Regularity of the Monge-Amp\`{e}re equation in Besov's space

Abstract

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.