\(JLip\) versus Sobolev spaces on a class of self-similar fractal foliages
type: Item
instance of: scholarly article
title: \(JLip\) versus Sobolev spaces on a class of self-similar fractal foliages
zbMATH Open document ID: 1239.46025
author: Yves Achdou
author: Thibaut Deheuvels
author: Nicoletta Tchou
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 2012-02-11T00:00:00Z
review text: The paper deals with special self-similar \(d\)-sets \(\Gamma\) in the plane \(\mathbb R^2\) which originate from two peculiar similitudes. Roughly speaking \(\Gamma\) is the boundary of a domain \(\Omega\). Let \(W^1_p (\Omega)\) be the corresponding Sobolev space and \(B^s_{p,p} (\Gamma)\) its trace, \(s= 1 - \frac{2-d}{p}\), \(1\leq p <\infty\), normed by \[ \| f \, | L_p (\Gamma, \mu) \| + \Big( \int_{\Gamma \times \Gamma} \frac{|f(x)-f(y)|^p}{|x-y|^{d + sp}} \, d \mu (x) \, d \mu (y) \Big)^{1/p} \] where \(\mu\) is the related Hausdorff measure on \(\Gamma\). Let \(\mathrm{Lip} (t,p,\Gamma)\) be Lipschitz spaces, defined via expansions by Haar functions on \(\Gamma\). The paper deals with the trace of \(W^1_p (\Omega)\) on \(\Gamma\) in terms of Besov spaces and a comparison of these Besov spaces with corresponding Lipschitz spaces.
reviewed by: Hans Triebel
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 6005648
zbMATH Keywords: \(d\)-sets
zbMATH Keywords: fractals
zbMATH Keywords: Besov spaces
zbMATH Keywords: trace theorem
MaRDI profile type: Publication
full work available at URL: j.matpur.2011.07.002
OpenAlex ID: W2005915569
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DOI: 10.1016/J.MATPUR.2011.07.002
description: scientific article; zbMATH DE number 6005648