1 Introduction

All symbols used in this introduction can be found in Sect. 2.

In the one-dimensional case, the weak (1,1)-type inequality for the maximal operator \(\sigma ^{*}\) of Fejér means \(\sigma _n\) with respect to the Walsh system

$$\begin{aligned} \sigma ^{*}f:=\sup _{n\in \mathbb {N}}\left| \sigma _{n}f\right| \end{aligned}$$

can be found in Schipp [21] and Pál, Simon [16] (see also [2]). Fujii [7] and Simon [23] proved that \(\sigma ^{*}\) is bounded from \(H_{1}\) to \(L_{1}\). Weisz [29] generalized this result and proved the boundedness of \(\sigma ^{*}\) from the martingale space \(H_{p}\) to the Lebesgue space \(L_{p}\) for \(p>1/2\). Simon [22] gave a counterexample, which shows that boundedness does not hold for \(0<p<1/2\). A counterexample for \(p=1/2\) was given by Goginava [10]. Moreover, in [11] (see also [19]) he proved that there exists a martingale \(F\in H_{p}\) \(\left( 0<p\le 1/2\right) \), such that

$$\begin{aligned} \sup _{n\in \mathbb {N}}\left\| \sigma _{n}F\right\| _{p}=+\infty . \end{aligned}$$

Weisz [32] proved that the maximal operator \(\sigma ^{*}\) of the Fejér means is bounded from the Hardy space \(H_{1/2}\) to the space \(weak-L_{1/2}\).

For \(0<p<1/2\) in [26] the weighted maximal operator \(\overset{\sim }{\sigma }^{*,p}\), defined by

$$\begin{aligned} \overset{\sim }{\sigma }^{*,p}F:=\sup _{n\in \mathbb {N}}\frac{\left| \sigma _{n}F\right| }{\left( n+1\right) ^{1/p-2} }, \end{aligned}$$
(1)

was investigated, and it was proved that the following estimate holds:

$$\begin{aligned} \left\| \overset{\sim }{\sigma }^{*,p}F\right\| _{p}\le c_{p}\left\| F\right\| _{H_{p}}. \end{aligned}$$
(2)

Moreover, it was proved that the rate of sequence \(\left( n_k+1\right) ^{ 1/p-2}\) given in the denominator of (1) cannot be improved. In the case \(p=1/2\) analogical results for the maximal operator \(\overset{\sim }{\sigma }^{*}\) defined by

$$\begin{aligned} \overset{\sim }{\sigma }^{*}F:=\sup _{n\in \mathbb {N}}\frac{\left| \sigma _{n}F\right| }{\log ^2\left( n+1\right) } \end{aligned}$$

were proved in [25].

To study the convergence of subsequences of Fejér means and their restricted maximal operators on the martingale Hardy spaces \(H_p(G)\) for \(0<p\le 1/2\), the central role is played by the fact that any natural number \(n\in \mathbb {N}\) can be uniquely expressed as

$$\begin{aligned} n=\sum _{k=0}^{\infty }n_{j}2^{j}, \ \ n_{j}\in Z_{2} ~(j\in \mathbb {N}), \end{aligned}$$

where only a finite number of \(n_{j}\) differs from zero and their important characters \(\left[ n\right] \), \(\left| n\right| \), \(\rho \left( n\right) \), and V(n) are defined by

$$\begin{aligned} \left[ n\right] :=\min \{j\in \mathbb {N},n_{j}\ne 0\}, \ \ \left| n\right| :=\max \{j\in \mathbb {N},n_{j}\ne 0\}, \ \ \rho \left( n\right) =\left| n\right| -\left[ n\right] \end{aligned}$$

and

$$\begin{aligned} V\left( n\right) :=n_{0}+\overset{\infty }{\underset{k=1}{\sum }}\left| n_{k}-n_{k-1}\right| , \ \text {for} \ \ \text {all}\,\,n\in \mathbb {N} \end{aligned}$$

Weisz [31] (see also [30]) also proved that for any \(F\in H_p(G)\) \((p>0)\), the maximal operator \( \underset{n\in \mathbb {N}}{\sup }\left| \sigma _{2^n}F\right| \) is bounded from the Hardy space \(H_{p}\) to the Lebesgue space \(L_{p}\). Persson and Tephnadze [18] generalized this result and proved that if \(0<p\le 1/2\) and \(\left\{ n_{k}:k\ge 0\right\} \) is a sequence of positive integers, such that

$$\begin{aligned} \sup _{k\in \mathbb {N}}\rho \left( n_{k}\right) \le c<\infty , \end{aligned}$$
(3)

then the maximal operator \({\widetilde{\sigma }}^{*,\vartriangle }\), defined by

$$\begin{aligned} {\widetilde{\sigma }}^{*,\vartriangle }F:=\underset{k\in \mathbb {N}}{\sup } \left| \sigma _{n_{k}}F\right| , \end{aligned}$$
(4)

is bounded from the Hardy space \(H_{p}(G)\) to the space \(L_{p}(G)\). Moreover, if \(0<p<1/2\) and \(\left\{ n_{k}:k\ge 0\right\} \) is a sequence of positive numbers, such that

$$\begin{aligned} \sup _{k\in \mathbb {N}}\rho \left( n_{k}\right) =\infty , \end{aligned}$$
(5)

then there exists a martingale \(F\in H_{p}\) such that

$$\begin{aligned} \underset{k\in \mathbb {N}}{\sup }\left\| \sigma _{n_{k}}F\right\| _{p}=\infty . \end{aligned}$$

From these facts, it follows that if \(0<p<1/2\), \(f\in H_{p}\), and \(\left\{ n_{k}:k\ge 0\right\} \) is any sequence of positive numbers, then the maximal operator defined by (4) is bounded from the Hardy space \(H_{p}\) to the Lebesgue space \(L_{p}\) if and only if the condition (3) is fulfilled.

