{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:647LT4VHUFLUI7OWM2AQILW353","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":"b7ff70c1e92d9cb6b02062b50d71ecdfb96f1547a270a4581b403e5783a0ffe7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-09-25T09:36:47Z","title_canon_sha256":"960617f41e51b5126099c7673e0af000c8771e6a4866f7e080e3a1772db36398"},"schema_version":"1.0","source":{"id":"1709.08399","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1709.08399","created_at":"2026-05-18T00:34:25Z"},{"alias_kind":"arxiv_version","alias_value":"1709.08399v1","created_at":"2026-05-18T00:34:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.08399","created_at":"2026-05-18T00:34:25Z"},{"alias_kind":"pith_short_12","alias_value":"647LT4VHUFLU","created_at":"2026-05-18T12:31:03Z"},{"alias_kind":"pith_short_16","alias_value":"647LT4VHUFLUI7OW","created_at":"2026-05-18T12:31:03Z"},{"alias_kind":"pith_short_8","alias_value":"647LT4VH","created_at":"2026-05-18T12:31:03Z"}],"graph_snapshots":[{"event_id":"sha256:13e756d92d77e5ebf4ce5d39d3947fb8d21544da26ded93a27fa98249ba9218c","target":"graph","created_at":"2026-05-18T00:34:25Z","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":"The first goal of this paper is to study necessary and sufficient conditions to obtain the attainability of the \\textit{fractional Hardy inequality } $$\\Lambda_{N}\\equiv\\Lambda_{N}(\\Omega):=\\inf_{\\{\\phi\\in \\mathbb{E}^s(\\Omega, D), \\phi\\neq 0\\}} \\dfrac{\\frac{a_{d,s}}{2} \\displaystyle\\int_{\\mathbb{R}^d} \\int_{\\mathbb{R}^d} \\dfrac{|\\phi(x)-\\phi(y)|^2}{|x-y|^{d+2s}}dx dy} {\\displaystyle\\int_\\Omega \\frac{\\phi^2}{|x|^{2s}}\\,dx}, $$ where $\\Omega$ is a bounded domain of $\\mathbb{R}^d$, $0<s<1$, $D\\subset \\mathbb{R}^d\\setminus \\Omega$ a nonempty open set and $$\\mathbb{E}^{s}(\\Omega,D)=\\left\\{ u \\in H^","authors_text":"Abdelrazek Dieb, Ahmed Attar, Boumediene Abdellaoui, Ireneo Peral","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-09-25T09:36:47Z","title":"Attainability of the fractional Hardy constant with nonlocal mixed boundary conditions. Applications"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.08399","kind":"arxiv","version":1},"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:cf1c0a1dd848854731285e1ad3c1d3911a45396ab126e38308eb8767af81629c","target":"record","created_at":"2026-05-18T00:34:25Z","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":"b7ff70c1e92d9cb6b02062b50d71ecdfb96f1547a270a4581b403e5783a0ffe7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-09-25T09:36:47Z","title_canon_sha256":"960617f41e51b5126099c7673e0af000c8771e6a4866f7e080e3a1772db36398"},"schema_version":"1.0","source":{"id":"1709.08399","kind":"arxiv","version":1}},"canonical_sha256":"f73eb9f2a7a157447dd66681042edbeec803ff2adc7afa960bdfcfde8af1168f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f73eb9f2a7a157447dd66681042edbeec803ff2adc7afa960bdfcfde8af1168f","first_computed_at":"2026-05-18T00:34:25.725947Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:34:25.725947Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"VMSr+kEYHSwDN3ReF5UGpzGiXc4bKaP4D0pnnsRk8R+uODGT9tgzfVDQSx7vtOyvyFK/eJg86dxFDkWj5rOpCg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:34:25.726369Z","signed_message":"canonical_sha256_bytes"},"source_id":"1709.08399","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cf1c0a1dd848854731285e1ad3c1d3911a45396ab126e38308eb8767af81629c","sha256:13e756d92d77e5ebf4ce5d39d3947fb8d21544da26ded93a27fa98249ba9218c"],"state_sha256":"6d45ebe478042732cecc7cf5cc1d167b52654fdbb5d20de3443662567769032a"}