Interior $W^{2,p}$ estimate for small perturbations to the complex Monge-Ampere equation

TL;DR

This paper proves that small perturbations of the complex Monge-Ampère equation ensure solutions have interior second-order derivatives in L^p, extending real Monge-Ampère regularity results to the complex setting.

Contribution

It establishes interior W^{2,p} estimates for solutions to perturbed complex Monge-Ampère equations, generalizing Caffarelli's real case to complex variables.

Findings

Solutions are in W^{2,p} for small perturbations.

The estimates depend only on dimension and p.

Extension of real Monge-Ampère regularity to complex case.

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 $L^{\infty}(B_1)$ with the following property: for any continuous, plurisubharmonic function $u$ in this neighborhood solving $1-\eps \le MA(u)\le 1+\eps$, one has $u\in W^{2,p}(B_{\frac{1}{2}})$, as long as $\eps>0$ is small enough depending only on $n$ and $p$. This partially generalizes Caffarelli's interior $W^{2,p}$ estimates for real Monge-Ampere to the complex version.