{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:PZ6FRM4G5B65NCDHNN5YL5T2LL","short_pith_number":"pith:PZ6FRM4G","schema_version":"1.0","canonical_sha256":"7e7c58b386e87dd688676b7b85f67a5ad9c73c8e0f30c2846708b375fcde203d","source":{"kind":"arxiv","id":"1112.0676","version":4},"attestation_state":"computed","paper":{"title":"Logarithmic bump conditions and the two weight boundedness of Calder\\'on-Zygmund operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.AP","authors_text":"Alexander Reznikov, Alexander Volberg, David Cruz-Uribe","submitted_at":"2011-12-03T17:28:42Z","abstract_excerpt":"We prove that if a pair of weights $(u,v)$ satisfies a sharp $A_p$-bump condition in the scale of log bumps and certain loglog bumps, then Haar shifts map $L^p(v)$ into $L^p(u)$ with a constant quadratic in the complexity of the shift. This in turn implies the two weight boundedness for all Calder\\'on-Zygmund operators. This gives a partial answer to a long-standing conjecture. We also give a partial result for a related conjecture for weak-type inequalities. To prove our main results we combine several different approaches to these problems; in particular we use many of the ideas developed to"},"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":"1112.0676","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-12-03T17:28:42Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"399648d043a1bfa59ffc8ba0888aa4baca431c7ba35f2f27ce4bf742989323e1","abstract_canon_sha256":"ce82b4bf1fd3f4a13e27baed4042c28dacc0e0d4dd79b0d6b83bbea1390f52fa"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:37:19.081422Z","signature_b64":"N0fcLyldSuH1hMBw83ik3/5VIGrmxxdb/rEq61JKRj9tsoIXty/r7zpMcy+BOA48sg3QOq9qN0nR4eDTAFOsBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7e7c58b386e87dd688676b7b85f67a5ad9c73c8e0f30c2846708b375fcde203d","last_reissued_at":"2026-05-18T03:37:19.080999Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:37:19.080999Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Logarithmic bump conditions and the two weight boundedness of Calder\\'on-Zygmund operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.AP","authors_text":"Alexander Reznikov, Alexander Volberg, David Cruz-Uribe","submitted_at":"2011-12-03T17:28:42Z","abstract_excerpt":"We prove that if a pair of weights $(u,v)$ satisfies a sharp $A_p$-bump condition in the scale of log bumps and certain loglog bumps, then Haar shifts map $L^p(v)$ into $L^p(u)$ with a constant quadratic in the complexity of the shift. This in turn implies the two weight boundedness for all Calder\\'on-Zygmund operators. This gives a partial answer to a long-standing conjecture. We also give a partial result for a related conjecture for weak-type inequalities. To prove our main results we combine several different approaches to these problems; in particular we use many of the ideas developed to"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.0676","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":"1112.0676","created_at":"2026-05-18T03:37:19.081058+00:00"},{"alias_kind":"arxiv_version","alias_value":"1112.0676v4","created_at":"2026-05-18T03:37:19.081058+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.0676","created_at":"2026-05-18T03:37:19.081058+00:00"},{"alias_kind":"pith_short_12","alias_value":"PZ6FRM4G5B65","created_at":"2026-05-18T12:26:39.201973+00:00"},{"alias_kind":"pith_short_16","alias_value":"PZ6FRM4G5B65NCDH","created_at":"2026-05-18T12:26:39.201973+00:00"},{"alias_kind":"pith_short_8","alias_value":"PZ6FRM4G","created_at":"2026-05-18T12:26:39.201973+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/PZ6FRM4G5B65NCDHNN5YL5T2LL","json":"https://pith.science/pith/PZ6FRM4G5B65NCDHNN5YL5T2LL.json","graph_json":"https://pith.science/api/pith-number/PZ6FRM4G5B65NCDHNN5YL5T2LL/graph.json","events_json":"https://pith.science/api/pith-number/PZ6FRM4G5B65NCDHNN5YL5T2LL/events.json","paper":"https://pith.science/paper/PZ6FRM4G"},"agent_actions":{"view_html":"https://pith.science/pith/PZ6FRM4G5B65NCDHNN5YL5T2LL","download_json":"https://pith.science/pith/PZ6FRM4G5B65NCDHNN5YL5T2LL.json","view_paper":"https://pith.science/paper/PZ6FRM4G","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1112.0676&json=true","fetch_graph":"https://pith.science/api/pith-number/PZ6FRM4G5B65NCDHNN5YL5T2LL/graph.json","fetch_events":"https://pith.science/api/pith-number/PZ6FRM4G5B65NCDHNN5YL5T2LL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PZ6FRM4G5B65NCDHNN5YL5T2LL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PZ6FRM4G5B65NCDHNN5YL5T2LL/action/storage_attestation","attest_author":"https://pith.science/pith/PZ6FRM4G5B65NCDHNN5YL5T2LL/action/author_attestation","sign_citation":"https://pith.science/pith/PZ6FRM4G5B65NCDHNN5YL5T2LL/action/citation_signature","submit_replication":"https://pith.science/pith/PZ6FRM4G5B65NCDHNN5YL5T2LL/action/replication_record"}},"created_at":"2026-05-18T03:37:19.081058+00:00","updated_at":"2026-05-18T03:37:19.081058+00:00"}