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arXiv · 1705.03641

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

Reinhard Stahn
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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.

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Reinhard Stahn. 2017-05-10. A quantified Tauberian theorem and local decay of $C_0$-semigroups. https://arxiv.org/abs/1705.03641

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