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Gaussian upper heat kernel bounds and Faber-Krahn inequalities on graphs

Christian Rose

TL;DR

The paper develops a comprehensive framework to characterize when FK inequalities, Gaussian upper bounds for the continuous-time heat kernel, and volume doubling hold simultaneously on graphs with potentially unbounded geometry. It extends Grigor'yan’s continuum results to discrete graphs by leveraging intrinsic metrics, mean-value inequalities, and Davies-type techniques, and it introduces a local regularity condition and a variable FK-dimension $n'(r)$ to handle unbounded vertex degree and nonuniform geometry. For the normalizing measure $m=\deg$, FK, $G$, and $V$ are shown to be equivalent on large scales up to constants; for counting or general measures, the analysis enriches FK with correction terms $\Psi,\Phi$ and allows a radius-dependent dimension that tracks the doubling dimension and degree growth. The work provides a robust analytic toolkit—mean-value inequalities, integrated maximum principles, and weighted $\ell^2$-norm controls—that yields sharp Gaussian heat-kernel bounds and a principled path from heat-kernel regularity back to Faber-Krahn inequalities on graphs.

Abstract

We investigate the equivalence of relative Faber-Krahn inequalities and the conjunction of Gaussian upper heat kernel bounds and volume doubling on large scales on graphs. For the normalizing measure, we obtain their equivalence up to constants by imposing comparability of small balls and the vertex degree at their centers. Removing this comparability assumption on the cost of a local regularity condition entering the equivalence and allowing for a variable dimension lead to a further generalization. The variable dimension converges to the doubling dimension for increasing ball radius. If the counting measure or arbitrary measures are considered, the local regularity condition contains the vertex degree. Furthermore, correction functions for the Gaussian, doubling, and Faber-Krahn dimension depending on the vertex degree are introduced. For the Gaussian and doubling, the variable correction functions always tend to one at infinity. Moreover, the variable Faber-Krahn dimension can be related to the doubling dimension and the vertex degree growth.

Gaussian upper heat kernel bounds and Faber-Krahn inequalities on graphs

TL;DR

The paper develops a comprehensive framework to characterize when FK inequalities, Gaussian upper bounds for the continuous-time heat kernel, and volume doubling hold simultaneously on graphs with potentially unbounded geometry. It extends Grigor'yan’s continuum results to discrete graphs by leveraging intrinsic metrics, mean-value inequalities, and Davies-type techniques, and it introduces a local regularity condition and a variable FK-dimension to handle unbounded vertex degree and nonuniform geometry. For the normalizing measure , FK, , and are shown to be equivalent on large scales up to constants; for counting or general measures, the analysis enriches FK with correction terms and allows a radius-dependent dimension that tracks the doubling dimension and degree growth. The work provides a robust analytic toolkit—mean-value inequalities, integrated maximum principles, and weighted -norm controls—that yields sharp Gaussian heat-kernel bounds and a principled path from heat-kernel regularity back to Faber-Krahn inequalities on graphs.

Abstract

We investigate the equivalence of relative Faber-Krahn inequalities and the conjunction of Gaussian upper heat kernel bounds and volume doubling on large scales on graphs. For the normalizing measure, we obtain their equivalence up to constants by imposing comparability of small balls and the vertex degree at their centers. Removing this comparability assumption on the cost of a local regularity condition entering the equivalence and allowing for a variable dimension lead to a further generalization. The variable dimension converges to the doubling dimension for increasing ball radius. If the counting measure or arbitrary measures are considered, the local regularity condition contains the vertex degree. Furthermore, correction functions for the Gaussian, doubling, and Faber-Krahn dimension depending on the vertex degree are introduced. For the Gaussian and doubling, the variable correction functions always tend to one at infinity. Moreover, the variable Faber-Krahn dimension can be related to the doubling dimension and the vertex degree growth.

Paper Structure

This paper contains 13 sections, 38 theorems, 265 equations.

Table of Contents

  1. Introduction and main results
  2. The set-up
  3. Global variants of the main results
  4. Strategy of the proofs
  5. Faber-Krahn, local regularity and doubling
  6. Mean value inequality
  7. Maximal inequalities for positive parts of subsolutions
  8. De Giorgi iteration steps
  9. The iteration steps and mean value inequality
  10. Integral maximum principle
  11. Weighted $\ell^2$-norm of the heat kernel
  12. Relative Faber-Krahn and heat kernel bounds
  13. From heat kernel bounds, regularity and doubling to Faber-Krahn

Key Result

Theorem 1.3

Let $m=\deg$, $\mathop{\mathrm{diam}}\nolimits X=\infty$, and $n>0$, $R\geq 300$ constants. If there exists a constant $a>0$ such that $FK(R,a,n)$ holds in $X$, then there exists a constant $C=C_{a,n,R_1}\geq 1$ such that $G(R,C,n)$ and $V(C,n)$ hold in $X$. Conversely, if there exists $C>0$ such th

Theorems & Definitions (84)

  • Definition 1.2
  • Remark
  • Remark
  • Theorem 1.3: normalizing measure, uniform version
  • Remark
  • Definition 1.4
  • Remark
  • Theorem 1.5: counting measure
  • Remark : behavior of the correction terms and dimension function
  • Theorem 1.6: general measure
  • ...and 74 more