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We show that the oscillation operator $\\mathcal{O}(R_{\\Delta_{\\lambda},\\ast})$ and variation operator $\\mathcal{V}_{\\rho}(R_{\\Delta_{\\lambda},\\ast})$ of the Riesz transform $R_{\\Delta_{\\lambda}}$ associated with\n  $\\Delta_\\lambda$ are both bounded on $L^p(\\mathbb R_+, dm_{\\lambda})$ for $p\\in(1,\\,\\infty)$, from $L^1(\\mathbb{R}_{+},dm_{\\lambda})$ to $L^{1,\\,\\infty}(\\mathbb{R}_{+},dm_{\\lambda})$, and from $L^{\\infty}(\\mathbb{R}_{+},dm_{\\lambda})$ to $BMO("},"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":"1605.01251","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-05-04T12:44:52Z","cross_cats_sorted":[],"title_canon_sha256":"b6ca13c1e548a1cf307bf29bc2ac3dce0e09b20104a1421791040f0d36ff16dc","abstract_canon_sha256":"539c3f11ef3148590f9fac3ba8cf8480ce7b108ae7b597cf01218c11b41a9a51"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:15:37.358934Z","signature_b64":"Ff3Fu7C4alGZaNZ+GWcsml5foQ6IKeiOwPVcmmTZ4qqENmvg+uDslH8poul5VTBU6Q6ICiw5wWUI76srOy/9Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8dc9ed1dc681295fb35a360ed276363f968b7b4497f1b3c86cbf515345043ec7","last_reissued_at":"2026-05-18T01:15:37.358165Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:15:37.358165Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Oscillation and variation for Riesz transform associated with Bessel operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Dongyong Yang, Huoxiong Wu, Jing Zhang","submitted_at":"2016-05-04T12:44:52Z","abstract_excerpt":"Let $\\lambda>0$ and $\\triangle_\\lambda:=-\\frac{d^2}{dx^2}-\\frac{2\\lambda}{x} \\frac d{dx}$ be the Bessel operator on $\\mathbb R_+:=(0,\\infty)$. We show that the oscillation operator $\\mathcal{O}(R_{\\Delta_{\\lambda},\\ast})$ and variation operator $\\mathcal{V}_{\\rho}(R_{\\Delta_{\\lambda},\\ast})$ of the Riesz transform $R_{\\Delta_{\\lambda}}$ associated with\n  $\\Delta_\\lambda$ are both bounded on $L^p(\\mathbb R_+, dm_{\\lambda})$ for $p\\in(1,\\,\\infty)$, from $L^1(\\mathbb{R}_{+},dm_{\\lambda})$ to $L^{1,\\,\\infty}(\\mathbb{R}_{+},dm_{\\lambda})$, and from $L^{\\infty}(\\mathbb{R}_{+},dm_{\\lambda})$ to $BMO("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.01251","kind":"arxiv","version":1},"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":"1605.01251","created_at":"2026-05-18T01:15:37.358289+00:00"},{"alias_kind":"arxiv_version","alias_value":"1605.01251v1","created_at":"2026-05-18T01:15:37.358289+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.01251","created_at":"2026-05-18T01:15:37.358289+00:00"},{"alias_kind":"pith_short_12","alias_value":"RXE62HOGQEUV","created_at":"2026-05-18T12:30:41.710351+00:00"},{"alias_kind":"pith_short_16","alias_value":"RXE62HOGQEUV7M22","created_at":"2026-05-18T12:30:41.710351+00:00"},{"alias_kind":"pith_short_8","alias_value":"RXE62HOG","created_at":"2026-05-18T12:30:41.710351+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/RXE62HOGQEUV7M22GYHNE5RWH6","json":"https://pith.science/pith/RXE62HOGQEUV7M22GYHNE5RWH6.json","graph_json":"https://pith.science/api/pith-number/RXE62HOGQEUV7M22GYHNE5RWH6/graph.json","events_json":"https://pith.science/api/pith-number/RXE62HOGQEUV7M22GYHNE5RWH6/events.json","paper":"https://pith.science/paper/RXE62HOG"},"agent_actions":{"view_html":"https://pith.science/pith/RXE62HOGQEUV7M22GYHNE5RWH6","download_json":"https://pith.science/pith/RXE62HOGQEUV7M22GYHNE5RWH6.json","view_paper":"https://pith.science/paper/RXE62HOG","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1605.01251&json=true","fetch_graph":"https://pith.science/api/pith-number/RXE62HOGQEUV7M22GYHNE5RWH6/graph.json","fetch_events":"https://pith.science/api/pith-number/RXE62HOGQEUV7M22GYHNE5RWH6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RXE62HOGQEUV7M22GYHNE5RWH6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RXE62HOGQEUV7M22GYHNE5RWH6/action/storage_attestation","attest_author":"https://pith.science/pith/RXE62HOGQEUV7M22GYHNE5RWH6/action/author_attestation","sign_citation":"https://pith.science/pith/RXE62HOGQEUV7M22GYHNE5RWH6/action/citation_signature","submit_replication":"https://pith.science/pith/RXE62HOGQEUV7M22GYHNE5RWH6/action/replication_record"}},"created_at":"2026-05-18T01:15:37.358289+00:00","updated_at":"2026-05-18T01:15:37.358289+00:00"}