{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:HBZJOI4JYWN6NOSDFK55TJKKHQ","short_pith_number":"pith:HBZJOI4J","canonical_record":{"source":{"id":"1612.09573","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-12-30T19:48:36Z","cross_cats_sorted":[],"title_canon_sha256":"59e6351ca74d0324e1b8e350dddcc5383e83be5f682645ec79a2add828ab2e9f","abstract_canon_sha256":"02cbb8ecfb4c08d0878647ce6abe9cd432020467aaaae546e6a8ebcbc785f538"},"schema_version":"1.0"},"canonical_sha256":"3872972389c59be6ba432abbd9a54a3c17f0b5a90b9a7444a1733e680a4faccd","source":{"kind":"arxiv","id":"1612.09573","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.09573","created_at":"2026-05-18T00:53:41Z"},{"alias_kind":"arxiv_version","alias_value":"1612.09573v1","created_at":"2026-05-18T00:53:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.09573","created_at":"2026-05-18T00:53:41Z"},{"alias_kind":"pith_short_12","alias_value":"HBZJOI4JYWN6","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_16","alias_value":"HBZJOI4JYWN6NOSD","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_8","alias_value":"HBZJOI4J","created_at":"2026-05-18T12:30:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:HBZJOI4JYWN6NOSDFK55TJKKHQ","target":"record","payload":{"canonical_record":{"source":{"id":"1612.09573","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-12-30T19:48:36Z","cross_cats_sorted":[],"title_canon_sha256":"59e6351ca74d0324e1b8e350dddcc5383e83be5f682645ec79a2add828ab2e9f","abstract_canon_sha256":"02cbb8ecfb4c08d0878647ce6abe9cd432020467aaaae546e6a8ebcbc785f538"},"schema_version":"1.0"},"canonical_sha256":"3872972389c59be6ba432abbd9a54a3c17f0b5a90b9a7444a1733e680a4faccd","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:53:41.190404Z","signature_b64":"eDY+dBWlVyYbCM/Zj+ywrEyYQNNgk3odiviqFMlAbb+z/rRZTiE1UrtlEeCsymR8EXOGQLSJlYbV04jWVbFNDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3872972389c59be6ba432abbd9a54a3c17f0b5a90b9a7444a1733e680a4faccd","last_reissued_at":"2026-05-18T00:53:41.189938Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:53:41.189938Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1612.09573","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":"2026-05-18T00:53:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8dewGUwTfyglKSNqlKHJ7i8Yq0RF6NGWQLA4D6P5ot9idKWa0AfdWIceH6jBvkbsZyufI0DnxnZeCa240VgXAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T06:14:35.764212Z"},"content_sha256":"2540aefebc52bbe83e2179f39a395ba5859185320bd0015d2b80d5633929ff30","schema_version":"1.0","event_id":"sha256:2540aefebc52bbe83e2179f39a395ba5859185320bd0015d2b80d5633929ff30"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:HBZJOI4JYWN6NOSDFK55TJKKHQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Endpoint Mapping properties of the Littlewood-Paley square function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Odysseas Bakas","submitted_at":"2016-12-30T19:48:36Z","abstract_excerpt":"In this note we give an alternative proof of a theorem due to Bourgain \\cite{Bourgain} concerning the growth of the constant in the Littlewood-Paley inequality on $\\mathbb{T}$ as $p \\rightarrow 1^+$. Our argument is based on the endpoint mapping properties of Marcinkiewicz multiplier operators, obtained by Tao and Wright in \\cite{TW}, and on Tao's converse extrapolation theorem \\cite{Tao}. Our method also establishes the growth of the constant in the Littlewood-Paley inequality on $\\mathbb{T}^n$ as $p \\rightarrow 1^+$. Furthermore, we obtain sharp weak-type inequalities for the Littlewood-Pale"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.09573","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":"2026-05-18T00:53:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2uwEO8F6i1hH5QQDeS4H76edTnmssCfgjXsweH6I0rGYggsJZcR9ryySq7tw4OcZnCUX2lK7a9GFSvnDUw0rCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T06:14:35.764655Z"},"content_sha256":"c128e10d9489c01c54febf781c4a56cb48a590c34d12751290b64f42a719476c","schema_version":"1.0","event_id":"sha256:c128e10d9489c01c54febf781c4a56cb48a590c34d12751290b64f42a719476c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HBZJOI4JYWN6NOSDFK55TJKKHQ/bundle.json","state_url":"https://pith.science/