  In this paper we study constant scalar curvature equation (CSCK), a nonlinear
fourth order elliptic equation, and its weak solutions on K\"ahler manifolds.
We first define a notion of weak solution of CSCK for an $L^\infty$ K\"ahler
metric. The main result is to show that such a weak solution (with uniform
$L^\infty$ bound) is smooth. As an application, this answers in part a
conjecture of Chen regarding the regularity of $K$-energy minimizers. The new
technical ingredient is a $W^{2, 2}$ regularity result for the Laplacian
equation $\Delta_g u=f$ on K\"ahler manifolds, where the metric has only
$L^\infty$ coefficients. It is well-known that such a $W^{2, 2}$ regularity
($W^{2, p}$ regularity for any $p>1$) fails in general (except for dimension
two) for uniform elliptic equations of the form $a^{ij}\partial^2_{ij}u=f$ for
$a^{ij}\in L^\infty$, without certain smallness assumptions on the local
oscillation of $a^{ij}$. We observe that the K\"ahler condition plays an
essential role to obtain a $W^{2, 2}$ regularity for elliptic equations with
only $L^\infty$ elliptic coefficients on compact manifolds.
