An Analogy of the Carleson–Hunt Theorem with Respect to Vilenkin Systems

Abstract

In this paper we discuss and prove an analogy of the Carleson–Hunt theorem with respect to Vilenkin systems. In particular, we use the theory of martingales and give a new and shorter proof of the almost everywhere convergence of Vilenkin–Fourier series of $f\in L_p(G_m)$ for $p>1$ in case the Vilenkin system is bounded. Moreover, we also prove sharpness by stating an analogy of the Kolmogorov theorem for $p=1$ and construct a function $f\in L_1(G_m)$ such that the partial sums with respect to Vilenkin systems diverge everywhere.

1 Introduction

In 1947 Vilenkin [61,62,63] investigated a group $G_m$, which is a direct product of the additive groups $Z_{m_k}:=\{0,1,\ldots ,m_k-1\}$ of integers modulo $m_k$, where $m:=(m_0,m_1,\ldots )$ are positive integers not less than 2, and introduced the Vilenkin systems $\{{\psi }_j\}_{j=0}^{\infty }$. These systems include as a special case the Walsh system.

The classical theory of Hilbert spaces (for details see e.g the books [58, 60]) says that if we consider the partial sums $S_nf:=\sum _{k=0}^{n-1}{\widehat{f}}\left( k\right) \psi _k,$ with respect to Vilenkin systems, then

$$\begin{aligned} \left\| S_n f\right\| _2\le \left\| f\right\| _2. \end{aligned}$$

In the same year Schipp [45], Simon [51] and Young [67] (see also the book [49]) generalized this inequality for $1<p<\infty$: there exists an absolute constant $c_p,$ depending only on p,  such that

$$\begin{aligned} \left\| S_n f\right\| _p\le c_p\left\| f\right\| _p, \, \text {when} \, f \in L_p(G_m). \end{aligned}$$

From this it follows that for every $f\in L_p(G_m)$ with $1<p \le \infty$,

$$\begin{aligned} \left\| S_{n}f-f\right\| _{p}\rightarrow 0, \, \text {as} \, n\rightarrow \infty . \end{aligned}$$

The boundedness does not hold for $p=1 ,$ but Watari [64] (see also Gosselin [23] and Young [67]) proved that there exists an absolute constant c such that, for $n =1,2,\ldots ,$ the weak type estimate holds:

$$\begin{aligned} y\mu \left\{ \vert S_n f\vert>y \right\} \le c\left\| f\right\| _{1},\quad \,\, f\in L_1(G_m),\quad \, \, y>0. \end{aligned}$$

The almost-everywhere convergence of Fourier series for $L_2$ functions was postulated by Luzin [39] in 1915 and the problem was known as Luzin’s conjecture. Carleson’s theorem is a fundamental result in mathematical analysis establishing the pointwise (Lebesgue) almost everywhere convergence of Fourier series of $L_2$ functions, proved by Carleson [10] in 1966. The name is also often used to refer to the extension of the result by Hunt [26] which was given in 1968 to $L_p$ functions for $p\in (1, \infty )$ (also known as the Carleson–Hunt theorem).

Carleson’s original proof is exceptionally hard to read, and although several authors have simplified the arguments there are still no easy proofs of his theorem. Expositions of the original Carleson’s paper were published by Kahane [28], Mozzochi [40], Jorsboe and Mejlbro [27] and Arias de Reyna [43]. Moreover, Fefferman [16] published a new proof of Hunt’s extension, which was done by bounding a maximal operator of partial sums

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

This, in turn, inspired a much simplified proof of the $L_2$ result by Lacey and Thiele [35], explained in more detail in Lacey [33]. The books Fremlin [17] and Grafakos [24] also give proofs of Carleson’s theorem. An interesting extension of Carleson–Hunt result much more closer to $L_1$ space then $L_p$ for any $p>1$ was done by Carleson’s student Sjölin [56] and later on, by Antonov [2]. Already in 1923, Kolmogorov [31] showed that the analogue of Carleson’s result for $L_1$ is false by finding such a function whose Fourier series diverges almost everywhere (improved slightly in 1926 to diverging everywhere). This result indeed inspired many authors after Carleson proved positive results in 1966. In 2000, Kolmogorov’s result was improved by Konyagin [32], by finding functions with everywhere-divergent Fourier series in a space smaller than $L_1$, but the candidate for such a space that is consistent with the results of Antonov and Konyagin is still an open problem.

The famous Carleson theorem was very important and surprising when it was proved in 1966. Since then this interest has remained and a lot of related research has been done. In fact, in recent years this interest has even been increased because of the close connections to e.g. scattering theory [41], ergodic theory [14, 15], the theory of directional singular integrals in the plane [4, 11, 13, 34] and the theory of operators with quadratic modulations [36]. We refer to [33] for a more detailed description of this fact. These connections have been discovered from various new arguments and results related to Carleson’s theorem, which have been found and discussed in the literature. We mean that these arguments share some similarities, but each of them has also a distinct new idea behind, which can be further developed and applied. It is also interesting to note that, for almost every specific application of Carleson’s theorem in the aforementioned fields, mainly only one of these new arguments was used.

The analogue of Carleson’s theorem for Walsh system was proved by Billard [5] for $p=2$ and by Sjölin [55] for $1<p<\infty$, while for bounded Vilenkin systems by Gosselin [23]. Schipp [46, 47, 49] investigated the so called tree martingales, i.e., martingales with respect to a stochastic basis indexed by a tree, and generalized the results about maximal function, quadratic variation and martingale transforms to these martingales (see also [48, 65]). Using these results, he gave a proof of Carleson’s theorem for Walsh–Fourier series. A similar proof for bounded Vilenkin systems can be found in Schipp and Weisz [48, 65]. In each proof, they show that the maximal operator of the partial sums is bounded on $L_p(G_m)$, i.e., there exists an absolute constant $c_p$ such that

$$\begin{aligned} \left\| S^{*}f\right\| _p\le c_p\left\| f\right\| _p,\quad \text { as } f\in L_p(G_m),\quad 1<p<\infty . \end{aligned}$$

Recent proof of almost everywhere convergence of Walsh–Fourier series was given by Demeter [12] in 2015. By using some methods of martingale Hardy spaces, almost everywhere convergence of subsequences of Vilenkin–Fourier series was considered in [8]. Antonov [3] proved that for $f\in L_1(\log ^+L)(\log ^+\log ^+\log ^+L)(G_m)$ its Walsh–Fourier series converges a.e. Similar result for the bounded Vilenkin systems was proved by Oniani [42]. However, there exists a function from $L_1(\log ^+\log ^+L)(G_m)$ whose Vilenkin–Fourier series diverges everywhere, where in this result $G_m$ is a general (not necessary “bounded”) Vilenkin group (see Tarkaev [59]).

Stein [57] constructed an integrable function whose Walsh–Fourier series diverges almost everywhere. Later Schipp [44, 49] proved that there exists an integrable function whose Walsh–Fourier series diverges everywhere. Kheladze [29, 30] proved that for any set of measure zero there exists a function in $f\in L_p(G_m)$ $(1<p<\infty )$ whose Vilenkin–Fourier series diverges on the set, while the result for continuous or bounded function was proved by Harris [25] or Bitsadze [6]. Moreover, Simon [53] constructed an integrable function such that its Vilenkin–Fourier series diverges everywhere. Bochkarev [9] considered rearrangements of Vilenkin–Fourier series of bounded type.

It is not known whether Carleson’s theorem holds for unbounded Vilenkin systems. However, some theorems were proved for unbounded Vilenkin systems by Gát [18,19,20,21], Simon [52, 53] and Tarkaev [59].

In this paper, we use the theory of martingales and give a new an shorter proof of the almost everywhere convergence of Vilenkin–Fourier series of $f\in L_p(G_m)$ for $p>1$ in the case the Vilenkin system is bounded. The positive results of this paper are derived in Sect. 3. In Theorem 2 we prove the boundedness of the maximal operator on $L_p$ $(1<p<\infty )$ spaces. By using this result, we derive the $L_p$ norm convergence of the partial sums of Vilenkin–Fourier series (Theorem 3) as well as the analogue of the Carleson–Hunt theorem, i.e., the almost everywhere convergence of the partial sums of $f \in L_p$ (Theorem 4), when $1<p<\infty$. The proof is built up by proving some new lemmas of independent interest. The corresponding sharpness and almost everywhere divergence are stated and proved in Sect. 4, see Theorems 5 and 6. Especially Theorem 6 is the Kolmogorov type result and also here the proof is built up by proving some lemmas of independent interest. In order not to disturb our discussion later, some necessary preliminaries are presented in Sect. 2.

2 Preliminaries

Denote by ${\mathbb {N}}_{+}$ the set of the positive integers, ${\mathbb {N}}:={\mathbb {N}}_{+}\cup \{0\}.$ Let $m:=(m_{0,}$ $m_{1},\ldots )$ be a sequence of the positive integers not less than 2. Denote by

$$\begin{aligned} Z_{m_{k}}:=\{0,1,\ldots ,m_{k}-1\} \end{aligned}$$

the additive group of integers modulo $m_{k}$.

