Almost Everywhere Convergence of Varying Parameter Setting Cesàro Means of Fourier Series With Respect to Walsh–Kaczmarz System

Adimasu, Anteneh Tilahun

doi:10.2478/amsil-2025-0001

Full Article

1.

Introduction

Let ℕ₊ denote the set of the positive integers, $ℕ : = ℕ_{+} \cup 0\}$ \mathbb{N}:={{\mathbb{N}}_{+}}\cup \left\{ 0 \right\} and ℝ denote the set of real numbers. In this paper, C denote absolute positive constants and C_q denote positive constants depending at most on q although not always the same in different occurrences.

The Walsh–Paley system (the detail briefs can be obtained in the books of [17] and [19]) is a special product generated by the so-called Rademacher functions r_n (n ∈ ℕ). For the definition let r be the function given on the interval [0, 1) by $r (x) = \begin{array}{l} 1, & if 0 \leq x < \frac{1}{2}, \\ - 1, & if \frac{1}{2} \leq x < 1, \end{array}$ r\left( x \right)=\left\{ \begin{array}{*{35}{l}} 1, & {\rm{if}}~0\le x<\frac{1}{2}, \\ -1, & {\rm{if}}~\frac{1}{2}\le x<1, \\\end{array} \right. and extended to the whole real line ℝ periodically by 1.

Now, define r_n(x) := r(2ⁿx) (x ∈ [0, 1), n ∈ ℕ). Then the usual product system (w_n, n ∈ ℕ) of $r_{n}^{'} s$ r_{n}^{'}s is obtained in the following way: $w_{n} (x) : = \prod_{k = 0}^{\infty} r_{k}^{n_{k}}, n \in ℕ,$ {{w}_{n}}\left( x \right):=\underset{k=0}{\overset{\infty }{\mathop \prod }}\,r_{k}^{{{n}_{k}}},~\,\,\,\,\,\,\,n\in \mathbb{N}, where $n = \sum_{k = 0}^{\infty} n_{k} 2^{k}$ n=\mathup{\sum }_{k=0}^{\infty }{{n}_{k}}{{2}^{k}} is the binary decomposition of n, i.e. n_k ∈ {0, 1} (k ∈ ℕ). It is well-known (for details see the book [19]) that (w_n, n ∈ ℕ) is a complete orthonormal system with respect to the Lebesgue measure of [0, 1).

Then a basic property of the Walsh–Dirichlet Kernel is (1.1) $D_{2^{n}} (x) = \begin{array}{l} 2^{n}, & if 0 \leq x < 2^{- n}, \\ 0, & if 2^{- n} \leq x < 1. \end{array}$ {{D}_{{{2}^{n}}}}\left( x \right)=\left\{ \begin{array}{*{35}{l}} {{2}^{n}}, & {\rm{if}}~0\le x<{{2}^{-n}}, \\ 0, & {\rm{if}}~{{2}^{-n}}\le x<1. \\\end{array} \right.

This interval [0, 1) can be treated as the so called dyadic group, i.e. the set of all sequences (x_k, k ∈ ℕ) where x_k = 0 ∨ 1. The group operation ∔ is the coordinate-wise addition modulo 2, i.e. if x = (x_k, k ∈ ℕ), y = (y_k, k ∈ ℕ) then x ∔ y := x_k ⊕ y_k, k ∈ ℕ, where a ⊕ b denotes the addition modulo 2 of a, b ∈ ℕ. For example the Rademacher functions can be computed in this sense r_n(x) = (−1)^x_n (x ∈ [0, 1), n ∈ ℕ). Furthermore, D_2ⁿ = 2ⁿχI_n (n ∈ ℕ) where I_n is the set of all (x_k, k ∈ ℕ) such that x₀ = x₁ = · · · = x_n−₁ = 0 and χI_n is its characteristic function.

In this work, we focus on summability methods of Walsh–Kaczmarz–Fourier series. For any $n = 2^{s} + \sum_{k = 0}^{s - 1} n_{k} 2^{k}$ n={{2}^{s}}+\mathup{\sum }_{k=0}^{s-1}{{n}_{k}}{{2}^{k}} , where 0 < n ∈ ℕ, s ∈ ℕ, the so-called Kaczmarz rearrangement (ψ_n, n ∈ ℕ) (called Walsh–Kaczmarz system) of Walsh–Paley system is defined in the following way $ψ_{n} : = r_{s} \prod_{k = 0}^{s - 1} r_{s - k - 1}^{n_{k}} and ψ_{0} : = w_{0},$ {{\psi }_{n}}:={{r}_{s}}\underset{k=0}{\overset{s-1}{\mathop \prod }}\,r_{s-k-1}^{{{n}_{k}}}\,\,\,\,\,\,\,{\rm and}\,\,\,\,\,\,\,{{\psi }_{0}}:={{w}_{0}}, and is called Walsh–Kaczmarz system. We commonly use the following notations. Let |n| := max {k ∈ ℕ : n_k ≠ 0} (that is, 2^|n| ≤ n < 2^|n|⁺¹) and $n^{(s)} : = \sum_{k = 0}^{s - 1} n_{k} 2^{k}$ {{n}^{\left( s \right)}}:=\mathup{\sum }_{k=0}^{s-1}{{n}_{k}}{{2}^{k}} .

If f ∈ L¹[0, 1), then we can define the Fourier coefficients, the partial sums of the Fourier series, the Dirichlet kernels with respect to the Walsh–Kaczmarz system in the usual manner: $\begin{array}{l} \hat{f} (k) & : = \int_{0, 1)} f ψ_{k} d μ, k \in ℕ, \\ S_{n} f & : = \sum_{k = 0}^{n - 1} \hat{f} (k) ψ_{k}, n \in ℕ_{+}, S_{0} f : = 0, \\ D_{n} & : = \sum_{k = 0}^{n - 1} ψ_{k}, n \in ℕ_{+} \cdot \end{array}$ \begin{align} & \hat{f}\left( k \right):=\mathup{\int }_{\left[ 0,1 \right)}^{{}}f{{\psi }_{k}}d\mu ,\,\,\,\,\,\,\,k\in \mathbb{N}, \\ & {{S}_{n}}f:=\sum\limits_{k=0}^{n-1}{\hat{f}\left( k \right){{\psi }_{k}},}\,\,\,\,\,\,\,\,n\in {{\mathbb{N}}_{+}},\,\,\,\,\,\,\,{{S}_{0}}f:=0, \\ & {{D}_{n}}:=\sum\limits_{k=0}^{n-1}{{{\psi }_{k}},~\,\,\,\,\,\,\,\,n\in {{\mathbb{N}}_{+}}}\cdot \\ \end{align}

It is known that (for details see [21]) ψ is a complete orthonormal system, $ψ_{2^{m}} = w_{2^{m}} = r_{m}$ {{\psi }_{{{2}^{m}}}}={{w}_{{{2}^{m}}}}={{r}_{m}} and $ψ_{k} : k = 2^{m}, \dots, 2^{m + 1} - 1\} = w_{k} : k = 2^{m}, \dots, 2^{m + 1} - 1\}, m \in ℕ .$ \left\{ {{\psi }_{k}}:k={{2}^{m}},\ldots ,{{2}^{m+1}}-1 \right\}=\left\{ {{w}_{k}}:k={{2}^{m}},\ldots ,{{2}^{m+1}}-1 \right\},\,\,\,\,\,\,\,\,\,\,m\in \mathbb{N}. Moreover, if we define $τ_{s} (x) : = (x_{s - 1}, x_{s - 2}, \dots, x_{1}, x_{0}, x_{s}, x_{s + 1}, \dots), x \in 0, 1),$ {{\tau }_{s}}\left( x \right):=({{x}_{s-1}},~{{x}_{s-2}},~\ldots ,{{x}_{1}},~{{x}_{0}},~{{x}_{s}},~{{x}_{s+1}},\ldots ),\,\,\,\,\,\,\,~x\in \left[ 0,1 \right), then (1.2) $ψ_{n} (x) = w_{n} (τ_{s} (x)) = r_{s} (x) w_{n - 2^{s}} (τ_{s} (x))$ {{\psi }_{n}}\left( x \right)={{w}_{n}}\left( {{\tau }_{s}}\left( x \right) \right)={{r}_{s}}\left( x \right){{w}_{n-{{2}^{s}}}}\left( {{\tau }_{s}}\left( x \right) \right) and $D_{2^{j}} (τ_{j} (x)) = D_{2^{j}} (x), j \in ℕ, x \in 0, 1) .$ {{D}_{{{2}^{j}}}}\left( {{\tau }_{j}}\left( x \right) \right)={{D}_{{{2}^{j}}}}\left( x \right)~,\,\,\,\,\,\,~j\in \mathbb{N},~x\in \left[ 0,1 \right).

The Fejér means and kernels with respect to the Walsh–Kaczmarz system are defined in the usual manner: $\begin{array}{l} σ_{n}^{1} f : = \frac{1}{n} \sum_{k = 1}^{n} S_{k} f, n \in ℕ_{+}, \\ K_{n} : = \frac{1}{n} \sum_{k = 1}^{n} D_{k} = \sum_{k = 0}^{n - 1} (1 - \frac{k}{n}) w_{k}, n \in ℕ_{+} \cdot \end{array}$ \begin{align} & \sigma _{n}^{1}f:=\frac{1}{n}\sum\limits_{k=1}^{n}{{{S}_{k}}f},\,\,\,\,\,\,\,~n\in {{\mathbb{N}}_{+}}, \\ & {{K}_{n}}:=\frac{1}{n}\sum\limits_{k=1}^{n}{{{D}_{k}}}=\sum\limits_{k=0}^{n-1}{\left( 1-\frac{k}{n} \right){{w}_{k}}},\,\,\,\,\,\,\,~n\in {{\mathbb{N}}_{+}}\cdot \\ \end{align}

Let K_o := 0. The next estimation with respect to K_n (see [21]) will be used often in this work: if x ∈ [0, 1), 0 < n ∈ ℕ then (1.3) $K_{n} (x)| \leq \sum_{j = 0}^{s} 2^{j - s - 1} \sum_{i = j}^{s} (D_{2^{i}} (x) + D_{2^{i}} (x ∔ 2^{- j - 1})), 2^{s} \leq n < 2^{s + 1} .$ \left| {{K}_{n}}\left( x \right) \right|\le \sum\limits_{j=0}^{s}{{{2}^{j-s-1}}}\sum\limits_{i=j}^{s}{\left( {{D}_{{{2}^{i}}}}\left( x \right)+{{D}_{{{2}^{i}}}}\left( x\dotplus {{2}^{-j-1}} \right) \right)},\,\,\,\,\,\,{{2}^{s}}\le n<{{2}^{s+1}}. From this it follows by (1.1) the uniform L₁− boundedness of K_n in which (1.4) $\sup_{n} {K_{n}‖}_{1} \leq \infty .$ \underset{n}{\mathop{\sup }}\,\|{{K}_{n}}{{\|}_{1}}\le \infty .

