{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:G3FX3LBOYOBFA7DXC3TZIHNIOG","short_pith_number":"pith:G3FX3LBO","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"},"canonical_sha256":"36cb7dac2ec382507c7716e7941da871bea69fef081858d28ca4465cecc27a0a","source":{"kind":"arxiv","id":"1410.1913","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1410.1913","created_at":"2026-05-18T00:22:30Z"},{"alias_kind":"arxiv_version","alias_value":"1410.1913v2","created_at":"2026-05-18T00:22:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.1913","created_at":"2026-05-18T00:22:30Z"},{"alias_kind":"pith_short_12","alias_value":"G3FX3LBOYOBF","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_16","alias_value":"G3FX3LBOYOBFA7DX","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_8","alias_value":"G3FX3LBO","created_at":"2026-05-18T12:28:28Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:G3FX3LBOYOBFA7DXC3TZIHNIOG","target":"record","payload":{"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"},"canonical_sha256":"36cb7dac2ec382507c7716e7941da871bea69fef081858d28ca4465cecc27a0a","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"},"source_kind":"arxiv","source_id":"1410.1913","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:22:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4zG+KsxEFxbjmcvROnY/eimGQ9p1w2bPOHbuxZIAOPoUH/rJNHFOQzJVN5nDu2EPzhtDNQXvKe1dpymvqmprAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T21:01:14.512970Z"},"content_sha256":"31dff810ac05d6d2b2d1acfc8e2b44abc1d03876b75d726a38aefb8dca2cd03c","schema_version":"1.0","event_id":"sha256:31dff810ac05d6d2b2d1acfc8e2b44abc1d03876b75d726a38aefb8dca2cd03c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:G3FX3LBOYOBFA7DXC3TZIHNIOG","target":"graph","payload":{"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:22:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9T2SRrVbg7YCIHpaVykAZ8VWGALkqJlYbV77uY/DZ+yJ/+phjJQqcPvtvA2PYmP/b71n3u/0R5hyS7vN+we+Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T21:01:14.513328Z"},"content_sha256":"35e9f3c0b9254f8321d3d608a274b45b8fcaa02ed31d92e1cceb5ae5ad12cfb2","schema_version":"1.0","event_id":"sha256:35e9f3c0b9254f8321d3d608a274b45b8fcaa02ed31d92e1cceb5ae5ad12cfb2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/G3FX3LBOYOBFA7DXC3TZIHNIOG/bundle.json","state_url":"https://pith.science/pith/G3FX3LBOYOBFA7DXC3TZIHNIOG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/G3FX3LBOYOBFA7DXC3TZIHNIOG/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-21T21:01:14Z","links":{"resolver":"https://pith.science/pith/G3FX3LBOYOBFA7DXC3TZIHNIOG","bundle":"https://pith.science/pith/G3FX3LBOYOBFA7DXC3TZIHNIOG/bundle.json","state":"https://pith.science/pith/G3FX3LBOYOBFA7DXC3TZIHNIOG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/G3FX3LBOYOBFA7DXC3TZIHNIOG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:G3FX3LBOYOBFA7DXC3TZIHNIOG","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"1b9ef2cf621f8bdaa307bb993ae2e4b915c77316b202a52422f7dfc66a361a97","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-10-07T21:08:58Z","title_canon_sha256":"712edc9cb90a989b9b6e10dc930408b45bd825cf0e7fb1a5937dcb6da111797a"},"schema_version":"1.0","source":{"id":"1410.1913","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1410.1913","created_at":"2026-05-18T00:22:30Z"},{"alias_kind":"arxiv_version","alias_value":"1410.1913v2","created_at":"2026-05-18T00:22:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.1913","created_at":"2026-05-18T00:22:30Z"},{"alias_kind":"pith_short_12","alias_value":"G3FX3LBOYOBF","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_16","alias_value":"G3FX3LBOYOBFA7DX","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_8","alias_value":"G3FX3LBO","created_at":"2026-05-18T12:28:28Z"}],"graph_snapshots":[{"event_id":"sha256:35e9f3c0b9254f8321d3d608a274b45b8fcaa02ed31d92e1cceb5ae5ad12cfb2","target":"graph","created_at":"2026-05-18T00:22:30Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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","authors_text":"Fr\\'ed\\'eric Robert, Nassif Ghoussoub","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-10-07T21:08:58Z","title":"On the Hardy-Schr\\\"odinger operator with a boundary singularity"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.1913","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:31dff810ac05d6d2b2d1acfc8e2b44abc1d03876b75d726a38aefb8dca2cd03c","target":"record","created_at":"2026-05-18T00:22:30Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"1b9ef2cf621f8bdaa307bb993ae2e4b915c77316b202a52422f7dfc66a361a97","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-10-07T21:08:58Z","title_canon_sha256":"712edc9cb90a989b9b6e10dc930408b45bd825cf0e7fb1a5937dcb6da111797a"},"schema_version":"1.0","source":{"id":"1410.1913","kind":"arxiv","version":2}},"canonical_sha256":"36cb7dac2ec382507c7716e7941da871bea69fef081858d28ca4465cecc27a0a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"36cb7dac2ec382507c7716e7941da871bea69fef081858d28ca4465cecc27a0a","first_computed_at":"2026-05-18T00:22:30.077946Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:22:30.077946Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Z3XG/spMikMN5S2jjDg6Vn/qYvW9V3PMGW3Y9HuZzxtNT0cC53qJrNUf3M16eayLJIohAfeqcT+Z5u6YXkhKCg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:22:30.078643Z","signed_message":"canonical_sha256_bytes"},"source_id":"1410.1913","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:31dff810ac05d6d2b2d1acfc8e2b44abc1d03876b75d726a38aefb8dca2cd03c","sha256:35e9f3c0b9254f8321d3d608a274b45b8fcaa02ed31d92e1cceb5ae5ad12cfb2"],"state_sha256":"c15d74ef98b27f1d76fd33cddde48c71b901347fa7d18a61b37e4047e1d47440"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"e4HyB/YRyC6IDIrlOLCCt9S9pZPGFD880gRGp/b/tPjSmuu6SNskvh733Mbqd09OuaLY0Ikp9KNwNvLuHZNLDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-21T21:01:14.515268Z","bundle_sha256":"172539195d49db4959bae1caf1016d2a7732756d14e00f7a77fc609776c8fa1e"}}