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On weighted integrability of functions defined by trigonometric series with p-bounded variation coefficients
Journal of Inequalities and Applications volume 2019, Article number: 275 (2019)
Abstract
In this paper we introduce new classes of p-bounded variation sequences and give a sufficient and necessary condition for weighted integrability of trigonometric series with coefficients belonging to these classes. This is a generalization of the results obtained by the first author [J. Inequal. Appl. 2010:1–19, 2010] and Dyachenko and Tikhonov [Stud. Math. 193(3):285–306, 2009].
1 Introduction
Let \(L^{s}\), \(1 \leqslant s< \infty \), be the space of all s-power integrable functions f of period 2π with the norm
Write
for those x, where the above series converge.
Denote by ϕ and \(\lambda _{n}\) either f or g and either \(a_{n}\) and \(b_{n}\), respectively.
Let \(\triangle _{r}a_{n}=a_{n}-a_{n+r}\) for a sequence of complex numbers \((a_{n})\) and \(r\in \mathbb{N} \).
Theorem 1
Let a nonnegative sequence \(( \lambda _{n} ) \in \Re \), \(1< s<\infty \) and \(1-s<\alpha <1\). Then
This theorem was proved for \(\Re =DS\), where DS denotes all decreasing sequences, in [1, 5, 14], and [2]. Later, Theorem 1 was showed in [7] for
and in [12] for
where \({}_{1}\beta _{n}= \vert a_{n} \vert \); C here and throughout the paper denotes a positive constant.
The proof in the case of class
where \({}_{2}\beta _{n}=\sum_{k= [ n/c ] }^{ [ cn ] }\frac{ \vert a_{k} \vert }{k}\) for some \(c>1\), is included in [13].
In [3] Dyachenko and Tikhonov extended this theorem to the class
where \({}_{3}\beta _{n}(\theta )=n^{\theta -1}\sum_{k= [ n/c ] }^{\infty }\frac{ \vert a_{k} \vert }{k ^{\theta }}<\infty \) for some \(c>1\) and \(\theta \in (0,1]\).
From the articles of Dyachenko and Tikhonov [3] and Leindler [7], it is well known that
for \(0<\theta _{1}\leqslant \theta _{2}\leqslant 1\).
Further, Szal defined a new class of sequences in the following way (see [9]):
Definition 1
Let \(\beta := ( \beta _{n} ) \) be a nonnegative sequence and r a natural number. The sequence of complex numbers \(a:= ( a _{n} ) \in \overline{GM} ( \beta ,r ) \) if the relation
holds for all \(n\in \mathbb{N}\).
Moreover, from [9] we know that
where \(r_{1}< r_{2}\), \(\theta \in (0,1]\) and \(r_{1}\mid r_{2}\).
Let \(r\in \mathbb{N} \) and \(\alpha \in \mathbb{R} \). We define on the interval \([ -\pi ,\pi ] \) an even function \(\omega _{\alpha ,r}\), which is given on the interval \([ 0,\pi ] \) by the formula
where \(U_{1}= \{ 0,1,\dots ,[r/2] \} \) if r is an odd number and \(U_{1}= \{ 0,1,\dots , [ r/2 ] -1 \} \) if r is an even number, \(U_{2}= \{ 0,1,\dots , [ r/2 ] -1 \} \) for \(r\geqslant 2\), and \(U_{3}= \{ 0,1, \dots , [ r/2 ] \} \) for \(r\geqslant 1\).
Theorem 1 was generalized for the class \(\overline{GM}({}_{3}\beta ( \theta ),r) \), where \(r\in \mathbb{N} \) and \(\theta \in (0,1]\), in [9]. We can formulate this result in the following way.
Theorem 2
([9, Theorem 5])
Let a nonnegative sequence \(( \lambda _{n} ) \in \overline{GM} ( {}_{3}\beta (\theta ),r ) \), where \(r\in \mathbb{N} \), \(\theta \in (0,1]\) and \(1\leqslant s<\infty \). If
then \(\omega _{\alpha ,r} \vert \phi \vert ^{s}\in L^{1}\) if and only if
Now, we define new classes of sequences.