Define the group $G_{m}$ as the complete direct product of the groups $Z_{m_{i}}$ with the product of the discrete topologies of $Z_{m_{j}}$’s. The direct product $\mu$ of the measures

$$\begin{aligned} \mu _{k}\left( \{j\}\right) :=1/m_{k}\quad (j\in Z_{m_{k}}) \end{aligned}$$

is the Haar measure on $G_{m}$ with $\mu \left( G_{m}\right) =1.$ In this paper we discuss bounded Vilenkin groups, i.e. the case when $\sup _{n}m_{n}<\infty .$

The elements of $G_{m}$ are represented by sequences

$$\begin{aligned} x:=\left( x_{0},x_{1},\ldots ,x_{j},\ldots \right) ,\; \left( x_{j}\in Z_{m_{j}}\right) . \end{aligned}$$

It is easy to give a base for the neighborhood of $G_{m}:$

$$\begin{aligned} I_{0}\left( x\right) :=G_{m}, \; I_{n}(x):=\{y\in G_{m}\mid y_{0}=x_{0},\ldots ,y_{n-1}=x_{n-1}\}, \end{aligned}$$

where $x\in G_{m}$, $n\in {\mathbb {N}}.$ Denote $I_{n}:=I_{n}\left( 0\right)$ for $n\in {\mathbb {N}}_{+},$ and $\overline{I_{n}}:=G_{m}$ $\backslash$ $I_{n}$.

If we define the so-called generalized number system based on m by

$$\begin{aligned} M_{0}:=1,\quad M_{k+1}:=m_{k}M_{k}\quad (k\in {\mathbb {N}}), \end{aligned}$$

then every $n\in {\mathbb {N}}$ can be uniquely expressed as $n=\sum _{j=0}^{\infty }n_{j}M_{j},$ where $n_{j}\in Z_{m_{j}}$ $(j\in {\mathbb {N}}_{+})$ and only a finite number of $n_{j}`$s differ from zero. For two natural numbers $n=\sum _{j=1}^{\infty }n_jM_j$ and $k=\sum _{j=1}^{\infty }k_jM_j$, we define that

$$\begin{aligned} n \oplus k:=\sum _{i=0}^{\infty } \left( \left( n_j+k_j\right) \pmod {m_i}\right) {M_{j}}, \quad n_j, k_j \in Z_{m_j}. \end{aligned}$$

Next, we introduce on $G_{m}$ an orthonormal system which is called the Vilenkin system. First, we define the complex-valued function $r_{k}\left( x\right) :G_{m}\rightarrow {\mathbb {C}},$ the generalized Rademacher functions, by

$$\begin{aligned} r_{k}\left( x\right) :=\exp \left( 2\pi \imath x_{k}/m_{k}\right) , \left( \imath ^{2}=-1, \; \; x\in G_{m}, \; \; k\in {\mathbb {N}}\right) . \end{aligned}$$

Now, define the Vilenkin system $\,\,\,\psi :=(\psi _{n}:n\in {\mathbb {N}})$ on $G_{m}$ as:

$$\begin{aligned} \psi _{n}(x):=\prod \limits _{k=0}^{\infty }r_{k}^{n_{k}}\left( x\right) ,\quad \left( n\in {\mathbb {N}}\right) . \end{aligned}$$

Specifically, we call this system the Walsh-Paley system when $m\equiv 2.$ The norms (or quasi-norms) of the spaces $L_{p}(G_{m})$ $\left( 0<p<\infty \right)$ is defined by

$$\begin{aligned} \left\| f\right\| _p^p:=\int \limits _{G_m}\left| f\right| ^pd\mu . \end{aligned}$$

The Vilenkin system is orthonormal and complete in $L_{2}\left( G_{m}\right)$ (for details see e.g. the books [1, 49]).

Now, we introduce analogues of the usual definitions in Fourier-analysis. If $f\in L_{1}\left( G_{m}\right)$, we can define the Fourier coefficients, the partial sums of the Fourier series, the Dirichlet kernels with respect to the Vilenkin system in the usual manner:

$$\begin{aligned}&{\widehat{f}}\left( n\right) :=\int \limits _{G_{m}}f\overline{\psi }_{n}d\mu ,\,\,\left( n\in {\mathbb {N}}\right) \\&S_{n}f:=\sum _{k=0}^{n-1}{\widehat{f}}\left( k\right) \psi _{k} \, \text {and} \, D_{n}:=\sum _{k=0}^{n-1}\psi _{k},\,\, \left( n\in {\mathbb {N}}_{+}\right) \end{aligned}$$

respectively. Recall that (see e.g. Simon [50, 54] and Golubov et al. [22])

$$\begin{aligned} \sum _{s=0}^{m_{k}-1}r^s_{k}(x)=\left\{ \begin{array}{ll} m_k, &{}\quad \text {if} \quad x_{k}=0, \\ 0, &{}\quad \text {if}\,\,x_{k}\ne 0. \end{array} \right. \end{aligned}$$

(1)

and

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

(2)

It is known that (for the details see e.g. [1, 7, 37, 38]) there exist absolute constants $C_1$ and $C_2$ such that

$$\begin{aligned} C_1 n\le \left\| D_{q_{n}}\right\| _1 \le C_2 n,\quad \text {for} \, q_{n}=M_{2n}+M_{2n-2}+M_2+M_0. \end{aligned}$$

(3)

A function P is called Vilenkin polynomial if $P=\sum _{k=0}^{n}c_k\psi _k.$ The spectra of the Vilenkin polynomial P is defined by

$$\begin{aligned} sp(P)=\left\{ n\in {\mathbb {N}}: {\widehat{P}}\left( n\right) \ne 0. \right\} \end{aligned}$$

3 Martingale Inequalities

We will also need some martingale inequalities. The $\sigma$-algebra generated by the intervals $\left\{ I_{n}\left( x\right) :x\in G_{m}\right\}$ will be denoted by ${\mathcal {F}}_{n}$ $(n\in {\mathbb {N}})$. If ${\mathcal {F}}$ denotes the set of Haar measurable subsets of $G_m$, then obviously ${\mathcal {F}}_n \subset {\mathcal {F}}$. By a Vilenkin interval we mean one of the form $I_n(x), \; \; n\in {\mathbb {N}}, \; \; x\in G_m.$ The conditional expectation operators relative to ${{\mathcal {F}}}_n$ are denoted by $E_n$. An integrable sequence $f = \left( f_{n}\right) _{n \in {{\mathbb {N}}}}$ is said to be a martingale if $f_{n}$ is ${\mathcal {F}}_{n}$-measurable for all $n \in {{\mathbb {N}}}$ and $E_{n} f_{m} = f_{n}$ in the case $n \le m$. We can see that if $f \in L_1(G_m)$, then $(E_nf)_{n \in {{\mathbb {N}}}}$ is a martingale. Martingales with respect to $({\mathcal {F}}_n, n \in {{\mathbb {N}}})$ are called Vilenkin martingales. It is easy to show (see e.g. Weisz [65, p. 11]) that the sequence $({\mathcal {F}}_n, n \in {{\mathbb {N}}})$ is regular, i.e.,

$$\begin{aligned} f_n\le Rf_{n-1} \quad (n \in {{\mathbb {N}}}) \end{aligned}$$

(4)

for all non-negative Vilenkin martingales $(f_n)$, where $R:=\max _{n\in {\mathbb {N}}} m_n$.

Using (2), we can show that $E_nf=S_{M_n} f$ for all $f\in L_p(G_m)$ with $1 \le p\le \infty$ (see e.g. [65, Sect. 1.2]). By the well known martingale theorems, this implies that

$$\begin{aligned} \left\| S_{M_n} f\right\| _p\le c_p\left\| f\right\| _p, \, \text {for all } \, f\in L_p(G_m) \, \text {when} \, 1 \le p<\infty \end{aligned}$$

and

$$\begin{aligned} \left\| S_{M_n} f-f\right\| _p\rightarrow 0, \, \text { as } \, n\rightarrow \infty \, \text { for all } \, f\in L_p(G_m) \, \text {when } \, p\ge 1. \end{aligned}$$

(5)

For a Vilenkin martingale $f = \left( f_{n}\right) _{n \in {{\mathbb {N}}}}$, the maximal function is defined by

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

For a martingale $f=(f_n)_{n\ge 0}$ let

$$\begin{aligned} d_{n}f=f_{n}-f_{n-1} \qquad (n\ge 0) \end{aligned}$$

denote the martingale differences, where $f_{-1}:=0$. The square function and the conditional square function of f are defined by

$$\begin{aligned} S(f)=\left( \sum _{n=0}^{\infty }|d_nf|^2\right) ^{1/2}, \quad s(f)=\left( |d_0f|^{2}+\sum \limits _{n=0}^{\infty }E_{n}|d_{n+1} f|^2\right) ^{1/2}. \end{aligned}$$

We have shown the following theorem in [65].

Theorem 1

If $0<p<\infty$, then

$$\begin{aligned} \left\| f^{*}\right\| _{p} \sim \left\| S(f)\right\| _{p} \sim \left\| s(f)\right\| _{p}. \end{aligned}$$

If in addition $1<p\le \infty$, then

$$\begin{aligned} \left\| f^{*}\right\| _{p} \sim \left\| f\right\| _{p}. \end{aligned}$$

We will use the following convexity, concavity theorem proved in [65].

