# Optimality of the quantified Ingham-Karamata theorem for operator semigroups with general resolvent growth

## TL;DR

This paper demonstrates the optimality of a generalized quantified Ingham-Karamata theorem for $C_0$-semigroups, showing that the bounds are sharp under mild resolvent growth conditions, extending previous results.

## Contribution

It establishes the sharpness of the generalized Ingham-Karamata theorem for operator semigroups, including the optimality of the Batty-Duyckaerts theorem for bounded semigroups with subpolynomial resolvent growth.

## Key findings

- The generalized Ingham-Karamata theorem is sharp under mild conditions.
- The Batty-Duyckaerts theorem is optimal for certain semigroups.
- The proof employs the open mapping theorem and a new technical lemma.

## Abstract

We prove that a general version of the quantified Ingham-Karamata theorem for $C_0$-semigroups is sharp under mild conditions on the resolvent growth, thus generalising the results contained in a recent paper by the same authors. It follows in particular that the well-known Batty-Duyckaerts theorem is optimal even for bounded $C_0$-semigroups whose generator has subpolynomial resolvent growth. Our proof is based on an elegant application of the open mapping theorem, which we complement by a crucial technical lemma allowing us to strengthen our earlier results.