from

Fe onto Fo,~. Moreover, by the compatibi l i ty of gi with the sets (3.45), the composi t ion of this homotopy with 7r satisfies conditions (2.3). Using the definition of the first set (3.45), we see also tha t

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

Stokes formula shows tha t 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 inequali ty (3.47) we deduce tha t 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 obta ined using a tr ivialization over M,, one knows (el. [Ha2] or [T]) tha t meas(Fe) is uniformly bounded for e E Mi. Since the s t ra ta Mi are in finite number , the inequali ty (3.48) implies tha t there is C > 0 such tha t 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 ~ ( ~ ' " ' ~ ) + : ' ~ -

Lett ing 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 funct ion 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~[(Im Y)2-c] • C


Proof. Because of (3.33), (3.44) we have

1 A/22 log [T2u(y, A, v)[

/ a l o g ~ ½(Im O ( ~ ) , _< supt,-T7 + v)~ + K log ¢ A/22

- - + +

Since k~(y, #, e) and #-2kr,(y, #, 0) are subanalytie, it follows from Lojaeiewiez inequalities (Theorem 1.4) tha t there is a posit ive/3 with

(3.50) kV(y,/2,e) _< kV(y,/2, 0) + cs te z _</2:[¢(y) + cst/2 ~] + c s t e z .

If one takes ~ =/22/~e - ~ 7 the result follows by the same argument as in the proof of Proposi t ion 3.4.

Proof of the second assertion of Theorem 3.2. Let (qo,q~) be a point of SS~'l(u) such that q0 = (to,to). The point Y0 = A- l (~ (q0 ,q~) ) corresponding through the identifications satisfies Rey0 = 0. Because of (3.33), (3.44), (3.50) we must have ¢(y0) = ½(Imy0) 2. Proposi t ion 3.7 then implies tha t ¢ is differentiable at y0 and then

2 0~ = _ Imy0. By Lemma 3.5, there exist a sequence (Ym,/2m)m with/2m > 0 for every i Oy m, converging to (Y0,0) such that for every m ~P(y,/2, 0) is real analyt ic in a neighborhood of (ym,/2m) and that ~ 2 og, 0 2 0,k ,m ~ b~y(Y",/2,,, ) converges to 7 -g~y (Y0) when m goes to infinity. Applying once again Lemma 3.5, we see tha t for every m, there is a sequence (ym,k,/2re,k, em,k)k with e,~,k > 0, converging to (gin,/2,~, 0) such tha t kV(y,/2, e) is real analytic in a ne ighborhood of (Ym,k,/2m,k,Sm,k) and that ) 2_ Og'(y,~,k,/2,,,k,em,k ) #z k i Oy

OqJ converges to ~ 2_ ~v(ym,/2m, 0) when k goes to infinity. t t~ 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). Theo rem 2.6 then implies tha t 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 ) = 2 0¢ i Oy -i N (ym'k'xm'k'zm'k'/2m'k)

(~m,~, zm,k; T ~ ( )' T ~(" s.m~ S*m,~ 6S*m,~

where ,.q~ is the strat if ication of ~2r,~ defined before Proposi t ion 2.1. The right hand side of the first formula in (3.51) is equal to " 2 z/2, . ,k(ym,k -- 2m,k)

and since i 2 o~ - -~-~v(ym,k,/2m,k,em,k) converges to - - Imy0 , one sees tha t Xm,k goes to i #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) tha t Xm,k, Zm,k, ~ ( Y m , k , ~2re,k) = 0 i . e .

(3 .52) i ( vm,k - ~ , ~ )

~@m~k + "'7~'--(M-i 1 Zm,k)] OM-1 -- -- k [ - glL(M--l(~m,k)) (Xm,k) -- T ( X ' r a , k ) • /2m,k


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

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

must be in

T* C N T* {}hl . . . . ,/2} U { h = e m . ~ } C N u T ~ C N U { ( z , ( ) E T * C N ; z e Z C, I z l - = r }

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

(Zm, i 1 (3.54) \

G 0-I(T•-I(0)Z C) U { (z,~) G T*cN; 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 m,k 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

_ _ 2 1 . tdM_ 1 o XL(Zm,--~m¢rn) (3.55) (Re Y0, - Im y0) = l i r n #~,

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 map from T*A to T*L deduced from M o XL. Since this map 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 natural ly 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 A C and * c * T~_~(o)Z respectively by A and T~_~(o)Z.

