$(H_{p}-L_{p})$-Type inequalities for subsequences of Nörlund means of Walsh–Fourier series

Abstract

We investigate the subsequence $\{t_{2^{n}}f \}$ of Nörlund means with respect to the Walsh system generated by nonincreasing and convex sequences. In particular, we prove that a large class of such summability methods are not bounded from the martingale Hardy spaces $H_{p}$ to the space $\mathit{weak-}L_{p}$ for $0< p<1/(1+\alpha )$, where $0<\alpha <1$. Moreover, some new related inequalities are derived. As applications, some well-known and new results are pointed out for well-known summability methods, especially for Nörlund logarithmic means and Cesàro means.

1 Introduction

The terminology and notations used in this introduction can be found in Sect. 2.

The fact that the Walsh system is the group of characters of a compact abelian group connects Walsh analysis with abstract harmonic analysis was discovered independently by Fine [7] and Vilenkin [28]. For general references to the Haar measure and harmonic analysis on groups see Pontryagin [22], Rudin [23], and Hewitt and Ross [14]. In particular, Fine investigated the group G, which is a direct product of the additive groups $Z_{2}=:\{0,1\}$ and introduced the Walsh system $\{{w}_{j}\}_{j=0}^{\infty}$.

It is well known (for details see, e.g., the books [21, 24], and [29]) that Walsh systems do not form bases in the space $L_{1}$. Moreover, there exists a martingale $f\in H_{p}$ ($0< p\leq 1$), such that $\sup_{n\in \mathbb{N}} \Vert S_{2^{n}+1}f \Vert _{p}=\infty$. On the other hand, by the definition of Hardy spaces, the subsequence $\{S_{2^{n}}\}$ of partial sums is bounded from the space $H_{p}$ to the space $H_{p}$, for all $p>0$.

Weisz [30] proved that the Fejér means of Vilenkin–Fourier series are bounded from the martingale Hardy space $H_{p}$ to the space $H_{p}$, for $p>1/2$. Goginava [11] (see also [19]) proved that there exists a martingale $f\in H_{1/2}$ such that

$$ \sup_{n\in \mathbb{N}} \Vert \sigma _{n}f \Vert _{1/2}=+ \infty . $$

However, Weisz [30] (see also [18]) proved that for every $f\in H_{p}$, there exists an absolute constant $c_{p}$, such that the following inequality holds:

$$ \Vert \sigma _{2^{n}}f \Vert _{H_{p}}\leq c_{p} \Vert f \Vert _{H_{p}}, \quad n\in \mathbb{N}, p>0. $$

(1)

Móricz and Siddiqi [17] investigated the approximation properties of some special Nörlund means of Walsh–Fourier series of $L_{p}$ functions in norm. Approximation properties for general summability methods can be found in [3, 4]. Fridli, Manchanda, and Siddiqi [8] improved and extended the results of Móricz and Siddiqi [17] to martingale Hardy spaces. The case when $\{ q_{k}=1/k:k\in \mathbb{N} \}$ was excluded, since the methods are not applicable to Nörlund logarithmic means. In [9] Gát and Goginava proved some convergence and divergence properties of the Nörlund logarithmic means of functions in the Lebesgue space $L_{1}$. In particular, they proved that there exists a function f in the space $L_{1}$, such that $\sup_{n\in \mathbb{N}} \Vert L_{n}f \Vert _{1}=\infty$. In [1] it was proved that there exists a martingale $f\in H_{p}$, ($0< p< 1$) such that

$$ \sup_{n\in \mathbb{N}} \Vert L_{2^{n}}f \Vert _{p}= \infty . $$

A counterexample for $p=1$ was proved in [20]. However, Goginava [10] proved that for every $f\in H_{1}$, there exists an absolute constant c, such that the following inequality holds:

$$ \Vert L_{2^{n}}f \Vert _{1}\leq c \Vert f \Vert _{H_{1}}, \quad n\in \mathbb{N}. $$

(2)

The convergence of subsequences of Nörlund logarithmic means of Walsh–Fourier series in martingale Hardy spaces was investigated by Goginava [13] and Memić [16].

In [19] it was proved that for any nondecreasing sequence $(q_{k},k\in \mathbb{N})$ satisfying the conditions

$$\begin{aligned} \frac{1}{Q_{n}}=O \biggl(\frac{1}{n^{\alpha}} \biggr), \quad \text{where } Q_{n} = \sum_{k=0}^{n-1} q_{k} \end{aligned}$$

(3)

and

$$\begin{aligned} q_{n}-q_{n+1}=O \biggl(\frac{1}{n^{2-\alpha}} \biggr) ,\quad\text{as } n\rightarrow \infty , \end{aligned}$$

(4)

then, for every $f\in H_{p}$, where $p>1/(1+\alpha )$, there exists an absolute constant $c_{p}$, depending only on p, such that the following inequality holds:

$$ \Vert t_{n}f \Vert _{H_{p}}\leq c_{p} \Vert f \Vert _{H_{p}}, \quad n\in \mathbb{N}. $$

(5)

