{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:FFSD6IJBSJ7JN7BJUD6X7PXOZ5","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":"6c0aa0f315da965e3aa161b70b8a69067b025a74efade37deebfb9a9de544fa6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-11-03T10:27:15Z","title_canon_sha256":"07b769f971cf2305a9924578c93594e57585405f18bb583950ed4eceaed1cc4c"},"schema_version":"1.0","source":{"id":"1311.0453","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1311.0453","created_at":"2026-05-18T03:08:05Z"},{"alias_kind":"arxiv_version","alias_value":"1311.0453v1","created_at":"2026-05-18T03:08:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1311.0453","created_at":"2026-05-18T03:08:05Z"},{"alias_kind":"pith_short_12","alias_value":"FFSD6IJBSJ7J","created_at":"2026-05-18T12:27:45Z"},{"alias_kind":"pith_short_16","alias_value":"FFSD6IJBSJ7JN7BJ","created_at":"2026-05-18T12:27:45Z"},{"alias_kind":"pith_short_8","alias_value":"FFSD6IJB","created_at":"2026-05-18T12:27:45Z"}],"graph_snapshots":[{"event_id":"sha256:58d9930b315cb9213c5197bf3a8019d9568ac781dc3798cd7c296d1605610e16","target":"graph","created_at":"2026-05-18T03:08:05Z","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 paper the notion of an abstract square function (estimate) is introduced as an operator X to gamma (H; Y), where X, Y are Banach spaces, H is a Hilbert space, and gamma(H; Y) is the space of gamma-radonifying operators. By the seminal work of Kalton and Weis, this definition is a coherent generalisation of the classical notion of square function appearing in the theory of singular integrals. Given an abstract functional calculus (E, F, Phi) on a Banach space X, where F (O) is an algebra of scalar-valued functions on a set O, we define a square function Phi_gamma(f) for certain H-valued","authors_text":"Bernhard Hermann Haak (IMB), Markus Haase (DIAM)","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-11-03T10:27:15Z","title":"Square Function Estimates and Functional Calculi"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.0453","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":"sha256:073a083ca5f6401df0f1ff7e53beceb996a7bcf57d77db5d50113f27f2a14506","target":"record","created_at":"2026-05-18T03:08:05Z","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":"6c0aa0f315da965e3aa161b70b8a69067b025a74efade37deebfb9a9de544fa6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-11-03T10:27:15Z","title_canon_sha256":"07b769f971cf2305a9924578c93594e57585405f18bb583950ed4eceaed1cc4c"},"schema_version":"1.0","source":{"id":"1311.0453","kind":"arxiv","version":1}},"canonical_sha256":"29643f2121927e96fc29a0fd7fbeeecf591bfd44b376c4e3c964a34eb7112ee3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"29643f2121927e96fc29a0fd7fbeeecf591bfd44b376c4e3c964a34eb7112ee3","first_computed_at":"2026-05-18T03:08:05.898962Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:08:05.898962Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"C4pHNWLoywsAZ6Zns8a7yE5AcJSblMhP2HKY36FkbcERdLejbv9OVk4M6xhgouiUG2ohyg2ejaBK3+J2dS/2Aw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:08:05.899470Z","signed_message":"canonical_sha256_bytes"},"source_id":"1311.0453","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:073a083ca5f6401df0f1ff7e53beceb996a7bcf57d77db5d50113f27f2a14506","sha256:58d9930b315cb9213c5197bf3a8019d9568ac781dc3798cd7c296d1605610e16"],"state_sha256":"b649d90c0812952ddc8dbae34343150eb7f489fc61ac5548b47d262aabe32d93"}