Let 0 < α ≤ 1, k ∈ ℕ, and f ∈ L¹[0, 1). Then, the n^th (C, α) Walsh–Kaczmarz Kernels and (C, α) Walsh–Kaczmarz means with respect to ψ will be defined respectively as follows $\begin{matrix} Θ_{n}^{α} : = \frac{1}{A_{n - 1}^{α}} \sum_{k = 0}^{n - 1} A_{n - k - 1}^{α} ψ_{k}, \\ σ_{n}^{α} f (x) : = \int_{0}^{1} f (t) Θ_{k}^{α} (x + t) d t, x \in 0, 1), n \in ℕ, \end{matrix}$ \begin{matrix} \Theta _{n}^{\alpha }:=\frac{1}{A_{n-1}^{\alpha }}\sum\limits_{k=0}^{n-1}{A_{n-k-1}^{\alpha }{{\psi }_{k}}}, \\ \sigma _{n}^{\alpha }f\left( x \right)~:=\int_{0}^{1}{f\left( t \right)\Theta _{k}^{\alpha }\left( x+t \right)dt},\,\,\,\,\,\,~x\in \left[ 0,1 \right),~n\in \mathbb{N}, \\\end{matrix} where $A_{k}^{α} : = \prod_{i = 1}^{k} \frac{α + i}{i} .$ A_{k}^{\alpha }:=\underset{i=1}{\overset{k}{\mathop \prod }}\,\frac{\alpha +i}{i}. It is well-known that (see [24]) $A_{n}^{α} = \sum_{k = 0}^{n} A_{n - k}^{α - 1}$ A_{n}^{\alpha }=\sum\limits_{k=0}^{n}{A_{n-k}^{\alpha -1}} and $A_{n}^{α} - A_{n - 1}^{α} = A_{n}^{α - 1} and A_{n}^{α} \sim n^{α} .$ A_{n}^{\alpha }-A_{n-1}^{\alpha }=A_{n}^{\alpha -1}\,\,\,\,\text{and}\,\,\,\,A_{n}^{\alpha }\sim {{n}^{\alpha }}. α may also be a sequence α = (α_n). In this case we have sequence of (C, α_n).

The maximal operator of (C, α_n) means is defined as $σ_{*, n}^{α} f : = \sup_{n} σ_{n}^{α} f| .$ \sigma _{*,n}^{\alpha }f:=\text{ }\!\!~\!\!\text{ }\underset{n}{\mathop{\sup }}\,\left| \sigma _{n}^{\alpha }f \right|.

Here, we give also the most important concepts with respect to the dyadic Hardy spaces. Let the maximal function of f ∈ L¹[0, 1) be given by $f^{*} (x) = sup_{n} 2^{n} \int_{x ∔ I_{n}} f (t) d μ (t)|, x \in 0, 1) .$ {{f}^{*}}\left( x \right)=\underset{n}{\mathop{\text{sup}}}\,\text{ }\!\!~\!\!\text{ }{{2}^{n}}\left| \int_{x\dotplus {{I}_{n}}}{f\left( t \right)d\mu \left( t \right)} \right|,\,\,\,\,\,~x\in \left[ 0,1 \right). Then, Hardy space on [0, 1) is defined as $H^{1} 0, 1) : = {f : {f‖}_{H_{1}} : = {f *‖}_{1} < \infty} .$ {{H}^{1}}\left[ 0,1 \right):=\{f:\,\,\|f{{\|}_{{{H}_{1}}}}:=\,\|{{f}^{*}}{{\|}_{1}}<\infty \}. A function a ∈ L^∞[0, 1) is called a 1-atom if either a is identically equal to 1 or there exists a dyadic interval I = x ∔ I_N for some N ∈ ℕ, x ∈ [0, 1) such that $supp a \subset I, {a‖}_{\infty} \leq 2^{N}$ \text{supp}\,a\subset I,\,\,\,\,\,\,{{\left\| a \right\|}_{\infty }}\le {{2}^{N}} and $\int_{0}^{1} a = 0$ \int_{0}^{1}{a}=0 . We shall say that a is supported on I.

Definition 1.1 ([19])

A sublinear operator T which maps H¹[0, 1) into the collection of measurable functions defined on [0, 1) is called 1-quasi-local if there exists a constant C such that $\int_{0, 1) \ I} T a| \leq C$ \int_{\left[ 0,1 \right)\backslash I}{\left| Ta \right|}\le C for every p-atom a supported on I.

Lemma 1.2.

Let 1-quasi-local operator T is L^∞-bounded, i.e., ${T f‖}_{\infty} \leq C {f‖}_{\infty} .$ {{\left\| Tf \right\|}_{\infty }}\le C{{\left\| f \right\|}_{\infty }}. Then T is bounded from H¹[0, 1) to L¹[0, 1).

Definition 1.3.

It is already defined in [2] that $P (n, α) : = \sum_{i = 0}^{\infty} n_{i} 2^{i α} for n \in ℕ, α \in ℝ .$ P\left( n,~\alpha \right):=\sum\limits_{i=0}^{\infty }{{{n}_{i}}{{2}^{i\alpha }}}\,\,\,\,\,\,{\rm for}\,\,n\in \mathbb{N},\alpha \in \mathbb{R}. For example P (n, 1) = n.

Moreover, for the set of sequences α = (α_n) and positive real number q, we consider the following subset of natural numbers: (1.5) $ℕ_{α_{n}, q} : = n \in ℕ : \frac{P (n, α_{n})}{n^{α_{n}}} \leq q\} .$ {{\mathbb{N}}_{{{\alpha }_{n}},q}}:=\left\{ n\in \mathbb{N}:\frac{P\left( n,{{\alpha }_{n}} \right)}{{{n}^{{{\alpha }_{n}}}}}\le q \right\}.

The first result on the a.e. convergence of the (C, 1) means of Walsh–Fourier series is due to Fine [8] and Schipp [18], if the Walsh functions are considered by Paley’s ordering. The analogical result in the case of Walsh–Kaczmarz system was also investigated by many authors. One of the Kaczmarz analogue of Schipp’s [18] results was given by Gát [10]. Besides, he proved also an (H¹, L¹)-like inequality for the maximal operator of Fejér means with respect to Walsh–Kaczmarz system ${\sup_{k \in ℕ} σ_{k}^{1} f|‖}_{1} \leq c {f‖}_{H^{1}}, f \in H^{1} .$ {{\left\| \underset{k\in \mathbb{N}}{\mathop{\sup }}\,\left| \sigma _{k}^{1}f \right| \right\|}_{1}}\le c{{\left\| f \right\|}_{{{H}^{1}}}},\,\,\,\,\,\,\,\,f\in {{H}^{1}}.

Convergence and summability of Cesàro means of the one and two dimensional cases in Lebesgue and martingale Hardy spaces were studied by a lot of authors. We mention Akhobadze [3], Blahota, Persson and Tephnadze [5], Blahota, Tephnadze and Toledo [7], Blahota, Tephnadze [6], Fridli [9], Gát [12], Nagy [15, 16], Simon [20], Weisz [23].

In 2007, Akhobadze [4] introduced the notion of Cesàro means of trigonometric Fourier series with variable parameter setting. The varying parameter settings of the (C, α) means of the Walsh–Paley–Fourier series for different situation were investigated in [1], [2], [13] and with respect to the character systems of the group of 2-adic integers in [22] (for the more general orthonormal system, i.e., with respect to Vilenkin system, in [14]). However, these problems with respect to Walsh–Kaczmarz orthonormal system have not been investigated yet.

Thus, in this paper, it is going to be proved that the maximal operator of Cesàro means of Walsh–Kaczmarz–Fourier series is of weak type (L¹, L¹). Moreover, the almost everywhere convergence of Cesàro means with varying parameter setting of integrable functions (i.e. $σ_{n}^{α_{n}} f \to f$ ~\sigma _{n}^{{{\alpha }_{n}}}f\to f , as n → ∞) is proved, for f ∈ L¹, for every sequence α = (α_n, n ∈ ℕ) where 0 < α_n < 1.

2.

Main results

Lemma 2.1.

