{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:IA6C4M6VSAOVWBR5UNOFELESOA","short_pith_number":"pith:IA6C4M6V","canonical_record":{"source":{"id":"2505.17707","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CA","submitted_at":"2025-05-23T10:21:41Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"4057184bfa78d1c743c6b285a6caa4785539e9cc638b7361ca9d336a6bcb6d3a","abstract_canon_sha256":"429d26064f794ee4ca91c110c77bb0d7b89aeb2e3c4ab1cdfa338046f5fb266b"},"schema_version":"1.0"},"canonical_sha256":"403c2e33d5901d5b063da35c522c92703b807e5672c7fb87fe33a9126c58d9bc","source":{"kind":"arxiv","id":"2505.17707","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2505.17707","created_at":"2026-06-02T02:04:06Z"},{"alias_kind":"arxiv_version","alias_value":"2505.17707v2","created_at":"2026-06-02T02:04:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2505.17707","created_at":"2026-06-02T02:04:06Z"},{"alias_kind":"pith_short_12","alias_value":"IA6C4M6VSAOV","created_at":"2026-06-02T02:04:06Z"},{"alias_kind":"pith_short_16","alias_value":"IA6C4M6VSAOVWBR5","created_at":"2026-06-02T02:04:06Z"},{"alias_kind":"pith_short_8","alias_value":"IA6C4M6V","created_at":"2026-06-02T02:04:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:IA6C4M6VSAOVWBR5UNOFELESOA","target":"record","payload":{"canonical_record":{"source":{"id":"2505.17707","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CA","submitted_at":"2025-05-23T10:21:41Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"4057184bfa78d1c743c6b285a6caa4785539e9cc638b7361ca9d336a6bcb6d3a","abstract_canon_sha256":"429d26064f794ee4ca91c110c77bb0d7b89aeb2e3c4ab1cdfa338046f5fb266b"},"schema_version":"1.0"},"canonical_sha256":"403c2e33d5901d5b063da35c522c92703b807e5672c7fb87fe33a9126c58d9bc","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-02T02:04:06.240312Z","signature_b64":"nWj9/8AAB11OyTLKnfLFTe/FH8DAqDjA2fN1rf77fA9DS/lHqTjV15XHALcAyBybcVnGmpW6U6wwyM8hZsNqBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"403c2e33d5901d5b063da35c522c92703b807e5672c7fb87fe33a9126c58d9bc","last_reissued_at":"2026-06-02T02:04:06.239874Z","signature_status":"signed_v1","first_computed_at":"2026-06-02T02:04:06.239874Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2505.17707","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":"2026-06-02T02:04:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0LOSJ5LdtHdDi7bo5mfvO+ITKON+6ZaHikKGClE3QwLgYeCY3ETADJdyK9xhBSdcfc8SZu4t5FQyquugvHlZDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T15:16:08.496028Z"},"content_sha256":"4e284fee52342ad0b63a31ce901b5d149a11aa3b30d2cd3a34861eb7d333eca1","schema_version":"1.0","event_id":"sha256:4e284fee52342ad0b63a31ce901b5d149a11aa3b30d2cd3a34861eb7d333eca1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:IA6C4M6VSAOVWBR5UNOFELESOA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Sharp weak bounds for the Hardy-Littlewood-P\\'{o}lya operator and the weak bounds for the multilinear integral operator","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.CA","authors_text":"Tianyang He","submitted_at":"2025-05-23T10:21:41Z","abstract_excerpt":"In this paper, we first obtain the operator norms of the $n$-dimensional Hardy-Littlewood-P\\'{o}lya operator $\\mathcal{H}$ from weighted Lebesgue spaces $L^p( \\mathbb{R} ^n,| x |^{\\beta} ) $ to weighted weak Lebesgue spaces $L^{q,\\infty}(\\mathbb{R} ^n,|x|^{\\gamma})$. Next, we obtain the weak bounds for the $m$-linear $n$-dimensional integral operator."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.17707","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/2505.17707/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":"2026-06-02T02:04:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"A2s8b9zy7WhM9qO7liPZiPQgbjrhcFH7I6W0XGG2zkwkY8QYhhIRgh+i/FogdQyP+6P/W18IZ4akKHr4ZqLiAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T15:16:08.496701Z"},"content_sha256":"212660f4306d703636fd1ef993eadaeb6b6dac144e2cbe6658eff23eb8aba117","schema_version":"1.0","event_id":"sha256:212660f4306d703636fd1ef993eadaeb6b6dac144e2cbe6658eff23eb8aba117"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/IA6C4M6VSAOVWBR5UNOFELESOA/bundle.json","state_url":"https://pith.science/pith/IA6C4M6VSAOVWBR5UNOFELESOA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/IA6C4M6VSAOVWBR5UNOFELESOA/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":"2026-06-05T15:16:08Z","links":{"resolver":"https://pith.science/pith/IA6C4M6VSAOVWBR5UNOFELESOA","bundle":"https://pith.science/pith/IA6C4M6VSAOVWBR5UNOFELESOA/bundle.json","state":"https://pith.science/pith/IA6C4M6VSAOVWBR5UNOFELESOA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/IA6C4M6VSAOVWBR5UNOFELESOA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:IA6C4M6VSAOVWBR5UNOFELESOA","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":"429d26064f794ee4ca91c110c77bb0d7b89aeb2e3c4ab1cdfa338046f5fb266b","cross_cats_sorted":["math.FA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CA","submitted_at":"2025-05-23T10:21:41Z","title_canon_sha256":"4057184bfa78d1c743c6b285a6caa4785539e9cc638b7361ca9d336a6bcb6d3a"},"schema_version":"1.0","source":{"id":"2505.17707","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2505.17707","created_at":"2026-06-02T02:04:06Z"},{"alias_kind":"arxiv_version","alias_value":"2505.17707v2","created_at":"2026-06-02T02:04:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2505.17707","created_at":"2026-06-02T02:04:06Z"},{"alias_kind":"pith_short_12","alias_value":"IA6C4M6VSAOV","created_at":"2026-06-02T02:04:06Z"},{"alias_kind":"pith_short_16","alias_value":"IA6C4M6VSAOVWBR5","created_at":"2026-06-02T02:04:06Z"},{"alias_kind":"pith_short_8","alias_value":"IA6C4M6V","created_at":"2026-06-02T02:04:06Z"}],"graph_snapshots":[{"event_id":"sha256:212660f4306d703636fd1ef993eadaeb6b6dac144e2cbe6658eff23eb8aba117","target":"graph","created_at":"2026-06-02T02:04:06Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2505.17707/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, we first obtain the operator norms of the $n$-dimensional Hardy-Littlewood-P\\'{o}lya operator $\\mathcal{H}$ from weighted Lebesgue spaces $L^p( \\mathbb{R} ^n,| x |^{\\beta} ) $ to weighted weak Lebesgue spaces $L^{q,\\infty}(\\mathbb{R} ^n,|x|^{\\gamma})$. Next, we obtain the weak bounds for the $m$-linear $n$-dimensional integral operator.","authors_text":"Tianyang He","cross_cats":["math.FA"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CA","submitted_at":"2025-05-23T10:21:41Z","title":"Sharp weak bounds for the Hardy-Littlewood-P\\'{o}lya operator and the weak bounds for the multilinear integral operator"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.17707","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":"sha256:4e284fee52342ad0b63a31ce901b5d149a11aa3b30d2cd3a34861eb7d333eca1","target":"record","created_at":"2026-06-02T02:04:06Z","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":"429d26064f794ee4ca91c110c77bb0d7b89aeb2e3c4ab1cdfa338046f5fb266b","cross_cats_sorted":["math.FA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CA","submitted_at":"2025-05-23T10:21:41Z","title_canon_sha256":"4057184bfa78d1c743c6b285a6caa4785539e9cc638b7361ca9d336a6bcb6d3a"},"schema_version":"1.0","source":{"id":"2505.17707","kind":"arxiv","version":2}},"canonical_sha256":"403c2e33d5901d5b063da35c522c92703b807e5672c7fb87fe33a9126c58d9bc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"403c2e33d5901d5b063da35c522c92703b807e5672c7fb87fe33a9126c58d9bc","first_computed_at":"2026-06-02T02:04:06.239874Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-02T02:04:06.239874Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"nWj9/8AAB11OyTLKnfLFTe/FH8DAqDjA2fN1rf77fA9DS/lHqTjV15XHALcAyBybcVnGmpW6U6wwyM8hZsNqBQ==","signature_status":"signed_v1","signed_at":"2026-06-02T02:04:06.240312Z","signed_message":"canonical_sha256_bytes"},"source_id":"2505.17707","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4e284fee52342ad0b63a31ce901b5d149a11aa3b30d2cd3a34861eb7d333eca1","sha256:212660f4306d703636fd1ef993eadaeb6b6dac144e2cbe6658eff23eb8aba117"],"state_sha256":"093edf53c5160ff9bfc0e0133c6503332a327140960b21b020fefbb0d3f99eee"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Oc6t315Fd1TXt4sCWQPV2M14qKOZvdY8elr4i6Phx1fOqBwBRrL0o0lXN41p2U8bRRvz+sJR5mxE4MIgxVKJBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T15:16:08.499968Z","bundle_sha256":"f96206792457a377acd330aa71b29d35da6b30fc9175df51ea1cb7b684795d16"}}