Lipschitz continuity, global smooth approximations and extension theorems for Sobolev functions in Carnot-Carathéodory spaces
type: Item
instance of: scholarly article
title: Lipschitz continuity, global smooth approximations and extension theorems for Sobolev functions in Carnot-Carathéodory spaces
zbMATH Open document ID: 0906.46026
DOI: 10.1007/BF02819446
author: Nicola Garofalo
author: D. M. Nhieu
published in: Journal d'Analyse Mathématique
publication date: 1999-02-18T00:00:00Z
review text: Let \(X= \{X_1,\dots, X_n\}\) be a system of locally Lipschitz real-valued vector fields in \(\mathbb{R}^n\), \(d(x, y)\) be the Carnot-Carathéodory distance associated to \(X\), \(B(x_0, R)= \{x\in\mathbb{R}^n\mid d(x, x_0)< R\}\). The authors introduce the following basic assumptions: the openness of the balls \(B(x_0, R)\) in the Euclidean topology, the so-called ``doubling condition'', and some Poincaré type inequality. Under these not very restrictive conditions some extension properties of Sobolev spaces are investigated and approximation of Sobolev functions by functions which are smooth up to the boundary of a domain is obtained. In particular, for the so-called \((\varepsilon, \delta)\)-domain \(\Omega\subset\mathbb{R}^n\) with the radius of \(\Omega>0\) the following extension result is established. Let \(1\leq p<\infty\), let \({\mathcal L}^{1,p}(\Omega)= \{f\in L^p(\Omega)\mid X_jf\in L^p(\Omega), j= 1,\dots,m\}\) be the weak Sobolev space endowed with the norm \(\| f\|_{{\mathcal L}^{1,p}(\Omega)}= \| f\|_{L^p(\Omega)}+ \sum^m_{j= 1}\| X_jf\|_{L^p(\Omega)}\). Then there exists a linear operator \({\mathcal E}: {\mathcal L}^{1,p}(\Omega)\to {\mathcal L}^{1,p}(\mathbb{R}^n)\) such that for some \(C>0\) one has for \(f\in {\mathcal L}^{1,p}(\Omega)\), (i) \({\mathcal E}f(x)= f(x)\) for a.e. \(x\in\Omega\), (ii) \(\|{\mathcal E}f\|_{{\mathcal L}^{1,p}(\mathbb{R}^n)}\leq C\| f\|_{{\mathcal L}^{1,p}(\Omega)}\).
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 35S15
zbMATH DE Number: 1187469
zbMATH Keywords: boundary value problem
zbMATH Keywords: doubling condition
zbMATH Keywords: Carnot-Carathéodory distance
zbMATH Keywords: Poincaré type inequality
zbMATH Keywords: extension properties
zbMATH Keywords: approximation of Sobolev functions
zbMATH Keywords: weak Sobolev space
MaRDI profile type: Publication
cites work: Q1923542
cites work: An embedding theorem and the harnack inequality for nonlinear subelliptic equations
cites work: Capacitary estimates and the local behavior of solutions of nonlinear subelliptic equations
cites work: Q4870772
cites work: Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung
cites work: The extension problem for certain function spaces involving fractional orders of differentiability
cites work: Q4695263
cites work: Q2548398
cites work: Regularity at the boundary for solutions of nonlinear subelliptic equations
cites work: A Fefferman-Phong type inequality and applications to quasilinear subelliptic equations
cites work: Q4705750
cites work: Les espaces du type de Beppo Levi
cites work: Q3134551
cites work: Fundamental solutions for second order subelliptic operators
cites work: Hardy Spaces on Homogeneous Groups. (MN-28), Volume 28
cites work: Q3345173
cites work: Une condition géométrique pour l'inégalité de Harnack. (A geometric condition for Harnack's inequality)
cites work: Q3129869
cites work: Q4359612
cites work: The Identity of Weak and Strong Extensions of Differential Operators
cites work: Isoperimetric and Sobolev inequalities for Carnot-Carath�odory spaces and the existence of minimal surfaces
cites work: The Dirichlet problem for sublaplacians on nilpotent Lie groups - geometric criteria for regularity
cites work: Q3933274
cites work: Q5570370
cites work: Hypoelliptic second order differential equations
cites work: The Poincaré inequality for vector fields satisfying Hörmander's condition
cites work: Boundary behavior of harmonic functions in non-tangentially accessible domains
cites work: Quasiconformal mappings and extendability of functions in Sobolev spaces
cites work: Embedding theorems into Lipschitz and BMO spaces and applications to quasilinear subelliptic differential equations
cites work: Injectivity theorems in plane and space
cites work: H = W
cites work: Q5518064
cites work: Balls and metrics defined by vector fields. I: Basic properties
cites work: Q5621610
cites work: Q5547857
cites work: Q3993586
cites work: Q4002250
cites work: On extension of functions of bounded mean oscillation from domains in a space of homogeneous type with intrinsic metric
full work available at URL: https://doi.org/10.1007/bf02819446
OpenAlex ID: W2325182507
description: scientific article; zbMATH DE number 1187469