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 ${ℰ}_{\chi }(X,\omega )$ 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 ${ℰ}_{\chi }(X,\omega )$. This is done by establishing a priori estimates on the capacity of sublevel sets of the solutions. Our result extends those of U. Cegrell’s and S. Kolodziej’s and puts them into a unifying frame. It also gives a simple proof of S. T. Yau’s celebrated a priori ${𝒞}^0$-estimate.