A priori estimates for weak solutions of complex Monge-Amp\`ere equations

Abstract

Let $X$ be a compact K\"ahler manifold and $\om$ a smooth closed form of bidegree $(1,1)$ which is nonnegative and big. We study the classes ${\mathcal E}_{\chi}(X,\om)$ of $\om$-plurisubharmonic functions of finite weighted Monge-Amp\`ere 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-Amp\`ere capacity, then it belongs to the range of the Monge-Amp\`ere operator on some class ${\mathcal E}_{\chi}(X,\om)$. This is done by establishing a priori estimates on the capacity of sublevel sets of the solutions. Our result extends U.Cegrell's and S.Kolodziej's results and puts them into a unifying frame. It also gives a simple proof of S.T.Yau's celebrated a priori ${\mathcal C}^0$-estimate.