H\"older and Sobolev regularity of optimal transportation potentials with rough measures

TL;DR

This paper extends classical optimal transportation regularity results to measures that are not fully absolutely continuous, establishing H"older and Sobolev regularity of Kantorovich potentials under natural assumptions relevant for numerical approximations.

Contribution

It generalizes the regularity theory of optimal transportation to measures comparable to Lebesgue measure on large enough scales, accommodating rough measures used in numerical computations.

Findings

Established H"older regularity of potentials up to a critical scale.

Proved Sobolev regularity results for Kantorovich potentials.

Extended classical theory to measures with less regularity.

Abstract

We consider a Kantorovich potential associated to an optimal transportation problem between measures that are not necessarily absolutely continuous with respect to the Lebesgue measure, but are comparable to the Lebesgue measure when restricted to balls with radius greater than some $\delta>0$. Our main results extend the classical regularity theory of optimal transportation to this framework. In particular, we establish both H\"older and Sobolev regularity results for Kantorovich potentials up to some critical length scale depending on $\delta$. Our assumptions are very natural in the context of the numerical computation of optimal maps, which often involves approximating by sums of Dirac masses some measures that are absolutely continuous with densities bounded away from zero and infinity on their supports.