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Here $T_n^1$ is $T_n$ acting on the first variable, $T_m^2$ is $T_m$ acting on the second variable, $A_p$ stands for the bi-parameter weights of $\\mathbb{R}^n \\times \\mathbb{R}^m$ and $\\operatorname{BMO}_{\\textrm{prod}}(\\nu)$ is a weighte"},"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":"1810.09303","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-10-22T14:10:56Z","cross_cats_sorted":[],"title_canon_sha256":"b35271967425543516f3e9393e048abdf7bb75db3dfbc05f50ce300173825ff4","abstract_canon_sha256":"c6d37cfc830475c18ef6ace5f2806c2d53e270d68b01d39f4d751dd5d1c9dc8b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:49:03.188977Z","signature_b64":"c/caw6gUgqgab3PKWIZoFJKHaV1XMEneviuyoBF1GKgji/AY1K0I2h49MLEY5nJvRHiexOQ6hKiw5ikAME/sCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"51b93c088a682e25b47ff7b872aee8a95e16dcb269329bf646a0d1179c3bd3f1","last_reissued_at":"2026-05-17T23:49:03.188587Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:49:03.188587Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Bloom type upper bounds in the product BMO setting","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Emil Vuorinen, Henri Martikainen, Kangwei Li","submitted_at":"2018-10-22T14:10:56Z","abstract_excerpt":"For a bounded singular integral $T_n$ in $\\mathbb{R}^n$ and a bounded singular integral $T_m$ in $\\mathbb{R}^m$ we prove that $$ \\| [T_n^1, [b, T_m^2]] \\|_{L^p(\\mu) \\to L^p(\\lambda)} \\lesssim_{[\\mu]_{A_p}, [\\lambda]_{A_p}} \\|b\\|_{\\operatorname{BMO}_{\\textrm{prod}}(\\nu)}, $$ where $p \\in (1,\\infty)$, $\\mu, \\lambda \\in A_p$ and $\\nu := \\mu^{1/p}\\lambda^{-1/p}$. Here $T_n^1$ is $T_n$ acting on the first variable, $T_m^2$ is $T_m$ acting on the second variable, $A_p$ stands for the bi-parameter weights of $\\mathbb{R}^n \\times \\mathbb{R}^m$ and $\\operatorname{BMO}_{\\textrm{prod}}(\\nu)$ is a weighte"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.09303","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":"1810.09303","created_at":"2026-05-17T23:49:03.188636+00:00"},{"alias_kind":"arxiv_version","alias_value":"1810.09303v2","created_at":"2026-05-17T23:49:03.188636+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1810.09303","created_at":"2026-05-17T23:49:03.188636+00:00"},{"alias_kind":"pith_short_12","alias_value":"KG4TYCEKNAXC","created_at":"2026-05-18T12:32:33.847187+00:00"},{"alias_kind":"pith_short_16","alias_value":"KG4TYCEKNAXCLND7","created_at":"2026-05-18T12:32:33.847187+00:00"},{"alias_kind":"pith_short_8","alias_value":"KG4TYCEK","created_at":"2026-05-18T12:32:33.847187+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/KG4TYCEKNAXCLND7664HFLXIVF","json":"https://pith.science/pith/KG4TYCEKNAXCLND7664HFLXIVF.json","graph_json":"https://pith.science/api/pith-number/KG4TYCEKNAXCLND7664HFLXIVF/graph.json","events_json":"https://pith.science/api/pith-number/KG4TYCEKNAXCLND7664HFLXIVF/events.json","paper":"https://pith.science/paper/KG4TYCEK"},"agent_actions":{"view_html":"https://pith.science/pith/KG4TYCEKNAXCLND7664HFLXIVF","download_json":"https://pith.science/pith/KG4TYCEKNAXCLND7664HFLXIVF.json","view_paper":"https://pith.science/paper/KG4TYCEK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1810.09303&json=true","fetch_graph":"https://pith.science/api/pith-number/KG4TYCEKNAXCLND7664HFLXIVF/graph.json","fetch_events":"https://pith.science/api/pith-number/KG4TYCEKNAXCLND7664HFLXIVF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KG4TYCEKNAXCLND7664HFLXIVF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KG4TYCEKNAXCLND7664HFLXIVF/action/storage_attestation","attest_author":"https://pith.science/pith/KG4TYCEKNAXCLND7664HFLXIVF/action/author_attestation","sign_citation":"https://pith.science/pith/KG4TYCEKNAXCLND7664HFLXIVF/action/citation_signature","submit_replication":"https://pith.science/pith/KG4TYCEKNAXCLND7664HFLXIVF/action/replication_record"}},"created_at":"2026-05-17T23:49:03.188636+00:00","updated_at":"2026-05-17T23:49:03.188636+00:00"}