{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:GFXBXOMFKLRA4FEZTGLLE3BXDP","short_pith_number":"pith:GFXBXOMF","canonical_record":{"source":{"id":"1506.03187","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-06-10T06:58:09Z","cross_cats_sorted":[],"title_canon_sha256":"774c12afa1fe7e32fdae6d9763130aed3a14bcc8a514fe9142f8b014265d1df4","abstract_canon_sha256":"07d1766493e1a793536e2ea000bad55de3af8356acd45238bc360a951392b4ea"},"schema_version":"1.0"},"canonical_sha256":"316e1bb98552e20e14999996b26c371bd947ec0efb99bf583c4d601551a8bd5b","source":{"kind":"arxiv","id":"1506.03187","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1506.03187","created_at":"2026-05-18T01:26:57Z"},{"alias_kind":"arxiv_version","alias_value":"1506.03187v2","created_at":"2026-05-18T01:26:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1506.03187","created_at":"2026-05-18T01:26:57Z"},{"alias_kind":"pith_short_12","alias_value":"GFXBXOMFKLRA","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_16","alias_value":"GFXBXOMFKLRA4FEZ","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_8","alias_value":"GFXBXOMF","created_at":"2026-05-18T12:29:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:GFXBXOMFKLRA4FEZTGLLE3BXDP","target":"record","payload":{"canonical_record":{"source":{"id":"1506.03187","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-06-10T06:58:09Z","cross_cats_sorted":[],"title_canon_sha256":"774c12afa1fe7e32fdae6d9763130aed3a14bcc8a514fe9142f8b014265d1df4","abstract_canon_sha256":"07d1766493e1a793536e2ea000bad55de3af8356acd45238bc360a951392b4ea"},"schema_version":"1.0"},"canonical_sha256":"316e1bb98552e20e14999996b26c371bd947ec0efb99bf583c4d601551a8bd5b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:26:57.581888Z","signature_b64":"i6XBS7D3jNMl1AS7UgrJshuah0XdRepAu05HxTVIVZB7eJvgZmEgAxQcoE8fv8u7Lf3njkv66lZRShffeymYDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"316e1bb98552e20e14999996b26c371bd947ec0efb99bf583c4d601551a8bd5b","last_reissued_at":"2026-05-18T01:26:57.581228Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:26:57.581228Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1506.03187","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":"2026-05-18T01:26:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PN8jrFp50svvxfJH3yv20xpbj3pl7u9FPcW7zzyUVT/fwmsFyYIhbkxlfx7Ksan5g/tf4cOJHQRMrnsbAqy1Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-20T20:01:25.620805Z"},"content_sha256":"7c3ccf7cdd46a29791bce7beaf1ae770430596bb7324705ec8a75144688e5bdd","schema_version":"1.0","event_id":"sha256:7c3ccf7cdd46a29791bce7beaf1ae770430596bb7324705ec8a75144688e5bdd"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:GFXBXOMFKLRA4FEZTGLLE3BXDP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Boundedness of commutators generated by m-th Calder\\'on-Zygmund type singular integrals and local Campanato functions on generalized local Morrey spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Hongyang Xue, Huixia Mo","submitted_at":"2015-06-10T06:58:09Z","abstract_excerpt":"Let $T_m$ be the $m$-th Calder\\'on-Zygmund type singular integral. In the paper, we consider the boundedness of $T_m$ on the generalized product local Morrey spaces $LM_{p_1,\\varphi_1}^{\\{x_0\\}}\\times LM_{p_2,\\varphi_2}^{\\{x_0\\}}\\times\\dots\\times LM_{p_m,\\varphi_m}^{\\{x_0\\}}.$ And, the boundedness of the commutators of $T_m$ with local Campanato functions is obtained, also."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.03187","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":"2026-05-18T01:26:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9SlZF3rDtck9SgfjfSSx84/YYsO0leeuFKW/sw61LKRJQv+v/0QTfSTtnu4+pGoDrp4snv9y91/OvGlIlskAAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-20T20:01:25.621174Z"},"content_sha256":"a531785cfccac9dea26b2b55ae4df3ab91c6a35a2f96c00528a9196a22384196","schema_version":"1.0","event_id":"sha256:a531785cfccac9dea26b2b55ae4df3ab91c6a35a2f96c00528a9196a22384196"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GFXBXOMFKLRA4FEZTGLLE3BXDP/bundle.json","state_url":"https://pith.science