In [27], it was proved that if \(F\in H_{1/2}\), then there exists an absolute constant c, such that

$$\begin{aligned} \left\| \sigma _{n}F\right\| _{H_{1/2}}\le cV^{2}\left( n\right) \left\| F\right\| _{H_{1/2}}. \end{aligned}$$

Moreover, the rate of sequence \(V^{2}\left( n\right) \) cannot be improved.

In [27], it was also proved that if \(0<p<1/2\) and \(F\in H_{p}\), then there exists an absolute constant \(c_p\), depending only on p, such that

$$\begin{aligned} \left\| \sigma _{n}F\right\| _{H_{p}}\le c_{p}2^{\rho \left( n\right) \left( 1/p-2\right) }\left\| F\right\| _{H_{p}}. \end{aligned}$$

Moreover, if \(0<p<1/2\) and \(\{\Phi _n \}\) is any nondecreasing sequence, such that

$$\begin{aligned} \sup _{k\in \mathbb {N}}\rho \left( n_{k}\right) =\infty ,\overline{\underset{k\rightarrow \infty }{\lim }} \frac{2^{\rho \left( n_{k}\right) \left( 1/p-2\right) }}{\Phi _{n_{k}} }=\infty , \end{aligned}$$

then there exists a martingale \(F\in H_{p}\), such that

$$\begin{aligned} \underset{k\in \mathbb {N}}{\sup }\left\| \frac{\sigma _{n_{k}}F}{\Phi _{ n_{k}}}\right\| _{\text {weak}-L_{p}}=\infty . \end{aligned}$$

Convergence and summability of Fejér means of Walsh–Fourier series can be found in [1, 3,4,5,6, 8, 9, 14, 15, 17, 28], and [29].

One main aim of this paper is to generalize the estimate (2) for \(f\in H_p(G)\), \(0<p<1/2\). Our main idea is to investigate much more general maximal operators by replacing the weights \(\left( n+1\right) ^{1/p-2} \) in (1) by more general “optimal” weights \(2^{\rho \left( n\right) \left( 1/p-2\right) } (\varphi (\rho \left( n\right) ))\), where \(\varphi :\mathbb {N}_+\rightarrow \mathbb {R}_+\) is any nonnegative and nondecreasing function satisfying the condition

$$\begin{aligned} \sum _{n=1}^{\infty }{1}/{\varphi ^p(n)}<c<\infty \end{aligned}$$

and prove that it is bounded from the martingale Hardy space \(H_p(G)\) to the Lebesgue space \(L_p(G)\), for \(0<p<1/2\). As a consequence, we obtain some new and well-known results. In particular, we prove that the maximal operator \(\widetilde{\sigma }^{*,\nabla }\), defined by

$$\begin{aligned} {\widetilde{\sigma }}^{*,\nabla ,\varepsilon }F:=\underset{n\in \mathbb {N}}{\sup } \frac{\left| \sigma _{n}F\right| }{2^{\rho \left( n\right) \left( 1/p-1\right) } \left( (\rho \left( n\right) )\right) ^{({1+\varepsilon })/p}}, \ \ \text {where} \ \ 0<p<1/2, \ \ \varepsilon \ge 0, \end{aligned}$$

is bounded from the Hardy space \({{H}_{p}}(G)\) to the Lebesgue space \({{L}_{p}}(G)\) for any \(\varepsilon >0\) but is not bounded from the Hardy space \({{H}_{p}}(G)\) to the Lebesgue space \({{L}_{p}}(G)\) when \(\varepsilon =0\).

This paper is organized as follows: In order not to disturb our discussions later on some definitions and notations are presented in Sect. 2. The main results and some of their consequences can be found in Sect. 3. For the proofs of the main results, we need some auxiliary lemmas, which are presented in Sect. 4. Detailed proofs are given in Sect. 5.

2 Definitions and Notations

Let \(\mathbb {N}_{+}\) denote the set of 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 \(Z_{2}\). The elements of G are represented by sequences

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

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

$$\begin{aligned} I_{0}\left( x\right) :=G,I_{n}(x):=\{y\in G:y_{0}=x_{0}, \ldots ,y_{n-1}=x_{n-1}\}(n\in \mathbb {N}). \end{aligned}$$

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

$$\begin{aligned} e_{n}:=\left( 0, \ldots ,0,x_{n}=1,0, \ldots \right) \in G, \ \ \text { for } \ \ n\in \mathbb {N}. \end{aligned}$$

Then it is easy to show that

$$\begin{aligned} \overline{I_{M}}=\underset{i=0}{\bigcup \limits ^{M-1}}I_{i}\backslash I_{i+1}=\left( \overset{M-2}{\underset{k=0}{\bigcup }}\overset{M-1}{\underset{l=k+1}{\bigcup }}I_{l+1}\left( e_{k}+e_{l}\right) \right) \bigcup \left( \underset{k=0}{\bigcup \limits ^{M-1}}I_{M}\left( e_{k}\right) \right) . \end{aligned}$$
(6)

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

$$\begin{aligned} \left\| f\right\| _{p}^{p}:=\int _{G}\left| f\right| ^{p}d\mu \end{aligned}$$

and

$$\begin{aligned} \left\| f\right\| _{weak-L_{p }(G)}^{p}:=\sup _{\lambda>0}\lambda ^{p}\mu \left( f>\lambda \right) <+\infty . \end{aligned}$$

The k-th Rademacher function \(r_{k}\left( x\right) \) is defined by

$$\begin{aligned} r_{k}\left( x\right) :=\left( -1\right) ^{x_{k}} {\qquad }\left( x\in G,\text { }k\in \mathbb {N}\right) . \end{aligned}$$

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

$$\begin{aligned} w_{n}(x):=\overset{\infty }{\underset{k=0}{\Pi }}r_{k}^{n_{k}}\left( x\right) =\left( -1\right) ^{\underset{k=0}{\overset{\left| n\right| }{\sum }}n_{k}x_{k}} \qquad \left( n\in \mathbb {N}\right) . \end{aligned}$$

The Walsh system is orthonormal and complete in \(L_{2}\left( G\right) \) (see [13] and [20]).

