1 Introduction

It is well-known that the Walsh system does not form a basis in the space \(L_{1}\) (see e.g. [2, 27]). Moreover, there exists a function f in the dyadic Hardy space \(H_{1},\) such that the partial sums of f are not bounded in \(L_{1}\)-norm, but the partial sums \(S_{n}\) of the Walsh–Fourier series of a function \(f\in L_{1}\) convergence in measure (see [9, 12]). Uniform and pointwise convergence and some approximation properties of partial sums in \(L_{1}\) norm were investigated by Onneweer [16], Goginava [10], Goginava and Tkebuchava [11], Nagy [15], Avdispahić and Memić [1], Persson et al. [18]. Fine [6] obtained sufficient conditions for the uniform convergence, which are complete analogies with the Dini-Lipschits conditions. Guličev [13] estimated the rate of uniform convergence of a Walsh–Fourier series by using Lebesgue constants

$$\begin{aligned} L(n):=\Vert D_n\Vert _1 \end{aligned}$$

and modulus of continuity. These problems for Vilenkin groups were considered by Blahota [3], Fridli [7] and Gát [8].

Above, and in the sequel, all used notations can be found in Sect. 2. For example, the notations \(D_n\) and \(S_n\) are given in (6).

To study convergence of subsequences of partial sums in the martingale Hardy spaces \(H_p(G)\) for \(0<p\le 1,\) the central role plays uniquely expression of any natural number \(n\in \mathbb {N}\)

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

where only a finite 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

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

and

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

In particular, (see [5, 14, 19])

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

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

$$\begin{aligned} \left\| S_n F\right\| _1\le c{V\left( n\right) }\left\| F\right\| _1. \end{aligned}$$
(2)

Moreover, for any \(f\in H_1,\)

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

In [23] and [24] it was proved that if \(0<p<1\) and \(F\in H_{p},\) then there exists an absolute constant \(c_{p},\) depending only on p,  such that

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

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 \(F\in H_{p},\) such that

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

For \(0<p<1\) in [21, 22] 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 estimate holds:

$$\begin{aligned} \left\| \overset{\sim }{S }^{*}F\right\| _{p}\le c_{p}\left\| F\right\| _{H_{p}}. \end{aligned}$$
(4)

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

For \(p=1\) analogical results for the maximal operator \(\overset{\sim }{S}^{*},\) defined by

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

was proved in [22].

One main aim of this paper is to generalize the estimate (4) for \(f\in H_p(G),\) \(0<p<1.\) Our main idea is to investigate much more general maximal operators by replacing the weights \(\left( n+1\right) ^{1/p-1} \) in (3) by more general “optimal” weights

$$\begin{aligned} 2^{\rho \left( n\right) \left( 1/p-1\right) } (\varphi (\rho \left( n\right) )), \end{aligned}$$

where \(\varphi :\mathbb {R}_+\rightarrow \mathbb {R}_+\) 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 \(0<p<1.\) As a consequence we obtain some new and well-known results. In particular, we prove that the maximal operator \({\widetilde{S}}^{*,\nabla },\) defined by

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

is bounded from the Hardy space \({{H}_{p}}(G)\) to the Lebesgue space \({{L}_{p}}(G)\) for any \(\varepsilon >0\) and 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 its consequences can be found in Sect. 3. The detailed proofs 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 \(x:=(x_{0},x_{1},...,x_{j},...),\) where \( x_{k}=0\vee 1.\)

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

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

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

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

Then it is easy to show that

$$\begin{aligned} \overline{I_{M}}=\overset{M-1}{\underset{s=0}{\bigcup }}I_{s}\backslash I_{s+1}. \end{aligned}$$
(5)

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

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

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

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

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

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

The Walsh system is orthonormal and complete in \(L_{2}\left( G\right) \) (see e.g. [19]).

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:

$$\begin{aligned} \widehat{f}\left( k\right):= & {} \int _{G}fw_{k}d\mu \,\,\,\,\left( k\in \mathbb {N }\right) , \nonumber \\ S_{n}f:= & {} \sum _{k=0}^{n-1}\widehat{f}\left( k\right) w_{k},\\ D_{n}:= & {} \sum _{k=0}^{n-1}w_{k\text { }}\,\,\,\left( n\in \mathbb {N}_{+}\right) . \nonumber \end{aligned}$$
(6)

Recall that (see [17, 19])

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

and

$$\begin{aligned} D_{n}=w_{n}\overset{\infty }{\underset{k=0}{\sum }}n_{k}r_{k}D_{2^{k}}=w_{n} \overset{\infty }{\underset{k=0}{\sum }}n_{k}\left( D_{2^{k+1}}-D_{2^{k}}\right) ,\text { for \ }n=\overset{\infty }{\underset{i=0 }{\sum }}n_{i}2^{i}. \end{aligned}$$
(8)