/pith/GFXBXOMFKLRA4FEZTGLLE3BXDP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GFXBXOMFKLRA4FEZTGLLE3BXDP/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":"2026-05-20T20:01:25Z","links":{"resolver":"https://pith.science/pith/GFXBXOMFKLRA4FEZTGLLE3BXDP","bundle":"https://pith.science/pith/GFXBXOMFKLRA4FEZTGLLE3BXDP/bundle.json","state":"https://pith.science/pith/GFXBXOMFKLRA4FEZTGLLE3BXDP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GFXBXOMFKLRA4FEZTGLLE3BXDP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:GFXBXOMFKLRA4FEZTGLLE3BXDP","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":"07d1766493e1a793536e2ea000bad55de3af8356acd45238bc360a951392b4ea","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-06-10T06:58:09Z","title_canon_sha256":"774c12afa1fe7e32fdae6d9763130aed3a14bcc8a514fe9142f8b014265d1df4"},"schema_version":"1.0","source":{"id":"1506.03187","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1506.03187","created_at":"2026-05-18T01:26:57Z"},{"alias_kind":"arxiv_version","alias_value":"1506.03187v2","created_at":"2026-05-18T01:26:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1506.03187","created_at":"2026-05-18T01:26:57Z"},{"alias_kind":"pith_short_12","alias_value":"GFXBXOMFKLRA","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_16","alias_value":"GFXBXOMFKLRA4FEZ","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_8","alias_value":"GFXBXOMF","created_at":"2026-05-18T12:29:22Z"}],"graph_snapshots":[{"event_id":"sha256:a531785cfccac9dea26b2b55ae4df3ab91c6a35a2f96c00528a9196a22384196","target":"graph","created_at":"2026-05-18T01:26:57Z","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 $T_m$ be the $m$-th Calder\\'on-Zygmund type singular integral. In the paper, we consider the boundedness of $T_m$ on the generalized product local Morrey spaces $LM_{p_1,\\varphi_1}^{\\{x_0\\}}\\times LM_{p_2,\\varphi_2}^{\\{x_0\\}}\\times\\dots\\times LM_{p_m,\\varphi_m}^{\\{x_0\\}}.$ And, the boundedness of the commutators of $T_m$ with local Campanato functions is obtained, also.","authors_text":"Hongyang Xue, Huixia Mo","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-06-10T06:58:09Z","title":"Boundedness of commutators generated by m-th Calder\\'on-Zygmund type singular integrals and local Campanato functions on generalized local Morrey spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.03187","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":"sha256:7c3ccf7cdd46a29791bce7beaf1ae770430596bb7324705ec8a75144688e5bdd","target":"record","created_at":"2026-05-18T01:26:57Z","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":"07d1766493e1a793536e2ea000bad55de3af8356acd45238bc360a951392b4ea","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-06-10T06:58:09Z","title_canon_sha256":"774c12afa1fe7e32fdae6d9763130aed3a14bcc8a514fe9142f8b014265d1df4"},"schema_version":"1.0","source":{"id":"1506.03187","kind":"arxiv","version":2}},"canonical_sha256":"316e1bb98552e20e14999996b26c371bd947ec0efb99bf583c4d601551a8bd5b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"316e1bb98552e20e14999996b26c371bd947ec0efb99bf583c4d601551a8bd5b","first_computed_at":"2026-05-18T01:26:57.581228Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:26:57.581228Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"i6XBS7D3jNMl1AS7UgrJshuah0XdRepAu05HxTVIVZB7eJvgZmEgAxQcoE8fv8u7Lf3njkv66lZRShffeymYDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:26:57.581888Z","signed_message":"canonical_sha256_bytes"},"source_id":"1506.03187","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7c3ccf7cdd46a29791bce7beaf1ae770430596bb7324705ec8a75144688e5bdd","sha256:a531785cfccac9dea26b2b55ae4df3ab91c6a35a2f96c00528a9196a22384196"],"state_sha256":"a8f8d56c0922bcbe395cc6c6680e4890eff90880b6cd3c631cd745b77876b2a0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vJSi5bkfF98Rj1zN3Rb/XqjCOptIkwdI21ogYdQbh7ozoENgEhImEx7O2yWFfK7aPk+/fS0KG5MuBX09xNO/AA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-20T20:01:25.624399Z","bundle_sha256":"ec37d67060192e64f53abe57e72498567d1765c6f6c2d9a29d90383369be9056"}}