If \(f\in L_{1}\left( G\right) \), we can define the Fourier coefficients, partial sums of Fourier series, Fejér means, and Dirichlet and Fejér kernels in the usual manner:

$$\begin{aligned} {\widehat{f}}\left( n\right):= & {} \int _{G}fw_{n}d\mu ,\,\,\left( n\in \mathbb {N}\right) ,\\ S_{n}f:= & {} \sum _{k=0}^{n-1}{\widehat{f}}\left( k\right) w_{k}, \left( n\in \mathbb {N}_{+},S_{0}f:=0\right) ,\\ \sigma _{n}f:= & {} \frac{1}{n}\sum _{k=1}^{n}S_{k}f, \\ D_{n}:= & {} \sum _{k=0}^{n-1}w_{k},\\ K_{n}:= & {} \frac{1}{n}\overset{n}{ \underset{k=1}{\sum }}D_{k},\left( n\in \mathbb {N}_{+}\right) . \end{aligned}$$

Recall that (see [13] and [20]) for any \(t,n\in \mathbb {N}\),

$$\begin{aligned} D_{2^{n}}\left( x\right) =\left\{ \begin{array}{ll} 2^{n} &{} \,\text {if} x\in I_{n}, \\ 0\, &{} \ \,\text {if}\,\,x\notin I_{n}. \end{array} \right. \end{aligned}$$
(7)

and

$$\begin{aligned} K_{2^{n}}\left( x\right) =\left\{ \begin{array}{c} 2^{t-1},\text {if }x\in I_{n}\left( e_{t}\right) ,n>t, x\in I_{t}\backslash I_{t+1}, \\ \left( 2^{n}+1\right) /2,\text {if }x\in I_{n}, \\ 0,\ \text {otherwise.} \end{array} \right. \end{aligned}$$
(8)

Let

$$\begin{aligned} n=\sum _{i=1}^{r}2^{n^{i}}, \ \ \ n^{1}>n^{2}> \cdots >n^{r}\ge 0 \end{aligned}$$

and

$$\begin{aligned} {{n}^{\left( k \right) }}:={{2}^{{{n}^{k+1}}}}+{{2}^{{{n}^{k+2}}}}+\cdots +{{2}^{{{n}^{r}}}}. \end{aligned}$$

Then (see [13] and [20]), for any \(n\in \mathbb {N}\),

$$\begin{aligned} nK_{n}=\sum _{A=1}^{r}\left( \underset{j=1}{\overset{A-1}{\prod }} w_{2^{n^{j}}}\right) \left( 2^{n^{A}}K_{2^{n^{A}}}+n^{\left( A\right) }D_{2^{n^{A}}}\right) . \end{aligned}$$
(9)

The \(\sigma \)-algebra, generated by the intervals \(\left\{ I_{n}\left( x\right) :x\in G\right\} \) will be denoted by \(\zeta _{n}\) \(\left( n\in \mathbb {N}\right) \). Denote by \(F=\left( F_{n},n\in \mathbb {N}\right) \) a martingale with respect to \(\zeta _{n}\) \(\left( n\in \mathbb {N}\right) \) (see e.g., [30]).

The maximal function \(F^{*}\) of a martingale F is defined by

$$\begin{aligned} F^{*}:=\sup _{n\in \mathbb {N}}\left| F_{n}\right| . \end{aligned}$$

In the case \(f\in L_{1}\left( G\right) \) the maximal function \(f^{*}\) is given by

$$\begin{aligned} f^{*}\left( x\right) :=\sup \limits _{n\in \mathbb {N}}\left( \frac{1}{\mu \left( I_{n}\left( x\right) \right) }\left| \int _{I_{n}\left( x\right) }f\left( u\right) d\mu \left( u\right) \right| \right) . \end{aligned}$$

For \(0<p<\infty \), the Hardy martingale spaces \(H_{p}\left( G\right) \) consist of all martingales for which

$$\begin{aligned} \left\| F\right\| _{H_{p}}:=\left\| F^{*}\right\| _{p}<\infty . \end{aligned}$$

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

$$\begin{aligned} \text { supp}\left( a\right) \subset I, \ \ \ \int _{I}ad\mu =0,\left\| a\right\| _{\infty }\le \mu \left( I\right) ^{-1/p}. \end{aligned}$$

It is easy to check that for every martingale \(F=\left( F_{n},n\in \mathbb {N}\right) \) and every \(k\in \mathbb {N}\) the limit

$$\begin{aligned} {\widehat{F}}\left( k\right) :=\lim _{n\rightarrow \infty }\int _{G}F_{n}\left( x\right) w_{k}\left( x\right) d\mu \left( x\right) \end{aligned}$$

exists, and it is called the k-th Walsh–Fourier coefficients of F.

