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Sharp \(H_{p}\)-\(L_{p}\) type inequalities of weighted maximal operators of Vilenkin-Nörlund means and its applications
Journal of Inequalities and Applications volume 2016, Article number: 242 (2016)
Abstract
We prove and discuss some new \(H_{p}\)-\(L_{p}\) type inequalities of weighted maximal operators of Vilenkin-Nörlund means with monotone coefficients. It is also proved that these inequalities are the best possible in a special sense. We also apply these results to prove strong summability for such Vilenkin-Nörlund means. As applications, both some well-known and new results are pointed out.
1 Introduction
The definitions and notations used in this introduction can be found in our next section. In the one-dimensional case the weak \((1,1)\)-type inequality for maximal operator of Fejér means \(\sigma^{\ast }f:=\sup_{n\in \mathbb{N}}\vert \sigma_{n}f\vert \) can be found in Schipp [1] for Walsh series and in Pál, Simon [2] for bounded Vilenkin series. Fujji [3] and Simon [4] verified that \(\sigma ^{\ast}\) is bounded from \(H_{1}\) to \(L_{1}\). Weisz [5] generalized this result and proved boundedness of \(\sigma^{\ast}\) from the martingale space \(H_{p}\) to the Lebesgue space \(L_{p}\) for \(p>1/2\). Simon [6] gave a counterexample, which shows that boundedness does not hold for \(0< p<1/2\). In the case \(p=1/2\) a counterexample with respect to Walsh system was given by Goginava [7] and for the bounded Vilenkin system was proved by Tephnadze [8]. Weisz [9] proved that the maximal operator of the Fejér means \(\sigma^{\ast }\) is bounded from the Hardy space \(H_{1/2}\) to the space weak-\(L_{1/2}\).
Weisz [10] proved that the maximal operator of Cesàro means \(\sigma^{\alpha,\ast}f:=\sup_{n\in\mathbb{N}}\vert \sigma _{n}^{\alpha}f \vert \) is bounded from the martingale space \(H_{p}\) to the space \(L_{p}\) for \(p>1/ ( 1+\alpha ) \). Goginava [11] gave a counterexample, which shows that boundedness does not hold for \(0< p\leq1/ ( 1+\alpha ) \). Simon and Weisz [12] showed that the maximal operator \(\sigma^{\alpha,\ast}\) (\(0<\alpha<1\)) of the \(( C,\alpha ) \) means is bounded from the Hardy space \(H_{1/ ( 1+\alpha ) }\) to the space weak-\(L_{1/ ( 1+\alpha ) }\). In [13] and [14] it was also proved that the maximal operator
is bounded from the Hardy space \(H_{p}\) to the Lebesgue space \(L_{p}\), where \(0< p\leq1/ ( 1+\alpha ) \). Moreover, the rate of the weights \(\{ ( n+1 ) ^{1/p-\alpha-1}\log^{ ( 1+\alpha ) [ p+\alpha ( 1+\alpha ) ] } ( n+1 ) \} _{n=1}^{\infty}\) in nth Cesàro mean is given exactly.
It is well known that Vilenkin systems do not form bases in the space \(L_{1} ( G_{m} ) \). Moreover, there is a function in the Hardy space \(H_{1} ( G_{m} ) \), such that the partial sums of f are not bounded in \(L_{1}\)-norm. Simon [15] (for unbounded Vilenkin systems in the case when \(p=1\) see [16] and for \(0< p<1\) another proof was pointed out in [17]) proved that there exists an absolute constant \(c_{p}\), depending only on p, such that
for all \(f\in H_{p}\) and \(n\in\mathbb{N}_{+}\), where \([ p ] \) denotes the integer part of p. In [18] for Walsh system and in [19] with respect to bounded Vilenkin system it was proved that sequence \(\{ 1/k^{2-p} \} _{k=1}^{\infty}\) (\(0< p<1\)) in (1) cannot be improved.
