Some interior regularity estimates for solutions of complex Monge-Amp\`ere equations on a ball

TL;DR

This paper establishes interior regularity estimates for solutions of the complex Monge-Ampère equation on a ball, extending previous results to less regular data and providing new interior Hölder estimates.

Contribution

The paper generalizes Bedford-Taylor interior regularity estimates to solutions with less regular boundary data in the complex Monge-Ampère equation setting.

Findings

Proves interior ^{1,\u03b1} estimates for solutions with ^{1,} data.

Establishes interior ^{0,} estimates for solutions with ^{0,} data.

Extends classical Bedford-Taylor estimates to broader regularity classes.

Abstract

In this paper, we consider the Dirichlet problem of a complex Monge-Amp\`ere equation on a ball in $\mathbb C^n$. With $\mathcal C^{1,\alpha}$ (resp. $\mathcal C^{0,\alpha}$) data, we prove an interior $\mathcal C^{1,\alpha}$ (resp. $\mathcal C^{0,\alpha}$) estimate for the solution. These estimates are generalized versions of the Bedford-Taylor interior $\mathcal C^{1,1}$ estimate.