Boundedness does not hold from $H_{p}$ to $\mathit{weak-}L_{p}$, for $0< p< 1/ (1+\alpha )$. As a consequence, (for details see [31]) we obtain that the Cesàro means $\sigma _{n}^{\alpha}$ is bounded from $H_{p}$ to $L_{p}$, for $p>1/(1+\alpha )$, but they are not bounded from $H_{p}$ to $\mathit{weak-}L_{p}$, for $0< p< 1/ (1+\alpha )$. In the endpoint case $p=1/ (1+\alpha )$, Weisz and Simon [26] (see also [25]) proved that the maximal operator $\sigma ^{\alpha ,\ast }$ of Cesàro means defined by

$$ \sigma ^{\alpha ,\ast }f:=\sup_{n\in \mathbb{N}} \bigl\vert \sigma ^{ \alpha}_{n}f \bigr\vert $$

is bounded from the Hardy space $H_{1/ (1+\alpha ) }$ to the space $\mathit{weak-}L_{1/ (1+\alpha )}$. Goginava [12] gave a counterexample, which shows that boundedness does not hold for $0< p\leq 1/ (1+\alpha )$.

In this paper we develop some methods considered in [1, 2, 15] (see also the new book [21]) and prove that for any $0< p<1$, there exists a martingale $f\in H_{p}$ such that

$$ \sup_{n\in \mathbb{N}} \Vert t_{2^{n}}f \Vert _{\mathit{weak-}L_{p}}= \infty . $$

Moreover, we prove that a class of subsequence $\{t_{2^{n}}f \}$ of Nörlund means with respect to the Walsh system generated by nonincreasing and convex sequences are not bounded from the martingale Hardy spaces $H_{p}$ to the space $\mathit{weak-}L_{p}$ for $0< p<1/(1+\alpha )$, where $0<\alpha <1$. Moreover, some new related inequalities are derived. As applications, some well-known and new results are pointed out for well-known summability methods, especially for Nörlund logarithmic means and Cesàro means.

The main results in this paper are presented and proved in Sect. 4. Section 3 is used to present some auxiliary results, where, in particular, Lemma 2 is new and of independent interest. In order not to disturb our discussions later some definitions and notations are given in Sect. 2.

2 Definitions and notations

Let $\mathbb{N}_{+}$ denote the set of the positive integers, $\mathbb{N}:=\mathbb{N}_{+}\cup \{0\}$. Denote by $Z_{2}$ the discrete cyclic group of order 2, that is $Z_{2}:=\{0,1\}$, where the group operation is the modulo 2 addition and every subset is open. The Haar measure on $Z_{2}$ is given so that the measure of a singleton is 1/2.

Define the group G as the complete direct product of the group $Z_{2}$, with the product of the discrete topologies of $Z_{2}$s.

The elements of G are represented by sequences

$$ x:=(x_{0},x_{1},\ldots,x_{j},\ldots), \quad \text{where } x_{k}=0\vee 1. $$

It is easy to give a base for the neighborhood of $x\in G$ namely:

$$ I_{0} ( x ) :=G,\qquad I_{n}(x):=\{y\in G:y_{0}=x_{0},\ldots,y_{n-1}=x_{n-1} \} \quad(n\in \mathbb{N}). $$

Denote $I_{n}:=I_{n} ( 0 )$, $\overline{I_{n}}:=G \backslash I_{n}$ and

$$ e_{n}:= ( 0,\ldots,0,x_{n}=1,0,\ldots ) \in G, \quad \text{for } n \in \mathbb{N}. $$

If $n\in \mathbb{N}$, then every n can be uniquely expressed as $n=\sum_{k=0}^{\infty }n_{j}2^{j}$, where $n_{j}\in Z_{2}$ ($j\in \mathbb{N}$) and only a finite number of $n_{j}$s differ from zero. Let

$$ \vert n \vert :=\max \{k\in \mathbb{N}: n_{k}\neq 0\}. $$

The norms (or quasinorms) of the spaces $L_{p}(G)$ and $\mathit{weak-}L_{p} ( G )$, ($0< p<\infty$) are, respectively, defined by

$$ \Vert f \Vert _{p}^{p}:= \int _{G} \vert f \vert ^{p}\,d\mu \quad \text{and} \quad \Vert f \Vert _{\mathit{weak-}L_{p}}^{p}:=\sup _{\lambda >0} \lambda ^{p}\mu ( f>\lambda ) . $$

The kth Rademacher function is defined by

$$ r_{k} ( x ) := ( -1 ) ^{x_{k}}\quad ( x\in G,k\in \mathbb{N} ) . $$

Now, define the Walsh system $w:=(w_{n}:n\in \mathbb{N})$ on G as:

$$ w_{n}(x):=\prod_{k=0}^{\infty }r_{k}^{n_{k}} ( x ) =r_{ \vert n \vert } ( x ) ( -1 ) ^{ \sum _{k=0}^{ \vert n \vert -1}n_{k}x_{k}} \quad ( n\in \mathbb{N} ) . $$

It is well known that (see, e.g., [24]) the Walsh system is orthonormal and complete in $L_{2} ( G )$. Moreover, for any $n\in \mathbb{N}$,

$$\begin{aligned} w_{n} ( x+y ) =&w_{n} ( x )w_{n} ( y ). \end{aligned}$$

(6)

