{"paper":{"title":"Nonlocal Hardy type inequalities with optimal constants and remainder terms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jean Van Schaftingen, Vitaly Moroz","submitted_at":"2012-08-31T10:08:52Z","abstract_excerpt":"Using a groundstate transformation, we give a new proof of the optimal Stein-Weiss inequality of Herbst [\\int_{\\R^N} \\int_{\\R^N} \\frac{\\varphi (x)}{\\abs{x}^\\frac{\\alpha}{2}} I_\\alpha (x - y) \\frac{\\varphi (y)}{\\abs{y}^\\frac{\\alpha}{2}}\\dif x \\dif y \\le \\mathcal{C}_{N,\\alpha, 0}\\int_{\\R^N} \\abs{\\varphi}^2,] and of its combinations with the Hardy inequality by Beckner [\\int_{\\R^N} \\int_{\\R^N} \\frac{\\varphi (x)}{\\abs{x}^\\frac{\\alpha + s}{2}} I_\\alpha (x - y) \\frac{\\varphi (y)}{\\abs{y}^\\frac{\\alpha + s}{2}}\\dif x \\dif y \\le \\mathcal{C}_{N, \\alpha, 1} \\int_{\\R^N} \\abs{\\nabla \\varphi}^2,] and with t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.6447","kind":"arxiv","version":1},"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"}