{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:G3FX3LBOYOBFA7DXC3TZIHNIOG","short_pith_number":"pith:G3FX3LBO","schema_version":"1.0","canonical_sha256":"36cb7dac2ec382507c7716e7941da871bea69fef081858d28ca4465cecc27a0a","source":{"kind":"arxiv","id":"1410.1913","version":2},"attestation_state":"computed","paper":{"title":"On the Hardy-Schr\\\"odinger operator with a boundary singularity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Fr\\'ed\\'eric Robert, Nassif Ghoussoub","submitted_at":"2014-10-07T21:08:58Z","abstract_excerpt":"We investigate the Hardy-Schr\\\"odinger operator $L_\\gamma=-\\Delta -\\frac{\\gamma}{|x|^2}$ on domains $\\Omega\\subset\\rn$, whose boundary contain the singularity $0$. The situation is quite different from the well-studied case when $0$ is in the interior of $\\Omega$. For one, if $0\\in\\Omega$, then $L_\\gamma$ is positive if and only if $\\gamma<\\frac{(n-2)^2}{4}$, while if $0\\in\\partial\\Omega$ the operator $L_{\\gamma}$ could be positive for larger value of $\\gamma$, potentially reaching the maximal constant $\\frac{n^2}{4}$ on convex domains.\n  We prove optimal regularity and a Hopf-type Lemma for v"},"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":"1410.1913","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-10-07T21:08:58Z","cross_cats_sorted":[],"title_canon_sha256":"712edc9cb90a989b9b6e10dc930408b45bd825cf0e7fb1a5937dcb6da111797a","abstract_canon_sha256":"1b9ef2cf621f8bdaa307bb993ae2e4b915c77316b202a52422f7dfc66a361a97"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:22:30.078643Z","signature_b64":"Z3XG/spMikMN5S2jjDg6Vn/qYvW9V3PMGW3Y9HuZzxtNT0cC53qJrNUf3M16eayLJIohAfeqcT+Z5u6YXkhKCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"36cb7dac2ec382507c7716e7941da871bea69fef081858d28ca4465cecc27a0a","last_reissued_at":"2026-05-18T00:22:30.077946Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:22:30.077946Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Hardy-Schr\\\"odinger operator with a boundary singularity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Fr\\'ed\\'eric Robert, Nassif Ghoussoub","submitted_at":"2014-10-07T21:08:58Z","abstract_excerpt":"We investigate the Hardy-Schr\\\"odinger operator $L_\\gamma=-\\Delta -\\frac{\\gamma}{|x|^2}$ on domains $\\Omega\\subset\\rn$, whose boundary contain the singularity $0$. The situation is quite different from the well-studied case when $0$ is in the interior of $\\Omega$. For one, if $0\\in\\Omega$, then $L_\\gamma$ is positive if and only if $\\gamma<\\frac{(n-2)^2}{4}$, while if $0\\in\\partial\\Omega$ the operator $L_{\\gamma}$ could be positive for larger value of $\\gamma$, potentially reaching the maximal constant $\\frac{n^2}{4}$ on convex domains.\n  We prove optimal regularity and a Hopf-type Lemma for v"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.1913","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1410.1913","created_at":"2026-05-18T00:22:30.078046+00:00"},{"alias_kind":"arxiv_version","alias_value":"1410.1913v2","created_at":"2026-05-18T00:22:30.078046+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.1913","created_at":"2026-05-18T00:22:30.078046+00:00"},{"alias_kind":"pith_short_12","alias_value":"G3FX3LBOYOBF","created_at":"2026-05-18T12:28:28.263976+00:00"},{"alias_kind":"pith_short_16","alias_value":"G3FX3LBOYOBFA7DX","created_at":"2026-05-18T12:28:28.263976+00:00"},{"alias_kind":"pith_short_8","alias_value":"G3FX3LBO","created_at":"2026-05-18T12:28:28.263976+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/G3FX3LBOYOBFA7DXC3TZIHNIOG","json":"https://pith.science/pith/G3FX3LBOYOBFA7DXC3TZIHNIOG.json","graph_json":"https://pith.science/api/pith-number/G3FX3LBOYOBFA7DXC3TZIHNIOG/graph.json","events_json":"https://pith.science/api/pith-number/G3FX3LBOYOBFA7DXC3TZIHNIOG/events.json","paper":"https://pith.science/paper/G3FX3LBO"},"agent_actions":{"view_html":"https://pith.science/pith/G3FX3LBOYOBFA7DXC3TZIHNIOG","download_json":"https://pith.science/pith/G3FX3LBOYOBFA7DXC3TZIHNIOG.json","view_paper":"https://pith.science/paper/G3FX3LBO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1410.1913&json=true","fetch_graph":"https://pith.science/api/pith-number/G3FX3LBOYOBFA7DXC3TZIHNIOG/graph.json","fetch_events":"https://pith.science/api/pith-number/G3FX3LBOYOBFA7DXC3TZIHNIOG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/G3FX3LBOYOBFA7DXC3TZIHNIOG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/G3FX3LBOYOBFA7DXC3TZIHNIOG/action/storage_attestation","attest_author":"https://pith.science/pith/G3FX3LBOYOBFA7DXC3TZIHNIOG/action/author_attestation","sign_citation":"https://pith.science/pith/G3FX3LBOYOBFA7DXC3TZIHNIOG/action/citation_signature","submit_replication":"https://pith.science/pith/G3FX3LBOYOBFA7DXC3TZIHNIOG/action/replication_record"}},"created_at":"2026-05-18T00:22:30.078046+00:00","updated_at":"2026-05-18T00:22:30.078046+00:00"}