If \(F:=\) \(\left( S_{2^n}f:n\in \mathbb {N}\right) \) is a regular martingale, generated by \(f\in L_{1}\left( G\right) \), then (see e.g., [19, 24], and [30])

$$\begin{aligned} {\widehat{F}}\left( k\right) ={\widehat{f}}\left( k\right) , \ k\in \mathbb {N}. \end{aligned}$$

3 The Main Results

Our first main result reads:

Theorem 1

Let \(0<p<1/2\), \(f\in {{H}_{p}}\left( G \right) \), and \(\varphi :\mathbb {N}_+\rightarrow \mathbb {R}\) be any nonnegative and nondecreasing function satisfying the condition

$$\begin{aligned} \sum _{n=1}^{\infty }\frac{1}{\varphi ^p(n)}<c<\infty . \end{aligned}$$
(10)

Then the weighted maximal operator \({\widetilde{\sigma }}^{*,\nabla }\), defined by

$$\begin{aligned} {\widetilde{\sigma }}^{*,\nabla }F:=\underset{n\in \mathbb {N}}{\sup } \frac{\left| \sigma _{n}F\right| }{2^{\rho \left( n\right) \left( 1/p-2\right) }\varphi (\rho \left( n\right) )}, \end{aligned}$$

is bounded from the Hardy space \({{H}_{p}}(G)\) to the Lebesgue space \({{L}_{p}}(G)\).

We also state and prove the sharpness of Theorem 1:

Theorem 2

Let \(0<p<1/2\), \(\left\{ n_{k}:k\ge 0\right\} \) be a sequence of positive numbers and \(\varphi :\mathbb {N}_+\rightarrow \mathbb {R}\) is any nonnegative and nondecreasing function satisfying the condition

$$\begin{aligned} \sum _{n=1}^{\infty }\frac{1}{\varphi ^p(n)}=\infty . \end{aligned}$$
(11)

Then there exist p-atoms \(f_{n_k}\), such that

$$\begin{aligned} \sup _{k\in \mathbb {N}}\frac{\left\| \sup _{n\in \mathbb {N}} \frac{\left| \sigma _{n}f_{n_k}\right| }{2^{\rho \left( n\right) \left( 1/p-2\right) }\varphi (\rho \left( n\right) )}\right\| _{p}}{\left\| f_{n_k}\right\| _{H_p}}= \infty . \end{aligned}$$

As we will point out (see Remark 1) Theorem 1 can be of special interest even if we restrict it to subsequences.

Corollary 1

Let \(0<p<1/2\), \(f\in {{H}_{p}}\left( G \right) \), \(\varphi :\mathbb {N}_+\rightarrow \mathbb {R}\) be any nonnegative and nondecreasing function satisfying the condition (10), and \(\left\{ n_{k}:k\ge 0\right\} \) be any sequence of positive numbers. Then the weighted maximal operator \(\widetilde{\sigma }^{*,\nabla }\), defined by

$$\begin{aligned} {\widetilde{\sigma }}^{*,\nabla }F:=\underset{k\in \mathbb {N}}{\sup } \frac{\left| \sigma _{n_k}F\right| }{2^{\rho \left( n_k\right) \left( 1/p-2\right) }\varphi (\rho \left( n_k\right) )}, \end{aligned}$$
(12)

is bounded from the Hardy space \({{H}_{p}}(G)\) to the Lebesgue space \({{L}_{p}}(G)\).

If we take \(\varphi (n)=n^{({1+\varepsilon })/p}, \ \text { for any} \ \varepsilon >0\), we get that the condition (10) is fulfilled. On the other hand, if we take \(\varphi (n)=n^{1/p}\), then the condition (11) holds. Hence, Theorem 1 and Theorem 2 imply the following sharp result:

Corollary 2

a) Let \(0<p<1/2\) and \(f\in {{H}_{p}}\left( G \right) \). Then the weighted maximal operator \({\widetilde{\sigma }}^{*,\nabla ,\varepsilon }\), defined by

$$\begin{aligned} {\widetilde{\sigma }}^{*,\nabla ,\varepsilon }F:=\underset{n\in \mathbb {N}}{\sup } \frac{\left| \sigma _{n}F\right| }{2^{\rho \left( n\right) \left( 1/p-2\right) } \left( \rho \left( n\right) \right) ^{{(1+\varepsilon )}/p}}, \ \ \ \varepsilon >0, \end{aligned}$$

is bounded from the Hardy space \({{H}_{p}}(G)\) to the Lebesgue space \({{L}_{p}}(G)\).

b) The weighted maximal operator \({\widetilde{\sigma }}^{*,\nabla ,0 }\), defined by

$$\begin{aligned} {\widetilde{\sigma }}^{*,\nabla ,0 }F:=\underset{n\in \mathbb {N}}{\sup } \frac{\left| \sigma _{n}F\right| }{2^{\rho \left( n\right) \left( 1/p-2\right) }\left( \rho \left( n\right) \right) ^{1/p}}, \end{aligned}$$

is not bounded from the Hardy space \({{H}_{p}}(G)\) to the Lebesgue space \({{L}_{p}}(G)\).

Remark 1

Suppose that \(\left\{ n_{k}:k\ge 0\right\} \) is a sequence of positive numbers, such that

$$\begin{aligned} \sup _{k\in \mathbb {N}}\left[ n_k\right]<c<\infty . \end{aligned}$$

Then

$$\begin{aligned}{} & {} \sup _{k\in \mathbb {N}}\varphi (\left[ n_k\right] )<\varphi (c)<\infty ,\\{} & {} 2^{\rho \left( n_k\right) \left( 1/p-2\right) }\sim 2^{\left| n_k\right| \left( 1/p-2\right) }\sim n_k^{ 1/p-1} \sim (n_k+1)^{ 1/p-2} \end{aligned}$$

and the maximal operator \({\widetilde{\sigma }}^{*,\nabla },\) defined by (12), can be estimated by

$$\begin{aligned} {\widetilde{\sigma }}^{*,\nabla }F\le \underset{k\in \mathbb {N}}{\sup } \frac{\left| \sigma _{n_k}F\right| }{\left( n_k+1\right) ^{ 1/p-2}}. \end{aligned}$$