Let 0 < α_n < 1, n ∈ ℕ. Then, $Θ_{n}^{α_{n}} = \sum_{t = 1}^{6} β_{t},$ \Theta _{n}^{{{\alpha }_{n}}}=\sum\limits_{t=1}^{6}{{{\beta }_{t}}}, where $\begin{array}{l} β_{1} : = 1 + \frac{1}{A_{n - 1}^{α_{n}}} \sum_{j = 0}^{n_{1} - 1} A_{n - 2^{j} - 1}^{α_{n}} (D_{2^{j + 1}} (x) - D_{2^{j}} (x)), \\ β_{2} : = - \frac{1}{A_{n - 1}^{α_{n}}} \sum_{j = 0}^{n_{1} - 1} w_{2^{j + 1} - 1} (τ_{j} (x)) (2^{j} - 1) A_{n - 2^{j - 1}}^{α - 1} K_{2^{j - 1}} (τ_{j} (x)), \\ β_{3} : = \frac{1}{A_{n - 1}^{α_{n}}} \sum_{j = 0}^{n_{1} - 1} w_{2^{j + 1} - 1} (τ_{j} (x)) \sum_{k = 1}^{2^{j} - 2} k A_{n - 2^{j + 1} + k + 1}^{α_{n} - 2} K_{k} (τ_{j} (x)), \\ β_{4} : = \frac{1}{A_{n - 1}^{α_{n}}} \sum_{k = 2}^{q} w_{n - n^{(k)} - 1} (τ_{n_{1}} (x)) A_{n^{(k - 1)} - 1}^{α_{n}} D_{2^{n_{k}}} (τ_{n_{1}} (x)), \\ β_{5} : = \frac{1}{A_{n - 1}^{α_{n}}} \sum_{k = 2}^{q} w_{n - n^{(k)} - 1} (τ_{n_{1}} (x)) A_{n^{(k - 1)} - 1}^{α_{n} - 1} (2^{n_{k}} - 1) K_{2^{n_{k}} - 1} (τ_{n_{1}} (x)), \\ β_{6} : = \frac{1}{A_{n - 1}^{α_{n}}} \sum_{k = 2}^{q} w_{n - n^{(k)} - 1} (τ_{n_{1}} (x)) \sum_{j = 1}^{2^{n_{k}} - 2} A_{n^{(k)} + j + 1}^{α_{n} - 2} j K_{j} (τ_{n_{1}} (x)) . \end{array}$ \begin{array}{l} {\beta _1}: = 1 + \frac{1}{{A_{n - 1}^{{\alpha _n}}}}\sum\limits_{j = 0}^{{n_1} - 1} {A_{n - {2^j} - 1}^{{\alpha _n}}\left( {{D_{{2^{j + 1}}}}\left( x \right) - {D_{{2^j}}}\left( x \right)} \right)} ,\\ {\beta _2}: = - \frac{1}{{A_{n - 1}^{{\alpha _n}}}}\sum\limits_{j = 0}^{{n_1} - 1} {{w_{{2^{j + 1}} - 1}}\left( {{\tau _j}\left( x \right)} \right)\left( {{2^j} - 1} \right)A_{n - {2^{j - 1}}}^{\alpha - 1}{K_{{2^{j - 1}}}}\left( {{\tau _j}\left( x \right)} \right)} ,\\ {\beta _3}: = \frac{1}{{A_{n - 1}^{{\alpha _n}}}}\sum\limits_{j = 0}^{{n_1} - 1} {{w_{{2^{j + 1}} - 1}}\left( {{\tau _j}\left( x \right)} \right)} \sum\limits_{k = 1}^{{2^j} - 2} {kA_{n - {2^{j + 1}} + k + 1}^{{\alpha _n} - 2}{K_k}\left( {{\tau _j}\left( x \right)} \right)} ,\\ {\beta _4}: = \frac{1}{{A_{n - 1}^{{\alpha _n}}}}\sum\limits_{k = 2}^q {{w_{n - {n^{\left( k \right)}} - 1}}\left( {{\tau _{{n_1}}}\left( x \right)} \right)A_{{n^{\left( {k - 1} \right)}} - 1}^{{\alpha _n}}{D_{{2^{{n_k}}}}}\left( {{\tau _{{n_1}}}\left( x \right)} \right)} ,\\ {\beta _5}: = \frac{1}{{A_{n - 1}^{{\alpha _n}}}}\sum\limits_{k = 2}^q {{w_{n - {n^{\left( k \right)}} - 1}}\left( {{\tau _{{n_1}}}\left( x \right)} \right)A_{{n^{\left( {k - 1} \right)}} - 1}^{{\alpha _n} - 1}\left( {{2^{{n_k}}} - 1} \right){K_{{2^{{n_k}}} - 1}}\left( {{\tau _{{n_1}}}\left( x \right)} \right)} ,\\ {\beta _6}: = \frac{1}{{A_{n - 1}^{{\alpha _n}}}}\sum\limits_{k = 2}^q {{w_{n - {n^{\left( k \right)}} - 1}}\left( {{\tau _{{n_1}}}\left( x \right)} \right)\sum\limits_{j = 1}^{{2^{{n_k}}} - 2} {A_{{n^{\left( k \right)}} + j + 1}^{{\alpha _n} - 2}j{K_j}\left( {{\tau _{{n_1}}}\left( x \right)} \right)} } . \end{array}

