Stability of Hölder regularity and weighted functional inequalities
Soobin Cho, Panki Kim
TL;DR
This work develops a general framework for Hölder regularity of caloric and harmonic functions tied to symmetric Dirichlet forms on metric measure spaces that may include both local and jump parts, including blow-up jump kernels. A new weighted tail condition is introduced through an admissible weight ${\Theta}$ and a scale function ${\phi}$, enabling weighted tail estimates, Caccioppoli and mean-value inequalities, and a suite of weighted Sobolev/capacity inequalities that remain stable under singular jump behavior. The authors prove stable, equivalent characterizations linking near-diagonal lower bounds, parabolic and elliptic Hölder regularity, energy conditions, and weighted cut-off inequalities; these results apply to trace processes and non-local Neumann problems, yielding Hölder continuity up to boundaries with blow-up. Overall, the paper unifies probabilistic and analytic techniques to obtain sharp heat-kernel estimates, exit-time controls, and Hölder regularity in broad nonlocal settings, significantly extending prior unweighted and non-blow-up theories.
Abstract
We study symmetric Dirichlet forms on metric measure spaces, which may possess both strongly local and pure-jump parts. We introduce a new formulation of a tail condition for jump measures and weighted functional inequalities. Our framework accommodates Dirichlet forms with singular jump measures and those associated with trace processes of mixed-type stable processes. Using these new weighted functional inequalities, we establish stable, equivalent characterizations of Hölder regularity for caloric and harmonic functions. As an application of our main result, we prove the Hölder continuity of caloric functions for a large class of symmetric Markov processes exhibiting boundary blow-up behavior, among other results.