An abstract approach to optimal decay of functions and operator semigroups

TL;DR

This paper presents a new, shorter proof of optimal decay theorems for functions and semigroups, using the open mapping theorem, and extends these results to more general settings.

Contribution

It introduces a novel, more general proof technique for decay theorems, improving upon existing literature with a shorter and more versatile approach.

Findings

Shorter proof of optimal decay theorems

Extension to vector-valued functions and $C_0$-semigroups

More general applicability of the results

Abstract

We provide a new and significantly shorter optimality proof of recent quantified Tauberian theorems, both in the setting of vector-valued functions and of $C_0$-semigroups, and in fact our results are also more general than those currently available in the literature. Our approach relies on a novel application of the open mapping theorem.