{"paper":{"title":"Equivalent and attained version of Hardy's inequality in $\\mathbb{R}^n$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Bernhard Ruf, Cristina Tarsi, Daniele Cassani","submitted_at":"2017-11-10T10:35:19Z","abstract_excerpt":"We investigate connections between Hardy's inequality in the whole space $\\mathbb{R}^n$ and embedding inequalities for Sobolev-Lorentz spaces. In particular, we complete previous results due to [A. Alvino, Sulla diseguaglianza di Sobolev in spazi di Lorentz, (1977)] and [G. Talenti, An inequality between $u^*$ and $|{\\rm{grad}} u^*|$, (1992)] by establishing optimal embedding inequalities for the Sobolev-Lorentz quasinorm $\\|\\nabla\\,\\cdot\\,\\|_{p,q}$ also in the range $p < q<\\infty$, which remained essentially open since the work of Alvino. Attainability of the best embedding constants is also "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.03763","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"}