$L^p$-Liouville theorems on complete smooth metric measure spaces

TL;DR

This paper establishes $L^p$-Liouville theorems and Gaussian bounds for the $f$-heat kernel on complete smooth metric measure spaces with lower Bakry-Émery Ricci curvature bounds, generalizing classical results.

Contribution

It introduces new $L^p$-Liouville theorems and heat kernel bounds on metric measure spaces with Bakry-Émery curvature, extending classical Ricci curvature results.

Findings

Derived a Moser's parabolic Harnack inequality for the $f$-heat equation.

Established Gaussian bounds on the $f$-heat kernel.

Proved $L^p$-Liouville theorems under curvature and function bounds.

Abstract

We study some function-theoretic properties on a complete smooth metric measure space $(M,g,e^{-f}dv)$ with Bakry-\'{E}mery Ricci curvature bounded from below. We derive a Moser's parabolic Harnack inequality for the $f$-heat equation, which leads to upper and lower Gaussian bounds on the $f$-heat kernel. We also prove $L^p$-Liouville theorems in terms of the lower bound of Bakry-\'{E}mery Ricci curvature and the bound of function $f$, which generalize the classical Ricci curvature case and the $N$-Bakry-\'{E}mery Ricci curvature case.