A sharp inequality for transport maps in W^{1,p}(R) via approximation

Jean Louet, Filippo Santambrogio

For $f$ convex and increasing, we prove the inequality $\int f(|U^{\prime }|)\ge \int f(n{T}^{\prime })$, every time that $U$ is a Sobolev function of one variable and $T$ is the non-decreasing map…