Let $\Omega $ be an open set in $R_n $ and let $\mathcal{E}(\Omega )$ denote the space of infinitely differentiable functions on $\Omega $. Necessary and sufficient conditions are exhibited for a family $\{ \Omega _i \} _{i = 1}^N $ of open sets in $R_n$ and a family $\{ S_i \} _{i = 1}^N \subset \mathcal{E}'(R_n )$ in order that the convolution equation \[ \sum _{i = 1}^N {S_i * u_i = f} \] have a solution $(u_1 ,u_2 , \cdots ,u_N )$ in $ \oplus _{i = 1}^N \mathcal{E}(\Omega _i )$ for every f in $\mathcal{E}(\Omega )$.A simple example and a geometrical interpretation of the condition on the family $\{ \Omega _i \} _{i = 1}^N $ is provided.