If $f\in L_{1} ( G )$ we define the Fourier coefficients, partial sums, and Dirichlet kernel by

$$\begin{aligned}& \widehat{f} ( k ) := \int _{G}fw_{k}\,d\mu \quad ( k \in \mathbb{N}\mathbbm{ } ) , \\& S_{n}f := \sum_{k=0}^{n-1} \widehat{f} ( k ) w_{k},\qquad D_{n}:=\sum _{k=0}^{n-1}w_{k}\quad ( n\in \mathbb{N}_{+} ). \end{aligned}$$

Recall that (for details see, e.g., [24]):

$$ D_{2^{n}} ( x ) =\textstyle\begin{cases} 2^{n}, & \text{if }x\in I_{n}, \\ 0, & \text{if } x\notin I_{n}\end{cases} $$

(7)

and

$$ D_{n}=w_{n}\sum_{k=0}^{\infty }n_{k}r_{k}D_{2^{k}}=w_{n}\sum_{k=0}^{\infty }n_{k} ( D_{2^{k+1}}-D_{2^{k}} ),\quad\text{for }n=\sum _{i=0}^{\infty }n_{i}2^{i}. $$

(8)

Let $\{ q_{k}, k\geq 0 \}$ be a sequence of nonnegative numbers. The Nörlund means for the Fourier series of f are defined by

$$ t_{n}f:=\frac{1}{Q_{n}}\sum_{k=1}^{n}q_{n-k}S_{k}f, \quad \text{where } Q_{n}:=\sum_{k=0}^{n-1}q_{k}. $$

In this paper we consider convex $\{ q_{k}, k\geq 0 \}$ sequences, that is

$$ q_{n-1}+q_{n+1}-2q_{n}\geq 0, \quad \text{for all } n \in \mathbb{N}\mathbbm{ }. $$

If the function $\psi (x)$ is any real-valued and convex function (for example $\psi (x)=x^{\alpha -1}$, $0\leq \alpha \leq 1$), then the sequence $\{\psi (n), n\in \mathbb{N}\mathbbm{ }\}$ is convex.

Since $q_{n-2}-q_{n-1}\geq q_{n-1}-q_{n}\geq q_{n}-q_{n+1}\geq q_{n+1}-q_{n+2}$ we find that

$$ q_{n-2}+q_{n+2}\geq q_{n-1}+q_{n+1} $$

and we also obtain that

$$ q_{n-2}+q_{n+2}-2q_{n}\geq 0, \quad \text{for all } n\in \mathbb{N}\mathbbm{ }. $$

(9)

In the special case when $\{q_{k}=1, k\in \mathbb{N}\}$, we have the Fejér means

$$ \sigma _{n}f:=\frac{1}{n}\sum_{k=1}^{n}S_{k}f. $$

Moreover, if $q_{k}={1}/{(k+1)}$, then we obtain the Nörlund logarithmic means:

$$ L_{n}f:=\frac{1}{l_{n}}\sum _{k=1}^{n}\frac{S_{k}f}{n+1-k}, \quad \text{where } l_{n}:=\sum_{k=1}^{n} \frac{1}{k}. $$

(10)

The Cesàro means $\sigma _{n}^{\alpha}$ (sometimes also denoted $(C,\alpha )$) is also a well-known example of Nörlund means defined by

$$ \sigma _{n}^{\alpha}f=:\frac{1}{A_{n}^{\alpha}} \sum _{k=1}^{n}A_{n-k}^{\alpha -1}S_{k}f, $$

where

$$ A_{0}^{\alpha}:=0,\qquad A_{n}^{\alpha}:= \frac{ (\alpha +1 )\ldots (\alpha +n )}{n!}, \quad \alpha \neq -1,-2,\ldots . $$

It is well known that

$$ A_{n}^{\alpha}=\sum_{k=0}^{n}A_{n-k}^{\alpha -1}, \qquad A_{n}^{\alpha}-A_{n-1}^{\alpha}=A_{n}^{\alpha -1} \quad \text{and} \quad A_{n}^{\alpha }\sim n^{\alpha }. $$

(11)

We also define $U_{n}^{\alpha}$ means as

$$ U^{\alpha}_{n}f:=\frac{1}{Q_{n}}\sum _{k=1}^{n}{(n+1-k)}^{( \alpha -1)} S_{k}f, \quad \text{where } Q_{n}:=\sum_{k=1}^{n}k^{ \alpha -1}. $$

Let us also define $V_{n}^{\alpha}$ means as

$$ V_{n}f:=\frac{1}{Q_{n}}\sum_{k=1}^{n}{ \ln (n+1-k)}S_{k}f, \quad \text{where } Q_{n}:=\sum _{k=1}^{n}\frac{1}{\ln (k+1)}. $$

The σ-algebra generated by the intervals $\{ I_{n} ( x ) :x\in G \}$ will be denoted by $\digamma _{n}$ ($n\in \mathbb{N}$). Denote by $f:= ( f^{ ( n ) },n\in \mathbb{N} )$ the martingale with respect to $\digamma _{n}$ ($n\in \mathbb{N}$) (for details see, e.g., [29]).

