{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:GK2BNRH2IXSHYUDLQKBI7PYT2G","short_pith_number":"pith:GK2BNRH2","canonical_record":{"source":{"id":"1803.01301","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-03-04T05:07:03Z","cross_cats_sorted":[],"title_canon_sha256":"15c1f62763d860f2b7cfe108be36882ecc348b1910ac78c16677acacd2cab624","abstract_canon_sha256":"113f1adae948caecaf9db47889e4cca33e6c16e17cfd0075c15c6c2acf2aef8d"},"schema_version":"1.0"},"canonical_sha256":"32b416c4fa45e47c506b82828fbf13d1bfc3b6b4145663d7a915e311315f7f58","source":{"kind":"arxiv","id":"1803.01301","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1803.01301","created_at":"2026-05-18T00:22:01Z"},{"alias_kind":"arxiv_version","alias_value":"1803.01301v1","created_at":"2026-05-18T00:22:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.01301","created_at":"2026-05-18T00:22:01Z"},{"alias_kind":"pith_short_12","alias_value":"GK2BNRH2IXSH","created_at":"2026-05-18T12:32:25Z"},{"alias_kind":"pith_short_16","alias_value":"GK2BNRH2IXSHYUDL","created_at":"2026-05-18T12:32:25Z"},{"alias_kind":"pith_short_8","alias_value":"GK2BNRH2","created_at":"2026-05-18T12:32:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:GK2BNRH2IXSHYUDLQKBI7PYT2G","target":"record","payload":{"canonical_record":{"source":{"id":"1803.01301","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-03-04T05:07:03Z","cross_cats_sorted":[],"title_canon_sha256":"15c1f62763d860f2b7cfe108be36882ecc348b1910ac78c16677acacd2cab624","abstract_canon_sha256":"113f1adae948caecaf9db47889e4cca33e6c16e17cfd0075c15c6c2acf2aef8d"},"schema_version":"1.0"},"canonical_sha256":"32b416c4fa45e47c506b82828fbf13d1bfc3b6b4145663d7a915e311315f7f58","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:22:01.128078Z","signature_b64":"G7An1xG9oNWZP9rQxpiDSX5UwxGf6opL2dl39DREMrEjqLbVcQIB6Ppa1J/KlxTJ8r/USy9B3iTm6LJCkYR+Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"32b416c4fa45e47c506b82828fbf13d1bfc3b6b4145663d7a915e311315f7f58","last_reissued_at":"2026-05-18T00:22:01.127494Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:22:01.127494Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1803.01301","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:22:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"c3mA+Y8efbg758T2VwlQlfEj8WXPKCMzSEm2oeSsJwoZTeGhVnwbgK+chJGIwn45C1SL55RAK1JsJ21hLySvAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T03:38:53.074368Z"},"content_sha256":"d59aa268c677c16bde829b072a3f4a6b87fd500b94941744e35e457d0d2384af","schema_version":"1.0","event_id":"sha256:d59aa268c677c16bde829b072a3f4a6b87fd500b94941744e35e457d0d2384af"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:GK2BNRH2IXSHYUDLQKBI7PYT2G","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Lower bound of Riesz transform kernels revisited and commutators on stratified Lie groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Brett D. Wick, Hong-Quan Li, Ji Li, Qingyan Wu, Xuan Thinh Duong","submitted_at":"2018-03-04T05:07:03Z","abstract_excerpt":"Let $\\mathcal G$ be a stratified Lie group and $\\{\\X_j\\}_{1 \\leq j \\leq n}$ a basis for the left-invariant vector fields of degree one on $\\mathcal G$. Let $\\Delta = \\sum_{j = 1}^n \\X_j^2 $ be the sub-Laplacian on $\\mathcal G$ and the $j^{\\mathrm{th}}$ Riesz transform on $\\mathcal G$ is defined by $R_j:= \\X_j (-\\Delta)^{-\\frac{1}{2}}$,\n  $1 \\leq j \\leq n$. In this paper we give a new version of the lower bound of the kernels of Riesz transform $R_j$ and then establish the Bloom-type two weight estimates as well as a number of endpoint characterisations for the commutators of the Riesz transfor"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.01301","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:22:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ysNCeWwtQ4d8UG3UYzZkNK+fpilye7GtNCFZiClE+WX3NUvY2oc1pplgtv3Fa5gylnSznNa37MgOU8Sn4rWjCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T03:38:53.074733Z"},"content_sha256":"d74ef46d0a6fc1b59ca927ce6c0b17491ac98fa09ba8e7fce5fd3badc74b8071","schema_version":"1.0","event_id":"sha256:d74ef46d0a6fc1b59ca927ce6c0b17491ac98fa09ba8e7fce5fd3badc74b8071"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GK2BNRH2IXSHYUDLQKBI7PYT2G/bundle.json","state_url":"https://pith.science/pith/GK2BNRH2IXSHYUDLQKBI7PYT2G/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GK2BNRH2IXSHYUDLQKBI7PYT2G/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":"2026-06-01T03:38:53Z","links":{"resolver":"https://pith.science/pith/GK2BNRH2IXSHYUDLQKBI7PYT2G","bundle":"https://pith.science/pith/GK2BNRH2IXSHYUDLQKBI7PYT2G/bundle.json","state":"https://pith.science/pith/GK2BNRH2IXSHYUDLQKBI7PYT2G/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GK2BNRH2IXSHYUDLQKBI7PYT2G/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:GK2BNRH2IXSHYUDLQKBI7PYT2G","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":"113f1adae948caecaf9db47889e4cca33e6c16e17cfd0075c15c6c2acf2aef8d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-03-04T05:07:03Z","title_canon_sha256":"15c1f62763d860f2b7cfe108be36882ecc348b1910ac78c16677acacd2cab624"},"schema_version":"1.0","source":{"id":"1803.01301","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1803.01301","created_at":"2026-05-18T00:22:01Z"},{"alias_kind":"arxiv_version","alias_value":"1803.01301v1","created_at":"2026-05-18T00:22:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.01301","created_at":"2026-05-18T00:22:01Z"},{"alias_kind":"pith_short_12","alias_value":"GK2BNRH2IXSH","created_at":"2026-05-18T12:32:25Z"},{"alias_kind":"pith_short_16","alias_value":"GK2BNRH2IXSHYUDL","created_at":"2026-05-18T12:32:25Z"},{"alias_kind":"pith_short_8","alias_value":"GK2BNRH2","created_at":"2026-05-18T12:32:25Z"}],"graph_snapshots":[{"event_id":"sha256:d74ef46d0a6fc1b59ca927ce6c0b17491ac98fa09ba8e7fce5fd3badc74b8071","target":"graph","created_at":"2026-05-18T00:22:01Z","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 $\\mathcal G$ be a stratified Lie group and $\\{\\X_j\\}_{1 \\leq j \\leq n}$ a basis for the left-invariant vector fields of degree one on $\\mathcal G$. Let $\\Delta = \\sum_{j = 1}^n \\X_j^2 $ be the sub-Laplacian on $\\mathcal G$ and the $j^{\\mathrm{th}}$ Riesz transform on $\\mathcal G$ is defined by $R_j:= \\X_j (-\\Delta)^{-\\frac{1}{2}}$,\n  $1 \\leq j \\leq n$. In this paper we give a new version of the lower bound of the kernels of Riesz transform $R_j$ and then establish the Bloom-type two weight estimates as well as a number of endpoint characterisations for the commutators of the Riesz transfor","authors_text":"Brett D. Wick, Hong-Quan Li, Ji Li, Qingyan Wu, Xuan Thinh Duong","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-03-04T05:07:03Z","title":"Lower bound of Riesz transform kernels revisited and commutators on stratified Lie groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.01301","kind":"arxiv","version":1},"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:d59aa268c677c16bde829b072a3f4a6b87fd500b94941744e35e457d0d2384af","target":"record","created_at":"2026-05-18T00:22:01Z","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":"113f1adae948caecaf9db47889e4cca33e6c16e17cfd0075c15c6c2acf2aef8d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-03-04T05:07:03Z","title_canon_sha256":"15c1f62763d860f2b7cfe108be36882ecc348b1910ac78c16677acacd2cab624"},"schema_version":"1.0","source":{"id":"1803.01301","kind":"arxiv","version":1}},"canonical_sha256":"32b416c4fa45e47c506b82828fbf13d1bfc3b6b4145663d7a915e311315f7f58","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"32b416c4fa45e47c506b82828fbf13d1bfc3b6b4145663d7a915e311315f7f58","first_computed_at":"2026-05-18T00:22:01.127494Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:22:01.127494Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"G7An1xG9oNWZP9rQxpiDSX5UwxGf6opL2dl39DREMrEjqLbVcQIB6Ppa1J/KlxTJ8r/USy9B3iTm6LJCkYR+Dw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:22:01.128078Z","signed_message":"canonical_sha256_bytes"},"source_id":"1803.01301","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d59aa268c677c16bde829b072a3f4a6b87fd500b94941744e35e457d0d2384af","sha256:d74ef46d0a6fc1b59ca927ce6c0b17491ac98fa09ba8e7fce5fd3badc74b8071"],"state_sha256":"ce2f5e6cccae4afedcd7aff38390be5c79cae91b559455bd7933214b4f83a84a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HzzQbKRDRHTliyeQU4XKKDQm6qDZ95oEsGuxPAyeY/iWK52wLpHFZs/vY9MN6MLF/MoVR4+xwh3OLIcyTWJgAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T03:38:53.076674Z","bundle_sha256":"de0b81fe45f325ab806365eeb684d1987528e48c57a66ff60952084e75fc915f"}}