{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:H4LOY6P2XCVFDQFDRH6K3JGA27","short_pith_number":"pith:H4LOY6P2","canonical_record":{"source":{"id":"1705.02548","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-05-07T02:11:30Z","cross_cats_sorted":[],"title_canon_sha256":"51b6f435ec941d85bdef5418ca5b93aa7aa1ba1229abfca4262fc418886649f8","abstract_canon_sha256":"d3af64b6a64321727ec9410f034ca2f5203137eb12795ce3b248472064057bf9"},"schema_version":"1.0"},"canonical_sha256":"3f16ec79fab8aa51c0a389fcada4c0d7d1b9eaf39e212b9176a6443480d4da59","source":{"kind":"arxiv","id":"1705.02548","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1705.02548","created_at":"2026-05-18T00:28:06Z"},{"alias_kind":"arxiv_version","alias_value":"1705.02548v2","created_at":"2026-05-18T00:28:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1705.02548","created_at":"2026-05-18T00:28:06Z"},{"alias_kind":"pith_short_12","alias_value":"H4LOY6P2XCVF","created_at":"2026-05-18T12:31:18Z"},{"alias_kind":"pith_short_16","alias_value":"H4LOY6P2XCVFDQFD","created_at":"2026-05-18T12:31:18Z"},{"alias_kind":"pith_short_8","alias_value":"H4LOY6P2","created_at":"2026-05-18T12:31:18Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:H4LOY6P2XCVFDQFDRH6K3JGA27","target":"record","payload":{"canonical_record":{"source":{"id":"1705.02548","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-05-07T02:11:30Z","cross_cats_sorted":[],"title_canon_sha256":"51b6f435ec941d85bdef5418ca5b93aa7aa1ba1229abfca4262fc418886649f8","abstract_canon_sha256":"d3af64b6a64321727ec9410f034ca2f5203137eb12795ce3b248472064057bf9"},"schema_version":"1.0"},"canonical_sha256":"3f16ec79fab8aa51c0a389fcada4c0d7d1b9eaf39e212b9176a6443480d4da59","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:28:06.997661Z","signature_b64":"C9DvSqUVrseK0/3+fctyF7iHpU53hoH2FPtmdJuw8nR/+0VJqF5LLswTcrBPMnzkoONcWHkv/GTXTzo+WZIqAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3f16ec79fab8aa51c0a389fcada4c0d7d1b9eaf39e212b9176a6443480d4da59","last_reissued_at":"2026-05-18T00:28:06.997101Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:28:06.997101Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1705.02548","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":"2026-05-18T00:28:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Famv7vpl2Jycl3VA7QawKwLDKeCMmoOJIVqPfTmAAVKBWjDEQvCwlFR2LMK8LXaW9Nl3NvjVl1rLoi3qwvMOCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-20T17:56:02.146481Z"},"content_sha256":"12c0fd8a344bc292f4ff4fbf1d390be0f0d7c0c271db9f5aae3e8e1d8f99b4aa","schema_version":"1.0","event_id":"sha256:12c0fd8a344bc292f4ff4fbf1d390be0f0d7c0c271db9f5aae3e8e1d8f99b4aa"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:H4LOY6P2XCVFDQFDRH6K3JGA27","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The multi-parameter Hausdorff operators on $H^1$ and $L^p$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Duong Quoc Huy, Luong Dang Ky","submitted_at":"2017-05-07T02:11:30Z","abstract_excerpt":"In the present paper, we characterize the nonnegative functions $\\varphi$ for which the multi-parameter Hausdorff operator $\\mathcal H_\\varphi$ generated by $\\varphi$ is bounded on the multi-parameter Hardy space $H^1(\\mathbb R\\times\\cdots\\times\\mathbb R)$ or $L^p(\\mathbb R^n)$, $p\\in [1,\\infty]$. The corresponding operator norms are also obtained. Our results improve some recent results in \\cite{FZ, LM2, LM3, We} and give an answer to an open question posted by Liflyand \\cite{Li}."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.02548","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":""},"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":"2026-05-18T00:28:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ezPLJrkdxRixNZ/J4f4RUCOP4CvTCVhlaj8JKoB/sWcOQlkMsMK/uVEfX0EnCQShMVQZl4WoU+4T/j5N+826Bw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-20T17:56:02.146828Z"},"content_sha256":"c2a8a4c27a7857210f349ac9f29ec34985b1596cea0aed3cd79ddf1837cf03f7","schema_version":"1.0","event_id":"sha256:c2a8a4c27a7857210f349ac9f29ec34985b1596cea0aed3cd79ddf1837cf03f7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/H4LOY6P2XCVFDQFDRH6K3JGA27/bundle.json","state_url