A Remark On C^{2,\alpha}-Regularity of the Complex Monge-Amp\`ere Equation

TL;DR

This paper establishes the $C^{2,eta}$ regularity of solutions to the complex Monge-Ampère equation under certain boundedness and regularity conditions on the data, resolving a longstanding regularity issue.

Contribution

It proves the $C^{2,eta}$ regularity of solutions assuming an upper bound on the Laplacian, advancing the understanding of regularity in complex Monge-Ampère equations.

Findings

Proves $C^{2,eta}$ regularity under bounded Laplacian assumption.

Resolves the regularity problem posed in previous works.

Connects regularity to bounds on the Laplacian of solutions.

Abstract

We prove the $C^{2,\alpha}$-regularity of the solution $u$ of the equation [\det(u_{\bar{k} j}) = f, \quad f^{1/n} \in C^{\alpha}, \quad f \geq \lambda] under the assumption in upper bound of $\Delta u$. Our result settles down the regularity problem related to the paper \cite{Tian} (also see \cite{Zhang}).