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Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $ε$-range and it's application

Wen-Qi Li, Zhikai Zhang

TL;DR

The paper develops a comprehensive framework for parabolic and spectral analysis on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds in the $\varepsilon$-range. By combining Lu–Ohta’s volume comparison with Fujitani’s local Sobolev inequality, it obtains a local $φ$-heat kernel estimate, from which Gaussian upper and lower bounds for the $φ$-heat kernel are deduced via Davies’ estimates and Li–Yau-type inequalities. These bounds yield a suite of consequences, including an $L^{1}_{φ}$ Liouville theorem for $φ$-subharmonic functions, $L^{1}_{φ}$-uniqueness of the $φ$-heat equation, and eigenvalue estimates for the weighted Laplacian $Δ_{φ}$, alongside a Li–Yau gradient estimate under a weighted $L^{p}$-norm bound on $| abla φ|^{2}$. The results address open questions about negative-dimension regimes in $N$-Ricci curvature and unify several curvature-bound frameworks through the $ε$-range, offering new tools for analysis on smooth metric measure spaces.

Abstract

In this paper, we establish a parabolic Harnack inequality for positive solutions of the $φ$-heat equation and prove Gaussian upper and lower bounds for the $φ$-heat kernel on weighted Riemannian manifolds under lower $N$-Ricci curvature bound with $\varepsilon$-range. Building on these results, we demonstrate: The $L^1_φ$-Liouville theorem for $φ$-subharmonic functions, $L^1_φ$-uniqueness property for solutions of the $φ$-heat equation and lower bounds for eigenvalues of the weighted Laplacian $Δ_φ$. Furthermore, leveraging the Gaussian upper bound of the weighted heat kernel, we construct a Li-Yau-type gradient estimate for the positive solution of weighted heat equation under a weighted $L^p(μ)$-norm constraint on $|\nablaφ|^2$.

Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $ε$-range and it's application

TL;DR

The paper develops a comprehensive framework for parabolic and spectral analysis on weighted Riemannian manifolds under lower -Ricci curvature bounds in the -range. By combining Lu–Ohta’s volume comparison with Fujitani’s local Sobolev inequality, it obtains a local -heat kernel estimate, from which Gaussian upper and lower bounds for the -heat kernel are deduced via Davies’ estimates and Li–Yau-type inequalities. These bounds yield a suite of consequences, including an Liouville theorem for -subharmonic functions, -uniqueness of the -heat equation, and eigenvalue estimates for the weighted Laplacian , alongside a Li–Yau gradient estimate under a weighted -norm bound on . The results address open questions about negative-dimension regimes in -Ricci curvature and unify several curvature-bound frameworks through the -range, offering new tools for analysis on smooth metric measure spaces.

Abstract

In this paper, we establish a parabolic Harnack inequality for positive solutions of the -heat equation and prove Gaussian upper and lower bounds for the -heat kernel on weighted Riemannian manifolds under lower -Ricci curvature bound with -range. Building on these results, we demonstrate: The -Liouville theorem for -subharmonic functions, -uniqueness property for solutions of the -heat equation and lower bounds for eigenvalues of the weighted Laplacian . Furthermore, leveraging the Gaussian upper bound of the weighted heat kernel, we construct a Li-Yau-type gradient estimate for the positive solution of weighted heat equation under a weighted -norm constraint on .

Paper Structure

This paper contains 7 sections, 25 theorems, 185 equations.

Table of Contents

  1. Introduction
  2. preliminaries
  3. Moser's Harnack inequality for $\phi-$Heat equation
  4. Upper and Lower Bounds of $\phi-$Heat Kernel
  5. $L_{\phi}^{1}$ Liouville theorem
  6. Eigenvalue estimates
  7. Li-Yau's Gradient estimate for $\phi-$Heat equation

Key Result

Theorem 1.1

Let $(M^n, g, \mu)$ be a weighted Riemannian manifold with a lower $N$-Ricci curvature bound and $\varepsilon$-range. Assume $\phi$ satisfies For all $o \in M^n$, $R > 0$, and $\varepsilon > 0$, the $\phi$-heat kernel satisfies where $E_{2}^{'}, D_{2}$ are constants depending on $a, b, n, \nu$, $C_{i}^{'}$ is a constant depends on $n,\ \nu,a,b,c,K$ and $\lim_{\varepsilon \to 0} C(\varepsilon) =

Theorems & Definitions (43)

  • Theorem 1.1: Weighted Heat Kernel Estimates
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Remark 2.2
  • Theorem 2.3
  • Corollary 2.4
  • proof
  • Corollary 2.5
  • Theorem 2.6
  • ...and 33 more