Analytic Distribution Kernels

Abstract

Introduction. In his study of the structure of distribution kernels (centering around his celebrated kernel theorem), L. Schwartz [11; 13] has studied the subclasses of regular and very regular distribution kernels, which may be characterized roughly by the fact that they carry infinitely differentiable functions into infinitely differentiable functions. It is our object in the present paper to extend this study to kernels which are analytically regular, i.e., which, again roughly, carry analytic functions into analytic functions. The motivation for such a study is provided by the fundamental solutions of elliptic equations with analytic coefficients. One consequence of our results is that for such equations the standard theorems on the analyticity of regular solutions imply the analyticity of the fundamental solution. The principal tool in our investigation is the general theory of topological vector spaces (particularly in certain forms given to it by A. Grothendieck) and especially the theory of topological tensor products. In ?1.1, we summarize the definitions from the theory of topological vector spaces which we shall use. In ?1.2, we give a brief summary of the definitions and principal results of Grothendieck'stheory of topological tensor products. In ?I.3, we summarize the results of Schwartz's study of distribution kernels, at least insofar as they relate to the generalizations to be given to analytic kernels. ?II is devoted to an intensive consideration of the space Gt(K) of real analytic functions on a compact subset of Rn. ?III states and proves our principal results on analytic distribution kernels. The writer should like to thank Professors L. Nachbin and F. E. Browder for discussion, suggestions, and criticism. The results of the present paper constitute a portion of the writer's Doctoral Dissertation [1] at the University of Sao Paulo, prepared while in residence at the Institute for Pure and Applied Mathematics, Rio de Janeiro.