1 Introduction

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

It is well-known that the Walsh system does not form a basis in the space \(L_1(G)\) (see e.g. [2, 30]). Moreover, there exists a function in the dyadic Hardy space \(H_{1}(G),\) such that the partial sums of f are not bounded in the \(L_{1}\)-norm. Uniform and pointwise convergence and some approximation properties of partial sums in \(L_{1}(G)\) norms were investigated by Avdispahić and Memić [1], Gát, Goginava and Tkebuchava [12, 13], Nagy [17], Onneweer [18] and Persson, Schipp, Tephnadze and Weisz [21]. Fine [9] obtained sufficient conditions for the uniform convergence which are completely analogous to the Dini–Lipschits conditions. Gulic̆ev [15] estimated the rate of uniform convergence of a Walsh–Fourier series2 by using Lebesgue constants and modulus of continuity. These problems for Vilenkin groups were investigated by Blatota, Nagy, Persson and Tephnadze [7] (see also [4,5,6]), Fridli [10] and Gát [11].

To study 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

$$\begin{aligned} n=\sum _{k=0}^{\infty }n_{j}2^{j}, \quad n_{j}\in Z_{2} ~(j\in \mathbb {N}), \end{aligned}$$
(1)

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

$$\begin{aligned} \left[ n\right] :=\min \{j\in \mathbb {N},n_{j}\ne 0\}, \quad \left| n\right| :=\max \{j\in \mathbb {N},n_{j}\ne 0\}, \quad \rho \left( n\right) =\left| n\right| -\left[ n\right] \nonumber \\ \end{aligned}$$
(2)

and

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

In particular, (see [8, 16, 22])

$$\begin{aligned} \frac{V\left( n\right) }{8}\le \Vert D_n\Vert _1\le V\left( n\right) \end{aligned}$$

from which it follows that, for any \(F\in L_1(G),\) there exists an absolute constant c such that the following inequality holds:

$$\begin{aligned} \left\| S_n F\right\| _1\le c{V\left( n\right) }\left\| F\right\| _1. \end{aligned}$$

Moreover, for any \(f\in H_1(G)\) (see [26])

$$\begin{aligned} \left\| S_{n}F\right\| _{H_{1}}\le c{V\left( n\right) }\left\| F\right\| _{H_{1}}. \end{aligned}$$

For \(0<p<1\) in Refs. [24, 25], the weighted maximal operator \(\overset{\sim }{S }^{*,p},\) defined by

$$\begin{aligned} \overset{\sim }{S }^{*,p}F:=\sup _{n\in \mathbb {N}}\frac{\left| S _{n}F\right| }{\left( n+1\right) ^{1/p-1} } \end{aligned}$$
(3)

was investigated and it was proved that the following inequalities hold:

$$\begin{aligned} \left\| \overset{\sim }{S }^{*}F\right\| _{p}\le c_{p}\left\| F\right\| _{H_{p}} \end{aligned}$$

Moreover, it was also proved that the rate of the sequence \(\{\left( n+1\right) ^{ 1/p-1}\}\) given in the denominator of (3) can not be improved.

In Refs. [26, 27] (see also [3]), it was proved that if \(F\in H_{p}(G),\) then there exists an absolute constant \(c_{p},\) depending only on p,  such that

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

which implies that

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

Moreover, if \(0<p<1,\) \(\left\{ n_{k}:\text { }k\ge 0\right\} \) is any increasing sequence of positive integers such that

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

and \(\Phi :\mathbb {N}_{+}\rightarrow [1,\infty )\) is any nondecreasing function, satisfying the condition

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

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

$$\begin{aligned} \underset{k\in \mathbb {N}}{\sup }\left\| \frac{S_{n_{k}}F}{\Phi \left( n_{k}\right) }\right\| _{\text {weak}-L_p}=\infty . \end{aligned}$$

In this paper, we prove that the weighted maximal operator of the partial sums of the Walsh–Fourier defined by

$$\begin{aligned} \sup _{n\in \mathbb {N}}\frac{\left| S_{n}F\right| }{2^{\rho \left( n\right) \left( 1/p-1\right) }} \end{aligned}$$

is bounded from the martingale Hardy space \(H_p(G)\) to the space \(\textrm{weak}-L_p(G),\) for \(0<p<1.\) We also prove the sharpness of this result (see Theorem 2). As a consequence, we obtain both some new and well-known results.

