A local regularity for the complex Monge-Amp\`ere equation

TL;DR

This paper establishes a local regularity result for solutions to the complex Monge-Ampère equation, showing that under certain integrability conditions, solutions are regular, with optimal assumptions on the Sobolev norm.

Contribution

It provides the first optimal regularity result for plurisubharmonic solutions assuming a $W^{2,p}$-norm bound, extending previous work with new methods.

Findings

Proves local regularity under $W^{2,p}$ control for $p > n(n-1)$

Establishes an a priori estimate for solutions of the complex Monge-Ampère equation

Demonstrates the optimality of the integrability condition

Abstract

We prove a local regularity (and a corresponding a priori estmate) for plurisubharmonic solutions of the nondegenerate complex Monge-Amp\'ere equation assuming that their $W^{2,p}$-norm is under control for some $p>n(n-1)$. This condition is optimal. We use in particular some methods developed by Trudinger and an $L^q$-estimate for the complex Monge-Amp\'ere equation due to Ko{\l}odziej.