$(p, q)$-Sobolev inequality and Nash inequality on forward complete Finsler metric measure manifolds

TL;DR

This paper investigates $(p, q)$-Sobolev and Nash inequalities on forward complete Finsler manifolds with Ricci curvature bounded below, deriving sharp inequalities and establishing their interrelations under geometric conditions.

Contribution

It introduces new global $(p, q)$-Sobolev inequalities with sharp constants and connects them to Nash inequalities on Finsler manifolds with Ricci curvature bounds.

Findings

Established a global $p$-Poincaré inequality.

Derived a $(p, q)$-Sobolev inequality from the Poincaré inequality.

Proved a Nash inequality as an application.

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 ${\rm Ric}_{\infty} \geq -K$ for some $K \geq 0$. We first obtain a global $p$-Poincar\'{e} 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 $p$-Poincar\'{e} inequality, we prove a Nash inequality.