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

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  • 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) $.

    Mathematics Subject Classification: 35B65, 46N10, 49N60.


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