{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:ETG2MQMC6VBVV6E6F7EKT4I2VL","short_pith_number":"pith:ETG2MQMC","schema_version":"1.0","canonical_sha256":"24cda64182f5435af89e2fc8a9f11aaad21a4ffd5f08249d0892309712746ede","source":{"kind":"arxiv","id":"1601.03193","version":2},"attestation_state":"computed","paper":{"title":"Three observations on commutators of Singular Integral Operators with BMO functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Carlos P\\'erez, Israel P. Rivera-R\\'ios","submitted_at":"2016-01-13T10:35:15Z","abstract_excerpt":"This paper contains three observations on commutators of Singular Integral Operators with BMO functions:\n  1) The subgaussian local decay for the commutator, namely \\[\\frac{1}{|Q|}\\left|\\left\\{x\\in Q\\, : \\, |[b,T](f\\chi_Q)(x)|>M^2f(x)t\\right\\}\\right|\\leq c e^{-\\sqrt{ct\\|b\\|_{BMO}}} \\] is sharp, that is, it is subgaussian and not better.\n  2) It is not possible to obtain a pointwise control of the commutator by a finite sum of sparse operators defined with $L\\log L$ averages.\n  3) If $w\\in A_p\\setminus A_1$ then $\\left\\| wM\\left(\\frac{f}{w}\\right)\\right\\|_{L^1(\\mathbb{R}^n)\\rightarrow L^{1,\\inf"},"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":"1601.03193","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-01-13T10:35:15Z","cross_cats_sorted":[],"title_canon_sha256":"9d73371ba61cc204a92bb327fb894061f12445b9ea39a7ad420e0e68d91e64d5","abstract_canon_sha256":"b10652b4d47b0c5c0059383c03b6fbcac21245db13beb9f5b9d9d1c792cea658"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:30:55.560652Z","signature_b64":"9sAap2Ikou6jBVT38EyXIXHTJtEkT2kR93d00VL2hkmrZmi4Nm1D5Yg2M3BkJscnXuZlTFTJ5IqjdTpFV3BdBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"24cda64182f5435af89e2fc8a9f11aaad21a4ffd5f08249d0892309712746ede","last_reissued_at":"2026-05-18T00:30:55.559941Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:30:55.559941Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Three observations on commutators of Singular Integral Operators with BMO functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Carlos P\\'erez, Israel P. 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