{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:24464Y6KLC647AU4TF3IFVHMA4","short_pith_number":"pith:24464Y6K","schema_version":"1.0","canonical_sha256":"d739ee63ca58bdcf829c997682d4ec07209cb547a32858489bfe9f50f78a648d","source":{"kind":"arxiv","id":"1701.08929","version":2},"attestation_state":"computed","paper":{"title":"Factorizations and Hardy-Rellich-Type Inequalities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Fritz Gesztesy, Lance Littlejohn","submitted_at":"2017-01-31T06:49:27Z","abstract_excerpt":"The principal aim of this note is to illustrate how factorizations of singular, even-order partial differential operators yield an elementary approach to classical inequalities of Hardy-Rellich-type. More precisly, introducing the two-parameter $n$-dimensional homogeneous scalar differential expressions $T_{\\alpha,\\beta} := - \\Delta + \\alpha |x|^{-2} x \\cdot \\nabla + \\beta |x|^{-2}$, $\\alpha, \\beta \\in \\mathbb{R}$, $x \\in \\mathbb{R}^n \\backslash \\{0\\}$, $n \\in \\mathbb{N}$, $n \\geq 2$, and its formal adjoint, denoted by $T_{\\alpha,\\beta}^+$, we show that nonnegativity of $T_{\\alpha,\\beta}^+ T_{"},"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":"1701.08929","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-01-31T06:49:27Z","cross_cats_sorted":[],"title_canon_sha256":"8c934d936a4a61ce65c6586e1eb7f4ae2aa28f6aa45cc785df418dfcff90f2ab","abstract_canon_sha256":"b598c2527aec0463b4d399745175faac9d2f4d9097c3aa5ecb13a00a5e2df0fa"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:46:18.539745Z","signature_b64":"eswOEGVlarR4ReOpksnQcUnmaG3ipBB7qwrFO8dLaOXh0WSEUGQhQod9EynHctXCBS16BshHSOCH0VY/uI+mDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d739ee63ca58bdcf829c997682d4ec07209cb547a32858489bfe9f50f78a648d","last_reissued_at":"2026-05-18T00:46:18.539269Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:46:18.539269Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Factorizations and Hardy-Rellich-Type Inequalities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Fritz Gesztesy, Lance Littlejohn","submitted_at":"2017-01-31T06:49:27Z","abstract_excerpt":"The principal aim of this note is to illustrate how factorizations of singular, even-order partial differential operators yield an elementary approach to classical inequalities of Hardy-Rellich-type. More precisly, introducing the two-parameter $n$-dimensional homogeneous scalar differential expressions $T_{\\alpha,\\beta} := - \\Delta + \\alpha |x|^{-2} x \\cdot \\nabla + \\beta |x|^{-2}$, $\\alpha, \\beta \\in \\mathbb{R}$, $x \\in \\mathbb{R}^n \\backslash \\{0\\}$, $n \\in \\mathbb{N}$, $n \\geq 2$, and its formal adjoint, denoted by $T_{\\alpha,\\beta}^+$, we show that nonnegativity of $T_{\\alpha,\\beta}^+ T_{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.08929","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":"1701.08929","created_at":"2026-05-18T00:46:18.539349+00:00"},{"alias_kind":"arxiv_version","alias_value":"1701.08929v2","created_at":"2026-05-18T00:46:18.539349+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1701.08929","created_at":"2026-05-18T00:46:18.539349+00:00"},{"alias_kind":"pith_short_12","alias_value":"24464Y6KLC64","created_at":"2026-05-18T12:30:55.937587+00:00"},{"alias_kind":"pith_short_16","alias_value":"24464Y6KLC647AU4","created_at":"2026-05-18T12:30:55.937587+00:00"},{"alias_kind":"pith_short_8","alias_value":"24464Y6K","created_at":"2026-05-18T12:30:55.937587+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/24464Y6KLC647AU4TF3IFVHMA4","json":"https://pith.science/pith/24464Y6KLC647AU4TF3IFVHMA4.json","graph_json":"https://pith.science/api/pith-number/24464Y6KLC647AU4TF3IFVHMA4/graph.json","events_json":"https://pith.science/api/pith-number/24464Y6KLC647AU4TF3IFVHMA4/events.json","paper":"https://pith.science/paper/24464Y6K"},"agent_actions":{"view_html":"https://pith.science/pith/24464Y6KLC647AU4TF3IFVHMA4","download_json":"https://pith.science/pith/24464Y6KLC647AU4TF3IFVHMA4.json","view_paper":"https://pith.science/paper/24464Y6K","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1701.08929&json=true","fetch_graph":"https://pith.science/api/pith-number/24464Y6KLC647AU4TF3IFVHMA4/graph.json","fetch_events":"https://pith.science/api/pith-number/24464Y6KLC647AU4TF3IFVHMA4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/24464Y6KLC647AU4TF3IFVHMA4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/24464Y6KLC647AU4TF3IFVHMA4/action/storage_attestation","attest_author":"https://pith.science/pith/24464Y6KLC647AU4TF3IFVHMA4/action/author_attestation","sign_citation":"https://pith.science/pith/24464Y6KLC647AU4TF3IFVHMA4/action/citation_signature","submit_replication":"https://pith.science/pith/24464Y6KLC647AU4TF3IFVHMA4/action/replication_record"}},"created_at":"2026-05-18T00:46:18.539349+00:00","updated_at":"2026-05-18T00:46:18.539349+00:00"}