On the Solvability of Convolution Equations in Beurling's Distributions

Abstract

Let \mathscr{D}_w' be the space of Beurling's generalized distributions on Rⁿ and \mathscr{E}_w' the spaces of generalized distributions which has compact support. We show that, for S ∈ \mathscr{E}_w', S * \mathscr{D}_w'=\mathscr{D}_w' is equivalent to the following: Every generalized distribution u ∈ \mathscr{E}_w' with S * u ∈ \mathscr{D}_w is in \mathscr{D}_w.