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 \(W^{\gamma ,1}_{loc}\). 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.
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This study was carried out within “The National Research University Higher School of Economics” Academic Fund Program in 2012-2013, research Grant No. 11-01-0175. The first author was supported by RFBR projects 10-01-00518, 11-01-90421-Ukr-f-a, and the program SFB 701 at the University of Bielefeld. The second author was partially supported by MTM 2011-27637, 2009 SGR 1303, RFFI 12-01-00169, and NSH-979.2012.1.
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Kolesnikov, A.V., Tikhonov, S.Y. Regularity of the Monge–Ampère equation in Besov’s spaces. Calc. Var. 49, 1187–1197 (2014). https://doi.org/10.1007/s00526-013-0617-5
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DOI: https://doi.org/10.1007/s00526-013-0617-5