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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1203.3457","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-03-15T19:49:01Z","cross_cats_sorted":[],"title_canon_sha256":"6d1187b4f393a99401b4065666d35eca96c08a901f92addc676434ea558449d6","abstract_canon_sha256":"b44615a0bad6a4f5b3e28a5481b4ae2e20616068e1914dfbaecd3cdc265adb4a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:36:06.482936Z","signature_b64":"n1g1JeW+Eu10vKKee+oyI17lVEsLf7DrXxz47OE+/Jq3TSq5Aoin2h0utSEzrS/6V0cbSW1R4mt0Y7UKZ8LkBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"92cb3deb9b0de534d2b2e77ce94b6da198c3ad09545b9e8d67bc38fc6a3371eb","last_reissued_at":"2026-05-18T03:36:06.482447Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:36:06.482447Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $p>1$. 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