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

TL;DR

This paper establishes a new inequality involving Sobolev functions and transport maps, which can be applied to variational problems in mass transport and volume constraints, enhancing understanding of transport map regularity.

Contribution

It introduces a sharp inequality for transport maps in Sobolev spaces, linking the derivatives of a function and its associated transport map with a pre-image counting function.

Findings

Proves the inequality for convex, increasing functions of derivatives.

Applicable to variational problems in mass transport.

Provides a tool for analyzing transport map regularity.

Abstract

For $f$ convex and increasing, we prove the inequality $ \int f(|U'|) \geq \int f(nT')$, every time that $U$ is a Sobolev function of one variable and $T$ is the non-decreasing map defined on the same interval with the same image measure as $U$, and the function $n(x)$ takes into account the number of pre-images of $U$ at each point. This may be applied to some variational problems in a mass-transport framework or under volume constraints.