Proof

Consider the binary expansion of 0 < n ∈ ℕ, where n_k ∈ ℕ, k = 1, ..., q and n_k ≥ n_k₊₁, k = 1, ..., q − 1. Then, $\begin{array}{l} Θ_{n}^{α} & = \frac{1}{A_{n - 1}^{α_{n}}} \sum_{k = 0}^{n - 1} A_{n - k - 1}^{α_{n}} ψ_{k} = \frac{1}{A_{n - 1}^{α_{n}}} \sum_{k = 0}^{2^{n_{1}} - 1} A_{n - k - 1}^{α_{n}} ψ_{k} + \frac{1}{A_{n - 1}^{α_{n}}} \sum_{k = 2^{n_{1}}}^{n - 1} A_{n - k - 1}^{α_{n}} ψ_{k} \\ = : Θ_{n_{1}}^{α_{n}} + Θ_{n_{2}}^{α_{n}} . \end{array}$ \begin{align} {\Theta _{n}^{\alpha }} \ & {=\frac{1}{A_{n-1}^{{{\alpha }_{n}}}}\sum\limits_{k=0}^{n-1}{A_{n-k-1}^{{{\alpha }_{n}}}{{\psi }_{k}}}=\frac{1}{A_{n-1}^{{{\alpha }_{n}}}}\sum\limits_{k=0}^{{{2}^{{{n}_{1}}}}-1}{A_{n-k-1}^{{{\alpha }_{n}}}{{\psi }_{k}}+\frac{1}{A_{n-1}^{{{\alpha }_{n}}}}}\sum\limits_{k={{2}^{{{n}_{1}}}}}^{n-1}{A_{n-k-1}^{{{\alpha }_{n}}}{{\psi }_{k}}}} \\{} & {=:\Theta _{{{n}_{1}}}^{{{\alpha }_{n}}}+\Theta _{{{n}_{2}}}^{{{\alpha }_{n}}}.} \\ \end{align} Let x ∈ [0, 1), thus by applying (1.2) we get $\begin{array}{l} Θ_{n_{1}}^{α_{n}} (x) & = 1 + \frac{1}{A_{n - 1}^{α}} \sum_{j = 0}^{n_{1} - 1} \sum_{k = 0}^{2^{j} - 1} A_{n - 1 - (2^{j + 1} - 1 - k)}^{α} ψ_{2^{j + 1} - 1 - k} (x) \\ = 1 + \frac{1}{A_{n - 1}^{α_{n}}} \sum_{j = 0}^{n_{1} - 1} \sum_{k = 0}^{2^{j} - 1} A_{n - 2^{j + 1} + k}^{α_{n}} w_{2^{j + 1} - 1 - k} (τ_{j} (x)) \\ = 1 + \frac{1}{A_{n - 1}^{α_{n}}} \sum_{j = 0}^{n_{1} - 1} \sum_{k = 0}^{2^{j} - 1} A_{n - 2^{j + 1} + k}^{α_{n}} w_{2^{j + 1} - 1} (τ_{j} (x)) w_{k} (τ_{j} (x)) \\ = 1 + \frac{1}{A_{n - 1}^{α_{n}}} \sum_{j = 0}^{n_{1} - 1} w_{2^{j + 1} - 1} (τ_{j} (x)) \\ \times (\sum_{k = 0}^{2^{j} - 1} A_{n - 2^{j + 1} + k}^{α_{n}} (D_{k + 1} (τ_{j} (x)) - D_{k} (τ_{j} (x))) . \end{array}$ \begin{array}{*{20}{l}} {\Theta _{{n_1}}^{{\alpha _n}}\left( x \right)}&{ = 1 + \frac{1}{{A_{n - 1}^\alpha }}\sum\limits_{j = 0}^{{n_1} - 1} {\sum\limits_{k = 0}^{{2^j} - 1} {A_{n - 1 - \left( {{2^{j + 1}} - 1 - k} \right)}^\alpha {\psi _{{2^{j + 1}} - 1 - k}}\left( x \right)} } }\\ {}&{ = 1 + \frac{1}{{A_{n - 1}^{{\alpha _n}}}}\sum\limits_{j = 0}^{{n_1} - 1} {\sum\limits_{k = 0}^{{2^j} - 1} {A_{n - {2^{j + 1}} + k}^{{\alpha _n}}{w_{{2^{j + 1}} - 1 - k}}\left( {{\tau _j}\left( x \right)} \right)} } }\\ {}&{ = 1 + \frac{1}{{A_{n - 1}^{{\alpha _n}}}}\sum\limits_{j = 0}^{{n_1} - 1} {\sum\limits_{k = 0}^{{2^j} - 1} {A_{n - {2^{j + 1}} + k}^{{\alpha _n}}{w_{{2^{j + 1}} - 1}}\left( {{\tau _j}\left( x \right)} \right){w_k}\left( {{\tau _j}\left( x \right)} \right)} } }\\ {}&{ = 1 + \frac{1}{{A_{n - 1}^{{\alpha _n}}}}\sum\limits_{j = 0}^{{n_1} - 1} {{w_{{2^{j + 1}} - 1}}\left( {{\tau _j}\left( x \right)} \right)} }\\ {}&{\,\,\,\,\, \times \left( {\sum\limits_{k = 0}^{{2^j} - 1} {A_{n - {2^{j + 1}} + k}^{{\alpha _n}}({D_{k + 1}}\left( {{\tau _j}\left( x \right)} \right) - {D_k}\left( {{\tau _j}\left( x \right)} \right)} } \right).} \end{array} Applying Abel’s transformation, we get the following $\begin{array}{l} Θ_{n_{1}}^{α_{n}} (x) & = 1 + \frac{1}{A_{n - 1}^{α_{n}}} \sum_{j = 0}^{n_{1} - 1} w_{2^{j + 1} - 1} (τ_{j} (x)) \\ \times (\sum_{k = 1}^{2^{j}} A_{n - 2^{j + 1} + k - 1}^{α_{_{n}}} D_{k} (τ_{j} (x)) - \sum_{k = 0}^{2^{j} - 1} A_{n - 2^{j + 1} + k}^{α_{n}} D_{k} (τ_{j} (x))) \\ = 1 + \frac{1}{A_{n - 1}^{α_{n}}} \sum_{j = 0}^{n_{1} - 1} w_{2^{j + 1} - 1} (τ_{j} (x)) A_{n - 2^{j} - 1}^{α_{n}} D_{2^{j}} (τ_{j} (x)) \\ - \frac{1}{A_{n - 1}^{α_{n}}} \sum_{j = 0}^{n_{1} - 1} w_{2^{j + 1} - 1} (τ_{j} (x)) (\sum_{k = 1}^{2^{j} - 1} A_{n - 2^{j + 1} + k - 1}^{α_{n}} - A_{n - 2^{j + 1} + k}^{α_{n}}) D_{k} (τ_{j} (x)) \\ = 1 + \frac{1}{A_{n - 1}^{α_{n}}} \sum_{j = 0}^{n_{1} - 1} w_{2^{j + 1} - 1} (τ_{j} (x)) A_{n - 2^{j} - 1}^{α_{n}} D_{2^{j}} (τ_{j} (x)) \\ - \frac{1}{A_{n - 1}^{α_{n}}} \sum_{j = 0}^{n_{1} - 1} w_{2^{j + 1} - 1} (τ_{j} (x)) \sum_{k = 1}^{2^{j} - 1} A_{n - 2^{j + 1} + k}^{α_{n}} D_{k} (τ_{j} (x)) \\ : = Θ_{n_{1}}^{α_{n, 1}} + Θ_{n_{2}}^{α_{n, 2}} . \end{array}$ \begin{array}{*{20}{l}}{\Theta _{{n_1}}^{{\alpha _n}}\left( x \right)}&{ = 1 + \frac{1}{{A_{n - 1}^{{\alpha _n}}}}\sum\limits_{j = 0}^{{n_1} - 1} {{w_{{2^{j + 1}} - 1}}\left( {{\tau _j}\left( x \right)} \right)} }\\{}&{\,\,\,\,\, \times \left( {\sum\limits_{k = 1}^{{2^j}} {A_{n - {2^{j + 1}} + k - 1}^{{\alpha _{_n}}}{D_k}\left( {{\tau _j}\left( x \right)} \right)} - \sum\limits_{k = 0}^{{2^j} - 1} {A_{n - {2^{j + 1}} + k}^{{\alpha _n}}{D_k}\left( {{\tau _j}\left( x \right)} \right)} } \right)}\\{}&{ = 1 + \frac{1}{{A_{n - 1}^{{\alpha _n}}}}\sum\limits_{j = 0}^{{n_1} - 1} {{w_{{2^{j + 1}} - 1}}\left( {{\tau _j}\left( x \right)} \right)A_{n - {2^j} - 1}^{{\alpha _n}}{D_{{2^j}}}\left( {{\tau _j}\left( x \right)} \right)} }\\{}&{\,\,\,\,\, - \frac{1}{{A_{n - 1}^{{\alpha _n}}}}\sum\limits_{j = 0}^{{n_1} - 1} {{w_{{2^{j + 1}} - 1}}\left( {{\tau _j}\left( x \right)} \right)} \left( {\sum\limits_{k = 1}^{{2^j} - 1} {A_{n - {2^{j + 1}} + k - 1}^{{\alpha _n}} - A_{n - {2^{j + 1}} + k}^{{\alpha _n}}} } \right){D_k}\left( {{\tau _j}\left( x \right)} \right)}\\{}&{ = 1 + \frac{1}{{A_{n - 1}^{{\alpha _n}}}}\sum\limits_{j = 0}^{{n_1} - 1} {{w_{{2^{j + 1}} - 1}}\left( {{\tau _j}\left( x \right)} \right)A_{n - {2^j} - 1}^{{\alpha _n}}{D_{{2^j}}}\left( {{\tau _j}\left( x \right)} \right)} }\\{}&{\,\,\,\,\, - \frac{1}{{A_{n - 1}^{{\alpha _n}}}}\sum\limits_{j = 0}^{{n_1} - 1} {{w_{{2^{j + 1}} - 1}}\left( {{\tau _j}\left( x \right)} \right)} \sum\limits_{k = 1}^{{2^j} - 1} {A_{n - {2^{j + 1}} + k}^{{\alpha _n}}{D_k}\left( {{\tau _j}\left( x \right)} \right)} }\\{}&{: = \Theta _{{n_1}}^{{\alpha _{n,1}}} + \Theta _{{n_2}}^{{\alpha _{n,2}}}.}\end{array} By considering $D_{k} = k K_{k} - (k - 1) K_{k - 1}, 0 < k \in ℕ,$ {D_k} = k{K_k} - \left( {k - 1} \right){K_{k - 1}},\;\,\,\,\,\,\,0 < k \in \mathbb{N}, we can transform $Θ_{n_{1}}^{α_{n, 2}}$ \Theta _{{{n}_{1}}}^{{{\alpha }_{n,2}}} as follows: $\begin{array}{l} Θ_{n_{1}}^{α_{n, 2}} & = - \frac{1}{A_{n - 1}^{α_{n}}} \sum_{j = 0}^{n_{1} - 1} w_{2^{j + 1} - 1} (τ_{j} (x)) \sum_{k = 1}^{2^{j} - 1} A_{n - 2^{j + 1} + k}^{α_{n} - 1} \\ \times (k K_{k} (τ_{j} (x)) - (k - 1) K_{k - 1} (τ_{j} (x))) \\ = - \frac{1}{A_{n - 1}^{α_{n}}} \sum_{j = 0}^{n_{1} - 1} w_{2^{j + 1} - 1} (τ_{j} (x)) \sum_{k = 1}^{2^{j} - 1} A_{n - 2^{j + 1} + k}^{α_{n} - 1} k K_{k} (τ_{j} (x)) \\ + \frac{1}{A_{n - 1}^{α_{n}}} \sum_{j = 0}^{n_{1} - 1} w_{2^{j + 1} - 1} (τ_{j} (x)) \sum_{k = 1}^{2^{j} - 1} A_{n - 2^{j + 1} + k}^{α_{n} - 2} (k - 1) K_{k - 1} (τ_{j} (x)) \\ = - \frac{1}{A_{n - 1}^{α_{n}}} \sum_{j = 0}^{n_{1} - 1} w_{2^{j + 1} - 1} (τ_{j} (x)) \sum_{k = 1}^{2^{j} - 1} A_{n - 2^{j + 1} + k}^{α_{n} - 1} k K_{k} (τ_{j} (x)) \\ + \frac{1}{A_{n - 1}^{α_{n}}} \sum_{j = 0}^{n_{1} - 1} w_{2^{j + 1} - 1} (τ_{j} (x)) \sum_{k = 0}^{2^{j} - 2} A_{n - 2^{j + 1} + k}^{α_{n} - 2} k K_{k} (τ_{j} (x)) \\ = - \frac{1}{A_{n - 1}^{α_{n}}} \sum_{j = 0}^{n_{1} - 1} w_{2^{j + 1} - 1} (τ_{j} (x)) (2^{j} - 1) A_{n - 2^{j} - 1}^{α_{n} - 1} K_{2^{j} - 1} (τ_{j} (x)) \\ + \frac{1}{A_{n - 1}^{α_{n}}} \sum_{j = 0}^{n_{1} - 1} w_{2^{j + 1} - 1} (τ_{j} (x)) \sum_{k = 1}^{2^{j} - 2} k A_{n - 2^{j + 1} + k + 1}^{α_{n} - 2} K_{k} (τ_{j} (x)) \\ = : β_{2} + β_{3} . \end{array}$ \begin{array}{*{35}{l}} \Theta _{{{n}_{1}}}^{{{\alpha }_{n,2}}} & =-\frac{1}{A_{n-1}^{{{\alpha }_{n}}}}\sum\limits_{j=0}^{{{n}_{1}}-1}{{{w}_{{{2}^{j+1}}-1}}\left( {{\tau }_{j}}\left( x \right) \right)}\sum\limits_{k=1}^{{{2}^{j}}-1}{A_{n-{{2}^{j+1}}+k}^{{{\alpha }_{n}}-1}} \\ {} & \,\,\,\,\,\times \left( k{{K}_{k}}\left( {{\tau }_{j}}\left( x \right) \right)-\left( k-1 \right){{K}_{k-1}}\left( {{\tau }_{j}}\left( x \right) \right) \right) \\ {} & =-\frac{1}{A_{n-1}^{{{\alpha }_{n}}}}\sum\limits_{j=0}^{{{n}_{1}}-1}{{{w}_{{{2}^{j+1}}-1}}}\left( {{\tau }_{j}}\left( x \right) \right)\sum\limits_{k=1}^{{{2}^{j}}-1}{A_{n-{{2}^{j+1}}+k}^{{{\alpha }_{n}}-1}k{{K}_{k}}\left( {{\tau }_{j}}\left( x \right) \right)} \\ {} & \,\,\,\,+\frac{1}{A_{n-1}^{{{\alpha }_{n}}}}\sum\limits_{j=0}^{{{n}_{1}}-1}{{{w}_{{{2}^{j+1}}-1}}\left( {{\tau }_{j}}\left( x \right) \right)}\sum\limits_{k=1}^{{{2}^{j}}-1}{A_{n-{{2}^{j+1}}+k}^{{{\alpha }_{n}}-2}\left( k-1 \right){{K}_{k-1}}\left( {{\tau }_{j}}\left( x \right) \right)} \\ {} & =-\frac{1}{A_{n-1}^{{{\alpha }_{n}}}}\sum\limits_{j=0}^{{{n}_{1}}-1}{{{w}_{{{2}^{j+1}}-1}}\left( {{\tau }_{j}}\left( x \right) \right)\sum\limits_{k=1}^{{{2}^{j}}-1}{A_{n-{{2}^{j+1}}+k}^{{{\alpha }_{n}}-1}k{{K}_{k}}\left( {{\tau }_{j}}\left( x \right) \right)}} \\ {} & \,\,\,\,+\frac{1}{A_{n-1}^{{{\alpha }_{n}}}}\sum\limits_{j=0}^{{{n}_{1}}-1}{{{w}_{{{2}^{j+1}}-1}}\left( {{\tau }_{j}}\left( x \right) \right)\sum\limits_{k=0}^{{{2}^{j}}-2}{A_{n-{{2}^{j+1}}+k}^{{{\alpha }_{n}}-2}k{{K}_{k}}\left( {{\tau }_{j}}\left( x \right) \right)}} \\ {} & =-\frac{1}{A_{n-1}^{{{\alpha }_{n}}}}\sum\limits_{j=0}^{{{n}_{1}}-1}{{{w}_{{{2}^{j+1}}-1}}\left( {{\tau }_{j}}\left( x \right) \right)\left( {{2}^{j}}-1 \right)A_{n-{{2}^{j}}-1}^{{{\alpha }_{n}}-1}{{K}_{{{2}^{j}}-1}}\left( {{\tau }_{j}}\left( x \right) \right)} \\ {} & \,\,\,\,+\frac{1}{A_{n-1}^{{{\alpha }_{n}}}}\sum\limits_{j=0}^{{{n}_{1}}-1}{{{w}_{{{2}^{j+1}}-1}}\left( {{\tau }_{j}}\left( x \right) \right)}\sum\limits_{k=1}^{{{2}^{j}}-2}{kA_{n-{{2}^{j+1}}+k+1}^{{{\alpha }_{n}}-2}{{K}_{k}}\left( {{\tau }_{j}}\left( x \right) \right)} \\ {} & =:{{\beta }_{2}}+{{\beta }_{3}}. \\\end{array}

