{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:2E254W77ZZP3LTOWNF3VFZVIZF","short_pith_number":"pith:2E254W77","canonical_record":{"source":{"id":"2508.13913","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CA","submitted_at":"2025-08-19T15:12:51Z","cross_cats_sorted":[],"title_canon_sha256":"81813dffa3ef07f297c1885a1e3cca9e8581b7daa3c40e16fc5194b4558e7686","abstract_canon_sha256":"c45add2bfbac81765fbf4cc65987267dc6a7ef7bd8df8c207088490bb118be89"},"schema_version":"1.0"},"canonical_sha256":"d135de5bffce5fb5cdd6697752e6a8c9534a01601e2dac36b8b09a9cbd1b47d9","source":{"kind":"arxiv","id":"2508.13913","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2508.13913","created_at":"2026-07-05T12:00:28Z"},{"alias_kind":"arxiv_version","alias_value":"2508.13913v2","created_at":"2026-07-05T12:00:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2508.13913","created_at":"2026-07-05T12:00:28Z"},{"alias_kind":"pith_short_12","alias_value":"2E254W77ZZP3","created_at":"2026-07-05T12:00:28Z"},{"alias_kind":"pith_short_16","alias_value":"2E254W77ZZP3LTOW","created_at":"2026-07-05T12:00:28Z"},{"alias_kind":"pith_short_8","alias_value":"2E254W77","created_at":"2026-07-05T12:00:28Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:2E254W77ZZP3LTOWNF3VFZVIZF","target":"record","payload":{"canonical_record":{"source":{"id":"2508.13913","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CA","submitted_at":"2025-08-19T15:12:51Z","cross_cats_sorted":[],"title_canon_sha256":"81813dffa3ef07f297c1885a1e3cca9e8581b7daa3c40e16fc5194b4558e7686","abstract_canon_sha256":"c45add2bfbac81765fbf4cc65987267dc6a7ef7bd8df8c207088490bb118be89"},"schema_version":"1.0"},"canonical_sha256":"d135de5bffce5fb5cdd6697752e6a8c9534a01601e2dac36b8b09a9cbd1b47d9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T12:00:28.853671Z","signature_b64":"6RxAtsV0n5wGjKkjXC1xxGyNh4ujDd2yntMA8F7jqSHTx8v5Har/GPtluPC5WfLePALTBqlNlxGmBu1XKWvEAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d135de5bffce5fb5cdd6697752e6a8c9534a01601e2dac36b8b09a9cbd1b47d9","last_reissued_at":"2026-07-05T12:00:28.853177Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T12:00:28.853177Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2508.13913","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":"2026-07-05T12:00:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DH2Wy6m6Wz9bf6T/4x7Vv8+J6Az4zPKAQIJLJ5+jBSiDof3gIW30xcpEeokJLI2E100GuaQimkTIHOJHs5vkCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-23T00:08:32.004210Z"},"content_sha256":"a92fd4aade614278cfb0ea915111041ddd77ffd0e606a35fe60ca3ac60de3702","schema_version":"1.0","event_id":"sha256:a92fd4aade614278cfb0ea915111041ddd77ffd0e606a35fe60ca3ac60de3702"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:2E254W77ZZP3LTOWNF3VFZVIZF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Estimates for Schr\\\"{o}dinger Groups and Imaginary Power Operators on Weak Hardy Spaces Associated with Non-negative Self-adjoint Operators and Ball Quasi-Banach Function Spaces","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Wenhua Wang, Xiong Liu","submitted_at":"2025-08-19T15:12:51Z","abstract_excerpt":"Let $(\\mathbb{X},d,\\mu)$ be a doubling metric measure space, $L$ a non-negative self-adjoint operator on $L^2(\\mathbb{X})$ satisfying the Davies-Gaffney estimate, and $X(\\mathbb{X})$ a ball quasi-Banach function space on $\\mathbb{X}$ satisfying some mild assumptions with $p\\in(0,\\infty)$ and $s_0\\in(0,\\min\\{p,1\\}]$. In this article, the authors study the weak Hardy space $WH_{X,L}(\\mathbb{X})$ associated with $L$ and $X(\\mathbb{X})$, and then give the atomic and molecular decompositions of $WH_{X,L}(\\mathbb{X})$. As applications, the authors establish the boundedness estimate of Schr\\\"{o}dinge"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2508.13913","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2508.13913/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":"2026-07-05T12:00:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Jjf6QsRseZBXgR2EV/HOvI6NsjZviQ4RkZ5FRu9AZNdnCXEr2NoutUhUOpym/3i0MCaO3zxrXx3lkkNlHSEwCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-23T00:08:32.004551Z"},"content_sha256":"c043310174e9af948e6611fb5b6d85b6141def9d894351ffe4d9d933201bb02b","schema_version":"1.0","event_id":"sha256:c043310174e9af948e6611fb5b6d85b6141def9d894351ffe4d9d933201bb02b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2E254W77ZZP3LTOWNF3VFZVIZF/bundle.json","state_url":"https://pith.science/pith