Association between temperate distributions and analytical functions in the context of wave-front sets

Abstract

Let B be a translation invariant Banach function space (BF-space). In this paper we prove that every temperate distribution f can be associated with a function F analytic in the convex tube Omega={z in C^d; |Im z|<1} such that the wave-front set of f of Fourier BF-space types in intersection with R^d \times S^{d-1} consists of the points (x,\xi) such that F does not belong to the Fourier BF-space at x-i\xi.