Elliptic Harnack inequality and its applications on Finsler metric measure spaces

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.