":"https://pith.science/pith/H4LOY6P2XCVFDQFDRH6K3JGA27/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/H4LOY6P2XCVFDQFDRH6K3JGA27/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":"2026-05-20T17:56:02Z","links":{"resolver":"https://pith.science/pith/H4LOY6P2XCVFDQFDRH6K3JGA27","bundle":"https://pith.science/pith/H4LOY6P2XCVFDQFDRH6K3JGA27/bundle.json","state":"https://pith.science/pith/H4LOY6P2XCVFDQFDRH6K3JGA27/state.json","well_known_bundle":"https://pith.science/.well-known/pith/H4LOY6P2XCVFDQFDRH6K3JGA27/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:H4LOY6P2XCVFDQFDRH6K3JGA27","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":"d3af64b6a64321727ec9410f034ca2f5203137eb12795ce3b248472064057bf9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-05-07T02:11:30Z","title_canon_sha256":"51b6f435ec941d85bdef5418ca5b93aa7aa1ba1229abfca4262fc418886649f8"},"schema_version":"1.0","source":{"id":"1705.02548","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1705.02548","created_at":"2026-05-18T00:28:06Z"},{"alias_kind":"arxiv_version","alias_value":"1705.02548v2","created_at":"2026-05-18T00:28:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1705.02548","created_at":"2026-05-18T00:28:06Z"},{"alias_kind":"pith_short_12","alias_value":"H4LOY6P2XCVF","created_at":"2026-05-18T12:31:18Z"},{"alias_kind":"pith_short_16","alias_value":"H4LOY6P2XCVFDQFD","created_at":"2026-05-18T12:31:18Z"},{"alias_kind":"pith_short_8","alias_value":"H4LOY6P2","created_at":"2026-05-18T12:31:18Z"}],"graph_snapshots":[{"event_id":"sha256:c2a8a4c27a7857210f349ac9f29ec34985b1596cea0aed3cd79ddf1837cf03f7","target":"graph","created_at":"2026-05-18T00:28:06Z","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":"In the present paper, we characterize the nonnegative functions $\\varphi$ for which the multi-parameter Hausdorff operator $\\mathcal H_\\varphi$ generated by $\\varphi$ is bounded on the multi-parameter Hardy space $H^1(\\mathbb R\\times\\cdots\\times\\mathbb R)$ or $L^p(\\mathbb R^n)$, $p\\in [1,\\infty]$. The corresponding operator norms are also obtained. Our results improve some recent results in \\cite{FZ, LM2, LM3, We} and give an answer to an open question posted by Liflyand \\cite{Li}.","authors_text":"Duong Quoc Huy, Luong Dang Ky","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-05-07T02:11:30Z","title":"The multi-parameter Hausdorff operators on $H^1$ and $L^p$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.02548","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":"sha256:12c0fd8a344bc292f4ff4fbf1d390be0f0d7c0c271db9f5aae3e8e1d8f99b4aa","target":"record","created_at":"2026-05-18T00:28:06Z","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":"d3af64b6a64321727ec9410f034ca2f5203137eb12795ce3b248472064057bf9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-05-07T02:11:30Z","title_canon_sha256":"51b6f435ec941d85bdef5418ca5b93aa7aa1ba1229abfca4262fc418886649f8"},"schema_version":"1.0","source":{"id":"1705.02548","kind":"arxiv","version":2}},"canonical_sha256":"3f16ec79fab8aa51c0a389fcada4c0d7d1b9eaf39e212b9176a6443480d4da59","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3f16ec79fab8aa51c0a389fcada4c0d7d1b9eaf39e212b9176a6443480d4da59","first_computed_at":"2026-05-18T00:28:06.997101Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:28:06.997101Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"C9DvSqUVrseK0/3+fctyF7iHpU53hoH2FPtmdJuw8nR/+0VJqF5LLswTcrBPMnzkoONcWHkv/GTXTzo+WZIqAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:28:06.997661Z","signed_message":"canonical_sha256_bytes"},"source_id":"1705.02548","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:12c0fd8a344bc292f4ff4fbf1d390be0f0d7c0c271db9f5aae3e8e1d8f99b4aa","sha256:c2a8a4c27a7857210f349ac9f29ec34985b1596cea0aed3cd79ddf1837cf03f7"],"state_sha256":"536dce4795619c64d23cabff6eb4f8acaeb900a54b511bb8c774f19b2e723b18"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mQsa8cBfxXt4nrAv2IGRA3jDQa/PJlJbXihUUly4bafC09B8QljFrvB6W5wVPNCPqunibgy8b7ezYvH7ik9YCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-20T17:56:02.149031Z","bundle_sha256":"14ebea66a1d135a23b0f774d624191410902020db4530640cc639ebaab7501e5"}}