To see that, let us take on Z = R 3 the real analytic function

h ( x , y , t ) = t + x 6 + x4y 2 . (3.57)

The distribution

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

is of the f o r m (3.2) a n d thus, the inclusions (3.4) are valid for it. Let N b e the s u b m a n i f o l d of IR 3 w i th equat ion t -- 0. D e n o t e b y (x, y, T) the coordi-

nates 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


(3.59) uo(x , y) = x 2 ( x 2 + y2)½ .

A straightforward computa t ion shows tha t

+ =

and thus T~[~ 3 is contained in SS(u0). Let us consider in par t icular the point (x = 0, y -- 0; ~ = 0, r] = 1). Because of the second inclusion (3.6) of Chap te r II, ei ther 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 . 6 2 ) (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 character izat ion (1.10) of Whi tney ' s normal cone (with the canonical trans- format ion X( X, y , t; ~, r h T) = ( x , y, T; ~, r], - t ) we see that there are sequences (z,~, y,~, tn;

• 3 ~,,~,,,T,~) C T~_~(0)C and un ~ 0 with

x~ -~ 0, y~ --* 0, 7~ --* To, ~" * 7/~ y . t~ ~_. - - - - * x = 0 , - - - - * = 1 , - - - - * = 0 . /t n Un Un

By (3.63), this means

3 2 (3.6~) 1 ( 6 ~ + 4~.yn) --* 0 Un

1 ( 2 x 4 ~ ) -~ ± It n TO

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

Writ ing the first relation (3.65) as

! ( 2 x ~ y ~ ) ( 3 ~ + 2 y" ) --* 0 Un Yn X n

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) mus t verify

3Xn + 2Yn ___, 0 . Yn X n

But the funct ion s ---+ 3s + ~ never vanishes on ll~, and so the quotients *~- must be imag- Y~ inary numbers. Thus we cannot replace the right hand side of (3.64) by CA(T~_~(o)It~3).

I V . S e m i l i n e a r C a u c h y p r o b l e m

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 , . . . ,Xd) and let

02 02 ~ 02 (1.1) C-- at 2 A- - at 2 ~ ~ 'Ox~

be the wave operator. Let 12 be an open subset of N TM 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

d given. Let P ( t , x , u ) m j Sobolev space H~(Nd), with a > = ~ o p j ( t , x )u 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

(1.2) Ou = P(t, x, u)

Ul t=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 analyt ic submanifold of w. We will assume:

(1.3) u0 and ul are C ~ classicM conormM distr ibutions along V.

Let us recall the meaning of the words "classical conormal". Remind first tha t if U is an open subset of V on which exists a sys tem 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 [ ( 1 + I~l)-r+lZllD:,D{a(x',~)l] < +oc . z' EU, .kEN

Such a symbol is said classical if for every k E N, there is a funct ion ak(x', X) smooth on g x (N - {0}), positively homogeneous of degree r - k in X, such tha t

N

(1.5) VN E N, sup a(x' ,a) - E ak(x', a) X N+I < +c~ [~1 -~1 o z'EU

Then (1.3) means that uo, ul are C °o outside V and tha t every point of V has a ne ighborhood W, endowed with a system of local coordinates x = (z ' , x") E I~ d-1 × R in which V M W = {x" = 0}, such tha t u0[w and ul]w may be wri t ten on the form

f iXtt. II (1.6) e ~ aj(x' , ~") d~" j = O, 1

for convenient C ~ classical symbols a0, al on ( W M V) × N. We want to est imate the Coo wave front set of the solution of (1.2) by an object

built f rom 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. ~2 m = r m

D e f i n i t i o n 1.1. We shall say tha t a set g of sequences fulfilling conditions i), ii), iii) above is admissible if it contains every subsequence of any of its elements and if it satisfies the four following axioms:

A.I : E contains every sequence (zm ;(m)m fulfilling i), ii), iii) and such tha t (,~ --+ 0.

is a sequence of C l+d such tha t lim(zm Zlm) ----- 0 A.2: If (zm; ~m)m E g and if z m 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 if (Z'm)m is a sequence of C l + d , s u c h t ha t limz~m is on the b o u n d a r y of the open ha l f light cone with ver tex a t l im z, , which does not meet t = 0, and such tha t for every m, (Zm; (m) and (z~; (,n) belong to the same complex

Z t • bicharacter is t ic of 0, there is a subsequence of ( m, (m)m belonging to g.

A.4: If (Zm; (Jm)m, j = 1, . . . , g are g sequences in g wi th a same base point for every m, and if (Zr,; ~-*),n is a sequence fulfilling condit ions i), ii), iii) and such tha 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 tha t there is an integer N and N sequences (Zm, ~Jm)m in g, j = 1 , . . . , N , with same base point , such tha t z = l imzm,

= l im(4~ + . . - + 4N).