Proposition 1

Let ${{\mathbb {T}}}$ be a countable index set and $({\mathcal {A}}_t, t \in {{\mathbb {T}}})$ be an arbitrary (not necessarily monotone) sequence of sub-$\sigma$-algebras of ${\mathcal {F}}$. Suppose that for all $h \in L_p(G_m)$ and all $1<p<\infty$ Doob’s inequality

$$\begin{aligned} \left\| \sup _{t \in {{\mathbb {T}}}} |E_th| \right\| _p \le C_p\Vert h\Vert _p \end{aligned}$$

holds where $E_t$ denotes the conditional expectation operator relative to ${\mathcal {A}}_t$. If $(f_t, t \in {{\mathbb {T}}})$ is a sequence of non-negative measurable functions, then for all $1 \le p <\infty$,

$$\begin{aligned} \int \limits _{G_m}\left( \sum _{t \in {{\mathbb {T}}}} E_tf_t\right) ^p \, d \mu \le C_p \int \limits _{G_m} \left( \sum _{t \in {{\mathbb {T}}}} f_t\right) ^p \, d \mu \end{aligned}$$

and for all $0<q \le 1$,

$$\begin{aligned} \int \limits _{G_m} \left( \sum _{t \in {{\mathbb {T}}}} f_t\right) ^{q} \, d \mu \le C_q \int \limits _{G_m}\left( \sum _{t \in {{\mathbb {T}}}} E_tf_t\right) ^{q} \, d \mu . \end{aligned}$$

4 A.E. Convergence of Vilenkin–Fourier Series

We introduce some notations. For $j,k\in {{\mathbb {N}}}$ we define the following subsets of ${\mathbb {N}}:$

$$\begin{aligned} I_{jM_k}^{k} := [jM_k, jM_k+M_k)\cap {{\mathbb {N}}}\end{aligned}$$

and

$$\begin{aligned} {\mathcal {I}}:= \left\{ I_{jM_k}^{k} : j,k \in {{\mathbb {N}}}\right\} . \end{aligned}$$

We introduce also the partial sums taken in these intervals:

$$\begin{aligned} s_{I_{jM_k}^{k}} f := \sum _{i\in I_{jM_k}^{k}} {\widehat{f}}(i) \psi _i. \end{aligned}$$

For simplicity, we suppose that ${\widehat{f}}(0)=0$. The last author has proved in [66] that, for an arbitrary $n \in I_{jM_k}^{k}$,

$$\begin{aligned} s_{I_{jM_k}^{k}} f = \psi _{n} E_k\left( f {\overline{\psi }}_n\right) . \end{aligned}$$

(6)

For

$$\begin{aligned} n=\sum _{j=0}^{\infty } n_jM_j \quad (0 \le n_j <m_j), \end{aligned}$$

we introduce

$$\begin{aligned} n(k) := \sum ^\infty _{j=k} n_j M_j, \quad I_{n(k)}^{k} =\bigl [n(k),n(k) +M_k \bigr ) \quad (n\in {{\mathbb {N}}}). \end{aligned}$$

(7)

For $I= I_{n(k)}^{k}$, let

$$\begin{aligned} T^If:=T^{I_{n(k)}^{k}}f:= \sum _{\begin{array}{c} [n(k+1),n(k)) \supset J \in {\mathcal {I}}\\ |J|=M_k \end{array}} s_Jf. \end{aligned}$$

(8)

Since $I_{n(k)}^{k}=I_{{\tilde{n}}(k)}^{k}$ implies $n(k+1)=\tilde{n}(k+1)$, the operators $T^I$ $(I \in {\mathcal {I}})$ are well defined. Note that there are $n_k$ summands in (8).

Lemma 1

For all $n \in {{\mathbb {N}}}$, we have

$$\begin{aligned} S_nf = \sum _{k=0}^\infty T^{I_{n(k)}^{k}}f = \psi _n \sum _{k=0}^\infty \sum _{l=0}^{n_k-1} {\overline{r}}_k^{n_k-l} E_k \left( d_{k+1}\left( f {\overline{\psi }}_{n}\right) r_k^{n_k-l} \right) , \end{aligned}$$

where $I_{n(k)}^{k}$ is defined in (7).

Proof

We sketch the proof, only. It is proved in [66] that

$$\begin{aligned} T^{I_{n(k)}^{k}}f= & {} \sum _{j \in [n(k+1),n(k))} {\widehat{f}}(j) \psi _j \nonumber \nonumber \\= & {} \psi _n \sum _{l=0}^{n_k-1} {\overline{r}}_k^{n_k-l} E_k \left( d_{k+1}\left( f {\overline{\psi }}_{n}\right) r_k^{n_k-l} \right) . \end{aligned}$$

(9)

Moreover, n is contained in $I_{n(k)}^{k}$ and $I_{n(k)}^{k} \subset I_{n(k+1)}^{k+1}$. Since

$$\begin{aligned}{}[0,n) = \bigcup ^\infty _{k=0} \bigl [ n(k+1), n(k) \bigr ), \end{aligned}$$

we get that

$$\begin{aligned} S_nf = \sum _{k=0}^\infty T^{I_{n(k)}^{k}}f. \end{aligned}$$

This finishes the proof of Lemma 1. $\square$

Lemma 2

For all $k,n \in {{\mathbb {N}}}$, we have

$$\begin{aligned} \left| T^{I_{n(k)}^{k}}f\right| \le R E_k\left( \left| s_{I_{n(k+1)}^{k+1}} f-s_{I_{n(k)}^{k}} f\right| \right) , \end{aligned}$$

where $R:=\max (m_n, n \in {{\mathbb {N}}})$.

Proof

Equalities (9) and (6) imply

$$\begin{aligned} |T^{I_{n(k)}^{k}}f|&\le m_k E_k\left( \left| d_{k+1}\left( f {\overline{\psi }}_{n}\right) \right| \right) \\&\le R E_k\left( \left| \psi _nE_{k+1}\left( f {\overline{\psi }}_{n}\right) -\psi _nE_{k}\left( f {\overline{\psi }}_{n}\right) \right| \right) \nonumber \\&= R E_k\left( \left| s_{I_{n(k+1)}^{k+1}} f-s_{I_{n(k)}^{k}} f\right| \right) ,\nonumber \end{aligned}$$

(10)

which shows the lemma. $\square$

Lemma 3

For all $n \in {{\mathbb {N}}}$, $\left( {\overline{\psi }}_{n}T^{I_{n(k)}^{k}}f\right) _{k \in {{\mathbb {N}}}}$ is a martingale difference sequence with respect to $({\mathcal {F}}_{k+1})_{k \in {{\mathbb {N}}}}$.

Proof

First, ${\overline{\psi }}_{n}T^{I_{n(k)}^{k}}f$ is ${\mathcal {F}}_{k+1}$ measurable because of (9) and the fact that $r_k$ is ${\mathcal {F}}_{k+1}$ measurable. Since $E_k(r_k^{i})=0$ for $i=1,\ldots ,m_n-1$, we can see that

$$\begin{aligned} E_k\left( {\overline{\psi }}_{n}T^{I_{n(k)}^{k}}f\right) = E_k\left( \sum _{l=0}^{n_k-1} {\overline{r}}_k^{n_k-l} E_k \left( d_{k+1}(f {\overline{\psi }}_{n}) r_k^{n_k-l} \right) \right) =0, \end{aligned}$$

hence

$$\begin{aligned} \left( \sum _{l=0}^{n_k-1} {\overline{r}}_k^{n_k-l} E_k \left( d_{k+1}(f {\overline{\psi }}_{n}) r_k^{n_k-l} \right) \right) _{k \in {{\mathbb {N}}}} \end{aligned}$$

is a martingale difference sequence. $\square$

Before proving our main theorem, we need some further notations and lemmas. In what follows, I, J, K denote some elements of ${\mathcal {I}}$. Let

$$\begin{aligned} {\mathcal {F}}_K:={\mathcal {F}}_n \quad \text{ and } \quad E_K:=E_n \quad \text{ if } |K|=M_n. \end{aligned}$$

Assume that $\epsilon =(\epsilon _K, K \in {\mathcal {I}})$ is a sequence of functions such that $\epsilon _K$ is ${\mathcal {F}}_K$ measurable. Set

$$\begin{aligned} T_{\epsilon ;I,J}f := \sum _{I \subset K \subsetneq J} \epsilon _K T^Kf \end{aligned}$$

and

$$\begin{aligned} T_{\epsilon ;I}^{*}f := \sup _{I \subset J} |T_{\epsilon ;I,J}f|, \qquad T_{\epsilon }^{*}f := \sup _{I \in {\mathcal {I}}} |T_{\epsilon ;I}^{*}f|. \end{aligned}$$

If $\epsilon _K(t)=1$ for all $K \in {\mathcal {I}}$ and $t \in G_m$, then we omit the notation $\epsilon$ and we write simply $T_{I,J}f$, $T_{I}^{*}f$ and $T^{*}f$.

For $I \in {\mathcal {I}}$ with $|I|=M_n$, let $I^{+} \in {\mathcal {I}}$ such that $I \subset I^{+}$ and $|I^{+}|=M_{n+1}$. Moreover, let $I^{-} \in {\mathcal {I}}$ denote one of the sets $I^{-} \subset I$ with $|I^{-}|=M_{n-1}$. Note that ${\mathcal {F}}_{I^{-}}={\mathcal {F}}_{n-1}$ and $E_{I^{-}}=E_{n-1}$ are well defined. We introduce the maximal functions

$$\begin{aligned} s_{I}^{*}f := \sup _{K \subset I} E_{K^{-}}|s_{K}f| \end{aligned}$$

and

$$\begin{aligned} s^{*}f := \sup _{I \in {\mathcal {I}}} s_I^{*}f. \end{aligned}$$

Since $|s_{I^{+}}f|$ is ${\mathcal {F}}_{I^{+}}$ measurable, by the regularity condition (4),

$$\begin{aligned} |s_{I^{+}}f| \le R E_I|s_{I^{+}}f| \le R s_{I^{+}}^{*}f. \end{aligned}$$

(11)

Lemma 4

For any real number $x>0$ and $K \in {\mathcal {I}}$, let

$$\begin{aligned} \epsilon _K := \chi _{\{t \in G_m: x<s_{K^{+}}^{*}f(t) \le 2x\}} \end{aligned}$$

and

$$\begin{aligned} \alpha _K := \chi _{\{t \in G_m: s_K^{*}f(t) >x, s_I^{*}f(t) \le x, I \subsetneq K\}}. \end{aligned}$$

Then

$$\begin{aligned} T_\epsilon ^* f \le 2 \sup _{K\in {\mathcal {I}}} \alpha _K T_{\epsilon ;K}^* f + 4R^{2} x \chi _{\{t \in G_m: s^{*}f(t) > x\}}. \end{aligned}$$

(12)