This paper is organized as follows: In order not to disturb our discussions later on some preliminaries are presented in Sect. 2. The main results and some of its consequences can be found in Sect. 3. The detailed proofs of the main results are given in Sect. 4.

2 Preliminaries

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

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

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

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

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

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

Then it is easy to prove that

$$\begin{aligned} \overline{I_{M}}=\overset{M-1}{\underset{s=0}{\bigcup }}I_{s}\backslash I_{s+1}. \end{aligned}$$
(4)

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

$$\begin{aligned} \left\| f\right\| _{p}^{p}:=\int _{G}\left| f\right| ^{p}\textrm{d}\mu \quad \text {and} \quad \left\| f\right\| _{\textrm{weak}-L_{p}}^{p}:=\sup _{\lambda>0}\lambda ^{p}\mu \left( f>\lambda \right) . \end{aligned}$$

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

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

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

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

The Walsh system is orthonormal and complete in \(L_{2}\left( G\right) \) (see e.g. [14, 22]).

If \(f\in L_{1}\left( G\right) \) we can establish the Fourier coefficients, the partial sums of the Fourier series, the Dirichlet kernels with respect to the Walsh system in the usual manner:

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

Recall that (see [19, 20, 22])

$$\begin{aligned} D_{2^{n}}\left( x\right) =\left\{ \begin{array}{ll} 2^{n}, &{} \,\text {if}{ }x\in I_{n} \\ 0, &{} \text {if}\,\, \ x\notin I_{n} \end{array} \right. \end{aligned}$$
(5)

and

$$\begin{aligned} D_{n}=w_{n}\overset{\infty }{\underset{k=0}{\sum }}n_{k}r_{k}D_{2^{k}}=w_{n} \overset{\infty }{\underset{k=0}{\sum }}n_{k}\left( D_{2^{k+1}}-D_{2^{k}}\right) , \quad \text {for } \ n=\overset{\infty }{\underset{i=0 }{\sum }}n_{i}2^{i}. \end{aligned}$$
(6)

Moreover, we have the following lower estimate (see [20]):

Lemma 1

Let \(n\in \mathbb {N}\) and \(\left[ n\right] \ne \left| n\right| .\) Then,

$$\begin{aligned} \left| D_n(x)\right| =\left| D_{n-2^{\left| n\right| }}(x)\right| \ge \frac{2^{\left[ n\right] }}{4}, \quad \textrm{for} \ \ x\in I_{\left[ n\right] +1}\left( e_{\left[ n\right] }\right) . \end{aligned}$$

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 \(F=\left( F_{n},n\in \mathbb {N}\right) \) the martingale with respect to \(\digamma _{n}\) \(\left( n\in \mathbb {N}\right) \) (see e.g. [28]).

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

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

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

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

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

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

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. [20, 23, 28])

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

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

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

The dyadic Hardy martingale spaces \(H_{p}(G)\) for \(0<p\le 1\) have an atomic characterization. Namely, the following holds (see [20, 28, 29]):

Lemma 2

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

$$\begin{aligned} \sum _{k=0}^{\infty }\mu _{k}S_{2^{n}}a_{k}=F_{n}, \quad \sum _{k=0}^{\infty }\left| \mu _{k}\right| ^{p}<\infty , \end{aligned}$$
(7)

Moreover, \( \left\| F\right\| _{H_{p}}\backsim \inf \left( \sum _{k=0}^{\infty }\left| \mu _{k}\right| ^{p}\right) ^{1/p}, \) where the infimum is taken over all decomposition of F of the form (7).

