$L^\infty$-estimates in optimal transport for non quadratic costs

TL;DR

This paper establishes $L^ Infty$-estimates for optimal transport maps with non-quadratic costs, specifically for homogeneous functions of degree $p \,\geq\, 2$, advancing understanding of stability and regularity in such problems.

Contribution

It provides new $L^ Infty$-bounds for transport maps under non-quadratic costs, extending regularity results beyond the quadratic case.

Findings

$L^ Infty$-estimates for $T x - x$ on balls

Estimates for interpolating maps $T_t$ derived from these bounds

Results applicable to costs with homogeneous functions of degree $p \geq 2$

Abstract

For cost functions $c(x,y)=h(x-y)$ with $h\in C^2$ homogeneous of degree $p\geq 2$, we show $L^\infty$-estimates of $Tx-x$ on balls, where $T$ is an $h$-monotone map. Estimates for the interpolating mappings $T_t=t(T-I)+I$ are deduced from this.