Let

$$\begin{aligned} \sup _{k\in \mathbb {N}}\left[ n_k\right] =\infty . \end{aligned}$$

Then we have the following estimation:

$$\begin{aligned} \underset{k\in \mathbb {N}}{\sup } \frac{\left| \sigma _{n_k}F\right| }{\left( n_k+1\right) ^{ 1/p-1}}\le {\widetilde{\sigma }}^{*,\nabla }F. \end{aligned}$$

In particular, we find that from Theorem 1, Remark 1, and the theorem proved in [26] follows immediately the following result:

Corollary 3

Let \(0<p<1/2\), \(f\in {{H}_{p}}\left( G \right) \), and \(\varphi :\mathbb {N}_+\rightarrow \mathbb {R}_+\) be any nonnegative and nondecreasing function satisfying the condition (10). Then the weighted maximal operator \({\widetilde{\sigma }}^{*,\nabla }\), defined by

$$\begin{aligned} {\widetilde{\sigma }}^{*,\nabla }F:=\underset{n\in \mathbb {N}}{\sup } \frac{\left| \sigma _{n}F\right| }{\min \{2^{\rho \left( n\right) \left( 1/p-2\right) }\varphi (\rho \left( n\right) ), (n+1)^{1/p-2}\}}, \end{aligned}$$

is bounded from the Hardy space \({{H}_{p}}(G)\) to the Lebesgue space \({{L}_{p}}(G)\).

From Theorem 1 and Theorem 2 follows immediately the following result given in [18]:

Corollary 4

a) Let \(0<p\le 1/2\) and \(\left( n_{k},k\in \mathbb {N}\right) \) be a subsequence of positive numbers such that condition (3) is fulfilled. Then the maximal operator \({\widetilde{\sigma }}^{*,\vartriangle }\), defined by (4), is bounded from the Hardy space \(H_{p}(G)\) to the Lebesgue space \(L_{p}(G)\).

b) Let \(0<p<1\) and \(\left( n_{k},k\in \mathbb {N}\right) \) be a subsequence of positive numbers satisfying the condition (5). Then the maximal operator \({\widetilde{\sigma }}^{*,\vartriangle }\), defined by (4), is not bounded from the Hardy space \(H_{p}(G)\) to the Lebesgue space \(L_{p}(G)\).

4 Auxiliary Results

The dyadic Hardy martingale spaces \(H_{p}\) for \(0<p\le 1\) have an atomic characterization. Namely, the following holds (see [19, 24, 30], and [31]):

Lemma 1

A martingale \(F=\left( F_{n}, n\in \mathbb {N}\right) \) belongs to \(H_{p}\left( 0<p\le 1\right) \) if and only if there exists a sequence \(\left( a_{k},k\in \mathbb {N}\right) \) of p-atoms and a sequence \(\left( \mu _{k},k\in \mathbb {N}\right) \) of real numbers, such that, for every \(n\in \mathbb {N},\)

$$\begin{aligned} \qquad \sum _{k=0}^{\infty }\mu _{k}S_{2^{n}}a_{k}=F_{n}, \sum _{k=0}^{\infty }\left| \mu _{k}\right| ^{p}<\infty . \end{aligned}$$
(13)

Moreover,

$$\begin{aligned} \left\| F\right\| _{H_{p}}\backsim \inf \left( \sum _{k=0}^{\infty }\left| \mu _{k}\right| ^{p}\right) ^{1/p}, \end{aligned}$$

where the infimum is taken over all decompositions of F of the form (13).

From this result follows the following important lemma proved by Weisz [30]:

Lemma 2

Suppose that an operator T is \(\sigma \)-sublinear and

$$\begin{aligned} \int \limits _{{\overline{I}}}\left| Ta\right| ^{p}d\mu \le c_{p}<\infty ,\left( 0<p\le 1\right) \end{aligned}$$

for every p-atom a, where I denotes the support of the atom. If T is bounded from \(L_{\infty }\) to \(L_{\infty }\), then, for \(0<p\le 1,\)

$$\begin{aligned} \left\| TF\right\| _{p}\le c_{p}\left\| F\right\| _{H_{p}}. \end{aligned}$$

The proof of the next lemma can be found in Persson and Tephnadze [18]:

Lemma 3

Let \(n\in \mathbb {N}\), \(\left[ n\right] \ne \left| n\right| \), and \(x\in I_{\left[ n\right] +1}\left( e_{\left[ n\right] -1}+e_{\left[ n\right] }\right) \). Then

$$\begin{aligned} \left| nK_n(x)\right| =\left| \left( n-2^{\left| n\right| }\right) K_{n-2^{\left| n\right| }}(x)\right| \ge \frac{2^{2\left[ n\right] }}{4}. \end{aligned}$$

We note that if \([n]=0\), we have the set \(I_{2}\left( e_{0}\right) \).

We also need the following lemma (see [12]):

Lemma 4

Let \(n\ge 2^{M}\) and \(x\in I_{M}(e_{k}+e_{l}), \ k=0, \ldots ,M-1, \ l=k+1, \ldots ,M\). Then

$$\begin{aligned} \int _{I_{M}}\left| K_{n}\left( x+t\right) \right| d\mu \left( t\right) \le \frac{c2^{k+l}}{2^{2M}}. \end{aligned}$$

5 Proofs of the Theorems

Proof of Theorem 1

Since \(\sigma _{n}\) is bounded from \(L_{\infty }\) to \(L_{\infty }\) by Lemma 2, the proof will be complete, if we prove that

$$\begin{aligned} \int \limits _{{\overline{I}}}\left( \sup _{n\in \mathbb {N}} \frac{\left| \sigma _{n}a(x) \right| }{2^{\rho \left( n\right) \left( 1/p-2\right) }\varphi (\rho \left( n\right) )} \right) ^{p}d\mu \le c_{p}<\infty , \end{aligned}$$
(14)

for every p-atom a. We may assume that a be an arbitrary p-atom with support I, \(\mu \left( I\right) =2^{-M}\), and \(I=I_{M}\). It is easy to see that

$$\begin{aligned} \sigma _{n}a\left( x\right) =0, \ \text { when } \ n< 2^{M}. \end{aligned}$$

