On $L^p$-Liouville property for smooth metric measure spaces

Abstract

In this short paper we study $L_f^p$-Liouville property with $0<p<1$ for nonnegative $f$-subharmonic functions on a complete noncompact smooth metric measure space $(M,g,e^{-f}dv)$ with ${\mathrm{Ric}}_f^m$ bounded below for $0<m\le \infty$. We prove a sharp $L_f^p$-Liouville theorem when $0<m<\infty$. We also prove an $L_f^p$-Liouville theorem when ${\mathrm{Ric}}_f\ge 0$ and $|f(x)|\le \delta (n)\ln r(x)$.

Preliminary version, all comments are welcome!