We say that this martingale belongs to the Hardy martingale spaces $H_{p} ( G )$, where $0< p<\infty$, if

$$ \Vert f \Vert _{H_{p}}:= \bigl\Vert f^{*} \bigr\Vert _{p}< \infty , \quad \text{with } f^{\ast }:=\sup _{n\in \mathbb{N}} \bigl\vert f^{(n)} \bigr\vert . $$

When $f\in L_{1} (G )$, the maximal functions are also given by

$$ M(f) ( x ) :=\sup_{n\in \mathbb{N}} \biggl( \frac{1}{\mu ( I_{n} ( x ) ) } \biggl\vert \int _{I_{n} ( x ) }f ( u ) \,d\mu ( u ) \biggr\vert \biggr) . $$

If $f\in L_{1} ( G )$, then it is easy to show that the sequence $F= ( S_{2^{n}}f :n\in \mathbb{N} )$ is a martingale and $F^{*}=M(f)$.

If $f= ( f^{ ( n ) }, n\in \mathbb{N} )$ is a martingale, then the Walsh–Fourier coefficients must be defined in a slightly different manner:

$$ \widehat{f} ( i ) :=\lim_{k\rightarrow \infty } \int _{G}f^{ ( k ) } ( x )w_{i} ( x ) \,d\mu ( x ) . $$

A bounded measurable function a is p-atom, if there exists an interval I, such that

$$ \operatorname{supp} ( a ) \subset I, \qquad \int _{I}a\,d\mu =0 \quad\text{and}\quad \Vert a \Vert _{\infty }\leq \mu ( I ) ^{-1/p}. $$

3 Auxiliary results

The Hardy martingale space $H_{p} ( G )$ has an atomic characterization (see Weisz [29, 30]):

Lemma 1

A martingale $f= ( f^{ ( n ) }, n\in \mathbb{N} )$ is in $H_{p}$ ($0< p\leq 1$) if and only if there exist a sequence $( a_{k},k\in \mathbb{N} )$ of p-atoms and a sequence $( \mu _{k},k\in \mathbb{N} )$ of real numbers such that for every $n\in \mathbb{N}$:

$$ \sum_{k=0}^{\infty}\mu _{k}S_{2^{n}}a_{k}=f^{ ( n ) }, \quad \textit{where } \sum_{k=0}^{\infty } \vert \mu _{k} \vert ^{p}< \infty . $$

(12)

Moreover, the following two-sided inequality holds

$$ \Vert f \Vert _{H_{p}}\backsim \inf \Biggl( \sum _{k=0}^{ \infty } \vert \mu _{k} \vert ^{p} \Biggr) ^{1/p}, $$

where the infimum is taken over all decompositions of f of the form (12).

We also state and prove the following new lemma of independent interest:

Lemma 2

Let $k\in \mathbb{N}$, $\{ q_{k}:k\in \mathbb{N} \}$ be any convex and nonincreasing sequence and $x\in I_{2}(e_{0}+e_{1})\in I_{0}\backslash I_{1}$. Then, for any $\{\alpha _{k}\}$, the following inequality holds:

$$\begin{aligned} \Biggl\vert \sum_{j=2^{2\alpha _{k}}}^{2^{2\alpha _{k}+1}}q_{2^{2 \alpha _{k}+1}-j}{D_{j}} \Biggr\vert \geq q_{1}-\frac{3}{2}q_{3}. \end{aligned}$$

Proof

Let $x\in I_{2}(e_{0}+e_{1})\in I_{0}\backslash I_{1}$. According to (7) and (8) we obtain that

$$ D_{j} ( x ) =\textstyle\begin{cases} -w_{j}, & \text{if }j \text{ is an odd number,} \\ 0, & \text{if } j \text{ is an even number}\end{cases} $$

and

$$\begin{aligned} \sum_{j=2^{2\alpha _{k}}}^{2^{2\alpha _{k}+1}-1}q_{2^{2\alpha _{k}+1}-j}D_{j}=- \sum_{j=2^{2\alpha _{k}-1}}^{2^{2\alpha _{k}}-1}q_{2^{2\alpha _{k}+1}-2j-1}{w_{2j+1}}=-w_{1} \sum_{j=2^{2\alpha _{k}-1}}^{2^{2\alpha _{k}}-1}q_{ 2^{2\alpha _{k}+1}-2j-1}{w_{2j}}. \end{aligned}$$

By using (9) we find that

$$\begin{aligned} &\sum_{j=2^{2\alpha _{k}-2}+1}^{2^{2\alpha _{k}-1}-1} \vert q_{2^{2 \alpha _{k}+1}-4j+3}-q_{2^{2\alpha _{k}+1}-4j+1} \vert \\ &\quad =\sum_{j=2^{2\alpha _{k}-2}+1}^{2^{2\alpha _{k}-1}-1} ( q_{ 2^{2 \alpha _{k}+1}-4j+1}-q_{2^{2\alpha _{k}+1}-4j+3} ) \\ &\quad = (q_{2^{2\alpha _{k}}-3}-q_{2^{2\alpha _{k}}-1} )+ ( q_{ 2^{2\alpha _{k}}-7}-q_{2^{2\alpha _{k}}-5} )+ \cdots + ( q_{5}-q_{7} ) \\ &\quad \leq \frac{1}{2} ( q_{2^{2\alpha _{k}}-3}-q_{2^{2\alpha _{k}}-1} )+ \frac{1}{2} ( q_{2^{2\alpha _{k}}-5}-q_{2^{2\alpha _{k}}-3} ) \\ &\qquad {}+\frac{1}{2} ( q_{2^{2\alpha _{k}}-7}-q_{2^{2\alpha _{k}}-5} ) + \frac{1}{2} ( q_{2^{2\alpha _{k}}-9}-q_{2^{2\alpha _{k}}-7} ) \\ &\qquad {}+\ldots +\frac{1}{2} ( q_{5}-q_{7} )+ \frac{1}{2} ( q_{3}-q_{5} )\leq \frac{1}{2}q_{3}-\frac{1}{2}q_{2^{2\alpha _{k}}-1}. \end{aligned}$$

