On the solvability of linear partial differential equations in spaces of hyperfunctions

Abstract

It is well known from the theory of linear partial differential equations in spaces of smooth functions and distributions, see HSrmander [11], [12], that the solvability of a differential equation is related to the non-existence of a solution of the homogeneous adjoint equation with compact singular support, and that this may be used to obtain semi-global existence results from the microlocal study of the adjoint o1> erator. In this paper we show that a similar strateg5 ~ is possible in the framework of hyperfunctions. Actually, we shall consider in this paper the more general case of a system of differential equations without compatibility conditions in the framework of hyperfunctions on a maximally real manifold in C ~ with low regularity.The first section of the paper may be considered as a continuation of Sehapira [26], [27], in which it was shown how functional analysis can be used in the hyperfunction theory of differential operators. We first recall the fact that hyperfunetion solvability is insensitive to the geometry of the boundary of the domain (Theorem 1.2) and show that finite dimensional obstruction to solvability never occurs (Theorem 1.3). Then we characterize the hyperfunction solvability of a differential operator in terms of the validity of an a priori inequality for the adjoint operator (Theorem 1.4). The main result of this section is perhaps Theorem 1.6 which states that the non-confinement of analytic singularities for the adjoint operator is a sufficient condition for the hyperfunetion solvability. This is similar to Theorem 1.2.4 of H6rmander [11].In Section 2 we give several examples of how the functional analysis statements of Section 1 apply to obtain seemingly new existence theorems or new proofs of classical existence theorems, as corollaries of already available, sometimes deep, microlocal results. Such topics as holonomic systems, hypo-analytic structures or analytic differential equations of principal type on R" are touched on. Theorem 2.2(1) Partially supported by CNPq, Brasil.

Global s-solvability and global s-hypoellipticity for certain perturbations of zero order of systems of constant real vector fields

“…Next we show that (5.1) implies that the transpose of the system R j is globally solvable in the Sobolev spaces. More precisely, similarly as in [19], we associate to the operators R 1 , . .…”

Hyperfunctions and (analytic) hypoellipticity

“…On the other hand t P : B( ) → B( ) is surjective [4] and consequently there is u ∈ B( ) such that t Pu .…”

Regularity and solvability of linear differential operators in Gevrey spaces

“…This reveals the nice properties that these spaces enjoy, leading us to pursue these matters as a purely functional analytic affair (Section ): here, everything is greatly simplified (in contrast, for instance, with the analysis developed in and ) by the intense use of results due to Komatsu and a very interesting result that we learned from P. D. Cordaro (Lemma ), together with some remarkable consequences of the Homomorphism Theorem for Fréchet–Montel spaces (Lemma ). This approach seems to be novel, although inspired by the work of Cordaro and Trépreau , which deals with solvability in hyperfunctions.…”