{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:FPGJXB3BXPUR7JRDQMNWQKOQFO","short_pith_number":"pith:FPGJXB3B","canonical_record":{"source":{"id":"2003.11843","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2020-03-26T11:35:01Z","cross_cats_sorted":[],"title_canon_sha256":"5b2d836e309b1b00a17dbcfbe7ae8e539a62a60c3ac4da242edbf9f1502b834a","abstract_canon_sha256":"a1b47622c4919490dd1fd7ef9d3b6eba0d71ea1a66bc7e534027a741fe7245a0"},"schema_version":"1.0"},"canonical_sha256":"2bcc9b8761bbe91fa623831b6829d02bbebbee6ad695fa225b03086e5a7e36ca","source":{"kind":"arxiv","id":"2003.11843","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2003.11843","created_at":"2026-07-05T03:02:07Z"},{"alias_kind":"arxiv_version","alias_value":"2003.11843v3","created_at":"2026-07-05T03:02:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2003.11843","created_at":"2026-07-05T03:02:07Z"},{"alias_kind":"pith_short_12","alias_value":"FPGJXB3BXPUR","created_at":"2026-07-05T03:02:07Z"},{"alias_kind":"pith_short_16","alias_value":"FPGJXB3BXPUR7JRD","created_at":"2026-07-05T03:02:07Z"},{"alias_kind":"pith_short_8","alias_value":"FPGJXB3B","created_at":"2026-07-05T03:02:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:FPGJXB3BXPUR7JRDQMNWQKOQFO","target":"record","payload":{"canonical_record":{"source":{"id":"2003.11843","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2020-03-26T11:35:01Z","cross_cats_sorted":[],"title_canon_sha256":"5b2d836e309b1b00a17dbcfbe7ae8e539a62a60c3ac4da242edbf9f1502b834a","abstract_canon_sha256":"a1b47622c4919490dd1fd7ef9d3b6eba0d71ea1a66bc7e534027a741fe7245a0"},"schema_version":"1.0"},"canonical_sha256":"2bcc9b8761bbe91fa623831b6829d02bbebbee6ad695fa225b03086e5a7e36ca","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:02:07.702883Z","signature_b64":"r0umtX/ZV0/eJOV82que35m20I4C22aCP0DJBgi4ewYXQCc8kOQQYXVHHJQfsybatmmz/IYPIl1zZqYjHCDjAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2bcc9b8761bbe91fa623831b6829d02bbebbee6ad695fa225b03086e5a7e36ca","last_reissued_at":"2026-07-05T03:02:07.702468Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:02:07.702468Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2003.11843","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T03:02:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"T28Owp+Vhc0IVsBK/SUR0LPkoORF1h2QXdDuahk8nO6qZY3/0HRT7l+qUzYdQeiOmM+lLZ+YU60yZ+JoI5ccCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T09:33:37.531293Z"},"content_sha256":"3876a742b41091cf5393904fad276dc94f2a09dec94e7e770538b0a8f9ae2c6b","schema_version":"1.0","event_id":"sha256:3876a742b41091cf5393904fad276dc94f2a09dec94e7e770538b0a8f9ae2c6b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:FPGJXB3BXPUR7JRDQMNWQKOQFO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Dimension-Free Square Function Estimates for Dunkl Operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Huaiqian Li, Mingfeng Zhao","submitted_at":"2020-03-26T11:35:01Z","abstract_excerpt":"Dunkl operators may be regarded as differential-difference operators parameterized by finite reflection groups and multiplicity functions. In this paper, the Littlewood--Paley square function for Dunkl heat flows in $\\mathbb{R}^d$ is introduced by employing the full \"gradient\" induced by the corresponding carr\\'{e} du champ operator and then the $L^p$ boundedness is studied for all $p\\in(1,\\infty)$. For $p\\in(1,2]$, we successfully adapt Stein's heat flows approach to overcome the difficulty caused by the difference part of the Dunkl operator and establish the $L^p$ boundedness, while for $p\\i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2003.11843","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2003.11843/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T03:02:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"c6Bzf2qX40frGCdm+VFhpHxJIsB1Uh6nn4mZ7vsP6vggN+snXuOUGU0SeBwrJDBdkmMsvjK3DObP4HnXYbxOBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T09:33:37.531814Z"},"content_sha256":"0fc8702eb4d8020c296eb7f31c07f957b7b191aac8211cc0d42031faaf5c4265","schema_version":"1.0","event_id":"sha256:0fc8702eb4d8020c296eb7f31c07f957b7b191aac8211cc0d42031faaf5c4265"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FPGJXB3BXPUR7JRDQMNWQKOQFO/bundle.json","state_url":"https://pith.science/pith/FPGJXB3BXPUR7JRDQMNWQKOQFO