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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 "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1110.2888","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-10-13T10:37:16Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"431e3c55afcb670b5cc1e72548e982dd5883d91f754184a402bbaa7169ce2c44","abstract_canon_sha256":"5df609aec9c16f8a761b5e17020f0b6f8ce329a1045fdf9866f0d58dbc645cbe"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:44:33.851664Z","signature_b64":"2iz3io6f1lBmkiRq4ie2leaLoW9faLXQR4YYPvp5B3iNfYWjlciTa2IXaqJXfj5Y1uysVBG7LgYtU9vWE3vXAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f18ef02e881705b74fab2058539764c8a22f76245dbea16feb024c90a301e804","last_reissued_at":"2026-05-18T03:44:33.850979Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:44:33.850979Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $\\nabla w/w$. 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