Proof

Let us fix $I \subsetneq J$ in ${\mathcal {I}}$ and t in $G_m$. Set

$$\begin{aligned} \tau _K := \chi _{\{t \in G_m: s_K^{*}f(t)>x\}} \qquad (K\in {\mathcal {I}}). \end{aligned}$$

Therefore $\epsilon _K = \tau _{K^+}\epsilon _K$. Consequently, if the set

$$\begin{aligned} \{K\in {\mathcal {I}}: I \subset K \subsetneq J, \tau _{K^+}(t) =1\} \end{aligned}$$

is empty then $T_{\epsilon ;I,J}f(t) =0$ or else let $K_1$ be its minimum element. Moreover, denote by $K_0$ one of the minimum elements of the set

$$\begin{aligned} \{K\in {\mathcal {I}}: K \subset K^+_1, \tau _K(t) =1\}. \end{aligned}$$

This means that if $L \subsetneq K_0$, then $\tau _L(t) =0$. Thus $\alpha _{K_0}(t) =1$ and

$$\begin{aligned} T_{\epsilon ;I,J}f(t)&= T_{\epsilon ;K_1,J}f(t) = \epsilon _{K_1} T^{K_1}f(t) + T_{\epsilon ;K_1^+,J}f(t) \\&= \epsilon _{K_1}T^{K_1}f(t) + \alpha _{K_0}(t) \left( T_{\epsilon ;K_0,J}f(t) - T_{\epsilon ;K_0,K^+_1}f(t) \right) . \end{aligned}$$

By Lemma 2 and (11),

$$\begin{aligned} \epsilon _{K_1}|T^{K_1}f|&\le R \epsilon _{K_1} E_{K_1}\left( |s_{K_1^{+}} f-s_{K_1} f| \right) \\&\le 2R^{2} \epsilon _{K_1} E_{K_1}\left( \epsilon _{K_1} s_{K_1^{+}}^{*} f \right) \le 4R^{2} x \chi _{\{t \in G_m: s^*f(t) >x\}}. \end{aligned}$$

On the other hand,

$$\begin{aligned} \left| T_{\epsilon ;K_0,J}f(t) - T_{\epsilon ;K_0,K^+_1}f(t) \right| \le 2 T_{\epsilon ;K_0}^*f(t). \end{aligned}$$

Taking the supremum over all $I \subsetneq J$, we get (12). $\square$

Now we introduce the quasi-norm $\Vert \cdot \Vert _{{p,q}}$ $(0<p,q<\infty )$ by

$$\begin{aligned} \Vert f\Vert _{{p,q}}:= \sup _{x>0} x \left( \int \limits _{G_m}\left( \sum _{I\in {\mathcal {I}}} \alpha _I \right) ^{p/q} \, d \mu \right) ^{1/p}, \end{aligned}$$

where $\alpha _I$ is defined in Lemma 4. Observe that $\alpha _I$ can be rewritten as

$$\begin{aligned} \alpha _I := \chi _{\{t \in G_m: E_{I^{-}}|s_{I}f(t)| >x, E_{J^{-}}|s_{J}f(t)| \le x, J \subsetneq I\}}. \end{aligned}$$

(13)

Denote by $P^{p,q}$ the set of functions $f \in L_1$ which satisfy $\Vert f\Vert _{{p,q}}<\infty$. For $q=\infty$, define

$$\begin{aligned} \Vert f\Vert _{{p,\infty }} := \sup _{x>0} x \left( \int \limits _{G_m} \left( \sup _{I\in {\mathcal {I}}} \alpha _I \right) ^{p} \, d \mu \right) ^{1/p} \qquad (0<p<\infty ). \end{aligned}$$

It is easy to see that

$$\begin{aligned} \Vert f\Vert _{{p,\infty }} \le \Vert f\Vert _{{p,q}} \qquad (0<q<\infty ) \end{aligned}$$

and

$$\begin{aligned} \Vert f\Vert _{{p,\infty }} = \sup _{x>0} x \mu (s^*f >x)^{1/p}. \end{aligned}$$

Lemma 5

Let $\max (1,p)<q<\infty$, $f \in P^{p,q}$ and $x,z>0$. Then

$$\begin{aligned} \mu \left( \sup _{I\in {\mathcal {I}}} \alpha _I T_{\epsilon ;I}^* f> zx \right) \le C_{p,q} z^{-q} x^{-p}\Vert f\Vert _{{p,q}}^{p}, \end{aligned}$$

where $\alpha _I$ is defined in Lemma 4.

Proof

Equality (9) implies that

$$\begin{aligned} \xi T^K(f) = T^K(f \xi ) \end{aligned}$$

for any ${\mathcal {F}}_K$ measurable function $\xi$. By Lemma 3, for a suitable $n \in I$, $\left( {\overline{\psi }}_nT^Kf\right) _{I \subset K}$ is a martingale difference sequence relative to $({\mathcal {F}}_{K^+})_{I \subset K}$. We have

$$\begin{aligned} T^*_{\epsilon ;I}f = \sup _{I \subset J} \left| \sum _{I \subset K \subsetneq J} \epsilon _K {\overline{\psi }}_n T^Kf \right| = \sup _{I \subset J} \left| \sum _{I \subset K \subsetneq J} {\overline{\psi }}_n T^K(f\epsilon _K) \right| . \end{aligned}$$

Using Burkholder–Gundy’s inequality (see Theorem 1) together with (10), we obtain

$$\begin{aligned} E_I \left( |T^*_{\epsilon ;I}f|^{p_0}\right)&\le C_{p_0} E_I \left( \sum _{I \subset K} |{\overline{\psi }}_nT^K(f\epsilon _K)|^2\right) ^{p_0/2}\\&\le C_{p_0} E_I \left( \sum _{I \subset K} E_K|d_{K^{+}}(f\epsilon _K {\overline{\psi }}_{n})|^2\right) ^{p_0/2}, \end{aligned}$$

where $p_0>1$. Applying again Theorem 1, one can establish that

$$\begin{aligned} E_I \left( |T^*_{\epsilon ;I}f|^{p_0}\right)&\le C_{p_0} E_I \left| \sum _{I \subset K} d_{K^{+}}(f\epsilon _K {\overline{\psi }}_{n})\right| ^{p_0}\\&= C_{p_0} E_I \left| \sum _{I \subset K} \epsilon _K d_{K^{+}}(f {\overline{\psi }}_{n})\right| ^{p_0}. \end{aligned}$$

For fixed I and $t \in G_m$ let us denote by $K_0(t) \in {\mathcal {I}}$ (resp. $K_1(t) \in {\mathcal {I}}$) the smallest (resp. largest) interval $K \supset I$ for which $\epsilon _{K_0(t)}=1$ (resp. $\epsilon _{K_1(t)}=1$). Then

$$\begin{aligned} \psi _n(t)&\sum _{I \subset K} \epsilon _K(t) d_{K^{+}}(f {\overline{\psi }}_{n})(t) \\&= \psi _n(t) \sum _{K_0(t) \subset K \subset K_1(t)} \epsilon _K(t) d_{K^{+}}(f {\overline{\psi }}_{n})(t)\\&= \epsilon _{K_1(t)}(t) \left( \psi _n(t) E_{K_1(t)^{+}}(f {\overline{\psi }}_{n})(t)-\psi _n(t) E_{K_0(t)}(f {\overline{\psi }}_{n})(t)\right) \\&= \epsilon _{K_1(t)}(t) \left( s_{K_1(t)^{+}}f(t)-s_{K_0(t)}f(t)\right) . \end{aligned}$$

By (11) and by the definition of $\epsilon _K$,

$$\begin{aligned}&\epsilon _{K_1(t)}(t) \left| s_{K_1(t)^{+}}f(t)-s_{K_0(t)}f(t)\right| \nonumber \\&\quad \le R \epsilon _{K_1(t)}(t) \left( s_{K_1(t)^{+}}^{*}f(t) + s_{K_0(t)}^{*}f(t)\right) \\&\quad \le 2R \epsilon _{K_1(t)}(t) s_{K_1(t)^{+}}^{*}f(t) \le 4Rx.\nonumber \end{aligned}$$

(14)

Hence

$$\begin{aligned} E_I \left( |T^*_{\epsilon ;I}f|^{p_0}\right) \le C_{p_0} x^{p_0}. \end{aligned}$$

By Tsebisev’s inequality and the concavity theorem (see Proposition 1), for $p_0\ge q>1$, one can see that

$$\begin{aligned} \mu \left( \sup _{I\in {\mathcal {I}}} \alpha _I T_{\epsilon ;I}^* f> zx \right)&\le (zx)^{-q} \int \limits _{G_m} \left( \sup _{I\in {\mathcal {I}}} \alpha _I T_{\epsilon ;I}^* f\right) ^{q} \, d \mu \\&\le (zx)^{-q} \int \limits _{G_m} \left( \sum _{I\in {\mathcal {I}}} \alpha _I T_{\epsilon ;I}^* f^{p_0}\right) ^{q/p_0} \, d \mu \\&\le C_{p_0,q} (zx)^{-q} \int \limits _{G_m} \left( \sum _{I\in {\mathcal {I}}} \alpha _I E_I\left( T_{\epsilon ;I}^* f^{p_0}\right) \right) ^{q/p_0} \, d \mu \\&\le C_{p_0,q} z^{-q} \int \limits _{G_m}\left( \sum _{I\in {\mathcal {I}}} \alpha _I \right) ^{q/p_0} \, d \mu . \end{aligned}$$

