Lecture Notes in Mathematics Editors: A. Dold, Heidelberg F. Takens, Groningen B. Teissier, Paris 1737 Springer Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore Tokyo Seiichiro Wakabayashi Classical Microlocal Analysis in the Space of Hyperfunctions Springer Author Seiichiro Wakabayashi Institute of Mathematics University of Tsukuba Tsukuba-shi, Ibaraki 305-8571, Japan E-mail: wkbysh @ math.tsukuba.ac.jp Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einlaeitsaufnahme Wakabayashi, Seiichiro: Classical microloca analysis in the space of hyperfunctions / Seiichiro Wakabayashi. - Berlin ; Heidelberg ; New York ; Barcelona ; Hong Kong ; London ; Milan ; Pads ; Singapore ; Tokyo : Springer, 2000 (Lecture notes in mathematics ; 1737) ISBN 3-540-67603- l Mathematics Subject Classification (2000): 35-02, 35S05, 35S30, 35A27, 35A20, 35A07, 35HI0, 35A21 ISSN 0075- 8434 ISBN 3-540-67603-1 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. 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Typesetting: Camera-ready TEX output by the author Printed on acid-free paper SPIN: 10724347 41/3143/du 543210 Preface Many author have studied the theory of hyperfunctions from the view- point of "Algebraic Analysis," which is not necessarily accessible to us, studying partial differential equations (P.D.E.) in the framework of distri- butions. The treatment there is considably different from ours. Although we think that it is natural to work in the space of hyperfunctions for the purpose of studying P.D.E. with analytic coefficients, we do not think that "Algebraic Analysis" is indispensable for this purpose. We want to apply various methods in the framework of distributions to the studies on P.D.E. with analytic coefficients. In so doing the major difficulty is not to be able to use the "cut-off" technique. For there is obviously no non-trivial real analytic function with compact support. We shall use here "cut-off" operators ( pseudodifferential operators) instead of "cut- off" functions, which map real analytic functions and hyperfunctions to real analytic functions and hyperfunctions, respectively. In this lecture notes we attempt to establish "Classical Microlocal Analysis" in the space of hyperfunctions ( or in a rather wider class of functions) which makes it possible to apply the methods in the C °O_ distribution category to the studies on P.D.E. in the hyperfunction cat- egory. Here "Classical Microlocal Analysis" means that it does not use "Algebraic Analysis" and that it is very similar to microlocal analysis in the Coo-distribution category. Our main tool is, in some sense, inte- gration by parts, which is equivalent to the fundamental theorem of the infinitesimal calculus. In our direction there are two books. One of them is HSrmander's book [Hr5] which gives a short introduction to the theory of hyperfunctions. The other is Treves' book [Tr2]. Treves developed in [Tr2] the theory of analytic pseudodifferential operators in the framework of distributions, which had been studied by Boutet de Monvel and Kree [BK]. On the basis of the methods in these two books, we shall establish "Classical Microlocal Analysis" in the space of hyperfunctions. Some parts of this lecture notes are simple generalizations of the re- vi suits obtained in joint work with Prof. Kajitani, and I would like to thank him for many useful discussions. Contents 1 Hyperfunctions 3 4 5 1.1 Function spaces 5 1.2 Supports 13 1.3 Localization 23 1.4 Hyperfunctions 28 1.5 Further applications of the Runge approximation theorem • 34 Basic calculus of Fourier integral operators and pseudo- differential operators 41 2.1 Preliminary lemmas 41 2.2 Symbol classes 52 2.3 Definition of Fourier integral operators 57 2.4 Product formula of Fourier integral operators I 65 2.5 Product formula of Fourier integral operators II 87 2.6 Pseudolocal properties 93 2.7 Pseudodifferential operators in B 107 2.8 Parametrices of elliptic operators 112 Analytic wave front sets and microfunctions 3.1 3.2 3.3 3.4 3.5 3.6 3.7 115 Analytic wave front sets 115 Action of Fourier integral operators on wave front sets • • • 130 The boundary values of analytic functions 155 Operations on hyperfunctions 165 Hyperfunctions supported by a half-space 183 Microfunctions 192 Formal