Stability of the Lipschitz extension property under metric transforms
type: Item
instance of: scholarly article
title: Stability of the Lipschitz extension property under metric transforms
zbMATH Open document ID: 1011.46037
author: R. Smith
published in: Geometric and Functional Analysis. GAFA
publication date: 2002-06-17T00:00:00Z
review text: Let \((\mathcal M, d)\) be a metric space and let \(X\) denote a Banach space with the norm \(||\cdot||\). The Lipschitz space \(\text{Lip}(\mathcal M, X)\) is the linear space of mappings \(F:\mathcal M\to X\) defined by the finiteness of the seminorm \[ |f|_{\text{Lip}(\mathcal M, X)}:=\sup_{x\neq y}\frac{||f(x)-f(y)||}{d(x,y)}. \] (a) A metric space \(\mathcal M\) satisfies the Lipschitz extension property with respect to \(X\), if there is a constant \(\lambda >0\) such that for every \(\mathcal M'\subset\mathcal M\) there exists an extension operator \(E_{\mathcal M'}:\text{Lip}(\mathcal M',X)\to\text{Lip}(\mathcal M, X)\) with norm \(\leq\lambda\). (b) \(\mathcal M\) satisfies the linear Lipschitz extension property with respect to \(X\), if the operator \(E_{\mathcal M'}\) can be chosen to be linear. The optimal choice of \(\lambda\) is denoted by \(e(\mathcal M,X)\) in case (a) and by \(e_l(\mathcal M,X)\) in case (b). Let \(w:\mathbb R_+\to\mathbb R_+\) be a concave non-decreasing function with \(\omega(0)=0\). The operation of superposition \(d_{\omega}:=\omega\circ d\) defines a new metric space \((\mathcal M, d_{\omega})\). This space is called the metric transform of \(\mathcal M\) by \(\omega\) and denoted by the symbol \(\omega(\mathcal M)\). The main theorem of the paper, which implies that linear and nonlinear Lipschitz extension properties of \(\mathcal M\) do not change when \(d\) is replaced by a new metric \(d_{\omega}\), is the following: Theorem. (a) \(e(\omega(\mathcal M), X)\leq C e(\mathcal M,X)^2\). (b) The same inequality holds for \(e_l\). \(C\) means an absolute constant.
reviewed by: Rauno Aulaskari
Mathematics Subject Classification ID: 46E99
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 54E35
zbMATH DE Number: 1756798
zbMATH Keywords: Lipschitz extension property
zbMATH Keywords: metric transform
MaRDI profile type: Publication
full work available at URL: s00039-002-8237-9
OpenAlex ID: W2055469646
DOI: 10.1007/S00039-002-8237-9
description: scientific article; zbMATH DE number 1756798