{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:WRINWGSTIER3S7NKHSC6HJF4SV","short_pith_number":"pith:WRINWGST","canonical_record":{"source":{"id":"1601.05242","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-01-20T11:32:26Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"2e78c8c53fb00911570877977fa69f93bad74669e159d5c97ea2f943c86410cd","abstract_canon_sha256":"1dcd84de1b75618ebd00ad5249e088ed1b50b822a6e844df1ab7fd9e8dd484e5"},"schema_version":"1.0"},"canonical_sha256":"b450db1a534123b97daa3c85e3a4bc9560205973570383ba0860b6eeae5ac050","source":{"kind":"arxiv","id":"1601.05242","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.05242","created_at":"2026-05-18T01:22:05Z"},{"alias_kind":"arxiv_version","alias_value":"1601.05242v2","created_at":"2026-05-18T01:22:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.05242","created_at":"2026-05-18T01:22:05Z"},{"alias_kind":"pith_short_12","alias_value":"WRINWGSTIER3","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_16","alias_value":"WRINWGSTIER3S7NK","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_8","alias_value":"WRINWGST","created_at":"2026-05-18T12:30:51Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:WRINWGSTIER3S7NKHSC6HJF4SV","target":"record","payload":{"canonical_record":{"source":{"id":"1601.05242","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-01-20T11:32:26Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"2e78c8c53fb00911570877977fa69f93bad74669e159d5c97ea2f943c86410cd","abstract_canon_sha256":"1dcd84de1b75618ebd00ad5249e088ed1b50b822a6e844df1ab7fd9e8dd484e5"},"schema_version":"1.0"},"canonical_sha256":"b450db1a534123b97daa3c85e3a4bc9560205973570383ba0860b6eeae5ac050","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:22:05.303083Z","signature_b64":"gVotMPzLf24s4iM52uKEAD3bFZ8rGdhT0zWjs1cKJMDukjE3wjNDsnc8+dDbV6zWNost3CU2gWz/s7L4ZjUVCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b450db1a534123b97daa3c85e3a4bc9560205973570383ba0860b6eeae5ac050","last_reissued_at":"2026-05-18T01:22:05.302384Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:22:05.302384Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1601.05242","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":"2026-05-18T01:22:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jeMb+pSDiA2VzLorwMTwHAcXx3TYHL3QcGTl8H4Rawri1cOSFeFHqP6CZcMtrtvjF+3PKH8h/IYZWsUyFXgJAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T01:30:00.451964Z"},"content_sha256":"2893a7c3088b5ffa2c4dede481fb69b5b389cf344d732b6c3c0844bff51e9380","schema_version":"1.0","event_id":"sha256:2893a7c3088b5ffa2c4dede481fb69b5b389cf344d732b6c3c0844bff51e9380"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:WRINWGSTIER3S7NKHSC6HJF4SV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Littlewood-Paley Characterizations of Anisotropic Hardy-Lorentz Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.CA","authors_text":"Dachun Yang, Jun Liu, Wen Yuan","submitted_at":"2016-01-20T11:32:26Z","abstract_excerpt":"Let $p\\in(0,1]$, $q\\in(0,\\infty]$ and $A$ be a general expansive matrix on $\\mathbb{R}^n$. Let $H^{p,q}_A(\\mathbb{R}^n)$ be the anisotropic Hardy-Lorentz spaces associated with $A$ defined via the non-tangential grand maximal function. In this article, the authors characterize $H^{p,q}_A(\\mathbb{R}^n)$ in terms of the Lusin-area function, the Littlewood-Paley $g$-function or the Littlewood-Paley $g_\\lambda^*$-function via first establishing an anisotropic Fefferman-Stein vector-valued inequality in the Lorentz space $L^{p,q}(\\mathbb{R}^n)$. All these characterizations are new even for the clas"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.05242","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":"2026-05-18T01:22:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OiXu/Gu0a3Qft5X/U0Ih946cBvrss9/xlPcOeMIQLb9DGcFJFqlQdpQUXj8SxqdQ8hi/a0i9CgaDKG0d+yTCDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T01:30:00.452667Z"},"content_sha256":"f85b218743f2f8fcb2ad087f3c17e154c394bfe1c0669988a60159dd10cb8f11","schema_version":"1.0","event_id":"sha256:f85b218743f2f8fcb2ad087f3c17e154c394bfe1c0669988a60159dd10cb8f11"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WRINWGSTIER3S7NKHSC6HJF4SV/bundle.json","state_url":"https://pith.science