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Weak parabolic Harnack inequality and Hölder regularity for non-local Dirichlet forms

Guanhua Liu

Abstract

In this paper we give equivalent conditions for the weak parabolic Harnack inequality for general regular Dirichlet forms without killing part, in terms of local heat kernel estimates or growth lemmas. With a tail estimate on the jump measure, we obtain from these conditions the Hölder continuity of caloric and harmonic functions. Our results generalize the theory of Chen, Kumagai and Wang, in the sense that the upper jumping smoothness condition is canceled. We also derive the complete forms of Harnack inequalities from the globally non-negative versions, and obtain continuity of caloric functions with worse tails.

Weak parabolic Harnack inequality and Hölder regularity for non-local Dirichlet forms

Abstract

In this paper we give equivalent conditions for the weak parabolic Harnack inequality for general regular Dirichlet forms without killing part, in terms of local heat kernel estimates or growth lemmas. With a tail estimate on the jump measure, we obtain from these conditions the Hölder continuity of caloric and harmonic functions. Our results generalize the theory of Chen, Kumagai and Wang, in the sense that the upper jumping smoothness condition is canceled. We also derive the complete forms of Harnack inequalities from the globally non-negative versions, and obtain continuity of caloric functions with worse tails.

Paper Structure

This paper contains 8 sections, 31 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Notions and main results
  3. Proof of Theorem \ref{['M1']}
  4. Proof of Theorem \ref{['M2']}
  5. Caloric extension and tails in Harnack inequalities
  6. Continuity of caloric functions with a weaker tail condition
  7. Joint measurabilty
  8. The stochastic counterpart

Key Result

Theorem 2.1

Assume that $(M,d,\mu)$ satisfies $(\mathrm{VD})$ and $(\mathrm{RVD})$, and $(\mathcal{E},\mathcal{F})$ is a regular Dirichlet form such that $dj(x,y)=J(x,dy)d\mu(x)$ and $\kappa\equiv 0$. Then the following are equivalent, each implying $(\mathrm{S}^-)$:

Theorems & Definitions (61)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Proposition 2.3
  • Example 2.4
  • Example 2.5
  • Corollary 2.6
  • Lemma 3.1
  • proof
  • ...and 51 more