3 The Main Results with Applications

Our first main result reads:

Theorem 1

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

$$\begin{aligned} \widetilde{S }^{*,\nabla }F=\underset{n\in \mathbb {N}}{\sup } \frac{\left| S_{n}F\right| }{2^{\rho \left( n\right) \left( 1/p-1\right) }}, \end{aligned}$$
(8)

is bounded from the martingale Hardy space \({{H}_{p}}(G)\) to the space \(\text {weak}-L_p(G).\)

Our second main result shows that Theorem 1 can not be improved in general, because it is sharp in some special senses:

Theorem 2

(a) Let \(0<p<1,\) n be defined by (1), \(\rho \left( n\right) \) be defined by (2) and \(\widetilde{S }^{*,\nabla }\) is defined by (8). Then, there exists a sequence \(\{f_n, n\in \mathbb {N}\}\) of p-atoms, such that

$$\begin{aligned} \sup _{n\in \mathbb {N}}\frac{\left\| \widetilde{S }^{*,\nabla }f_n\right\| _{p}}{\left\| f_n \right\| _{H_p(G)}}=\infty . \end{aligned}$$

(b) Let \(0<p<1\), n be defined by (1) and \(\rho \left( n\right) \) be defined by (2). If \(\varphi :\mathbb {N}\rightarrow [1,\) \(\infty )\) is a nondecreasing function, satisfying the condition

$$\begin{aligned} \overline{\lim _{n\rightarrow \infty }}\frac{2^{\rho (n){(1/p-1)}}}{\varphi \left( n\right) }=\infty , \end{aligned}$$
(9)

then there exists a sequence \(\{f_n, n\in \mathbb {N}\}\) of p-atoms, such that

$$\begin{aligned} \sup _{n\in \mathbb {N}}\frac{\left\| \sup _{k\in \mathbb {N}}\frac{\vert S_kf_n\vert }{\varphi \left( k\right) }\right\| _{\text {weak}-L_p(G)}}{\left\| f_n \right\| _{H_p(G)}}=\infty . \end{aligned}$$

Theorem 1 implies the following result of Weisz [29] (see also [28]):

Corollary 1

Let \(0<p<1\) and \(f\in {{H}_{p}}\left( G \right) \). Then the maximal operator \(S ^{*,\triangle }\) defined by

$$\begin{aligned} S ^{*,\triangle }F:= \underset{n\in \mathbb {N}}{\sup }{\left| S_{2^{n}}F\right| } \end{aligned}$$

is bounded from the Hardy space \({{H}_{p}(G)}\) to the Lebesgue space \(\text {weak}-L_p(G)\) (and, thus, to the Lebesgue space \(L_p(G)\)).

Moreover, Theorems 1 and 2 imply the following results (see [20]):

Corollary 2

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

$$\begin{aligned} S^{*,\nabla }F:=\underset{k\in \mathbb {N}}{\sup } \left| S_{n_{k}}F\right| \end{aligned}$$

is bounded from the Hardy space \({{H}_{p}(G)}\) to the Lebesgue space \(\text {weak}-L_p(G)\) if and only if condition

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

is fulfilled.

Remark 1

The statement in Corollary 2 holds also if the space \(\textrm{weak}-L_p(G)\) is replaced by \(L_p(G)\).

Corollary 3

(a) Let \(0<p<1\) and \(f\in {{H}_{p}}\left( G \right) \). Then, the weighted maximal operator defined by

$$\begin{aligned} \underset{n\in \mathbb {N}}{\sup } \frac{\left| S_{2^{n}+2^{n/2}}F\right| }{2^{\frac{n}{2}\left( 1/p-1\right) }} \end{aligned}$$

is bounded from the Hardy space \({{H}_{p}}(G)\) to the Lebesgue space \(\text {weak}-L_p(G).\)