Hence, if we apply

$$ w_{4k+2}=w_{2}w_{4k}=-w_{4k}, \quad \text{for } x\in I_{2}(e_{0}+e_{1}), $$

we find that

$$\begin{aligned} & \Biggl\vert \sum_{j=2^{2\alpha _{k}}}^{2^{2\alpha _{k}+1}-1}q_{2^{2 \alpha _{k}+1}-j}{D_{j}} \Biggr\vert \\ &\quad = \Biggl\vert q_{1}w_{2^{2\alpha _{k}+1}-2}+q_{3}{w_{2^{2\alpha _{k}+1}-4}}+ \sum_{j=2^{2\alpha _{k}-1}}^{2^{2\alpha _{k}}-3}q_{2^{2\alpha _{k}+1}-2j-1}{w_{2j}} \Biggr\vert \\ &\quad = \Biggl\vert (q_{3}-q_{1}){2w_{2^{2\alpha _{k}+1}-4}} +\sum_{j=2^{2 \alpha _{k}-2}+1}^{2^{2\alpha _{k}-1}-1} ( q_{2^{2\alpha _{k}+1}-4j+3}{w_{4j-4}}-q_{2^{2 \alpha _{k}+1}-4j+1}{w_{4j-4}} ) \Biggr\vert \\ &\quad \geq q_{1}-q_{3}-\sum _{j=2^{2\alpha _{k}-2}+1}^{2^{2\alpha _{k}-1}-1} \vert q_{2^{2\alpha _{k}+1}-4j+3}-q_{2^{2\alpha _{k}+1}-4j+1} \vert \\ &\quad \geq q_{1}-q_{3}-\frac{1}{2}(q_{3}-q_{2^{2\alpha _{k}}-1}) \geq q_{1}- \frac{3}{2}q_{3}. \end{aligned}$$

The proof is complete. □

4 The main result

In previous sections we have discussed a number of inequalities and sometimes their sharpness. Our main result is the following new sharpness result:

Theorem 1

Let $0\leq \alpha \leq 1$, β be any nonnegative real number and $t_{n}$ be Nörlund means with a convex and nonincreasing sequence $\{ q_{k}:k\in \mathbb{N} \}$ satisfying the condition

$$ \frac{q_{1}-({3}/{2})q_{3}}{Q_{n}}\geq \frac{C}{ n^{\alpha}\ln ^{\beta}n}, $$

(13)

for some positive constant C. Then, for any $0< p<1/(1+\alpha )$ there exists a martingale $f\in H_{p}$ such that

$$ \sup_{n\in \mathbb{N}} \Vert t_{2^{n}}f \Vert _{\mathit{weak-}L_{p}}= \infty . $$

Proof

Let $0< p<1/(1+\alpha )$. Under condition (13) there exists a sequence $\{ n _{k}:k\in \mathbb{N} \}$ such that

$$ \frac{2^{2n_{k}(1/p-1)}}{n_{k}Q_{2^{2n_{k}+1}}}\geq \frac{2^{2n_{k}(1/p-1-\alpha )}}{n_{k}^{\beta +1}}\to \infty , \quad \text{as } k\to \infty . $$

Let $\{ \alpha _{k}:k\in \mathbb{N} \}\subset \{ n _{k}:k \in \mathbb{N} \}$ be an increasing sequence of positive integers such that

$$\begin{aligned}& \sum_{k=0}^{\infty }\alpha _{k}^{-p/2}< \infty , \end{aligned}$$

(14)

$$\begin{aligned}& \sum_{\eta =0}^{k-1} \frac{ (2^{2\alpha _{\eta}} )^{1/p}}{\sqrt{ \alpha _{\eta}}}< \frac{ (2^{2\alpha _{k}} )^{1/p}}{\sqrt{\alpha _{k}}} \end{aligned}$$

(15)

and

$$ \frac{ ( 2^{2\alpha _{k-1}} ) ^{1/p}}{\sqrt{\alpha _{k-1}}}< \frac{q_{1}-({3}/{2})q_{3}}{Q_{2^{2\alpha _{k}+1}}} \frac{2^{2\alpha _{k}(1/p-1)-3}}{\alpha _{k}}. $$

(16)

Let

$$ f^{ ( n ) } :=\sum_{ \{ k;2\alpha _{k}< n \} }\lambda _{k}a_{k}, $$

where

$$ \lambda _{k}=\frac{1}{\sqrt{\alpha _{k}}} \quad \text{and} \quad a_{k}={2^{2\alpha _{k}(1/p-1)}} ( D_{2^{2\alpha _{k}+1}}-D_{2^{2 \alpha _{k}}} ) . $$

From (14) and Lemma 1 we find that $f \in H_{p}$.

