The Poincaré inequality for \(C^{1,\alpha}\)-smooth vector fields
type: Item
instance of: scholarly article
title: The Poincaré inequality for \(C^{1,\alpha}\)-smooth vector fields
zbMATH Open document ID: 1310.53028
DOI: 10.1134/S0037446614020049
author: S. G. Basalaev
published in: Siberian Mathematical Journal
publication date: 2014-12-17T00:00:00Z
review text: Let \(M\) be a connected Riemannian \(C^\infty\)-manifold \(M\). Assume \(g\in M\), and \(r_0>0\). The author establishes the Poincaré inequality for John domains in the form \[ \| f- f_\Omega\|_{p,\Omega}\leq C\Biggl({b\over a}\Biggr)^\nu\text{\,diam}(\Omega)\|\nabla_H f\|_{p,\Omega}, \] with \(1\leq p<\infty\), where \(\Omega\subset B(g,r_0)\subset M\) is a John domain with inner diameter \(a\) and outer diameter \(b\), while \(f\in C^\infty(\Omega)\), and \(\nu\) is the Hausdorff dimension of \(M\). Moreorer, the constant \(C\) is independent of \(\Omega\) and \(f\), \(\nabla_H\) is the horizontal gradient. Equivalently, in the paper under review, the Poincaré inequality is obtained for the equiregular Carnot-Carathéodory spaces spanned by vector fields with Hölder class derivatives.
reviewed by: Grigori N. Milstein
Mathematics Subject Classification ID: 53C17
Mathematics Subject Classification ID: 57R25
zbMATH DE Number: 6381078
zbMATH Keywords: Carnot-Carathéodory space
zbMATH Keywords: Poincaré inequality
zbMATH Keywords: embedding theorems
zbMATH Keywords: John domain
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 6381078