{"paper":{"title":"Regularity of the Monge-Amp\\`{e}re equation in Besov's space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Alexander V. Kolesnikov, Sergey Yu. Tikhonov","submitted_at":"2012-03-15T19:49:01Z","abstract_excerpt":"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 no"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.3457","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}