Some weak type inequalities and almost everywhere convergence of Vilenkin–Nörlund means

Abstract

We prove and discuss some new weak type $(1,1 )$ inequalities of maximal operators of Vilenkin–Nörlund means generated by monotone coefficients. Moreover, we use these results to prove a.e. convergence of such Vilenkin–Nörlund means. As applications, both some well-known and new inequalities are pointed out.

1 Introduction

In 1947 Vilenkin [31] actually introduced a large class of compact groups (now called Vilenkin groups) and the corresponding characters. In particular, Vilenkin investigated the 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 $\{{\psi}_{j}\}_{j=0}^{\infty}$ as follows:

$$ \psi _{n}(x):=\prod_{k=0}^{\infty }r_{k}^{n_{k}} ( x ), \qquad r_{k} ( x ) :=\exp ( 2\pi ix_{k}/m_{k} ),\quad \bigl(i^{2}=-1,x\in G_{m},k\in \mathbb{N} \bigr), $$

where $\mathbb{N}_{+}$ denotes the set of positive integers and $\mathbb{N}:=\mathbb{N}_{+}\cup \{0\}$. In this paper we discuss bounded Vilenkin groups only, that is, $\sup_{n\in \mathbb{N}}m_{n}<\infty$. The Vilenkin system is orthonormal and complete in $L^{2} ( G_{m} )$ (see [31]). Specifically, we call this system the Walsh–Paley system when $m\equiv 2$.

It is well known (see e.g. the books [1] and [27]) that if $f\in L^{1}(G_{m})$ and the Vilenkin series $T (x )=\sum_{j=0}^{\infty}c_{j}\psi _{j} (x )$ converges to f in $L^{1}$-norm, then

$$ c_{j}= \int _{G_{m}}f\overline{\psi }_{j}\,d\mu :=\widehat{f} ( j ), \quad j=0,1,2,\ldots, $$

where $c_{j}$ is called the jth Vilenkin–Fourier coefficient and μ is the Haar measure on the locally compact abelian groups $G_{m}$, which coincide with the direct product of measures $\mu _{k} ( \{j\} ) :=1/m_{k}$ ($j\in Z_{m_{k}}$).

The classical theory of Hilbert spaces (for details, see e.g. the books [1, 27]) implies that if we consider the partial sums $S_{n}$, defined by

$$ S_{n}f:=\sum_{k=0}^{n-1} \widehat{f} (k )\psi _{k}, $$

with respect to any orthonormal systems and among them to Vilenkin systems, then the inequality $\Vert S_{n} f \Vert _{2}\leq \Vert f \Vert _{2}$ holds. It follows that for every $f\in L^{2}$,

$$ \Vert S_{n}f-f \Vert _{2}\to 0 \quad \text{as } n\to \infty . $$

Since

$$ S_{n}f (x )= \int _{G_{m}}f (t )D_{n} (x-t )\,d\mu (t ) $$

and the Dirichlet kernels

$$ D_{n} :=\sum_{k=0}^{n-1}\psi _{k } \quad ( n\in \mathbb{N}_{+} ) $$

are not uniformly bounded in $L^{1}(G_{m})$, the boundedness of partial sums does not hold from $L^{1}(G_{m})$ to $L^{1}(G_{m})$.

The analogue of Carleson’s theorem for the Walsh system was proved by Billard [4] for $p=2$ and by Sjölin [29] for $1 < p<\infty$, while for bounded Vilenkin systems it was proved by Gosselin [13]. 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

$$ \bigl\Vert S^{\ast }f \bigr\Vert _{p}\leq C_{p} \Vert f \Vert _{p},\quad \text{when }f\in L^{p}, 1< p< \infty . $$

A recent proof of almost everywhere convergence of subsequences of Walsh–Fourier series was given by Demeter [7] in 2015. Hence, if $f\in L^{p}(G_{m})$ for $p>1$, then

$$ S_{n}f\to f \quad \text{a.e. on } G_{m}. $$

Persson, Schipp, Tephnadze, and Weisz [22] (see also [25]) gave a new and shorter proof of almost everywhere convergence of Vilenkin–Fourier series of $f\in L^{p}(G_{m})$, which was based on the theory of martingales.

The nth Nörlund mean $L_{n}$ is defined by

$$\begin{aligned} L_{n}f :=&\frac{1}{l_{n}}\sum_{k=0}^{n-1} \frac{S_{k}f}{n-k}, \quad \text{where } l_{n}:=\sum _{k=1}^{n}\frac{1}{k}. \end{aligned}$$

In [9] Gát and Goginava proved some properties of the Nörlund logarithmic means of integrable functions in $L^{1}$ norm. Moreover, in [10] they proved that weak type $(1,1)$ inequality does not hold for the maximal operator of Nörlund logarithmic means $L^{\ast}$, defined by

$$\begin{aligned} L^{\ast}f :=&\sup_{n\in \mathbb{N}} \vert L_{n}f \vert , \end{aligned}$$

but there exists an absolute constant $c_{p}$ such that the inequality

$$ \bigl\Vert L^{\ast }f \bigr\Vert _{p}\leq c_{p} \Vert f \Vert _{p}\quad \text{when }f\in L^{p}, p>1 $$

holds.

