Limits of Besov norms
type: Item
instance of: scholarly article
title: Limits of Besov norms
zbMATH Open document ID: 1222.46028
author: Hans Triebel
published in: Archiv der Mathematik
publication date: 2011-03-31T00:00:00Z
review text: Let \(\Delta_{h}^{m}f(x)\) with \(x \in\mathbb{R}^n\), \(h\in \mathbb{R}^n\), \(m\in \mathbb{N}\), be the usual differences of complex-valued Lebesgue-measurable functions on \(\mathbb{R}^n\) and let \(\omega_{m}(f, t)_{p}=\sup_{|h| \leq t}\|\Delta_{h}^{m}f|L_{p}(\mathbb{R}^n)\|\), \(0< p \leq \infty\), \(t >0\), be the related moduli of smoothness. Let \(0 < p, q \leq \infty\) and \(0 < s < m \in \mathbb{N}\). Then \[ \|f|\mathbf{B}^{s}_{p,q}(\mathbb{R}^n)\|_{m}=\|f|L_{p}(\mathbb{R}^n)\|+ \biggl(\int_{|h| \leq 1}|h|^{-sq}\|\Delta_{h}^{m}f|L_{p}(\mathbb{R}^n)\|^{q} \displaystyle{\frac{dh}{|h|^n}} \biggr)^{1/q} \] and \[ \|f|\mathbf{B}^{s}_{p,q}(\mathbb{R}^n)\|^{*}_{m}=\|f|L_{p}(\mathbb{R}^n)\|+ \biggl(\int_{0}^{1}t^{-sq}\omega_{m}^{q}(f,t)_{p} \displaystyle{\frac{dt}{t}} \biggr)^{1/q}\tag{1} \] (obviously modified if \(q=\infty\)) are equivalent quasi-norms of the Besov spaces \(\mathbf{B}^{s}_{p,q}(\mathbb{R}^n)\) consisting of those measurable functions for which (1) is finite. Let \(0 < p \leq \infty\) and \(m \in \mathbb{N}\). Then \(\mathbf{W}_{p}^{m}(\mathbb{R}^n)\) is the collection of all \(f \in L_{p}(\mathbb{R}^n)\) for which there exists a function \(0 \leq g \in L_{p}(\mathbb{R}^n)\) such that for all \(h \in \mathbb{R}^n\), \(0<|h| \leq 1,\) \[ |h|^{-m}|\Delta^{m}_{h}f(x)| \leq \sum_{l=0}^{m}g(x+lh) \,\, {\text{ a.e. \;\;in}}\;\;\mathbb{R}^n\tag{2} \] Furthermore, \[ \|f|\mathbf{W}^{m}_{p}(\mathbb{R}^n)\| =\|f|L_{p}(\mathbb{R}^n)\|+\inf\|g|L_{p}(\mathbb{R}^n)\| \] where the infimum is taken over all \(g\) with (2). If \(1 < p< \infty\), then \(\mathbf{W}^{m}_{p}(\mathbb{R}^n)=W^{m}_{p}(\mathbb{R}^n)\) are the classical Sobolev spaces and \[ \|f|\mathbf{W}^{m}_{p}(\mathbb{R}^n)\|\sim \|f|W^{m}_{p}(\mathbb{R}^n)\|= \sum_{|\alpha| \leq m}\|D^{\alpha}f|L_{p}(\mathbb{R}^n)\| \] are equivalent norms. The paper deals with the question what happens if \(s \uparrow m\) and how the outcome is related to the Sobolev spaces \(\mathbf{W}^{m}_{p}(\mathbb{R}^n)\). We …
reviewed by: Koichi Saka
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 42B35
zbMATH DE Number: 5872601
zbMATH Keywords: Besov spaces
zbMATH Keywords: Sobolev spaces
zbMATH Keywords: limiting embeddings
MaRDI profile type: Publication
full work available at URL: s00013-010-0214-1
OpenAlex ID: W2015611221
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DOI: 10.1007/S00013-010-0214-1
description: scientific article; zbMATH DE number 5872601