Moser-Trudinger inequalities and complex Monge-Ampère equation

Our aim is to give a version of the Moser-Trudinger inequality in setting complex geometry. As very particular case, our result already gives new for functions Sobolev space $W^{1,2}$ domain $R^2$. We also deduce necessary condition Monge-Ampere equation given measure on compact Kahler manifold admit Holder continuous solution.