$L^p$ Liouville type theorems for harmonic functions on gradient Ricci solitons

In this paper we consider $L^p$ Liouville type theorems for harmonic functions on gradient Ricci solitons. In particular, assume that $(M,g)$ is a gradient shrinking or steady Kähler-Ricci soliton, then we prove that any pluriharmonic function $u$ on $M$ with $\mathrm{\nabla }u\in L^p(M)$ for some $1<p\le 2$ is a constant function.