Therefore, we can suppose that \(n\ge 2^{M}\). Since \(\left\| a\right\| _{\infty }\le 2^{M/p}\), we find that

$$\begin{aligned}{} & {} \frac{\left| \sigma _{n}a\left( x\right) \right| }{2^{\rho \left( n\right) \left( 1/p-2\right) }\varphi (\rho \left( n\right) )} \nonumber \\\le & {} \frac{1}{2^{\rho \left( n\right) \left( 1/p-2\right) } \varphi (\rho \left( n\right) )} \left\| a\right\| _{\infty }\int _{I_{M}}\left| K_{n}\left( x+t\right) \right| d\mu \left( t\right) \nonumber \\\le & {} \frac{1}{2^{\rho \left( n\right) \left( 1/p-2\right) }\varphi (\rho \left( n\right) )} 2^{M/p}\int _{I_{M}}\left| K_{n}\left( x+t\right) \right| d\mu \left( t\right) . \end{aligned}$$
(15)

Let \(x\in I_{l+1}\left( e_{k}+e_{l}\right) ,\,0\le k<l< \left[ n\right] \le M\). Then \(x+t\in I_{l+1}\left( e_{k}+e_{l}\right) \) and if we apply (7), (8), and (9), then we get that

$$\begin{aligned} K_{n}\left( x+t\right) =0, \ \ \ \text { for } \ \ \ t\in I_{M} \end{aligned}$$

and from (15) it follows that

$$\begin{aligned} \frac{1}{2^{\rho \left( n\right) \left( 1/p-2\right) }\varphi (\rho \left( n\right) )} \left| \sigma _{n}a\left( x\right) \right| =0. \end{aligned}$$
(16)

Let

$$\begin{aligned} x\in I_{l+1}\left( e_{k}+e_{l}\right) ,\,\left[ n\right] \le k<l< M \text { or } \ k< \left[ n\right] \le l< M. \end{aligned}$$

Since \(\vert n\vert \ge M\) by using (15) and Lemma 4 we can conclude that

$$\begin{aligned} \frac{1}{2^{\rho \left( n\right) \left( 1/p-2\right) }\varphi (\rho \left( n\right) )} \left| \sigma _{n}a\left( x\right) \right|\le & {} \frac{c2^{M\left( 1/p-2\right) +k+l}}{2^{\rho \left( n\right) \left( 1/p-2\right) }\varphi (\rho \left( n\right) )} \nonumber \\= & {} \frac{c2^{[ n] \left( 1/p-2\right) }2^{M\left( 1/p-2\right) +k+l}}{2^{\left| n\right| \left( 1/p-2\right) }\varphi (\rho \left( n\right) )} \nonumber \\\le & {} \frac{c2^{M\left( 1/p-2\right) }}{2^{\left| n\right| \left( 1/p-2\right) }}\frac{2^{\left[ n\right] \left( 1/p-2\right) +k+l}}{\varphi (M-l)}\nonumber \\\le & {} \frac{c2^{M\left( 1/p-2\right) }}{2^{\left| n\right| \left( 1/p-2\right) }}\frac{2^{k+l\left( 1/p-1\right) }}{\varphi (M-l)}\nonumber \\\le & {} \frac{c2^{k+l\left( 1/p-1\right) }}{\varphi (M-l)}. \end{aligned}$$
(17)

By applying (16) and (17) for any \(x\in I_{l+1}\left( e_{k}+e_{l}\right) ,\,0\le k<l< M\) we find that

$$\begin{aligned}{} & {} \sup _{n\in \mathbb {N}}\frac{\left| \sigma _{n}a\left( x\right) \right| }{2^{\rho \left( n\right) \left( 1/p-2\right) }\varphi (\rho \left( n\right) )} \le \frac{c2^{k+l\left( 1/p-1\right) }}{\varphi (M-l)}. \end{aligned}$$
(18)

Let \(x\in I_{M}\left( e_{k}\right) ,\) \(0\le k< M.\) By using again (15) and Lemma 4 for \(k=l\) we can conclude that

$$\begin{aligned} \frac{\left| \sigma _{n}a\left( x\right) \right| }{2^{\rho \left( n\right) \left( 1/p-2\right) }\varphi (\rho \left( n\right) )}\le & {} c2^{M/p}\int _{I_{M}}\left| K_{n}\left( x+t\right) \right| d\mu \left( t\right) \\\le & {} c2^{M/p}\frac{2^k}{2^M}=c2^{k+M(1/p-1)}. \end{aligned}$$

and

$$\begin{aligned}{} & {} \sup _{n\in \mathbb {N}}\frac{\left| \sigma _{n}a\left( x\right) \right| }{2^{\rho \left( n\right) \left( 1/p-2\right) }\varphi (\rho \left( n\right) )} \le c2^{k+M(1/p-1)}. \end{aligned}$$
(19)

