  Let $\mu = e^{-V} \ dx$ be a probability measure and $T = \nabla \Phi$ be the
optimal transportation mapping pushing forward $\mu$ onto a log-concave
compactly supported measure $\nu = e^{-W} \ dx$. In this paper, we introduce a
new approach to the regularity problem for the corresponding Monge--Amp{\`e}re
equation $e^{-V} = \det D^2 \Phi \cdot e^{-W(\nabla \Phi)}$ in the Besov spaces
$W^{\gamma,1}_{loc}$. We prove that $D^2 \Phi \in W^{\gamma,1}_{loc}$ provided
$e^{-V}$ belongs to a proper Besov class and $W$ is convex. In particular, $D^2
\Phi \in L^p_{loc}$ for some $p>1$. Our proof does not rely on the previously
known regularity results.
