The complex Monge-Ampere equation on compact Kaehler manifolds

TL;DR

This paper establishes regularity estimates and existence results for solutions to the complex Monge-Ampère equation on compact Kähler manifolds when the right-hand side has weak regularity, specifically in certain Sobolev spaces.

Contribution

It proves gradient and second-order estimates for solutions with right-hand side in W^{1,p} spaces and demonstrates the existence of classical solutions under these conditions.

Findings

Gradient estimates hold for F in W^{1,p_0} with p_0 > 2n.

Existence of classical solutions in W^{3,p_0} for F in W^{1,p_0}.

Regularity results extend the solvability theory to weaker data.

Abstract

We consider the complex Monge-Amp\`ere equation on a compact K\"ahler manifold $(M, g)$ when the right hand side $F$ has rather weak regularity. In particular we prove that estimate of $\t\phi$ and the gradient estimate hold when $F$ is in $W^{1, p_0}$ for any $p_0>2n$. As an application, we show that there exists a classical solution in $W^{3, p_0}$ for the complex Monge-Amp\`ere equation when $F$ is in $W^{1, p_0}$.