Maximal Regularity: Positive Counterexamples on UMD-Banach Lattices and Exact Intervals for the Negative Solution of the Extrapolation Problem

TL;DR

This paper constructs positive analytic semigroups on certain Banach lattices that lack maximal regularity and demonstrates the worst-case behavior of the extrapolation problem for maximal regularity across intervals.

Contribution

It provides the first examples of positive semigroups on UMD-Banach lattices without maximal regularity and characterizes the exact intervals where maximal regularity holds for semigroups.

Findings

Existence of positive analytic semigroups on rm-Banach lattices without maximal regularity

Maximal regularity behavior can be precisely controlled across intervals

Counterexamples show the extrapolation problem can behave in the worst possible way

Abstract

Using methods from Banach space theory, we prove two new structural results on maximal regularity. The first says that there exist positive 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 $p \in I$.