An isoperimetric inequality and the geometric Sobolev embedding for vector fields
type: Item
instance of: scholarly article
title: An isoperimetric inequality and the geometric Sobolev embedding for vector fields
zbMATH Open document ID: 0861.46019
author: Luca Capogna
author: D. Danielli
author: Nicola Garofalo
published in: Mathematical Research Letters
publication date: 1996-01-02T00:00:00Z
review text: Let \(X_1,\dots, X_m\) be \(C^\infty\) vector fields in \(\mathbb{R}^n\) satisfying Hörmander's condition for hypoellipticity and let \(d(x,y)\) be the Carnot-Carathéodory distance associated to \((X_1,\dots, X_m)\). It is well known that for a given \(U\subseteq \mathbb{R}^n\) there exists \(c_1>0\) and \(R_0>0\) such that for any \(x\in U\), \(0<R\leq R_0\) and \(0<t<1\) the inequality \(|B(x,R)|\leq c_1t^{-Q}|B(x,tR)|\) holds, where \(Q={{\log c_1}\over {\log 2}}\) and \(B(x,R)= \{y\in\mathbb{R}^n: d(x,y)<R\}\). The authors call the number \(Q\) the local homogeneous dimension and they prove the Gagliardo-Nirenberg type inequality for vector fields: \((-\hskip-.9em\int_{B(x,R)}|u|^k)^{1/k}\leq c_2R -\hskip-.9em\int_{B(x,R)} [\sum^m_{j=1}(X_j u)^2]^{1/2}\), where \(c_2\) is a constant, \(x\in U\), \(0<R\leq R_0\) and \(1\leq k\leq {Q\over {Q-1}}\). The exponent \({Q\over {Q-1}}\) is shown to be optimal. The above inequality is a consequence of an isoperimetric inequality proven in the paper, which involves a suitable definition of perimeter relative to the vector fields.
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 817694
zbMATH Keywords: Hörmander's condition for hypoellipticity
zbMATH Keywords: Carnot-Carathéodory distance
zbMATH Keywords: homogeneous dimension
zbMATH Keywords: Gagliardo-Nirenberg type inequality for vector fields
zbMATH Keywords: isoperimetric inequality
reviewed by: Vincenzo Ferone
MaRDI profile type: Publication
full work available at URL: mrl.1994.v1.n2.a14
OpenAlex ID: W2051856264
DOI: 10.4310/MRL.1994.V1.N2.A14
description: scientific article; zbMATH DE number 817694