Analytic structure of distributions and essential support theory

The notion of essential support of a distribution f, defined on Rsup(n) or on a real analytic manifold M, is presented. The essential support of f at a given real point X (of Rsup(n) or of M) is the set of directions along which a certain generalized Fourier transform of f at X does not decrease exponentially. This set is a set of 'singular directions' of f at X in the following sense. The distribution f is shown to be analytic at X, respectively to be at X the boundary value of an analytic function f from the directions of a cone GAMMA, if and only if its essential support at X is empty, respectively is contained in the closed convex salient dual cone C of GAMMA. More generally, the notion of essential support allow one to characterize possible decomposition of f into sums of boundary values of analytic functions, from directions which may depend on the real point X considered (decomposition theorems). Local edge-of-the-wedge and generalized edge-of-the-wedge theorems are obtained as corollaries of these results. Finally a number of useful results on products of distributions, partial integrals and restriction to submanifolds are presented in the general framework of essential support theory. (Auth.)