A quantified Tauberian theorem and local decay of $C_0$-semigroups

TL;DR

This paper establishes a new quantified Tauberian theorem with generalized conditions on Laplace transforms, providing optimal decay rates for functions and improving understanding of wave energy decay in odd-dimensional exterior domains.

Contribution

It introduces a novel Tauberian condition involving functions M and K, generalizing previous results and demonstrating optimal decay rates for a broad class of functions.

Findings

Proved a generalized quantified Tauberian theorem.

Established optimal decay rates for functions under new conditions.

Improved decay estimates for local energy of waves in odd-dimensional exterior domains.

Abstract

We prove a quantified Tauberian theorem for functions under a new kind of Tauberian condition. In this condition we assume in particular that the Laplace transform of the considered function extends to a domain to the left of the imaginary axis, given in terms of an increasing function $M$ and is bounded at infinity within this domain in terms of a different increasing function $K$. Our result generalizes a result of Batty, Borichev and Tomilov (2016). We also prove that the obtained decay rates are optimal for a very large class of functions $M$ and $K$. Finally we explain in detail how our main result improves known decay rates for the local energy of waves in odd-dimensional exterior domains.