{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:SCKUX5XUSOQBAUIKNR6V56EICN","short_pith_number":"pith:SCKUX5XU","schema_version":"1.0","canonical_sha256":"90954bf6f493a010510a6c7d5ef88813495352cd3af38e5ef69232bedf4042da","source":{"kind":"arxiv","id":"1511.08075","version":1},"attestation_state":"computed","paper":{"title":"On weak$^*$-convergence in the localized Hardy spaces $H^1_\\rho(\\mathcal X)$ and its application","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.CA","authors_text":"Dinh Thanh Duc, Ha Duy Hung, Luong Dang Ky","submitted_at":"2015-11-25T14:43:02Z","abstract_excerpt":"Let $(\\mathcal X, d, \\mu)$ be a complete RD-space. Let $\\rho$ be an admissible function on $\\mathcal X$, which means that $\\rho$ is a positive function on $\\mathcal X$ and there exist positive constants $C_0$ and $k_0$ such that, for any $x,y\\in \\mathcal X$, $$\\rho(y)\\leq C_0 [\\rho(x)]^{1/(1+k_0)} [\\rho(x)+d(x,y)]^{k_0/(1+k_0)}.$$\n  In this paper, we define a space $VMO_\\rho(\\mathcal X)$ and show that it is the predual of the localized Hardy space $H^1_\\rho(\\mathcal X)$ introduced by Yang and Zhou \\cite{YZ}. Then we prove a version of the classical theorem of Jones and Journ\\'e \\cite{JJ} on we"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1511.08075","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-11-25T14:43:02Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"7f5ea78f8a37276cc767b4267f741fe80901495d4b38191a986c661a19d42b72","abstract_canon_sha256":"088f71e4fd6462f3377d6dc1ac1495bd4564ac2120e3833f740d6e7de19159c7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:50:57.590296Z","signature_b64":"XnIL0Sp9FetV/vbY0SCCWtLd1AaUQFdiCghSD7pyCNKdLAQeLC/eZ8ezao2j3TzO2C6QcT50RJtuyixErtadDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"90954bf6f493a010510a6c7d5ef88813495352cd3af38e5ef69232bedf4042da","last_reissued_at":"2026-05-18T00:50:57.589570Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:50:57.589570Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On weak$^*$-convergence in the localized Hardy spaces $H^1_\\rho(\\mathcal X)$ and its application","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.CA","authors_text":"Dinh Thanh Duc, Ha Duy Hung, Luong Dang Ky","submitted_at":"2015-11-25T14:43:02Z","abstract_excerpt":"Let $(\\mathcal X, d, \\mu)$ be a complete RD-space. Let $\\rho$ be an admissible function on $\\mathcal X$, which means that $\\rho$ is a positive function on $\\mathcal X$ and there exist positive constants $C_0$ and $k_0$ such that, for any $x,y\\in \\mathcal X$, $$\\rho(y)\\leq C_0 [\\rho(x)]^{1/(1+k_0)} [\\rho(x)+d(x,y)]^{k_0/(1+k_0)}.$$\n  In this paper, we define a space $VMO_\\rho(\\mathcal X)$ and show that it is the predual of the localized Hardy space $H^1_\\rho(\\mathcal X)$ introduced by Yang and Zhou \\cite{YZ}. Then we prove a version of the classical theorem of Jones and Journ\\'e \\cite{JJ} on we"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.08075","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1511.08075","created_at":"2026-05-18T00:50:57.589684+00:00"},{"alias_kind":"arxiv_version","alias_value":"1511.08075v1","created_at":"2026-05-18T00:50:57.589684+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.08075","created_at":"2026-05-18T00:50:57.589684+00:00"},{"alias_kind":"pith_short_12","alias_value":"SCKUX5XUSOQB","created_at":"2026-05-18T12:29:39.896362+00:00"},{"alias_kind":"pith_short_16","alias_value":"SCKUX5XUSOQBAUIK","created_at":"2026-05-18T12:29:39.896362+00:00"},{"alias_kind":"pith_short_8","alias_value":"SCKUX5XU","created_at":"2026-05-18T12:29:39.896362+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/SCKUX5XUSOQBAUIKNR6V56EICN","json":"https://pith.science/pith/SCKUX5XUSOQBAUIKNR6V56EICN.json","graph_json":"https://pith.science/api/pith-number/SCKUX5XUSOQBAUIKNR6V56EICN/graph.json","events_json":"https://pith.science/api/pith-number/SCKUX5XUSOQBAUIKNR6V56EICN/events.json","paper":"https://pith.science/paper/SCKUX5XU"},"agent_actions":{"view_html":"https://pith.science/pith/SCKUX5XUSOQBAUIKNR6V56EICN","download_json":"https://pith.science/pith/SCKUX5XUSOQBAUIKNR6V56EICN.json","view_paper":"https://pith.science/paper/SCKUX5XU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1511.08075&json=true","fetch_graph":"https://pith.science/api/pith-number/SCKUX5XUSOQBAUIKNR6V56EICN/graph.json","fetch_events":"https://pith.science/api/pith-number/SCKUX5XUSOQBAUIKNR6V56EICN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SCKUX5XUSOQBAUIKNR6V56EICN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SCKUX5XUSOQBAUIKNR6V56EICN/action/storage_attestation","attest_author":"https://pith.science/pith/SCKUX5XUSOQBAUIKNR6V56EICN/action/author_attestation","sign_citation":"https://pith.science/pith/SCKUX5XUSOQBAUIKNR6V56EICN/action/citation_signature","submit_replication":"https://pith.science/pith/SCKUX5XUSOQBAUIKNR6V56EICN/action/replication_record"}},"created_at":"2026-05-18T00:50:57.589684+00:00","updated_at":"2026-05-18T00:50:57.589684+00:00"}