An abstract approach to optimal decay of functions and operator semigroups

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.

arXiv commentTo appear in the Israel Journal of Mathematics

References in corpus (7)

• Optimal rates of decay for operator semigroups on Hilbert spaces
• Quantified versions of Ingham's theorem
• On the non-triviality of certain spaces of analytic functions. Hyperfunctions and ultrahyperfunctions of fast growth
• Local decay of $C_0$-semigroups with a possible singularity of logarithmic type at zero
• Complex Tauberian theorems for Laplace transforms with local pseudofunction boundary behavior
• Optimal Tauberian constant in Ingham's theorem for Laplace transforms
• Note on the absence of remainders in the Wiener-Ikehara theorem

Cited by in corpus (6)

• Semi-uniform stability of operator semigroups and energy decay of damped waves
• Optimality of the quantified Ingham-Karamata theorem for operator semigroups with general resolvent growth
• On the absence of remainders in the Wiener-Ikehara and Ingham-Karamata theorems: a constructive approach
• An asymptotic analysis of the Fourier-Laplace transforms of certain oscillatory functions
• Optimal decay of semi-uniformly stable operator semigroups with empty spectrum
• On a space of functions with entire Laplace transforms and its connection with the optimality of the Ingham-Karamata theorem