Remarks on a result of Chen-Cheng

In their seminal work (\cite{CC}, \cite{CC2}), Chen and Cheng proved apriori estimates for the constant scalar curvature metrics on compact Kähler manifolds. They also prove $C^{3,\alpha }$-estimate for the potential of the Kähler metrics under boundedness assumption on the scalar curvature and the entropy. The goal of this paper is to replace the uniform boundedness of the scalar curvature to the $L^p$-boundedness of the scalar curvature.