## Classical microlocal analysis in the space of hyperfunctions.(English)Zbl 0949.35003

Lecture Notes in Mathematics. 1737. Berlin: Springer. viii, 367 p. (2000).
The author gives an important contribution to the study of partial differential equations (P.D.E.) with analytic coefficients in the space of hyperfunctions by methods of “Classical Microlocal Analysis”, i.e. integration by parts and cut-off functions (versus the methods of “Algebraic Analysis”). He continues directions started by Hörmander, Treves, Boutet de Monvel and Kree (among others). The main difficulty is, of course, the lack of “cut-off” techniques in the framework of analytic functions.
The starting point is the formula $$p(x,D)u:=(2\pi)^{-n} \int e^{i\langle x,\xi\rangle}p(x,\xi)\widehat u(\xi) d\xi,$$ where $$u\in{\mathcal A}'$$ is an analytic functional, and $$\widehat u(\xi):=u(e^{-i \langle z,\xi\rangle})$$ denotes its Fourier transform. Since, fortunately or unfortunately, $$p(x,D)u$$ is not in general an element of $${\mathcal A}'$$, when $$p(x,\xi)$$ is not a polynomial, the author defines the spaces, corresponding to the Schwartz spaces $${\mathcal S}$$ and $${\mathcal S}'$$, that play a key role in the calculus: with $$\varepsilon\geq 0,$$ and upon denoting by $$F$$ the Fourier transform, he defines \begin{aligned} {\mathcal S}_{-\varepsilon}&:= ^t F^{-1}\bigl( \{v\in C^\infty({\mathbb R}^n);\;e^{-\varepsilon\sqrt{1+|\xi|^2}}v(\xi)\in{\mathcal S}\}\bigr), \\ {\mathcal S}_{\varepsilon}&:=F^{-1}\bigl( \{v\in C^\infty({\mathbb R}^n);\;e^{\varepsilon\sqrt{1+|\xi|^2}}v(\xi)\in{\mathcal S}\}\bigr)\end{aligned} and $${\mathcal S}'_\varepsilon$$ as the dual space of $${\mathcal S}_\varepsilon.$$ It turns out that $${\mathcal S}_\varepsilon$$ is dense in $${\mathcal S}$$ for $$\varepsilon\geq 0,$$ and, upon writing $${\mathcal E}_0:= \bigcap_{\varepsilon>0}{\mathcal S}_{-\varepsilon}$$ and $${\mathcal F}_0:= \bigcap_{\varepsilon>0}{\mathcal S}'_{\varepsilon},$$ that $${\mathcal E}'\subset{\mathcal A}'\subset{\mathcal E}_0\subset{\mathcal F}_0.$$ Now, for $$\varepsilon\geq 0$$ and $$K\subset{\mathbb R}^n$$ a closed set, he defines $${\mathcal A}'_\varepsilon(K)=\{u\in {\mathcal A}'({\mathbb C}^n)\cap \bigcap_{\delta>\varepsilon}{\mathcal S}'_ \delta$$; $$\text{supp} u\subset K\},$$ and for $$X\subset{\mathbb R}^n$$ a bounded open set $${\mathcal B}_\varepsilon(X):={\mathcal A}'_\varepsilon\left(\overline{X}\right)/ {\mathcal A}'_\varepsilon(\partial X).$$ The space $${\mathcal B}(X)$$ of hyperfunctions on $$X$$ is then the space $${\mathcal B}_0(X).$$ When $$X$$ is any open set of $${\mathbb R}^n$$ he gives a sheaf-theoretical definition of $${\mathcal B}_\varepsilon(X)$$ by using an open covering of $$X$$ ($$\varepsilon\geq 0$$). He then introduces classes of symbols that include elements with compact supports, whose corresponding pseudodifferential operators (defined as above) are well-behaved when acting on the spaces $${\mathcal S}_\delta$$ and $${\mathcal S}_\varepsilon'$$. In this context, he gets “cut-off” operators, and is able to define pseudodifferential operators and Fourier integral operators on the spaces (or the sheaves) of hyperfunctions and microfunctions. He obtains a symbol calculus with the same properties as the usual one. He then introduces, in this context, the analytic wave-front-set $$WF_A(u),$$ and, in the last chapters, produces general criteria for microlocal uniqueness, that yield results on the propagation of analytic singularities, analytic hypoellipticity and on local solvability in the hyperfunctions setting by means of suitable energy (a-priori) estimates.
(See also [J. Sjöstrand, Asterisque 95 (1982; Zbl 0524.35007)] and [O. Liess, Rodino, Luigi (ed.), Microlocal analysis and spectral theory. Proceedings of the NATO Advanced Study Institute, Il Ciocco, Castelvecchio Pascoli (Lucca), Italy, 23 September–3 October 1996. Dordrecht: Kluwer Academic Publishers. NATO ASI Ser., Ser. C, Math. Phys. Sci. 490, 61-90 (1997; Zbl 0880.35146)] for alternative approaches.)
This book will be very useful to everyone who wishes to work in the area of analytic coefficient partial differential equations by using methods from the usual distribution theory.

### MSC:

 35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations 35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs 46F15 Hyperfunctions, analytic functionals 35S05 Pseudodifferential operators as generalizations of partial differential operators 35S30 Fourier integral operators applied to PDEs 35A20 Analyticity in context of PDEs 35A07 Local existence and uniqueness theorems (PDE) (MSC2000) 35H10 Hypoelliptic equations

### Citations:

Zbl 0524.35007; Zbl 0880.35146
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