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

Abstract

We study some function-theoretic properties on a complete smooth metric measure space $(M,g,e^{-f}dv)$ with Bakry-Émery 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-Émery Ricci curvature and the bound of function $f$, which generalize the classical Ricci curvature case and the $N$-Bakry-Émery Ricci curvature case.

Final version, to appear in Bulletin des sciences mathematiques

References in corpus (1)

• Upper bounds on the first eigenvalue for a diffusion operator via Bakry-Émery Ricci curvature II

Cited by in corpus (3)

• Heat kernel on smooth metric measure spaces and applications
• Elliptic gradient estimates and Liouville theorems for a weighted nonlinear parabolic equation
• The Liouville theorem for $p$-harmonic functions and quasiminimizers with finite energy