{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:A67R6VDDK4QGK62HJ75GMJLH4P","short_pith_number":"pith:A67R6VDD","canonical_record":{"source":{"id":"1908.03291","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2019-08-09T03:14:06Z","cross_cats_sorted":["math.AP","math.FA"],"title_canon_sha256":"6650f2ddc9774b9b3b1eb876530ec7a6d2d90f8612c3d7a8075cd6014b99e713","abstract_canon_sha256":"972b0c9348f8de084a1977487f05e2ba9297e696130d8d991cc9fde74e5830d7"},"schema_version":"1.0"},"canonical_sha256":"07bf1f54635720657b474ffa662567e3eb4424005fdbd8ff07e115bd388f9b1d","source":{"kind":"arxiv","id":"1908.03291","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.03291","created_at":"2026-07-04T23:52:38Z"},{"alias_kind":"arxiv_version","alias_value":"1908.03291v1","created_at":"2026-07-04T23:52:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.03291","created_at":"2026-07-04T23:52:38Z"},{"alias_kind":"pith_short_12","alias_value":"A67R6VDDK4QG","created_at":"2026-07-04T23:52:38Z"},{"alias_kind":"pith_short_16","alias_value":"A67R6VDDK4QGK62H","created_at":"2026-07-04T23:52:38Z"},{"alias_kind":"pith_short_8","alias_value":"A67R6VDD","created_at":"2026-07-04T23:52:38Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:A67R6VDDK4QGK62HJ75GMJLH4P","target":"record","payload":{"canonical_record":{"source":{"id":"1908.03291","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2019-08-09T03:14:06Z","cross_cats_sorted":["math.AP","math.FA"],"title_canon_sha256":"6650f2ddc9774b9b3b1eb876530ec7a6d2d90f8612c3d7a8075cd6014b99e713","abstract_canon_sha256":"972b0c9348f8de084a1977487f05e2ba9297e696130d8d991cc9fde74e5830d7"},"schema_version":"1.0"},"canonical_sha256":"07bf1f54635720657b474ffa662567e3eb4424005fdbd8ff07e115bd388f9b1d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T23:52:38.390356Z","signature_b64":"4p69wKEi4le/0XqkcRHffzO3p1zpr1iY1FPzGIMg2Ll/GfBN2y4KOWmje7Ah+LP/54yrlsvx9t4pUqMrl0KXCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"07bf1f54635720657b474ffa662567e3eb4424005fdbd8ff07e115bd388f9b1d","last_reissued_at":"2026-07-04T23:52:38.389996Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T23:52:38.389996Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1908.03291","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":"2026-07-04T23:52:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"m7hgAhBHeDoUED4vLVmJn+xT80EaweV6SCtxSfwBfp8ViouQ7zI4GURoENu6+noaKTbkrNw6FY3ozvnx3DSZDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T17:07:40.940223Z"},"content_sha256":"4ebfb6b8359fce1e20c4a573ddef0f42c8489e3951a65915f1aa80d995174930","schema_version":"1.0","event_id":"sha256:4ebfb6b8359fce1e20c4a573ddef0f42c8489e3951a65915f1aa80d995174930"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:A67R6VDDK4QGK62HJ75GMJLH4P","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On Function Spaces with Mixed Norms --- A Survey","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.FA"],"primary_cat":"math.CA","authors_text":"Dachun Yang, Long Huang","submitted_at":"2019-08-09T03:14:06Z","abstract_excerpt":"The targets of this article are threefold. The first one is to give a survey on the recent developments of function spaces with mixed norms, including mixed Lebesgue spaces, iterated weak Lebesgue spaces, weak mixed-norm Lebesgue spaces and mixed Morrey spaces as well as anisotropic mixed-norm Hardy spaces. The second one is to provide a detailed proof for a useful inequality about mixed Lebesgue norms and the Hardy--Littlewood maximal operator and also to improve some known results on the maximal function characterizations of anisotropic mixed-norm Hardy spaces and the boundedness of Calder\\'"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.03291","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.03291/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":"2026-07-04T23:52:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"e6L0wsAgG9NxBqhH5fR/8dgA4gpxhpWavcv9ZAgYckXeS2ulzvWksjsuLOSQ+TZq3vjkJfu9xKL5YM8f1umNDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T17:07:40.940765Z"},"content_sha256":"3e3c3754f845c9a280236bfd71b1cd55522c8256fccaef9a2d65b0fccc2fc071","schema_version":"1.0","event_id":"sha256:3e3c3754f845c9a280236bfd71b1cd55522c8256fccaef9a2d65b0fccc2fc071"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/A67R6VDDK4QGK62HJ75GMJLH4P/bundle.json","state_url":"https://pith.science/pith/A67R6VDDK4QGK62HJ75GMJLH4P/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/A67R6VDDK4QGK62HJ75GMJLH4P/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":"2026-08-16T17:07:40Z","links":{"resolver":"https://pith.science/pith/A67R6VDDK4QGK62HJ75GMJLH4P","bundle":"https://pith.science/pith/A67R6VDDK4QGK62HJ75GMJLH4P/bundle.json","state":"https://pith.science/pith/A67R6VDDK4QGK62HJ75GMJLH4P/state.json","well_known_bundle":"https://pith.science/.well-known/pith/A67R6VDDK4QGK62HJ75GMJLH4P/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:A67R6VDDK4QGK62HJ75GMJLH4P","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":"972b0c9348f8de084a1977487f05e2ba9297e696130d8d991cc9fde74e5830d7","cross_cats_sorted":["math.AP","math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2019-08-09T03:14:06Z","title_canon_sha256":"6650f2ddc9774b9b3b1eb876530ec7a6d2d90f8612c3d7a8075cd6014b99e713"},"schema_version":"1.0","source":{"id":"1908.03291","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.03291","created_at":"2026-07-04T23:52:38Z"},{"alias_kind":"arxiv_version","alias_value":"1908.03291v1","created_at":"2026-07-04T23:52:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.03291","created_at":"2026-07-04T23:52:38Z"},{"alias_kind":"pith_short_12","alias_value":"A67R6VDDK4QG","created_at":"2026-07-04T23:52:38Z"},{"alias_kind":"pith_short_16","alias_value":"A67R6VDDK4QGK62H","created_at":"2026-07-04T23:52:38Z"},{"alias_kind":"pith_short_8","alias_value":"A67R6VDD","created_at":"2026-07-04T23:52:38Z"}],"graph_snapshots":[{"event_id":"sha256:3e3c3754f845c9a280236bfd71b1cd55522c8256fccaef9a2d65b0fccc2fc071","target":"graph","created_at":"2026-07-04T23:52:38Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1908.03291/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The targets of this article are threefold. The first one is to give a survey on the recent developments of function spaces with mixed norms, including mixed Lebesgue spaces, iterated weak Lebesgue spaces, weak mixed-norm Lebesgue spaces and mixed Morrey spaces as well as anisotropic mixed-norm Hardy spaces. The second one is to provide a detailed proof for a useful inequality about mixed Lebesgue norms and the Hardy--Littlewood maximal operator and also to improve some known results on the maximal function characterizations of anisotropic mixed-norm Hardy spaces and the boundedness of Calder\\'","authors_text":"Dachun Yang, Long Huang","cross_cats":["math.AP","math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2019-08-09T03:14:06Z","title":"On Function Spaces with Mixed Norms --- A Survey"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.03291","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":"sha256:4ebfb6b8359fce1e20c4a573ddef0f42c8489e3951a65915f1aa80d995174930","target":"record","created_at":"2026-07-04T23:52:38Z","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":"972b0c9348f8de084a1977487f05e2ba9297e696130d8d991cc9fde74e5830d7","cross_cats_sorted":["math.AP","math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2019-08-09T03:14:06Z","title_canon_sha256":"6650f2ddc9774b9b3b1eb876530ec7a6d2d90f8612c3d7a8075cd6014b99e713"},"schema_version":"1.0","source":{"id":"1908.03291","kind":"arxiv","version":1}},"canonical_sha256":"07bf1f54635720657b474ffa662567e3eb4424005fdbd8ff07e115bd388f9b1d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"07bf1f54635720657b474ffa662567e3eb4424005fdbd8ff07e115bd388f9b1d","first_computed_at":"2026-07-04T23:52:38.389996Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T23:52:38.389996Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4p69wKEi4le/0XqkcRHffzO3p1zpr1iY1FPzGIMg2Ll/GfBN2y4KOWmje7Ah+LP/54yrlsvx9t4pUqMrl0KXCA==","signature_status":"signed_v1","signed_at":"2026-07-04T23:52:38.390356Z","signed_message":"canonical_sha256_bytes"},"source_id":"1908.03291","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4ebfb6b8359fce1e20c4a573ddef0f42c8489e3951a65915f1aa80d995174930","sha256:3e3c3754f845c9a280236bfd71b1cd55522c8256fccaef9a2d65b0fccc2fc071"],"state_sha256":"5d103e5786480e6db88a1a46b13363abed51ef6f92a935e4cc55d0632a8cb681"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gomfDEvjsJSzrFvXsy81KQ0zMXS8GUnn5eF6HT0s1QnMumhnyu0INdyIOmtFP7/C6cLJxwh1SDNv/Me78XDxDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-16T17:07:40.946437Z","bundle_sha256":"dbeab4eed216dd65be37565dc4a2894937694f80db55980cf3181e76566b3c9d"}}