Partial regularity for optimal transport with p-cost away from fixed points

TL;DR

This paper proves partial regularity of optimal transport maps with p-costs for H"older marginals, showing that the transport map is $C^{1,eta}$ smooth away from fixed points.

Contribution

It establishes a $C^{1,eta}$ partial regularity result for optimal transport maps with p-costs, extending regularity theory to non-quadratic costs.

Findings

Regularity holds away from fixed points where $T(x)=x$

Optimal transport maps are $C^{1,eta}$ on the set where $T(x) eq x$

Results apply to H"older continuous marginals

Abstract

We consider maps $T$ solving the optimal transport problem with a cost $c(x-y)$ modeled on the $p$-cost. For H\"older continuous marginals, we prove a $C^{1,\alpha}$-partial regularity result for $T $in the set $\{|T(x)-x|>0\}$.