Lack of regularity of the transport density in the monge problem

In this paper, we provide a family of counterexamples to the regularity of the transport density in the classical Monge-Kantorovich problem. We prove that the W^{1,p} regularity of the source and target measures f ^\pm does not imply that the transport density $\sigma$ is W^{1,p} , that the BV regularity of f ^\pm does not imply that $\sigma$ is BV and that f^\pm $\in$ C^\infty does not imply that $\sigma$ is W^{1,p} , for large p.