(b) (Sharpness) Let \(\varphi :\mathbb {N}\rightarrow [1,\) \(\infty )\) be a nondecreasing function, satisfying the condition

$$\begin{aligned} \overline{\lim _{n\rightarrow \infty }}\frac{2^{\frac{n}{2}\left( 1/p-1\right) }}{\varphi \left( n\right) }=\infty . \end{aligned}$$

Then, there exists sequence \(\{f_n, n\in \mathbb {N}\}\) of p-atoms, such that

$$\begin{aligned} \sup _{n\in \mathbb {N}}\frac{\left\| \frac{S_{2^{n}+2^{n/2}}f_n}{\varphi \left( 2^{n}+2^{n/2}\right) } \right\| _{\text {weak}-L_p(G)}}{\left\| f_n \right\| _{H_p(G)}}=\infty . \end{aligned}$$

Corollary 4

(a) Let \(0<p<1\) and \(f\in {{H}_{p}}\left( G \right) .\) Then, the weighted maximal operator defined by

$$\begin{aligned} \underset{n\in \mathbb {N}}{\sup } \frac{\left| S_{2^{n}+1}F\right| }{2^{n\left( 1/p-2\right) }} \end{aligned}$$

is bounded from the Hardy space \({{H}_{p}}(G)\) to the Lebesgue space \(\text {weak}-L_p(G).\)

(b) (Sharpness) Let \(\varphi :\mathbb {N}\rightarrow [1,\) \(\infty )\) is a nondecreasing function, satisfying the condition

$$\begin{aligned} \overline{\lim _{n\rightarrow \infty }}\frac{2^{n\left( 1/p-1\right) }}{\varphi \left( n\right) }=\infty . \end{aligned}$$

Then, there exists sequence \(\{f_n, n\in \mathbb {N}\}\) of p-atoms, such that

$$\begin{aligned} \sup _{n\in \mathbb {N}}\frac{\left\| \frac{S_{2^n+1}f_n}{\varphi \left( 2^n+1\right) } \right\| _{\text {weak}-L_p(G)}}{\left\| f_n \right\| _{H_p(G)}}=\infty . \end{aligned}$$

Finally, we note that Theorem 1 implies the following result of Tephnadze [27]:

Corollary 5

(a) Let \(0<p<1\) and \(f\in H_p(G).\) Then, the weighted maximal operator \(\overset{\sim }{S }^{*,p},\) defined by (3) is bounded from the martingale Hardy space \(H_p(G)\) to the Lebesgue space \(\text {weak}-L_p(G).\)

(b) Let \(\{\varphi _n\}\) be any nondecreasing sequence satisfying the condition

$$\begin{aligned} \overline{\lim _{n\rightarrow \infty }}\frac{\left( n+1\right) ^{1/p-1}}{ \varphi _n}=\infty . \end{aligned}$$

Then, there exists a martingale \(f\in H_{p}(G),\) such that

$$\begin{aligned} \sup _{n\in \mathbb {N}}\left\| \frac{S _{n}f}{\varphi _n }\right\| _{p}=\infty . \end{aligned}$$

4 Proofs of the Theorems

Proof of Theorem 1

Since \(\sigma _{n}\) is bounded from \(L_{\infty }\) to \( L_{\infty },\) by Lemma 2, the proof of Theorem 1 will be complete, if we prove that

$$\begin{aligned} t^p\mu \left\{ x\in \overline{I_M}:\widetilde{S }^{*,\nabla }a(x) \ge t\right\} \le c_p<\infty , \quad t\ge 0 \end{aligned}$$
(10)

for every p-atom a. In this paper, \(c_p\) (or \(C_p\)) denotes a positive constant depending only on p but which can be different in different places.

