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

Abstract

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