Some remarks about metric spaces, spherical mappings, functions and their derivatives
type: Item
instance of: scholarly article
title: Some remarks about metric spaces, spherical mappings, functions and their derivatives
zbMATH Open document ID: 0927.46021
DOI: 10.5565/PUBLMAT_40296_12
author: Stephen Semmes
published in: Publicacions Matemàtiques
publication date: 1999-09-14T00:00:00Z
full work available at URL: https://eudml.org/doc/41266
review text: Given a metric \(d\) in \(\mathbb{R}^{n}\) and \(p\in \mathbb{R}^{n}\), the author investigates the existence of a function \(\theta _{p}:\mathbb{R}^{n}\backslash \{p\}\rightarrow S^{n-1}\) satisfying a ``Lipschitz'' condition \[ \left| \theta _{p}(x)-\theta _{p}(y)\right| \leq C\;\frac{d(x,y)}{\min (d(x,p),d(y,p))} \] and approximating the radial projection \(\pi _{p}:\mathbb{R} ^{n}\backslash \{p\}\rightarrow S^{n-1}\), \(\pi _{p}(x)=\frac{x-p}{\left| x-p\right| }.\) The existence of such mappings is used to derive Sobolev-type inequalities. The author constructs \(\theta _{p}\) when \(d\) satisfies the condition \[ d(x,y)\leq td(x,z)\text{ implies }\left| x-y\right| \leq \eta (t)\left| x-z\right| ,\text{ }(x,y\in \mathbb{R}^{n},t>0) \] where \(\eta :[0,\infty)\rightarrow [0,\infty)\) is a homeomorphism, and obtains an estimate for the values of a \(C^{1}\) function in terms of its gradient when \(d\) is a quasimetric equivalent to a metric , \(d\) being generated by a ``doubling measure'' \(\mu \) on \(\mathbb{R}^{n}\) (i.e. \(\mu (B(x,2r))\leq C\mu (B(x,r))\))
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 54E35
zbMATH DE Number: 999992
zbMATH Keywords: spherical mapping
zbMATH Keywords: metric doubling measure
zbMATH Keywords: partition of unity
zbMATH Keywords: Sobolev inequality
reviewed by: Valeriu Anisiu
MaRDI profile type: Publication
OpenAlex ID: W2137651612
description: scientific article; zbMATH DE number 999992