Some inequalities and gradient estimates for harmonic functions on Finsler measure spaces

In this paper, we study functional and geometric inequalities on complete Finsler measure spaces under the condition that the weighted Ricci curvature ${\rm Ric}_\infty$ has a lower bound. We first obtain some local uniform Poincar\'{e} inequalities and Sobolev inequalities. Further, we give a mean value inequality for nonnegative subsolutions of elliptic equations. Finally, we obtain local and global Harnack inequalities, and then, establish a global gradient estimate for positive harmonic functions on forward complete non-compact Finsler measure spaces. Besides, as a by-product of the mean value inequality, we prove a Liouville type theorem.