{"paper":{"title":"Derivative Formulae and Poincar\\'e Inequality for Kohn-Laplacian Type Semigroups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Feng-Yu Wang","submitted_at":"2012-08-25T02:52:50Z","abstract_excerpt":"As a generalization to the heat semigroup on the Heisenberg group, the diffusion semigroup generated by the subelliptic operator $L:=\\ff 1 2 \\sum_{i=1}^m X_i^2$ on $\\R^{m+d}:= \\R^m\\times\\R^d$ is investigated, where $$X_i(x,y)= \\sum_{k=1}^m \\si_{ki} \\pp_{x_k} + \\sum_{l=1}^d (A_l x)_i\\pp_{y_l},\\ \\ (x,y)\\in\\R^{m+d}, 1\\le i\\le m$$ for $\\si$ an invertible $m\\times m$-matrix and $\\{A_l\\}_{1\\le l\\le d}$ some $m\\times m$-matrices such that the H\\\"ormander condition holds. We first establish Bismut-type and Driver-type derivative formulae with applications on gradient estimates and the coupling/Liouvil"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.5093","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"}