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Stable characterization of diagonal heat kernel upper bounds for symmetric Dirichlet forms

Soobin Cho

TL;DR

The paper develops a robust, density-free framework for stable on-diagonal heat-kernel upper bounds p(t,x,x) of symmetric Dirichlet forms on metric measure spaces with volume doubling. By coupling tail/integrated jump bounds TJ(φ) and IJ_{2,γ}(φ) with cutoff Sobolev inequalities CS(φ) and Faber–Krahn-type inequalities FK_ν(φ)/GFK_{ν,b}(φ), it establishes a precise equivalence: DUE(φ) together with SE(φ) follows from FK_ν(φ) + IJ_{2,γ}(φ) under a quantitative (1−ν)γ < 1+ν constraint, and further improves under density-based TJ_q(φ). The work also constructs near-optimal counterexamples, analyzes the special case φ(x,r)=r^β via truncation and metric-change techniques, and demonstrates applications to variable-order non-local operators and singular jump kernels, thereby broadening the scope of stable heat-kernel estimates in nonlocal settings.

Abstract

We present a stable characterization of on-diagonal upper bounds for heat kernels associated with regular Dirichlet forms on metric measure spaces satisfying the volume doubling property. Our conditions include integral bounds on the jump kernel outside metric balls, a variant of the Faber-Krahn inequality, a cutoff Sobolev inequality, and an integral control of inverse square volumes of balls with respect to the jump kernel. Crucially, we do not assume that the jump kernel has a density, and we show that these assumptions are essentially optimal.

Stable characterization of diagonal heat kernel upper bounds for symmetric Dirichlet forms

TL;DR

The paper develops a robust, density-free framework for stable on-diagonal heat-kernel upper bounds p(t,x,x) of symmetric Dirichlet forms on metric measure spaces with volume doubling. By coupling tail/integrated jump bounds TJ(φ) and IJ_{2,γ}(φ) with cutoff Sobolev inequalities CS(φ) and Faber–Krahn-type inequalities FK_ν(φ)/GFK_{ν,b}(φ), it establishes a precise equivalence: DUE(φ) together with SE(φ) follows from FK_ν(φ) + IJ_{2,γ}(φ) under a quantitative (1−ν)γ < 1+ν constraint, and further improves under density-based TJ_q(φ). The work also constructs near-optimal counterexamples, analyzes the special case φ(x,r)=r^β via truncation and metric-change techniques, and demonstrates applications to variable-order non-local operators and singular jump kernels, thereby broadening the scope of stable heat-kernel estimates in nonlocal settings.

Abstract

We present a stable characterization of on-diagonal upper bounds for heat kernels associated with regular Dirichlet forms on metric measure spaces satisfying the volume doubling property. Our conditions include integral bounds on the jump kernel outside metric balls, a variant of the Faber-Krahn inequality, a cutoff Sobolev inequality, and an integral control of inverse square volumes of balls with respect to the jump kernel. Crucially, we do not assume that the jump kernel has a density, and we show that these assumptions are essentially optimal.
Paper Structure (22 sections, 68 theorems, 346 equations)

This paper contains 22 sections, 68 theorems, 346 equations.

Table of Contents

  1. Introduction
  2. Framework and main results
  3. Preliminaries
  4. Faber-Krahn type inequalities
  5. $L^2$-mean value inequality and resolvent estimates
  6. The case of special scale function
  7. Tail estimates for truncated Dirichlet form
  8. On-diagonal upper estimate for truncated Dirichlet form
  9. Some off-diagonal estimates
  10. Diagonal upper heat kernel estimate
  11. Upper estimates for the semigroup
  12. Change of metric
  13. Proofs of Proposition \ref{['p:conservative-TE']} and Theorem \ref{['t:NDU-main']}
  14. Proofs of main results
  15. Proofs of Theorem \ref{['t:main1']}, Corollary \ref{['c:main1']} and Theorem \ref{['t:main2']}
  16. ...and 7 more sections

Key Result

Theorem 1.1

Assume that $(M,d,\mu)$ satisfies VD, and $({\mathcal{E}},{\mathcal{F}})$ satisfies CS$(\beta)$ and TJ$(\beta)$. For any $\nu>0$ and $\gamma \ge 0$ satisfying we have

Theorems & Definitions (86)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • ...and 76 more