{"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"}