Set $p_0:= q^2/p\ge q>1$ and observe that

$$\begin{aligned} \mu \left( \sup _{I\in {\mathcal {I}}} \alpha _I T_{\epsilon ;I}^* f> zx \right)&\le C_{p,q} z^{-q} \int \limits _{G_m} \left( \sum _{I\in {\mathcal {I}}} \alpha _I \right) ^{p/q} \, d \mu \\&\le C_{p,q} z^{-q} x^{-p}\Vert f\Vert _{{p,q}}^{p}, \end{aligned}$$

which shows the lemma. $\square$

Lemma 6

Let $\max (1,p)<q<\infty$ and $f \in P^{p,q}$. Then

$$\begin{aligned} \sup _{y>0} y^{p}\mu \Big (T^* f> (2+8R^{2})y \Big ) \le C_{p,q} \Vert f\Vert _{{p,q}}. \end{aligned}$$

Proof

First we define a decomposition generated by the sequences $\epsilon ^k = (\epsilon ^k_K,K \in {\mathcal {I}})$, where

$$\begin{aligned} \epsilon ^k_K:= \chi _{\{t \in G_m: 2^k <s_{K^{+}}^{*}f(t) \le 2^{k+1}\}} \qquad (k \in {{\mathbb {Z}}}). \end{aligned}$$

Notice that (10) and (14) imply

$$\begin{aligned} \chi _{\{t \in G_m: s^*f(t)=0\}} T^Kf = \chi _{\{t \in G_m: s^*f(t=0\}}\chi _{\{t \in G_m: s_{K^{+}}^*f(t)=0\}} T^Kf =0. \end{aligned}$$

Henceforth

$$\begin{aligned} T^Kf= \chi _{\{t \in G_m: s^*f(t)>0\}} T^Kf = \sum _{k\in {{\mathbb {Z}}}} \epsilon ^k_K T^Kf \end{aligned}$$

and

$$\begin{aligned} T^*f \le \sum _{k\in {{\mathbb {Z}}}} T^*_{\epsilon ^k}f. \end{aligned}$$

Let us apply Lemma 4 to $\epsilon ^k$ and $x=2^k$ to write

$$\begin{aligned} T^*_{\epsilon ^k}f \le 2 \sup _{K\in {\mathcal {I}}} \alpha _K^{k} T_{\epsilon ^{k};K}^* f + 2^{k+2}R^{2} \chi _{\{t \in G_m: s^{*}f(t) > 2^{k}\}}, \end{aligned}$$

where

$$\begin{aligned} \alpha _K^{k} := \chi _{\{t \in G_m: s_K^{*}f(t) >2^{k}, s_I^{*}f(t) \le 2^{k}, I \subsetneq K\}} \qquad (K\in {\mathcal {I}}). \end{aligned}$$

Choosing $j\in {{\mathbb {Z}}}$ such that $2^j<y\le 2^{j+1}$, we get that

$$\begin{aligned} \chi _{\{t \in G_m: s^{*}f(t) \le y\}} T^*f&\le 2 \sum _{k\le j} \sup _{K\in {\mathcal {I}}} \alpha _K^{k} T_{\epsilon ^{k};K}^* f + \sum _{k\le j} 2^{k+2}R^{2} \chi _{\{t \in G_m: s^{*}f(t) > 2^{k}\}} \\&\le 2 \sum _{k\le j} \sup _{K\in {\mathcal {I}}} \alpha _K^{k} T_{\epsilon ^{k};K}^* f + 8R^{2} y. \end{aligned}$$

By Lemma 5, for any $k\in {{\mathbb {Z}}}$ and $z_k>0$, we have

$$\begin{aligned} \mu \left( \sup _{K\in {\mathcal {I}}} \alpha _K^{k} T_{\epsilon ^{k};K}^* f> z_k2^k \right) \le C_{p,q} z_k^{-q} 2^{-pk} \Vert f\Vert ^p_{{p,q}}. \end{aligned}$$

(15)

Consequently,

$$\begin{aligned} y^{p}\mu \Big (T^* f\!>\! (2+8R^{2})y \Big )&\le y^p \mu (s^*f \!>\!y) \!+\! y^{p}\mu \Big (T^* f> (2+8R^{2})y, s^*f \!\le \! y \Big ) \\&\le \Vert f\Vert ^p_{{p,\infty }}+ y^p \mu \left( \sum _{k\le j} \sup _{K\in {\mathcal {I}}} \alpha _K^{k} T_{\epsilon ^{k};K}^* f>y \right) \\&\le \Vert f\Vert ^p_{{p,q}}+ y^p \mu \left( \sum _{k\le j} \sup _{K\in {\mathcal {I}}} \alpha _K^{k} T_{\epsilon ^{k};K}^* f>2^j \right) . \end{aligned}$$

To use (15), observe that $c_\beta \sum _{k\le j}2^{\beta (k-j)} =1$ if $\beta >0$ and $c_\beta =1-2^{-\beta }$. Set

$$\begin{aligned} 2^j c_\beta 2^{\beta (k-j)}= c_\beta 2^{(\beta -1)(k-j)} 2^k =: z_k 2^k. \end{aligned}$$

Then for $\beta =(q-p)/(2q)$, we get

$$\begin{aligned} z_k^{-q}2^{-pk} \le C_{p,q} 2^{-pj} 2^{p(j-k)+q(\beta -1)(j-k)} \le C_{p,q} y^{-p} 2^{(q-p)(k-j)/2}. \end{aligned}$$

Thus, by (15),

$$\begin{aligned} \mu \left( \sum _{k\le j} \sup _{K\in {\mathcal {I}}} \alpha _K^{k} T_{\epsilon ^{k};K}^* f>2^j \right)&\le \sum _{k\le j} \mu \left( \sup _{K\in {\mathcal {I}}} \alpha _K^{k} T_{\epsilon ^{k};K}^* f>z_k2^k \right) \\&\le C_{p,q} \sum _{k\le j} z_k^{-q} 2^{-pk} \Vert f\Vert ^p_{{p,q}} \\&\le C_{p,q} y^{-p} \Vert f\Vert ^p_{{p,q}} \sum _{k\le j} 2^{(q-p)(k-j)/2} \\&\le C_{p,q} y^{-p} \Vert f\Vert ^p_{{p,q}}, \end{aligned}$$

so the lemma is proved. $\square$

Let $\Delta$ denote the closure of the triangle in ${{\mathbb {R}}}^2$ with vertices (0, 0), (1/2, 1/2) and (1, 0) except the points $(x,1-x)$, $1/2<x \le 1$.

Lemma 7

Suppose that $1< p, q<\infty$ satisfy $(1/p, 1/q) \in \Delta$. Then, for all $f\in L_p$,

$$\begin{aligned} \Vert f\Vert _{{p,q}} \le C_{p,q} \Vert f\Vert _p. \end{aligned}$$

Proof

For an arbitrary $x>0$, let us use the definition of $\alpha _I$ given in (13). Then $\alpha _I$ is ${\mathcal {F}}_{I^{-}}$ measurable and, obviously,

$$\begin{aligned} \alpha _I\alpha _J=0 \quad \mathrm{if} \quad I \subsetneq J \quad \mathrm{or} \quad J \subsetneq I. \end{aligned}$$

For all $I \in {\mathcal {I}}$, introduce the projections $F_I:= \alpha _I s_I$ and observe that $s_I \circ s_J =0$ for every incomparable I and J. Therefore, we get for every $g\in L_1$ and $I,J \in {\mathcal {I}}$ that

$$\begin{aligned} F_I(F_Jg) = \alpha _I s_I (s_J (\alpha _J g)) = s_I (\alpha _I \alpha _J s_J g) = \delta _{I,J} F_I g, \end{aligned}$$

where $\delta _{I,J}$ is the Kronecker symbol. Thus the projections $F_I$ are orthogonal and Bessel’s inequality implies for any $g\in L_2$ that

$$\begin{aligned} \left\| (F_Ig, I \in {\mathcal {I}}) \right\| _{L_2(l_2)}^2 = \sum _{I \in {\mathcal {I}}} \Vert F_I g\Vert ^2_2 \le \Vert g\Vert ^2_2. \end{aligned}$$

Let us introduce the operators

$$\begin{aligned} G_Ig := E_{I^{-}} (\eta _I F_Ig) \qquad (g \in L_1, I \in {\mathcal {I}}), \end{aligned}$$

where $(\eta _I, I \in {\mathcal {I}})$ is a fixed sequence of functions satisfying $\Vert \eta _I\Vert _\infty \le 1$ for each $I \in {\mathcal {I}}$. Then

$$\begin{aligned} \Vert (G_I g, I \in {\mathcal {I}})\Vert _{L_2(l_2)}^2&\le \int \limits _{G_m} \sum _{I \in {\mathcal {I}}} E_{I^{-}} |F_Ig|^2 \, d \mu \\&= \int \limits _{G_m}\sum _{I \in {\mathcal {I}}} |F_Ig|^2 \, d \mu \le \Vert g\Vert ^2_2. \end{aligned}$$

Furthermore, by Doob’s inequality,

$$\begin{aligned} \left\| (G_I g, I \in {\mathcal {I}}) \right\| _{L_s(l_\infty )} \le \left\| \sup _{I \in {\mathcal {I}}} E_{I^{-}} |g|\right\| _s \le C_s \Vert g\Vert _s \end{aligned}$$

for any $1 <s \le \infty$ and $g\in L_s$. It follows by interpolation that

$$\begin{aligned} \left\| (G_I g, t \in {\mathcal {I}}) \right\| _{L_p(l_q)}\le C_{p,q} \Vert g\Vert _p \quad (g\in L_p) \end{aligned}$$

where $1/p=(1-t)/2+t/s$ and $1/q=(1-t)/2$ for any $0\le t \le 1$. Setting $g:=f$ and $\eta _t := \mathrm{sign} \; s_If$, we have

$$\begin{aligned} \left( \int \limits _{G_m}\left( \sum _{I \in {\mathcal {I}}} (\alpha _I E_{I^{-}} |s_If|)^q \right) ^{p/q} \, d \mu \right) ^{1/p} \le C_{p,q} \Vert f\Vert _p. \end{aligned}$$

Using the fact that

$$\begin{aligned} \alpha _I E_{I^{-}} |s_If| > x \alpha _I, \end{aligned}$$

we can see that

$$\begin{aligned} x \left( \int \limits _{G_m} \left( \sum _{I \in {\mathcal {I}}} \alpha _I \right) ^{p/q} \, d \mu \right) ^{1/p} \le C_{p,q} \Vert f\Vert _p, \end{aligned}$$

which finishes the proof. $\square$

Now we are ready to formulate our first main result.

