Necessary conditions for deterministic and stochastic maximal regularity

Abstract: We study the role of Banach space geometry in deterministic and stochastic maximal regularity. We first construct an example showing that the UMD assumption in Weis' characterisation of maximal $L<sup>p$-regularity in terms of $R$-sectoriality cannot be omitted. Combining this construction with an equivalence between stochastic maximal regularity and deterministic maximal regularity on the $2$-concavification of the underlying space, we obtain an operator on a UMD Banach function space of type $2$ that has a bounded $H<sup>\infty$-calculus of angle zero, but fails stochastic maximal $L<sup>p$-regularity (SMR$_p$) for every $p\in[2,\infty)$. Motivated by this example, we study the Banach space geometry hypothesis underlying SMR$_p$ more closely. This is an $R$-boundedness condition $(S_p)$ for stochastic convolution operators. For UMD spaces $X$ of type $2$, we show that this condition is not only sufficient, but also necessary for two canonical test operators: a diagonal multiplier on a Rademacher space and, for $q&gt;2$, the Laplacian on $L^q(\mathbb R<sup>d;X)$. Finally, we prove that its interval-kernel and exponential-kernel formulations are equivalent and that, at the endpoint $p=2$, condition $(S_2)$ holds if and only if $X$ is isomorphic to a Hilbert space.