{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:6KL3MD2EM7FXYTSUWLZVU6NEXG","short_pith_number":"pith:6KL3MD2E","canonical_record":{"source":{"id":"1603.02699","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-03-08T21:06:33Z","cross_cats_sorted":[],"title_canon_sha256":"b59c8b7eb305244f44314c8b298c02b41e67c709be89cd65313a02fd1de292aa","abstract_canon_sha256":"26343df3ef35de572aef0ff5c7752dd5a9d84ecd4994dcd112b7d8451fa89b26"},"schema_version":"1.0"},"canonical_sha256":"f297b60f4467cb7c4e54b2f35a79a4b9ba65c23a71bec1a6fafc1bc165e688a0","source":{"kind":"arxiv","id":"1603.02699","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1603.02699","created_at":"2026-05-18T00:44:38Z"},{"alias_kind":"arxiv_version","alias_value":"1603.02699v2","created_at":"2026-05-18T00:44:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.02699","created_at":"2026-05-18T00:44:38Z"},{"alias_kind":"pith_short_12","alias_value":"6KL3MD2EM7FX","created_at":"2026-05-18T12:30:01Z"},{"alias_kind":"pith_short_16","alias_value":"6KL3MD2EM7FXYTSU","created_at":"2026-05-18T12:30:01Z"},{"alias_kind":"pith_short_8","alias_value":"6KL3MD2E","created_at":"2026-05-18T12:30:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:6KL3MD2EM7FXYTSUWLZVU6NEXG","target":"record","payload":{"canonical_record":{"source":{"id":"1603.02699","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-03-08T21:06:33Z","cross_cats_sorted":[],"title_canon_sha256":"b59c8b7eb305244f44314c8b298c02b41e67c709be89cd65313a02fd1de292aa","abstract_canon_sha256":"26343df3ef35de572aef0ff5c7752dd5a9d84ecd4994dcd112b7d8451fa89b26"},"schema_version":"1.0"},"canonical_sha256":"f297b60f4467cb7c4e54b2f35a79a4b9ba65c23a71bec1a6fafc1bc165e688a0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:44:38.979928Z","signature_b64":"DJhuYaqnP3BUQDyhQ6PtiyzppKSf5I/oB9hiDws2UeF+Dh0duoCIdvXwglQTpDbLOjzWOLV0V8+noDqvThB2BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f297b60f4467cb7c4e54b2f35a79a4b9ba65c23a71bec1a6fafc1bc165e688a0","last_reissued_at":"2026-05-18T00:44:38.979499Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:44:38.979499Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1603.02699","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:44:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"K75LRpqmLoQhKaOLGMSb0/ZbPIakYNSbzg0Fe2KBqre2qR7PCl9/YTkJS7mGqVh9kRInnCT6XpH87YLXuNPvDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T23:08:00.926824Z"},"content_sha256":"c62aad3f77dd3e3468443ebcc4532b9907a553821993b13e42bd2dcb4cce64c8","schema_version":"1.0","event_id":"sha256:c62aad3f77dd3e3468443ebcc4532b9907a553821993b13e42bd2dcb4cce64c8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:6KL3MD2EM7FXYTSUWLZVU6NEXG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Weak Factorizations of the Hardy space $H^1(\\mathbb{R}^n)$ in terms of Multilinear Riesz Transforms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Brett D. Wick, Ji Li","submitted_at":"2016-03-08T21:06:33Z","abstract_excerpt":"This paper provides a constructive proof of the weak factorizations of the classical Hardy space $H^1(\\mathbb{R}^n)$ in terms of multilinear Riesz transforms. As a direct application, we obtain a new proof of the characterization of ${\\rm BMO}(\\mathbb{R}^n)$ (the dual of $H^1(\\mathbb{R}^n)$) via commutators of the multilinear Riesz transforms."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.02699","kind":"arxiv","version":2},"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":"2026-05-18T00:44:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cEgH8418ZgYBopN3W2VcWZJ66NU2JcAUg4yNXQldZo9iIQs6Ds5QD0VQoKe9EPyidocTS07dchCY8sqGTcxtDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T23:08:00.927221Z"},"content_sha256":"c1e30736c07661bf2b4adbbac356c4e9d22935fe670f6dc4a5c74e280f95fda0","schema_version":"1.0","event_id":"sha256:c1e30736c07661bf2b4adbbac356c4e9d22935fe670f6dc4a5c74e280f95fda0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6KL3MD2EM7FXYTSUWLZVU6NEXG/bundle.json","state_url":"https://pith