Theorem 2

Let $f\in L_p(G_m)$, where $1<p<\infty$. Then

$$\begin{aligned} \left\| S^{*}f\right\| _p\le c_p\left\| f\right\| _p, \end{aligned}$$

where

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

Proof

It is easy to see that Lemma 1 implies $S^{*}f \le T^{*}f$. It follows from Lemmas 6 and 7 that

$$\begin{aligned} \sup _{y>0} y^{p}\mu \left( S^* f> y \right) \le C_{p} \Vert f\Vert _p \end{aligned}$$

for $1<p<\infty$. Now the proof of the theorem follows by the Marcinkiewicz interpolation theorem. $\square$

The next norm convergence result in $L_p$ spaces for $1<p<\infty$ follow from the density of the Vilenkin polynomials in $L_p(G_m)$ and from Theorem 2.

Theorem 3

Let $f\in L_p(G_m)$, where $1<p<\infty$. Then

$$\begin{aligned} \Vert S_nf-f\Vert _p\rightarrow 0, \, \text {as} \, n\rightarrow \infty . \end{aligned}$$

Our announced Carleson–Hunt type theorem reads:

Theorem 4

Let $f\in L_p(G_m)$, where $p>1$. Then

$$\begin{aligned} S_{n}f\rightarrow f, \quad \text {a.e.,} \, \mathrm{as } \, n\rightarrow \infty . \end{aligned}$$

The proof follows directly by using Theorem 2 and the fact that the Vilinkin polynomials are dense in $L_p.$

5 Almost Everywhere Divergence of Vilenkin–Fourier Series

A set $E\subset G_m$ is called a set of divergence for $L_p(G_m)$ if there exists a function $f\in L_p(G_m)$ whose Vilenkin–Fourier series diverges on E.

Lemma 8

If E is a set of divergence for $L_1(G_m),$ then there is a function $f\in L_1(G_m)$ such that $S^*f=\infty$ on E.

Proof

We claim that given any $g\in L_1(G_m),$ there is an unbounded monotone increasing sequence $\lambda = \left( \lambda _j , j \in {\mathbb {N}}\right)$ of positive real numbers and a function $f \in L_1(G_m)$ such that

$$\begin{aligned} {\hat{f}}(j)=\lambda _j{\hat{g}}(j) \qquad (j\in {\mathbb {N}}). \end{aligned}$$

(16)

To prove this claim use (5) for $p=1$ to choose a strictly increasing sequence of positive integer $n_1, n_2, \dots$ such that

$$\begin{aligned} \Vert S_{M_{n_k}}g-g \Vert _1 <M_{k}^{-1} \qquad (k\in {\mathbb {N}}_+). \end{aligned}$$

(17)

Consider the function f defined by

$$\begin{aligned} f:=g+\sum _{k=1}^{\infty }\left( g-S_{M_{n_k}}g\right) . \end{aligned}$$

By (17), the series converges in the norm of $L_1(G_m)$. In particular, f belongs to $L_1(G_m)$ and

$$\begin{aligned} {\hat{f}}(j) ={\hat{g}}(j)+\sum _{k=1}^{\infty }\int \limits _{G_m}\left( g-S_{M_{n_k}}g\right) {\overline{\psi }}_j d\mu \end{aligned}$$

for $j\in {\mathbb {N}}.$ Therefore, the claim follows from orthogonality if we set

$$\begin{aligned} \lambda _j:=1+\sum _{k\in {\mathbb {N}}_+: M_{n_k}\le j}1\quad ( j\in {\mathbb {N}}). \end{aligned}$$

To prove the theorem, suppose that $g\in L_1(G_m)$ is a function whose Vilenkin–Fourier series diverges on E. Use the claim to choose a monotone increasing, unbounded sequence $\lambda$ which satisfies (16). By Abel’s transformation,

$$\begin{aligned} S_n g -S_m g= & {} \sum _{j=m}^{n-1}\left( S_{j+1}f - S_j f\right) \frac{1}{\lambda _j} \\= & {} \frac{S_n f}{\lambda _{n-1}}-\frac{S_m f}{\lambda _m} + \sum _{j=m+1}^{n-1} \left( \frac{1}{\lambda _{j-1}} - \frac{1}{\lambda _j}\right) S_jf \end{aligned}$$

for any integers $n,m \in N$ with $n>m.$ Since $\lambda$ is increasing, it follows that

$$\begin{aligned} \left| S_n g - S_m g\right| \le \frac{2}{\lambda _m} S^*f\quad (n,m \in {\mathbb {N}}, n>m). \end{aligned}$$

Since $\lambda$ is unbounded, it follows that $S_n g$ converge at x when $S^*f(x)$ is finite. In particular, $(S^*f)(x)=\infty$ for all $x \in E.$ $\square$

Lemma 9

A set $E \subseteq G_m$ is a set of divergence for $L_1(G_m)$ if and only if there exist Vilenkin polynomials $P_1 , P_2 , \dots$such that

$$\begin{aligned} \sum _{j=1}^{\infty }\Vert P_j\Vert _1 < \infty \end{aligned}$$

(18)

and

$$\begin{aligned} \sup _{j\in {\mathbb {N}}_+} S^* P_j(x)=\infty \quad (x\in E). \end{aligned}$$

(19)

Proof

Suppose first that E is a set of divergence for $L_1(G_m).$ Let $g \in L_1(G_m)$ be a function whose Vilenkin–Fourier series diverges on E. By repeating the proof of Lemma 8, we can choose an unbounded, monotone increasing positive sequence $(\lambda _j,j\in {\mathbb {N}})$ and a function $f \in L_1(G_m)$ such that

$$\begin{aligned} S_n g -S_m g = \frac{S_n f}{\lambda _{n-1}}-\frac{S_m f}{\lambda _m} + \sum _{j=m+1}^{n-1} \left( \frac{1}{\lambda _{j-1}} - \frac{1}{\lambda _j}\right) S_jf \end{aligned}$$

for all integers $n, m \in {\mathbb {N}}, m < n.$

Let $(\omega _j,j \in {\mathbb {N}})$ be an unbounded sequence of positive, increasing numbers which satisfy

$$\begin{aligned} \sum _{j=1}^{\infty } \left( \frac{1}{\lambda _j} - \frac{1}{\lambda _{j+1}}\right) \omega _j < \infty . \end{aligned}$$

For example, let

$$\begin{aligned} \omega _j:=\frac{1}{\frac{1}{\sqrt{\lambda _j}}+\frac{1}{\sqrt{\lambda _{j+1}}}}. \end{aligned}$$

Indeed, then

$$\begin{aligned} \sum _{j=1}^{\infty } \left( \frac{1}{\lambda _{j}} - \frac{1}{\lambda _{j+1}}\right) \omega _j \le \sum _{j=1}^{\infty } \left( \frac{1}{\sqrt{\lambda _j}}-\frac{1}{\sqrt{\lambda _{j+1}}}\right) = \frac{1}{\sqrt{\lambda _1}}. \end{aligned}$$

Fix $x\in E$. If

$$\begin{aligned} \vert S_jf(x)\vert =O(\omega _j), \; \; \text {as} \; \; j\rightarrow \infty , \end{aligned}$$

then

$$\begin{aligned} \vert S_n g(x) -S_m g(x)\vert \rightarrow 0, \; \; \text {as} \; \; n, \; m\rightarrow \infty \end{aligned}$$

and we get that $S_n g(x)$ is a convergent series for any $x \in E,$ which is contradiction. Consequently, the inequality

$$\begin{aligned} \vert S_n f(x)\vert >\omega _n \end{aligned}$$

(20)

holds for infinitely many integers $n \in {\mathbb {N}}.$

Use (5) for $p=1$ to choose strictly increasing sequences of positive integers $(n_j,j \in {\mathbb {N}})$ and $(\alpha _j,j \in {\mathbb {N}})$ which satisfy $n_j < \alpha _j + 1,$

$$\begin{aligned} \Vert f-S_{M_{n_j}} f\Vert _1 < M_j^{-1} \end{aligned}$$

(21)

and

$$\begin{aligned} \Vert S^*(S_{M_{n_j}} f)\Vert _{\infty } < \frac{\omega _{\alpha _j}}{2} \quad (j \in {\mathbb {N}}). \end{aligned}$$

(22)

Consider the functions defined by

$$\begin{aligned} P_j := S_{M_{\alpha _{j+1}}} (f-S_{M_{n_j}} f) \qquad (j\in {\mathbb {N}}_+). \end{aligned}$$

Clearly, these functions are Vilenkin polynomials. We will show that they satisfy (18) and (19). Since $\Vert S_{M_n}h \Vert _1 \le \Vert h\Vert _1, \; \text { for } \; n \in {\mathbb {N}} \; \text { and } \; h \in L_1(G_m),$ (18) is a direct consequence of (21). To verify (19), fix $x \in E$ and choose an $n \in {\mathbb {N}}$ satisfying (20) which is large enough so that $\alpha _j < n \le \alpha _{j+1}$ for some $j \in {\mathbb {N}}_+.$ Since the definition of $P_j$ implies

$$\begin{aligned} S_n P_j = S_n f -S_n (S_{M_{n_j}} f), \end{aligned}$$

we have by (20) and (22) that

$$\begin{aligned} \vert (S_n P_j)(x)\vert \ge \vert (S_n f)(x)\vert -\frac{\omega _{\alpha _j}}{2}\ge \frac{1}{2}\omega _{\alpha _j}. \end{aligned}$$