We may assume that a is an arbitrary p-atom, with support \(I, \ \mu \left( I\right) =2^{-M}\) and \(I=I_{M}.\) It is easy to see that \(S _{n}a\left( x\right) =0, \ \text { when } \ n< 2^{M}.\) Therefore, we can suppose that \(n\ge 2^{M}.\) Since \(\left\| a\right\| _{\infty }\le 2^{M/p},\) we obtain that

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

Let \(x\in I_{s}\backslash I_{s+1},\,0\le s< \left[ n\right] \le M\) or \(0\le s\le M<\left[ n\right] .\) Then, it is easy to see that \(x+t\in I_{s}\backslash I_{s+1}\) for \(t\in I_M\) and if we combine (5) and (6) we get that \(D_{n}\left( x+t\right) =0, \ \text { for } \ t\in I_{M}\) so that

$$\begin{aligned} \frac{\left| S_{n}a\left( x\right) \right| }{2^{\rho \left( n\right) \left( 1/p-1\right) }}=0. \end{aligned}$$
(11)

Let \(I_{s}\backslash I_{s+1},\,\left[ n\right] \le s\le M\) or \( \left[ n\right] \le s\le M.\) Then, it is easy to see that \(x+t\in I_{s}\backslash I_{s+1}\) for \(t\in I_M\) and if we again combine (5) and (6), we find that \(D_{n}\left( x+t\right) \le c2^s, \ \text { for } \ t\in I_{M}\) and

$$\begin{aligned} \frac{\left| S_{n}a\left( x\right) \right| }{2^{\rho \left( n\right) \left( 1/p-1\right) }}\le & {} c_p 2^{M/p}\frac{2^{s-M}}{2^{\rho \left( n\right) \left( 1/p-1\right) }} \nonumber \\ {}\le & {} c_p \frac{2^{[n]\left( 1/p-1\right) +s+M(1/p-1)}}{2^{\left| n\right| \left( 1/p-1\right) }} \le c_{p}2^{\left[ n\right] \left( 1/p-1\right) +s}\le c_{p}2^{s}. \end{aligned}$$
(12)

By applying (11) and (12) for any \(x\in I_{s}\backslash I_{s+1},\,0\le s< M,\) we find that

$$\begin{aligned} \widetilde{S }^{*,\nabla }a\left( x\right) =\sup _{n\in \mathbb {N}}\left( \frac{\left| S_{n}a\left( x\right) \right| }{2^{\rho \left( n\right) \left( 1/p-1\right) }} \right) \le C_{p}2^{s/p}. \end{aligned}$$
(13)

It immediately follows that for \(s\le M\), we have the following estimate

$$\begin{aligned} \widetilde{S}^{*,\nabla }a\left( x\right) \le C_p2^{M/p}\text { for any }x\in I_{s}\backslash I_{s+1}, \quad s=0,1,\ldots , M \end{aligned}$$

and also that

$$\begin{aligned} \mu \left\{ x\in I_{s}\backslash I_{s+1}:\widetilde{S }^{*,\nabla }a\left( x\right) > C_p2^{k/p}\right\} =0, \quad k=M, M+1,\ldots \end{aligned}$$
(14)

By combining (4) and (13), we get that

$$\begin{aligned} \left\{ x\in \overline{I_{N}}:\widetilde{S }^{*,\nabla }a\left( x\right) \ge C_p2^{k/p}\right\} \subset \bigcup _{s=k}^{M-1}\left\{ x\in I_{s}\backslash I_{s+1}:\widetilde{S }^{*,\nabla }a\left( x\right) \ge C_p 2^{k/p}\right\} \end{aligned}$$

and

$$\begin{aligned} \mu \left\{ x\in \overline{I_{M}}:\widetilde{S }^{*,\nabla }a\left( x\right) \ge C_p 2^{k/p}\right\} \le \overset{M-1}{\underset{s=k}{\sum }}\frac{1}{2^{s}}\le \frac{2}{2^{k}}. \end{aligned}$$
(15)

In view of (14) and (15), we can conclude that

$$\begin{aligned} 2^{k}\mu \left\{ x\in \overline{I_{N}}:\widetilde{S }^{*,\nabla }a\left( x\right) \ge {C_p}{2^{k/p}}\right\}<c_p<\infty , \end{aligned}$$

which shows that (10) holds and the proof of is complete. \(\square \)

