{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2005:XXGZGNNYDL7JCA3YQPVQ4LJRC7","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"6fbd725d9deca522bdcc13541c1404af5949c7335bd586f875b245744eda9c2e","cross_cats_sorted":[],"license":"","primary_cat":"math.CA","submitted_at":"2005-02-16T05:55:37Z","title_canon_sha256":"b3210745ae443d6084b029dc0c40c96c925466dca5ffc1e50d07cb4727f57566"},"schema_version":"1.0","source":{"id":"math/0502334","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0502334","created_at":"2026-05-18T03:56:10Z"},{"alias_kind":"arxiv_version","alias_value":"math/0502334v2","created_at":"2026-05-18T03:56:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0502334","created_at":"2026-05-18T03:56:10Z"},{"alias_kind":"pith_short_12","alias_value":"XXGZGNNYDL7J","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"XXGZGNNYDL7JCA3Y","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"XXGZGNNY","created_at":"2026-05-18T12:25:53Z"}],"graph_snapshots":[{"event_id":"sha256:77bc457623bebc699762ec242625dd6caba904ca32c0af199b7eec97446ec6da","target":"graph","created_at":"2026-05-18T03:56:10Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"For multiparameter bilinear paraproduct operators $B$ we prove the estimate $$\n  B: L^p X L^q --> L^r, 1<p,q\\le{}\\infty. $$ Here, $1/p+1/q=1/r$ and special attention is paid to the case of $0<r<1$. (Note that the families of multiparameter paraproducts are much richer than in the one parameter case.) These estimates are the essential step in the version of the multiparameter Coifman-Meyer theorem proved by C. Muscalu, J. Pipher, T. Tao, and C. Thiele. We offer a different proof of these inequalities.","authors_text":"Jason Metcalfe, Michael T Lacey","cross_cats":[],"headline":"","license":"","primary_cat":"math.CA","submitted_at":"2005-02-16T05:55:37Z","title":"Paraproducts in One and Several Parameters"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0502334","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:3b2cb9538547e3710f065d8012381d04264a4b050c01c9bbf3a79ae3d5b5a013","target":"record","created_at":"2026-05-18T03:56:10Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"6fbd725d9deca522bdcc13541c1404af5949c7335bd586f875b245744eda9c2e","cross_cats_sorted":[],"license":"","primary_cat":"math.CA","submitted_at":"2005-02-16T05:55:37Z","title_canon_sha256":"b3210745ae443d6084b029dc0c40c96c925466dca5ffc1e50d07cb4727f57566"},"schema_version":"1.0","source":{"id":"math/0502334","kind":"arxiv","version":2}},"canonical_sha256":"bdcd9335b81afe91037883eb0e2d3117e7b8b81067ab87084ebbf1dc3cd41686","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bdcd9335b81afe91037883eb0e2d3117e7b8b81067ab87084ebbf1dc3cd41686","first_computed_at":"2026-05-18T03:56:10.996864Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:56:10.996864Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8EvcTWhBqrYIHFCFVJ3e/nPNikWsCQUbR/Hy0bLELmyYDBA/eDRjZod28IMiv4eXjtgRF0MTLYly1WYRUhOVAg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:56:10.997665Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0502334","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3b2cb9538547e3710f065d8012381d04264a4b050c01c9bbf3a79ae3d5b5a013","sha256:77bc457623bebc699762ec242625dd6caba904ca32c0af199b7eec97446ec6da"],"state_sha256":"3e6d230b29de4bbc0a653ead16fb2124fbef8a48bfeed21eb5b4c961d4003c89"}