We will denote by w c a small enough ne ighborhood of ~o in C d and by V c the complexif icat ion of V in w c. We put

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

Zm = (0, Xm), = (Tm, m) 'T'* a)C 2 2 (Zm,s¢,,,) • - v c and T m = ~rn }

The aim of this chapter is to prove:

T h e o r e m 1.3. I f u is a solution of (1.2) with Cauchy data satisfying (1.3), we have

(1.8) 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 detai led and complete demons t ra t ion for the general case will be done in the next sections.

~r 5 Let us take the space dimension d be equal to 3 and let u E Hlo¢(~ ) wi th 2 < a < 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 rob lem ob ta ined when one replaces u 2 by 0 in the right hand side of the first equat ion (1.9) and let f = u - v. Let e+ be the e lementa ry solution of the wave opera to r suppor ted in the forward light cone and denote by E+ the opera to r of convolut ion by e+. We will use the following fact: if g • H1:c(12), then E+(llt>__0}g ) • Hl~oc(Y2) and is suppor t ed in {t >_ 0}. To see tha t , wri te 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 suppor t ed close to ( = 0, equal to 1 on a conic ne ighborhood 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 s tudy 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 te rm 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~+2 and 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 te rm in the right hand side depends only on v and so is essentially known. The two other terms have a regularity which is not bet ter than H~¢ and moreover the unknown function f+ is involved in their expression. The solution v of the linear problem may be writ ten as a linear combination of integrals of the form

(1.14)

where

(1.15)

f +(z ° - z l ) w ( z l ) d z ' j = 0 , 1

w0(z) = u0(x) ® w l ( z ) = u l (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 ' t l ) ~ _ 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


( 1 . 1 7 ) ( z O , z t l , z t t l , z 2 ; ~ g , O , O , O )

¢ 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 zo o 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 - z t l ) ( ) ~ e + ) ( z 0 - z t t l ) ( x e + ) ( z t t l - 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)< ,1 de it1 de 2

and because of (1.18), we have

CN(1 + [¢o + (,1 + (-1,1 Jr ¢2D-N (1.21) [~(¢0 ¢,1, cttl , (2 ) ] < (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 ^ tl ^ 2 ,1 d ( 2 ¢ ,¢ )wi(-( )wj(-( )de < 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) (1 + [¢o[)~,+1 ~ CN(1 -~- ]¢ 0 "~ ¢t'1 _~_ ¢,,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+ It° + r"l + r2l)-' dr~ (1 + I~-,11)(1 + 1~-21) - 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 + IClY" [ i ( 1 + 1~° + p +~"' + ¢21)-N+4(1+ I('1) - ' ' + ' (1.25)

x (1 + 1~2 I)-2"+~(1 + IC ''~ I) -2" d4 '1 d~ '2 dC "1]

IS +l.o +."' i)). x ( 1 + ! ~ ° + { " + { " 1 + { l) ~, x+b_o+~_. , i

X t~1(~11)2/+ ( ( H 1 ) 2 W2(~- 2 ) 2 d f ,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) ¢ W F ( ~ + ( 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 . S o b o l e v s p a c e s a n d i n t e g r a t i o n s b y p a r t

In this scction, we will give the first step in the proof of Theorem 1.3. We follow closely the reference [L4].

5 First of all, to get rid of the limitation a < 7 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 ~ > ~.

De f in i t i on 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

i (i 12 (2.1) II~ull~, clef (1 + I~'1) =" I ~0 - , ~)1 d~" de < +oo

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.


ii) Let vo E H~(Rd), V l e H~-~(Rd). Then the solution v of the

[3v = O, vlt=o = Vo, Otvlt=o = v~ is in A(NI+4).

iii) I ra E L~(N ,H~(Nd) ) then E+(l{t___0}a) is in A(R~+d).

Cauchy problem

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 119zuII A whence the C~(~ ) -mo 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 <_ f f 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} d t ' )d t

1 fo +~ = ~)e [o(t ,~ - I~1) - o ( t ' , , + I~1)] dr' 2gill 82(t,, -it '~ ,

where O(t',r) = e i''~ f t +~ 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.