Moreover, we have the following lower estimate (see [17]):

Lemma 1

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

$$\begin{aligned} \left| S_n(x)\right| =\left| S_{n-2^{\left| n\right| }}(x)\right| \ge {2^{\left[ n\right] }}, \ \ \ \text {for } \ \ \ x\in I_{[n]}\backslash I_{[n]+1}. \end{aligned}$$

The \(\sigma \)-algebra generated by the intervals \(\left\{ I_{n}\left( x\right) :x\in G\right\} \) will be denoted by \(\zeta _{n}\left( n\in \mathbb { N}\right) .\) It is easy to see that

$$\begin{aligned} I_{n+1}\left( x\right) \subset I_{n}\left( x\right) \ \text { and } \ I_{n}\left( x\right) =\bigcup _{x_n=0}^{m_n-1} I_{n+1}\left( x\right) , \ \text { for any } \ x\in G \ \text { and } \ n\in \mathbb {N}. \end{aligned}$$

It follows that

$$\begin{aligned} \zeta _{n}\subset \zeta _{n+1}\left( n\in \mathbb {N}\right) . \end{aligned}$$

Denote by \(F=\left( F_{n},n\in \mathbb {N}\right) \) the martingale with respect to \(\digamma _{n}\) \(\left( n\in \mathbb {N}\right) \) (for details see e.g. [25]).

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

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

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

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

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 (for details see e.g. [17, 20] and [25])

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

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

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

The dyadic Hardy martingale spaces \(H_{p}\) for \(0<p\le 1\) have an atomic characterization. Namely, the following theorem holds (see [17, 25, 26]):

Lemma 2

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 \( \left( a_{k},\text { }k\in \mathbb {N}\right) \) of p-atoms and a sequence \( \left( \mu _{k},k\in \mathbb {N}\right) \) of real numbers such that for every \(n\in \mathbb {N}\)

$$\begin{aligned} \qquad \sum _{k=0}^{\infty }\mu _{k}S_{2^{n}}a_{k}=F_{n}, \end{aligned}$$
(9)

where

$$\begin{aligned} \sum _{k=0}^{\infty }\left| \mu _{k}\right| ^{p}<\infty . \end{aligned}$$

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 (9).

3 The main results

Our first main result reads:

Theorem 1

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

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

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

Theorem 1 can be of special interest even if we restrict to subsequences.

Corollary 1

Let \(0<p<1\), \(f\in {{H}_{p}}\left( G \right) \), \(\varphi :\mathbb {N}_+\rightarrow \mathbb {R}_+\) 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 \({\widetilde{S}}^{*,\nabla },\) defined by

$$\begin{aligned} {\widetilde{S}}^{*,\nabla }F=\underset{k\in \mathbb {N}}{\sup } \frac{\left| S_{n_k}F\right| }{2^{\rho \left( n_k\right) \left( 1/p-1\right) }\varphi (\rho \left( n_k\right) )}, \end{aligned}$$
(11)

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

We also prove sharpness of Theorem 1:

Theorem 2

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

$$\begin{aligned} \sum _{n=1}^{\infty }\frac{1}{\varphi ^p(n)}=\infty . \end{aligned}$$
(12)

Then there exists p-atoms \(a_{k},\) such that

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

If we take

$$\begin{aligned} \varphi (n)=\left( n\log ^{1+\varepsilon } n\right) ^{1/p}, \ \text { for any} \ \varepsilon >0, \end{aligned}$$

we get that condition (10) is fulfilled, on the other hand, if we take

$$\begin{aligned} \varphi (n)=\left( n\log n\right) ^{1/p}, \end{aligned}$$

then condition (12) holds true. Hence, Theorems 1 and 2 imply the following sharp result:

Corollary 2

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

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

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

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

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

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

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

and

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

Let

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

Then, the maximal operator (11) can not be estimated by

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

Hence, Theorem 1 and Remark 1 and Theorem proved in [21, 22] follows that if \(0<p<1\), \(f\in {{H}_{p}}\left( G \right) \) and \(\varphi :\mathbb {N}_+\rightarrow \mathbb {R}_+\) be any nonnegative and nondecreasing function satisfying the condition (10), then the weighted maximal operator \({\widetilde{S}}^{*,\nabla },\) defined by

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

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

Now, we formulate a result proved in [22], which follows from Theorems 1 and 2:

