Gradient estimate for complex Monge-Ampere equation with continuous right hand side

TL;DR

This paper establishes gradient estimates for solutions to the complex Monge-Ampère equation on compact Kähler manifolds, linking the regularity of the solution's gradient to the continuity of the right-hand side function.

Contribution

It provides new $L^p$ and $L^{ abla}$ gradient estimates for solutions based on the continuity properties of the right-hand side.

Findings

Gradient estimates depend on the continuity of the right-hand side

Established $L^p$ and $L^{ abla}$ bounds for solutions

Results applicable to complex Monge-Ampère equations on Kähler manifolds

Abstract

In this note, we consider complex Monge-Ampere equation posed on a compact K\"ahler manifold. We show how to get $L^p$($p<\infty$) and $L^{\infty}$ estimate for the gradient of the solution in terms of the continuity of the right hand side.