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

摘要

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 holomor-phic 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(Σ_m~∞=_1 a_mf_m) = Σ _m~∞=_1 2~ma_mf_m. Moreover, we show that maximal regularity does not extrapolate: we construct consistent holomorphic semigroups (T_p(t))_(t >0) on L~p(?) 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.