# L^{p,q} estimates on the transport density

## TL;DR

This paper establishes new regularity results for the transport density in optimal mass transport, showing it belongs to certain Lebesgue spaces when the source and target densities are in those spaces.

## Contribution

The paper proves that the transport density inherits the L^{p,q} regularity from the source and target densities in the Monge-Kantorovich problem.

## Key findings

- Transport density belongs to L^{p,q} when source and target densities are in L^{p,q}.
- Regularity results extend the understanding of optimal transport solutions.
- Provides new tools for analyzing regularity in mass transport problems.

## Abstract

In this paper, we show a new regularity result on the transport density {\sigma} in the classical Monge-Kantorovich optimal mass transport problem between two measures, {\mu} and {\nu}, having some summable densities, f^+ and f^-. More precisely, we prove that the transport density {\sigma} belongs to L^{p,q}({\Omega}) as soon as f^+, f^- \in L^{p,q}({\Omega}).