Regularity of the Monge-Amp`ere equation in Besov'\''s spaces

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–Ampère 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.