Proof of Theorem 2

a) Set

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

It is evident that

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

Then, we have that

$$\begin{aligned} S_{i}f_{n_{k}}\left( x\right) =\left\{ \begin{array}{ll} D_{i}\left( x\right) -D_{2^{n_{k}}}\left( x\right) , &{} \text {if } \ i=2^{n_{k}},\ldots ,2^{n_{k}+1}-1, \\ f_{n_{k}}\left( x\right) ,&{} \text {if } \ i\ge 2^{n_{k}+1}, \\ 0,&{} \text {otherwise}. \end{array} \right. \end{aligned}$$
(16)

Since

$$\begin{aligned} D_{j+2^{n_{k}}}\left( x\right) -D_{2^{n_{k}}}\left( x\right) =w_{2^{n_{k}}}D_{j}(x),\quad \,j=1,2,\ldots ,2^{n_{k}}, \end{aligned}$$
(17)

from (5), it follows that

$$\begin{aligned} \left\| f_{n_{k}}\right\| _{H_{p}}= & {} \left\| \sup \limits _{n\in \mathbb {N}}S_{2^{n}}f_{n_{k}} \right\| _p =\left\| D_{2^{n_{k}+1}}-D_{2^{n_{k}}}\right\| _p\nonumber \\= & {} \left\| D_{2^{n_{k}}}\right\| _p \le 2^{n_{k}(1-1/p)}. \end{aligned}$$
(18)

Let \(q_{n_k}^s\in \mathbb {N}\) be such that

$$\begin{aligned} 2^{n_k}< q_{n_k}^s< 2^{n_k+1}, \quad \text {where } \ [q_{n_k}^s]=s, \quad s=0,\ldots , n_k-1. \end{aligned}$$

By combining (16) and (17), we can conclude that

$$\begin{aligned} \left| S _{q_{n_k}^s}f_{n_{k}}\left( x\right) \right| =\left| D_{q_{n_k}^s}\left( x\right) -D_{2^{n_{k}}}\left( x\right) \right| =\left| D_{q_{n_k}^s-2^{n_k}-1}\left( x\right) \right| \end{aligned}$$

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

$$\begin{aligned} \left| S_{q_{n_k}^s}f_{n_k}\left( x\right) \right| \ge c2^{s} \end{aligned}$$
(19)

so that

$$\begin{aligned} \frac{\left| S_{q_{n_k}^s}f_{n_{k}}\left( x\right) \right| }{2^{{(1/p-1)}\rho \left( q_{n_k}^s\right) }}\ge \frac{c2^{s/p}}{2^{{n_k}{(1/p-1)} }}. \end{aligned}$$

Hence,

$$\begin{aligned}{} & {} \int _{G}\left( \sup _{n\in \mathbb {N}}\frac{\left| S_{n}f_{n_{k}}\left( x\right) \right| }{2^{{(1/p-1)}\rho \left( n\right) }}\right) ^{p}\textrm{d}\mu \left( x\right) \nonumber \\{} & {} \quad \ge \overset{n_{k}-1}{\underset{s=0}{\sum }} \int _{I_{s+1}\left( e_{s}\right) }\left( \frac{\left| S _{q_{n^s_{k}}}f_{n_{k}}\left( x\right) \right| }{2^{{(1/p-1)}\rho \left( q_{n_k}^{s}\right) } }\right) ^{p}\textrm{d}\mu \left( x\right) \nonumber \\ {}{} & {} \quad \ge c_p\overset{n_{k}-1}{\underset{s=0}{\sum }}\frac{1}{2^{s}}\frac{2^{s}}{2^{n_{k}(1-p)}}\ge \frac{C_p }{2^{n_{k}(1-p)}}\overset{n_{k}}{\underset{s=1}{\sum }}1\ge \frac{C_p n_k }{2^{n_{k}(1-p)}}. \end{aligned}$$
(20)

