Pointwise Schauder estimates for optimal transport maps of rough densities

TL;DR

This paper establishes pointwise Schauder estimates for optimal transport potentials when the involved densities are nearly constant in an $L^p$ sense, advancing understanding of regularity under rough density conditions.

Contribution

It provides the first pointwise $C^{2,\alpha}$ regularity estimates for optimal transport potentials with densities close to constant in $L^p$, under minimal regularity assumptions.

Findings

Proves $C^{2,\alpha}$ regularity for transport potentials with rough densities.

Establishes estimates under $L^p$ closeness to constant densities.

Extends regularity theory to less smooth density scenarios.

Abstract

We prove a pointwise $C^{2,\,\alpha}$ estimate for the potential of the optimal transport map in the case that the densities are only close to constant in a certain $L^p$ sense.