A quantified Tauberian theorem and local decay of C₀-semigroups

2017/05/10 by Stahn, Reinhard
#35B40 (Secondary) #40E05 (Primary) #47D06 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1705.03641

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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