If we define the so-called generalized number system based on m in the following way:

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

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. Moreover, if we consider the following restricted maximal operator $\widetilde{L}_{\#}^{\ast}$, defined by

$$\begin{aligned} \widetilde{L}_{\#}^{\ast}f :=&\sup_{n\in \mathbb{N}} \vert L_{M_{n}}f \vert , \end{aligned}$$

then

$$\begin{aligned} y \mu \bigl\{ \widetilde{L}_{\#}^{\ast}f>y \bigr\} \leq c \Vert f \Vert _{1},\quad f\in L^{1}(G_{m}), y>0. \end{aligned}$$

Hence, if $f\in L^{1}(G_{m})$, then $L_{M_{n}}f\to f$ a.e. on $G_{m}$.

If we consider the Fejér means $\sigma _{n}$ and Fejér kernels $K_{n}$, defined by

$$ \sigma _{n}f:=\frac{1}{n}\sum_{k=1}^{n}S_{k}f \quad \text{and}\quad K_{n}:= \frac{1}{n}\sum_{k=0}^{n-1}D_{k}, $$

it is obvious that

$$\begin{aligned} \sigma _{n}f (x ) = (f\ast K_{n} ) (x )= \int _{G_{m}}f (t ) K_{n} (x-t )\,d\mu (t ). \end{aligned}$$

Since $\Vert K_{n} \Vert _{1} \leq c<\infty$, we obtain that the Fejér means are bounded from the space $L^{p}$ to the space $L^{p}$ for $1\leq p\leq \infty$. The a.e. convergence of Fejér means is due to Schipp [26] for Walsh series and Pál, Simon [21] (see also Simon, Weisz [28] and Weisz [28, 32–34]) for bounded Vilenkin series proved that the maximal operator of Fejér means $\sigma ^{\ast }$, defined by

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

is of weak type $(1,1)$, from which the a.e. convergence follows by standard argument (see [14]). Another well-known summability method is the so-called $(C,\alpha )$-means (denoted by $\sigma _{n}^{\alpha}$), which are defined by

$$ \sigma _{n}^{\alpha}f:=\frac{1}{A_{n}^{\alpha}} \sum _{k=1}^{n}A_{n-k}^{\alpha -1}S_{k}f,\qquad A_{0}^{\alpha}:=0, \qquad A_{n}^{\alpha}:= \frac{ (\alpha +1 )\cdots (\alpha +n )}{n!},\quad \alpha \neq -1,-2,\ldots. $$

It is well known that for $\alpha =1$ this summability method coincides with the Fejér summation and for $\alpha =0$ we just have the partial sums of the Vilenkin–Fourier series. Moreover, if we consider the maximal operator of the Cesáro means $\sigma ^{\alpha ,\ast}$, defined by

$$\begin{aligned} \sigma ^{\alpha ,\ast}f :=&\sup_{n\in \mathbb{N}} \bigl\vert \sigma _{n}^{ \alpha }f \bigr\vert \quad \text{for } 0< \alpha \leq 1, \end{aligned}$$

then the following weak type inequality holds (for details, see [23]):

$$\begin{aligned} y \mu \bigl\{ \sigma ^{\alpha ,\ast}f>y \bigr\} \leq c \Vert f \Vert _{1},\quad f\in L^{1}(G_{m}), y>0. \end{aligned}$$

The boundedness of the maximal operator of the Cesáro means does not hold from $L^{1}(G_{m})$ to the space $L^{1}(G_{m})$. However,

$$ \bigl\Vert \sigma _{n}^{\alpha}f-f \bigr\Vert _{p}\rightarrow 0, \quad \text{when }n\rightarrow \infty , \bigl(f\in L^{p}(G_{m}), 1\leq p< \infty \bigr). $$

The nth Nörlund mean $t_{n}$ for the Fourier series of f is defined by

$$ t_{n}f:=\frac{1}{Q_{n}}\sum_{k=1}^{n}q_{n-k}S_{k}f, $$

where $\{q_{k}:k\in \mathbb{N}\}$ is a sequence of nonnegative numbers and $Q_{n}:=\sum_{k=0}^{n-1}q_{k}$.

If we assume that $q_{0}>0$ and $\lim_{n\rightarrow \infty }Q_{n}=\infty$, then it is well known (see [15]) that the summability method generated by $\{q_{k}:k\geq 0\}$ is regular if and only if $\lim_{n\rightarrow \infty }\frac{q_{n-1}}{Q_{n}}=0$. The representation

$$ t_{n}f (x )= \int_{G_{m}}f (t )F_{n} (x-t )\,d\mu (t ),\quad \text{where } F_{n}:= \frac{1}{Q_{n}}\sum_{k=1}^{n}q_{n-k}D_{k} $$

plays a central role in the sequel. The Nörlund means are generalizations of the Fejér, Cesàro, and Nörlund logarithmic means.

Móricz and Siddiqi [16] investigated the approximation properties of some special Nörlund means of Walsh–Fourier series of $L^{p}$ functions in norm. Similar problems for the two-dimensional case can be found in papers by Nagy [17–20] (see also [5]).

Let us define the maximal operator $t^{\ast}$ of Nörlund means by

$$\begin{aligned} t^{\ast}f:=\sup_{n\in \mathbb{N}} \vert t_{n}f \vert , \end{aligned}$$

and if $\{q_{k}:k\in \mathbb{N}\}$ is nonincreasing and satisfying the condition

$$ \frac{1}{Q_{n}}=O \biggl( \frac{1}{n} \biggr)\quad \text{as } n \rightarrow \infty , $$

(1)

then in [23] it was proved that the weak type inequality

$$\begin{aligned} y \mu \bigl\{ t^{\ast}f>y \bigr\} \leq c \Vert f \Vert _{1},\quad f\in L^{1}(G_{m}), y>0 \end{aligned}$$

(2)

holds. When the sequence $\{q_{k}:k\in \mathbb{N}\}$ is nonincreasing, then the weak type $(1,1)$ inequality (2) holds for every maximal operator of Nörlund means. The boundedness of the maximal operator of the Nörlund means does not hold from $L^{1}(G_{m})$ to the space $L^{1}(G_{m})$. However,

$$ \Vert t_{n}f-f \Vert _{p}\rightarrow 0 \quad \text{as }n \rightarrow \infty \ \bigl(f\in L^{p}(G_{m}), 1 \leq p< \infty \bigr). $$

Moreover, if $\{q_{k}:k\in \mathbb{N}\}$ is nondecreasing and satisfying the condition

$$ \frac{q_{n-1}}{Q_{n}}=O \biggl( \frac{1}{n} \biggr) \quad \text{as } n\rightarrow \infty , $$

(3)

or $\{q_{k}:k\in \mathbb{N}\}$ is nonincreasing, then for any $f\in L^{1}(G_{m})$ we have that

$$ \lim_{n\rightarrow \infty }t_{n}f(x)=f(x) $$

for all Vilenkin–Lebesgue points of f.

In this paper we investigate a wider class of Nörlund means and prove that if $\{q_{k}:k\in \mathbb{N}\}$ is nondecreasing and satisfying the conditions

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

(4)

then the weak type inequality (2) holds. In particular, from this result follows almost everywhere convergence of such Nörlund means.

The paper is organized as follows: In Sect. 3 we present and prove the main results. Moreover, in order not to disturb our discussions in this section, some preliminaries are given in Sect. 2. Also some of these results are new and of independent interest.

2 Preliminaries

Lemma 1

(see [1, 12])

Let $n\in \mathbb{N}$. Then

$$ D_{M_{n}} (x )=\textstyle\begin{cases} M_{n}, & x\in I_{n}, \\ 0, & x\notin I_{n}. \end{cases} $$

Moreover, if $n\in \mathbb{N}$ and $1\leq s_{n}\leq m_{n}-1$, then

$$ D_{s_{n}M_{n}}=D_{M_{n}}\sum_{k=0}^{s_{n}-1} \psi _{kM_{n}}=D_{M_{n}} \sum_{k=0}^{s_{n}-1}r_{n}^{k} $$

and

$$ D_{n}=\psi _{n} \Biggl(\sum_{j=0}^{\infty}D_{M_{j}} \sum_{k=m_{j}-n_{j}}^{m_{j}-1}r_{j}^{k} \Biggr) \quad \textit{for } n=\sum_{i=0}^{\infty}n_{i}M_{i}, $$

where $n=\sum_{i=0}^{\infty}n_{i}M_{i}$. We note that $\sum_{k=m_{j}-n_{j}}^{m_{j}-1}r_{j}^{k}\equiv 0$ for all $n_{j}=0$.

Lemma 2

(see [8])

Let $n>t$, $t,n\in \mathbb{N}$. Then

$$ K_{M_{n}} (x )=\textstyle\begin{cases} \frac{M_{t}}{1-r_{t} (x ) },& x\in I_{t}\backslash I_{t+1}, x-x_{t}e_{t}\in I_{n}, \\ \frac{M_{n}+1}{2}, & x\in I_{n}, \\ 0, & \textit{otherwise. } \end{cases} $$

Lemma 3

(see [3, 6, 23, 24, 30])

If $n\geq M_{N}$ and $\{q_{k}:k\in \mathbb{N}\}$ is a sequence of nondecreasing numbers, then there exists an absolute constant c such that

$$ \Biggl\vert \frac{1}{Q_{n}}\sum_{j=M_{N}}^{n} q_{n-j}D_{j} \Biggr\vert \leq \frac{c}{M_{N}} \Biggl\{ \sum_{j=0}^{ \vert n \vert }M_{j} \vert K_{M_{j}} \vert \Biggr\} . $$

If the sequence $\{q_{k}:k\in \mathbb{N}\}$ is either nondecreasing and satisfying condition (3) or nonincreasing and satisfying condition (1), then the inequality

$$ \vert F_{n} \vert \leq \frac{c}{n} \Biggl\{ \sum _{j=0}^{ \vert n \vert }M_{j} \vert K_{M_{j}} \vert \Biggr\} $$

holds. On the other hand, if $\{q_{k}:k\in \mathbb{N}\}$ is a sequence of nonincreasing numbers satisfying (4) for $0<\alpha <1$, then there exists a constant $c_{\alpha}$, depending only on α, such that the following inequality holds:

$$ \vert F_{n} \vert \leq \frac{c_{\alpha}}{n^{\alpha}} \Biggl\{ \sum_{j=0}^{ \vert n \vert }M_{j}^{\alpha} \vert K_{M_{j}} \vert \Biggr\} . $$

(5)

Lemma 4

(see [3, 6, 23, 24])

Let $\{q_{k}:k\in \mathbb{N}\}$ be either a sequence of nondecreasing numbers or nonincreasing numbers satisfying condition (1) or nonincreasing numbers satisfying the conditions in (4). Then, for any $n, N\in \mathbb{N_{+}}$,

$$\begin{aligned} & \int _{G_{m}} F_{n}(x)\,d\mu (x)=1, \\ &\sup_{n\in \mathbb{N}} \int _{G_{m}} \bigl\vert F_{n}(x) \bigr\vert \,d\mu (x)\leq c< \infty , \\ &\sup_{n\in \mathbb{N}} \int _{G_{m} \backslash I_{N}} \bigl\vert F_{n}(x) \bigr\vert \,d\mu (x)\rightarrow 0\quad \textit{as } n\rightarrow \infty , \end{aligned}$$

where

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

for any $x\in G_{m}$, $n\in \mathbb{N}$.

