The $\mathcal C^{2,\alpha}$ estimate of complex Monge-Ampere equation

TL;DR

This paper proves that solutions to the complex Monge-Ampère equation with certain regularity conditions are actually twice differentiable with Hölder continuous second derivatives, improving understanding of solution regularity.

Contribution

It establishes $ ext{C}^{2,eta}$ regularity for solutions of the complex Monge-Ampère equation under minimal regularity assumptions on the data.

Findings

Solutions with $ ext{C}^{1,1}$ regularity are actually in $ ext{C}^{2,eta}$ for some $eta ext{ in }(0,1)$.

The regularity result applies to equations with Hölder continuous right-hand side.

Provides a key regularity estimate for complex Monge-Ampère equations.

Abstract

We prove that any $\mathcal C^{1,1}$ solution to complex Monge-Amp\`ere equation $det(u_{i\bar{j}})=f$ with $0<f\in\mathcal C^{\alpha}$ is in $\mathcal C^{2,\alpha}$ for $\alpha\in (0,1)$.