Let still 17 be an open subset of R l+a which is a domain of de te rmina t ion 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 12 k x I7 wi th for every j = 1 , . . . , k , zj = ( t j , x j ) and 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 compac t ly suppor ted dis t r ibut ion on g2 k x f2, let us set

k

(2.2) I l v l l ~ ( , . , ) - - / ] - [ ( l + l ~ j l ) -~" sup I~((rj ,¢j) j=l ..... k~W)12 d~ . j - - j=1 (rl ..... rk)

D e f i n i t i o n 2.3. One says tha t a d is t r ibut ion u E D ' ( O k × f2) is in the space M~(qo)

if and only if:

i) Supp(u ) C { ( z ,w) E ~k x ~ ; Vj = 1 , . . . , k , w - zj E -P },

it) The re is 9 E C~'~(~), T - 1 close to w0 and for every q~ E C ~ ( ~ k) an integer N E N such tha t (1 + IwI)-NN~ i @ ~PUIIM(W) E L 2 ( ~ l + d ) ,

iii) There is T E C~( J2 ) , ~ -= 1 close to w0 and 3' open conic ne ighborhood of w0 in R TM - {0} such tha t for every • E C~(J2k) , (1 + Iw[)~]l~ • ~uliM(w ) E L2(7).

We will make use of the following proper t ies of the space we jus t defined:

L e m m a 2.4 . i) The space M[(qo) is a C~( f2k+l) -module . I f u E M[(qo) is compactly

supported in z, then v = f~k u ( z , w ) d z is in the space H ~ (i.e. v is microlocalty g ~ qo

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 proper t ies i) and it) of Definit ion 2.3 for 0u to the reader. To check iii) let us wri te

0 ~ ( ( , ~ ) = f ~ ( ¢ - ¢ ' , ~ - J ) ~ ( C J ) d ( ' d~ '

wi th ~2 = q5 ® Tu. Wri t ing ( = (~-, ~), { ' = (~-', ( ' ) we thus have

k 1 ,

sup,. Io~(O-, ~), ~)1 _< c N 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, mul t ip ly ing by 11j(1 + [~j[)-2~

(1 + ]w[) 2v, in tegra t ing in ~j E R d, j -- 1 , . . . , k, and in w E "y small conic ne ighborhood of wo, we get by Cauchy-Schwarz inequali ty

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~ <~ . ,,r.#

1

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 sum/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 may 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 may 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)l i IgII(T:'4II)IdT~) d4tl "

Since ~ > d/2, the inequality

I1(1 + 1~1 I ) - 'V * g(6)llL~<d~) <-- cst I1(1 + 16 I ) -VI IL~ I1(1 + 141)~gllL~

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

k P

(2.3) E-~%)(z ,w) = / 1-I e_(zj - z})~(z~,... ,4,~)dz',." .dz'~. d j = l

Because of condition i) of Definition 2.3, the integrand in (2.3) is compactly supported in z t • ~2 k.

We saw in Section 1 that the fundamental point, in the proof of Theorem 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 • ~2 k, (z, ~ = 0, q0) • WE(u). Then E~_k(u) • M; +1 (qo).

Proo]. We may 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

k P

(2.4) E~_k,¢(u) J I I ( ¢ e - ) ( z j -- 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


k def ~ H '~' '~ ( ¢J = E~_k,¢(u)=- ¢ e _ rj ,~j) . f i

j = l

the first c laim of the l e m m a follows. The a s sumpt ion of ii) implies t ha t there is e > 0 such tha t fi((, w) is rapidly decreas-

ing for I(~l--< elwl and 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 han (1 + ]w I) -~ sup , 1~((, w)l. The second assert ion of the l e m m a follows.

In the sequel, we will have to make use of the following kind of opera tors . If ( / 1 , . . . , lk) is an element of (N*) k, we denote by

k

(2.5) ~ -~ I I ~'~ (~) j = l

the ope ra to r sending the dis t r ibut ion u E ~D'(~ k+l) on the d is t r ibut ion on $2 h+ ' ' '+ l~+l (with coordinates 3 J w ( (Zl) j=l ..... l l , ' ' ' , (Zk)j=l ..... l , , )) given by:

1 ZO) @ [~'(Z~ -- Zl 2) @ ' ' " @ ~(Z/, t - 1 -- Z[1)] @ ' ' " (2.6) u(z~,. . . ,Zk,

. ® [~ (d - z~) ® . . . ® ~(z~ ~-~ - z ~ ) l .

We have

L e m m a 2.6 . The operator (2.5) maps M[(qo) into M~+...+tk(qo ).

Proof. By induct ion on ( l l , . . . , Ik) , it is enough to prove the l e m m a when 11 = 2, 12 . . . . . lk = 1. If u is a compac t ly suppor ted element of M~(qo) and if v is its image by (2.5) we have

^ I ~(~, ¢1, ¢~,---, Ck, ~,) = ~.(¢1 + ¢I, ¢~,--., ~ , w)

d i = z12. Using tha t , since n > 7 where we denote by (~ the dual variable of z 1

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 example of an element belonging to one of the spaces M~(qo) we just defined and studied, is the d is t r ibut ion 5(z - w), which is in M~°(qo) for every u0 < - 1 and every q0 E T * Q - 0.

