{"paper":{"title":"An Optimal Sobolev Embedding for $L^1$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.FA","authors_text":"Daniel Spector","submitted_at":"2018-06-20T07:30:33Z","abstract_excerpt":"In this paper we establish an optimal Lorentz space estimate for the Riesz potential acting on curl-free vectors: There is a constant $C=C(\\alpha,d)>0$ such that \\[ \\|I_\\alpha F \\|_{L^{d/(d-\\alpha),1}(\\mathbb{R}^d;\\mathbb{R}^d)} \\leq C \\|F\\|_{L^1(\\mathbb{R}^d;\\mathbb{R}^d)} \\] for all fields $F \\in L^1(\\mathbb{R}^d;\\mathbb{R}^d)$ such that $\\operatorname*{curl} F=0$ in the sense of distributions. This is the best possible estimate on this scale of spaces and completes the picture in the regime $p=1$ of the well-established results for $p>1$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.07588","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"}