Definition 2
Let \(\beta := ( \beta _{n} ) \) be a nonnegative sequence, p a positive real number, \(r\in \mathbb{N} \). One says that a sequence \(a=(a_{n})\) of complex numbers belongs to \(GM(p,\beta ,r)\) if the relation
holds for all \(n\in \mathbb{N} \).
Moreover, we say that a sequence \((a_{n})\in \overline{GM}(p,\beta ,r)\) if the relation
holds for all \(n\in \mathbb{N} \).
The class \(GM(p,\beta ,1)\) was defined by Tikhonov and Liflyand in [8].
In this paper we present some properties of the classes \(\overline{GM} ( p,{}_{3}\beta (\theta ),r ) \) and \(GM ( p, {}_{3}\beta (\theta ),r ) \). Moreover, we will generalize Theorem 2 for the class \(GM ( p,{}_{3}\beta (\theta ),r ) \) with \(0<\theta < \frac{1}{s}\) and \(r\in \mathbb{N} \).
We will write \(I_{1}\ll I_{2}\) if there exists a positive constant C such that \(I_{1}\leqslant CI_{2}\).
2 Main results
We formulate our results as follows:
Theorem 3
Let \(r\in \mathbb{N} \), \(\theta \in (0,1)\), and p be a positive real number. Then
Theorem 4
Let \(r\in \mathbb{N} \), \(\theta \in (0,1)\), and \(p_{1}\), \(p_{2}\) be two positive real numbers such that \(0< p_{1}<p_{2}\). Then
Theorem 5
Let \(r_{1},r_{2}\in \mathbb{N} \), \(r_{1}< r_{2}\), \(\theta \in (0,1]\) and \(p\geqslant 1\). If \(r_{1}|r_{2}\), then
Theorem 6
Let \((b_{n})\in GM ( p,{}_{3}\beta (\theta ),r ) \), where \(r\in \mathbb{N} \), \(p\geqslant 1\), \(0<\theta <\frac{1}{p}\) and \(1\leqslant s<\infty \). If
and
then \(\omega _{\alpha .r}|\phi |^{s}\in L^{1}\).
Theorem 7
Let a nonnegative sequence \((b_{n})\) belong to \(GM ( p,{}_{3} \beta (\theta ),r ) \), where \(r\in \mathbb{N} \), \(p\geqslant 1\), \(0<\theta <\frac{1}{p}\) and \(1\leqslant s<\infty \). If
and \(\omega _{\alpha .r}|\phi |^{s}\in L^{1}\) then
Remark 1
If we take \(p=1\), then the result of Szal [9] (Theorem 2) follows from our Theorem 6 and 7. Moreover, by the embedding relations (1) and (2), we can also derive from Theorem 6 and 7 the result of Dyachenko and Tikhonov [3] and all the results mentioned before.
3 Auxiliary results
For \(n\in \mathbb{N} \) and \(k=0,1,2,\ldots \) , denote by
the Dirichlet-type kernels.
Lemma 1
([10, Lemma 3.1] and [11, Lemma 17])
Let \(r\in \mathbb{N} \), \(l\in \mathbb{Z} \), and \((a_{n})\subset \mathbb{C} \). If \(x\neq \frac{2l\pi }{r}\), then for all \(m\geqslant n\)
Lemma 2
([6, Corollary 1])
Let \(p\geqslant 1,\gamma _{n}>0\), and \(a_{n}\geqslant 0\) for \(n\in \mathbb{N} \). Then
Lemma 3
([4, Theorem 19])
If \(a_{n}\geqslant 0\) for \(n\in \mathbb{N} \) and \(0< p_{1}\leqslant p_{2}<\infty \), then
Lemma 4
([4])
Let \(a_{k}\geqslant 0\) for \(k\in \mathbb{N} \) and \(p\geqslant 1\). Then
Lemma 5
Let \((a_{k})\subset \mathbb{C} \), \(p\geqslant 1\), \(r,n\in \mathbb{N} \) and \(d\in \mathbb{N} _{0}=\mathbb{N} \) \(\cup \{ 0 \} \). Then
Proof
From Lemma 4 we have
Hence
and this ends our proof. □
Lemma 6
Let \((a_{k})\in GM ( p,{}_{3}\beta (\theta ),r ) \), \(p\geqslant 1\), \(r\in \mathbb{N} \), \(d\in \mathbb{N} _{0}\), and \(0<\theta <\frac{1}{p}\). Then
Proof
We have
Using Hölder inequality with \(p>1\), we get
When \(p=1\), we have
If \(\theta -\frac{1}{p} <0\), then
and our proof is complete. □
4 Proofs
4.1 Proof of Theorem 3
Let \((a_{n})\in GM ( p,{}_{3}\beta (\theta ),r ) \), where \(p>0\), \(r\in \mathbb{N,} \) and \(\theta \in (0,1)\). Then
If \(0<\theta <1\) then \((\theta -1)p<0\), and we have
So \((a_{n})\in \overline{GM} ( p,{}_{3}\beta (\theta ),r ) \).