analytic symbols 201 Microlocal uniqueness 205 4.1 Preliminary lemmas 205 4.2 General results 222 viii CONTENTS A B 4.3 Microhyperbolic operators 231 4.4 Canonical transformation 239 4.5 Hypoellipticity 244 Local solvability 259 5.1 Preliminaries 259 5.2 Necessary conditions on locM solvability and hypoellipticity 268 5.3 Sufficient conditions on local solvability 272 5.4 Some examples 285 Proofs of product formulae A.1 Proof of Theorem 2.4.4 A.2 Proof of Corollary 2.4.5 A.3 Proof of Theorem 2.4.6 A.4 Proof of Corollary 2.4.7 A.5 Proof of Theorem 2.5.3 295 295 323 328 336 338 A priori estimates 351 B.1 Gru~in operators 351 B.2 A class of operators with double characteristics 355 Introduction Let .4 be the space of entire analytic functions on C ~. An analytic functional is a continuous linear functional on ,4 with usual topology. We say that an analytic functional u is carried by a compact subset K of C ~, i.e., u E .4t(K), if for any neighborhood w of K in C n there is C~ :> 0 such that lu(@l < C~osupb,(z)l for ~ e A. zEw We denote .4' := U .4'(K). K(::I:R '~ The space A' of analytic functionals carried by R n is very similar to the space :D t of distributions, particularly to ~F. One can identify ~-t with a subspace of A r, and anlytic functionals have compact "supports." For u E A r we can define supp u by supp u = N{K; u E At(K)}, which is called the support of u. The concept of "support" is relating to restriction mappings and sheaves 4 t defines the sheaf B of hyperfunctions while C r does the sheaf l) r of distributions. In order to study partial differential equations (P.D.E.) in the space of hyperfunctions it is usually sufficient to consider problems in A t ( or T0 defined below). For u E A t we can define the Fourier transform fi(~) of u by - 7[u](,,) := Therefore, we can formally define pseudodifferential operators p(x, D) with appropriate symbols as n)u := (2~r) -~ f ei*4p(x,()~(~) d~. (0.1) p(x, However, p(x, D)u does not always belong to ¢4 r if p(x,~) is not a polyno- mial of ~. In microlocal analysis pseudodifferential operators play essen- tial roles. So we need the corresponding spaces to the Schwartz spaces S 2 INTROD UCTION ands ~. ForeERweput We introduce the topology to S~ in a standard way. Then the dual space ge I of S~ is given by g,'= {~(~) c v'; e-~<~)~(~) e s'}. If ¢ _> 0, then S+ is a dense subset of S and we can define S, := .T-I[$+] ( C S). By duality we can define the transposed operators tic, tic-1 . S~' ~ S~' for ¢ _> 0. We rewrite t j- = ~- since tU = 5 on S'. Noting that ~' S_efor _>0, we e D ¢ define S_~ := : 1[~_~] for e > 0. Moreover, SJ E is defined as the dual space of S_~ when ¢ _> 0. We define ~'o := N~>o S_~ and 9% := N~>o S/. From estimates of the Fourier transforms of analytic functoinals we see that £' c A' c £o c Fo. Let V be an open conic subset of R n x (R n \ {0}). We say that p(x,~) E C°°(F) is an analytic symbol in F if p(x,~) satisfies the estimates O~ D~p(x,~) I <_ CAJ"'+l~llo, l!lfll!<~> '~-I~i for (~,~) ~ r with I~1 > R and ~,/~ e Z~. Ifp(~,~) is an analytic symbol in R n x (R~ \{0}), p(x, D) defined by (0.1) maps C0 and ~'0 to E0 and .To, respectively. However, we can not define p(x, D) as an operator on £0 and 9% by (0.1) ifp(z,~) is an analytic symbol in F and F does not coincide with R n x (R~\{0}). We do not want to abandon (0.1) as the definition of p(x, D). So we introduce some symbol classes which contain symbols with compact supports. We say that a symbol a(~, y, 71) E C°~(R ~ x R n x R n) belongs to S m~'m2'~'~2 (R, A) if a(~, y, T/) satisfies "-"u '-'n ~',, - Clal+l~l+l.~l x (A/R)I,~1+ I~' I+W I+M (,~)~, -I,~ I+1~' I (r/>,,,2-1"~1+1~2 I x exp[(~l(~> -4- (~2(~>] if <¢> _> R(lal + I/~11) and (r/> _> R(lfl21 + b'l). For a(,~,y, rl) ~ S ~,'~2'~''~2 (R, A) we define the pseudodifferential operator a(D~, y, D~) by [...]