### A priori estimates for weak solutions of complex Monge-Ampère equations

Let $X$ be a compact Kähler manifold and $\omega $ be a smooth closed form of bidegree $(1,1)$ which is nonnegative and big. We study the classes ${\mathcal{E}}_{\chi}(X,\omega )$ of $\omega $-plurisubharmonic functions of finite weighted Monge-Ampère energy. When the weight $\chi $ has fast growth at infinity, the corresponding functions are close to be bounded. We show that if a positive Radon measure is suitably dominated by the Monge-Ampère capacity, then it belongs to the range of the Monge-Ampère operator on some class ${\mathcal{E}}_{\chi}(X,\omega )$. This is done by establishing...