By combining (6), (18), and (19), we obtain that

$$\begin{aligned}{} & {} \int _{\overline{I_{M}}}\left( \sup _{n\in \mathbb {N}}\frac{\left| \sigma _{n}a\left( x\right) \right| }{2^{\rho \left( n\right) \left( 1/p-2\right) }\varphi (\rho \left( n\right) )} \right) ^{p}d\mu \left( x\right) \nonumber \\= & {} \overset{M-2}{\underset{k=0}{\sum }} \overset{M-1}{\underset{l=k+1}{\sum }} \int _{I_{l+1}\left( e_{k}+e_{l}\right) } \left( \sup _{n\in \mathbb {N}}\frac{\left| \sigma _{n}a\left( x\right) \right| }{2^{\rho \left( n\right) \left( 1/p-2\right) }\varphi (\rho \left( n\right) )} \right) ^{p}d\mu \left( x\right) \nonumber \\+ & {} \overset{M-1}{\underset{k=0}{\sum }}\int _{I_{M}\left( e_{k}\right) } \left( \sup _{n\in \mathbb {N}}\frac{\left| \sigma _{n}a\left( x\right) \right| }{2^{\rho \left( n\right) \left( 1/p-2\right) }\varphi (\rho \left( n\right) )} \right) ^{p}d\mu \left( x\right) \nonumber \\\le & {} c_{p}\overset{M-2}{\underset{k=0 }{\sum }}\overset{M-1}{\underset{l=k+1}{\sum }}\frac{1}{2^l}\frac{2^{pk+l\left( 1-p\right) }}{\varphi ^p(M-l)} +c_{p}\overset{M}{\underset{k=0}{\sum }}\frac{1}{2^M}2^{pk+M\left( 1-p\right) }\nonumber \\:= & {} I+II. \end{aligned}$$
(20)

Hence,

$$\begin{aligned} I\le & {} c_{p}\overset{M-2}{\underset{k=0 }{\sum }}2^{pk}\overset{M-1}{\underset{l=k+1}{\sum }}\frac{1}{2^{pl}\varphi ^p(M-l)} \end{aligned}$$
(21)
$$\begin{aligned}= & {} c_{p}\overset{M-2}{\underset{k=0 }{\sum }}2^{pk} \overset{[(k+M)/2]}{\underset{l=k+1}{\sum }}\frac{1}{2^{pl}\varphi ^p(M-l)} +c_{p}\overset{M-2}{\underset{k=0 }{\sum }}2^{pk} \overset{M-1}{\underset{l=[(k+M)/2]+1}{\sum }}\frac{1}{2^{pl}\varphi ^p(M-l)}\\:= & {} I_1+I_2. \end{aligned}$$

By using (10) for \(I_1\) we get that

$$\begin{aligned} I_1\le & {} c_{p}\overset{M-2}{\underset{k=0 }{\sum }}\frac{2^{pk}}{\varphi ^p([(M-k)/2])}\overset{[(k+M)/2]}{\underset{l=k+1}{\sum }}\frac{1}{2^{pl}}\\\le & {} c_{p}\overset{M-2}{\underset{k=0 }{\sum }}\frac{1}{\varphi ^p([(M-k)/2])}<c_p<\infty . \end{aligned}$$

For \(I_2\) we find that

$$\begin{aligned} I_2\le & {} c_{p}\overset{M-2}{\underset{k=0 }{\sum }}2^{pk} \overset{M-1}{\underset{l=[(k+M)/2]+1}{\sum }}\frac{1}{2^{pl}} \le c_{p}\overset{M-2}{\underset{k=0 }{\sum }}2^{pk} \frac{1}{2^{p[(k+M)/2]}}\\\le & {} c_{p}\overset{M-2}{\underset{k=0 }{\sum }} \frac{2^{pk/2}}{2^{pM/2}}<c_p<\infty . \end{aligned}$$

For II we can conclude that

$$\begin{aligned} II \le c_{p}\overset{M-2}{\underset{k=0 }{\sum }}\frac{2^{pk}}{2^{pM}}<c_{p}<\infty . \end{aligned}$$
(22)

By combining (20)-(22) we conclude that (14) holds so the proof is complete. \(\square \)

Proof of Theorem 2

In view of (11) we have that

$$\begin{aligned} \left( \overset{n_{k}-1}{\underset{s=1}{\sum }}\frac{1}{\varphi ^p \left( s\right) }\right) ^{1/p}\rightarrow \infty ,\quad \text {as} \quad k\rightarrow \infty . \end{aligned}$$
(23)

Set

$$\begin{aligned} f_{n_{k}}\left( x\right) =D_{2^{n_{k}+1}}\left( x\right) -D_{2^{{n_{k}}}}\left( x\right) ,{ \qquad }n_{k}\ge 3. \end{aligned}$$

It is evident that

$$\begin{aligned} {\widehat{f}}_{n_{k}}\left( i\right) =\left\{ \begin{array}{l} 1,\text { if }i=2^{n_{k}}, \ldots ,2^{n_{k}+1}-1, \\ 0, \ \text {otherwise}. \end{array} \right. \end{aligned}$$

Then we easily can derive that

$$\begin{aligned} S_{i}f_{n_{k}}\left( x\right) =\left\{ \begin{array}{l} D_{i}\left( x\right) -D_{2^{n_{k}}}\left( x\right) ,\text { if } i=2^{n_{k}}, \ldots ,2^{n_{k}+1}-1, \\ f_{n_{k}}\left( x\right) ,\text { if }i\ge 2^{n_{k}+1}, \\ 0, \qquad \text {otherwise}. \end{array} \right. \end{aligned}$$
(24)

Since

$$\begin{aligned} D_{j+2^{n_{k}}}\left( x\right) -D_{2^{n_{k}}}\left( x\right) =w_{2^{n_{k}}}D_{j}(x), { \qquad }\,j=1,2,..,2^{n_{k}}, \end{aligned}$$
(25)

from (7) it follows that

$$\begin{aligned} \left\| f_{n_{k}}\right\| _{H_{p}}= & {} \left\| \sup \limits _{n\in \mathbb {N}}S_{2^{n}}f_{n_{k}} \right\| _p =\left\| D_{2^{n_{k}+1}}-D_{2^{n_{k}}}\right\| _p\nonumber \\= & {} \left\| D_{2^{n_{k}}}\right\| _p = 2^{n_{k}(1-1/p)}. \end{aligned}$$
(26)