/pith/WRINWGSTIER3S7NKHSC6HJF4SV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WRINWGSTIER3S7NKHSC6HJF4SV/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":"2026-06-09T01:30:00Z","links":{"resolver":"https://pith.science/pith/WRINWGSTIER3S7NKHSC6HJF4SV","bundle":"https://pith.science/pith/WRINWGSTIER3S7NKHSC6HJF4SV/bundle.json","state":"https://pith.science/pith/WRINWGSTIER3S7NKHSC6HJF4SV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WRINWGSTIER3S7NKHSC6HJF4SV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:WRINWGSTIER3S7NKHSC6HJF4SV","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":"1dcd84de1b75618ebd00ad5249e088ed1b50b822a6e844df1ab7fd9e8dd484e5","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-01-20T11:32:26Z","title_canon_sha256":"2e78c8c53fb00911570877977fa69f93bad74669e159d5c97ea2f943c86410cd"},"schema_version":"1.0","source":{"id":"1601.05242","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.05242","created_at":"2026-05-18T01:22:05Z"},{"alias_kind":"arxiv_version","alias_value":"1601.05242v2","created_at":"2026-05-18T01:22:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.05242","created_at":"2026-05-18T01:22:05Z"},{"alias_kind":"pith_short_12","alias_value":"WRINWGSTIER3","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_16","alias_value":"WRINWGSTIER3S7NK","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_8","alias_value":"WRINWGST","created_at":"2026-05-18T12:30:51Z"}],"graph_snapshots":[{"event_id":"sha256:f85b218743f2f8fcb2ad087f3c17e154c394bfe1c0669988a60159dd10cb8f11","target":"graph","created_at":"2026-05-18T01:22:05Z","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":"Let $p\\in(0,1]$, $q\\in(0,\\infty]$ and $A$ be a general expansive matrix on $\\mathbb{R}^n$. Let $H^{p,q}_A(\\mathbb{R}^n)$ be the anisotropic Hardy-Lorentz spaces associated with $A$ defined via the non-tangential grand maximal function. In this article, the authors characterize $H^{p,q}_A(\\mathbb{R}^n)$ in terms of the Lusin-area function, the Littlewood-Paley $g$-function or the Littlewood-Paley $g_\\lambda^*$-function via first establishing an anisotropic Fefferman-Stein vector-valued inequality in the Lorentz space $L^{p,q}(\\mathbb{R}^n)$. All these characterizations are new even for the clas","authors_text":"Dachun Yang, Jun Liu, Wen Yuan","cross_cats":["math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-01-20T11:32:26Z","title":"Littlewood-Paley Characterizations of Anisotropic Hardy-Lorentz Spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.05242","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":"sha256:2893a7c3088b5ffa2c4dede481fb69b5b389cf344d732b6c3c0844bff51e9380","target":"record","created_at":"2026-05-18T01:22:05Z","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":"1dcd84de1b75618ebd00ad5249e088ed1b50b822a6e844df1ab7fd9e8dd484e5","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-01-20T11:32:26Z","title_canon_sha256":"2e78c8c53fb00911570877977fa69f93bad74669e159d5c97ea2f943c86410cd"},"schema_version":"1.0","source":{"id":"1601.05242","kind":"arxiv","version":2}},"canonical_sha256":"b450db1a534123b97daa3c85e3a4bc9560205973570383ba0860b6eeae5ac050","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b450db1a534123b97daa3c85e3a4bc9560205973570383ba0860b6eeae5ac050","first_computed_at":"2026-05-18T01:22:05.302384Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:22:05.302384Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gVotMPzLf24s4iM52uKEAD3bFZ8rGdhT0zWjs1cKJMDukjE3wjNDsnc8+dDbV6zWNost3CU2gWz/s7L4ZjUVCw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:22:05.303083Z","signed_message":"canonical_sha256_bytes"},"source_id":"1601.05242","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2893a7c3088b5ffa2c4dede481fb69b5b389cf344d732b6c3c0844bff51e9380","sha256:f85b218743f2f8fcb2ad087f3c17e154c394bfe1c0669988a60159dd10cb8f11"],"state_sha256":"be55e4cbdc8bdc5a8b048bbe09479542cad0ce0e3b818f7a02e90015562c14e3"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Y/Ndg15/+TALrIa3+lb8mPglmacRyVhUIoafZ6Pyfa0OcOBS5eLhizedMtBVG5MghcDS2NJYf1uFc4+zS6pNAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T01:30:00.456880Z","bundle_sha256":"a1ac0b8934dd47a33e7339a0b82834ec0ff983bd4c28038998a8792f302af696"}}