A note on perturbations of $C_0$-semigroups

TL;DR

This paper investigates a variation of constants inequality for $C_0$-semigroups on Banach spaces, linking it to generator inequalities, with applications to heat kernel estimates and bi-continuous semigroups.

Contribution

It introduces a new inequality characterization for semigroups and their generators, unifying estimates for heat kernels and extending to bi-continuous semigroups.

Findings

Derived a variation of constants inequality for semigroups

Established a link between semigroup inequalities and generator inequalities

Applied results to heat kernel estimates and bi-continuous semigroups

Abstract

This article deals with a variation of constants type inequality for semigroups acting consistently on a scale of Banach spaces. This inequality can be characterized by a corresponding (easy to verify) inequality for their generators. The results have applications to heat kernel estimates and provide a unified perspective to estimates of these type. Moreover, bi-continuous semigroups can be treated as well.