Embedding of Lipschitz classes into classes of functions of \(\Lambda \)-bounded variation
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. Wang}, Sci. Sin., Ser. A. 25, 149--160 (1982; Zbl 0482.26005)] for \(p=1\) and in [\textit{Yu. E. Kuprikov}, Mosc. Univ. Math. Bull. 52, No.~5, 46--49 (1997); translation from Vestn. Mosk. Univ., Ser. I 1997, No.~5, 59--62 (1997; Zbl 0912.26002)] for \(1<p<\infty\). Furthermore, in [Ukr. Mat. Zh. 57, No.~11, 1557--1562 (2005) and Ukr. Math. J. 57, No.~11, 1818--1824 (2005; Zbl 1096.26004)] \textit{U. Goginava} gave the necessary and sufficient conditon for the inclusion of the class \(\Lambda BV\) in the class \(H_p^\omega\): \(\Lambda BV\subset H_p^\omega\) if and only if \(\max_{1\leq m\leq n}\frac{m^{1/p}}{\sum^m_{k=1}1/ \lambda_k}=O(n^{1/p}\omega(\frac 1n))\), as \(n\to\infty\). Obviously, for \(1\leq p< \infty\), \(0<\alpha,\beta\leq 1\), we get that \(\{n^\beta\}BV\subset H_p^\alpha\) if and only if \(\max(1,n^{1/p-1+\beta})=O(n^{1/p-\alpha})\) if and only if \(\alpha\leq\min\{\tfrac 1p,1-\beta\}\). On the other hand, \textit{M. V. Medvedeva} showed in [(*) Math. Notes 64, No.~5, 616--621 (1998); translation from Mat. Zametki 64, No.~5, 713--719 (1998; Zbl 0959.45022)] that a sufficient and necessary condition for the embedding \(H^\omega\subset \Lambda BV\) is that \(\sum^\infty_{k=1}\frac{\omega(t_k)} {\lambda_k}<\infty\) for any sequence \(\{t_k\}^\infty_{k=1}\), \(t_k\geq 0\), \(\sum^\infty_{k=1}t_k\leq 2\i\). In [Sb. Math. 193, No.~7, 1049--1070 (2002); translation from Mat. Sb. 193, No.~7, 109--130 (2002; Zbl 1042.42013)], \textit{M. V. Medvedeva} also gave criteria for the embedding \(H^\omega\subset\Lambda BV\) under some restrictions imposed on \(\omega(t)\). In particular, for \(0<\alpha<1\), \(0<\beta\leq 1\), \[ H^\alpha\subset \Lambda BV\text{ if and only if }\sum^\infty_{k=1}\left(\frac{1}{\lambda_k} \right)^{\frac{1}{1-\alpha}}<\infty;\tag{2} \] \[ H^\alpha\subset\{n^\beta\}BV \text{ if and only if }\alpha+\beta>1.\tag{3} \] Conditions for the embedding \(H^\infty_p\subset\{n^\beta\}BV\), \(1\leq p<\infty\) are discussed. It follows from the embedding relation \(H^\alpha_p\subset H^{\alpha-\frac 1p}\), \(\alpha>\frac 1p\) (see [\textit{V. N. Temlyakov}, Approximation of periodic functions. Computational Mathematics and Analysis Series. Commack, NY: Nova Science Publishers, Inc. (1993; Zbl 0899.41001)]) and (*) that the embedding \(H^\alpha_p\subset \{n^\beta\}BV\) is true if \(\alpha+\beta >1+\frac 1p\). However, this condition is too strong and not necessary. The author's sufficient or necessary conditions for the embedding \(H^\omega_p\subset\Lambda BV\) (or \(H^\alpha_p\subset\Lambda BV)\) are given. In particular sufficient and necessary conditions for the embedding \(H^\alpha_p\subset\{n^\beta\}BV\), \(1<p<\infty\), is \(\alpha>\max \{\frac 1p,1-\beta\}\) are obtained. The author's method is completely different from the one in [(*)] and is based on a decomposition technique of functions. The main results are the following. Theorem 1. Let \(1\leq p<\infty\). If \(H^\omega_p\subset\Lambda BV\), then \(\omega(t)=O (t^{1/p})\) as \(t\to 0+\). If \(H^\alpha_p\subset\Lambda BV\), then \(\alpha>\frac 1p\). Remark 1. For \(p=1\), we have \(H^\omega \subset \Lambda BV\) if and only if \(\omega(t)=O(t)\) as \(t\to 0^+\). In this case, \(H^\omega_1=H^1_1=BV\). Theorem 2. Let \(1<p< \infty\). \(p^i=\frac{p}{p-1}\) and \(f\in H^\omega_p\). If there exists a concave modulus of continuity \(\omega\), satisfying the following two conditions (i) \(\sum^\infty_{k=1}\frac{ \omega_1(t_k)} {\lambda_k^{p'}}<+\infty\) for any sequence \(\{t_k\}^\infty_{k=1}\), \(t_k\geq 0\), \(\sum^\infty_{k=1}t_k\leq 2\pi\), (ii) (4)\(\quad \sum^\infty_{s=0} \frac{\omega(2^{-s})\cdot 2^{s/p}}{(\omega, (2^{-s}))^{1/p'}}<+\infty\), then \(f\in\Lambda BV\). Theorem 3. Let \(1<p<\infty\) and \(\alpha\in(\frac 1p,1)\). If there exists an \(\alpha_0 >0\) such that \(\alpha>\alpha_0\) and \[ \sum^\infty_{k=1} \left(\frac {1}{\lambda_k}\right)^{\frac{1}{1-\alpha_0}}<+\infty,\tag{5} \] then \(H^\alpha_p\subset\Lambda BV\). Theorem 4. Let \(1<p<\infty\) and \(0<\alpha, \beta\leq 1\). Then \[ H^\alpha_p\subset\{n^\beta\}BV\text{ if and only if }\alpha >\max\{\frac 1p,1-\beta\}.\tag{6} \] Remark 2. For \(1<p<\infty\) and \(0<\alpha\leq 1\), \(H^\alpha_p\subset\{n\}BV\) if and only if \(\alpha>\frac 1p\). Remark 3. For \(0<\beta<1\), \(\varepsilon >0\), we conclude from (1) and (6) that \(H^{1-\beta + \varepsilon}_{\frac{1}{1-\beta}}\subset\{n^\beta\}BV\subset H^{1-\beta}_{\frac {1}{1-\beta}}\). References are listed.
scientific article; zbMATH DE number 5555227