Corollary 3

a) Let \(0<p\le 1\) and \(\left\{ \alpha _{k},k\in \mathbb {N}\right\} \) be a subsequence of positive numbers such that \( \sup _{k\in \mathbb {N}}\rho \left( \alpha _{k}\right) <\infty . \) Then the maximal operator \({\widetilde{S}}^{*,\vartriangle }\) defined by

$$\begin{aligned} {\widetilde{S}}^{*,\vartriangle }f:=\sup _{k\in \mathbb {N}}\left| S_{\alpha _{k}}f\right| \end{aligned}$$
(13)

is bounded from the Hardy space \(H_{p}(G)\) to the Lebesgue space \(L_{p}(G).\)

b) Let \(0<p<1\) and \(\left( \alpha _{k},k\in \mathbb {N}\right) \) be a subsequence of positive numbers satisfying the condition \( \sup _{k\in \mathbb {N}}\rho \left( \alpha _{k}\right) =\infty . \) Then the maximal operator \({\widetilde{S}}^{*,\vartriangle }\) defined by (13) is not bounded from the Hardy space \(H_{p}(G)\) to the Lebesgue space \(L_{p}(G).\)

4 Proofs of the Theorems

Proof of Theorem 1

By using Lemma 2 the proof of Theorem 1 will be complete, if we prove that

$$\begin{aligned} \int _{G}\left| {\widetilde{S}}^{*,\nabla }a(x) \right| ^{p}d\mu \left( x\right) \le c<\infty , \end{aligned}$$
(14)

for every p-atom a,  with support I and \(\mu \left( I\right) =2^{-M}\). We may assume that this arbitrary p-atom a has support \(I=I_{M}.\) It is easy to see that \(S_{n}a\left( x\right) =0,\) when \(n\le 2^{M}\). Therefore, we can suppose that \(n>2^{M}\). Since \(\left\| a\right\| _{\infty }\le 2^{M/p}\) we find that

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

Let \(x\in I_{M}\). Since \(V\left( n\right) \le \rho \left( n\right) +2,\) by applying (2) we get that

$$\begin{aligned} \left| \frac{S_{n}a(x)}{ 2^{\left( 1/p-1\right) \rho \left( n\right) }\varphi (\rho \left( n\right) )} \right|\le & {} 2^{M/p}\frac{1}{ 2^{\left( 1/p-1\right) \rho \left( n\right) }\varphi (\rho \left( n\right) )}V\left( n\right) \\\le & {} 2^{M/p}\rho \left( n\right) \frac{1}{ 2^{\left( 1/p-1\right) \rho \left( n\right) }\varphi (\rho \left( n\right) )}\le c 2^{M/p} \end{aligned}$$

so that

$$\begin{aligned} \int _{I_{M}}\left( \sup _{n\in \mathbb {N}}\left| \frac{S_{n}a(x)}{ 2^{\left( 1/p-1\right) \rho \left( n\right) }\varphi (\rho \left( n\right) )}\right| \right) ^{p}d\mu (x)<c_{p}<\infty . \end{aligned}$$
(16)

Let \(t\in I_{M}\) and \(x\in I_{s}\backslash I_{s+1}, \ 0\le s\le M-1<[ n] \ \text { or } \ 0\le s<[ n] \le M-1.\) Then \(x+t\) \(\in I_{s}\backslash I_{s+1}\). By using (8) we get that \(D_{n}\left( x+t\right) =0\) and

$$\begin{aligned} \left| \frac{S_{n}a(x)}{ 2^{\left( 1/p-1\right) \rho \left( n\right) }\varphi (\rho \left( n\right) )} \right| =0. \end{aligned}$$
(17)

Let \(x\in I_{s}\backslash I_{s+1}, \ [ n] \le s\le M-1.\) Then \(x+t\in I_{s}\backslash I_{s+1},\) for \(t\in I_{M}\). By using (8) we find that

$$\begin{aligned} \left| D_{n}\left( x+t\right) \right| \le \sum _{j=0}^{s}n_{j}2^{j}\le c2^{s}. \end{aligned}$$

Hence, by applying (15) we get that

$$\begin{aligned}{} & {} \left| \frac{S_{n}a(x)}{ 2^{\left( 1/p-1\right) \rho \left( n\right) }\varphi (\rho \left( n\right) )} \right| \nonumber \\{} & {} \quad \le \frac{c}{ 2^{\left( 1/p-1\right) \rho \left( n\right) }\varphi (\rho \left( n\right) )}2^{M/p}\frac{2^{s} }{2^{M}} \le \frac{c2^{M(1/p-1)}}{2^{\vert n\vert (1/p-1)}}\frac{2^{[ n] \left( 1/p-1\right) }2^{s}}{\varphi (\rho \left( n\right) )}. \end{aligned}$$
(18)