... distributions Therefore, we can study P.D.E in the hyperfunction category in the almost same way as in the distribution category Our aim here is to provide microlocal analysis in the space of hyperfunctions in the same way as for distributions As applications we shall consider microlocal uniqueness and local solvability in the last two chapters These are still basic problems in the theory of linear partial... we think that his theory is different from usual microlocal analysis in the distribution category, although it is new and powerful So we will establish microlocal analysis in the space of hyperfunctions which is very similar to microlocal analysis in the framework of C °O and distributions Chapter 1 Hyperfunctions In this chapter we shall introduce the function spaces S~ and S~' corresponding to the. .. Schwartz spaces 8 and 8 ~, respectively These spaces play a key role in our calculus The spaces S_~ and S~~ ( ~ > 0) include the space A' of analytic functionals We shall define the supports and the restrictions of functions belonging to these spaces Hyperfunctions ( in a bounded open subset of R n) will be defined as residue classes of analytic functionals after the m a n n e r of HSrmander's book [Hr5] in. .. one "cut-off" symbol and consider a family of "cut-off" symbols depending on R This is a disadvantage in comparison with usual calculus in the ditribution category However, we can overcome this disadvantage in most cases Using "cut-off" operators we can define pseudodifferential operators and Fourier integral operators acting on the spaces ( or the sheaves) of hyperfunctions and microfunctions Since we... 1,2), sinceu-uj=vj (j=l,2) D 1.4 Hyperfunctions Following [Hr51, we shall define the space of hyperfunctions and s t u d y some properties of hyperfunctions D e f i n i t i o n 1.4 1 Let v _> 0 and X be a bounded open subset of R n (i) We define t3 (x) := a'.(X)lA'e(ox) We also write B(X) = Bo(X), which is called the space of hyperfunctions in X (ii) For an open subset Y of X and u ° E B~(X) the restriction... well-known in the framework of C °O and distributions that Carleman type estimates play an essential role in microlocal versions of the Holmgren uniqueness theorem This is also true in the framework of analytic INTROD UCTION functions and hyperfunctions General criteria on microlocal uniqueness will be given in Chapter 4 Microlocal uniqueness yields results not merely on propagation of analytic singularities... neighborhood F of ~0, R0 > 0 and {gn(~)}R>R0 C C°°(R '~) such t h a t gR(~) = 1 in F M {(~) >_ R}, Iog+ )l < R( if (~> >_ RI(~I, and gR(D)uis analytic at x° for R _> R0 The precise definition that gR(D)uis analytic at x° will be given in Definition 1.2.8 Our definition of WFA(u) ,of course, coincides with usual definitions Our definition of WFA(U)is very similar to the definition of the wave front set of distributions... homeomorphic S_~ denotes the dual space of S_¢ for E _> 0 Then we have S_~ = Sr[~_~] C S' C S~ for c _> 0 and j r = t jr on S ( So we write t j r as jr Note t h a t & is a Fr~chet space with the topology determined by the seminorms ]ul,s,,t := ]jr[u]l~,,t ( g E Z + ) We denote by A the space of entire analytic functions in CTM Let K be a compact subset of C ~, and denote by A ' ( K ) the space of analytic functionals... shall prove t h a t the presheaf B of hyperfunctions is a flabby sheaf We shall also prove flabbiness of the quotient sheaf B/`4 of B by the sheaf 4 of real analytic functions in Section 1.5 1.1 Function spaces Let ~ E R, and denote (~) = (1 + ]~[ = ()-~jn I~jl2) 1/2 We define 1 1 12)1/ , where (~1,"" ,~n) E R n and Here S denotes the Schwartz space We introduce a family of seminorms on S~ as follows:... u°iY E B~(Y) of u ° to Y is defined by the residue class [v] of v E ~4~(Y) which satisfies supp ( u - v ) C X \ Y, where the residue class of u E A~(X) is u ° in Be(X) ( see Theorem 1 3 3 ) (iii) For u ° e Be(X) we define supp u ° := supp u M X , where the residue class of u E ~4~(X) is u ° in Be(X) (iv) For u ° E 13(X) we define sing supp u ° : sing supp u M X , where the residue class of u E A ' . fundamental theorem of the infinitesimal calculus. In our direction there are two books. One of them is HSrmander's book [Hr5] which gives a short introduction to the theory of hyperfunctions. The. Monvel and Kree [BK]. On the basis of the methods in these two books, we shall establish " ;Classical Microlocal Analysis& quot; in the space of hyperfunctions. Some parts of this lecture notes. establish microlocal analysis in the space of hyperfunctions which is very similar to microlocal analysis in the framework of C °O and distributions. Chapter 1 Hyperfunctions In this chapter