In [20] it was proved that there exists an absolute constant \(c_{p}\), depending only on p, such that
An analogous result for \(( C,\alpha )\) (\(0<\alpha<1\)) means when \(p=1/ ( 1+\alpha ) \) was generalized in [13] and when \(0< p<1/ ( 1+\alpha ) \) it was proved in [14]. In particular, the following inequality:
holds.
Móricz and Siddiqi [21] investigated the approximation properties of some special Nörlund mean of the \(L_{p}\) function in norm. For more information on Nörlund means, see the paper of Blahota and Gát [22] and Nagy [23] (see also [24, 25], and [26]).
In [27] for \(p=1/ (1+\alpha) \) and in [28] for \(0< p<1/ (1+\alpha)\) there was proved that for every \(f\in H_{p} \) and for every Nörlund mean \(t_{n}f\), generated by the non-increasing sequence \(\{q_{n}:n\geq0\}\), satisfying the conditions
and
there exists an absolute constant \(c_{\alpha,p} \) such that
and
In [29] it was proved that in the endpoint case \(p=1/(1+\alpha) \) both (3) and (4) conditions are sharp in a special sense.
In this paper we investigate the case when \(0< p< 1/ ( 1+\alpha )\) and prove inequalities (5) and (6) for \(f\in H_{p} \) and Vilenkin-Nörlund means with non-increasing coefficients, but with weaker conditions than (3) and (4), which give possibility to prove analogous results for the wider class of Vilenkin-Nörlund means when \(0< p< 1/ ( 1+\alpha)\). As applications, both some well-known and new results are pointed out.
This paper is organized as follows: In order not to disturb our discussions later on some definitions and notations are presented in Section 2. The main results can be found in Section 3. For the proofs of the main results we need some lemmas, both well known, but also some new ones of independent interest. These results are presented in Section 4. The detailed proofs are given in Section 5. Some well-known and new consequences of our main results are presented in Section 6.
2 Definitions and notations
Denote by \(\mathbb{N} _{+}\) the set of the positive integers, \(\mathbb{N} :=\mathbb{N} _{+}\cup\{0\}\). Let \(m:=(m_{0},m_{1},\ldots)\) be a sequence of the positive integers not less than 2. Denote by
the additive group of integers modulo \(m_{k}\).
Define the group \(G_{m}\) as the complete direct product of the groups \(Z_{m_{i}}\) with the product of the discrete topologies of the \(Z_{m_{j}}\).
The direct product μ of the measures
is the Haar measure on \(G_{m}\) with \(\mu ( G_{m} ) =1\).
In this paper we discuss bounded Vilenkin groups, i.e. the case when \(\sup_{n}m_{n}<\infty\).
The elements of \(G_{m}\) are represented by the sequences
It is easy to give a base for the neighborhood of \(G_{m}\):
where \(x\in G_{m}\), \(n\in \mathbb{N} \).
Denote \(I_{n}:=I_{n} ( 0 ) \) for \(n\in \mathbb{N} _{+}\), and \(\overline{I_{n}}:=G_{m}\backslash I_{n}\).
If we define the so-called generalized number system based on m in the following way:
then every \(n\in \mathbb{N} \) can be uniquely expressed as \(n=\sum_{j=0}^{\infty}n_{j}M_{j}\), where \(n_{j}\in Z_{m_{j}}\) (\(j\in \mathbb{N} _{+}\)) and only a finite number of the \(n_{j}\) differ from zero.
Next, we introduce on \(G_{m}\) an orthonormal system which is called the Vilenkin system. At first, we define the complex-valued function \(r_{k} ( x ) :G_{m}\rightarrow \mathbb{C} \), the generalized Rademacher functions, by
Now, define the Vilenkin system \(\psi:=(\psi_{n}:n\in \mathbb{N} )\) on \(G_{m}\) as
Specifically, we call this system a Walsh-Paley system when \(m\equiv2\).
The norms (or quasi-norms) of the spaces \(L_{p}(G_{m})\) and weak- \(L_{p}(G_{m})\) (\(0< p<\infty\)) are, respectively, defined by
The Vilenkin system is orthonormal and complete in \(L_{2} ( G_{m} ) \) (see [30]).
