{"paper":{"title":"Geometric Hardy inequalities for the sub-elliptic Laplacian on convex domains in the Heisenberg group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.SP"],"primary_cat":"math.AP","authors_text":"Simon Larson","submitted_at":"2016-03-04T08:46:10Z","abstract_excerpt":"We prove geometric $L^p$ versions of Hardy's inequality for the sub-elliptic Laplacian on convex domains $\\Omega$ in the Heisenberg group $\\mathbb{H}^n$, where convex is meant in the Euclidean sense. When $p=2$ and $\\Omega$ is the half-space given by $\\langle \\xi, \\nu\\rangle > d$ this generalizes an inequality previously obtained by Luan and Yang. For such $p$ and $\\Omega$ the inequality is sharp and takes the form \\begin{equation}\n  \\int_\\Omega |\\nabla_{\\mathbb{H}^n}u|^2 \\, d\\xi \\geq \\frac{1}{4}\\int_{\\Omega} \\sum_{i=1}^n\\frac{\\langle X_i(\\xi), \\nu\\rangle^2+\\langle Y_i(\\xi), \\nu\\rangle^2}{\\tex"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.01379","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"}