The next lemma is very important to study problems concerning almost everywhere convergence.

Lemma 5

(see [32])

Suppose that the σ-sublinear operator V is bounded from $L^{p_{1}}$ to $L^{p_{1}}$ for some $1< p_{1}\leq \infty$ and

$$ \int _{\overline{I}} \vert Vf \vert \,d\mu \leq C \Vert f \Vert _{1} $$

for $f\in L^{1}$ and Vilenkin interval I, which satisfy

$$ \operatorname{supp} f\subset I,\qquad \int _{G_{m}}f\,d\mu =0. $$

(6)

Then the operator V is of weak type $( 1,1 )$, i.e., the following inequality holds:

$$ \sup_{y>0}y\mu \bigl( \{ Vf>y \} \bigr) \leq \Vert f \Vert _{1}. $$

Lemma 6

(see [14])

Let

$$ T,T_{n}:L^{p} ( G_{m} )\rightarrow L^{p} ( G_{m} ) $$

be sublinear operators for some $1\leq p<\infty$ with T bounded and

$$ T_{n}f\rightarrow Tf \quad \textit{a.e. on } G_{m} \textit{ as } n \rightarrow \infty , $$

for each $f\in X_{0}$, where $X_{0}$ is dense in $L^{p} ( G_{m} )$. Set

$$ T^{\ast }f:=\sup_{n\in \mathbb{N}} \vert T_{n}f \vert , \quad f\in X. $$

If there is a constant $C>0$, independent of f and n, such that the weak type inequalities

$$ y^{p} \mu \bigl( \bigl\{ \vert Tf \vert >y \bigr\} \bigr) \leq C \Vert f \Vert ^{p}_{X} $$

and

$$ y^{p}\mu \bigl( \bigl\{ T^{\ast }f>y \bigr\} \bigr) \leq C \Vert f \Vert ^{p}_{X} $$

hold for all $y>0$ and $f\in L^{p} ( G_{m} )$, then

$$ Tf=\lim_{n\rightarrow \infty }T_{n}f\quad \textit{a.e. on } G_{m} $$

for every $f\in L^{p} ( G_{m} )$.

Next we prove a new lemma of independent interest, which is very important to prove almost everywhere convergence of Nörlund means generated by nondecreasing sequences $\{q_{k}:k\in \mathbb{N}\}$.

Lemma 7

Let $n\in \mathbb{N}$ and $\{q_{k}:k\in \mathbb{N}\}$ be a sequence of nondecreasing numbers. Then

$$\begin{aligned}& \int _{\overline{I_{N}}}\sup_{n>M_{N}} \Biggl\vert \frac{1}{Q_{n}} \sum_{j=M_{N}}^{n}q_{n-j}D_{j} (x ) \Biggr\vert \,d\mu (x)\leq c< \infty , \end{aligned}$$

where c is an absolute constant.

Proof

If we define

$$\begin{aligned} I_{N}^{k,l}:=\textstyle\begin{cases} I_{N}(0,\ldots ,0,x_{k}\neq 0,0,\ldots,0,x_{l}\neq 0,x_{l+1},\ldots ,x_{N-1}, \ldots ), \\ \quad \text{for } k< l< N, \\ I_{N}(0,\ldots ,0,x_{k}\neq 0,x_{k+1}=0,\ldots ,x_{N-1}=0,x_{N}, \ldots ), \\ \quad \text{for } l=N. \end{cases}\displaystyle \end{aligned}$$

then we can decompose $\overline{I_{N}}:=G_{m} \backslash I_{N}$ as

$$ G_{m} \backslash I_{N}=\bigcup _{s=0}^{N-1}I_{s} \backslash I_{s+1}= \Biggl(\bigcup_{k=0}^{N-2} \bigcup_{l=k+1}^{N-1}I_{N}^{k,l} \Biggr)\cup \Biggl( \bigcup^{N-1}_{k=0}I_{N}^{k,N} \Biggr). $$

(7)

Let $n>M_{N}$ and

$$ x\in I_{N}^{k,l},\quad k=0,\dots ,N-2, l=k+1,\dots ,N-1. $$

By using Lemma 3, we get that

$$\begin{aligned} \Biggl\vert \frac{1}{Q_{n}}\sum_{j=M_{N}}^{n}q_{n-j}D_{j} (x ) \Biggr\vert \leq &\frac{c}{M_{N}}\sum_{i=0}^{l}M_{i} \bigl\vert K_{M_{i}} (x ) \bigr\vert \\ \leq &\frac{c}{M_{N}}\sum_{i=0}^{l}M_{i} M_{k} \\ \leq& \frac{cM_{l}M_{k}}{M_{N}} \end{aligned}$$

so that

$$\begin{aligned} \sup_{n>M_{N}} \Biggl\vert \frac{1}{Q_{n}} \sum_{j=M_{N}}^{n}q_{n-j}D_{j} (x ) \Biggr\vert \leq &\frac{c}{M_{N}}\sum_{i=0}^{ \vert n \vert }M_{i} \bigl\vert K_{M_{i}} (x ) \bigr\vert \\ \leq & \frac{cM_{l}M_{k}}{M_{N}}. \end{aligned}$$

(8)

