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

Fackler, Stephan

doi:10.1090/proc/13012

Maximal regularity: Positive counterexamples on UMD-Banach lattices and exact intervals for the negative solution of the extrapolation problem
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Proc. Amer. Math. Soc. 144 (2016), 2015-2028 Request permission

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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