Next, we introduce analogs of the usual definitions in Fourier-analysis. If \(f\in L_{1} ( G_{m} ) \) we can define the Fourier coefficients, the partial sums of the Fourier series, the Dirichlet kernels with respect to the Vilenkin system in the usual manner:
respectively.
Recall that
and
for \(n=\sum_{i=0}^{\infty}n_{i}M_{i}\).
The σ-algebra generated by the intervals \(\{ I_{n} ( x ) :x\in G_{m} \} \) will be denoted by \(\digamma_{n}\) (\(n\in \mathbb{N}\)). Denote by \(f= ( f^{ ( n ) },n\in \mathbb{N} ) \) a martingale with respect to \(\digamma_{n}\) (\(n\in \mathbb{N}\)) (for details see e.g. [31]).
The maximal function of a martingale f is defined by
For \(0< p<\infty\) the Hardy martingale spaces \(H_{p} ( G_{m} ) \) consist of all martingales for which
If \(f= ( f^{ ( n ) },n\in \mathbb{N} ) \) is a martingale, then the Vilenkin-Fourier coefficients must be defined in a slightly different manner:
Let \(\{q_{k}:k\geq0\}\) be a sequence of nonnegative numbers. The nth Nörlund means for a Fourier series of f is defined by
where
A representation
plays a central role in the sequel, where
is the so-called Nörlund kernel.
We say that the Nörlund mean \(t_{n}\) is of \(( N,\alpha ) \) type if
and for any \(\varepsilon>0\), we have
For our further investigation it is much more convenient to replace condition (12) by its equivalent one:
We always assume that \(q_{0}>0\) and \(\lim_{n\rightarrow\infty }Q_{n}=\infty\). In this case it is well known that the summability method generated by \(\{q_{k}:k\geq0\}\) is regular if and only if
Concerning this fact and related basic results, we refer to [32].
If \(q_{n}\equiv1\), then we get the nth Fejér mean and the Fejér kernel
respectively.
Let \(t,n\in\mathbb{N}\). It is well known that (see [33])
The \(( C,\alpha ) \)-means of the Vilenkin-Fourier series are defined by
where
For the martingale f we consider the following maximal operators:
We also consider the following weighted maximal operators:
A bounded measurable function a is a p-atom, if there exists an interval I, such that
3 The main results
Our sharp \(H_{p}\)-\(L_{p} \) inequality reads as follows.
Theorem 1
-
(a)
Let \(f\in H_{p}\), where \(0< p <1/ ( 1+\alpha )\) for some \(0<\alpha\leq1\), and \(\{q_{k}:k\in\mathbb{N}\} \) be a sequence of non-increasing numbers satisfying conditions (11) and (12). Then the maximal operator
$$ \overset{\sim}{t}_{p,\alpha}^{\ast}:=\sup_{n\in\mathbb{N}} \frac{\vert t_{n}f\vert }{( n+1 ) ^{1/p-1-\alpha}} $$is bounded from the martingale Hardy space \(H_{p}\) to the Lebesgue space \(L_{p}\), i.e. the following inequality holds:
$$ \Bigl\Vert \sup_{n\in\mathbb{N}}\vert t_{n}f \vert / \bigl( (n+1 )^{1/p-1-\alpha} \bigr) \Bigr\Vert _{p}\leq c_{\alpha,p} \Vert f\Vert _{H_{p}}. $$(15) -
(b)
Let \(0< p <1/ ( 1+\alpha )\) for some \(0<\alpha\leq1\), and \(\{ \Phi_{n}:n\in \mathbb{N}_{+} \} \) be any non-decreasing sequence, satisfying the condition
$$ \overline{\lim_{n\rightarrow\infty}}\frac{ ( n+1 ) ^{1/p-1-\alpha }}{\Phi_{n}}=\infty . $$(16)Then the inequality (15) is sharp in the sense that there exist a Nörlund mean with non-increasing sequence \(\{q_{k}:k\in\mathbb{N}\}\) satisfying the conditions (11) and (12) and a martingale \(f\in H_{P} \) such that
$$ \sup_{k\in\mathbb{N}}\frac{\Vert \frac {t_{M_{2n_{k}}+1}f_{k}}{\Phi _{M_{2n_{k}}+1}}\Vert _{\mathrm{weak}\text{-}L_{p}}}{\Vert f_{k}\Vert _{H_{p}}}=\infty. $$
Our new result concerning strong summability of Nörlund means with non-increasing sequences reads as follows.
