Local potential and H\"older estimates for the linearized Monge-Amp\`ere equation

TL;DR

This paper develops local potential and H"older estimates for solutions to linearized Monge-Ampère equations with measure data, providing new tools for understanding their regularity and applications to the Abreu equation.

Contribution

It introduces novel local potential and H"older estimates for linearized Monge-Ampère equations with measure right-hand sides, advancing regularity theory.

Findings

Interior H"older estimate for inhomogeneous equations

Applicable to equations with divergence-form right-hand sides

New approach for singular Abreu equation estimates

Abstract

In this paper, we establish local potential estimates and H\"older estimates for solutions of linearized Monge-Amp\`ere equations with the right-hand side being a signed measure, under suitable assumptions on the data. In particular, the interior H\"older estimate holds for an inhomogeneous linearized Monge-Amp\`ere equation with right-hand side being the nonnegative divergence of a bounded vector field in all dimensions. As an application, we give a new approach for the interior estimate of the singular Abreu equation.