{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:X32EQZFEOMLKOFW76R5SBJD2BK","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":"5c4e44dc2393484626a9d22822b13c1754197eec8d3ab9280a09d8f2945caa7e","cross_cats_sorted":["math.AP","math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2019-05-06T15:21:32Z","title_canon_sha256":"fd6500261e1644d0b06a8cbb5b2aafb7b0d99ada10b9ef9c1140093adcbbaf1d"},"schema_version":"1.0","source":{"id":"1905.02097","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1905.02097","created_at":"2026-05-17T23:42:02Z"},{"alias_kind":"arxiv_version","alias_value":"1905.02097v2","created_at":"2026-05-17T23:42:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1905.02097","created_at":"2026-05-17T23:42:02Z"},{"alias_kind":"pith_short_12","alias_value":"X32EQZFEOMLK","created_at":"2026-05-18T12:33:33Z"},{"alias_kind":"pith_short_16","alias_value":"X32EQZFEOMLKOFW7","created_at":"2026-05-18T12:33:33Z"},{"alias_kind":"pith_short_8","alias_value":"X32EQZFE","created_at":"2026-05-18T12:33:33Z"}],"graph_snapshots":[{"event_id":"sha256:360d16ef1d106cbfaa6d7177b9982073f7c561d45646b00f0b9f162537885dd0","target":"graph","created_at":"2026-05-17T23:42:02Z","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 $X$ be a ball quasi-Banach function space on ${\\mathbb R}^n$. In this article, the authors introduce the weak Hardy-type space $WH_X({\\mathbb R}^n)$, associated with $X$, via the radial maximal function. Assuming that the powered Hardy--Littlewood maximal operator satisfies some Fefferman--Stein vector-valued maximal inequality on $X$ as well as it is bounded on both the weak ball quasi-Banach function space $WX$ and the associated space, the authors then establish several real-variable characterizations of $WH_X({\\mathbb R}^n)$, respectively, in terms of various maximal functions, atoms a","authors_text":"Dachun Yang, Songbai Wang, Wen Yuan, Yangyang Zhang","cross_cats":["math.AP","math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2019-05-06T15:21:32Z","title":"Weak Hardy-Type Spaces Associated with Ball Quasi-Banach Function Spaces I: Decompositions with Applications to Boundedness of Calder\\'on--Zygmund Operators"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.02097","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":"sha256:1d207a427d57e6a96c8d6d34cba797e8e4265c804a81d84f30eb138229ddfa90","target":"record","created_at":"2026-05-17T23:42:02Z","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":"5c4e44dc2393484626a9d22822b13c1754197eec8d3ab9280a09d8f2945caa7e","cross_cats_sorted":["math.AP","math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2019-05-06T15:21:32Z","title_canon_sha256":"fd6500261e1644d0b06a8cbb5b2aafb7b0d99ada10b9ef9c1140093adcbbaf1d"},"schema_version":"1.0","source":{"id":"1905.02097","kind":"arxiv","version":2}},"canonical_sha256":"bef44864a47316a716dff47b20a47a0ab047c8a55dbdcd0606ba0e81c2fe070c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bef44864a47316a716dff47b20a47a0ab047c8a55dbdcd0606ba0e81c2fe070c","first_computed_at":"2026-05-17T23:42:02.226805Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:42:02.226805Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"KP8WXJ4u38UHL6FFJxvMoDK5xjnp43wgxIqorHrOKndlxuDe8WCk86LIbW3kNKmMg7SST3hvdlIjJoIF4GJfCg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:42:02.227295Z","signed_message":"canonical_sha256_bytes"},"source_id":"1905.02097","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1d207a427d57e6a96c8d6d34cba797e8e4265c804a81d84f30eb138229ddfa90","sha256:360d16ef1d106cbfaa6d7177b9982073f7c561d45646b00f0b9f162537885dd0"],"state_sha256":"73886cf79ade417fef842be13eaf0768a168de850eb5a44aea3208353a616545"}