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Robinson","submitted_at":"2010-09-26T05:56:06Z","abstract_excerpt":"Let $\\Omega$ be an open subset of $\\Ri^d$ with $0\\in \\Omega$.\n  Further let $H_\\Omega=-\\sum^d_{i,j=1}\\partial_i\\,c_{ij}\\,\\partial_j$ be a second-order partial differential operator with domain $C_c^\\infty(\\Omega)$ where the coefficients $c_{ij}\\in W^{1,\\infty}_{\\rm loc}(\\bar\\Omega)$ are real, $c_{ij}=c_{ji}$ and the coefficient matrix $C=(c_{ij})$ satisfies bounds $0<C(x)\\leq c(|x|) I$ for all $x\\in \\Omega$. If \\[ \\int^\\infty_0ds\\,s^{d/2}\\,e^{-\\lambda\\,\\mu(s)^2}<\\infty \\] for some $\\lambda>0$ where $\\mu(s)=\\int^s_0dt\\,c(t)^{-1/2}$ then we establish that $H_\\Omega$ is $L_1$-unique, i.e.\\ it has"},"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":"1009.5065","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-09-26T05:56:06Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"e808c5f56275af189a0ccf12c7b825e8b1a099087157bb30965b526f3ae25282","abstract_canon_sha256":"7036f31f944add93dae1cd6a9bbf0d9cea6279a02bc71acec3d894aaf1be68af"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:03:33.046953Z","signature_b64":"jwdq4CJhrcpNk7CEn8yW1PvTlEnQliDPjbEp2hPiOltz6QHpBptNTNO98simoGrWkOWVy55KFnxD3J4B5U23BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f59eb0ae06ed0dab72a47e6dd545afcd27b6276325d2aa3c5e20cf85fcb8cba3","last_reissued_at":"2026-05-18T03:03:33.046430Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:03:33.046430Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"$L_1$-uniqueness of degenerate elliptic operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.AP","authors_text":"Adam Sikora, Derek W. Robinson","submitted_at":"2010-09-26T05:56:06Z","abstract_excerpt":"Let $\\Omega$ be an open subset of $\\Ri^d$ with $0\\in \\Omega$.\n  Further let $H_\\Omega=-\\sum^d_{i,j=1}\\partial_i\\,c_{ij}\\,\\partial_j$ be a second-order partial differential operator with domain $C_c^\\infty(\\Omega)$ where the coefficients $c_{ij}\\in W^{1,\\infty}_{\\rm loc}(\\bar\\Omega)$ are real, $c_{ij}=c_{ji}$ and the coefficient matrix $C=(c_{ij})$ satisfies bounds $0<C(x)\\leq c(|x|) I$ for all $x\\in \\Omega$. If \\[ \\int^\\infty_0ds\\,s^{d/2}\\,e^{-\\lambda\\,\\mu(s)^2}<\\infty \\] for some $\\lambda>0$ where $\\mu(s)=\\int^s_0dt\\,c(t)^{-1/2}$ then we establish that $H_\\Omega$ is $L_1$-unique, i.e.\\ it has"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.5065","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":"1009.5065","created_at":"2026-05-18T03:03:33.046501+00:00"},{"alias_kind":"arxiv_version","alias_value":"1009.5065v1","created_at":"2026-05-18T03:03:33.046501+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1009.5065","created_at":"2026-05-18T03:03:33.046501+00:00"},{"alias_kind":"pith_short_12","alias_value":"6WPLBLQG5UG2","created_at":"2026-05-18T12:26:04.259169+00:00"},{"alias_kind":"pith_short_16","alias_value":"6WPLBLQG5UG2W4VE","created_at":"2026-05-18T12:26:04.259169+00:00"},{"alias_kind":"pith_short_8","alias_value":"6WPLBLQG","created_at":"2026-05-18T12:26:04.259169+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/6WPLBLQG5UG2W4VEPZW5KRNPZU","json":"https://pith.science/pith/6WPLBLQG5UG2W4VEPZW5KRNPZU.json","graph_json":"https://pith.science/api/pith-number/6WPLBLQG5UG2W4VEPZW5KRNPZU/graph.json","events_json":"https://pith.science/api/pith-number/6WPLBLQG5UG2W4VEPZW5KRNPZU/events.json","paper":"https://pith.science/paper/6WPLBLQG"},"agent_actions":{"view_html":"https://pith.science/pith/6WPLBLQG5UG2W4VEPZW5KRNPZU","download_json":"https://pith.science/pith/6WPLBLQG5UG2W4VEPZW5KRNPZU.json","view_paper":"https://pith.science/paper/6WPLBLQG","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1009.5065&json=true","fetch_graph":"https://pith.science/api/pith-number/6WPLBLQG5UG2W4VEPZW5KRNPZU/graph.json","fetch_events":"https://pith.science/api/pith-number/6WPLBLQG5UG2W4VEPZW5KRNPZU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6WPLBLQG5UG2W4VEPZW5KRNPZU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6WPLBLQG5UG2W4VEPZW5KRNPZU/action/storage_attestation","attest_author":"https://pith.science/pith/6WPLBLQG5UG2W4VEPZW5KRNPZU/action/author_attestation","sign_citation":"https://pith.science/pith/6WPLBLQG5UG2W4VEPZW5KRNPZU/action/citation_signature","submit_replication":"https://pith.science/pith/6WPLBLQG5UG2W4VEPZW5KRNPZU/action/replication_record"}},"created_at":"2026-05-18T03:03:33.046501+00:00","updated_at":"2026-05-18T03:03:33.046501+00:00"}