It is easy to prove that

$$ \widehat{f}(j)=\textstyle\begin{cases} \frac{2^{2\alpha _{k}(1/p-1)}}{\sqrt{\alpha _{k}}},& \text{if }j\in \{ 2^{2\alpha _{k}},\ldots, 2^{2\alpha _{k}+1}-1 \} , k\in \mathbb{N}, \\ 0,& \text{if }j\notin \bigcup_{k=1}^{\infty } \{ 2^{2 \alpha _{k}},\ldots,2^{2\alpha _{k}+1}-1 \}. \end{cases} $$

(17)

Moreover,

$$\begin{aligned} &t_{2^{2\alpha _{k}+1}}f \\ &\quad =\frac{1}{Q_{2^{2\alpha _{k}+1}}}\sum_{j=1}^{2^{2\alpha _{k}}-1}q_{2^{2 \alpha _{k}+1}-j}{S_{j}f}+ \frac{1}{Q_{2^{2\alpha _{k}+1}}} \sum_{j=2^{2 \alpha _{k}}}^{2^{2\alpha _{k}+1}}q_{{2^{2\alpha _{k}+1}-j}} {S_{j}f} \\ &\quad :=I+\mathit{II}. \end{aligned}$$

(18)

Let $j<2^{2\alpha _{k}}$. By combining (15), (16), and (17) we can conclude that

$$\begin{aligned} \vert S_{j}f \vert \leq &\sum_{\eta =0}^{k-1} \sum_{v=2^{2 \alpha _{\eta }}}^{2^{2\alpha _{\eta }+1}-1} \bigl\vert \widehat{f}(v) \bigr\vert \\ \leq & \sum_{\eta =0}^{k-1}\sum _{v=2^{2\alpha _{\eta }}}^{2^{2 \alpha _{\eta }+1}-1} \frac{ 2^{2\alpha _{\eta }(1/p-1)}}{\sqrt{\alpha _{\eta }}} \leq \sum _{ \eta =0}^{k-1}\frac{2^{2\alpha _{\eta }/p}}{\sqrt{\alpha _{\eta}}} \leq \frac{2^{2\alpha _{k-1}/p}}{\sqrt{\alpha _{k-1}}}. \end{aligned}$$

Hence,

$$\begin{aligned} \vert I \vert \leq &\frac{1}{Q_{2^{2\alpha _{k}+1}}} \sum _{j=1}^{2^{2\alpha _{k}}-1} q_{2^{2\alpha _{k}+1}-j}{ \vert S_{j}f \vert } \\ \leq &\frac{1}{Q_{2^{2\alpha _{k}+1}}} \frac{2^{2\alpha _{k-1}/p}}{\sqrt{\alpha _{k-1}}}\sum _{j=0}^{2^{2\alpha _{k}+1}-1}q_{j}\leq \frac{2^{2\alpha _{k-1}/p}}{\sqrt{\alpha _{k-1}}}. \end{aligned}$$

(19)

Let $2^{2\alpha _{k}}\leq j\leq 2^{2\alpha _{k}+1}$. Since

$$\begin{aligned} S_{j}f =&\sum_{\eta =0}^{k-1}\sum _{v=2^{2\alpha _{\eta }}}^{2^{2 \alpha _{\eta }+1}-1}\widehat{f}(v)w_{v}+ \sum_{v=2^{2\alpha _{k}}}^{j-1} \widehat{f}(v)w_{v} \\ =&\sum_{\eta =0}^{k-1} \frac{2^{{2\alpha _{\eta }} ( 1/p-1 ) }}{\sqrt{\alpha _{\eta }}} ( D_{2^{2\alpha _{\eta }+1}}-D_{2^{2 \alpha _{\eta }}} ) + \frac{2^{{2\alpha _{k}} ( 1/p-1 ) }}{\sqrt{\alpha _{k}}} ( D_{j}-D_{2^{{2\alpha _{k}}}} ), \end{aligned}$$

for II we can conclude that

$$\begin{aligned} \mathit{II} =& \frac{1}{Q_{2^{2\alpha _{k}+1}}} \sum _{j=2^{2\alpha _{k}}}^{2^{2\alpha _{k}+1}} q_{2^{2\alpha _{k}+1}-j} \Biggl( \sum _{ \eta =0}^{k-1} \frac{2^{2\alpha _{\eta } ( 1/p-1 ) }}{\sqrt{\alpha _{\eta}}} ( D_{2^{2\alpha _{\eta }+1}}-D_{2^{2\alpha _{\eta }}} ) \Biggr) \\ &{}+\frac{1}{Q_{2^{2\alpha _{k}+1}}} \frac{2^{2\alpha _{k} ( 1/p-1 ) }}{\sqrt{\alpha _{k}}}\sum_{j=2^{2\alpha _{k}}}^{2^{2\alpha _{k}+1}} q_{2^{2 \alpha _{k}+1}-j}{ (D_{j}-D_{2^{2\alpha _{k}}} )}. \end{aligned}$$