Hence (19) follows from the fact that $\omega _n \rightarrow \infty$ as $n \rightarrow \infty .$

Conversely, suppose that

$$\begin{aligned} P_j := \sum _{k=0}^{M_{\alpha _{j-1}}}c_k ^{(j)}\psi _k \quad (\alpha _j \in {\mathbb {N}}, j\in {\mathbb {N}}_+) \end{aligned}$$

is a sequence of Vilenkin polynomials which satisfies (18) and (19). Let $n_1 := \alpha _1$ and for $j > 1$ set $n_j := 1 + \max \{n_{j-1},\alpha _j\}.$ Then $(n_j,j \in {\mathbb {N}})$ is a strictly increasing sequence of integers and it is easy to see that

$$\begin{aligned} M_{n_{j+1}}\oplus k_1 > M_{n_j} \oplus k_0 \end{aligned}$$

(23)

for any choice of integers $k_0$ and $k_1$ which satisfy $0\le k_0 \le M_{\alpha _j}, 0\le k_1 \le M_{\alpha _{j+1}}$ and $j \in {\mathbb {N}}.$ Let

$$\begin{aligned} f:=\sum _{j=1}^{\infty }\psi _{M_{n_j}} P_j \end{aligned}$$

and observe by (18) that $f \in L_1(G_m).$ It is clear that the series defining f converges in $L_1(G_m)$ norm. Consequently, this series is the Vilenkin–Fourier series of f. Moreover, (23) can be used to see that

$$\begin{aligned} S_{M_{n_j}+k}f-S_{M_{n_j}}f=\psi _{M_{n_j}}S_kP_j \end{aligned}$$

for $0\le k < M_{n_{j+1}}-M_{n_j}, j\in {\mathbb {N}}_+.$ In particular, (19) implies the Vilenkin–Fourier series of f diverges at each $x \in E.$ $\square$

Corollary 1

If $E_1, E_2,\dots$ are sets of divergence for $L_1(G_m),$ then

$$\begin{aligned} E:=\cup _{n=1} ^{\infty }E_n \end{aligned}$$

is also a set of divergence for $L_1(G_m).$

Proof

Apply Lemma 9 to choose Vilenkin polynomials $P_1 ^{(n)}, P_2^{(n)}, \dots$ such that

$$\begin{aligned} \sum _{j=1}^{\infty } \Vert P_j ^{(n)}\Vert _1<\infty \end{aligned}$$

and

$$\begin{aligned} \sup _{j\in {\mathbb {N}}_+} (S^* P_j ^{(n)})(x)=\infty \quad \left( x\in E_n , n\in {\mathbb {N}}_+\right) . \end{aligned}$$

(24)

Thus there exist integers $\alpha _1< \alpha _2 < \dots$ such that

$$\begin{aligned} \sum _{j=\alpha _n}^{\infty } \left\| P_j ^{(n)}\right\| _1<\frac{1}{M_n}\quad (n \in {\mathbb {N}}_+). \end{aligned}$$

Let $(Q_j , j \in {\mathbb {N}}_+)$ be any enumeration of the polynomials

$$\begin{aligned} \left\{ P_j ^{(n)} :j \ge \alpha _n, n=1,2,\dots \right\} . \end{aligned}$$

e.g.,

$$\begin{aligned}&Q_1:=P_{\alpha _1}^{(1)}, \\&Q_2:=P_{\alpha _2}^{(2)}, \ \ \ \ Q_3:=P_{{\alpha _1}+1}^{(1)}, \\&Q_4:=P_{{\alpha _1}+2}^{(1)}, \ \ Q_5:=P_{{\alpha _2}+1}^{(2)}, \ \ Q_6:=P_{\alpha _3}^{(3)}, \\&Q_7:=P_{\alpha _4}^{(4)}, \ \ \ \ Q_8:=P_{\alpha _3+1}^{(3)}, \ \ Q_9:=P_{\alpha _2+2}^{(2)}, \ \ Q_{10}:=P_{\alpha _1+3}^{(1)} \\&Q_{11}:=P_{\alpha _1+4}^{(1)}, \ Q_{12}:=P_{\alpha _2+3}^{(2)}, \ Q_{13}:=P_{\alpha _3+2}^{(3)}, \ Q_{14}:=P_{\alpha _4+1}^{(4)} \ Q_{15}:=P_{\alpha _5}^{(5)} \dots . \end{aligned}$$

Each $Q_j$ is a Vilenkin polynomial and

$$\begin{aligned} \sum _{j=1}^{\infty } \Vert Q_j \Vert _1<\sum _{n=1}^{\infty }\frac{1}{M_n}<\infty . \end{aligned}$$

In particular, by Lemma 9 it suffices to show that

$$\begin{aligned} \sup _{j\in {\mathbb {N}}_+} (S^* Q_j )(x)=\infty , \ \ \text {for} \ \ x \in E. \end{aligned}$$

But this follows from the construction and from (24) since every $x \in E$ necessarily belongs to some $E_n.$ $\square$

Theorem 5

If $1\le p < \infty$ and $E\subseteq G_m$ is a set of Haar measure zero, then E is a set of divergence for $L_p(G_m).$

Proof

We begin with a general remark. If $A \subseteq G_m$ is a finite union of intervals $I_1, I_2, \dots , I_n$ for some $n \in {\mathbb {N}}_+$ and if N is any non-negative integer, then there exists a Vilenkin polynomial P such that, for some $i\ge N,$

$$\begin{aligned} P=\sum _{k=M_N}^{M_i-1}c_k\psi _k, \end{aligned}$$

which satisfies

$$\begin{aligned} \left| P(x)\right| =1, \ (x\in A) \ \ \ \ \ \ \ \ \text {and} \ \ \ \ \ \ \ \ \int \limits _{G_m}\left| P\right| ^p d\mu =\mu (A). \end{aligned}$$

Indeed, if $i := \max \{ M_N, 1/\mu (I_j) :1\le j \le n \},$ then, in view of (2), we find that $P:= \chi (A)\psi _{i}$ is such a polynomial.

To prove the theorem, suppose $E \subseteq G_m$ satisfies $\mu (E)= 0.$ Cover E with intervals $(I_k, k\in {\mathbb {N}})$ such that

$$\begin{aligned} \sum _{k=0}^{\infty }\mu (I_k)<1 \end{aligned}$$

and each $x \in E$ belongs to infinitely many of the sets $I_k.$ Set $n_0 := 0$ and choose integers $n_0< n_1 < n_2 \cdots$ such that

$$\begin{aligned} \sum _{k=n_j}^{\infty }\mu (I_j)<M_j^{-1} \ \ \ \ \ \ \ \ (j \in {\mathbb {N}}). \end{aligned}$$

Apply the general remark above successively to the sets

$$\begin{aligned} A_j := \bigcup _{k=n_j}^{n_{j+1}-1} I_k \ \ \ \ \ \ (j \in {\mathbb {N}}) \end{aligned}$$

to generate integers $\alpha _0 :=0< \alpha _1< \alpha _2 < \cdots$ and Vilenkin polynomials $P_0, P_1,\dots$ such that $sp(P_j)\subset \left[ M_{\alpha _j}, M_{\alpha _j+1} \right) :$

$$\begin{aligned} P_j=\sum _{k=M_{\alpha _j}}^{M_{\alpha _j+1}-1}c_k\psi _k, \end{aligned}$$

$$\begin{aligned} \Vert P_j \Vert _p^p=\mu (A_j)\le M_j^{-1} \end{aligned}$$

(25)

and

$$\begin{aligned} \left| P_j(x)\right| =1 \ \ \ x\in A_j, \ \ \ \text {for} \ \ \ j \in {\mathbb {N}}. \end{aligned}$$

(26)

Setting

$$\begin{aligned} f:=\sum _{j=1}^{\infty }P_j , \end{aligned}$$

we observe by (25) that this series converges in $L_p(G_m)$ norm. Hence $f \in L_p(G_m)$ and this series is the Vilenkin–Fourier series of f. Moreover, since the spectra of the polynomials $P_j$ are pairwise disjoint, we have

$$\begin{aligned} S_{M_{\alpha _j+1}}f-S_{M_{\alpha _j}}f=P_j \ \ \ \ (j \in {\mathbb {N}}_+). \end{aligned}$$

Since every $x \in E$ belongs to infinitely many of the sets $A_j,$ it follows from (26) that the Vilenkin–Fourier series of f diverges at every point $x \in E.$ $\square$

This theorem cannot be improved for $1< p < \infty$ and measurable sets with non-zero measure. Indeed, in this case the Vilenkin–Fourier series of an $f \in L_p(G_m)$ converges a.e. (see Theorem 4). However, it can be improved considerably for $p = 1.$

Theorem 6

There is a function $f \in L_1 (G_m)$ whose Vilenkin–Fourier series diverges everywhere.

