On the Cauchy problem for convolution equations

Volume 133 / 2013

Jan Kisyński Colloquium Mathematicum 133 (2013), 115-132 MSC: Primary 35E15, 47D06, 46F99, 42B99. DOI: 10.4064/cm133-1-8

Abstract

We consider one-parameter $(C_{0})$-semigroups of operators in the space $\mathcal S'({\mathbb R}^n;{\mathbb C}^m)$ with infinitesimal generator of the form $(G\,*)|_{\mathcal S'({\mathbb R}^n;{\mathbb C}^m)}$ where $G$ is an $M_{m\times m}$-valued rapidly decreasing distribution on ${\mathbb R}^n$. It is proved that the Petrovskiĭ condition for forward evolution ensures not only the existence and uniqueness of the above semigroup but also its nice behaviour after restriction to whichever of the function spaces $\mathcal S({\mathbb R}^n;{\mathbb C}^m)$, $\mathcal D_{L^{p}}({\mathbb R}^n;{\mathbb C}^m)$, $p\in [1,\infty ]$, $(\mathcal O_{a})({\mathbb R}^n;{\mathbb C}^m)$, $a\in \mathopen ]0,\infty \mathclose [$, or the spaces $\mathcal D'_{L^{q}}({\mathbb R}^n;{\mathbb C}^m)$, $q\in \mathopen ]1,\infty ]$, of bounded distributions.

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