$C^{2,\alpha}$-estimate for Monge-Ampere equations with H\"older-continuous right hand side

TL;DR

This paper introduces a new proof for the $C^{2,eta}$-estimate of solutions to Monge-Ampère equations, which relies only on the Hölder continuity of the right-hand side and avoids differentiating the equation.

Contribution

The authors provide a novel proof technique for $C^{2,eta}$-estimates that simplifies assumptions and applies to both real and complex Monge-Ampère equations.

Findings

Establishes $C^{2,eta}$-regularity without differentiating the equation

Applicable to both real and complex Monge-Ampère equations

Depends only on the Hölder norm of the right-hand side

Abstract

We present a somewhat new proof to the $C^{2,\alpha}$-aprori estimate for the uniform elliptic Monge-Ampere equations, in both the real and complex settings. Our estimates do not need to differentiate the equation, and only depends on the $C^{\alpha^{\prime}}-$norm of the right hand side of the equation, $0<\alpha<\alpha^{\prime}$.