Weighted cscK metric (I): a priori estimates

Let $X$ be a compact Kähler manifold. In this paper we study the existence of constant weighted scalar curvature Kähler (weighted cscK) metrics on $X$. More precisely, we establish a priori $C^k$-estimates ($k\ge 0$) for the Kähler potential associated with these metrics, thereby extending a result due to Chen and Cheng in the classical cscK setting.