An arXiv preprint reports that a constructed operator can pass a demanding calculus test and still fail a key form of stochastic regularity. In the example, the operator has a bounded H∞-calculus of angle zero but fails stochastic maximal L^p-regularity for every p≥2. The construction uses X=ℓ^2(ℓ^r) with r>2, a UMD Banach function space of type 2.
The finding is deliberately narrow, but important within the paper’s framework. It says that a bounded H∞-calculus alone is not enough for stochastic maximal regularity for this constructed operator; it does not say that every operator with such a calculus fails. The wider paper asks how Banach-space geometry constrains deterministic and stochastic maximal regularity—roughly, whether the mathematical system maintains the needed control when random input is present.
The diagonal test bed
The paper is a proof-driven analysis of a positive diagonal operator on mixed-norm sequence spaces X_{q,r}=ℓ^q(ℓ^r). A diagonal operator is specified coordinate by coordinate, allowing the authors to compare exact regularity conditions as the exponents p, q and r vary.
For the deterministic problem, the classification is exact for p,q,r≥1. Maximal L^p-regularity holds when p>1 and either q>1 or r=1; the all-one case p=q=r=1 also qualifies. Outside those cases, it fails for this diagonal operator.
A separate example shows the same gap between a favorable calculus property and maximal regularity in a deterministic setting. For r>1, the operator on X_{1,r} is R-sectorial, has a bounded H∞-calculus of angle zero, yet fails deterministic maximal L^p-regularity for every p≥1.
The stochastic version shifts the thresholds. For p,q,r≥2, stochastic maximal L^p-regularity holds exactly when p>2 and either q>2 or r=2, with the all-two case p=q=r=2 as an additional qualifying case.
A proof step called concavification links the two classifications: for p,q,r≥2, stochastic maximal L^p-regularity for the diagonal operator is equivalent to deterministic maximal L^{p/2}-regularity for a related construction on X_{q/2,r/2}.
A condition beyond the calculus
The authors then examine (S_p), a condition on stochastic-convolution kernels. For UMD Banach spaces of type 2 and p≥2, its interval-kernel and exponential-kernel formulations are equivalent.
At the endpoint p=2, (S_2) holds exactly when the space is isomorphic to a Hilbert space.
For p,q>2, the paper states an equivalence among (S_p), stochastic maximal L^p-regularity for the diagonal multiplier on Rad_p(X), and stochastic maximal L^p-regularity for the Laplacian on L^q(ℝ^d;X).
What the result does—and does not—say
A broader generalized diagonal construction separates two geometric requirements. On X(ℓ^2), R-sectoriality of angle zero is equivalent to finite cotype of the underlying space, while maximal L^p-regularity is equivalent to the UMD property.
The conclusions remain tied to specified constructions and parameter ranges. The full Laplacian equivalence is stated for p,q>2; when exactly one of p or q equals 2, the proof does not establish the converse implication. Likewise, the counterexample does not show that every operator with a bounded H∞-calculus fails stochastic maximal regularity.
The manuscript is identified as arXiv version 1, dated 20 August 2026. Its results are mathematical equivalences about the stated operators and Banach-space constructions, rather than findings from an empirical dataset.
Paper data and sources
Original title: Necessary conditions for deterministic and stochastic maximal regularity
Authors: Emiel Lorist, Jan van Neerven, Mark Veraar
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text