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Given $\\Omega\\subset\\mathbb{R}^n$, let $\\rho$ be a quasi-metric on $\\Omega$, and let $Q$ be an $n\\times n$ semi-definite matrix function defined on $\\Omega$. For an open set $\\Theta\\Subset\\Omega$, we give sufficient conditions to show that if the local weak Sobolev inequality %\n\\[ \\Big(\\fint_B\n  |f|^{p\\sigma}dx\\Big)^\\frac{1}{p\\sigma} \\leq C\\Big[ r(B)\\fint_B\n  |\\sqrt{Q}\\nabla f|^pdx + \\fint_B\n  |f|^pdx\\Big]^\\frac{1}{p} \\]\nholds for some"},"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":"1801.09610","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-01-29T16:42:10Z","cross_cats_sorted":[],"title_canon_sha256":"5a9a0b985bf195fd49b44fa696876a94c6813b75e2e0c84221bd2cbc95f11452","abstract_canon_sha256":"13c666090734d270aebd24c3e76cea60ebfbe7f89217f27dbd5a29dac92c42f4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:24:56.925808Z","signature_b64":"OGJC66U2jeH7Q7UzD3SGJF+1e1XkTL81znlshGhC6jMK+i456UOEPK7xUh/i2FijQDdSOyUjthyxhhGrIE6YAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"261edfe6b67416a09635ed0dd2ccf34bd8a0824bc810eb079601b546eccaefa9","last_reissued_at":"2026-05-18T00:24:56.925228Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:24:56.925228Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Global Sobolev inequalities and Degenerate P-Laplacian equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"David Cruz-Uribe, Emily Rosta, Scott Rodney","submitted_at":"2018-01-29T16:42:10Z","abstract_excerpt":"We prove that a local, weak Sobolev inequality implies a global Sobolev estimate using existence and regularity results for a family of $p$-Laplacian equations. Given $\\Omega\\subset\\mathbb{R}^n$, let $\\rho$ be a quasi-metric on $\\Omega$, and let $Q$ be an $n\\times n$ semi-definite matrix function defined on $\\Omega$. For an open set $\\Theta\\Subset\\Omega$, we give sufficient conditions to show that if the local weak Sobolev inequality %\n\\[ \\Big(\\fint_B\n  |f|^{p\\sigma}dx\\Big)^\\frac{1}{p\\sigma} \\leq C\\Big[ r(B)\\fint_B\n  |\\sqrt{Q}\\nabla f|^pdx + \\fint_B\n  |f|^pdx\\Big]^\\frac{1}{p} \\]\nholds for some"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.09610","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1801.09610","created_at":"2026-05-18T00:24:56.925333+00:00"},{"alias_kind":"arxiv_version","alias_value":"1801.09610v1","created_at":"2026-05-18T00:24:56.925333+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1801.09610","created_at":"2026-05-18T00:24:56.925333+00:00"},{"alias_kind":"pith_short_12","alias_value":"EYPN7ZVWOQLK","created_at":"2026-05-18T12:32:22.470017+00:00"},{"alias_kind":"pith_short_16","alias_value":"EYPN7ZVWOQLKBFRV","created_at":"2026-05-18T12:32:22.470017+00:00"},{"alias_kind":"pith_short_8","alias_value":"EYPN7ZVW","created_at":"2026-05-18T12:32:22.470017+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/EYPN7ZVWOQLKBFRV5UG5FTHTJP","json":"https://pith.science/pith/EYPN7ZVWOQLKBFRV5UG5FTHTJP.json","graph_json":"https://pith.science/api/pith-number/EYPN7ZVWOQLKBFRV5UG5FTHTJP/graph.json","events_json":"https://pith.science/api/pith-number/EYPN7ZVWOQLKBFRV5UG5FTHTJP/events.json","paper":"https://pith.science/paper/EYPN7ZVW"},"agent_actions":{"view_html":"https://pith.science/pith/EYPN7ZVWOQLKBFRV5UG5FTHTJP","download_json":"https://pith.science/pith/EYPN7ZVWOQLKBFRV5UG5FTHTJP.json","view_paper":"https://pith.science/paper/EYPN7ZVW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1801.09610&json=true","fetch_graph":"https://pith.science/api/pith-number/EYPN7ZVWOQLKBFRV5UG5FTHTJP/graph.json","fetch_events":"https://pith.science/api/pith-number/EYPN7ZVWOQLKBFRV5UG5FTHTJP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/EYPN7ZVWOQLKBFRV5UG5FTHTJP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/EYPN7ZVWOQLKBFRV5UG5FTHTJP/action/storage_attestation","attest_author":"https://pith.science/pith/EYPN7ZVWOQLKBFRV5UG5FTHTJP/action/author_attestation","sign_citation":"https://pith.science/pith/EYPN7ZVWOQLKBFRV5UG5FTHTJP/action/citation_signature","submit_replication":"https://pith.science/pith/EYPN7ZVWOQLKBFRV5UG5FTHTJP/action/replication_record"}},"created_at":"2026-05-18T00:24:56.925333+00:00","updated_at":"2026-05-18T00:24:56.925333+00:00"}