{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:GK2BNRH2IXSHYUDLQKBI7PYT2G","short_pith_number":"pith:GK2BNRH2","schema_version":"1.0","canonical_sha256":"32b416c4fa45e47c506b82828fbf13d1bfc3b6b4145663d7a915e311315f7f58","source":{"kind":"arxiv","id":"1803.01301","version":1},"attestation_state":"computed","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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"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"},"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"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1803.01301","created_at":"2026-05-18T00:22:01.127569+00:00"},{"alias_kind":"arxiv_version","alias_value":"1803.01301v1","created_at":"2026-05-18T00:22:01.127569+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.01301","created_at":"2026-05-18T00:22:01.127569+00:00"},{"alias_kind":"pith_short_12","alias_value":"GK2BNRH2IXSH","created_at":"2026-05-18T12:32:25.280505+00:00"},{"alias_kind":"pith_short_16","alias_value":"GK2BNRH2IXSHYUDL","created_at":"2026-05-18T12:32:25.280505+00:00"},{"alias_kind":"pith_short_8","alias_value":"GK2BNRH2","created_at":"2026-05-18T12:32:25.280505+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/GK2BNRH2IXSHYUDLQKBI7PYT2G","json":"https://pith.science/pith/GK2BNRH2IXSHYUDLQKBI7PYT2G.json","graph_json":"https://pith.science/api/pith-number/GK2BNRH2IXSHYUDLQKBI7PYT2G/graph.json","events_json":"https://pith.science/api/pith-number/GK2BNRH2IXSHYUDLQKBI7PYT2G/events.json","paper":"https://pith.science/paper/GK2BNRH2"},"agent_actions":{"view_html":"https://pith.science/pith/GK2BNRH2IXSHYUDLQKBI7PYT2G","download_json":"https://pith.science/pith/GK2BNRH2IXSHYUDLQKBI7PYT2G.json","view_paper":"https://pith.science/paper/GK2BNRH2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1803.01301&json=true","fetch_graph":"https://pith.science/api/pith-number/GK2BNRH2IXSHYUDLQKBI7PYT2G/graph.json","fetch_events":"https://pith.science/api/pith-number/GK2BNRH2IXSHYUDLQKBI7PYT2G/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GK2BNRH2IXSHYUDLQKBI7PYT2G/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GK2BNRH2IXSHYUDLQKBI7PYT2G/action/storage_attestation","attest_author":"https://pith.science/pith/GK2BNRH2IXSHYUDLQKBI7PYT2G/action/author_attestation","sign_citation":"https://pith.science/pith/GK2BNRH2IXSHYUDLQKBI7PYT2G/action/citation_signature","submit_replication":"https://pith.science/pith/GK2BNRH2IXSHYUDLQKBI7PYT2G/action/replication_record"}},"created_at":"2026-05-18T00:22:01.127569+00:00","updated_at":"2026-05-18T00:22:01.127569+00:00"}