Theorem 2
Let \(f\in H_{p}\), where \(0<\alpha<1\), \(0< p<1/ ( 1+\alpha) \), and let \(\{q_{n}:n\geq0\}\) be a sequence of non-increasing numbers, satisfying conditions (11) and (12). Then there exists an absolute constant \(c_{\alpha,p}\), depending only on α and p, such that the inequality
holds.
4 Lemmas
We need the following well-known lemma of Weisz [34].
Lemma 1
Suppose that an operator T is σ-linear and for some \(0< p\leq1\) and
for every p-atom a, where I denotes the support of the atom. If T is bounded from \(L_{\infty }\) to \(L_{\infty}\), then
The next results are due to Blahota, Persson, and Tephnadze [27].
Lemma 2
Let \(s_{n}M_{n}< r\leq ( s_{n}+1 ) M_{n}\), where \(1\leq s_{n}\leq m_{n}-1\). Then for every Nörlund mean, without any restriction on the generative sequence \(\{q_{k}:k\in\mathbb{N}\}\) we have the following equality:
We also need the following new lemmas of independent interest.
Lemma 3
Let \(0<\alpha\leq1 \) and \(\{q_{n}:n\geq0\}\) be a sequence of non-increasing numbers satisfying conditions (11) and (12). Then
where
Lemma 4
Let \(0<\alpha\leq1\) and \(\{q_{n}:n\geq0\}\) be a sequence of non-increasing numbers, satisfying conditions (11) and (12). If \(r\geq M_{N}\), then
where
and
where
5 Proofs
Proof of Lemma 3
Let \(0<\alpha\leq1\) and \(\{q_{k}:k\geq0\}\) satisfy the conditions (11) and (12). Since
we obtain
where \(\varphi_{n}\) satisfies condition (17).
By using an Abel transformation we get
and
Suppose that
for all \(j\in\mathbb{N} \), where \(\delta_{j}\) is any function, such that
Under condition (19) there exists an increasing sequence \(\{ \alpha_{k}:k\geq0 \} \), such that \(\alpha_{k+1}\geq2\alpha_{k}\) and
Hence,
By combining (18) and (22) we get
This is a contradiction with condition (13), that is,
It follows that
It is easy to see that
and
Let
and
By combining (24)-(26) and Lemma 2 we have
By repeating this process r times we get
By combining (8), (9), and (14) we find that
and
Moreover,
By applying (14) for \(\mathit{II}_{1}\) we get
By using (14) for \(\mathit{II}_{2}\) we have similarly
The proof is complete by combining the estimates above. □
Proof of Lemma 4
Let \(x\in I_{l+1} ( s_{k}e_{k}+s_{l}e_{l} )\), \(1\leq s_{k}\leq m_{k}-1\), \(1\leq s_{l}\leq m_{l}-1\). Then, by applying (14), we have
Suppose that \(k< n\leq l\). Moreover, by using (14) we get
Let \(n\leq k< l\). Then
If we now apply Lemma 3 and (14) we can conclude that
Let \(x\in I_{l+1} ( s_{k}e_{k}+s_{l}e_{l} ) \), for some \(0\leq k< l\leq N-1\). Since \(x-t\in I_{l+1} ( s_{k}e_{k}+s_{l}e_{l} ) \), for \(t\in I_{N}\) and \(r\geq M_{N}\) from (27) we obtain
Let \(x\in I_{N} ( s_{k}e_{k} ) \), \(k=0,\dots,N-1\). Then, by applying (8) and (9) we have
By combining (28) and (29) we complete the proof of Lemma 4. □
Proof of Theorem 1
According to Lemma 1 the proof of the first part of Theorem 1 will be complete, if we show that
for every \(1/ ( 1+\alpha-\varepsilon ) \)-atom a. We may assume that a is an arbitrary p-atom with support I, \(\mu ( I ) =M_{N}^{-1}\) and \(I=I_{N}\). It is easy to see that \(t_{n} ( a ) =0\), when \(n\leq M_{N}\). Therefore, we can suppose that \(n>M_{N}\).
