Duc-Viet Vu, Moser-Trudinger Inequalities and Complex Monge-Ampere Equations

I present a version of the Moser-Trudinger inequality in the setting of complex geometry. As a very particular case, the result already gives a new Moser-Trudinger inequality for functions in the Sobolev space W^1,2 of a domain in R². As an application, we deduce a new necessary condition for the complex Monge-Ampere equation for a given measure on a compact Kahler manifold to admit a Holder continuous solution. This is a joint work with Tien-Cuong Dinh and George Marinescu.