Some functional inequalities under lower Bakry-\'{E}mery-Ricci curvature bounds with $\varepsilon$-range

TL;DR

This paper extends classical geometric inequalities to weighted Riemannian manifolds with variable lower bounds on Bakry-Émery-Ricci curvature, broadening the scope of curvature conditions under which these inequalities hold.

Contribution

It proves Cheng type and local Sobolev inequalities under generalized lower Bakry-Émery-Ricci curvature bounds with psilon-range, extending results to a wider range of the parameter m.

Findings

Established Cheng type inequality under psilon-range bounds.

Proved local Sobolev inequality with variable curvature bounds.

Generalized inequalities from constant to variable curvature bounds.

Abstract

For $n$-dimensional weighted Riemannian manifolds, lower $m$-Bakry-\'{E}mery-Ricci curvature bounds with $\varepsilon$-range, introduced by Lu-Minguzzi-Ohta, integrate constant lower bounds and certain variable lower bounds in terms of weight functions. In this paper, we prove a Cheng type inequality and a local Sobolev inequality under lower $m$-Bakry-\'{E}mery-Ricci curvature bounds with $\varepsilon$-range. These generalize those inequalities under constant curvature bounds for $m \in (n,\infty)$ to $m\in(-\infty,1]\cup\{\infty\}$.