Now we assume \((a_{n})\in \overline{GM} ( p,{}_{3}\beta (1),r ) \), \(p>0\), \(r\in \mathbb{N} \). We have
This means \((a_{n})\in GM ( p,{}_{3}\beta (1),r )\). □
4.2 Proof of Theorem 4
Let \(r\in \mathbb{N} \), \(\theta \in (0,1]\), \(0< p_{1}\leqslant p_{2}\), and \((a_{n})\in GM ( p_{1},{}_{3}\beta (\theta ),r ) \). We will show that \(GM ( p_{1},{}_{3}\beta (\theta ),r ) \subseteq GM ( p_{2},{}_{3}\beta (\theta ),r ) \). Using Lemma 3, we have
This means that \((a_{n})\in GM ( p_{2},{}_{3}\beta (\theta ),r ) \).
Now we will show that \(GM ( p_{1},{}_{3}\beta (\theta ),r ) \neq GM ( p_{2},{}_{3}\beta (\theta ),r ) \) for \(0< p_{1}< p _{2}\). Let
We prove that \((a_{n}) \in GM (p_{2},{}_{3}\beta (\theta ),r )\). Suppose
Then
Moreover,
This means \((a_{n})\in GM ( p_{2},{}_{3}\beta (\theta ),r ) \). We will show that \((a_{n})\notin GM ( p_{1},{}_{3}\beta (\theta ),r ) \). We have
Let
On the other hand, we get
Therefore the inequality
cannot be satisfied because \(n^{\frac{1}{p_{1}}-\frac{1}{p_{2}}} \rightarrow \infty \) as \(n\rightarrow \infty \). □
4.3 Proof of Theorem 5
Let \(r_{1},r_{2}\in \mathbb{N}\), \(r_{1}\leqslant r_{2}\), \(r_{1}|r_{2}\), \(p\geqslant 1\) and \((a_{n})\in GM ( p,{}_{3}\beta (\theta ),r_{1} ) \).
If \(r_{1}|r_{2}\), then \(r_{2}=\alpha r_{1}\), where \(\alpha \in \mathbb{N}\). Using Hölder inequality with \(p>1\), we have
If \(p=1\) then
Hence \((a_{n})\in GM ( p,{}_{3}\beta (\theta ),r_{2} ) \).
