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

Abstract

We prove that a general version of the quantified Ingham-Karamata theorem for C0-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 C0-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.2010 Mathematics Subject Classification. 40E05, 47D06 (44A10, 34D05).

Semi-uniform stability of operator semigroups and energy decay of damped waves

“…The relevant theory was developed in a number of papers, arguably starting with the seminal paper [64]. Later, in [65], a finer distributional approach was developed, which has recently been extended to include quantitative aspects; see [66–69]. A good introduction to modern Tauberian theory may be found in [70], while applications to operator semigroups are discussed thoroughly in [1].…”

“…Observe, however, that Theorem 3.1 is best possible in the sense that one cannot expect any rate of decay for f if no further assumptions are imposed on the growth of ffalse^, even if ffalse^ extends to an entire function; see e.g. [69,74,75]. In order to obtain quantitative results, one assumes that ffalse^ extends analytically beyond iR to some precise domain and that this analytic extension satisfies an appropriate bound in this domain.…”

“…Optimality for p = ∞ and polynomial M = K was proved in [77, Theorem 3.8] by an explicit and fairly involved construction. By developing further the ideas from [77], this approach was then extended to all p and to some other functions M (and also to individual orbits of Hilbert space semigroups) in [66, Sections 5 and 7], and in [78, Theorem 4.1] for an even larger class of functions M and K ; see also [68,69] for an alternative abstract approach to optimality.…”

“…As in the case of Theorem 3.2, Theorem 3.4 is optimal in the sense that one cannot obtain better upper estimates for the semigroup orbits than the one given by (3.6). In the case of polynomial resolvent growth this goes back once again to [77], but more recently an elegant abstract approach for obtaining optimal lower bounds for semigroup orbits was found by Debruyne & Seifert [68,69]. Note, however, that the lower bounds contained in these optimality results hold only along a subsequence of double-struckR+.…”

An asymptotic analysis of the Fourier-Laplace transforms of certain oscillatory functions

“…The technique was then refined to show optimality for more general versions of Theorem 1 in [1] and the most general optimality results achieved via this technique can be found in [19]. The second approach only appeared very recently in [8] and crucially depends on a careful application of the open mapping theorem, see also [9] for the most general results obtained by this method. The question then remains whether one can find "simple" functions showing optimality results.…”