By using Lemma 3 we easily see that \(\overset{\sim}{t}^{\ast,p}\) is bounded from \(L_{\infty}\) to \(L_{\infty} \). Let \(x\in I_{N}\). Since \(\Vert a\Vert _{\infty}\leq M_{N}^{1/p}\) we obtain
Let \(x\in I_{l+1} ( s_{k}e_{k}+s_{l}e_{l} ) \), \(0\leq k< l< N\). From Lemma 4 we get
Let \(x\in I_{N} ( s_{k}e_{k} ) \), \(0\leq k< N\). According to Lemma 4 we have
By combining (7) and (30)-(31) we obtain
The proof of part (a) is complete.
Under condition (16) there exist positive integers \(n_{k}\) such that
Let \(t_{n} \) be Nörlund mean with non-increasing sequence \(\{q_{k}:k\in \mathbb{N}\}\) satisfying (11) and condition (12), but in the restricted form
Set
Then
and
Moreover,
where \(\lambda=\sup_{n}m_{n}\).
By using (33) we get
Hence,
By applying (35) we have
The proof is complete. □
Proof of Theorem 2
According to Lemma 1 the proof of Theorem 2 will be complete, if we show that
for every p-atom a. Analogously to the first part of Theorem 1 we can assume that \(n>M_{N}\) and a be an arbitrary p-atom, with support I, \(\mu (I ) =M_{N}\), and \(I=I_{N}\).
Let \(x\in I_{N}\). Since \(\Vert a\Vert _{\infty}\leq cM_{N}^{1/p}\) if we apply Lemma 3 we obtain
Hence
By combining (7), (30), and (31) analogously to first part of Theorem 1 we can write
which completes the proof. □
6 Applications and final remarks
Remark 1
We note that under the conditions (3) and (4) we see that the conditions (11) and (12) are also fulfilled.
Proof
Let \(0<\alpha\leq1\). We can write
First suppose that \(\alpha=1\). Then
and
Moreover, condition (12) automatically holds,
for any \(\varepsilon>0\).
Since the case \(q_{0}n/Q_{n}=O ( 1 ) \), as \(n\rightarrow\infty\), has already been considered, we can exclude it. Hence, we may assume that \(\{q_{k}:k\geq0\}\) satisfies conditions (3) and (4) and, in addition, satisfies the following condition:
It follows that
and
□
From Remark 1 we immediately see that the following is true.
Corollary 1
Conditions (3) and (4) provide a wider class of Nörlund means with non-increasing coefficients than conditions (11) and (12).
From the proof of Remark 1 for \(\alpha=1 \) we immediately have the following.
Remark 2
Let \(\alpha=1\) and \(\{q_{k}:k\in\mathbb{N}\}\) be a sequence of non-increasing numbers. Then condition (12) automatically holds,
for any \(\varepsilon>0\).
By applying Remark 2 and Theorem 1 we get the following.
