Elliptic Harnack inequality and its applications on Finsler metric measure spaces

TL;DR

The paper extends elliptic Harnack theory to forward complete Finsler metric measure spaces under the curvature-dimension condition ${\rm Ric}_{\infty}\geq -K$ and linear distortion growth $|\tau|\leq a r+b$. It develops a local $L^p$ mean value inequality and Sobolev-type inequalities, then establishes a sharp elliptic $p$-Harnack inequality via Moser iteration. Consequently, it deduces Hölder continuity, Liouville theorems for positive harmonic functions, and a gradient estimate for positive harmonic functions, thereby generalizing classical Riemannian results to nonreversible Finsler settings with nonpositive Ricci curvature. These results provide tools for geometric-analytic analysis on Finsler spaces and enhance understanding of harmonic function behavior in non-Riemannian contexts.

Abstract

In this paper, we study the elliptic Harnack inequality and its applications on forward complete Finsler metric measure spaces under the conditions that the weighted Ricci curvature ${\rm Ric}_{\infty}$ has non-positive lower bound and the distortion $τ$ is of linear growth, $|τ|\leq ar+b$, where $a,b$ are some non-negative constants, $r=d(x_0,x)$ is the distance function for some point $x_{0} \in M$. We obtain an elliptic $p$-Harnack inequality for positive harmonic functions from a local uniform Poincaré inequality and a mean value inequality. As applications of the Harnack inequality, we derive the Hölder continuity estimate and a Liouville theorem for positive harmonic functions. Furthermore, we establish a gradient estimate for positive harmonic functions.

Theorems & Definitions (28)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 3.1
• proof
• Theorem 3.2