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Maximal regularity: Positive counterexamples on UMD-Banach lattices and exact intervals for the negative solution of the extrapolation problem


Author: Stephan Fackler
Journal: Proc. Amer. Math. Soc. 144 (2016), 2015-2028
MSC (2010): Primary 47D06; Secondary 35B65, 46B15
DOI: https://doi.org/10.1090/proc/13012
Published electronically: January 20, 2016
MathSciNet review: 3460163
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Abstract: Using methods from Banach space theory, we prove two new structural results on maximal regularity. The first says that there exist positive bounded analytic semigroups on UMD-Banach lattices; namely, $ \ell _p(\ell _q)$ for $ p \neq q \in (1, \infty )$, without maximal regularity. In the second result we show that the extrapolation problem for maximal regularity behaves in the worst possible way: for every interval $ I \subset (1, \infty )$ with $ 2 \in I$ there exists a family of consistent bounded analytic semigroups $ (T_p(z))_{z \in \Sigma _{\pi /2}}$ on $ L_p(\mathbb{R})$ such that $ (T_p(z))$ has maximal regularity if and only if $ p \in I$.


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Additional Information

Stephan Fackler
Affiliation: Institute of Applied Analysis, University of Ulm, Helmholtzstr. 18, 89069 Ulm, Germany
Email: stephan.fackler@uni-ulm.de

DOI: https://doi.org/10.1090/proc/13012
Keywords: Maximal regularity, $H^{\infty}$-calculus, Schauder bases, counterexamples
Received by editor(s): October 8, 2014
Received by editor(s) in revised form: May 20, 2015
Published electronically: January 20, 2016
Additional Notes: The author was supported by a scholarship of the “Landesgraduiertenfarderung Baden-Wattemberg”.
Communicated by: Marius Junge
Article copyright: © Copyright 2016 American Mathematical Society