Monotonicity and best approximation in Orlicz--Sobolev spaces with the Luxemburg norm
type: Item
instance of: scholarly article
title: Monotonicity and best approximation in Orlicz--Sobolev spaces with the Luxemburg norm
zbMATH Open document ID: 1153.46014
author: S. Chen
author: Xin He
author: Henryk Hudzik
author: Anna Kamińska
published in: Journal of Mathematical Analysis and Applications
publication date: 2008-06-17T00:00:00Z
review text: Let \(\Omega\) be a bounded and connected open subset of \(\mathbb{R}^{n}\), and let \((\Omega,\Sigma,\mu)\) be a nonatomic finite measure space. The modular of a measurable function \(u\) on \(\Omega\) is defined by \(\rho_{A}(u)=\int_{\Omega }A(u(t))\,dt\), where \(A(u)\) is a given \(N\)-function. The Orlicz space is \(L_{A}(\Omega )=\{u(t):\) there\ exists \(\lambda >0\) such\ that \(\rho (\lambda u)<\infty \}\) and the Orlicz--Sobolev space is defined as \[ W_{m,A}=\{u\in L_{A}(\Omega):\partial^{\alpha }u\in L_{A}(\Omega),\;0\leq\left|\alpha \right|\leq m\}, \] where \(m\in \{0,1,2,\dots\}\), \(\alpha =(\alpha _{1},\dots,\alpha_{n})\), \( \left|\alpha\right|=\alpha_{1}+\cdots+\alpha_{n}\), \(\alpha_{i}\geq 0\) (\(i=1,2,\dots,n\)) are integers and \(\partial^{\alpha}u\) is the \(\alpha\)-th distributional derivative of \(u\). The space \(W_{m,A}\) is equipped with the norm \[ \left\| u\right\|_{m,A}=\left(\sum_{0\leq\left|\alpha\right| \leq m}\left\|\partial^{\alpha }u\right\|^{p}\right)^{1/p} \] (\(p\in(1,\infty)\) fixed), where \(\left\| u\right\| =\inf \{k>0:\rho_{A}(u/k)\leq 1\}\) is the Luxemburg norm in the Orlicz space \(L_{A}(\Omega)\). The space \(W_{m,A}\) is also equipped with the partial order \(u\leq \upsilon \) whenever \(\partial^{\alpha}u(t)\leq \partial^{\alpha }\upsilon(t)\) a.e.\ in \(\Omega\), for all \(\alpha=(\alpha_{1},\dots,\alpha _{n})\) with \(0\leq\left|\alpha\right|\leq m\). Thus \(W_{m,A}\) is a Banach lattice, and the authors study the monotonicity properties in \(W_{m,A}\), namely, strict monotonicity, upper (lower) local uniform monotonicity, uniform monotonicity, etc. The connections of these properties with some geometric properties of the unit ball of \(W_{m,A}\) are also established. Using these properties in the last section of the paper, the problem of best dominated approximation by elements of a convex set in \(W_{m,A}\) is studied (existence and unicity of the best dominated approximant, stability and continuity of the be…
reviewed by: Costică Mustăţa
Mathematics Subject Classification ID: 46B42
Mathematics Subject Classification ID: 41A50
zbMATH DE Number: 5289176
zbMATH Keywords: Orlicz-Sobolev spaces
zbMATH Keywords: strict monotonicity
zbMATH Keywords: upper (lower) local uniform monotonicity
zbMATH Keywords: uniform monotonicity
zbMATH Keywords: best approximation
MaRDI profile type: Publication
full work available at URL: j.jmaa.2008.02.015
OpenAlex ID: W2053287936
cites work: Equivalent Norms for Sobolev Spaces
cites work: Q3029575
cites work: Best approximation in inner product spaces
cites work: Some approximation properties in Orlicz-Sobolev spaces
cites work: Q5331549
cites work: Q4716146
cites work: Q4459025
cites work: Q4391750
cites work: Sobolev embedding theorems for spaces \(W^{k,p(x)}(\Omega)\)
cites work: Q3207595
cites work: Q3689692
cites work: Sobolev capacity of the space \(W^{1,p(\cdot)} (\mathbb{R}^n)\)
cites work: Banach lattices with order isometric copies of \(l^\infty\)
cites work: Q4165759
cites work: Q4176544
cites work: Q3866649
cites work: Monotonicity Properties of Lorentz Spaces
cites work: Monotonicity and rotundity properties in Banach lattices
cites work: Approximative compactness and full rotundity in Musielak--Orlicz spaces and Lorentz--Orlicz spaces
cites work: Monotonicity properties of Musielak-Orlicz spaces and dominated best approximation in Banach lattices
cites work: Points of monotonicity in Musielak--Orlicz function spaces endowed with the Luxemburg norm
cites work: Relationships between monotonicity and complex rotundity properties with some consequences
cites work: Best approximants in modular function spaces
cites work: Q3272715
cites work: On uniqueness of bestL1-approximation in sobolev spaces
cites work: Strictly and uniformly monotone Musielak-Orlicz spaces and applications to best approximation
cites work: Best approximants in L ? -spaces
cites work: Q4188284
cites work: Q5705063
cites work: Monotone coefficients and monotonicity of Orlicz spaces
cites work: Q4348185
cites work: Orlicz spaces and modular spaces
DOI: 10.1016/J.JMAA.2008.02.015
description: scientific article; zbMATH DE number 5289176