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In particular the following estimates are proved: %\n\\[ \\|T_\\Omega \\|_{L^p(w)}\\le c_{n,p}\\|\\Omega\\|_{L^\\infty} [w]_{A_1}^{\\frac{1}{p}}\\,[w]_{A_{\\infty}}^{1+\\frac{1}{p'}}\\|f\\|_{L^p(w)} \\] %\nand %\n\\[ \\| [b,T_{\\Omega}]f\\|_{L^{p}(w)}\\leq c_{n,p}\\|b\\|_{BMO}\\|\\Omega\\|_{L^{\\infty}} [w]_{A_1}^{\\frac{1}{p}}[w]_{A_{\\infty}}^{2+\\frac{1}{p'}}\\|f\\|_{L^{p}\\left(w\\right)}, \\] %\nfor $1<p<\\infty$ and $1/p+1/p'=1$."},"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":"1607.06432","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-07-21T19:07:27Z","cross_cats_sorted":[],"title_canon_sha256":"6cabda32890e0639a11ad19dfd71d065efdc67b9517df51848ac09640ab1677b","abstract_canon_sha256":"b093352fbe712fa7cae8473d4d4222bc75e8ba5db0b98325e40c7f0cf43f2e4e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:10:41.848582Z","signature_b64":"Oa49FkWQQUohdqRe1dPU9bXjzLb5ClB+kbE2Z3/MKx3VW3TcYfmttyTon7fzV7x41A9UODww/cDzDaKGW3c8Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6322987e4945e566271079d3a89d8ce0fede99929fbfa59a712ea25d7345964e","last_reissued_at":"2026-05-18T01:10:41.848157Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:10:41.848157Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"$A_1$ theory of weights for rough homogeneous singular integrals and commutators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"C. Perez, I. Rivera-Rios, L. Roncal","submitted_at":"2016-07-21T19:07:27Z","abstract_excerpt":"Quantitative $A_1-A_\\infty$ estimates for rough homogeneous singular integrals $T_{\\Omega}$ and commutators of $BMO$ symbols and $T_{\\Omega}$ are obtained. In particular the following estimates are proved: %\n\\[ \\|T_\\Omega \\|_{L^p(w)}\\le c_{n,p}\\|\\Omega\\|_{L^\\infty} [w]_{A_1}^{\\frac{1}{p}}\\,[w]_{A_{\\infty}}^{1+\\frac{1}{p'}}\\|f\\|_{L^p(w)} \\] %\nand %\n\\[ \\| [b,T_{\\Omega}]f\\|_{L^{p}(w)}\\leq c_{n,p}\\|b\\|_{BMO}\\|\\Omega\\|_{L^{\\infty}} [w]_{A_1}^{\\frac{1}{p}}[w]_{A_{\\infty}}^{2+\\frac{1}{p'}}\\|f\\|_{L^{p}\\left(w\\right)}, \\] %\nfor $1<p<\\infty$ and $1/p+1/p'=1$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.06432","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":"1607.06432","created_at":"2026-05-18T01:10:41.848229+00:00"},{"alias_kind":"arxiv_version","alias_value":"1607.06432v1","created_at":"2026-05-18T01:10:41.848229+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.06432","created_at":"2026-05-18T01:10:41.848229+00:00"},{"alias_kind":"pith_short_12","alias_value":"MMRJQ7SJIXSW","created_at":"2026-05-18T12:30:32.724797+00:00"},{"alias_kind":"pith_short_16","alias_value":"MMRJQ7SJIXSWMJYQ","created_at":"2026-05-18T12:30:32.724797+00:00"},{"alias_kind":"pith_short_8","alias_value":"MMRJQ7SJ","created_at":"2026-05-18T12:30:32.724797+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/MMRJQ7SJIXSWMJYQPHJ2RHMM4D","json":"https://pith.science/pith/MMRJQ7SJIXSWMJYQPHJ2RHMM4D.json","graph_json":"https://pith.science/api/pith-number/MMRJQ7SJIXSWMJYQPHJ2RHMM4D/graph.json","events_json":"https://pith.science/api/pith-number/MMRJQ7SJIXSWMJYQPHJ2RHMM4D/events.json","paper":"https://pith.science/paper/MMRJQ7SJ"},"agent_actions":{"view_html":"https://pith.science/pith/MMRJQ7SJIXSWMJYQPHJ2RHMM4D","download_json":"https://pith.science/pith/MMRJQ7SJIXSWMJYQPHJ2RHMM4D.json","view_paper":"https://pith.science/paper/MMRJQ7SJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1607.06432&json=true","fetch_graph":"https://pith.science/api/pith-number/MMRJQ7SJIXSWMJYQPHJ2RHMM4D/graph.json","fetch_events":"https://pith.science/api/pith-number/MMRJQ7SJIXSWMJYQPHJ2RHMM4D/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MMRJQ7SJIXSWMJYQPHJ2RHMM4D/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MMRJQ7SJIXSWMJYQPHJ2RHMM4D/action/storage_attestation","attest_author":"https://pith.science/pith/MMRJQ7SJIXSWMJYQPHJ2RHMM4D/action/author_attestation","sign_citation":"https://pith.science/pith/MMRJQ7SJIXSWMJYQPHJ2RHMM4D/action/citation_signature","submit_replication":"https://pith.science/pith/MMRJQ7SJIXSWMJYQPHJ2RHMM4D/action/replication_record"}},"created_at":"2026-05-18T01:10:41.848229+00:00","updated_at":"2026-05-18T01:10:41.848229+00:00"}