Interior H\"older estimate for the linearized complex Monge-Ampere equation

TL;DR

This paper establishes a H"older continuity estimate for solutions to the linearized complex Monge-Ampère equation, extending real Monge-Ampère estimates to the complex setting under small perturbations.

Contribution

It provides a partial generalization of Caffarelli's interior H"older estimate from the real to the complex Monge-Ampère equation for linearized cases.

Findings

Established H"older estimate for solutions to the linearized complex Monge-Ampère equation.

Extended real Monge-Ampère regularity results to the complex case.

Demonstrated dependence of regularity on small perturbations of the Monge-Ampère measure.

Abstract

Let $w_0$ be a bounded, $C^3$, strictly plurisubharmonic function defined on $B_1\subset \mathbb{C}^n$. Then $w_0$ has a neighborhood in $L^{\infty}(B_1)$. Suppose that we have a function $\phi$ in this neighborhood with $1-\epsilon \le MA(u)\le 1+\epsilon$ and there exists a function $u$ solving the linearized complex Monge-Ampere equation: $det(\phi_{k\bar{l}})\phi^{I\bar{j}}u_{I\bar{j}}=0$. Then one has an estimate on $|u|_{C^{\alpha}(B_{\frac{1}{2}})}$ for some $\alpha>0$ depending on $n$, as long as $\epsilon$ is small depending on $n$. This partially generalizes Caffarelli's estimate for linearized real Monge-Ampere equation to the complex version.