{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:4UXH5KM7ATYRSDJMYT3GBJASRR","short_pith_number":"pith:4UXH5KM7","canonical_record":{"source":{"id":"1604.07037","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-04-24T14:50:11Z","cross_cats_sorted":[],"title_canon_sha256":"7e0f0f53e35a9ded8989033b8b601f0de94eadd65af57479bb9d1f119d92699d","abstract_canon_sha256":"dc7a83209466861198ebef2015f1ba352eab56daa8d85ae73801d680f282f5d8"},"schema_version":"1.0"},"canonical_sha256":"e52e7ea99f04f1190d2cc4f660a4128c5a9d94fe1bd92b4ca4401e60fc7c5d60","source":{"kind":"arxiv","id":"1604.07037","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1604.07037","created_at":"2026-05-18T01:16:23Z"},{"alias_kind":"arxiv_version","alias_value":"1604.07037v1","created_at":"2026-05-18T01:16:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.07037","created_at":"2026-05-18T01:16:23Z"},{"alias_kind":"pith_short_12","alias_value":"4UXH5KM7ATYR","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_16","alias_value":"4UXH5KM7ATYRSDJM","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_8","alias_value":"4UXH5KM7","created_at":"2026-05-18T12:29:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:4UXH5KM7ATYRSDJMYT3GBJASRR","target":"record","payload":{"canonical_record":{"source":{"id":"1604.07037","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-04-24T14:50:11Z","cross_cats_sorted":[],"title_canon_sha256":"7e0f0f53e35a9ded8989033b8b601f0de94eadd65af57479bb9d1f119d92699d","abstract_canon_sha256":"dc7a83209466861198ebef2015f1ba352eab56daa8d85ae73801d680f282f5d8"},"schema_version":"1.0"},"canonical_sha256":"e52e7ea99f04f1190d2cc4f660a4128c5a9d94fe1bd92b4ca4401e60fc7c5d60","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:16:23.242846Z","signature_b64":"S2DpCHZohbhH/T65usg9qjnAc0xO19WtGdjDpIVhojNnRlFd4jOFj2a3u0+JZkLFrw50H11PRjx581k0QS0fDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e52e7ea99f04f1190d2cc4f660a4128c5a9d94fe1bd92b4ca4401e60fc7c5d60","last_reissued_at":"2026-05-18T01:16:23.242088Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:16:23.242088Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1604.07037","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:16:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QYIHUb/4ZDl14/EiOjjZp4lJ0rivnfF2YadonmKuhViQXGi2nzyaZO6RwYzXI2vz9kPqq+MizM0CXT7ICQA5Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T09:02:07.620659Z"},"content_sha256":"ed79a9143e6a4eb4fb644dec4b37447a3f3a04510ec2f6dcb55d00ac7fa7765c","schema_version":"1.0","event_id":"sha256:ed79a9143e6a4eb4fb644dec4b37447a3f3a04510ec2f6dcb55d00ac7fa7765c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:4UXH5KM7ATYRSDJMYT3GBJASRR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Non-homogeneous $Tb$ Theorem for Bi-parameter $g$-Function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Mingming Cao, Qingying Xue","submitted_at":"2016-04-24T14:50:11Z","abstract_excerpt":"The main result of this paper is a bi-parameter $Tb$ theorem for Littlewood-Paley $g$-function, where $b$ is a tensor product of two pseudo-accretive functions. Instead of the doubling measure, we work with a product measure $\\mu = \\mu_n \\times \\mu_m$, where the measures $\\mu_n$ and $\\mu_m$ are only assumed to be upper doubling. The main techniques of the proof include a bi-parameter $b$-adapted Haar function decomposition and an averaging identity over good double Whitney regions. Moreover, the non-homogeneous analysis and probabilistic methods are used again."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.07037","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:16:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UIeFtrK1wIh2PWf0dh28+bsM1LB7zP1l7P31sHrcDdbpR9TCCSHSmwwWfLav7gXHlz8rSaptxaVIcKOOpZn9Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T09:02:07.621248Z"},"content_sha256":"be8f3edb43bf2f4edc179903dd40366aa91c27c51ef0813b9d559be8dddc3a02","schema_version":"1.0","event_id":"sha256:be8f3edb43bf2f4edc179903dd40366aa91c27c51ef0813b9d559be8dddc3a02"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4UXH5KM7ATYRSDJMYT3GBJASRR/bundle.json","state_url":"https://pith.science/pith/4UXH5KM7ATYRSDJMYT3GBJASRR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4UXH5KM7ATYRSDJMYT3GBJASRR/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-11T09:02:07Z","links":{"resolver":"https://pith.science/pith/4UXH5KM7ATYRSDJMYT3GBJASRR","bundle":"https://pith.science/pith/4UXH5KM7ATYRSDJMYT3GBJASRR/bundle.json","state":"https://pith.science/pith/4UXH5KM7ATYRSDJMYT3GBJASRR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4UXH5KM7ATYRSDJMYT3GBJASRR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:4UXH5KM7ATYRSDJMYT3GBJASRR","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":"dc7a83209466861198ebef2015f1ba352eab56daa8d85ae73801d680f282f5d8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-04-24T14:50:11Z","title_canon_sha256":"7e0f0f53e35a9ded8989033b8b601f0de94eadd65af57479bb9d1f119d92699d"},"schema_version":"1.0","source":{"id":"1604.07037","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1604.07037","created_at":"2026-05-18T01:16:23Z"},{"alias_kind":"arxiv_version","alias_value":"1604.07037v1","created_at":"2026-05-18T01:16:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.07037","created_at":"2026-05-18T01:16:23Z"},{"alias_kind":"pith_short_12","alias_value":"4UXH5KM7ATYR","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_16","alias_value":"4UXH5KM7ATYRSDJM","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_8","alias_value":"4UXH5KM7","created_at":"2026-05-18T12:29:58Z"}],"graph_snapshots":[{"event_id":"sha256:be8f3edb43bf2f4edc179903dd40366aa91c27c51ef0813b9d559be8dddc3a02","target":"graph","created_at":"2026-05-18T01:16:23Z","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":"The main result of this paper is a bi-parameter $Tb$ theorem for Littlewood-Paley $g$-function, where $b$ is a tensor product of two pseudo-accretive functions. Instead of the doubling measure, we work with a product measure $\\mu = \\mu_n \\times \\mu_m$, where the measures $\\mu_n$ and $\\mu_m$ are only assumed to be upper doubling. The main techniques of the proof include a bi-parameter $b$-adapted Haar function decomposition and an averaging identity over good double Whitney regions. Moreover, the non-homogeneous analysis and probabilistic methods are used again.","authors_text":"Mingming Cao, Qingying Xue","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-04-24T14:50:11Z","title":"Non-homogeneous $Tb$ Theorem for Bi-parameter $g$-Function"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.07037","kind":"arxiv","version":1},"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:ed79a9143e6a4eb4fb644dec4b37447a3f3a04510ec2f6dcb55d00ac7fa7765c","target":"record","created_at":"2026-05-18T01:16:23Z","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":"dc7a83209466861198ebef2015f1ba352eab56daa8d85ae73801d680f282f5d8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-04-24T14:50:11Z","title_canon_sha256":"7e0f0f53e35a9ded8989033b8b601f0de94eadd65af57479bb9d1f119d92699d"},"schema_version":"1.0","source":{"id":"1604.07037","kind":"arxiv","version":1}},"canonical_sha256":"e52e7ea99f04f1190d2cc4f660a4128c5a9d94fe1bd92b4ca4401e60fc7c5d60","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e52e7ea99f04f1190d2cc4f660a4128c5a9d94fe1bd92b4ca4401e60fc7c5d60","first_computed_at":"2026-05-18T01:16:23.242088Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:16:23.242088Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"S2DpCHZohbhH/T65usg9qjnAc0xO19WtGdjDpIVhojNnRlFd4jOFj2a3u0+JZkLFrw50H11PRjx581k0QS0fDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:16:23.242846Z","signed_message":"canonical_sha256_bytes"},"source_id":"1604.07037","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ed79a9143e6a4eb4fb644dec4b37447a3f3a04510ec2f6dcb55d00ac7fa7765c","sha256:be8f3edb43bf2f4edc179903dd40366aa91c27c51ef0813b9d559be8dddc3a02"],"state_sha256":"c27f4f7d606b884609ef465c41d261d7a03d66d4ea9375b8d155511e1490077c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"C9UjcqhwrNIVqL6P4d41F81ThvHlJbtRqAKWSaeoc2mcKDDrsf6YXsU1fuK2qdcEluTf9rKT6Rg+oOOYsq5IAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T09:02:07.624567Z","bundle_sha256":"374e52d6a09afe11234c6ba6cb6696d7ac47124ab75b07a25acc4a0cddb9145c"}}