Quantitative linearization results for the Monge-Amp\`ere equation

TL;DR

This paper develops a large-scale regularity theory for the Monge-Ampère equation with rough data, showing that measures close to Lebesgue in Wasserstein distance lead to solutions with controlled linearization properties.

Contribution

It introduces a quantitative linearization framework for the Monge-Ampère equation using harmonic approximation and Campanato iteration, extending regularity results to rough data.

Findings

Displacement of optimal coupling is close to the gradient of a Poisson solution.

Harmonic approximation is effective for arbitrary measures in optimal transport.

Regularity results hold uniformly across scales for measures near Lebesgue.

Abstract

This paper is about quantitative linearization results for the Monge-Amp\`ere equation with rough data. We develop a large-scale regularity theory and prove that if a measure $\mu$ is close to the Lebesgue measure in Wasserstein distance at all scales, then the displacement of the macroscopic optimal coupling is quantitatively close at all scales to the gradient of the solution of the corresponding Poisson equation. The main ingredient we use is a harmonic approximation result for the optimal transport plan between arbitrary measures. This is used in a Campanato iteration which transfers the information through the scales.