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Also consider the Bessel operators $\\Delta_\\lambda:=-\\frac{d^2}{dx^2}-\\frac{2\\lambda}{x} \\frac d{dx}$, and $S_\\lambda:=-\\frac{d^2}{dx^2}+\\frac{\\lambda^2-\\lambda}{x^2}$ on $\\mathbb{R_+}$. The Hardy spaces $H^1_{\\Delta_\\lambda}$ and $H^1_{S_\\lambda}$ associated with $\\Delta_\\lambda$ and $S_\\lambda$ are defined via the Riesz transforms $R_{\\Delta_\\lambda}:=\\partial_x (\\Delta_\\lambda)^{-1/2}$ an"},"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":"1509.00079","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-08-31T21:24:14Z","cross_cats_sorted":[],"title_canon_sha256":"87bba3ce5e6c3d6bb63aa0fe90dbf2fd3f72b4ec5aa4628abc536409b3b337a1","abstract_canon_sha256":"dea389b5dd1d0d7f6cb5da594ad0dcdf6d5a055a027b02017402b6921809615e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:34:05.012779Z","signature_b64":"iufXRwcw6ihxPHWQBAVwvpMq0yDngilsk4OpN650QwsMrbf2HmT+K4tXGxWE07s4buy+RxRXtEz6aoVEFzc+CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"817b2974ea5322592289292df6485c0b88129f1efc34414aff84e580f73ab7bb","last_reissued_at":"2026-05-18T01:34:05.012324Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:34:05.012324Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Factorization for Hardy spaces and characterization for BMO spaces via commutators in the Bessel setting","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Brett D. Wick, Dongyong Yang, Ji Li, Xuan Thinh Duong","submitted_at":"2015-08-31T21:24:14Z","abstract_excerpt":"Fix $\\lambda>0$. Consider the Hardy space $H^1(\\mathbb{R}_+,dm_\\lambda)$ in the sense of Coifman and Weiss, where $\\mathbb{R_+}:=(0,\\infty)$ and $dm_\\lambda:=x^{2\\lambda}dx$ with $dx$ the Lebesgue measure. Also consider the Bessel operators $\\Delta_\\lambda:=-\\frac{d^2}{dx^2}-\\frac{2\\lambda}{x} \\frac d{dx}$, and $S_\\lambda:=-\\frac{d^2}{dx^2}+\\frac{\\lambda^2-\\lambda}{x^2}$ on $\\mathbb{R_+}$. The Hardy spaces $H^1_{\\Delta_\\lambda}$ and $H^1_{S_\\lambda}$ associated with $\\Delta_\\lambda$ and $S_\\lambda$ are defined via the Riesz transforms $R_{\\Delta_\\lambda}:=\\partial_x (\\Delta_\\lambda)^{-1/2}$ an"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.00079","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":"1509.00079","created_at":"2026-05-18T01:34:05.012387+00:00"},{"alias_kind":"arxiv_version","alias_value":"1509.00079v2","created_at":"2026-05-18T01:34:05.012387+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.00079","created_at":"2026-05-18T01:34:05.012387+00:00"},{"alias_kind":"pith_short_12","alias_value":"QF5SS5HKKMRF","created_at":"2026-05-18T12:29:37.295048+00:00"},{"alias_kind":"pith_short_16","alias_value":"QF5SS5HKKMRFSIUJ","created_at":"2026-05-18T12:29:37.295048+00:00"},{"alias_kind":"pith_short_8","alias_value":"QF5SS5HK","created_at":"2026-05-18T12:29:37.295048+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/QF5SS5HKKMRFSIUJFEW7MSC4BO","json":"https://pith.science/pith/QF5SS5HKKMRFSIUJFEW7MSC4BO.json","graph_json":"https://pith.science/api/pith-number/QF5SS5HKKMRFSIUJFEW7MSC4BO/graph.json","events_json":"https://pith.science/api/pith-number/QF5SS5HKKMRFSIUJFEW7MSC4BO/events.json","paper":"https://pith.science/paper/QF5SS5HK"},"agent_actions":{"view_html":"https://pith.science/pith/QF5SS5HKKMRFSIUJFEW7MSC4BO","download_json":"https://pith.science/pith/QF5SS5HKKMRFSIUJFEW7MSC4BO.json","view_paper":"https://pith.science/paper/QF5SS5HK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1509.00079&json=true","fetch_graph":"https://pith.science/api/pith-number/QF5SS5HKKMRFSIUJFEW7MSC4BO/graph.json","fetch_events":"https://pith.science/api/pith-number/QF5SS5HKKMRFSIUJFEW7MSC4BO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/QF5SS5HKKMRFSIUJFEW7MSC4BO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/QF5SS5HKKMRFSIUJFEW7MSC4BO/action/storage_attestation","attest_author":"https://pith.science/pith/QF5SS5HKKMRFSIUJFEW7MSC4BO/action/author_attestation","sign_citation":"https://pith.science/pith/QF5SS5HKKMRFSIUJFEW7MSC4BO/action/citation_signature","submit_replication":"https://pith.science/pith/QF5SS5HKKMRFSIUJFEW7MSC4BO/action/replication_record"}},"created_at":"2026-05-18T01:34:05.012387+00:00","updated_at":"2026-05-18T01:34:05.012387+00:00"}