Now, we will show that \(GM ( p,{}_{3}\beta (\theta ),r_{1} ) \varsubsetneq GM ( p,{}_{3}\beta (\theta ),r_{2} ) \), when \(r_{1}< r_{2}\). Let \(a_{n}=\frac{2+\alpha _{n}}{n^{2}}\), where \(\alpha _{n}= \bigl\{ \scriptsize{ \begin{array}{l@{\quad}l} -1,&\text{when }r_{1}|n, \\ 1,&\text{when }r_{1}\nmid n. \end{array} }\)
We will prove that \((a_{n})\in GM ( p,{}_{3}\beta (\theta ),r_{2} ) \) and \((a_{n})\notin GM ( p,{}_{3} \beta (\theta ),r_{1} ) \). Let
Then using Lemma 3 for \(p\geqslant 1\), we have
Moreover,
It means that \((a_{n})\in GM ( p,{}_{3}\beta (\theta ),r_{2} ) \). Furthermore,
If \(n\geqslant 5r_{1}\), then \(2n^{2}-2nr_{1}-r_{1}^{2}\geqslant (n+r _{1})^{2} \). Whence for \(n\geqslant 5r_{1}\),
On the other hand,
Therefore, the inequality
cannot be satisfied because \(n^{\frac{1}{p}}\rightarrow \infty \) as \(n\rightarrow \infty \). □
4.4 Proof of Theorem 6
We prove the theorem for the case when \(\phi (x)=g(x)\). We have
For an odd r,
(for \(r=1\) the last sum should be omitted), and for an even r,
Now, we estimate the following integral:
By Lemma 2, for \(\alpha <1\), we have
Using Lemma 1 when \(m\rightarrow \infty \) and the inequality
we get
Further by Hölder inequality with \(p>1\), we get
Applying Lemma 5, we have
From Lemma 6, we get
If \(\theta -\frac{1}{p} <0\), then
Now, we use Lemma 2 and get
For \(1+\frac{s}{p}-\theta s-s<\alpha <1+\frac{s}{p}\), we have
Now, we estimate the following integral:
By Lemma 2, for \(\alpha <1\), we have
Using Lemma 1 with \(m\rightarrow \infty \) and the inequality
we have
and similarly as in the case \(I_{2}\) we obtain
Finally, combining (3)–(6), we obtain that
The case when \(\phi (x)=\sum_{k=1}^{\infty }b_{k}\cos kx\) can be proved similarly. □
4.5 Proof of Theorem 7
We prove the theorem for the case where \(\phi (x)=\sum_{k=1} ^{\infty }b_{k}\sin kx\). We follow the method adopted by Tikhonov [9]. Note that if \(1-\theta s<\alpha <1+s\), then \(\phi \in L^{1}\). Namely, if \(s>1\) then using Hölder inequality, we have
We will show that \(\int _{0}^{\pi } ( \omega _{\alpha ,r}(x) ) ^{-\frac{1}{s-1}}\,dx<\infty \). We can write
when r is an even number, and
when r is an odd number.
Using integration by substitution, we get
when r is an even number, and
when r is an odd number.
If \(s=1\) then \(\alpha >0\) and
Further, integrating ϕ, we have
and consequently,
Since \((b_{n})\in GM ( p,{}_{3}\beta (\theta ),r ) \) and using Lemma 4, we get for \(\theta -\frac{1}{p}<0 \) that
Using (7) yields
Elementary calculations give
Using Lemma 2, for \(1-\theta s<\alpha <1+s\), we have
and
Therefore, for \(1-\theta s<\alpha <1+s\), we get
Denoting by \(d_{v}:=\int _{\frac{\pi }{v+1}}^{\frac{\pi }{v}} \vert \phi ( x ) \vert \,dx\), we get
By Lemma 2, for \(\alpha >1-s\), we obtain
Applying Hölder inequality when \(s>1\), we have
Finally, using the latter estimate, we get
The case when \(\phi (x)=\sum_{k=1}^{\infty }b_{k}\cos kx\) can by proved similarly. □
5 Conclusions
We have introduced two new classes of p-bounded variation sequences, \(\overline{GM}(p,\beta ,r)\) and \(GM(p,\beta ,r)\), where \(\beta := ( \beta _{n} ) \) is a nonnegative sequence, p a positive real number, \(r\in \mathbb{N} \), \(\theta \in (0,1]\). Moreover, we have studied properties of such classes and obtained a sufficient and necessary condition for weighted integrability of functions defined by trigonometric series with coefficients belonging to these classes. In particular, from our theorems we derive all related earlier results.
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Szal, B., Kubiak, M. On weighted integrability of functions defined by trigonometric series with p-bounded variation coefficients. J Inequal Appl 2019, 275 (2019). https://doi.org/10.1186/s13660-019-2225-1
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DOI: https://doi.org/10.1186/s13660-019-2225-1