Theorem 3
-
(a)
Let \(f\in H_{p}\), where \(0 < p<1/2\) and \(\{q_{k}:k\in\mathbb{N}\}\) be a sequence of non-increasing numbers satisfying condition (11) for \(\alpha=1 \). Then the maximal operator
$$ \overset{\sim}{t}_{p,1}^{\ast}:=\sup_{n\in\mathbb{N}} \frac{\vert t_{n}f\vert }{( n+1 ) ^{1/p-2}} $$is bounded from the martingale Hardy space \(H_{p}\) to the Lebesgue space \(L_{p}\), i.e. the inequality
$$ \Bigl\Vert \sup_{n\in\mathbb{N}}\vert t_{n}f \vert / \bigl( (n+1 )^{1/p-2} \bigr) \Bigr\Vert _{p}\leq c_{\alpha,p} \Vert f\Vert _{H_{p}} $$(36)holds.
-
(b)
Let \(\{ \Phi_{n}:n\in\mathbb{N}_{+} \} \) be any non-decreasing sequence, satisfying the condition
$$ \overline{\lim_{n\rightarrow\infty}}\frac{ ( n+1 ) ^{1/p-2 }}{\Phi_{n}}=\infty. $$Then the inequality (36) is sharp in the sense that there exists a Nörlund mean with non-increasing sequence \(\{q_{k}:k\in\mathbb{N}\}\) satisfying the condition (11) such that
$$ \sup_{k\in\mathbb{N}}\frac{\Vert \frac {t_{M_{2n_{k}}+1}f_{k}}{\Phi _{M_{2n_{k}}+1}}\Vert _{\mathrm{weak}\text{-}L_{p}}}{\Vert f_{k}\Vert _{H_{p}}}=\infty. $$
By applying Remark 2 and Theorem 2 we get the following.
Theorem 4
Let \(f\in H_{p}\), where \(0< p<1/2\) and \(\{q_{n}:n\geq 0\}\) be a sequence of non-increasing numbers, satisfying condition (11). Then there exists an absolute constant \(c_{\alpha,p}\), depending only on α and p, such that
From Theorem 3 we get the following result by Tephnadze [35].
Corollary 2
-
(a)
Let \(f\in H_{p}\), where \(0< p<1/2\). Then the maximal operator
$$ \overset{\sim}{\sigma}_{p}^{\ast}:=\sup_{n\in\mathbb{N}} \frac{\vert \sigma_{n}f\vert }{ ( n+1 ) ^{1/p-2}} $$is bounded from the martingale Hardy space \(H_{p}\) to the Lebesgue space \(L_{p}\).
-
(b)
Let \(\{ \Phi_{n}:n\in\mathbb{N}_{+} \} \) be any non-decreasing sequence, satisfying the condition
$$ \overline{\lim_{n\rightarrow\infty}}\frac{ ( n+1 ) ^{1/p-2}}{\Phi _{n}}=\infty. $$Then
$$ \sup_{k\in\mathbb{N}}\frac{\Vert \frac{\sigma _{M_{2n_{k}}+1}f_{k}}{\Phi_{M_{2n_{k}}+1}}\Vert _{\mathrm{weak}\text{-}L_{p}}}{\Vert f_{k}\Vert _{H_{p}}}=\infty. $$
Moreover, Theorem 4 implies the following result by Tephnadze [18].
Corollary 3
Let \(f\in H_{p}\), where \(0< p<1/2\). Then there exists an absolute constant \(c_{p}\), depending only on p, such that
Next we note that Theorem 1 and Remark 1 imply the following results of Blahota, Tephnadze [13] for \(0<\alpha<1\).
Corollary 4
-
(a)
Let \(f\in H_{p}\), where \(0< p<1/(1+\alpha)\) for some \(0<\alpha<1\). Then the maximal operator
$$ \overset{\sim}{\sigma}_{p}^{\ast,{\alpha}}:=\sup_{n\in\mathbb{N}} \frac{\vert \sigma^{\alpha}_{n}f\vert }{ (n+1 )^{1/p-1-\alpha}} $$is bounded from the martingale Hardy space \(H_{p}\) to the Lebesgue space \(L_{p}\).
