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 vectorvalued functions and of C0-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.2010 Mathematics Subject Classification. 40E05, 47D06 (44A10, 34D05).

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

“…The following result is a strengthened version of the second part of [7, Theorem 2.2], where a similar result is proved when M = K but with the additional assumption that M grows at least polynomially. The result is ancillary in nature for the purposes of the present paper, but in fact it shows optimality of a certain quantified Tauberian theorem for scalar-valued functions, as discussed in [7].…”

“…Furthermore, we improve on the value of the constant c appearing in our earlier optimality results [7, Theorems 2.2 and 2.4], which is significant now that M K is allowed to have subpolynomial growth. In fact, we shall obtain the value c = 1, which is best possible in view of the fact that (1.4) holds for all c ∈ (0, 1) when M = K. We achieve our results by refining the technique used in [7]. In particular, we first construct, in Lemma 2.1 below, a sequence of functions with certain properties, which we then use to prove an important preliminary result, Theorem 2.3, which can be viewed as proving optimality of a particular variant of the quantified Ingham-Karamata theorem for scalar-valued functions and is of considerable intrinsic interest.…”

“…At the heart of these proofs lies an extremely delicate construction of a certain complex measure. In our recent paper [7] we presented a significantly simpler optimality proof based on a striking application of the open mapping theorem. The results in [7] go beyond the polynomial case discussed above, and this generalisation had already been achieved in [14] by extending the existing more complicated technique.…”

“…In our recent paper [7] we presented a significantly simpler optimality proof based on a striking application of the open mapping theorem. The results in [7] go beyond the polynomial case discussed above, and this generalisation had already been achieved in [14] by extending the existing more complicated technique. However, in the semigroup case both the results in [7] and those in [14] essentially require the functions M and K to grow at least polynomially, and it was left open whether Theorem 1.1 is optimal for more slowly growing functions.…”

“…The aim of this paper, which can be viewed as a follow-up contribution to [7], is to prove a more general optimality result in the semigroup setting which in particular imposes no lower bound whatsoever on the functions M and K appearing in Theorem 1.1. Furthermore, we improve on the value of the constant c appearing in our earlier optimality results [7, Theorems 2.2 and 2.4], which is significant now that M K is allowed to have subpolynomial growth.…”