The Kalton-Lancien theorem revisited: Maximal regularity does not extrapolate

We give a new more explicit proof of a result by Kalton and Lancien stating that on each Banach space with an unconditional basis not isomorphic to a Hilbert space there exists a generator A of a holomorphic semigroup which does not have maximal regularity. In particular, we show that there always exists a Schauder basis (fm) such that A can be chosen of the form A(∑m=1∞amfm)=∑m=1∞2mamfm. Moreover, we show that maximal regularity does not extrapolate: we construct consistent holomorphic semigroups (Tp(t))t⩾0 on Lp(R) for p∈(1,∞) 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 and Lancien. We use the characterization of maximal regularity by R-sectoriality for our construction.