One-parameter convolution semigroups of rapidly decreasing distributions

Abstract

Let $\Mmm$ denote the set of $\mm$ matrices with complex entries, and let $\calG(\partial_1,...,\partial_n)$ be an $\mm$ matrix whose entries are partial differential operators on $\Rn$ with constant complex coefficients. It is proved that $\calG(\partial_1,...,\partial_n)\otimes \delta$ is the generating distribution of a smooth one-parameter convolution semigroup of $\Mmm$-valued rapidly decreasing distributions on $\Rn$ if and only if $$\sup_{(\xi_1,...,\xi_n)\in\Rn}\hRe\sigma(\calG(i\xi_1,...,i\xi_n))<\infty.$$ Applications to systems of partial differential operators with constant coefficients are considered.