pith/HBZJOI4JYWN6NOSDFK55TJKKHQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HBZJOI4JYWN6NOSDFK55TJKKHQ/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":"2026-05-26T06:14:35Z","links":{"resolver":"https://pith.science/pith/HBZJOI4JYWN6NOSDFK55TJKKHQ","bundle":"https://pith.science/pith/HBZJOI4JYWN6NOSDFK55TJKKHQ/bundle.json","state":"https://pith.science/pith/HBZJOI4JYWN6NOSDFK55TJKKHQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HBZJOI4JYWN6NOSDFK55TJKKHQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:HBZJOI4JYWN6NOSDFK55TJKKHQ","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":"02cbb8ecfb4c08d0878647ce6abe9cd432020467aaaae546e6a8ebcbc785f538","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-12-30T19:48:36Z","title_canon_sha256":"59e6351ca74d0324e1b8e350dddcc5383e83be5f682645ec79a2add828ab2e9f"},"schema_version":"1.0","source":{"id":"1612.09573","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.09573","created_at":"2026-05-18T00:53:41Z"},{"alias_kind":"arxiv_version","alias_value":"1612.09573v1","created_at":"2026-05-18T00:53:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.09573","created_at":"2026-05-18T00:53:41Z"},{"alias_kind":"pith_short_12","alias_value":"HBZJOI4JYWN6","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_16","alias_value":"HBZJOI4JYWN6NOSD","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_8","alias_value":"HBZJOI4J","created_at":"2026-05-18T12:30:19Z"}],"graph_snapshots":[{"event_id":"sha256:c128e10d9489c01c54febf781c4a56cb48a590c34d12751290b64f42a719476c","target":"graph","created_at":"2026-05-18T00:53:41Z","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":"In this note we give an alternative proof of a theorem due to Bourgain \\cite{Bourgain} concerning the growth of the constant in the Littlewood-Paley inequality on $\\mathbb{T}$ as $p \\rightarrow 1^+$. Our argument is based on the endpoint mapping properties of Marcinkiewicz multiplier operators, obtained by Tao and Wright in \\cite{TW}, and on Tao's converse extrapolation theorem \\cite{Tao}. Our method also establishes the growth of the constant in the Littlewood-Paley inequality on $\\mathbb{T}^n$ as $p \\rightarrow 1^+$. Furthermore, we obtain sharp weak-type inequalities for the Littlewood-Pale","authors_text":"Odysseas Bakas","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-12-30T19:48:36Z","title":"Endpoint Mapping properties of the Littlewood-Paley square function"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.09573","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":"sha256:2540aefebc52bbe83e2179f39a395ba5859185320bd0015d2b80d5633929ff30","target":"record","created_at":"2026-05-18T00:53:41Z","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":"02cbb8ecfb4c08d0878647ce6abe9cd432020467aaaae546e6a8ebcbc785f538","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-12-30T19:48:36Z","title_canon_sha256":"59e6351ca74d0324e1b8e350dddcc5383e83be5f682645ec79a2add828ab2e9f"},"schema_version":"1.0","source":{"id":"1612.09573","kind":"arxiv","version":1}},"canonical_sha256":"3872972389c59be6ba432abbd9a54a3c17f0b5a90b9a7444a1733e680a4faccd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3872972389c59be6ba432abbd9a54a3c17f0b5a90b9a7444a1733e680a4faccd","first_computed_at":"2026-05-18T00:53:41.189938Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:53:41.189938Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"eDY+dBWlVyYbCM/Zj+ywrEyYQNNgk3odiviqFMlAbb+z/rRZTiE1UrtlEeCsymR8EXOGQLSJlYbV04jWVbFNDA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:53:41.190404Z","signed_message":"canonical_sha256_bytes"},"source_id":"1612.09573","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2540aefebc52bbe83e2179f39a395ba5859185320bd0015d2b80d5633929ff30","sha256:c128e10d9489c01c54febf781c4a56cb48a590c34d12751290b64f42a719476c"],"state_sha256":"8c57b2ffb2af37bcb24aa35c3accd4410d7dbbd3ac628d9b7e9efab6f16be3e9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iV6iQ+OK0NPLHUDjuOZOL80//WtlkdQvffOqkgb5BUGm9s/q0GyiMGTrCc29lB48du9eZUBxspTF3ejiNnM9Bg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T06:14:35.768224Z","bundle_sha256":"c820248ab6fc85769d210cc179d1185e788c417c08b3dcbb9001131c32f4e6a9"}}