{"paper":{"title":"On Sobolev regularity of mass transport and transportation inequalities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Alexander V. Kolesnikov","submitted_at":"2010-07-07T10:48:17Z","abstract_excerpt":"We study Sobolev a priori estimates for the optimal transportation $T = \\nabla \\Phi$ between probability measures $\\mu=e^{-V} \\ dx$ and $\\nu=e^{-W} \\ dx$ on $\\R^d$. Assuming uniform convexity of the potential $W$ we show that $\\int \\| D^2 \\Phi\\|^2_{HS} \\ d\\mu$, where $\\|\\cdot\\|_{HS}$ is the Hilbert-Schmidt norm, is controlled by the Fisher information of $\\mu$. In addition, we prove similar estimate for the $L^p(\\mu)$-norms of $\\|D^2 \\Phi\\|$ and obtain some $L^p$-generalizations of the well-known Caffarelli contraction theorem. We establish a connection of our results with the Talagrand transp"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.1103","kind":"arxiv","version":3},"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"}