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A simplified characterization of stable-like heat kernel estimates

Mathav Murugan

TL;DR

The paper studies stable-like heat kernel estimates for symmetric pure jump processes on general metric measure spaces by developing a non-local Whitney blending technique. It shows that two-sided jump kernel bounds $J(\\phi)$ together with a capacity upper bound Cap$(\\phi)_{\\le}$ imply a cutoff Sobolev inequality CSJ$(\\phi)$, and that HK$(\\phi)$ can be characterized in terms of these conditions, affirming a Grigor'yan–Hu–Hu conjecture in this non-local setting. The main novelty is extending Whitney blending to non-local Dirichlet forms, allowing energy control via equilibrium potentials and a careful truncation analysis of the jump kernel. This yields a practical framework for verifying stable-like HK bounds in concrete examples and broadens the stability theory for heat kernels of jump processes on diverse spaces.

Abstract

We study heat kernel estimates for symmetric pure jump processes on general metric measure spaces. Building on recent progress in the local setting due to S.~Eriksson-Bique, we develop a non-local version of the Whitney blending technique and use it to relate stable-like heat kernel estimates to capacity upper bounds. Under two-sided stable-like bounds on the jump kernel, we show that a capacity upper bound across annuli implies a cutoff Sobolev inequality, and we obtain a characterization of stable-like heat kernel estimates in terms of these conditions. As a consequence, we give an affirmative answer to a conjecture of A. Grigor'yan, E. Hu, and J. Hu.

A simplified characterization of stable-like heat kernel estimates

TL;DR

The paper studies stable-like heat kernel estimates for symmetric pure jump processes on general metric measure spaces by developing a non-local Whitney blending technique. It shows that two-sided jump kernel bounds together with a capacity upper bound Cap imply a cutoff Sobolev inequality CSJ, and that HK can be characterized in terms of these conditions, affirming a Grigor'yan–Hu–Hu conjecture in this non-local setting. The main novelty is extending Whitney blending to non-local Dirichlet forms, allowing energy control via equilibrium potentials and a careful truncation analysis of the jump kernel. This yields a practical framework for verifying stable-like HK bounds in concrete examples and broadens the stability theory for heat kernels of jump processes on diverse spaces.

Abstract

We study heat kernel estimates for symmetric pure jump processes on general metric measure spaces. Building on recent progress in the local setting due to S.~Eriksson-Bique, we develop a non-local version of the Whitney blending technique and use it to relate stable-like heat kernel estimates to capacity upper bounds. Under two-sided stable-like bounds on the jump kernel, we show that a capacity upper bound across annuli implies a cutoff Sobolev inequality, and we obtain a characterization of stable-like heat kernel estimates in terms of these conditions. As a consequence, we give an affirmative answer to a conjecture of A. Grigor'yan, E. Hu, and J. Hu.
Paper Structure (11 sections, 11 theorems, 149 equations)

This paper contains 11 sections, 11 theorems, 149 equations.

Table of Contents

  1. Introduction
  2. Framework and results
  3. Outline of the proof
  4. Proofs
  5. Identification of the domain of the Dirichlet form
  6. Truncated Dirichlet form
  7. Estimates on the equilibrium potential
  8. Whitney cover
  9. Whitney blending
  10. Self-improvement of a simplified cutoff Sobolev inequality
  11. Characterization of stable-like heat kernel estimate

Key Result

Theorem 1.6

Let $(X,d,m)$ be a metric measure space that satisfies vd. Let $(\mathcal{E},\mathcal{F})$ be a pure jump Dirichlet form on $L^2(X,m)$ that satisfies jphi, and cap. Then $(\mathcal{E},\mathcal{F})$ satisfies the cutoff energy inequality csj.

Theorems & Definitions (28)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Remark 1.4
  • Definition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Remark 1.8
  • Proposition 2.1
  • Lemma 2.2
  • ...and 18 more