Integrability of absolutely continuous measure transformations and applications to optimal transportation

Given two Borel probability measures $\mu$ and $\nu$ on $\mathbf{R}^d$ such that $d\nu/d\mu =g$, we consider certain mappings of the form $T(x)=x+F(x)$ that transform $\mu$ into $\nu$. Our main results give estimates of the form $\int_{\mathbf{R}^d}\Phi_1(|F|)\,d\mu\leq\int_{\mathbf{R}^d}\Phi_2(g)\, d\mu$ for certain functions $\Phi_1$ and $\Phi_2$ under appropriate assumptions on $\mu$. Applications are given to optimal mass transportations in the Monge problem.