Regularity of the Monge-Ampère 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{è}re equation $e^{-V}=\det D^2\Phi \cdot e^{-W(\nabla \Phi )}$ in the Besov spaces $W_{loc}^{\gamma ,1}$. We prove that $D^2\Phi \in W_{loc}^{\gamma ,1}$ provided $e^{-V}$ belongs to a proper Besov class and $W$ is convex. In particular, $D^2\Phi \in L_{loc}^p$ for some $p>1$. Our proof does not rely on the previously known regularity results.

10 pages, minor changes