If x₀ = ... = x_j−₁ = 0, note that $w_{2^{j + 1} - 1} (τ_{j} (x)) = r_{j} (x)$ {{w}_{{{2}^{j+1}}-1}}\left( {{\tau }_{j}}\left( x \right) \right)={{r}_{j}}\left( x \right) , then by (1.1) we get $w_{2^{j + 1} - 1} (τ_{j} (x)) D_{2^{j}} (x) = r_{j} (x) D_{2^{j}} (x) = D_{2^{j + 1}} (x) - D_{2^{j}} (x) .$ {{w}_{{{2}^{j+1}}-1}}\left( {{\tau }_{j}}\left( x \right) \right){{D}_{{{2}^{j}}}}\left( x \right)={{r}_{j}}\left( x \right){{D}_{{{2}^{j}}}}\left( x \right)={{D}_{{{2}^{j+1}}}}\left( x \right)-{{D}_{{{2}^{j}}}}\left( x \right). Thus, $Θ_{n_{1}}^{α_{n, 1}} = 1 + \frac{1}{A_{n - 1}^{α_{n}}} \sum_{j = 0}^{n_{1} - 1} A_{n - 2^{j} - 1}^{α_{n}} (D_{2^{j + 1}} (x) - D_{2^{j}} (x)) = : β_{1} .$ \Theta _{{{n}_{1}}}^{{{\alpha }_{n,1}}}=1+\frac{1}{A_{n-1}^{{{\alpha }_{n}}}}\sum\limits_{j=0}^{{{n}_{1}}-1}{A_{n-{{2}^{j}}-1}^{{{\alpha }_{n}}}\left( {{D}_{{{2}^{j+1}}}}\left( x \right)-{{D}_{{{2}^{j}}}}\left( x \right) \right)}=:{{\beta }_{1}}. For x ∈ [0, 1), the situation for $Θ_{n_{2}}^{α_{n}} (x)$ \Theta _{{{n}_{2}}}^{{{\alpha }_{n}}}\left( x \right) becomes $\begin{matrix} Θ_{n_{2}}^{α_{n}} & = \frac{1}{A_{n - 1}^{α_{n}}} \sum_{k = 2^{n_{1}}}^{n - 1} A_{n - k - 1}^{α_{n}} ψ_{k} (x) = \frac{1}{A_{n - 1}^{α_{n}}} \sum_{k = 1}^{q - 1} \sum_{j = 2^{n_{1}} + .. + 2^{n_{k}}}^{2^{n_{1}} + \dots + 2^{n_{k + 1}} - 1} A_{n - j - 1}^{α_{n}} ψ_{j} (x) \\ = \frac{1}{A_{n - 1}^{α_{n}}} \sum_{k = 1}^{q - 1} \sum_{j = 0}^{2^{n_{k}} + 1^{- 1}} A_{n - 1 - (2^{n_{1}} + \dots + 2^{n_{k + 1}} - 1 - j)}^{α_{n}} ψ_{2^{n_{1}} + \dots + 2^{n_{k + 1}} - 1 - j} (x) \\ = \frac{1}{A_{n - 1}^{α_{n}}} \sum_{k = 1}^{q - 1} w_{2^{n_{1}} + \dots + 2^{n_{k + 1}} - 1} (τ_{n_{1}} (x)) \sum_{j = 0}^{2^{n_{k + 1}} - 1} A_{n - (2^{n_{1}} + \dots + 2^{n_{k + 1}}) + j}^{α_{n}} w_{j} (τ_{n_{1}} (x)) \\ = \frac{1}{A_{n - 1}^{α_{n}}} \sum_{k = 2}^{q} w_{2^{n_{1}} + \dots + 2^{n_{k}} - 1} (τ_{n_{1}} (x)) \sum_{j = 0}^{2^{n_{k}} - 1} A_{n - (2^{n_{1}} + \dots + 2^{n_{k}}) + j}^{α_{n}} w_{j} (τ_{n_{1}} (x)) \\ = \frac{1}{A_{n - 1}^{α_{n}}} \sum_{k = 2}^{q} w_{n - n^{(k)} - 1} (τ_{n_{1}} (x)) \sum_{j = 0}^{2^{n_{k}} - 1} A_{n^{(k)} + j}^{α_{n}} w_{j} (τ_{n_{1}} (x)) . \end{matrix}$ \begin{align} \Theta _{{{n}_{2}}}^{{{\alpha }_{n}}} & =\frac{1}{A_{n-1}^{{{\alpha }_{n}}}}\sum\limits_{k={{2}^{{{n}_{1}}}}}^{n-1}{A_{n-k-1}^{{{\alpha }_{n}}}{{\psi }_{k}}\left( x \right)}=\frac{1}{A_{n-1}^{{{\alpha }_{n}}}}\sum\limits_{k=1}^{q-1}{\sum\limits_{j={{2}^{{{n}_{1}}}}+..+{{2}^{{{n}_{k}}}}}^{{{2}^{{{n}_{1}}}}+\ldots +{{2}^{{{n}_{k+1}}}}-1}{A_{n-j-1}^{{{\alpha }_{n}}}{{\psi }_{j}}\left( x \right)}} \\ {} & =\frac{1}{A_{n-1}^{{{\alpha }_{n}}}}\sum\limits_{k=1}^{q-1}{\sum\limits_{j=0}^{{{2}^{{{n}_{k}}}}+{{1}^{-1}}}{A_{n-1-\left( {{2}^{{{n}_{1}}}}+\ldots +{{2}^{{{n}_{k+1}}}}-1-j \right)}^{{{\alpha }_{n}}}{{\psi }_{{{2}^{{{n}_{1}}}}+\ldots +{{2}^{{{n}_{k+1}}}}-1-j}}\left( x \right)}} \\ {} & =\frac{1}{A_{n-1}^{{{\alpha }_{n}}}}\sum\limits_{k=1}^{q-1}{{{w}_{{{2}^{{{n}_{1}}}}+\ldots +{{2}^{{{n}_{k+1}}}}-1}}\left( {{\tau }_{{{n}_{1}}}}\left( x \right) \right)}\sum\limits_{j=0}^{{{2}^{{{n}_{k+1}}}}-1}{A_{n-\left( {{2}^{{{n}_{1}}}}+\ldots +{{2}^{{{n}_{k+1}}}} \right)+j}^{{{\alpha }_{n}}}{{w}_{j}}\left( {{\tau }_{{{n}_{1}}}}\left( x \right) \right)} \\ {} & =\frac{1}{A_{n-1}^{{{\alpha }_{n}}}}\sum\limits_{k=2}^{q}{{{w}_{{{2}^{{{n}_{1}}}}+\ldots +{{2}^{{{n}_{k}}}}-1}}\left( {{\tau }_{{{n}_{1}}}}\left( x \right) \right)}\sum\limits_{j=0}^{{{2}^{{{n}_{k}}}}-1}{A_{n-\left( {{2}^{{{n}_{1}}}}+\ldots +{{2}^{{{n}_{k}}}} \right)+j}^{{{\alpha }_{n}}}{{w}_{j}}\left( {{\tau }_{{{n}_{1}}}}\left( x \right) \right)} \\ {} & =\frac{1}{A_{n-1}^{{{\alpha }_{n}}}}\sum\limits_{k=2}^{q}{{{w}_{n-{{n}^{\left( k \right)}}-1}}\left( {{\tau }_{{{n}_{1}}}}\left( x \right) \right)\sum\limits_{j=0}^{{{2}^{{{n}_{k}}}}-1}{A_{{{n}^{\left( k \right)}}+j}^{{{\alpha }_{n}}}{{w}_{j}}\left( {{\tau }_{{{n}_{1}}}}\left( x \right) \right).}} \\\end{align} Using Abel’s transformation, where $n^{(k)} : = \sum_{i = k + 1}^{\infty} 2^{n_{i}}$ {{n}^{\left( k \right)}}:=\mathup{\sum }_{i=k+1}^{\infty }{{2}^{{{n}_{i}}}} , k = 1, . . . , q, we get $\begin{array}{l} Θ_{n_{2}}^{α_{n}} (x) = \frac{1}{A_{n - 1}^{α_{n}}} \sum_{k = 2}^{q} w_{n - n^{(k)} - 1} (τ_{n_{1}} (x)) \sum_{j = 0}^{2^{n_{k}} - 1} A_{n^{(k)} + j}^{α_{n}} (D_{j + 1} (τ_{n_{1}} (x)) - D_{j} (τ_{n_{1}} (x))) \\ = \frac{1}{A_{n - 1}^{α_{n}}} \sum_{k = 2}^{q} w_{n - n^{(k)} - 1} (τ_{n_{1}} (x)) \\ \times \sum_{j = 1}^{2^{n_{k}}} A_{n^{(k)} + j - 1}^{α_{n}} D_{j} (τ_{n_{1}} (x)) - \sum_{j = 0}^{2^{n_{k}} - 1} A_{n^{(k)} + j}^{α_{n}} D_{j} (τ_{n_{1}} (x))] \\ = \frac{1}{A_{n - 1}^{α_{n}}} \sum_{k = 2}^{q} w_{n - n^{(k)} - 1} (τ_{n_{1}} (x)) A_{n^{(k)} + 2^{n_{k}} - 1}^{α_{n}} D_{2^{n_{k}}} (τ_{n_{1}} (x)) \\ - \sum_{j = 1}^{2^{n_{k}} - 1} A_{n^{(k)} + j}^{α_{n} - 1} D_{j} (τ_{n_{1}} (x))) \\ = \frac{1}{A_{n - 1}^{α_{n}}} \sum_{k = 2}^{q} w_{n - n^{(k)} - 1} (τ_{n_{1}} (x)) A_{n^{(k - 1)} - 1}^{α_{n}} D_{2^{n_{k}}} (τ_{n_{1}} (x)) \\ - \frac{1}{A_{n - 1}^{α_{n}}} \sum_{k = 2}^{q - 1} w_{n - n^{(k)} - 1} (τ_{n_{1}} (x)) \sum_{j = 1}^{2^{n_{k}} - 1} A_{n^{(k)} + j}^{α_{n} - 1} (j K_{j} (τ_{n_{1}} (x)) - (j - 1) K_{j - 1} (τ_{n_{1}} (x))) \\ = \frac{1}{A_{n - 1}^{α_{n}}} \sum_{k = 2}^{q} w_{n - n^{(k)} - 1} (τ_{n_{1}} (x)) A_{n^{(k - 1)} - 1}^{α_{n}} D_{2^{n_{k}}} (τ_{n_{1}} (x)) \\ - \frac{1}{A_{n - 1}^{α_{n}}} \sum_{k = 2}^{q} w_{n - n^{(k)} - 1} (τ_{n_{1}} (x)) A_{n^{(k - 1)} - 1}^{α_{n} - 1} (2^{n_{k}} - 1) K_{2^{n_{k}} - 1} (τ_{n_{1}} (x)) \\ + \frac{1}{A_{n - 1}^{α_{n}}} \sum_{k = 2}^{q} w_{n - n^{(k)} - 1} (τ_{n_{1}} (x)) \sum_{j = 1}^{2^{n_{k}} - 2} A_{n^{(k)} + j + 1}^{α_{n} - 2} j K_{j} (τ_{n_{1}} (x)) \\ = : β_{4} + β_{5} + β_{6} . \end{array}$ \begin{align} & \Theta _{{{n}_{2}}}^{{{\alpha }_{n}}}\left( x \right)=\frac{1}{A_{n-1}^{{{\alpha }_{n}}}}\sum\limits_{k=2}^{q}{{{w}_{n-{{n}^{\left( k \right)}}-1}}\left( {{\tau }_{{{n}_{1}}}}\left( x \right) \right)}\sum\limits_{j=0}^{{{2}^{{{n}_{k}}}}-1}{A_{{{n}^{\left( k \right)}}+j}^{{{\alpha }_{n}}}\left( {{D}_{j+1}}\left( {{\tau }_{{{n}_{1}}}}\left( x \right) \right)-{{D}_{j}}\left( {{\tau }_{{{n}_{1}}}}\left( x \right) \right) \right)} \\ & =\frac{1}{A_{n-1}^{{{\alpha }_{n}}}}\sum\limits_{k=2}^{q}{{{w}_{n-{{n}^{\left( k \right)}}-1}}\left( {{\tau }_{{{n}_{1}}}}\left( x \right) \right)} \\ & \,\,\,\,\times \left[ \sum\limits_{j=1}^{{{2}^{{{n}_{k}}}}}{A_{{{n}^{\left( k \right)}}+j-1}^{{{\alpha }_{n}}}{{D}_{j}}\left( {{\tau }_{{{n}_{1}}}}\left( x \right) \right)}-\sum\limits_{j=0}^{{{2}^{{{n}_{k}}}}-1}{A_{{{n}^{\left( k \right)}}+j}^{{{\alpha }_{n}}}{{D}_{j}}\left( {{\tau }_{{{n}_{1}}}}\left( x \right) \right)} \right] \\ & =\frac{1}{A_{n-1}^{{{\alpha }_{n}}}}\sum\limits_{k=2}^{q}{{{w}_{n-{{n}^{\left( k \right)}}-1}}\left( {{\tau }_{{{n}_{1}}}}\left( x \right) \right)\left( A_{{{n}^{\left( k \right)}}+{{2}^{{{n}_{k}}}}-1}^{{{\alpha }_{n}}}{{D}_{{{2}^{{{n}_{k}}}}}}\left( {{\tau }_{{{n}_{1}}}}\left( x \right) \right) \right.} \\ & \,\,\,\,\left. -\sum\limits_{j=1}^{{{2}^{{{n}_{k}}}}-1}{A_{{{n}^{\left( k \right)}}+j}^{{{\alpha }_{n}}-1}{{D}_{j}}\left( {{\tau }_{{{n}_{1}}}}\left( x \right) \right)} \right) \\ & =\frac{1}{A_{n-1}^{{{\alpha }_{n}}}}\sum\limits_{k=2}^{q}{{{w}_{n-{{n}^{\left( k \right)}}-1}}\left( {{\tau }_{{{n}_{1}}}}\left( x \right) \right)A_{{{n}^{\left( k-1 \right)}}-1}^{{{\alpha }_{n}}}{{D}_{{{2}^{{{n}_{k}}}}}}\left( {{\tau }_{{{n}_{1}}}}\left( x \right) \right)} \\ & -\frac{1}{A_{n-1}^{{{\alpha }_{n}}}}\sum\limits_{k=2}^{q-1}{{{w}_{n-{{n}^{\left( k \right)}}-1}}\left( {{\tau }_{{{n}_{1}}}}\left( x \right) \right)}\sum\limits_{j=1}^{{{2}^{{{n}_{k}}}}-1}{A_{{{n}^{\left( k \right)}}+j}^{{{\alpha }_{n}}-1}\left( j{{K}_{j}}\left( {{\tau }_{{{n}_{1}}}}\left( x \right) \right)-\left( j-1 \right){{K}_{j-1}}\left( {{\tau }_{{{n}_{1}}}}\left( x \right) \right) \right)} \\ & =\frac{1}{A_{n-1}^{{{\alpha }_{n}}}}\sum\limits_{k=2}^{q}{{{w}_{n-{{n}^{\left( k \right)}}-1}}\left( {{\tau }_{{{n}_{1}}}}\left( x \right) \right)A_{{{n}^{\left( k-1 \right)}}-1}^{{{\alpha }_{n}}}{{D}_{{{2}^{{{n}_{k}}}}}}\left( {{\tau }_{{{n}_{1}}}}\left( x \right) \right)} \\ & \,\,\,\,-\frac{1}{A_{n-1}^{{{\alpha }_{n}}}}\sum\limits_{k=2}^{q}{{{w}_{n-{{n}^{\left( k \right)}}-1}}\left( {{\tau }_{{{n}_{1}}}}\left( x \right) \right)A_{{{n}^{\left( k-1 \right)}}-1}^{{{\alpha }_{n}}-1}\left( {{2}^{{{n}_{k}}}}-1 \right){{K}_{{{2}^{{{n}_{k}}}}-1}}\left( {{\tau }_{{{n}_{1}}}}\left( x \right) \right)} \\ & \,\,\,\,+\frac{1}{A_{n-1}^{{{\alpha }_{n}}}}\sum\limits_{k=2}^{q}{{{w}_{n-{{n}^{\left( k \right)}}-1}}\left( {{\tau }_{{{n}_{1}}}}\left( x \right) \right)}\sum\limits_{j=1}^{{{2}^{{{n}_{k}}}}-2}{A_{{{n}^{\left( k \right)}}+j+1}^{{{\alpha }_{n}}-2}j{{K}_{j}}\left( {{\tau }_{{{n}_{1}}}}\left( x \right) \right)} \\ & =:{{\beta }_{4}}+{{\beta }_{5}}+{{\beta }_{6}}. \\ \end{align} Hence, the theorem follows.