/2E254W77ZZP3LTOWNF3VFZVIZF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2E254W77ZZP3LTOWNF3VFZVIZF/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":"2026-08-23T00:08:32Z","links":{"resolver":"https://pith.science/pith/2E254W77ZZP3LTOWNF3VFZVIZF","bundle":"https://pith.science/pith/2E254W77ZZP3LTOWNF3VFZVIZF/bundle.json","state":"https://pith.science/pith/2E254W77ZZP3LTOWNF3VFZVIZF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2E254W77ZZP3LTOWNF3VFZVIZF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:2E254W77ZZP3LTOWNF3VFZVIZF","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":"c45add2bfbac81765fbf4cc65987267dc6a7ef7bd8df8c207088490bb118be89","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CA","submitted_at":"2025-08-19T15:12:51Z","title_canon_sha256":"81813dffa3ef07f297c1885a1e3cca9e8581b7daa3c40e16fc5194b4558e7686"},"schema_version":"1.0","source":{"id":"2508.13913","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2508.13913","created_at":"2026-07-05T12:00:28Z"},{"alias_kind":"arxiv_version","alias_value":"2508.13913v2","created_at":"2026-07-05T12:00:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2508.13913","created_at":"2026-07-05T12:00:28Z"},{"alias_kind":"pith_short_12","alias_value":"2E254W77ZZP3","created_at":"2026-07-05T12:00:28Z"},{"alias_kind":"pith_short_16","alias_value":"2E254W77ZZP3LTOW","created_at":"2026-07-05T12:00:28Z"},{"alias_kind":"pith_short_8","alias_value":"2E254W77","created_at":"2026-07-05T12:00:28Z"}],"graph_snapshots":[{"event_id":"sha256:c043310174e9af948e6611fb5b6d85b6141def9d894351ffe4d9d933201bb02b","target":"graph","created_at":"2026-07-05T12:00:28Z","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/2508.13913/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $(\\mathbb{X},d,\\mu)$ be a doubling metric measure space, $L$ a non-negative self-adjoint operator on $L^2(\\mathbb{X})$ satisfying the Davies-Gaffney estimate, and $X(\\mathbb{X})$ a ball quasi-Banach function space on $\\mathbb{X}$ satisfying some mild assumptions with $p\\in(0,\\infty)$ and $s_0\\in(0,\\min\\{p,1\\}]$. In this article, the authors study the weak Hardy space $WH_{X,L}(\\mathbb{X})$ associated with $L$ and $X(\\mathbb{X})$, and then give the atomic and molecular decompositions of $WH_{X,L}(\\mathbb{X})$. As applications, the authors establish the boundedness estimate of Schr\\\"{o}dinge","authors_text":"Wenhua Wang, Xiong Liu","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CA","submitted_at":"2025-08-19T15:12:51Z","title":"Estimates for Schr\\\"{o}dinger Groups and Imaginary Power Operators on Weak Hardy Spaces Associated with Non-negative Self-adjoint Operators and Ball Quasi-Banach Function Spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2508.13913","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:a92fd4aade614278cfb0ea915111041ddd77ffd0e606a35fe60ca3ac60de3702","target":"record","created_at":"2026-07-05T12:00:28Z","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":"c45add2bfbac81765fbf4cc65987267dc6a7ef7bd8df8c207088490bb118be89","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CA","submitted_at":"2025-08-19T15:12:51Z","title_canon_sha256":"81813dffa3ef07f297c1885a1e3cca9e8581b7daa3c40e16fc5194b4558e7686"},"schema_version":"1.0","source":{"id":"2508.13913","kind":"arxiv","version":2}},"canonical_sha256":"d135de5bffce5fb5cdd6697752e6a8c9534a01601e2dac36b8b09a9cbd1b47d9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d135de5bffce5fb5cdd6697752e6a8c9534a01601e2dac36b8b09a9cbd1b47d9","first_computed_at":"2026-07-05T12:00:28.853177Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T12:00:28.853177Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"6RxAtsV0n5wGjKkjXC1xxGyNh4ujDd2yntMA8F7jqSHTx8v5Har/GPtluPC5WfLePALTBqlNlxGmBu1XKWvEAw==","signature_status":"signed_v1","signed_at":"2026-07-05T12:00:28.853671Z","signed_message":"canonical_sha256_bytes"},"source_id":"2508.13913","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a92fd4aade614278cfb0ea915111041ddd77ffd0e606a35fe60ca3ac60de3702","sha256:c043310174e9af948e6611fb5b6d85b6141def9d894351ffe4d9d933201bb02b"],"state_sha256":"8bc4c9a1eb6e288db68896f09f0bf438f6e09ddcf9cc0faf0025b0e2d4b8c9c8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"g1Q6qaXlTRY4CR+lwbAe6LLg4JlPXj84Y5tsePYm7Xfwp9jvOHSt3XT7jz4CCWluUbrzbC+OifV0AkJlohAZAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-23T00:08:32.007867Z","bundle_sha256":"bdcb14709a33d34a2c698bd66c2d5dd8dd7c3ff7b6abf4bc2317d670658ee467"}}