Proof

Fix $\alpha _n\in \left[ M_{n-1}, M_n\right) ,$ where $n\in {\mathbb {N}}_+$ is odd. By using the lower estimate for the Lebesgue constant in (3), we can conclude that there exists an absolute constant $C>0,$ which does not depend on n,  such that

$$\begin{aligned} \left\| \sum _{k=0}^{\alpha _n-1} \psi _k \right\| _1=\left\| D_{\alpha _n} \right\| _1 >Cn. \end{aligned}$$

Consider the function

$$\begin{aligned} g_n(x)=\left\{ \begin{array}{ll} \frac{ D_{\alpha _n}(x) }{\left| D_{\alpha _n}(x) \right| }, &{}\quad \text {if} \,\,\, D_{\alpha _n}(x)\ne 0, \\ 0, &{}\quad \text {if}\,\,D_{\alpha _n}(x)=0. \end{array} \right. \end{aligned}$$

It is constant on any set of the form $I_n\left( x\right) ,$ $x\in G_m.$ Hence $g_n$ is a Vilenkin polynomial of order at most $M_n.$ Moreover, since

$$\begin{aligned} \psi _k(x-t)=\psi _k(x){\overline{\psi }}_k(t)={\overline{\psi }}_k(t), \ \ \ \ 0\le k <M_n, \ \ \ x\in I_n (0). \end{aligned}$$

we get that

$$\begin{aligned} D_{\alpha _n}(x-t)={\overline{D}}_{\alpha _n}(t), \ \ \ x\in I_n (0). \end{aligned}$$

Hence, by the choice of $\alpha _n\in \left[ M_{n-1}, M_n\right)$ we have

$$\begin{aligned} \left( S_{\alpha _n}g_n\right) (x)=\int \limits _{G_m}g_n(t)D_{\alpha _n}(x-t)=\left\| D_{\alpha _n}\right\| _1>Cn, \ \ \ \ \ \ \ \ (x\in I_n (0)). \end{aligned}$$

For $k=\sum _{s=0}^{n-1}k_sM_s,$$(k_s\in Z_{m_s}),$ let us define the points $x_k ^{(n)} \in G_m,$$(0\le k <M_n)$ by $x_k^{(n)}:=(k_{0}, k_{1},\ldots ,k_{n-1},0,0,\ldots )$ and set

$$\begin{aligned} Q_n:=\prod _{k=0}^{M_n -1}\left( 1- \tau _{x_k^{(n)}} g_n \frac{\sum _{s=1}^{m_{n+k}-1}r^s_{n+k}}{m_{n+k}-1}\right) , \ \ \text {where} \ \ \tau _{x_k^{(n)}} g_n(x)=g_n(x-x_k^{(n)}). \end{aligned}$$

By using (1) we find that

$$\begin{aligned} \frac{\sum _{s=1}^{m_{n+k}-1}r^s_{n+k}(x)}{m_{n+k}-1}=\left\{ \begin{array}{ll} 1, &{}\quad \text {if} \,\, x_{n+k}=0, \\ \frac{-1}{m_{n+k}-1}, &{}\quad \text {if}\,\,x_{n+k}\ne 0 \end{array} \right. \end{aligned}$$

(27)

and

$$\begin{aligned} 1-\frac{\sum _{s=1}^{m_{n+k}-1}r^s_{n+k}(x)}{m_{n+k}-1}=\left\{ \begin{array}{ll} 0, &{}\quad \text {if} \quad x_{n+k}=0, \\ 1+\frac{1}{m_{n+k}-1}, &{}\quad \text {if}\,\,x_{n+k}\ne 0. \end{array} \right. \end{aligned}$$

(28)

It is easy to show that for any $x\in G_m$ there exists $x_j^{(n)}:=(j_0, j_1,\ldots ,j_{n-1},0,0,\ldots )$ such that $I_n(x)=I_n(x^{(n)}_j),$ that is

$$\begin{aligned} x_0=j_{0}, \ x_1=j_{1}, \ x_{n-1}=j_{n-1}. \end{aligned}$$

Consider the j-th term of the expression of $Q_n$ and let $x\in I_n(x^{(n)}_j).$ Since $\tau _{x_j^{(n)}} g_n(x_j^{(n)})=g_n(0)=1$, according to (28) we can conclude that

$$\begin{aligned} 1- \tau _{x_j^{(n)}} g_n (x) \frac{\sum _{s=1}^{m_{n+j}-1}r^s_{n+j}(x)}{m_{n+j}-1}=1- \frac{\sum _{s=1}^{m_{n+j}-1}r^s_{n+j}(x)}{m_{n+j}-1}=0 \ \ \text {if}\ \ x_{n+j}=0. \end{aligned}$$

Since $x\in I_n(x^{(n)}_j)$ for some $\ 0\le k\le M_n-1,$

$$\begin{aligned} Q_n(x)=\prod _{k=0}^{M_n -1}\left( 1- \tau _{x_k^{(n)}} g_n(x) \frac{\sum _{s=1}^{m_{n+k}-1}r^s_{n+k}(x)}{m_{n+k}-1}\right) =0 \ \ \text {if} \ \ x_{n+k}=0. \end{aligned}$$

On the other hand, (27) and $\vert \tau _{x_k^{(n)}} g_n(x)\vert \le 1$ imply that if $x_{n+k}\ne 0,$ we get that

$$\begin{aligned} \left| 1- \tau _{x_k^{(n)}} g_n \frac{\sum _{s=1}^{m_{n+k}-1}r^s_{n+k}}{m_{n+k}-1}\right|\le & {} 1+ \left| \tau _{x_k^{(n)}} g_n \frac{\sum _{s=1}^{m_{n+k}-1}r^s_{n+k}}{m_{n+k}-1}\right| \\\le & {} 1+\frac{1}{m_{n+k}-1}=\frac{m_{n+k}}{m_{n+k}-1}. \end{aligned}$$

It follows that

$$\begin{aligned} \vert Q_n(x)\vert \le \prod _{k=0}^{M_n -1}\frac{m_{n+k}}{m_{n+k}-1}, \ \ \text {if} \ \ x\in I_n(x) \ \ \text {and} \ \ x_{n+k}\ne 0, \ \text {for all} \ 0\le k\le M_n-1. \end{aligned}$$

Hence, we can conclude that $Q_n\in L_1{(G_m)}.$ Indeed,

$$\begin{aligned}&\int _{G_m}\vert Q_n\vert d\mu \\\le & {} \sum _{x_0=0}^{m_0-1}\cdots \sum _{x_{n-1}=0}^{m_{n-1}-1}\sum _{x_n=0}^{m_n-1}\cdots \sum _{x_{n+M_n-1}=1}^{m_{n+M_n-1}-1}\int _{I_{n+M_n}(x)}\left( \prod _{k=0}^{M_n -1}\frac{m_{n+k}}{m_{n+k}-1}\right) d\mu \\= & {} \sum _{x_0=0}^{m_0-1}\cdots \sum _{x_{n-1}=0}^{m_{n-1}-1}\sum _{x_n=0}^{m_n-1}\cdots \sum _{x_{n+M_n-1}=1}^{m_{n+M_n-1}-1}\frac{1}{M_{n+M_n}}\left( \prod _{k=0}^{M_n -1}\frac{m_{n+k}}{m_{n+k}-1}\right) \\= & {} \frac{1}{M_n \prod _{k=0}^{M_n-1}{m_{n+k}}}\left( \prod _{k=0}^{M_n -1}\frac{m_{n+k}}{m_{n+k}-1}\right) \sum _{x_0=0}^{m_0-1}\cdots \sum _{x_{n-1}=0}^{m_{n-1}-1}\sum _{x_n=0}^{m_n-1}\cdots \sum _{x_{n+M_n-1}=1}^{m_{n+M_n-1}-1}1\\= & {} \frac{1}{M_n \prod _{k=0}^{M_n-1}{m_{n+k}}}\left( \prod _{k=0}^{M_n -1}\frac{m_{n+k}}{m_{n+k}-1}\right) M_n \prod _{k=0}^{M_n-1}{\left( m_{n+k}-1\right) }= 1.\\ \end{aligned}$$

Clearly, $Q_n$ is a Vilenkin polynomial. Moreover, since the terms of the expanded product have pairwise disjoint spectra, by expanding the product used to define $Q_n,$ it is easy to see for $k=0,1,\dots , M_n -1$ that

$$\begin{aligned} S_{M_{n+k} + \alpha _n}Q_n - S_{M_{n+k}}Q_n=\frac{\pm 1}{m_{n+k}-1}r_{n+k}S_{\alpha _n}(\tau _{x_k^{(n)}}g_n ), \end{aligned}$$

where $+$ sign is if $M_n -1$ is even number and − sign is if $M_n -1$ is odd.

Since

$$\begin{aligned} \left| \left( S_{M_{n+k} }Q_n\right) (x)\right| \le \left\| D_{M_{n+k} } \right\| _1 \left\| Q_n\right\| _1\le 1, \end{aligned}$$

choice of the integers $\alpha _n$ therefore, for sufficiently large n imply

$$\begin{aligned} \left| \left( S_{M_{n+k} +\alpha _n}Q_n\right) (x)\right|> & {} Cn-\left| \left( S_{M_{n+k} }Q_n\right) (x)\right| \\ \nonumber> & {} \frac{C}{2}n, \ \ \ \ \ (x\in I_n(x_k^{(n)})). \end{aligned}$$

(29)

Let $n_1< n_2 < \cdots$ be positive integers chosen so that

$$\begin{aligned} \sum _{j=1}^{\infty }\frac{1}{\sqrt{n_j}}<\infty \ \ \ \text { and set } \ \ \ _j:=\frac{Q_{n_j}}{\sqrt{n_j}} \ \ \ \ \ \ (j \in {\mathbb {N}}_+). \end{aligned}$$

It is evident that

$$\begin{aligned} \sum _{j=1}^{\infty } \Vert P_j \Vert _1 < \infty . \end{aligned}$$

Moreover, for a fixed $x \in G_m$ it is possible to choose integers $0\le k_{(j)}<2^{n_j}$ such that $x\in I_{n_j}\left( x_{k_{(j)}}^{(n_j)}\right) \ (j\in {\mathbb {N}}_+).$ Hence, (29) implies

$$\begin{aligned} \left( S^*P_j\right) (x)\ge \frac{C}{2}\sqrt{n_j} \end{aligned}$$

for $j\in {\mathbb {N}}_+$ and $x\in G_m.$ Consequently, $G_m$ is a set of divergence for $L_1 (G_m)$ by Lemma 9. The proof is complete. $\square$