Let \(q_{n_k}^{s}\in \mathbb {N}\) be such that \(2^{n_k}\le q_{n_k}^{s}\le 2^{n_k+1}\) and \([q_{n_k}^{s}]=s\), where \(0\le s<n_k\). By applying (24) we can conclude that

$$\begin{aligned} \left| \sigma _{q_{n_k}^{s}}f_{n_{k}}\left( x\right) \right|= & {} \frac{1}{q_{n_k}^{s}}\left| \overset{q_{n_k}^{s}}{\underset{j=1}{\sum }}S_{j}f_{n_{k}}\left( x\right) \right| \\= & {} \frac{1}{q_{n_k}^{s}}\left| \overset{q_{n_k}^{s}}{ \underset{j=2^{n_{k}}+1}{\sum }}S_{j}f_{n_{k}}\left( x\right) \right| \\= & {} \frac{1}{ q_{n_k}^{s}}\left| \overset{q_{n_k}^{s}}{\underset{j=2^{n_{k}}+1}{\sum }}\left( D_{j}\left( x\right) -D_{2^{n_{k}}}\left( x\right) \right) \right| \\= & {} \frac{1}{q_{n_k}^{s}}\left| \overset{q_{n_k}^{s}-2^{n_k}}{\underset{j=1}{\sum }}\left( D_{j+2^{n_{k}}}\left( x\right) -D_{2^{n_{k}}}\left( x\right) \right) \right| . \end{aligned}$$

According to (25) we obtain that

$$\begin{aligned} \left| \sigma _{q_{n_k}^{s}}f_{n_{k}}\left( x\right) \right|= & {} \frac{1}{q_{n_k}^{s}}\left| \overset{q_{n_k}^{s}-2^{n_k}}{\underset{j=0}{\sum }}D_{j}\left( x\right) \right| \\= & {} \frac{q_{n_k}^{s}-2^{n_k}}{q_{n_k}^{s}} \left| K_{q_{n_k}^{s}-2^{n_k}}\left( x\right) \right| . \end{aligned}$$

Let \(x\in I_{s+1}\left( e_{s-1}+e_{s}\right) \). By using Lemma 3 we have that

$$\begin{aligned} \left| \sigma _{q_{n_k}^{s}}f_{n_k}\left( x\right) \right| \ge \frac{c2^{2s}}{2^{n_k}} \end{aligned}$$

and

$$\begin{aligned} \frac{\left| \sigma _{q_{n_k}^{s}}f_{n_{k}}\left( x\right) \right| }{2^{{(1/p-2)}\rho \left( q_{n_k}^{s}\right) }\varphi \left( \rho \left( q_{n_k}^{s}\right) \right) } \ge \frac{c_p2^{s/p}}{2^{n_{k}(1/p-1)}\varphi \left( n_{k}-s\right) }. \end{aligned}$$

Hence,

$$\begin{aligned}{} & {} \int _{G}\left( \sup _{k\in \mathbb {N}}\left| \frac{\left| \sigma _{q_{n_k}^{s}}f_{n_{k}}\left( x\right) \right| }{2^{{(1/p-2)}\rho \left( q_{n_k}^{s}\right) }\varphi \left( \rho \left( q_{n_k}^{s}\right) \right) }\right| \right) ^{p}d\mu \left( x\right) \\\ge & {} \frac{1}{2}\overset{n_{k}-1}{\underset{s=0}{\sum }} \int _{I_{s+1}\left( e_{s-1}+e_{s}\right) }\left( \frac{\left| \sigma _{q_{n^s_{k}}}f_{n_{k}}\left( x\right) \right| }{2^{{(1/p-2)}\rho \left( q_{n_k}^{s}\right) }\varphi \left( \rho \left( q_{n_k}^{s}\right) \right) }\right) ^{p}d\mu \left( x\right) \\\ge & {} c_p\overset{n_{k}-1}{\underset{s=0}{\sum }}\frac{1}{2^{s}}\frac{2^{s}}{2^{n_{k}(1-p)}\varphi ^p \left( n_k-s\right) }\\\ge & {} \frac{c_p }{2^{n_{k}(1-p)}}\overset{n_{k}}{\underset{s=1}{\sum }}\frac{1}{\varphi ^p \left( s\right) }. \end{aligned}$$

Finally, by using this estimate combined with (23) and (26) we find that

$$\begin{aligned}{} & {} \frac{\left( \int _{G}\left( \sup _{k\in \mathbb {N}}\sup _{0\le s<{n_k}}\left| \frac{\left| \sigma _{ q_{n_k}^{s}}f_{n_{k}}\left( x\right) \right| }{2^{{(1/p-2)}\rho \left( q_{n_k}^{s}\right) }\varphi \left( \rho ( q_{n_k}^{s})\right) }\right| \right) ^{p} d\mu \left( x\right) \right) ^{1/p}}{\left\| f_{n_{k}}\right\| _{H_p}} \\\ge & {} \frac{\left( \frac{c_p }{2^{n_{k}(1-p)}}\overset{n_{k}}{\underset{s=1}{\sum }}\frac{1}{\varphi ^p \left( s\right) }\right) ^{1/p}}{2^{n_k(1-1/p)}}\\\ge & {} c_p\left( \overset{n_{k}}{\underset{s=1}{\sum }}\frac{1}{\varphi ^p \left( s\right) }\right) ^{1/p}\rightarrow \infty ,\quad \text {as }\quad k\rightarrow \infty . \end{aligned}$$

The proof is complete. \(\square \)