$(p,q)$-Sobolev inequality and Nash inequality on forward complete 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 forward complete 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 such Finsler manifolds. Based on this, we can derive a $(p,q)$-Sobolev inequality. Furthermore, we establish a global optimal $(p,q)$-Sobolev inequality with a sharp Sobolev constant. Finally, as an application of the $p$-Poincaré inequality, we prove a Nash inequality.

any comments and suggestions are warmly welcome. arXiv admin note: text overlap with arXiv:2501.10773