Extension of differentiable functions beyond the boundary of the domain on Carnot groups
type: Item
instance of: scholarly article
title: Extension of differentiable functions beyond the boundary of the domain on Carnot groups
zbMATH Open document ID: 0917.31007
published in: Siberian Advances in Mathematics
publication date: 1999-02-24T00:00:00Z
review text: Let \(D\) be a domain of a Carnot group \(\mathbb G\), which is a special Lie group introduced by \textit{P. Pansu} [Ann. Math. (2) 129, 1--60 (1989; Zbl 0678.53042)]. The author investigates the problems of extension of functions in Sobolev spaces \(W_p^1(D)\) and \(L_p^1(D)\). It is shown that the \((\varepsilon,\delta)\)-condition on the domain \(D\) is sufficient for the existence of bounded extension operators from spaces \(W_p^1(D)\) and \(L_p^1(D)\) to spaces \(W_p^1(\mathbb G)\) and \(L_p^1(\mathbb G)\), respectively. The author also obtains some geometric conditions on the domain \(D\) which are necessary for the existence of a bounded extension operator. Furthermore the paper contains some results on the geometry of Carnot groups. For example, the author proves that a ball in the Carnot-Carathéodory metric on a Carnot group is a uniform domain.
Mathematics Subject Classification ID: 31C45
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 1047526
zbMATH Keywords: extension of functions
zbMATH Keywords: extension operator
zbMATH Keywords: Carnot group
zbMATH Keywords: Sobolev space
zbMATH Keywords: uniform domain
zbMATH Keywords: Carnot-Carathéodory metric
author: A. V. Greshnov
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1047526