-
(b)
Let \(\{ \Phi_{n}:n\in\mathbb{N}_{+} \} \) be any non-decreasing sequence, satisfying the condition
$$ \overline{\lim_{n\rightarrow\infty}}\frac{ ( n+1 ) ^{1/p-1-\alpha }}{\Phi_{n}}=\infty. $$Then
$$ \sup_{k\in\mathbb{N}}\frac{\Vert \frac{\sigma^{\alpha}_{M_{2n_{k}}+1}f_{k}}{\Phi_{M_{2n_{k}}+1}}\Vert _{\mathrm{weak}\text{-}L_{p}}}{\Vert f_{k}\Vert _{H_{p}}}=\infty. $$
Similarly, Theorem 2 and Remark 1 immediately imply the following result of Blahota, Tephnadze [13] for \(0<\alpha<1\).
Corollary 5
Let \(f\in H_{p}\), where \(0< p<1/ ( 1+\alpha )\), for some \(0<\alpha<1\). Then there exists an absolute constant \(c_{\alpha ,p}\), depending only on α and p, such that
Let \(0<\alpha\leq1\), \(\beta>0\), and \(\theta_{n}^{\alpha,\beta}\) denote the Nörlund mean, where
that is,
Remark 3
\(0<\alpha\leq1\) and \(\beta=0\). Then \(\theta _{n}^{\alpha,\beta} \) satisfy conditions (3) and (4) and also conditions (11) and (12).
Remark 4
\(0<\alpha\leq1\) and \(\beta>0\). Then \(\theta _{n}^{\alpha,\beta} \) satisfies conditions (11) and (12), but does not satisfy (3) and (4).
Finally, we also point out some new consequences of our results.
First we note that Theorem 1 and Remark 1 immediately imply the following new result.
Corollary 6
-
(a)
Let \(f\in H_{p}\), where \(0< p<1/(1+\alpha)\) for some \(0<\alpha\leq1\). Then for every \(\beta>0 \) the maximal operator
$$ \overset{\sim}{\theta}_{p}^{\ast,{\alpha,\beta}}:=\sup_{n\in \mathbb{N}} \frac{\vert \theta^{\alpha,\beta}_{n}f\vert }{ (n+1 )^{1/p-1-\alpha}} $$is bounded from the martingale Hardy space \(H_{p}\) to the Lebesgue space \(L_{p}\).
-
(b)
Let \(\{ \Phi_{n}:n\in\mathbb{N}_{+} \} \) be any non-decreasing sequence, satisfying the condition
$$ \overline{\lim_{n\rightarrow\infty}}\frac{ ( n+1 ) ^{1/p-1-\alpha }}{\Phi_{n}}=\infty. $$Then
$$ \sup_{k\in\mathbb{N}}\frac{\Vert \frac{\theta^{\alpha,\beta}_{M_{2n_{k}}+1}f_{k}}{\Phi_{M_{2n_{k}}+1}}\Vert _{\mathrm{weak}\text{-}L_{p}}}{\Vert f_{k}\Vert _{H_{p}}}=\infty. $$
In a similar way we see that Theorem 2 and Remark 1 immediately generates the following new result.
Corollary 7
Let \(f\in H_{p}\), where \(0< p<1/ ( 1+\alpha )\), for some \(0<\alpha\leq1\). Then for every \(\beta>0 \) there exists an absolute constant \(c_{\alpha,\beta,p}\), depending only on α, β, and p, such that
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Acknowledgements
The research was supported by Shota Rustaveli National Science Foundation grants no. DO/24/5-100/14 and YS15-2.1.1-47, by a Swedish Institute scholarship no. 10374-2015 and by target scientific research programs grant for the students of faculty of Exact and Natural Sciences. The authors would like to thank the referees for helpful suggestions.
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Baramidze, L., Persson, LE., Tephnadze, G. et al. Sharp \(H_{p}\)-\(L_{p}\) type inequalities of weighted maximal operators of Vilenkin-Nörlund means and its applications. J Inequal Appl 2016, 242 (2016). https://doi.org/10.1186/s13660-016-1182-1
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DOI: https://doi.org/10.1186/s13660-016-1182-1