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.
Hinweise
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All the authors contributed equally and significantly in writing this paper. All the authors read and approved the final manuscript.
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.
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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
$$ \frac{1}{\log^{ [ p ] }n}\overset{n}{\underset{k=1}{\sum}}\frac{ \Vert S_{k}f\Vert _{p}^{p}}{k^{2-p}} \leq c_{p}\Vert f\Vert _{H_{p}}^{p}\quad ( 0< p \leq1 ) $$
(1)
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:
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]).
Anzeige
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
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
$$ Z_{m_{k}}:=\{0,1,\ldots,m_{k}-1\} $$
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 \(Z_{m_{j}}\).
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
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:
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
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
A bounded measurable function a is a p-atom, if there exists an interval I, such that
$$ \int_{I}a\,d\mu=0, \quad\quad \Vert a\Vert _{\infty}\leq\mu( I ) ^{-1/p},\quad \quad \operatorname {supp}( a ) \subset I. $$
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
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
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 onαandp, such that the inequality
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:
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
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
$$\begin{aligned} \bigl\vert t_{n}a ( x ) \bigr\vert \leq& \int_{I_{N}} \bigl\vert a ( t ) \bigr\vert \bigl\vert F_{n} ( x-t ) \bigr\vert \,d\mu( t ) \\ \leq&\Vert a\Vert _{\infty} \int_{I_{N}} \bigl\vert F_{n} ( x-t ) \bigr\vert \,d\mu( t ) \leq M_{N}^{1/p} \int_{I_{N}} \bigl\vert F_{n} ( x-t ) \bigr\vert \,d\mu( t ) . \end{aligned}$$
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 \(t_{n} \) be Nörlund mean with non-increasing sequence \(\{q_{k}:k\in \mathbb{N}\}\) satisfying (11) and condition (12), but in the restricted form
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
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:
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
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
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 onαandp, such that
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 onαandp, such that
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 onα, β, andp, such that
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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Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All the authors contributed equally and significantly in writing this paper. All the authors read and approved the final manuscript.