The gap between strong tests and regularity
A theorem-based preprint shows, in a constructed setting, that strong operator tests can coexist with failure of stochastic maximal regularity, a precise condition on time-dependent problems. It also gives exact if-and-only-if rules for deterministic and stochastic maximal regularity for a constructed diagonal operator.
The work is not an empirical study. It uses a theorem-and-proof construction centered on a diagonal multiplier and explicitly built mathematical spaces and operators. Its main family is a collection of mixed-norm sequence spaces with a positive diagonal operator. On every member, the operator passes two technical tests: it is R-sectorial and has a bounded H-infinity calculus of angle zero.
That distinction matters because the deterministic result is exact. For the diagonal operator on the mixed-norm family, maximal regularity at exponent p holds when p, the time-integrability exponent, is greater than 1 and either q, the outer mixed-norm exponent, is greater than 1 or r, the inner exponent, equals 1. There is also a separate case in which p, q and r all equal 1. Outside those alternatives, the property does not hold.
The paper then broadens the construction to an order-continuous Banach function space X with an ell-two component. R-sectoriality with angle zero is equivalent to X having finite cotype, while maximal regularity at exponent p is equivalent to X being UMD. The mathematical message is that the two operator properties point to different geometric requirements in the underlying space.
Randomness changes the threshold
For the stochastic version, the authors reduce the problem to a deterministic one on a transformed, or concavified, space. In the stated range beginning at 2, stochastic maximal regularity on the original mixed-norm space is equivalent to deterministic maximal regularity after the three exponents p, q and r are each divided by 2 on the transformed space. This concavification step is the bridge between the two classifications.
That bridge yields a complete rule for the constructed diagonal operator. Stochastic maximal regularity holds exactly when p is greater than 2 and either q is greater than 2 or r equals 2, with a separate case in which all three exponents equal 2. The result is an exact classification of this operator family, not a universal statement about every operator or Banach space.
The paper also presents a counterexample that separates functional calculus from stochastic regularity. It constructs a UMD Banach function space of type 2 and a sectorial operator with a bounded H-infinity calculus of angle zero, yet stochastic maximal regularity fails throughout the stated range beginning at 2. In this exhibited setting, the bounded calculus property alone is insufficient.
One condition, several operators
The next test uses stochastic convolutions. The paper compares an interval-kernel condition with an exponential-kernel condition, two ways of weighting the operator's time response, and proves that they are equivalent for a UMD space of type 2 in the stated range beginning at 2. Above the endpoint, the condition is independent of the exponent: if it holds for one exponent above 2, it holds for every exponent above 2.
At the endpoint where p equals 2, the same condition becomes a geometric dividing line. It holds exactly when the underlying space X is isomorphic to a Hilbert space. In the theorem's language, isomorphic means that the two spaces have equivalent linear structure, rather than being literally identical.
Finally, the authors test the abstract condition on two canonical operators. For the diagonal multiplier on a Rademacher space, stochastic maximal regularity on a finite interval, on the positive half-line, and the interval-kernel condition are equivalent. For the Laplacian acting on X-valued functions with spatial exponent q, the same three-way equivalence holds when p and q are above 2. These results tie the abstract condition to both a sequence-space model and a differential operator.
The scope remains deliberately mathematical. There is no empirical sample; the evidence consists of theorems for explicitly specified spaces, operators and exponent ranges. The equivalences rely on assumptions named in the theorems, including UMD, type 2 and the particular operator or exponent range under study. The paper also leaves open whether a similar example without the H-infinity-calculus requirement can be constructed on ell-one or L-one spaces. The paper is an arXiv version 2 preprint dated 2 September 2026.
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