(p,q)-Sobolev inequality and Nash inequality on compact Finsler metric measure manifolds

Abstract

In this paper, we carry out in-depth research centering around the $(p,q)$-Sobolev inequality and Nash inequality on compact Finsler metric measure manifolds under the condition that ${\mathrm{Ric}}_{\infty }\ge -K$ for some $K\ge 0$. We first obtain a global p-Poincaré inequality on complete Finsler manifolds. Based on this, we can derive a $(p,q)$-Sobolev inequality. Furthermore, we establish a global optimal $(p,q)$-Sobolev inequality. Finally, as an application of the p-Poincaré inequality, we prove a Nash inequality.