  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 $p
\in I$.
