A note on Hajłasz-Sobolev spaces on fractals
type: Item
instance of: scholarly article
title: A note on Hajłasz-Sobolev spaces on fractals
zbMATH Open document ID: 1023.46033
DOI: 10.1016/S0022-247X(03)00039-8
author: Jiaxin Hu
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-05-25T00:00:00Z
review text: Let \(M \subset\mathbb{R}^n\) and let \(\mu\) be a Borel measure on \(M\). Then \(L_p (M,\mu)\) with \(1\leq p \leq \infty\) has the usual meaning. For \(\sigma >0\) the Hajłasz-Sobolev space \(H^\sigma_p (M,\mu)\) consists of all \(u \in L_p (M, \mu)\) such that \[ |u(x) - u(y)|\leq |x-y|^\sigma (v(x) + v(y)) \] for some \(v \in L_p (M, \mu)\). (\(\sigma >1\) makes sense if, for example, \(M\) is a disconnected fractal.) It is the main aim of this paper to compare these spaces with the well-known spaces \(B^\sigma_{p, \infty} (M, \mu)\) introduced by Jonsson and Wallin. In addition, some embedding theorems are proved.
reviewed by: Hans Triebel
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 28A80
zbMATH DE Number: 1915607
zbMATH Keywords: Sobolev spaces
zbMATH Keywords: Besov spaces
zbMATH Keywords: fractals
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1915607