Let $n>M_{N}$ and $x\in I_{N}^{k,N}$. By using Lemma 1, we can conclude that

$$ \bigl\vert D_{n}(x) \bigr\vert \leq cM_{k} $$

and

$$\begin{aligned} \Biggl\vert \frac{1}{Q_{n}}\sum_{j=M_{N}}^{n}q_{n-j}D_{j} (x ) \Biggr\vert \leq &\frac{c}{Q_{n}} \sum^{n }_{j=M_{N}}q_{n-j}M_{k} \\ \leq &\frac{cQ_{n-M_{N}}}{Q_{n}}M_{k}\leq cM_{k}, \end{aligned}$$

so that

$$\begin{aligned} \sup_{n>M_{N}} \Biggl\vert \frac{1}{Q_{n}} \sum_{j=M_{N}}^{n}q_{n-j}D_{j} (x ) \Biggr\vert \leq cM_{k}. \end{aligned}$$

(9)

Hence, if we apply estimates (8) and (9), then we get that

$$\begin{aligned}& \int _{\overline{I_{N}}}\sup_{n>M_{N}} \Biggl\vert \frac{1}{Q_{n}} \sum_{j=M_{N}}^{n}q_{n-j}D_{j} (x ) \Biggr\vert \,d\mu \\& \quad = \sum_{k=0}^{N-2} \sum _{l=k+1}^{N-1} \sum_{x_{j}=0,j\in \{l+1,\ldots,N-1 \}}^{m_{j-1}} \int _{I_{N}^{k,l}}\sup_{n>M_{N}} \Biggl\vert \frac{1}{Q_{n}}\sum_{j=M_{N}}^{n}q_{n-j}D_{j} (x ) \Biggr\vert \,d\mu \\& \qquad {} + \sum_{k=0}^{N-1} \int _{I_{N}^{k,N}}\sup_{n>M_{N}} \Biggl\vert \frac{1}{Q_{n}}\sum_{j=M_{N}}^{n}q_{n-j}D_{j} (x ) \Biggr\vert \,d\mu \\& \quad \leq c\sum_{k=0}^{N-2} \sum _{l=k+1}^{N-1}\frac{m_{l+1}\cdots m_{N-1}}{M_{N}} \frac{M_{l}M_{k}}{M_{N}}+c\sum_{k=0}^{N-1} \frac{M_{k}}{M_{N}} \\& \quad \leq c \sum_{k=0}^{N-2} \frac{(N-k)M_{k}}{M_{N}}+c< C< \infty . \end{aligned}$$

The proof is complete. □

We also need the following new lemmas.

Lemma 8

Let $\{q_{k}:k\in \mathbb{N}\}$ be a sequence of nonincreasing numbers satisfying condition (1). Then there exists an absolute constant c such that

$$\begin{aligned}& \int _{\overline{I_{N}}}\sup_{n>M_{N}} \vert F_{n} \vert \,d \mu \leq c< \infty . \end{aligned}$$

Proof

The proof is analogous to that of Lemma 7. Hence, we leave out the details. □

Lemma 9

Let $\{q_{k}:k\in \mathbb{N}\}$ be a sequence of nondecreasing numbers satisfying condition (3). Then there exists an absolute constant c such that

$$\begin{aligned}& \int _{\overline{I_{N}}}\sup_{n>M_{N}} \vert F_{n} \vert \,d \mu \leq c< \infty . \end{aligned}$$

Proof

Also in this case the proof is analogous to that of Lemma 7, so we leave out the details. □

Finally, we prove the following new estimate of independent interest.

Lemma 10

Let $n\in \mathbb{N}$ and $\{q_{k}:k\in \mathbb{N}\}$ be a sequence of nonincreasing numbers satisfying the conditions in (4). Then there exists an absolute constant c such that

$$\begin{aligned} \int _{\overline{I_{N}}} \sup_{n>M_{N}} \Biggl\vert \frac{1}{Q_{n}} \sum_{j=M_{N}}^{n}q_{n-j}D_{j} (x ) \Biggr\vert \,d\mu (x)\leq c< \infty . \end{aligned}$$

(10)

Proof

Let $n>M_{N}$ and $x\in I_{N}^{k,l}$, $k=0,\dots ,N-2$, $l=k+1,\dots ,N-1$. By combining Lemma 2 and (5) in Lemma 3, we get that

$$\begin{aligned} & \Biggl\vert \frac{1}{Q_{n}}\sum_{j=M_{N}}^{n}q_{n-j}D_{j} (x ) \Biggr\vert \leq \frac{cM_{l}^{\alpha }M_{k}}{M_{N}^{\alpha}}, \end{aligned}$$

so that

$$\begin{aligned} &\sup_{n>M_{N}} \Biggl\vert \frac{1}{Q_{n}} \sum_{j=M_{N}}^{n} q_{n-j}D_{j} (x ) \Biggr\vert \leq \frac{cM_{l}^{\alpha }M_{k}}{M_{N}^{\alpha}}. \end{aligned}$$

(11)

Let $n>M_{N}$ and $x\in I_{N}^{k,N}$. By using Lemma 1, we can conclude that

$$\begin{aligned} \Biggl\vert \frac{1}{Q_{n}}\sum_{j=M_{N}}^{n}q_{n-j}D_{j} (x ) \Biggr\vert \leq \frac{c}{Q_{n}} \sum^{n }_{j=M_{N}}q_{n-j}M_{k} \leq cM_{k}, \end{aligned}$$

so that

$$\begin{aligned} \sup_{n>M_{N}} \Biggl\vert \frac{1}{Q_{n}} \sum_{j=M_{N}}^{n}q_{n-j}D_{j} (x ) \Biggr\vert \leq cM_{k}. \end{aligned}$$

(12)

