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