.science/pith/6KL3MD2EM7FXYTSUWLZVU6NEXG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6KL3MD2EM7FXYTSUWLZVU6NEXG/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":"2026-05-31T23:08:00Z","links":{"resolver":"https://pith.science/pith/6KL3MD2EM7FXYTSUWLZVU6NEXG","bundle":"https://pith.science/pith/6KL3MD2EM7FXYTSUWLZVU6NEXG/bundle.json","state":"https://pith.science/pith/6KL3MD2EM7FXYTSUWLZVU6NEXG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6KL3MD2EM7FXYTSUWLZVU6NEXG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:6KL3MD2EM7FXYTSUWLZVU6NEXG","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":"26343df3ef35de572aef0ff5c7752dd5a9d84ecd4994dcd112b7d8451fa89b26","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-03-08T21:06:33Z","title_canon_sha256":"b59c8b7eb305244f44314c8b298c02b41e67c709be89cd65313a02fd1de292aa"},"schema_version":"1.0","source":{"id":"1603.02699","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1603.02699","created_at":"2026-05-18T00:44:38Z"},{"alias_kind":"arxiv_version","alias_value":"1603.02699v2","created_at":"2026-05-18T00:44:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.02699","created_at":"2026-05-18T00:44:38Z"},{"alias_kind":"pith_short_12","alias_value":"6KL3MD2EM7FX","created_at":"2026-05-18T12:30:01Z"},{"alias_kind":"pith_short_16","alias_value":"6KL3MD2EM7FXYTSU","created_at":"2026-05-18T12:30:01Z"},{"alias_kind":"pith_short_8","alias_value":"6KL3MD2E","created_at":"2026-05-18T12:30:01Z"}],"graph_snapshots":[{"event_id":"sha256:c1e30736c07661bf2b4adbbac356c4e9d22935fe670f6dc4a5c74e280f95fda0","target":"graph","created_at":"2026-05-18T00:44:38Z","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":"This paper provides a constructive proof of the weak factorizations of the classical Hardy space $H^1(\\mathbb{R}^n)$ in terms of multilinear Riesz transforms. As a direct application, we obtain a new proof of the characterization of ${\\rm BMO}(\\mathbb{R}^n)$ (the dual of $H^1(\\mathbb{R}^n)$) via commutators of the multilinear Riesz transforms.","authors_text":"Brett D. Wick, Ji Li","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-03-08T21:06:33Z","title":"Weak Factorizations of the Hardy space $H^1(\\mathbb{R}^n)$ in terms of Multilinear Riesz Transforms"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.02699","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:c62aad3f77dd3e3468443ebcc4532b9907a553821993b13e42bd2dcb4cce64c8","target":"record","created_at":"2026-05-18T00:44:38Z","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":"26343df3ef35de572aef0ff5c7752dd5a9d84ecd4994dcd112b7d8451fa89b26","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-03-08T21:06:33Z","title_canon_sha256":"b59c8b7eb305244f44314c8b298c02b41e67c709be89cd65313a02fd1de292aa"},"schema_version":"1.0","source":{"id":"1603.02699","kind":"arxiv","version":2}},"canonical_sha256":"f297b60f4467cb7c4e54b2f35a79a4b9ba65c23a71bec1a6fafc1bc165e688a0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f297b60f4467cb7c4e54b2f35a79a4b9ba65c23a71bec1a6fafc1bc165e688a0","first_computed_at":"2026-05-18T00:44:38.979499Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:44:38.979499Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"DJhuYaqnP3BUQDyhQ6PtiyzppKSf5I/oB9hiDws2UeF+Dh0duoCIdvXwglQTpDbLOjzWOLV0V8+noDqvThB2BA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:44:38.979928Z","signed_message":"canonical_sha256_bytes"},"source_id":"1603.02699","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c62aad3f77dd3e3468443ebcc4532b9907a553821993b13e42bd2dcb4cce64c8","sha256:c1e30736c07661bf2b4adbbac356c4e9d22935fe670f6dc4a5c74e280f95fda0"],"state_sha256":"2cd5b9a12371811c3f8660cb8eaa0bffb2547b86100464f1a1530163e553515d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AFmw/OgyknyhtksBw1G+/HN/Fu87p7aVljg7aZYfnvFRRBDjhmM6k+ZcPZjhbTRJJq1wWyk1QOfaT4TAm5BRDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T23:08:00.930035Z","bundle_sha256":"cfd7c6c63f1d33675f7745f6ec3cbf7520b0a6a8fdc8c98184d761bde5ad2f19"}}