{"paper":{"title":"Hardy-Sobolev inequalities for vector fields and canceling linear differential operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Jean Van Schaftingen, Pierre Bousquet","submitted_at":"2013-05-18T13:19:47Z","abstract_excerpt":"Given a homogeneous k-th order differential operator $A (D)$ on $\\mathbb{R}^n$ between two finite dimensional spaces, we establish the Hardy inequality $$\\int_{\\mathbb{R}^n} \\frac{\\lvert D^{k-1}u\\rvert}{\\lvert x \\rvert} \\,\\mathrm{d} x \\leq C \\int_{\\mathbb{R}^n} \\lvert A(D)u\\rvert $$ and the Sobolev inequality $$\\lVert D^{k-n} u\\rVert_{L^{\\infty}(\\mathbb{R}^n)}\\leq C \\int_{\\mathbb{R}^n} \\lvert A(D)u\\rvert $$ when $A(D)$ is elliptic and satisfies a recently introduced cancellation property. We also study the necessity of these two conditions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.4262","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"}