Stability of Hölder regularity and weighted functional inequalities

2025/03/01 by Cho, Soobin, Kim, Panki
#60J46 #60J76 #Analysis of PDEs (math.AP) #FOS: Mathematics #Primary 60J45 #Probability (math.PR) #Secondary 35K08

paper · doi:10.48550/arxiv.2503.00318

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.

Related

• Stability of heat kernel estimates for symmetric non-local Dirichlet forms
• Intrinsic Hölder continuity of harmonic functions
• Stability of parabolic Harnack inequalities for symmetric non-local Dirichlet forms
• Heat kernel estimates for general symmetric pure jump Dirichlet forms
• Characterization of subordinate symmetric Markov processes