By combining (7), (11), and (12), we can conclude that

$$\begin{aligned}& \int _{\overline{I_{N}}}\sup_{n>M_{N}} \Biggl\vert \frac{1}{Q_{n}} \sum_{j=M_{N}}^{n}q_{n-j}D_{j} \Biggr\vert \,d\mu \\& \quad = \sum_{k=0}^{N-2} \sum _{l=k+1}^{N-1} \sum_{x_{j}=0,j\in \{l+1,\ldots,N-1 \}}^{m_{j-1}} \int _{I_{N}^{k,l}}\sup_{n>M_{N}} \vert F_{n} \vert \,d\mu \\& \qquad {} +\sum_{k=0}^{N-1} \int _{I_{N}^{k,N}}\sup_{n>M_{N}} \vert F_{n} \vert \,d\mu \\& \quad \leq c\sum_{k=0}^{N-2} \sum _{l=k+1}^{N-1}\frac{m_{l+1}\cdots m_{N-1}}{M_{N}} \frac{M_{l}^{\alpha}M_{k}}{M^{\alpha}_{N}}+c \sum_{k=0}^{N-1} \frac{M_{k}}{M_{N}} \\& \quad \leq c\sum_{k=0}^{N-2} \sum _{l=k+1}^{N-1} \frac{M_{l}^{\alpha -1}M_{k}}{M^{\alpha}_{N}}+c \sum _{k=0}^{N-1}\frac{M_{k}}{M_{N}} \\& \quad \leq c\sum_{l=k+1}^{N-1} \frac{M^{\alpha}_{k}}{M^{\alpha}_{N}}+c \\& \quad < C< \infty , \end{aligned}$$

so (10) holds and the proof is complete. □

3 The main results

Our first main result reads as follows.

Theorem 1

Let $t_{n}$ be the Nörlund means and $F_{n}$ be the corresponding Nörlund kernels such that

$$ \int_{\overline{I_{N}}}\sup_{n>M_{N}} \Biggl\vert \frac{1}{Q_{n}}\sum_{k=M_{N}+1}^{n}q_{n-k}D_{k} (x ) \Biggr\vert \,d\mu (x )< c< \infty . $$

If the maximal operator $t^{*}$ of Nörlund means is bounded from $L^{p_{1}}$ to $L^{p_{1}}$ for some $1< p_{1}\leq \infty$, then the operator $t^{*}$ is of weak type $( 1,1 )$, i.e., for all $f\in L^{1}(G_{m})$, the following weak type inequality holds:

$$ \sup_{y>0}y\mu \bigl\{ t^{*}f>y \bigr\} \leq \Vert f \Vert _{1}. $$

Proof

In view of Lemma 5 we obtain that the proof is complete if we prove that

$$ \int _{\overline{I}} \bigl\vert t^{*}f(x) \bigr\vert \,d\mu (x) \leq c \Vert f \Vert _{1} $$

(13)

for every function f, which satisfies the conditions in (6), where I denotes the support of the function f.

Without loss of generality we may assume that f is a function with support I and $\mu ( I ) =M_{N}$. We may also assume that $I=I_{N}$. It is easy to see that

$$ t_{n}f =0\quad \text{when } n\leq M_{N}. $$

Therefore, we can suppose that $n>M_{N}$. Moreover,

$$ S_{n}f =0 \quad \text{for } n\leq M_{N}, $$

so that

$$ \frac{1}{Q_{n}} \Biggl(\sum_{k=0}^{M_{N}}q_{n-k}S_{k}f (x ) \Biggr)=0, $$

which implies that

$$ \int_{I_{N}} \frac{1}{Q_{n}} \Biggl( \sum _{k=0}^{M_{n}}q_{n-k}D_{k} (x-t ) \Biggr)f(t)\,d\mu (t )=0. $$

Hence,

$$\begin{aligned}& \bigl\vert t^{*}f(x) \bigr\vert \\& \quad \leq \sup_{n>M_{N}} \Biggl\vert \int_{I_{N}} \frac{1}{Q_{n}} \Biggl(\sum _{k=0}^{M_{N}}q_{n-k}D_{k} (x-t ) \Biggr)f(t)\,d\mu (t ) \Biggr\vert \\& \qquad {} + \sup_{n>M_{N}} \Biggl\vert \int_{I_{N}} \frac{1}{Q_{n}} \Biggl(\sum _{k=M_{N}+1}^{n}q_{n-k}D_{k} (x-t ) \Biggr)f(t)\,d\mu (t ) \Biggr\vert \\& \quad = \sup_{n>M_{N}} \Biggl\vert \int_{I_{N}} \frac{1}{Q_{n}} \Biggl(\sum _{k=M_{N}+1}^{n}q_{n-k}D_{k} (x-t ) \Biggr)f(t)\,d\mu (t ) \Biggr\vert . \end{aligned}$$

(14)

Let $t\in I_{N}$ and $x\in \overline{I_{N}}$. Then $x-t\in \overline{I_{N}}$ and (14) implies that

