This is an announcement for the paper "The Kalton-Lancien theorem revisited: Maximal regularity does not extrapolate" by Stephan Fackler.

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.

Archive classification: math.FA math.AP

Mathematics Subject Classification: 35K90, 47D06 (Primary) 46B15 (Secondary)

Remarks: 16 pages

Submitted from: stephan.fackler@uni-ulm.de

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1210.4333

or

http://arXiv.org/abs/1210.4333