We will now describe the m e t h o d of in tegra t ion by pa r t s which is one of the key s teps in the proof of T h e o r e m 1.3. Let v be the solution in Q of the linear Cauchy p rob lem

(2.7) Dv = 0

t~ I t=0 ~" u0

OtVlt=O ~ It 1


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 + = f l l + t > 0 } .

As in Section 1, we write

(2.9) f+ = E+[P(t,x,vl{t>_o} + f+)]

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) a o = 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

(2.11) f+ = E+[P(t, x, vl{t>0} + at + st)]

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 . k] v l{t>o}al] j,k

s,+, : F , ml ~k)v at s, ] , ,,>1 j,k

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* Vo n--{O} Mk (q) generated

by the distribution 5(z - w).We then define Vt' , 1 < i < l, by induction: V t' is the C-vector subspace of M generated by all the distributions of the form

k k

at_ i (zj)l{t,>_o}V"' (zi)p,j(zj)E2k(b(zl,... ,zk, w)) j = l j = l

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 < degP . 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 Vz i-1. Moreover, for every i, V z' 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


(2.14)

with b C V~'.

/ ~,_i( zl ) . . . ~,- i( zk )b( Zl , . . • , zk ,w)dz l ... dzk

Proof. For i ---- 0, we jus t write st(w) ---- f 6(z - -w)st(z)dz . Assume tha t 8t has been expressed in t e rms of s t - i + l using expressions like (2.14). By (2.12), s t - i+1 = E + ( ~ t - i + I ) where st-i+1 is a sum of monomia 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 pe r fo rm the in tegra t ion by par t since, on one hand, it follows f rom (2.13) t ha t for every fixed w • ~, z ---* b (z l , . . . , zk ,w) is compac t ly suppor t ed in D k and on the o ther hand ~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 obta in the expression we were looking for.

T h e expression (2.14) with i = t will play the same role tha t the expressions (1.16) in Section 1. For 0 < i < l let us define

(2.15) Z ~ = { q e T * Y e ; 3 b ( z , , . . . , z k , w ) • V t i, 3 ( z a , . . . , z , ) G Y e '

such tha t ( z l , . . . , z k , ¢ l = 0 , . . . , ¢ k -- 0;q) e WF(b) } .

We have

P r o p o s i t i o n 2.8. For every 1 e N, WF~0+t(u)[ ,> 0 C WF~0+/(at) t_J Z~ 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 tha t if q ~ Z ° U . . - U Z[ -1, then st is microlocal ly in H ~°+l a t q. Because of (2.14) wi th i = l and of L e m m a 2.4i), it is enough to see tha t if q ~ Z ° U - - - U Z[ -1 every element

b(za,. . . , zk, w) of Vt t is in M~,°+t(q). Let us show tha t every e lement b(z l , . . . , zk, w) of V~ i is in M~°+l(q) by increasing induct ion on i. If i = 0, this follows f rom the inclusion

~0 1 Assume tha t the result has been proved Vt ° C M~ (q) for every q when u0 < - 3 " at order i - 1. If b ( z l , . . . , Zk, w) E Vt i-1 and q ~/ Z~ -1 we see f rom L e m m a 2.5 ii) tha t

E~_k(b) E M~°+i-1)+l(q). It then follows f rom (2.13) and L e m m a 2.4 tha t every element of Vt i is in az~0+l ""k (q) (under our assumpt ion: q ¢ Z~ U . . . . . . Z[ -1).

The second step in the proof of Theo rem 1.3 is to ob ta in a geometr ic uppe r bound for WF,o+ l ( a l ) U Z ° U---U Z[ -x . To do so we will prove tha t for every e lement b(z l , . . . , Zk, w) of Vt i, E°_k(b) m a y be wr i t ten in t e rms of e+ and the Cauchy da t a th rough an expression


generalizing (1.17). Applying then the results of Chapters II and I I I we will get a geometric estimate which will provide the conclusion of Theorem 1.3.

3 . E n d o f the proof o f T h e o r e m 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 non- empty). 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 - z j ) jeI-{0}

{ e } ( ( z J ) J e l ) = H "da-~(')) .~(e(j)) "{tj=0} ® u~(j)(x j ) H ® 1 ~{ti=0} " j E J t j E J "

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 re-

places in the f irst formula (3.1) e+( t , x ) by ¢ ( t ) e + ( t , x ) . A s s u m e that Uo E H~oc(Rd),

Ul E loc ~,

(3.2)

3. End of the proof of Theorem 1.3 85

There exists ~ > 0 such that for any g E C~(12), with e; > ~.