(20)

Let $x\in I_{2}(e_{0}+e_{1})\in I_{0}\backslash I_{1}$. According to the fact that $\alpha _{0}\geq 1$ we obtain that $2\alpha _{k}\geq 2$, for all $k\in \mathbb{N}$ and if we use (7) we obtain that $D_{2^{2\alpha _{k}}}=0$ and if we use Lemma 2 we can also conclude that

$$\begin{aligned} \vert \mathit{II} \vert =&\frac{1}{Q_{2^{2\alpha _{k}+1}}} \frac{2^{2\alpha _{k}(1/p-1)}}{ \sqrt{\alpha _{k}}}\sum_{j=2^{2\alpha _{k}}}^{2^{2\alpha _{k}+1}} q_{2^{2 \alpha _{k}+1}-j}{D_{j}} \\ \geq & \frac{q_{1}-(3/2)q_{3}}{Q_{2^{2\alpha _{k}+1}}}\frac{2^{2\alpha _{k}(1/p-1)}}{\sqrt{\alpha _{k}}}. \end{aligned}$$

(21)

By combining (16), and (18)–(21) for $x\in I_{2}(e_{0}+e_{1})$ we have that

$$\begin{aligned} \bigl\vert t_{2^{2\alpha _{k}+1}}f (x ) \bigr\vert \geq & \vert \mathit{II} \vert - \vert I \vert \\ \geq &\frac{q_{1}-({3}/{2})q_{3}}{Q_{2^{2\alpha _{k}+1}}} \frac{2^{2\alpha _{k}(1/p-1)}}{ \sqrt{\alpha _{k}}}-\frac{q_{1}-(3/2)q_{3}}{Q_{2^{2\alpha _{k}+1}}} \frac{2^{2\alpha _{k}(1/p-1)-3}}{\alpha _{k}} \\ \geq &\frac{q_{1}-(3/2)q_{3}}{Q_{2^{2\alpha _{k}+1}}} \frac{2^{2\alpha _{k}(1/p-1)-3}}{ \sqrt{\alpha _{k}}}\geq \frac{C2^{2\alpha _{k}(1/p-1-\alpha )-3}}{ (\ln 2^{2\alpha _{k}+1}+1 )^{\beta}\sqrt{\alpha _{k}}} \\ \geq & \frac{C2^{2\alpha _{k}(1/p-1-\alpha )-3}}{{\alpha ^{\beta +1} _{k}}}. \end{aligned}$$

Hence, we can conclude that

$$\begin{aligned} & \Vert t_{2^{2\alpha _{k}+1}}f \Vert _{\mathit{weak-}L_{p}} \\ &\quad \geq \frac{C2^{2\alpha _{k}(1/p-1-\alpha )-3}}{\alpha ^{\beta +1}_{k}}\mu \biggl\{ x\in G: \vert t_{2^{2\alpha _{k}+1}}f \vert \geq \frac{C2^{2\alpha _{k}(1/p-1)-3}}{\alpha ^{\beta +1}_{k}} \biggr\} ^{1/p} \\ &\quad \geq \frac{C2^{2\alpha _{k}(1/p-1-\alpha )-3}}{\alpha ^{\beta +1}_{k}}\mu \biggl\{ x\in I_{2}(e_{0}+e_{1}): \vert t_{2^{2\alpha _{k}+1}}f \vert \geq \frac{C2^{2\alpha _{k}(1/p-1)-3}}{\alpha ^{\beta +1}_{k}} \biggr\} ^{1/p} \\ &\quad \geq \frac{C2^{2\alpha _{k}(1/p-1-\alpha )-3}}{\alpha ^{\beta +1}_{k}}\bigl( \mu \bigl( I_{2}(e_{0}+e_{1}) \bigr) \bigr)^{1/p} \\ &\quad >\frac{c2^{2\alpha _{k}(1/p-1-\alpha )}}{\alpha _{k}^{\beta +1}} \rightarrow \infty , \quad \text{as }k\rightarrow \infty . \end{aligned}$$

The proof is complete. □

In an actual case we obtain a result for Nörlund logarithmic means $\{ L_{n} \}$ proved in [1]:

Corollary 1

Let $0< p<1$. Then, there exists a martingale $f\in H_{p}$ such that

$$ \sup_{n\in \mathbb{N}} \Vert L_{2^{n}}f \Vert _{\mathit{weak-}L_{p}}= \infty . $$

Proof

It is easy to show that

$$ q_{1}-({3}/{2})q_{3}=\frac{1}{2}- \frac{3}{2}\cdot \frac{1}{4}= \frac{1}{8}>0, $$

and condition (13) holds true for $\alpha =\beta =0$. □

We also obtain a similar new result for the $V_{n}$ means:

Corollary 2

Let $0< p<1$. Then, there exists a martingale $f\in H_{p}$ such that

$$ \sup_{n\in \mathbb{N}} \Vert V_{2^{n}}f \Vert _{\mathit{weak-}L_{p}}= \infty . $$

Proof

It is easy to show that

$$ q_{1}-({3}/{2})q_{3}=\frac{1}{\ln 2}- \frac{3}{2}\cdot \frac{1}{\ln 4}= \log _{2}^{e}-(3/2) \frac{\log _{2}^{e}}{\log _{2}^{4}}=\log _{2}^{e} \biggl(1-\frac{3}{4} \biggr)>0, $$

and condition (13) holds true for $\alpha =\beta =0$. □

We also obtain a corresponding new result for the Cesàro means $\sigma ^{\alpha}_{2^{n}}$.