Define the maximal operator $σ_{*, n}^{α_{n}} f : = \sup_{n \in ℕ_{α_{n}, q}} σ_{n}^{α_{n}} f| = \sup_{n \in ℕ_{α_{n}, q}} \int_{I}^{} \frac{1}{A_{n - 1}^{α_{n}}} \sum_{k = 0}^{n - 1} A_{n - k - 1}^{α_{n}} ψ_{k} * f| .$ \sigma _{*,n}^{{{\alpha }_{n}}}f:=\text{ }\!\!~\!\!\text{ }\underset{n\in {{\mathbb{N}}_{{{\alpha }_{n}},q}}}{\mathop{\sup }}\,\left| \sigma _{n}^{{{\alpha }_{n}}}f \right|=\text{ }\!\!~\!\!\text{ }\underset{n\in {{\mathbb{N}}_{{{\alpha }_{n}},q}}}{\mathop{\sup }}\,\left| \mathop{\int }_{I}^{{}}\frac{1}{A_{n-1}^{{{\alpha }_{n}}}}\sum\limits_{k=0}^{n-1}{A_{n-k-1}^{{{\alpha }_{n}}}{{\psi }_{k}}*f} \right|.

Lemma 2.2.

Let α = (α_n, n ∈ ℕ) where 0 < α_n < 1. Then, the maximal operator $σ_{*, n}^{α_{n}} f$ \sigma _{*,n}^{{{\alpha }_{n}}}f is quasi-local.

Proof

By the definition of quasi-locality, let f ∈ L¹[0, 1) be such that $supp f \subset I_{N} (u), \int_{I_{N} (u)}^{} f d μ = 0$ \text{supp}\,\,f\subset {{I}_{N}}\left( u \right),\,\,\,\,\,\,\mathup{\int }_{{{I}_{N}}\left( u \right)}^{{}}fd\mu =0 for some dyadic interval I_N(u). Then, $\begin{array}{l} \int_{0, 1) \ I_{N} (u)} \sup_{n \in ℕ} \int_{I_{N} (u)} \frac{1}{A_{n - 1}^{α_{n}}} \sum_{k = 0}^{n - 1} A_{n - k - 1}^{α_{n}} ψ_{k} (x ∔ y) f (x) d μ (y)| d μ (x) \\ \leq C \int_{0, 1) \ I_{N} (u)} \sup_{n \in ℕ} \int_{I_{N} (u)} Θ_{n_{1}}^{α_{n}} (x ∔ y)| f (x)| d μ (y) d μ (x) \\ + C \int_{0, 1) \ I_{N} (u)} \sup_{n \in ℕ} \int_{I_{N} (u)} Θ_{n_{2}}^{α_{n}} (x ∔ y)| f (x)| d μ (y) d μ (x) \\ : = α_{1} + α_{2} . \end{array}$ \begin{align} & \int_{\left[ 0,1 \right)\backslash {{I}_{N}}\left( u \right)}{\text{ }\!\!~\!\!\text{ }\underset{n\in \mathbb{N}}{\mathop{\sup }}\,\text{ }\!\!~\!\!\text{ }\left| \int_{{{I}_{N}}\left( u \right)}{\frac{1}{A_{n-1}^{{{\alpha }_{n}}}}\sum\limits_{k=0}^{n-1}{A_{n-k-1}^{{{\alpha }_{n}}}{{\psi }_{k}}\left( x\dotplus y \right)f\left( x \right)d\mu \left( y \right)}} \right|}d\mu \left( x \right) \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\le C\int_{\left[ 0,1 \right)\backslash {{I}_{N}}\left( u \right)}{\underset{n\in \mathbb{N}}{\mathop{\sup }}\,\text{ }\!\!~\!\!\text{ }\int_{{{I}_{N}}\left( u \right)}{|\Theta _{{{n}_{1}}}^{{{\alpha }_{n}}}\left( x\dotplus y \right)|\left| f\left( x \right) \right|d\mu \left( y \right)d\mu \left( x \right)}}\text{ }\!\!~\!\!\text{ } \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+C\int_{\left[ 0,1 \right)\backslash {{I}_{N}}\left( u \right)}{\underset{n\in \mathbb{N}}{\mathop{\sup }}\,}\int_{{{I}_{N}}\left( u \right)}{|\Theta _{{{n}_{2}}}^{{{\alpha }_{n}}}\left( x\dotplus y \right)|\left| f\left( x \right) \right|d\mu \left( y \right)d\mu \left( x \right)} \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,:={{\alpha }_{1}}+{{\alpha }_{2}}. \\ \end{align} Since for n ∈ ℕ, n ≤ 2^N and x ∈ I_N(u) we have $σ_{n}^{α_{n}} f = 0$ \sigma _{n}^{{{\alpha }_{n}}}f=0 , thus $σ_{*, n}^{α_{n}} f = \sup_{n > 2^{N}, n \in ℕ_{α_{n}, q}} σ_{n}^{α_{n}} f| .$ \sigma _{*,n}^{{{\alpha }_{n}}}f=\underset{n>{{2}^{N}},n\in {{\mathbb{N}}_{{{\alpha }_{n}},q}}}{\mathop{\sup }}\,\left| \sigma _{n}^{{{\alpha }_{n}}}f \right|. From the proof of Lemma 2.1, we have the decomposition $σ_{n}^{α_{n}} f = {\tilde{σ}}_{n}^{α_{n}} f + {\bar{σ}}_{n}^{α_{n}} f,$ \sigma _{n}^{{{\alpha }_{n}}}f=\tilde{\sigma }_{n}^{{{\alpha }_{n}}}f+\bar{\sigma }_{n}^{{{\alpha }_{n}}}f, where $\begin{array}{l} {\tilde{σ}}_{n}^{α_{n}} f (y) : = \int_{0}^{1} Θ_{n_{1}}^{α_{n}} (x ∔ y) f (x) d μ (x), \\ {\tilde{σ}}_{n}^{α_{n}} f (y) : = \int_{0}^{1} Θ_{n_{2}}^{α_{n}} (x ∔ y) f (x) d μ (x) . \end{array}$ \begin{align} & \tilde{\sigma }_{n}^{{{\alpha }_{n}}}f\left( y \right):=\mathop{\int }_{0}^{1}\Theta _{{{n}_{1}}}^{{{\alpha }_{n}}}\left( x\dotplus y \right)f\left( x \right)d\mu \left( x \right), \\ & \tilde{\sigma }_{n}^{{{\alpha }_{n}}}f\left( y \right):=\mathop{\int }_{0}^{1}\Theta _{{{n}_{2}}}^{{{\alpha }_{n}}}\left( x\dotplus y \right)f\left( x \right)d\mu \left( x \right). \\ \end{align} Again, from the proof of Lemma 2.1, we have $\begin{array}{l} {\tilde{σ}}_{n}^{α_{n}} f (y) & = \int_{0}^{1} β_{1} (x ∔ y) f (x) d μ (x) \\ + \int_{0}^{1} β_{2} (x ∔ y) f (x) d μ (x) + \int_{0}^{1} β_{3} (x ∔ y) f (x) d μ (x) \\ = : I + I I + I I I . \end{array}$ \begin{align} {\tilde{\sigma }_{n}^{{{\alpha }_{n}}}f\left( y \right)} \ & {=\mathop{\int }_{0}^{1}{{\beta }_{1}}\left( x\dotplus y \right)f\left( x \right)d\mu \left( x \right)} \\{} & {\,\,~+\int_{0}^{1}{{{\beta }_{2}}\left( x\dotplus y \right)f\left( x \right)d\mu \left( x \right)+\mathop{\int }_{0}^{1}{{\beta }_{3}}\left( x\dotplus y \right)f\left( x \right)d\mu \left( x \right)}} \\ {} & {=:I+II+III.} \\ \end{align} If x ∈ [0, 1) \ I_N then by (1.1), we get $\int_{0}^{1} f (x) D_{t} (y ∔ x) d μ (x) = 0$ \int_{0}^{1}{f\left( x \right){{D}_{t}}\left( y\dotplus x \right)d\mu \left( x \right)}=0 for all t = 0, . . . , 2ⁿ^₁. From the proof of Lemma 2.1, we have I = 0. By Lemma 2.1 in [11], II = 0. The situation for III: with respect to x and for any 0 ≤ j < 2^N, we have that the Fejér kernel K_j(y ∔ x) depends only on the coordinates x₀, x₁, . . . , x_N−₁. This implies that, $\int_{I_{N}} f (x) K_{j} (y ∔ x)| d μ (x) = K_{j} (y)| \int_{I_{N}} f (x) d μ (x) = 0.$ \int_{{{I}_{N}}}{f\left( x \right)\left| {{K}_{j}}\left( y\dotplus x \right)\left| d\mu \left( x \right)= \right|{{K}_{j}}\left( y \right) \right|}\int_{{{I}_{N}}}{f\left( x \right)d\mu \left( x \right)=0}. Thus, we can re-write $\begin{array}{l} \int_{0, 1) \ I_{N} (u)} \sup_{k \in ℕ} \int_{I_{N} (u)} β_{3} (y ∔ x) f (x) d μ (x)| d μ (y) \\ = \int_{0, 1) \ I_{N} (u)} \sup_{k \geq 2^{N}, k \in ℕ} \int_{I_{N} (u)} β_{3} (y ∔ x) f (x) d μ (x)| d μ (y) . \end{array}$ \begin{align} & \int_{\left[ 0,1 \right)\backslash {{I}_{N}}\left( u \right)}{\underset{k\in \mathbb{N}}{\mathop{\sup }}\,\left| \int_{{{I}_{N}}\left( u \right)}{{{\beta }_{3}}\left( y\dotplus x \right)f\left( x \right)d\mu \left( x \right)} \right|}\text{ }\!\!~\!\!\text{ }d\mu \left( y \right) \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\int_{\left[ 0,1 \right)\backslash {{I}_{N}}\left( u \right)}{\underset{k\ge {{2}^{N}},~k\in \mathbb{N}}{\mathop{\sup }}\,\left| \int_{{{I}_{N}}\left( u \right)}{{{\beta }_{3}}\left( y\dotplus x \right)f\left( x \right)d\mu \left( x \right)} \right|d\mu \left( y \right).}\text{ }\!\!~\!\!\text{ } \\ \end{align} So, using Lemma 3 in [11], we get $\begin{array}{l} \int_{0, 1) \ I_{N} (u)} \sup_{k \geq 2^{N}, k \in ℕ} \int_{I_{N} (u)} β_{3} (y ∔ x) f (x) d μ (x)| d μ (y) \\ \leq C \int_{I_{N} (u)} f (x)| \int_{0, 1) \ I_{N} (u)} \sum_{j = 0}^{n_{1}} \sum_{k = 2^{N}}^{2^{j} - 2} \sup_{k \geq 2^{N}} k K_{k} (τ_{j} (y ∔ x))| d μ (x) \\ \leq C \int_{I_{N} (u)} f (x)| d μ (x) \leq C {f‖}_{1} . \end{array}$ \begin{align} & \int_{\left[ 0,1 \right)\backslash {{I}_{N}}\left( u \right)}{\underset{k\ge {{2}^{N}},~k\in \mathbb{N}}{\mathop{\sup }}\,\left| \int_{{{I}_{N}}\left( u \right)}{{{\beta }_{3}}\left( y\dotplus x \right)f\left( x \right)d\mu \left( x \right)} \right|d\mu \left( y \right)} \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\le C\int_{{{I}_{N}}\left( u \right)}{\left| f\left( x \right) \right|}\int_{\left[ 0,1 \right)\backslash {{I}_{N}}\left( u \right)}{\sum\limits_{j=0}^{{{n}_{1}}}{\sum\limits_{k={{2}^{N}}}^{{{2}^{j}}-2}{\underset{k\ge {{2}^{N}}}{\mathop{\sup }}\,k\left| {{K}_{k}}\left( {{\tau }_{j}}\left( y\dotplus x \right) \right) \right|d\mu \left( x \right)}}\text{ }\!\!~\!\!\text{ }} \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\le C\int_{{{I}_{N}}\left( u \right)}{|f\left( x \right)|d\mu \left( x \right)\le C\|f{{\|}_{1}}}. \\ \end{align} Hence, $\int_{0, 1) \ I_{N} (u)} \sup_{k \in ℕ} \int_{I_{N} (u)} (β_{3} + β_{2} + β_{1}) (y ∔ x) f (x) d μ (x)| d μ (y) \leq C {f‖}_{1} .$ \int_{\left[ 0,1 \right)\backslash {{I}_{N}}\left( u \right)}{\underset{k\in \mathbb{N}}{\mathop{\sup }}\,\left| \int_{{{I}_{N}}\left( u \right)}{({{\beta }_{3}}+{{\beta }_{2}}+{{\beta }_{1}})\left( y\dotplus x \right)f\left( x \right)d\mu \left( x \right)} \right|d\mu \left( y \right)\le C\|f{{\|}_{1}}}. Note that $\int_{0, 1) \ I_{N} (u)} \sup_{k \geq 2^{N}, k \in ℕ} \int_{I_{N} (u)} β_{4} (y ∔ x) f (x) d μ (x)| d μ (y) = 0,$ \int_{\left[ 0,1 \right)\backslash {{I}_{N}}\left( u \right)}{\underset{k\ge {{2}^{N}},~k\in \mathbb{N}}{\mathop{\sup }}\,\left| \int_{{{I}_{N}}\left( u \right)}{{{\beta }_{4}}\left( y\dotplus x \right)f\left( x \right)d\mu \left( x \right)} \right|d\mu \left( y \right)}=0, since $f * D_{2^{n_{k}}} = 0$ f*{{D}_{{{2}^{{{n}_{k}}}}}}=0 for n_l < n_s ≤ n_k because of the A_{n_k} measurablity of $D_{2^{n_{k}}}$ {{D}_{{{2}^{{{n}_{k}}}}}} and ∫ f = 0. Moreover, $D_{2^{n_{k}}} (y ∔ x) = 0$ {{D}_{{{2}^{{{n}_{k}}}}}}\left( y\dotplus x \right)=0 for n_s > n_k, y ∔ x ∉ I_N.

