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

Xinyue Cheng, Qihui Ni

TL;DR

This work extends Sobolev and Nash-type inequalities to forward complete Finsler metric measure manifolds under a lower bound on the infinite Ricci curvature ${\rm Ric}_{\infty} \ge -K$. The authors first derive a global $p$-Poincaré inequality using volume doubling and covering arguments, then obtain a global $(p,q)$-Sobolev inequality via Xia's linearization technique, followed by a Nash inequality as an application of the Poincaré bound and volume comparison. The results provide explicit dependence of constants on geometric data such as the reversibility $\Lambda$, diameter, total measure, and distortion, thereby enabling large-scale analysis on irreversible Finsler spaces with curvature bounds. Overall, the paper broadens the functional-analytic toolkit available for Finsler geometry, offering global, optimal-type inequalities that parallel classical Riemannian results in a non-reversible setting.

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 $p$-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.

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

TL;DR

This work extends Sobolev and Nash-type inequalities to forward complete Finsler metric measure manifolds under a lower bound on the infinite Ricci curvature . The authors first derive a global -Poincaré inequality using volume doubling and covering arguments, then obtain a global -Sobolev inequality via Xia's linearization technique, followed by a Nash inequality as an application of the Poincaré bound and volume comparison. The results provide explicit dependence of constants on geometric data such as the reversibility , diameter, total measure, and distortion, thereby enabling large-scale analysis on irreversible Finsler spaces with curvature bounds. Overall, the paper broadens the functional-analytic toolkit available for Finsler geometry, offering global, optimal-type inequalities that parallel classical Riemannian results in a non-reversible setting.

Abstract

In this paper, we carry out in-depth research centering around the -Sobolev inequality and Nash inequality on forward complete Finsler metric measure manifolds under the condition that for some . We first obtain a global -Poincaré inequality on such Finsler manifolds. Based on this, we can derive a -Sobolev inequality. Furthermore, we establish a global optimal -Sobolev inequality with a sharp Sobolev constant. Finally, as an application of the -Poincaré inequality, we prove a Nash inequality.

Paper Structure

This paper contains 5 sections, 9 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A global $p$-Poincaré inequality
  4. A $(p, q)$-Sobolev inequality
  5. The Nash inequality

Key Result

Theorem 1.1

Let $(M, F, m)$ be an $n$-dimensional $(n \geq 2)$ forward complete Finsler manifold with finite reversibility $\Lambda$. Assume that ${\rm Ric}_{\infty} \geq -K$ for some $K \geq 0$, $d:= {\rm Diam}(M)< \infty$ and $m_{0}:= m(M)> 0$. Then, for any $\nu > n+1$ and $q\in [1, 2]$, there exists a posit where $\frac{1}{p} = \frac{1}{q}-\frac{1}{\nu}$.

Theorems & Definitions (12)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Theorem 3.4
  • proof
  • Lemma 4.1
  • Theorem 4.2
  • proof
  • ...and 2 more