{"paper":{"title":"Comparison between $W_2$ distance and $\\dot{H}^{-1}$ norm, and localisation of Wasserstein distance","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"R\\'emi Peyre","submitted_at":"2011-04-24T13:30:03Z","abstract_excerpt":"It is well known that the quadratic Wasserstein distance $W_2 (\\mathord{\\boldsymbol{\\cdot}}, \\mathord{\\boldsymbol{\\cdot}})$ is formally equivalent, for infinitesimally small perturbations, to some weighted $H^{-1}$ homogeneous Sobolev norm. In this article I show that this equivalence can be integrated to get non-asymptotic comparison results between these distances. Then I give an application of these results to prove that the $W_2$ distance exhibits some localisation phenomenon: if $\\mu$ and $\\nu$ are measures on $\\mathbf{R}^n$ and $\\varphi \\colon \\mathbf{R}^n \\to \\mathbf{R}_+$ is some bump "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.4631","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"}