A $\mathcal C^{2,\alpha}$ estimate of the complex Monge-Amp\`ere equation

TL;DR

This paper establishes a $ ext{C}^{2, ext{alpha}}$ regularity estimate for solutions to the complex Monge-Ampère equation, assuming the solution is initially in $ ext{C}^{1,eta}$ and the right-hand side is $ ext{C}^{ ext{alpha}}$.

Contribution

It provides a new $ ext{C}^{2, ext{alpha}}$ estimate for solutions under weaker regularity assumptions on the solution.

Findings

Proves $ ext{C}^{2, ext{alpha}}$ regularity for solutions with $ ext{C}^{1,eta}$ initial regularity.

Establishes estimates depending on the dimension $n$ and the Hölder exponent $ ext{alpha}$.

Extends regularity theory for complex Monge-Ampère equations under less restrictive conditions.

Abstract

In this paper, we prove a $\mathcal C^{2,\alpha}$-estimate for the solution to the complex Monge-Amp\`ere equation $\det(u_{i\bar{j}})=f$ with $0< f\in \mathcal C^{\alpha}$, under the assumption that $u\in \mathcal C^{1,\beta }$ for some $\beta <1$ which depends on $n$ and $\alpha $.