Hyperfunctions

Summary

We defined D’,(X) as the space of continuous linear forms on C₀∞(X). This is by no means the most general concept of its kind, for a larger space of distributions is obtained if C₀∞(X) is replaced by a dense subspace with a stronger topology. An example is the space of elements of compact support in C^L (defined in Section 8.4) provided that it does not contain just the 0 function, that is, $$\sum 1/L_k infty .$$.

The study of the dual space of distributions is then fairly similar to that of D’,(X).

The situation is rather different in the quasi-analytic case where $$\sum 1/L_k = \infty .$$.

No analogue of C₀∞(X) is then available but we may regard C^L(X) as a substitute for C∞(X). The dual space of C^L(X) can be taken as the elements of compact support in a distribution theory preserving many of the features of D’,(X) but differing in some respects. The largest space of distributions is obtained when C^L is the real analytic class. It was introduced in a different way by Sato who coined the term hyperfunction for its elements. In this chapter we shall give an introduction to the theory of hyperfunctions in a manner which follows Schwartz distribution theory as closely as possible.

Section 9.1 is devoted to the study of hyperfunctions of compact support. In particular we give an elementary proof of the crucial and non-trivial fact that there is a good notion of support. The general definition of hyperfunctions can then be given in Section 9.2 along lines first proposed by Martineau. Section 9.3 is devoted to the wave front set with respect to analytic functions of a hyperfunction and the definition of operations such as multiplication. This is done rather quickly for most proofs in Sections 8.4 and 8.5 were chosen so that they are applicable to hyperfunctions after a few basic facts have been established.

Section 9.4 is devoted to the existence of analytic solutions of analytic differential equations. In addition to the classical Cauchy-Kovalevsky theorem precise information on bounds and existence domains is given. These are applied in Section 9.5 to prove some basic facts on hyperfunction solutions of analytic differential equations. In Section 9.6 finally we present the Bros-Iagolnitzer definition of WF_A(u) and prove as an application a theorem of Kashiwara on the relation between supp u and WF_A(u) similar to Holmgren’s uniqueness theorem.