Moser-Trudinger inequalities and complex Monge-Ampere equation

TL;DR

This paper extends Moser-Trudinger inequalities to complex geometry and derives new conditions for solutions to the complex Monge-Ampère equation on Kähler manifolds, connecting functional inequalities with complex PDEs.

Contribution

It introduces a novel version of the Moser-Trudinger inequality in complex geometry and establishes a new necessary condition for the existence of Hölder continuous solutions to the complex Monge-Ampère equation.

Findings

New Moser-Trudinger inequality for functions in Sobolev space W^{1,2}

Derived necessary condition for complex Monge-Ampère equation solutions

Connections between functional inequalities and complex geometric PDEs

Abstract

Our aim is to give a version of the Moser-Trudinger inequality in the setting of complex geometry. As a very particular case, our result already gives a new Moser-Trudinger inequality for functions in the Sobolev space $W^{1,2}$ of a domain in $R^2$. We also 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.