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 the characterisation of maximal Lp-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∞-calculus of angle zero, but fails stochastic maximal Lp-regularity (SMRp) for every p∈[2,∞). Motivated by this example, we study the Banach space geometry hypothesis underlying SMRp more closely. This is an R-boundedness condition (Sp) 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>2, the Laplacian on Lq(ℝd;X). Finally, we prove that its interval-kernel and exponential-kernel formulations are equivalent and that, at the endpoint p=2, condition (S2) holds if and only if X is isomorphic to a Hilbert space.
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BibTeX
@misc{lorist2026necessary,
title = {{Necessary conditions for deterministic and stochastic maximal regularity}},
author = {Emiel Lorist and Jan van Neerven and Mark Veraar and Lutz Weis},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.20266},
eprint = {2608.20266},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.20266},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Necessary conditions for deterministic and stochastic maximal regularity
AU - Emiel Lorist
AU - Jan van Neerven
AU - Mark Veraar
AU - Lutz Weis
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2608.20266
UR - https://arxiv.org/abs/2608.20266
AB - 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 the characterisation of maximal L^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^∞-calculus of angle zero, but fails stochastic maximal L^p-regularity (SMR_p) for every p∈[2,∞). 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>2, the Laplacian on L^q(ℝ^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.
ER -