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In this paper, the author prove that, the composite operator $T_1T_2$ satisfies the following estimate: for $\\lambda>0$ and weight $w\\in A_1(\\mathbb{R}^n)$, \\begin{eqnarray*}&&w\\big(\\{x\\in\\mathbb{R}^n:\\,|T_{1} T_2f(x)|>\\lambda\\}\\big)\\\\ &&\\quad\\lesssim [w]_{A_1}[w]_{A_{\\infty}}\\log ({\\rm e}+[w]_{A_{\\infty}}\\big) \\int_{\\mathbb{R}^n}\\frac{|f(x)|}{\\lambda}\\log \\Big({\\rm e}+\\frac{|f(x)|}{\\lambda}\\Big)w(x)dx,\\nonumber \\end{eqnarray*} and the composite operator $"},"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":"1806.00289","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-06-01T11:18:47Z","cross_cats_sorted":[],"title_canon_sha256":"af9649972bba3e49c4d4eb80f9861d994d6bcb56ae64698f2f8486d3059e207c","abstract_canon_sha256":"0a2580e0e814d1c51a6ee01085f34a32c9cf089e4e069a67811d56ec258cec3b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:09:52.216162Z","signature_b64":"8kBt0l/DrnuDy5xJ11XO4yzqtSn87HAvmTiV4AjXASJEvzeq9nraFLg1O4B26lT/xnN+eyrzSVdWnvNFgkWsBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9d479dfde69aec3cedd881b7647f32861fe55c92773f144d7451d5c3f31c352a","last_reissued_at":"2026-05-18T00:09:52.215477Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:09:52.215477Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Weighted weak type endpoint estimates for the composition of Calderon-Zygmund operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Guoen Hu","submitted_at":"2018-06-01T11:18:47Z","abstract_excerpt":"Let $T_1$, $T_2$ be two Calder\\'on-Zygmund operators and $T_{1,\\,b}$ be the commutator of $T_1$ with symbol $b\\in {\\rm BMO}(\\mathbb{R}^n)$. In this paper, the author prove that, the composite operator $T_1T_2$ satisfies the following estimate: for $\\lambda>0$ and weight $w\\in A_1(\\mathbb{R}^n)$, \\begin{eqnarray*}&&w\\big(\\{x\\in\\mathbb{R}^n:\\,|T_{1} T_2f(x)|>\\lambda\\}\\big)\\\\ &&\\quad\\lesssim [w]_{A_1}[w]_{A_{\\infty}}\\log ({\\rm e}+[w]_{A_{\\infty}}\\big) \\int_{\\mathbb{R}^n}\\frac{|f(x)|}{\\lambda}\\log \\Big({\\rm e}+\\frac{|f(x)|}{\\lambda}\\Big)w(x)dx,\\nonumber \\end{eqnarray*} and the composite operator $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.00289","kind":"arxiv","version":4},"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":"1806.00289","created_at":"2026-05-18T00:09:52.215577+00:00"},{"alias_kind":"arxiv_version","alias_value":"1806.00289v4","created_at":"2026-05-18T00:09:52.215577+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.00289","created_at":"2026-05-18T00:09:52.215577+00:00"},{"alias_kind":"pith_short_12","alias_value":"TVDZ37PGTLWD","created_at":"2026-05-18T12:32:56.356000+00:00"},{"alias_kind":"pith_short_16","alias_value":"TVDZ37PGTLWDZ3OY","created_at":"2026-05-18T12:32:56.356000+00:00"},{"alias_kind":"pith_short_8","alias_value":"TVDZ37PG","created_at":"2026-05-18T12:32:56.356000+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/TVDZ37PGTLWDZ3OYQG3WI7ZSQY","json":"https://pith.science/pith/TVDZ37PGTLWDZ3OYQG3WI7ZSQY.json","graph_json":"https://pith.science/api/pith-number/TVDZ37PGTLWDZ3OYQG3WI7ZSQY/graph.json","events_json":"https://pith.science/api/pith-number/TVDZ37PGTLWDZ3OYQG3WI7ZSQY/events.json","paper":"https://pith.science/paper/TVDZ37PG"},"agent_actions":{"view_html":"https://pith.science/pith/TVDZ37PGTLWDZ3OYQG3WI7ZSQY","download_json":"https://pith.science/pith/TVDZ37PGTLWDZ3OYQG3WI7ZSQY.json","view_paper":"https://pith.science/paper/TVDZ37PG","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1806.00289&json=true","fetch_graph":"https://pith.science/api/pith-number/TVDZ37PGTLWDZ3OYQG3WI7ZSQY/graph.json","fetch_events":"https://pith.science/api/pith-number/TVDZ37PGTLWDZ3OYQG3WI7ZSQY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TVDZ37PGTLWDZ3OYQG3WI7ZSQY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TVDZ37PGTLWDZ3OYQG3WI7ZSQY/action/storage_attestation","attest_author":"https://pith.science/pith/TVDZ37PGTLWDZ3OYQG3WI7ZSQY/action/author_attestation","sign_citation":"https://pith.science/pith/TVDZ37PGTLWDZ3OYQG3WI7ZSQY/action/citation_signature","submit_replication":"https://pith.science/pith/TVDZ37PGTLWDZ3OYQG3WI7ZSQY/action/replication_record"}},"created_at":"2026-05-18T00:09:52.215577+00:00","updated_at":"2026-05-18T00:09:52.215577+00:00"}