Corollary 3

Let $0< p<1/(1+\alpha )$, for some $0< \alpha \leq 0.56$. Then, there exists a martingale $f\in H_{p}$ such that

$$ \sup_{n\in \mathbb{N}} \bigl\Vert \sigma ^{\alpha}_{2^{n}}f \bigr\Vert _{\mathit{weak-}L_{p}}=\infty . $$

Proof

By a routine calculation we find that

$$ q_{1}-({3}/{2})q_{3}=\alpha -\frac{\alpha (\alpha +1)(\alpha +2)}{4}= \alpha \cdot \frac{2-3\alpha -\alpha ^{2}}{4}. $$

It is easy to show that when $0< \alpha <0.56$ this expression is positive. Hence, condition (13) holds true for $\beta =0$ and $0<\alpha <1$. □

Corollary 4

Let $0< p<1/(1+\alpha )$, for some $0< \alpha \leq 0.41$. Then, there exists a martingale $f\in H_{p}$ such that

$$ \sup_{n\in \mathbb{N}} \bigl\Vert U^{\alpha}_{2^{n}}f \bigr\Vert _{\mathit{weak-}L_{p}}= \infty . $$

Proof

By a straightforward calculation, we find that

$$ q_{1}-({3}/{2})q_{3}=2^{\alpha -1}-({3}/{2})4^{\alpha -1}=2^{\alpha -1} \bigl(1-3/ 2^{2-\alpha} \bigr). $$

It is easy to show that when $0< \alpha <0.41$ this expression is positive. Hence, condition (13) holds true for $\beta =0$ and $0<\alpha <1$. □

5 Open questions and final remarks

Remark 1

This article can be regarded as a complement to the new book [21]. In this book a number of open problems are also raised. Also, this new investigation implies some corresponding open questions.

Open Problem 1

Let $0< p<1/(1+\alpha )$, for some $0.56< \alpha <1$. Does there exist a martingale $f\in H_{p}$ such that

$$ \sup_{n\in \mathbb{N}} \bigl\Vert \sigma ^{\alpha}_{2^{n}}f \bigr\Vert _{\mathit{weak-}L_{p}}=\infty ? $$

Open Problem 2

Let $0< p<1/(1+\alpha )$, for some $0.41< \alpha <1$. Does there exist a martingale $f\in H_{p}$ such that

$$ \sup_{n\in \mathbb{N}} \bigl\Vert U^{\alpha}_{2^{n}}f \bigr\Vert _{\mathit{weak-}L_{p}}= \infty ? $$

We also can investigate similar problems for more general summability methods:

Open Problem 3

Let $0< p<1/(1+\alpha )$, for some $0.56< \alpha <1$ and $t_{n}$ be Nörlund means of Walsh–Fourier series with nonincreasing and convex sequence $\{ q_{k}:k\in \mathbb{N} \}$, satisfying the condition (13).

Does there exist a martingale $f\in H_{1/(1+\alpha )}$ ($0< p<1$), such that

$$ \sup_{n\in \mathbb{N}} \Vert t_{2^{n}}f \Vert _{H_{1/(1+ \alpha )}}= \infty ? $$

Open Problem 4

Let $f\in H_{1/(1+\alpha )}$, where $0< \alpha <1$. Does there exist an absolute constant $C_{\alpha}$, such that the following inequality holds

$$ \bigl\Vert \sigma ^{\alpha}_{2^{n}}f \bigr\Vert _{1/(1+\alpha )} \leq C_{\alpha } \Vert f \Vert _{H_{1/(1+\alpha )}}? $$

Open Problem 5

Let $f\in H_{1/(1+\alpha )}$, where $0< \alpha <1$. Does there exist an absolute constant $C_{\alpha}$, such that the following inequality holds

$$ \bigl\Vert U^{\alpha}_{2^{n}}f \bigr\Vert _{1/(1+\alpha )}\leq C_{ \alpha } \Vert f \Vert _{H_{1/(1+\alpha )}}? $$

Open Problem 6

Let $f\in H_{1/(1+\alpha )}$, where $0< \alpha <1$ and $t_{n}$ are Nörlund means of Walsh–Fourier series with a nonincreasing and convex sequence $\{ q_{k}:k\in \mathbb{N} \}$, satisfying the condition (13). Does there exist an absolute constant $C_{\alpha}$, such that the following inequality holds

$$ \bigl\Vert t^{\alpha}_{2^{n}}f \bigr\Vert _{1/(1+\alpha )}\leq C_{ \alpha } \Vert f \Vert _{H_{1/(1+\alpha )}}? $$

Remark 2

There is an important relation between Walsh–Fourier series and wavelet theory, see, e.g., [21] and the papers [5] and [6]. This is of special interest also for applications as described in the recent PhD thesis of K. Tangrand [27].