{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:32Y2REZBQH2RGBRV2DB4EFW7KV","short_pith_number":"pith:32Y2REZB","schema_version":"1.0","canonical_sha256":"deb1a8932181f5130635d0c3c216df5576ec53ae63d5450912c0cc3b1ac36ec4","source":{"kind":"arxiv","id":"1505.01804","version":2},"attestation_state":"computed","paper":{"title":"Borderline Weak Type Estimates for Singular Integrals and Square Functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Carlos Domingo-Salazar, Guillermo Rey, Michael T. Lacey","submitted_at":"2015-05-07T18:25:16Z","abstract_excerpt":"For any Calder\\'on-Zygmund operator $ T$, any weight $ w$, and $ \\alpha >1$, the operator $ T$ is bounded as a map from $ L ^{1} (M _{ L \\log\\log L (\\log\\log\\log L) ^{\\alpha } } w )$ into weak-$L^1(w)$. The interest in questions of this type goes back to the beginnings of the weighted theory, with prior results, due to Coifman-Fefferman, P\\'erez, and Hyt\\\"onen-P\\'erez, on the $ L (\\log L) ^{\\epsilon }$ scale. Also, for square functions $ S f$, and weights $ w \\in A_p$, the norm of $ S$ from $ L ^p (w)$ to weak-$L^p (w)$, $ 2\\leq p < \\infty $, is bounded by $ [w] _{A_p}^{1/2} (1+\\log [w] _{A_ \\"},"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":"1505.01804","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-05-07T18:25:16Z","cross_cats_sorted":[],"title_canon_sha256":"8c0dc39f7db85c82039ccdaf4a6572ff10ec279854e2d9d6ab6732ab2cd171d0","abstract_canon_sha256":"93024c016f8a589d2ff9d093979e790a61e24b5d8d508ec077a9c5cc74d9bbda"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:01:39.479905Z","signature_b64":"tSvYBFN1brNHQKZpPyQvIZqKSlQ78NhTnWpS0cCHlfZhIJtChFnlQOuPGZmAnKzie5k7XyXrg3jFLv18xfPIAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"deb1a8932181f5130635d0c3c216df5576ec53ae63d5450912c0cc3b1ac36ec4","last_reissued_at":"2026-05-18T00:01:39.479439Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:01:39.479439Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Borderline Weak Type Estimates for Singular Integrals and Square Functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Carlos Domingo-Salazar, Guillermo Rey, Michael T. Lacey","submitted_at":"2015-05-07T18:25:16Z","abstract_excerpt":"For any Calder\\'on-Zygmund operator $ T$, any weight $ w$, and $ \\alpha >1$, the operator $ T$ is bounded as a map from $ L ^{1} (M _{ L \\log\\log L (\\log\\log\\log L) ^{\\alpha } } w )$ into weak-$L^1(w)$. The interest in questions of this type goes back to the beginnings of the weighted theory, with prior results, due to Coifman-Fefferman, P\\'erez, and Hyt\\\"onen-P\\'erez, on the $ L (\\log L) ^{\\epsilon }$ scale. Also, for square functions $ S f$, and weights $ w \\in A_p$, the norm of $ S$ from $ L ^p (w)$ to weak-$L^p (w)$, $ 2\\leq p < \\infty $, is bounded by $ [w] _{A_p}^{1/2} (1+\\log [w] _{A_ \\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.01804","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":"1505.01804","created_at":"2026-05-18T00:01:39.479511+00:00"},{"alias_kind":"arxiv_version","alias_value":"1505.01804v2","created_at":"2026-05-18T00:01:39.479511+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.01804","created_at":"2026-05-18T00:01:39.479511+00:00"},{"alias_kind":"pith_short_12","alias_value":"32Y2REZBQH2R","created_at":"2026-05-18T12:29:02.477457+00:00"},{"alias_kind":"pith_short_16","alias_value":"32Y2REZBQH2RGBRV","created_at":"2026-05-18T12:29:02.477457+00:00"},{"alias_kind":"pith_short_8","alias_value":"32Y2REZB","created_at":"2026-05-18T12:29:02.477457+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/32Y2REZBQH2RGBRV2DB4EFW7KV","json":"https://pith.science/pith/32Y2REZBQH2RGBRV2DB4EFW7KV.json","graph_json":"https://pith.science/api/pith-number/32Y2REZBQH2RGBRV2DB4EFW7KV/graph.json","events_json":"https://pith.science/api/pith-number/32Y2REZBQH2RGBRV2DB4EFW7KV/events.json","paper":"https://pith.science/paper/32Y2REZB"},"agent_actions":{"view_html":"https://pith.science/pith/32Y2REZBQH2RGBRV2DB4EFW7KV","download_json":"https://pith.science/pith/32Y2REZBQH2RGBRV2DB4EFW7KV.json","view_paper":"https://pith.science/paper/32Y2REZB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1505.01804&json=true","fetch_graph":"https://pith.science/api/pith-number/32Y2REZBQH2RGBRV2DB4EFW7KV/graph.json","fetch_events":"https://pith.science/api/pith-number/32Y2REZBQH2RGBRV2DB4EFW7KV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/32Y2REZBQH2RGBRV2DB4EFW7KV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/32Y2REZBQH2RGBRV2DB4EFW7KV/action/storage_attestation","attest_author":"https://pith.science/pith/32Y2REZBQH2RGBRV2DB4EFW7KV/action/author_attestation","sign_citation":"https://pith.science/pith/32Y2REZBQH2RGBRV2DB4EFW7KV/action/citation_signature","submit_replication":"https://pith.science/pith/32Y2REZBQH2RGBRV2DB4EFW7KV/action/replication_record"}},"created_at":"2026-05-18T00:01:39.479511+00:00","updated_at":"2026-05-18T00:01:39.479511+00:00"}