Summability estimates on the transport density in the Import-Export transport problemwith Riemannian cost

In this paper, we consider a mass transportation problem with transport cost given by a Riemannian metric in a bounded domain $ \Omega $, where a mass $ f^+ $ is sent to a location $ f^- $ in $ \Omega $ with the possibility of importing or exporting masses from or to the boundary $ \partial\Omega $. First, we study the $ L^p $ summability of the transport density $ \sigma $ in the Monge-Kantorovich problem with Riemannian cost between two diffuse measures $ f^+ $ and $ f^- $. Using some technical geometrical estimates on the transport rays, we will show that $ \sigma $ belongs to $ L^p(\Omega) $ as soon as the source measure $ f^+ $ and the target one $ f^- $ are both in $ L^p(\Omega) $, for all $ p \in [1, \infty] $. Moreover, we will prove that the transport density between a diffuse measure $ f^+ $ and its Riemannian projection onto the boundary (so, the target measure is singular) is in $ L^p(\Omega) $ provided that $ f^+ \in L^p(\Omega) $ and $ \Omega $ satisfies a uniform exterior ball condition. Finally, we will extend the $ L^p $ estimates on the transport density $ \sigma $ to the case of a transport problem with import-export taxes.