{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:MKIYIU6DY5L46WH57XX4U2ARZQ","short_pith_number":"pith:MKIYIU6D","canonical_record":{"source":{"id":"1805.07778","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-05-20T15:33:32Z","cross_cats_sorted":["math.AP","math.FA"],"title_canon_sha256":"66aec561c7ac01db3121a1559c88351b78d352734f480aba14108823fc4f8248","abstract_canon_sha256":"37196024df42a5f7902723d27ead7b2fc4d0692757b26a475b20f50aba18ac79"},"schema_version":"1.0"},"canonical_sha256":"62918453c3c757cf58fdfdefca6811cc3be93e040c64f8f0ab543d7cf63bc4cb","source":{"kind":"arxiv","id":"1805.07778","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1805.07778","created_at":"2026-05-18T00:15:32Z"},{"alias_kind":"arxiv_version","alias_value":"1805.07778v1","created_at":"2026-05-18T00:15:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.07778","created_at":"2026-05-18T00:15:32Z"},{"alias_kind":"pith_short_12","alias_value":"MKIYIU6DY5L4","created_at":"2026-05-18T12:32:37Z"},{"alias_kind":"pith_short_16","alias_value":"MKIYIU6DY5L46WH5","created_at":"2026-05-18T12:32:37Z"},{"alias_kind":"pith_short_8","alias_value":"MKIYIU6D","created_at":"2026-05-18T12:32:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:MKIYIU6DY5L46WH57XX4U2ARZQ","target":"record","payload":{"canonical_record":{"source":{"id":"1805.07778","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-05-20T15:33:32Z","cross_cats_sorted":["math.AP","math.FA"],"title_canon_sha256":"66aec561c7ac01db3121a1559c88351b78d352734f480aba14108823fc4f8248","abstract_canon_sha256":"37196024df42a5f7902723d27ead7b2fc4d0692757b26a475b20f50aba18ac79"},"schema_version":"1.0"},"canonical_sha256":"62918453c3c757cf58fdfdefca6811cc3be93e040c64f8f0ab543d7cf63bc4cb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:15:32.588140Z","signature_b64":"hCXNVwzmlp+jCEZSV1Q3kCNPHQcH+jDYVUZWQltGRvrCygQrEd3VQKYUWtUoOYqtlNfn6FFeeKWfcpWlEf4JCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"62918453c3c757cf58fdfdefca6811cc3be93e040c64f8f0ab543d7cf63bc4cb","last_reissued_at":"2026-05-18T00:15:32.587544Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:15:32.587544Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1805.07778","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":"2026-05-18T00:15:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1FhZn6uu7D703XPJ6DBgpC+3ZMmxEjVdKCRRTz+L24bQdKOe2bQicEYANS2lcTQDkMaqTY21Fm9FHj4ygxf0Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T23:05:16.715077Z"},"content_sha256":"db66cd18c95202e93468489ebc75c643823f6cc3da486854d17032c1b0bf9fc5","schema_version":"1.0","event_id":"sha256:db66cd18c95202e93468489ebc75c643823f6cc3da486854d17032c1b0bf9fc5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:MKIYIU6DY5L46WH57XX4U2ARZQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Variable Weak Hardy Spaces $W\\!H_L^{p(\\cdot)}({\\mathbb R}^n)$ Associated with Operators Satisfying Davies-Gaffney Estimates","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.FA"],"primary_cat":"math.CA","authors_text":"Ciqiang Zhuo, Dachun Yang","submitted_at":"2018-05-20T15:33:32Z","abstract_excerpt":"Let $p(\\cdot):\\ \\mathbb R^n\\to(0,1]$ be a variable exponent function satisfying the globally log-H\\\"older continuous condition and $L$ a one to one operator of type $\\omega$ in $L^2({\\mathbb R}^n)$, with $\\omega\\in[0,\\,\\pi/2)$, which has a bounded holomorphic functional calculus and satisfies the Davies-Gaffney estimates. In this article, the authors introduce the variable weak Hardy space $W\\!H_L^{p(\\cdot)}(\\mathbb R^n)$ associated with $L$ via the corresponding square function. Its molecular characterization is then established by means of the atomic decomposition of the variable weak tent s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.07778","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":""},"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":"2026-05-18T00:15:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xkR37IvmVVdV93vm12TTw1jkCd29Ki6XvVRm1IiCjXawGePj+HMdQiyw0iBjD9bcCXT59WXreEpU3RRmrnRjCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T23:05:16.715811Z"},"content_sha256":"49fe25d470ae1aa237ae3bc6fd4e9f2291dded353096d1694b1d25cf1c8ec046","schema_version":"1.0","event_id":"sha256:49fe25d470ae1aa237ae3bc6fd4e9f2291dded353096d1694b1d25cf1c8ec046"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MKIYIU6DY5L46WH57XX4U2ARZQ/bundle.json","state