By now using (18) for \(0<[n]<s/2\) we can conclude that

$$\begin{aligned} \left| \frac{S_{n}a(x)}{ 2^{\left( 1/p-1\right) \rho \left( n\right) }\varphi (\rho \left( n\right) )} \right|\le & {} \frac{c2^{M(1/p-1)}}{2^{\vert n\vert (1/p-1)}}\frac{2^{[ n] \left( 1/p-1\right) }2^{s}}{\varphi (\rho \left( n\right) )} \nonumber \\ {}\le & {} \frac{2^{ (s/2)\left( 1/p-1\right) }2^{s}}{\varphi (\rho \left( n\right) )}\le 2^{(s/2)(1/p+1)}. \end{aligned}$$
(19)

Moreover, according to (18) for \(s/2\le [n]\le s\) we have that

$$\begin{aligned} \left| \frac{S_{n}a(x)}{ 2^{\left( 1/p-1\right) \rho \left( n\right) }\varphi (\rho \left( n\right) )} \right|\le & {} \frac{c2^{M(1/p-1)}}{2^{\vert n\vert (1/p-1)}}\frac{2^{s/p}}{\varphi (\rho \left( n\right) )} \nonumber \\ {}\le & {} \frac{2^{s/p}}{\varphi (\rho \left( n\right) )}\le \frac{2^{s/p}}{\varphi (M-s)}. \end{aligned}$$
(20)

By combining (17), (19) and (20), for all \( x\in I_{s}\backslash I_{s+1}, \ 0 \le s\le M-1\) we get that

$$\begin{aligned} \ \ \ {\widetilde{S}}^{*,\nabla }a(x)= \sup _{n\in \mathbb {N}}\left| \frac{S_{n}a(x)}{ 2^{\left( 1/p-1\right) \rho \left( n\right) }\varphi (\rho (n))} \right| \le c2^{(s/2)(1/p+1)}+\frac{c2^{s/p}}{\varphi ( M-s)}. \end{aligned}$$
(21)

By now combining (5) and (21) we obtain that

$$\begin{aligned}{} & {} \int _{\overline{I_{M}}}\left| {\widetilde{S}}^{*,\nabla }a(x) \right| ^{p}d\mu \left( x\right) \nonumber \\{} & {} \quad \le c_p\overset{M-1}{\underset{s=0 }{\sum }} \int _{I_{s}\backslash I_{s+1}}\left| 2^{(s/2)(1/p+1)}+ \frac{2^{s/p}}{\varphi (M-s)}\right| ^{p}d\mu \left( x\right) \nonumber \\ {}{} & {} \quad \le c_p\overset{M-1}{\underset{s=0 }{\sum }} \int _{I_{s}\backslash I_{s+1}}\left| 2^{(s/2)(1/p+1)}\right| ^{p}d\mu \left( x\right) +c_p\overset{M-1}{\underset{s=0 }{\sum }} \int _{I_{s}\backslash I_{s+1}}\left| \frac{2^{s/p}}{\varphi (M-s)}\right| ^{p}d\mu \left( x\right) \nonumber \\ {}{} & {} \quad \le c_p\overset{M-1}{\underset{s=0 }{\sum }} \frac{1}{2^s} 2^{(s/2)(1+p)}+c_p\overset{M-1}{\underset{s=0 }{\sum }} \frac{1}{2^s} \frac{2^{s}}{\varphi ^p(M-s)} \nonumber \\ {}{} & {} \quad \le c_p\overset{M-1}{\underset{s=0 }{\sum }} 2^{(s/2)(p-1)}+c_p\overset{M-1}{\underset{s=0 }{\sum }}\frac{1}{\varphi ^p(M-s)}\le c_{p}<\infty . \end{aligned}$$
(22)

By combining (16) and (22) we obtain that (14) holds and the proof is complete.   \(\square \)

Proof of Theorem 2

In view of the condition (12) we have that

$$\begin{aligned} \left( \overset{n_{k}-1}{\underset{s=0}{\sum }}\frac{1}{\varphi ^p \left( s\right) }\right) ^{1/p}\rightarrow \infty ,\quad \text {as \quad }k\rightarrow \infty . \end{aligned}$$
(23)

Set

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

It is evident that

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

Then we easily can derive that

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

Since

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

from (7) it follows that

$$\begin{aligned} \left\| f_{n_{k}}\right\| _{H_{p}}= & {} \left\| \sup \limits _{n\in \mathbb {N}}S_{2^{n}}f_{n_{k}} \right\| _p =\left\| D_{2^{n_{k}+1}}-D_{2^{n_{k}}}\right\| _p \nonumber \\= & {} \left\| D_{2^{n_{k}}}\right\| _p \le 2^{n_{k}(1-1/p)}. \end{aligned}$$
(25)

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

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

By applying (24) we can conclude that

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

Let \(x\in I_{s}\backslash I_{s+1}\). By using Lemma 1 we obtain that

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

and

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

Hence,

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

Finally, by combining (23) and (25) we find that

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

The proof is complete. \(\square \)