On the Solvability of Convolution Equations in Beurling's Distributions

Abstract

Let ${\mathcal{D}}_w^{\prime }$ be the space of Beurling's generalized distributions on ${\mathbf{R}}^n$ and ${\mathcal{E}}_w^{\prime }$ the spaces of generalized distributions which has compact support. We show that, for $S\in {\mathcal{E}}_w^{\prime }$, $S\ast {\mathcal{D}}_w^{\prime }={\mathcal{D}}_w^{\prime }$ is equivalent to the following: Every generalized distribution $u\in {\mathcal{E}}_w^{\prime }$ with $S\ast u\in {\mathcal{D}}_w$ is in ${\mathcal{D}}_w$.