Mean Value Inequalities for Jump Processes

Abstract

Parabolic Harnack inequalities are one of the most important inequalities in analysis and PDEs, partly because they imply Holder regularity of the solutions of heat equations. Mean value inequalities play an important role in deriving parabolic Harnack inequalities. In this paper, we first survey the recent results obtained in Chen et al. (Stability of heat kernel estimates for symmetric non-local Dirichlet forms, 2016, [15]; Stability of parabolic Harnack inequalities for symmetric non-local Dirichlet forms, 2016, [16]) on the study of stability of heat kernel estimates and parabolic Harnack inequalities for symmetric jump processes on general metric measure spaces. We then establish the \(L^p\)-mean value inequalities for all \(p\in (0, 2]\) for these processes.