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 1-I~e~_j ~(z~)1-IieJ,, 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

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

I t I t zj = - z j + z~(j) j E I - {O}, z 0 =z 0

the dual variables are related by the formulas

E k~f-~(j)

E

j e I - {0),

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 ' j E J


in tegra ted with respect to d ~ , j E J (where we used formula (12) of the in t roduct ion to A

es t imate ICe+ ].) To conclude the proof it is enough to show tha t it is less or equal than cat I l j~ j (1 + I~1) ~' log(2 + I~1) with 6' small relat ively to ~; - d. To do so, decompose the domain of in tegra t ion into I~1 >> (1 + I~1) and I~'1 < (1 + I~1) for every j and in tegra te first with respect to the indices j for which lr~[ >> (1 + I~1), using tha t 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+~ .

Then in tegra te 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 obta ins an es t imat ion by cst I - [ je j (1 + l~t) ~' log(2 + t~l) where 6' is a cons tant mult iple of 6. If 6 is taken small enough, we get the conclusion.

L e m m a 3.2 implies tha t the product

(3.3) ]D] = [D]. {D}

is well defined. The explicit d is t r ibut ions whose wave front set will allow one to get an uppe r bound for the quant i ty 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 d is t r ibut ions over Y2 x Y2 k genera ted by all d is t r ibut ions of the form

(3.4) a(z0, z') = f [Dl(z0, z', z")~(z0, z', z") dz"

for all d i ag ram D = ( I , J ' , J " , e ) , all ~ E C~(/21II) , such t ha 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 emark tha t if (z0, z ~) s tays in a compac t subset of ~2 x f2 k, then z" --~ iD[(zo,z',z") is compac t ly suppor ted in 12]/"1: in fact , for every j E I - {0} we have z j 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 s tays in a compac t by assumpt ion , and if 3 E J , zj = (tj, xj) with tj = 0. This implies tha t zj s tays inside a compac t subset of Y2 (see the figure).

Zo


We have:

L e m m a 3 .4 . i) The space Mo is an algebra, all of whose elements are supported inside { z z (t, z); t > o }, stable under the action of E+.

ii) I f a(zo,z') E Adk, a j (z j ) E Ado, lj E N for j E I' , we have

(3.5) E-*r'~ [H ~'~ H ~,(zk)a(zo,(zj)j~,,)] e AdZ,~ j E I ~ k E I '

Proof. i) An element a E .A40 is a dis t r ibut ion of the form

(3.6) a(zo ) = / [Dl(z0, z")~(Zo, z") dz" .

It is an evidence tha t the produc t of two expressions like (3.6) gives an expression of the same form. On the o ther hand since I ' = 0, it follows f rom the discussion before the last figure tha t a is suppor t ed inside {t _> 0}. Lastly, since E+(a)(zo) = f e+(zo-Z'o)a(4) dz'o we see tha t E+(a) E .£40.

ii) The assert ion (3.5) is an immedia te consequence of the definition of the opera tors

E_~ Z~ b and 1-IjeI, ~b"

We will use also the following lemma:

L e m m a 3 .5 . For every integer l, the distribution al defined by (2.12) is in Ado. More- over for 1 < i < l, for every b (z l , . . . , zk ,w) C VL E°-k(b) ~ Adk

Proof. Write if w = (s, y)

= [ e+(w - z)~,=01 ® 1 ~z 1{~>0}

- f e + ( w - z ) [~ ,=0/® u0 + ~ .=0 /® u,] d z ?3(w)l {s>0/

This shows tha t these two dis t r ibut ions belong to Ado. Using (2.12) and L e m m a 3.4. i) we see t ha t at E .Ado.


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 Vt i, 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 Vl i, 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

(3.7) UWF(/ indexed by diagrams D, and smooth functions ~ (were we denoted z 0 = w a n d z ' =

(z~ , . . . , zk) ) . By (2.13) every element b C Vi z 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 J2 Ill-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 every 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 = 0 i f j f f J, zj = (O, xj) if j c J, T* C d * C d ( j = ( T i , ~ j ) a n d ( x j , ~ j ) E V ~ UT~d i f j c J ' ,

(j = (ri ,~j) and (x j ,~j) < T ~ C d i f j E J " }

where V c is the complexification in w c of V C ~v, and

(3.11)

3. End of the proof of Theorem 1.3

AID] = { ((zj)jEx, (~j)jEI) C T*(Cl+d)JII;

there exist :~-j E C TM, j E I - {0} such tha t

(zj - zi(j) , ~ j ) E Ao j E I - {O}

¢0= }--~ ~= jEY- ~(0)

z, ,

ke f - l ( j )

where we denoted by AD the set

89

(3.12) Acl = { ( t , x ;A t , -Ax) ; ( t ,x) E C l+d, t 2 = Z 2, /~ e C }

U T~o}C TM U T~I+~C TM •

Let us now int roduce a notat ion. 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 tha t 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) WF( IDI ) 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 hand side of (3.14) being closed, we just have to see tha t it contains W F , ( I D I ) for every integer v.

