Sharp extensions for convoluted solutions of abstract Cauchy problems

TL;DR

This paper develops sharp extension results for convoluted solutions of abstract Cauchy problems in Banach spaces, utilizing algebraic structures and introducing $k$-distribution semigroups to advance the theory.

Contribution

It introduces new algebraic methods and the concept of $k$-distribution semigroups to extend solutions of abstract Cauchy problems in Banach spaces.

Findings

Established sharp extension results for convoluted solutions

Defined algebra homomorphisms from new test-function classes

Extended the concept of distribution semigroups with $k$-distribution semigroups

Abstract

In this paper we give sharp extension results for convoluted solutions of abstract Cauchy problems in Banach spaces. The main technique is the use of algebraic structure (for usual convolution product $\ast$) of these solutions which are defined by a version of the Duhamel formula. We define algebra homomorphisms from a new class of test-functions and apply our results to concrete operators. Finally, we introduce the notion of $k$-distribution semigroups to extend previous concepts of distribution semigroups.