One-parameter convolution semigroups of rapidly decreasing distributions

TL;DR

This paper characterizes when certain matrix-valued distributions generate smooth one-parameter convolution semigroups based on spectral bounds, with applications to constant coefficient PDE systems.

Contribution

It provides a necessary and sufficient condition for matrix-valued distributions to generate convolution semigroups, extending the theory to systems of PDEs with constant coefficients.

Findings

Characterization of generating distributions via spectral bounds.

Condition involving the supremum of the real part of the spectrum.

Applications to systems of PDEs with constant coefficients.

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.