The Kalton-Lancien Theorem Revisited: Maximal Regularity does not extrapolate

TL;DR

This paper provides a new proof of a theorem about Banach spaces and holomorphic semigroups, showing that maximal regularity does not extrapolate across different L^p spaces, solving two open problems.

Contribution

It offers a more explicit proof of existing results and constructs semigroups with maximal regularity only at p=2, addressing open questions in the field.

Findings

Existence of generators without maximal regularity on certain Banach spaces.

Construction of semigroups with maximal regularity only at p=2.

Maximal regularity does not extrapolate across L^p spaces.

Abstract

We give a new more explicit proof of a result by Kalton & Lancien stating that on each Banach space with an unconditional basis not isomorphic to a Hilbert space there exists a generator of a holomorphic semigroup which does not have maximal regularity. In particular, we show that there always exists a Schauder basis (f_m) such that the generator is a Schauder multiplier associated to the sequence (2^m). Moreover, we show that maximal regularity does not extrapolate: we construct consistent holomorphic semigroups (T_p(t)) on L^p for p in (1, \infty) which have maximal regularity if and only if p = 2. These assertions were both open problems. Our approach is completely different than the one of Kalton & Lancien. We use the characterization of maximal regularity by R-sectoriality for our construction.