A theorem about local smoothness
An arXiv preprint reports a conditional theorem about the regularity of weak solutions to a class of nonlocal equations. Under the paper’s assumptions, any solution that is locally bounded has a locally Hölder-continuous representative. In plain language, its values vary in a controlled way within small regions rather than changing without bound from point to point. The theorem says the Hölder exponent lies between 0 and 1, but it does not provide a numerical value.
The paper studies no empirical participants or observational records. It is a proof-based analysis of weak solutions on abstract doubling metric measure spaces, mathematical settings where the measure of a ball is controlled as the ball is enlarged. The equations use non-standard (p,q)-Orlicz growth, meaning their growth is described by a lower index p and an upper index q, with p greater than 1, p no greater than q, and q finite. The operator’s differential order lies between 0 and 1.
The setting allows general, non-translation-invariant interaction measures, so the interaction need not be described by a translation-invariant pattern. The assumptions highlighted by the paper are symmetry, a suitable nonlocal Poincaré inequality and a tail bound. The last condition matters because the local Hölder estimate contains a tail term that records how the solution behaves outside the domain.
The machinery behind the result
Those conditions feed into a proof built from De Giorgi, Nash and Moser theory, test-function energy estimates and De Giorgi iteration. The paper establishes Caccioppoli and logarithmic energy estimates for weak supersolutions. It also derives a dilated Sobolev–Poincaré inequality from the interaction-measure Poincaré assumption and the doubling property. That inequality provides an integrability gain used by the iteration.
The proof’s central turning point is a critical-mass step. It tests whether the low-value part of a supersolution is small enough inside a ball. More precisely, the set where u is at most 2a in a ball of radius 2r must occupy no more than a threshold γ times the ball’s measure. This threshold is a structural ingredient of the argument, not a measurement taken from a dataset.
A related expansion-of-positivity lemma starts from the opposite pattern. If the set where u is at least a occupies at least an ε fraction of a ball with radius 4r, the argument introduces a positive δ determined by ε and other structural constants. It then gives an expansion conclusion involving either the solution’s tail or a lower bound. The constants are described through their dependence on the assumptions, rather than as universal numerical values.
Together, the estimates and positivity lemmas control the solution’s oscillation, the gap between its high and low values on a region. Repeating that decay across smaller regions is the final analytic step leading to Hölder regularity. The paper identifies expansion of positivity as the key ingredient in this oscillation-decay argument.
A conditional result with clear boundaries
The result has a separate corollary for globally bounded local weak solutions. It also states local Hölder regularity, with an exponent that depends on structural parameters. The main theorem itself, however, begins with local boundedness as an assumption.
That starting point is a significant boundary on the claim. The authors state that local boundedness cannot be removed in general for singular kernels, and that a priori local boundedness and Harnack’s inequality fail in that setting. In other words, the preprint gives a regularity theorem for solutions already known to be locally bounded; it does not show that boundedness follows automatically.
The conclusions are conditional on the geometry and interaction structure: doubling of the underlying space, symmetry, a suitable Poincaré condition and a tail bound. Because the work is theoretical, it uses no participant sample or observational dataset. It also does not report a numerical Hölder exponent.
The author presents the result as the first to combine non-standard Orlicz growth, general non-translation-invariant interaction measures and abstract doubling metric measure spaces. That is an author-level novelty claim, not an independently verified comparison in the supplied analysis. The document is an arXiv version 1 preprint dated 28 Aug 2026. No funding source is reported in the supplied text; the acknowledgment thanks advisor Moritz Kassmann for suggesting the problem and for discussions.
Paper data and sources
Original title: Hölder regularity for nonlocal equations governed by measures and non-standard growth
Authors: Luke Schleef
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-28
DOI: Not available
Original paper · Full text