Stability of Hölder regularity and weighted functional inequalities

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 H\"older regularity for caloric and harmonic functions. As an application of our main result, we prove the H\"older continuity of caloric functions for a large class of symmetric Markov processes exhibiting boundary blow-up behavior, among other results.