Embedding of Lipschitz classes into classes of functions of \(\Lambda \)-bounded variation
type: Item
instance of: scholarly article
title: Embedding of Lipschitz classes into classes of functions of \(\Lambda \)-bounded variation
zbMATH Open document ID: 1179.26033
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-05-19T00:00:00Z
review text: Introduction and results: Let \(\Lambda=\{\lambda_n\}\) be a non-decreasing sequence of positive numbers such that \(\sum 1/ \lambda_n\) diverges. We say that a real-valued function \(f\) on \([a,b]\) is of \(\Lambda\)-bounded variation \((\Lambda BV)\) if for any sequence of non-overlapping intervals \([a_n,b_n]\) in \([a,b]\), \[ \sum^\infty_{n=1} \frac{|f(b_n)-f(a_n)|}{\lambda_n}<+\infty. \] This is equivalent requiring that the terms are uniformly bounded (see [\textit{D. Waterman}, Stud. Math. 57, 33--45 (1976; Zbl 0341.26008)]). It is obvious that if all \(\lambda_n=1\), \(\{1\}BV\) coincides with the class \(BV\) of functions of the usual bounded variation. In the sequel, we always suppose that \([a,b]=[0,2\pi]\). Let \(\omega(t)\) be a modulus of continuity, i.e., a continuous and non-decreasing function on \([0,+\infty)\) satisfying \(\omega(0)=0\) and \(\omega(t_1+t_2)\leq\omega(t_1)+ \omega(t_2)\) for nonnegative \(t_1\) and \(t_2\). For \(1\leq p\leq\infty\), denote by \(H^\omega_p\equiv H_p^{\omega(t)}\) the class of \(2\pi\)-periodic functions for which \(\|f\|_{H^\omega_p}:=\|f\|_p+ \sup_{t>0} \frac {\omega(f,t)p}{\omega(t)}<\infty\), where \[ \omega(f;t)_p:= \begin{cases}\sup_{0\leq h\leq t}\left\{\int_0^{2\pi}|f(x+h)-f(x)|^pdx\right \}^{1/p},\;1\leq p<\infty,\\ \sup_{0 \leq h\leq t}\sup_{x\in[0,2\pi]}|f(x+h) -f(x)|,\;p=\infty\end{cases} \] is the \(L_p\) modulus of continuity of \(f\). We write \(H^\omega\) instead of \(H^\omega_\infty\) and \(H^\alpha_p(0< \alpha\leq 1)\) instead of \(H_p^{t^\alpha}\equiv\text{Lip} (\alpha,p)\), the Lipschitz class, for brevity. We always assume that \(\omega\) is a concave modulus of continuity and \(0<\alpha<1\). In recent years, much attention is drawn on the relationship of the class \(\Lambda BV\) and the Lipschitz class \(H_p^\omega\). It is well known that \(H_p'\subset H_1'=BV\subset \Lambda BV\). For \(f\in\Lambda BV\), the estimates of \(L_p\) modulus of continuity of \(f\) had been given in [\textit{S. W…
Mathematics Subject Classification ID: 26A45
Mathematics Subject Classification ID: 42B30
zbMATH DE Number: 5555227
zbMATH Keywords: \(\Lambda\)-bounded variation
zbMATH Keywords: Lipschitz classes
zbMATH Keywords: embedding
zbMATH Keywords: modulus of continuity of \(f\)
author: Heping Wang
MaRDI profile type: Publication
full work available at URL: j.jmaa.2009.01.033
OpenAlex ID: W2023595528
cites work: On the imbedding of the Waterman class in the class h p ω
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cites work: On convergence of Fourier series of functions of generalized bounded variation
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DOI: 10.1016/J.JMAA.2009.01.033
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description: scientific article; zbMATH DE number 5555227