_url":"https://pith.science/pith/MKIYIU6DY5L46WH57XX4U2ARZQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MKIYIU6DY5L46WH57XX4U2ARZQ/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":"2026-06-05T23:05:16Z","links":{"resolver":"https://pith.science/pith/MKIYIU6DY5L46WH57XX4U2ARZQ","bundle":"https://pith.science/pith/MKIYIU6DY5L46WH57XX4U2ARZQ/bundle.json","state":"https://pith.science/pith/MKIYIU6DY5L46WH57XX4U2ARZQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MKIYIU6DY5L46WH57XX4U2ARZQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:MKIYIU6DY5L46WH57XX4U2ARZQ","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":"37196024df42a5f7902723d27ead7b2fc4d0692757b26a475b20f50aba18ac79","cross_cats_sorted":["math.AP","math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-05-20T15:33:32Z","title_canon_sha256":"66aec561c7ac01db3121a1559c88351b78d352734f480aba14108823fc4f8248"},"schema_version":"1.0","source":{"id":"1805.07778","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1805.07778","created_at":"2026-05-18T00:15:32Z"},{"alias_kind":"arxiv_version","alias_value":"1805.07778v1","created_at":"2026-05-18T00:15:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.07778","created_at":"2026-05-18T00:15:32Z"},{"alias_kind":"pith_short_12","alias_value":"MKIYIU6DY5L4","created_at":"2026-05-18T12:32:37Z"},{"alias_kind":"pith_short_16","alias_value":"MKIYIU6DY5L46WH5","created_at":"2026-05-18T12:32:37Z"},{"alias_kind":"pith_short_8","alias_value":"MKIYIU6D","created_at":"2026-05-18T12:32:37Z"}],"graph_snapshots":[{"event_id":"sha256:49fe25d470ae1aa237ae3bc6fd4e9f2291dded353096d1694b1d25cf1c8ec046","target":"graph","created_at":"2026-05-18T00:15:32Z","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 $p(\\cdot):\\ \\mathbb R^n\\to(0,1]$ be a variable exponent function satisfying the globally log-H\\\"older continuous condition and $L$ a one to one operator of type $\\omega$ in $L^2({\\mathbb R}^n)$, with $\\omega\\in[0,\\,\\pi/2)$, which has a bounded holomorphic functional calculus and satisfies the Davies-Gaffney estimates. In this article, the authors introduce the variable weak Hardy space $W\\!H_L^{p(\\cdot)}(\\mathbb R^n)$ associated with $L$ via the corresponding square function. Its molecular characterization is then established by means of the atomic decomposition of the variable weak tent s","authors_text":"Ciqiang Zhuo, Dachun Yang","cross_cats":["math.AP","math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-05-20T15:33:32Z","title":"Variable Weak Hardy Spaces $W\\!H_L^{p(\\cdot)}({\\mathbb R}^n)$ Associated with Operators Satisfying Davies-Gaffney Estimates"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.07778","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:db66cd18c95202e93468489ebc75c643823f6cc3da486854d17032c1b0bf9fc5","target":"record","created_at":"2026-05-18T00:15:32Z","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":"37196024df42a5f7902723d27ead7b2fc4d0692757b26a475b20f50aba18ac79","cross_cats_sorted":["math.AP","math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-05-20T15:33:32Z","title_canon_sha256":"66aec561c7ac01db3121a1559c88351b78d352734f480aba14108823fc4f8248"},"schema_version":"1.0","source":{"id":"1805.07778","kind":"arxiv","version":1}},"canonical_sha256":"62918453c3c757cf58fdfdefca6811cc3be93e040c64f8f0ab543d7cf63bc4cb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"62918453c3c757cf58fdfdefca6811cc3be93e040c64f8f0ab543d7cf63bc4cb","first_computed_at":"2026-05-18T00:15:32.587544Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:15:32.587544Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"hCXNVwzmlp+jCEZSV1Q3kCNPHQcH+jDYVUZWQltGRvrCygQrEd3VQKYUWtUoOYqtlNfn6FFeeKWfcpWlEf4JCA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:15:32.588140Z","signed_message":"canonical_sha256_bytes"},"source_id":"1805.07778","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:db66cd18c95202e93468489ebc75c643823f6cc3da486854d17032c1b0bf9fc5","sha256:49fe25d470ae1aa237ae3bc6fd4e9f2291dded353096d1694b1d25cf1c8ec046"],"state_sha256":"0402ef563b530274421fad6b2116463e705d80083b5da688c67e139082aa6b30"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"waPzn0FpHyVy0AiSusrR+9oXEuQ0DLhHiPKKA7/ZRJCGlQOazxRSHvY9wbStgbac1YPa6O6EIW/VGBgnOIRTBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T23:05:16.719782Z","bundle_sha256":"69c820e77a1a7dd38fd8452e9c157de8bffa2d8e2ce025a4b8c257225495a8c9"}}