Partial $W^{2,p}$ regularity for optimal transport maps

TL;DR

This paper proves that optimal transport potentials are locally in $W^{2,p}$ outside a measure-zero singular set and establishes global $W^{2,p}$ estimates for costs close to quadratic, advancing regularity theory.

Contribution

It introduces new regularity results for optimal transport maps with general costs, including global estimates for costs near quadratic, even in the classical quadratic case.

Findings

Potential functions are in $W^{2,p}_{loc}$ outside measure-zero sets.

Global $W^{2,p}$ estimates hold for costs close to quadratic.

Results extend regularity understanding in optimal transport theory.

Abstract

We prove that, in the optimal transportation problem with general costs and positive continuous densities, the potential function is always of class $W^{2,p}_{loc}$ for any $p \geq 1$ outside of a closed singular set of measure zero. We also establish global $W^{2,p}$ estimates when the cost is a small perturbation of the quadratic cost. The latter result is new even when the cost is exactly the quadratic cost.