Close to a point of V, let us choose a local coordinate sys tem (x ~, xd) f lat tening V to Xd = 0. Since uj, j = 0,1, is classical conormal, its associated symbol aJ(x', ~a) (see (1.5)) has an asymptot ic development which may be wri t ten as

(3.15) E j , -k • ak(x' 1)('~a)+J + E a~(x',--1)(--{d)+ j -k .

kEN 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 and ei~d'¢d X(~d)(--~a)-td~a oo

with l > 0 are ramified over C - {0}. If we decompose the restrictions of these functions to N as sums of functions suppor ted in 4-x~ > O, we deduce from (3.15) tha t there is for every integer v a decomposit ion

90 IV. Semilinear Cnuchy problem

(3.17) u j (x ) = ~ ~ ( x ) g ~ ( x ) + gJ(x)

aEA(v)

where:

j=0 ,1

• 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) l l l . 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 wr~ ' l ( [D] ® {D}). The elementary solution e+( t ,x) of [] satisfies

(3.20) e+( t ,x) = cst E]klll~l<t} if d = 2k + 1

e+(t, x) = cst Elk(t 2 -- x2)1/21{1~1<0 if d = 2k.

Since for any distribution U, Wf([ ]kV) C WE(U), WF2'I([3kU) C WFA4(U), it is enough to estimate WF([])] N {D}) and WF~'~([/)] ® {D}) where

(3.21) [b l ( ( z j ) ke I ) = I I l{l*,u)-*sl-<t,U)-'J} if d is odd jGI -{0}

[b]((zj)kei) = H l{l~m)-~Jl<tm)-ti}[(t/(J) - tJ) 2 - (xy(j) - xj)2] 1/2 i f d is even. jel-{0}

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 ,k j ) . 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


(3.23) = 1-I [ ( t j ( s ) - t s )2 - (x s ( j ) -x j ) I I jeJ-{0} 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


= o(xj)}

for j = 0 , . . . , d. Denote by aj,k(z, ~) = 2-~la(Aj,k)(z, ~). It follows from (3.26) tha t

d4xj)( (s)) = }2 k

If ~4 is the mat r ix ..4 = (aj,k(O(s)))j,k and ~(s) = t(~r(X0)(O(s)) , . . . , (r(Xa)(O(s))), • is a solution of the differential equat ion d~qli(s) = A(s)~(s) and so there is a continuous funct ion 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 characterist ic 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 ne ighborhood of a point of V in X2 a system of real analytic coordinates (yo ,g l , . - . , 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 hand 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 extract ing a subsequenee, ]Y~I -< 1 X m plY?[ or [y•[ _< lly~,[ or 7]Y~[ < [Yl [ < 2[Y~I" Let us t rea t the second case. If

x-,d b o X = z_,0 Jb--~vj is a vector field with lipschitz coefficients tangent to V+ c, V c, one has

bl(yo,O, y2 , . . . , yd) -- 0 and bo(O,O, y2 , . . . , yd) -- O. Then, if 0 E C ~ ( N ) is such that O ( s ) = _ l i f ] d < ~ , O ( s ) _ O i f l s l > } , thevec tor f i e ldO( 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). The two other cases are similm'. To prove iii), it is enough, because of

(3.28) $ ( V f , V c)

= 0}, v c)

ii), to show tha t

c g(v ) u g(v c) c g(v+ c) u

Since V~ is characterist ic and since {t = 0} is non-characteris t ic , the principal symbol of [] in the chosen system of coordinates may be wri t ten

(3.29) ayo~7g -~ b7]o7]1 ~- CylT] 2 + 7logo(y; 7] t) -~ I l l /1 (y ; 7] t) -k q(y; rl' )

where a, b, c are real analyt ic in y, b(O, O, Y2, . . . , Yd) • O, lj(y, .), j = 0, 1, (resp. q(y, ")) is a linear form (resp. a quadrat ic form) in r/' = ( r /2 , . . . , rid), real analyt ic in y. Let then


(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 yp than

cst Ib(ym),~ '~ + o(1)1 < lu~ ,F(b(y '~ ) ,~ + e (ym)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 ~(V c , V_ c) C $(V+ c) U g(V_ c) which follows from iii).

