Luke Schleef | Searcharxiv

arXiv2026

H\"older regularity for nonlocal equations governed by measures and non-standard growth

On doubling metric measure spaces, we study nonlocal operators with non-standard $(p,q)$-Orlicz growth ($1<p\leq q<\infty$), where the interaction kernel is given by a general, non-translation-invariant measure. Under natural assumptions on the interaction measure - namely symmetry, a suitable nonlocal Poincar\'{e} inequality, and a tail bound - we prove that every weak solution to the corresponding homogeneous nonlocal equation admits a locally H\"older continuous representative. These regularity results are new even in the Euclidean setting.

math.AP↗

arXiv2026

Partial H\"older regularity for fully nonlinear nonlocal parabolic equations with integrable kernels

In this work, we consider solutions to (fully nonlinear) parabolic integro-differential equations with integrable interaction kernels. A typical equation would be that obtained by starting with, for $s\in(0,1)$, the $s$-fractional heat equation, but replacing the interaction kernel in the integro-differential term with one which has been truncated, for $\rho>0$, at the value $\rho^{-d-2s}$, hence integrable. We show that solutions to these equations have a partial regularity estimate which captures differences of the solution up to the scale at which the kernel has a truncation in its singularity. The estimates we provide are robust with respect to the truncation parameter, and they include the existing results for the original operators without truncation. There are some earlier results for linear and elliptic cases of this situation of integrable interaction kernels, and so our work is a generalization of those to the nonlinear and parabolic setting.

math.AP↗