{"paper":{"title":"Uniqueness of weighted Sobolev spaces with weakly differentiable weights","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.FA","authors_text":"Jonas M. T\\\"olle","submitted_at":"2011-10-13T10:37:16Z","abstract_excerpt":"We prove that weakly differentiable weights $w$ which, together with their reciprocals, satisfy certain local integrability conditions, admit a unique associated first-order $p$-Sobolev space, that is \\[H^{1,p}(\\mathbb{R}^d,w\\,\\d x)=V^{1,p}(\\mathbb{R}^d,w\\,\\d x)=W^{1,p}(\\mathbb{R}^d,w\\,\\d x),\\] where $d\\in\\N$ and $p\\in [1,\\infty)$. If $w$ admits a (weak) logarithmic gradient $\\nabla w/w$ which is in $L^q_{\\text{loc}}(w\\,\\d x;\\R^d)$, $q=p/(p-1)$, we propose an alternative definition of the weighted $p$-Sobolev space based on an integration by parts formula involving $\\nabla w/w$. We prove that "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.2888","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"}