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FPGJXB3BXPUR7JRDQMNWQKOQFO/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":"2026-08-14T09:33:37Z","links":{"resolver":"https://pith.science/pith/FPGJXB3BXPUR7JRDQMNWQKOQFO","bundle":"https://pith.science/pith/FPGJXB3BXPUR7JRDQMNWQKOQFO/bundle.json","state":"https://pith.science/pith/FPGJXB3BXPUR7JRDQMNWQKOQFO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FPGJXB3BXPUR7JRDQMNWQKOQFO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:FPGJXB3BXPUR7JRDQMNWQKOQFO","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":"a1b47622c4919490dd1fd7ef9d3b6eba0d71ea1a66bc7e534027a741fe7245a0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2020-03-26T11:35:01Z","title_canon_sha256":"5b2d836e309b1b00a17dbcfbe7ae8e539a62a60c3ac4da242edbf9f1502b834a"},"schema_version":"1.0","source":{"id":"2003.11843","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2003.11843","created_at":"2026-07-05T03:02:07Z"},{"alias_kind":"arxiv_version","alias_value":"2003.11843v3","created_at":"2026-07-05T03:02:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2003.11843","created_at":"2026-07-05T03:02:07Z"},{"alias_kind":"pith_short_12","alias_value":"FPGJXB3BXPUR","created_at":"2026-07-05T03:02:07Z"},{"alias_kind":"pith_short_16","alias_value":"FPGJXB3BXPUR7JRD","created_at":"2026-07-05T03:02:07Z"},{"alias_kind":"pith_short_8","alias_value":"FPGJXB3B","created_at":"2026-07-05T03:02:07Z"}],"graph_snapshots":[{"event_id":"sha256:0fc8702eb4d8020c296eb7f31c07f957b7b191aac8211cc0d42031faaf5c4265","target":"graph","created_at":"2026-07-05T03:02:07Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2003.11843/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Dunkl operators may be regarded as differential-difference operators parameterized by finite reflection groups and multiplicity functions. In this paper, the Littlewood--Paley square function for Dunkl heat flows in $\\mathbb{R}^d$ is introduced by employing the full \"gradient\" induced by the corresponding carr\\'{e} du champ operator and then the $L^p$ boundedness is studied for all $p\\in(1,\\infty)$. For $p\\in(1,2]$, we successfully adapt Stein's heat flows approach to overcome the difficulty caused by the difference part of the Dunkl operator and establish the $L^p$ boundedness, while for $p\\i","authors_text":"Huaiqian Li, Mingfeng Zhao","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2020-03-26T11:35:01Z","title":"Dimension-Free Square Function Estimates for Dunkl Operators"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2003.11843","kind":"arxiv","version":3},"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":"sha256:3876a742b41091cf5393904fad276dc94f2a09dec94e7e770538b0a8f9ae2c6b","target":"record","created_at":"2026-07-05T03:02:07Z","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":"a1b47622c4919490dd1fd7ef9d3b6eba0d71ea1a66bc7e534027a741fe7245a0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2020-03-26T11:35:01Z","title_canon_sha256":"5b2d836e309b1b00a17dbcfbe7ae8e539a62a60c3ac4da242edbf9f1502b834a"},"schema_version":"1.0","source":{"id":"2003.11843","kind":"arxiv","version":3}},"canonical_sha256":"2bcc9b8761bbe91fa623831b6829d02bbebbee6ad695fa225b03086e5a7e36ca","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2bcc9b8761bbe91fa623831b6829d02bbebbee6ad695fa225b03086e5a7e36ca","first_computed_at":"2026-07-05T03:02:07.702468Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T03:02:07.702468Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"r0umtX/ZV0/eJOV82que35m20I4C22aCP0DJBgi4ewYXQCc8kOQQYXVHHJQfsybatmmz/IYPIl1zZqYjHCDjAw==","signature_status":"signed_v1","signed_at":"2026-07-05T03:02:07.702883Z","signed_message":"canonical_sha256_bytes"},"source_id":"2003.11843","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3876a742b41091cf5393904fad276dc94f2a09dec94e7e770538b0a8f9ae2c6b","sha256:0fc8702eb4d8020c296eb7f31c07f957b7b191aac8211cc0d42031faaf5c4265"],"state_sha256":"9c6b4e705e786b2367c8658b11def7983c6da25e487a542b9a14ba29746b924b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"swfW6/pUaUzlgl1o37LKVZ4Nc9lBEajB2PwSDbSyyxwsBBWn5rOKXOCuAkX8sOKUGFz+glT31mlqhjuVmDyIAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-14T09:33:37.535531Z","bundle_sha256":"1c8b1e773de87b252154b69b374b1ab56d9448da384076c95e4bebd2db8f2062"}}