Finally, by combining (18) and (20), we obtain that

$$\begin{aligned} \frac{\left( \int _{G}\left( \sup _{n\in \mathbb {N}}\frac{\left| S_{n}f_{n_{k}}\left( x\right) \right| }{2^{{(1/p-1)}\rho \left( n\right) }}\right) ^{p}\textrm{d}\mu \left( x\right) \right) ^{1/p}}{\left\| f_{n_{k}}\right\| _{H_p}} \ge \frac{{\left( \frac{C_pn_k}{2^{n_{k}(1-p)}}\right) }^{1/p}}{2^{n_k(1-1/p)}}\ge c_p n^{1/p}_k\rightarrow \infty ,\quad \text {as}\quad k\rightarrow \infty , \end{aligned}$$

so the proof of part a) is complete.

(b) Under condition (9), we can choose \(q^{s_k}_{n_k}\in \mathbb {N}\) for some \(0\le s_k <n_k\) such that

$$\begin{aligned} 2^{n_k}< q_{n_k}^{s_k}< 2^{n_k+1}, \quad \text {where } \ [q_{n_k}]=s_k \end{aligned}$$

and

$$\begin{aligned} {\lim _{k\rightarrow \infty }}\frac{2^{\rho (q_{n_k}^{s_k}){(1/p-1)}}}{\varphi \left( q_{n_k}^{s_k}\right) }=\infty . \end{aligned}$$

Let \(x\in I_{s_k+1}\left( e_{s_k}\right) \). By using (19) we get that

$$\begin{aligned} \left| S_{q^{s_k}_{n_k}}f_{n_k}\left( x\right) \right| \ge c2^{s_k} \end{aligned}$$

so that

$$\begin{aligned} \frac{\left| S_{q^{s_k}_{n_k}}f_{n_{k}}\left( x\right) \right| }{ \varphi \left( q^{s_k}_{n_k}\right) }\ge \frac{c2^{s_k}}{\varphi \left( q^{s_k}_{n_k}\right) }. \end{aligned}$$

Hence, we find that

$$\begin{aligned}{} & {} \mu \left\{ x\in G:\frac{\left| S_{q^{s_k}_{n_k}}f_{{n_{k}}}\left( x\right) \right| }{\varphi \left( q^{s_k}_{n_k}\right) }\ge \frac{c2^{s_k}}{\varphi \left( q^{s_k}_{n_k}\right) }\right\} \ge \mu \left( I_{s_k+1}(e_{s_k})\right) >c/2^{s_k}. \end{aligned}$$
(21)

By combining (18) and (21), we get that

$$\begin{aligned}{} & {} \frac{\frac{c2^{s_k}}{\varphi \left( q_{n_k}^{s_k}\right) }\left( \mu \left\{ x\in G:\frac{\left| S_{q^{s_k}_{n_k}}f_{{n_{k}}}\left( x\right) \right| }{\varphi \left( q_{n_k}^{s_k}\right) }\ge \frac{c2^{s_k}}{\varphi \left( q_{n_k}^{s_k}\right) }\right\} \right) ^{1/p}}{\left\| f_{n_{k}} \right\| _{H_{p}(G)}} \\{} & {} \quad \ge \frac{c_p2^{s_k}}{\varphi \left( q_{n_k}^{s_k}\right) 2^{n_k{(1-1/p)}}}\frac{1}{2^{s_k/p}}\\{} & {} \quad = \frac{c_p2^{{n_k}{(1/p-1)}}}{2^{{s_k}{(1/p-1)}}\varphi \left( q_{n_k}^{s_k}\right) }\\{} & {} \quad = \frac{c_p2^{\rho \left( q_{n_k}^{s_k}\right) (1/p-1)}}{\varphi \left( q_{n_k}^{s_k}\right) } \rightarrow \infty \,\quad \hbox { as }\quad k\rightarrow \infty , \end{aligned}$$

so also part (b) is proved and the proof is complete. \(\square \)