E n d of the proof 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 WF, (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, (z j ) 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 , z j(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) - z l( j)(1 , k), ~j(1, k)) E AD

4o(1, k) = E Zj(1, k) jE f - l (O)

4j(1, k ) = - Z j ( 1 , k ) + E ~t(1, k), j E [ - { 0 } .


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 (z j (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 2(v+C,v_C,{t = 0}) = ~(V c) 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).

• 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 v a r i o u s e x t e n s i o n s 95

4. T h e swa l low- ta i l ' s t h e o r e m and var ious e x t e n s i o n s

Theorem 1.3, we finished to prove in the preceding section, gives an upper b o u n d for the C °~ wave-front set of the solution of (1.2) with Cauchy da t a classical conormal along a real analyt ic submanifold V, in terms of any admissible set of sequences g, satisfying axioms A.1 to A.4 and related to V by the condit ion A v C g (with the no ta t ion (1.7)).

For any given geometr ic da ta V, if one wishes to get an "explicit" geometr ic upper bound for WF(u) , one is thus reduced to the const ruct ion of a set g, satisfying the different conditions recalled above, and such that Z (g ) can be es t imated explicitly. This has been done by Lebeau in [L4] when V is a curve close to parabo la in two space dimension (d =- 2). In this last section, we will describe this result, wi thout proof, and ment ion extensions of tha t theorem.

Let us consider problem (1.2) with d = 2 and assume that the Cauchy da ta u0, ul are classical conormal along a real analyt ic curve V of ll{ 2 which has at a unique point a non-degenerate min imum of its curvature radius (for instance, V may be a parabola) . Let A be the union of all null bicharacteristics of [] issued f rom T~N 3 N Char []. Then A is a smoo th lagrangian submanifold of T*R a. If 7r : T*R 3 ~ N 3 is the project ion, 7r(A) is a singular analyt ic hypersurface of R 3 which is the union of two irreducible components 17+ and V_. One of them, for instance V_, is smooth in t > 0. The other one V+ is smooth 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 na tura l stratification:

- the singular point O,

- the curve of cusp points C,

- the curve of t ransverse self-intersection points T,

- the set of smooth points,

(since we imposed to the s t ra ta of a stratif ication to be connected, one should in fact take the components of the previous subsets).

The 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 Cauchy problem, with da ta (classical) conormal along V, one knows that u is (classical) lagrangian along A (see [H] for the definition of tha t last not ion and for a proof of this assertion). In part icular , it follows that u is C ~ on R 3 - 7r(A) = N 3 - (V_ U V+) and even tha t u is (classical) conormal along the smooth points of V_ U 17+.

W h e n one studies the solution to a semilinear problem, because of the phenomenon of interact ion of singularities we recalled in the introduct ion, one expects new singularities produced 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 tha t the solution will be smoo th 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 / '


- $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

(4.1) 5

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 parametrizat ion 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 ~ra--1,-I-oo (see [91]). O11 the other the notations of the introduction) u0 E ~ v , ul E **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 arbi trary 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

/

© Z

c~ 0

P

0

0

e ~

O"Q

t~ b~

II 0

B i b l i o g r a p h y

[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 non- lin~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 stricte- ment 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 Mathemati- schen 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-Meyer- Schwartz, 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.

I n d e x

A, 77 Av, 74 [D], {D}, 84 ID[, 86 E_ ~k, 80

E_%, s0 A{D}, 88 A[D], 88 M~, 86 M;~qo), 79 II-IIM(¢'-'), 79 l ~ & ~ , 81 +~ 89 V~', 82 Z(g), 74 Z~, 83

Admissible set of sequences, 73 Analytic wave front set (SS(.)),

Characterization of WF~(-), 27 Classical conormal distribution, 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 WFa. (-),

12

73

12

Good contour, 20, 38

HS-wave front set, 8

Inversion formula, 12

Lojaciewiecz inequalities, 48

Phase of FBI transform, 16 - - of second kind, 37

Phase of quantized canonical transformation, 19

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

- - o f a m a p , 49 - - o f a s e t , 48

Subanalytic - - map, 48 - - set, 47

Symbol (formal ~,d, classical cod), 18

Totally real submanifold, 33 Trace theorem, 42 Tree, 84

Upper bounds for microsupports, 59

Whitney's normal cone, 51

Printing: Druckhaus Beltz, Hemsbach Binding: Buchbinderei Schfiffer, Grfinstadt

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F.B.I. Transformation: 2nd Microlocalization and Semilinear Caustics

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