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

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 $p \in I$.