$$\begin{aligned}& \int _{\overline{I_{N}}} \bigl\vert t^{*}f(x) \bigr\vert \,d \mu (x) \\& \quad \leq \int _{\overline{I_{N}}}{\sup_{n>M_{N}}} \int_{I_{N}} \Biggl\vert \frac{1}{Q_{n}} \Biggl( \sum _{k=M_{N}+1}^{n}q_{n-k}D_{k} (x-t ) \Biggr)f(t) \Biggr\vert \,d\mu (t )\,d\mu (x ) \\& \quad \leq \int _{\overline{I_{N}}} \int_{I_{N}}{\sup_{n>M_{N}}} \Biggl\vert \frac{1}{Q_{n}} \Biggl( \sum_{k=M_{N}+1}^{n}q_{n-k}D_{k} (x-t ) \Biggr)f(t) \Biggr\vert \,d\mu (t )\,d\mu (x ) \\& \quad \leq \int _{I_{N}} \int_{\overline{I_{N}}}{\sup_{n>M_{N}}} \Biggl\vert \frac{1}{Q_{n}} \Biggl( \sum_{k=M_{N}+1}^{n}q_{n-k}D_{k} (x-t ) \Biggr)f(t) \Biggr\vert \,d\mu (x )\,d\mu (t ) \\& \quad \leq \int _{I_{N}} \int_{\overline{I_{N}}}{\sup_{n>M_{N}}} \Biggl\vert \frac{1}{Q_{n}} \Biggl( \sum_{k=M_{N}+1}^{n}q_{n-k}D_{k} (x ) \Biggr)f(t) \Biggr\vert \,d\mu (x )\,d\mu (t ) \\& \quad \leq \int _{I_{N}} \bigl\vert f(t) \bigr\vert \,d\mu (t ) \int_{\overline{I_{N}}}{\sup_{n>M_{N}}} \Biggl\vert \frac{1}{Q_{n}} \Biggl( \sum_{k=M_{N}+1}^{n}q_{n-k}D_{k} (x ) \Biggr) \Biggr\vert \,d\mu (x ) \\& \quad = \Vert f \Vert _{1} \int_{\overline{I_{N}}}{ \sup_{n>M_{N}}} \Biggl\vert \frac{1}{Q_{n}} \Biggl( \sum_{k=M_{N}+1}^{n}q_{n-k}D_{k} (x ) \Biggr) \Biggr\vert \,d\mu (x ) \\& \quad \leq c \Vert f \Vert _{1}. \end{aligned}$$

Thus (13) holds, so the proof is complete. □

By using the same technique of proof, we obtain in a similar way the following result.

Theorem 2

Let $t_{n}$ be Nörlund means and $F_{n}$ be the corresponding Nörlund kernels such that

$$ \int_{\overline{I_{N}}}\sup_{n>M_{N}} \bigl\vert F_{n} (t ) \bigr\vert \,d\mu (t )< c< \infty . $$

If the maximal operator $t^{*}$ of the Nörlund means is bounded from $L^{p_{1}}$ to $L^{p_{1}}$ for some $1< p_{1}\leq \infty$, then the operator $t^{*}$ is of weak type $( 1,1 )$, i.e., the following weak type inequality

$$ \sup_{y>0}y\mu \bigl\{ t^{*}f>y \bigr\} \leq \Vert f \Vert _{1} $$

holds for all $f\in L^{1}(G_{m})$.

Next, we present a new related result concerning almost everywhere convergence of some summability methods. The study of almost everywhere convergence is one of the most difficult topics in Fourier analysis.

Theorem 3

Let $f\in L^{1}(G_{m})$ and $t_{n}$ be the regular Nörlund means with nondecreasing sequences $\{q_{k}:k\in \mathbb{N}\}$. Then

$$ t_{n}f\rightarrow f\quad \textit{a.e. as } n\rightarrow \infty . $$

Proof

Since

$$ S_{n}P=P\quad \text{for every }P\in \mathcal{P} $$

according to the regularity of Nörlund means with nondecreasing sequence $\{q_{k}:k\in \mathbb{N}\}$, we obtain that

$$ t_{n}P\rightarrow P \quad \text{a.e. as } n \rightarrow \infty , $$

where $P\in \mathcal{P}$ is dense in the space $L^{1}$.

On the other hand, by combining Lemma 4, Lemma 7, and Theorem 1, we obtain that the maximal operator $t^{\ast}$ of the Nörlund means with nondecreasing sequence $\{q_{k}:k\in \mathbb{N}\}$ is bounded from the space $L^{1}$ to the space $weak-L^{1}$, that is, the following weak type inequality holds:

$$ \sup_{y>0}y \mu \bigl\{ x\in G_{m}: \bigl\vert t^{\ast} f (x ) \bigr\vert >y \bigr\} \leq \Vert f \Vert _{1}. $$

Hence, according to Lemma 6, we obtain the claimed almost everywhere convergence of Nörlund means with nondecreasing sequence $\{q_{k}:k\in \mathbb{N}\}$:

$$ t_{n}f\rightarrow f\quad \text{a.e. as }n\rightarrow \infty . $$

The proof is complete. □

Theorem 4

Let $f\in L^{1}$ and $t_{n}$ be the Nörlund means with nondecreasing sequence $\{q_{k}:k\geq 0\}$ satisfying the conditions in (3). Then

$$ t_{n}f\rightarrow f,\quad \textit{a.e., as }n\rightarrow \infty . $$

Proof

The proof is similar to the proof of Theorem 3 if we instead apply Lemma 4, Lemma 9, and Theorem 1, so we omit the details. □

Next we consider almost everywhere convergence of Nörlund means with nonincreasing sequence $\{q_{k}:k\in \mathbb{N}\}$.

