{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:MHPTP3MLJHRZ7KJJZTLMTUFYMP","short_pith_number":"pith:MHPTP3ML","canonical_record":{"source":{"id":"1407.0280","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-07-01T15:29:28Z","cross_cats_sorted":[],"title_canon_sha256":"4615dc62b15143ef75d64a1798a42de61fa83ec101b8e2793e6337b0d6d7428f","abstract_canon_sha256":"e2dea71136857ee9a82306a67829ece99eca131de8950d3a8e765ef1e6b37871"},"schema_version":"1.0"},"canonical_sha256":"61df37ed8b49e39fa929ccd6c9d0b863c29991cf237b6bce36ec9e756e84c76a","source":{"kind":"arxiv","id":"1407.0280","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1407.0280","created_at":"2026-05-18T02:19:16Z"},{"alias_kind":"arxiv_version","alias_value":"1407.0280v3","created_at":"2026-05-18T02:19:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.0280","created_at":"2026-05-18T02:19:16Z"},{"alias_kind":"pith_short_12","alias_value":"MHPTP3MLJHRZ","created_at":"2026-05-18T12:28:38Z"},{"alias_kind":"pith_short_16","alias_value":"MHPTP3MLJHRZ7KJJ","created_at":"2026-05-18T12:28:38Z"},{"alias_kind":"pith_short_8","alias_value":"MHPTP3ML","created_at":"2026-05-18T12:28:38Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:MHPTP3MLJHRZ7KJJZTLMTUFYMP","target":"record","payload":{"canonical_record":{"source":{"id":"1407.0280","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-07-01T15:29:28Z","cross_cats_sorted":[],"title_canon_sha256":"4615dc62b15143ef75d64a1798a42de61fa83ec101b8e2793e6337b0d6d7428f","abstract_canon_sha256":"e2dea71136857ee9a82306a67829ece99eca131de8950d3a8e765ef1e6b37871"},"schema_version":"1.0"},"canonical_sha256":"61df37ed8b49e39fa929ccd6c9d0b863c29991cf237b6bce36ec9e756e84c76a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:19:16.733541Z","signature_b64":"a9P4lPIfjFbHtlfpx8dPXE52ZmrDF4wsAGLDZF+pSTUlrGU/zo/oYBQi/NQL/I1h+gIOwyw78CkisWRwax1cCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"61df37ed8b49e39fa929ccd6c9d0b863c29991cf237b6bce36ec9e756e84c76a","last_reissued_at":"2026-05-18T02:19:16.732917Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:19:16.732917Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1407.0280","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:19:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"v8tp3NXE2vrONoBsGj0A/K4di3hbNL/OwgzXpXb/FtIC3A0AZ/S4odq0t9xZouqQIOFTmEMriJDtiMZkSwd1Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-21T21:48:19.693534Z"},"content_sha256":"c8e398a1fe917d773d7bac0bcf01c787d4f80c95318746442199a87683925dab","schema_version":"1.0","event_id":"sha256:c8e398a1fe917d773d7bac0bcf01c787d4f80c95318746442199a87683925dab"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:MHPTP3MLJHRZ7KJJZTLMTUFYMP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the product of functions in $BMO$ and $H^1$ over spaces of homogeneous type","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Luong Dang Ky","submitted_at":"2014-07-01T15:29:28Z","abstract_excerpt":"Let $\\mathcal X$ be an RD-space, which means that $\\mathcal X$ is a space of homogeneous type in the sense of Coifman-Weiss with the additional property that a reverse doubling property holds in $\\mathcal X$. The aim of the present paper is to study the product of functions in $BMO$ and $H^1$ in this setting. Our results generalize some recent results in \\cite{Feu} and \\cite{LP}."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.0280","kind":"arxiv","version":3},"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":"2026-05-18T02:19:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YC/Hn3qiMHgfg7B3KcnQAIOJ2l075VZ4WXIpFuecalrq/rdIgqkBBRf3CG6q+ylo/0V7KghTVMsdIaQgLroYCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-21T21:48:19.693886Z"},"content_sha256":"2d8c4b9826abf90dc7dfb07c17475ac7f262aaca311ab0fde573c531e7223f4e","schema_version":"1.0","event_id":"sha256:2d8c4b9826abf90dc7dfb07c17475ac7f262aaca311ab0fde573c531e7223f4e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MHPTP3MLJHRZ7KJJZTLMTUFYMP/bundle.json","state_url":"https://pith.science/pith/MHPTP3MLJHRZ7KJJZTLMTUFYMP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MHPTP3MLJHRZ7KJJZTLMTUFYMP/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":"2026-05-21T21:48:19Z","links":{"resolver":"https://pith.science/pith/MHPTP3MLJHRZ7KJJZTLMTUFYMP","bundle":"https://pith.science/pith/MHPTP3MLJHRZ7KJJZTLMTUFYMP/bundle.json","state":"https://pith.science/pith/MHPTP3MLJHRZ7KJJZTLMTUFYMP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MHPTP3MLJHRZ7KJJZTLMTUFYMP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:MHPTP3MLJHRZ7KJJZTLMTUFYMP","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":"e2dea71136857ee9a82306a67829ece99eca131de8950d3a8e765ef1e6b37871","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-07-01T15:29:28Z","title_canon_sha256":"4615dc62b15143ef75d64a1798a42de61fa83ec101b8e2793e6337b0d6d7428f"},"schema_version":"1.0","source":{"id":"1407.0280","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1407.0280","created_at":"2026-05-18T02:19:16Z"},{"alias_kind":"arxiv_version","alias_value":"1407.0280v3","created_at":"2026-05-18T02:19:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.0280","created_at":"2026-05-18T02:19:16Z"},{"alias_kind":"pith_short_12","alias_value":"MHPTP3MLJHRZ","created_at":"2026-05-18T12:28:38Z"},{"alias_kind":"pith_short_16","alias_value":"MHPTP3MLJHRZ7KJJ","created_at":"2026-05-18T12:28:38Z"},{"alias_kind":"pith_short_8","alias_value":"MHPTP3ML","created_at":"2026-05-18T12:28:38Z"}],"graph_snapshots":[{"event_id":"sha256:2d8c4b9826abf90dc7dfb07c17475ac7f262aaca311ab0fde573c531e7223f4e","target":"graph","created_at":"2026-05-18T02:19:16Z","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 $\\mathcal X$ be an RD-space, which means that $\\mathcal X$ is a space of homogeneous type in the sense of Coifman-Weiss with the additional property that a reverse doubling property holds in $\\mathcal X$. The aim of the present paper is to study the product of functions in $BMO$ and $H^1$ in this setting. Our results generalize some recent results in \\cite{Feu} and \\cite{LP}.","authors_text":"Luong Dang Ky","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-07-01T15:29:28Z","title":"On the product of functions in $BMO$ and $H^1$ over spaces of homogeneous type"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.0280","kind":"arxiv","version":3},"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:c8e398a1fe917d773d7bac0bcf01c787d4f80c95318746442199a87683925dab","target":"record","created_at":"2026-05-18T02:19:16Z","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":"e2dea71136857ee9a82306a67829ece99eca131de8950d3a8e765ef1e6b37871","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-07-01T15:29:28Z","title_canon_sha256":"4615dc62b15143ef75d64a1798a42de61fa83ec101b8e2793e6337b0d6d7428f"},"schema_version":"1.0","source":{"id":"1407.0280","kind":"arxiv","version":3}},"canonical_sha256":"61df37ed8b49e39fa929ccd6c9d0b863c29991cf237b6bce36ec9e756e84c76a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"61df37ed8b49e39fa929ccd6c9d0b863c29991cf237b6bce36ec9e756e84c76a","first_computed_at":"2026-05-18T02:19:16.732917Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:19:16.732917Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"a9P4lPIfjFbHtlfpx8dPXE52ZmrDF4wsAGLDZF+pSTUlrGU/zo/oYBQi/NQL/I1h+gIOwyw78CkisWRwax1cCg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:19:16.733541Z","signed_message":"canonical_sha256_bytes"},"source_id":"1407.0280","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c8e398a1fe917d773d7bac0bcf01c787d4f80c95318746442199a87683925dab","sha256:2d8c4b9826abf90dc7dfb07c17475ac7f262aaca311ab0fde573c531e7223f4e"],"state_sha256":"12a78ae57cadb1c2dd11afd018fb6f13f700a9f5a6eae80d5ccd836bbf82ff12"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"k75VKA9X7ASfhU8sYQzMjjn5VLP8tPBE9oOWeSPxKfomQrmO31ZbGwJxSDENR3RUHQ+hlbmUZriQdvTIv33JCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-21T21:48:19.696274Z","bundle_sha256":"0683b0f1436baccf4f7930766df565e519b624a9a589583ea26fcd22143a62e0"}}