On convolvability conditions for distributions

Abstract

We investigate several conditions of the convolvability and \({\mathcal{S}'}\) -convolvability of distributions and we show their equivalence by characterizing the partial summability of distribution kernels by multiplicative properties. More generally, partial summability to the power p and the partial vanishing at infinity of kernels are characterized by multiplicative properties. As an application we state several sufficient equivalent conditions ensuring the validity of the equation, $$ (\partial_jS)\ast T=S\ast (\partial_j T).$$ Furthermore, it is shown that the Chevalley condition for the convolvability of two distributions \({S,T\in\mathcal{D}'}\) , i.e., $$ (\varphi\ast S)(\psi\ast\check T)\in L^1\quad\text{for all }\varphi,\psi\in\mathcal D,$$ is equivalent with $$S(x-y)T(y-z)\in\mathcal D'_{xz}\hat\otimes L^1_y.$$