# American Institute of Mathematical Sciences

November  2019, 18(6): 3001-3009. doi: 10.3934/cpaa.2019134

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

 Laboratoire de Mathématiques d'Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay Cedex, France

Received  September 2018 Revised  March 2019 Published  May 2019

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 $f^+,\,f^- \in L^{p,q}(\Omega)$.

Citation: Samer Dweik. $L^{p, q}$ estimates on the transport density. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3001-3009. doi: 10.3934/cpaa.2019134
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