Elliptic Harnack inequality and its applications on Finsler metric measure spaces

TL;DR

This paper establishes elliptic Harnack inequalities and related regularity results for harmonic functions on Finsler metric measure spaces with specific curvature and distortion conditions.

Contribution

It introduces new elliptic p-Harnack inequalities and applications such as Hölder continuity, Liouville theorem, and gradient estimates in Finsler spaces.

Findings

Elliptic p-Harnack inequality for positive harmonic functions.

Hölder continuity and Liouville theorem derived from Harnack inequality.

Gradient estimates for positive harmonic functions.

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 $\tau$ is of linear growth, $|\tau|\leq ar+b$, 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 Poincar\'{e} inequality and a mean value inequality. As applications of the Harnack inequality, we derive the H\"{o}lder continuity estimate and a Liouville theorem for positive harmonic functions. Furthermore, we establish a gradient estimate for positive harmonic functions.