Theorem 5

Let $f\in L^{1}$ and $t_{n}$ be the Nörlund means with nonincreasing sequence $\{q_{k}:k\in \mathbb{N}\}$ satisfying condition (1). Then

$$ t_{n}f\rightarrow f\quad \textit{a.e. as } n\rightarrow \infty . $$

Proof

The proof is quite analogous to that of Theorem 3 if we apply Lemma 4, Lemma 8, and Theorem 1, so we omit the details. □

Theorem 6

Let $f\in L^{1}$ and $t_{n}$ be Nörlund means with nonincreasing sequence $\{q_{k}:k\in \mathbb{N}\}$ satisfying the conditions in (4). Then

$$ t_{n}f\rightarrow f\quad \textit{a.e. as } n\rightarrow \infty . $$

Proof

The proof is similar to the proof of Theorem 3 if we instead apply Lemma 4, Lemma 10, and Theorem 1, so we omit the details. □

Theorem 7

Let $f\in L^{1}$ and $t_{n}$ be Nörlund means with nonincreasing sequence $\{q_{k}:k\in \mathbb{N}\}$. Then

$$ t_{M_{n}}f\rightarrow f\quad \textit{a.e. as }n\rightarrow \infty . $$

Proof

If we apply the fact that (see [8–10], and [25])

$$\begin{aligned} F_{M_{n}}(x)=D_{M_{n}}(x)-\psi _{M_{n}-1}(x) \overline{F^{-1}}_{M_{n}}(x), \end{aligned}$$

we can prove that if $\{q_{k}:k\in \mathbb{N}\}$ is a sequence of nonincreasing numbers, then, for any $N\in \mathbb{N_{+}}$,

$$\begin{aligned} & \int _{G_{m}} F_{M_{n}}(x)\,d\mu (x)=1, \\ &\sup_{n\in \mathbb{N}} \int _{G_{m}} \bigl\vert F_{M_{n}}(x) \bigr\vert \,d\mu (x)\leq c< \infty , \\ &\sup_{n\in \mathbb{N}} \int _{G_{m} \backslash I_{N}} \bigl\vert F_{M_{n}}(x) \bigr\vert \,d\mu (x)\rightarrow 0 \quad \text{as } n\rightarrow \infty , \end{aligned}$$

and

$$\begin{aligned} & \int _{\overline{I_{N}}}\sup_{n>N} \Biggl\vert \frac{1}{Q_{n}} \sum_{j=M_{N}}^{M_{n}}q_{j}D_{n-j} (x ) \Biggr\vert \,d\mu (x)\leq c< \infty , \end{aligned}$$

and also in this case the proof is absolutely analogous to that of Theorem 3, so we can omit the details. □

A number of special cases of our results are of particular interest and give both well-known and new information. We just give the following examples of such corollaries.

In particular, since $\sigma _{n}$ and $\sigma _{n}^{\alpha }$ are regular Nörlund means with nondecreasing sequence $\{q_{k}:k\in \mathbb{N}\}$, we have the following consequences of our Theorems:

Corollary 1

(see [23] and [32])

Let $f\in L^{1}$. Then

$$\begin{aligned} \sigma _{n}f \rightarrow &f,\quad \textit{a.e., as }n \rightarrow \infty \end{aligned}$$

and

$$\begin{aligned} \sigma _{n}^{\alpha }f \rightarrow &f,\quad \textit{a.e., as }n\rightarrow \infty ,\textit{when }0< \alpha < 1. \end{aligned}$$

Corollary 2

(see [2] and [11])

Let $f\in L^{1}$. Then

$$ L_{M_{n}}f\rightarrow f\quad \textit{a.e. as }n\rightarrow \infty . $$

We also give the following examples of new consequences.

Corollary 3

Let $f\in L^{1}$ and the summability method $V_{n}^{\alpha}$ be defined by

$$ V_{n}^{\alpha}f:=\frac{1}{Q_{n}}\sum _{k=1}^{n} (n-k-1 )^{\alpha -1}S_{k}f. $$

Then

$$ V_{n}^{\alpha}f\rightarrow f\quad \textit{a.e. as }n \rightarrow \infty ,\textit{as } 0< \alpha < 1. $$

Proof

Since $V_{n}^{\alpha }$ are Nörlund means with nonincreasing sequences $\{q_{k}:k\in \mathbb{N}\}$ satisfying the conditions in (4). Hence, the proof is complete by just using Theorem 6. □

Corollary 4

Let $f\in L^{1}$ and the summability method $\beta _{n}^{\alpha}$ be defined by

$$ \beta _{n}^{\alpha}f:=\frac{1}{Q_{n}}\sum _{k=1}^{n}\log ^{\alpha} ( n-k-1 )S_{k}f. $$

Then

$$ \beta _{n}^{\alpha }f\rightarrow f \quad \textit{a.e. as } n \rightarrow \infty . $$

Proof

We note that $\beta _{n}^{\alpha}$ are Nörlund means with nondecreasing sequences $\{q_{k}:k\in \mathbb{N}\}$. Hence, the proof is complete by just using Theorem 3. □

Corollary 5

Let $f\in L^{1}$ and $B_{n}$ be the Nörlund means with monotone and bounded sequence $\{ q_{k}:k\in \mathbb{N} \}$. Then

$$ B_{n}f\rightarrow f \quad \textit{a.e. as } n\rightarrow \infty . $$

Proof

The proof follows from Theorems 4 and 5. □

Corollary 6

Let $f\in L^{1}$ and the summability method $U_{n}^{\alpha}$ be defined by

$$ U_{n}^{\alpha }f:=\frac{1}{Q_{n}}\sum _{k=1}^{n} \frac{S_{k}f}{ (n-k-3 )\ln ^{\alpha} (n-k-3 )}. $$

Then

$$ U^{\alpha}_{M_{n}}f\rightarrow f\quad \textit{a.e., as }n \rightarrow \infty . $$

Proof

Obviously, $U^{\alpha}_{n}$ are regular Nörlund means with nonincreasing sequences $\{q_{k}:k\in \mathbb{N}\}$, the proof follows from Theorem 7. □