From Lemma 1.1 of [14] (see also [4]), we have $\frac{A_{n^{(k)} + j + 1}^{α_{n} - 1}}{A_{n - 1}^{α_{n}}} \leq C \frac{{(n^{(k)} + j)}^{α_{n}}}{{(n)}^{α_{n}}}, j = 1, \dots, 2^{n_{k}} - 1, k = 2, \dots, q - 1.$ \frac{A_{{{n}^{\left( k \right)}}+j+1}^{{{\alpha }_{n}}-1}}{A_{n-1}^{{{\alpha }_{n}}}}\le C\frac{{{({{n}^{\left( k \right)}}+j)}^{{{\alpha }_{n}}}}}{{{(n)}^{{{\alpha }_{n}}}}},\,\,\,\,\,\,j=1,\ldots ,{{2}^{{{n}_{k}}}}-1,\,\,k=2,\ldots ,q-1. Thus, by the fact that n ∈ ℕ_{α_n,q}, we have (see (1.5)) $\begin{array}{l} \sum_{k = 2}^{q - 1} \sum_{j = 1}^{2^{n_{k}} - 1} \frac{A_{n^{(k)} + j + 1}^{α - 2}}{A_{n - 1}^{α}} j \leq C \sum_{k = 2}^{q - 1} \sum_{t = 0}^{n_{k} - 1} \sum_{j = 2^{l}}^{2^{l + 1} - 1} \frac{{(n^{(k)} + j)}^{α_{n}}}{{(n)}^{α_{n}}} j \\ \leq C \sum_{k = 2}^{q - 1} \sum_{t = 0}^{n_{k} - 1} \frac{{(n^{(k)} + 2^{l})}^{α_{n}}}{{(n)}^{α_{n}}} \sum_{j = 2^{l}}^{2^{l + 1} - 1} j \leq C \sum_{k = 2}^{q - 1} \frac{2^{k α_{n}}}{n^{α_{n}}} \leq C_{q} . \end{array}$ \begin{align} & \sum\limits_{k=2}^{q-1}{\sum\limits_{j=1}^{{{2}^{{{n}_{k}}}}-1}{\frac{A_{{{n}^{\left( k \right)}}+j+1}^{\alpha -2}}{A_{n-1}^{\alpha }}j\le C\sum\limits_{k=2}^{q-1}{\sum\limits_{t=0}^{{{n}_{k}}-1}{\sum\limits_{j={{2}^{l}}}^{{{2}^{l+1}}-1}{\frac{{{({{n}^{\left( k \right)}}+j)}^{{{\alpha }_{n}}}}}{{{(n)}^{{{\alpha }_{n}}}}}j}}}}} \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\le C\sum\limits_{k=2}^{q-1}{\sum\limits_{t=0}^{{{n}_{k}}-1}{\frac{{{({{n}^{\left( k \right)}}+{{2}^{l}})}^{{{\alpha }_{n}}}}}{{{(n)}^{{{\alpha }_{n}}}}}\sum\limits_{j={{2}^{l}}}^{{{2}^{l+1}}-1}{j\le C}\sum\limits_{k=2}^{q-1}{\frac{{{2}^{k{{\alpha }_{n}}}}}{{{n}^{{{\alpha }_{n}}}}}\le {{C}_{q}}}}}. \\ \end{align} Consequently, using (1.4), we can estimate $\int_{0, 1) \ I_{N} (u)} \sup_{n} \int_{I_{N} (u)} β_{5} (y ∔ x) + β_{6} (y ∔ x) f (x)] d μ (x)| d μ (y) \leq C_{q} {f‖}_{1} .$ \int_{\left[ 0,1 \right)\backslash {{I}_{N}}\left( u \right)}{\underset{n}{\mathop{\sup }}\,\left| \int_{{{I}_{N}}\left( u \right)}{\left[ {{\beta }_{5}}\left( y\dotplus x \right)+{{\beta }_{6}}\left( y\dotplus x \right)f\left( x \right) \right]d\mu \left( x \right)} \right|d\mu \left( y \right)\le {{C}_{q}}\|f{{\|}_{1}}}. Hence, the lemma is proved.

Lemma 2.3.

Let α = (α_n, n ∈ ℕ), where 0 < α_n < 1 satisfy condition (1.5). Then:

(I)
${Θ_{n}^{α_{n}}‖}_{1} \leq C_{q}$ {{\left\| \Theta _{n}^{{{\alpha }_{n}}} \right\|}_{1}}\le {{C}_{q}} ,
(II)
there exists an absolute constant C_q such that ${σ_{n}^{α_{n}} f‖}_{1} \leq C_{q} {f‖}_{1}$ {{\left\| \sigma _{n}^{{{\alpha }_{n}}}f \right\|}_{1}}\le {{C}_{q}}{{\left\| f \right\|}_{1}} ,
(III)
the maximal operator $σ_{*, n}^{α_{n}}$ \sigma _{*,n}^{{{\alpha }_{n}}} is of type (L^∞, L^∞).

Proof

To prove (I) we use Lemma 2.1 and estimation (1.3). That is, $\begin{array}{l} β_{3}| \leq C n^{- α_{n}} \frac{1}{A_{n - 1}^{α_{n}} - 1} \sum_{j = 1}^{n_{1}} \sum_{s = 1}^{j - 1} \sum_{l = 2^{s - 1}}^{2^{s} - 1} A_{n - 2^{j} + l + 1}^{α_{n} - 2}| \\ \times \sum_{i = 0}^{s - 1} \sum_{m = 0}^{i} 2^{m} (D_{2^{i}} (τ_{j - 1}) (x) + D_{2^{i}} (τ_{j - 1}) (x ∔ e_{m})) \\ \leq C n^{- α_{n}} \frac{1}{A_{n - 1}^{α_{n}} - 1} \sum_{j = 1}^{n_{1}} \sum_{i = 0}^{j - 2} \sum_{i = i + 1}^{j - 1} \sum_{l = 2^{s - 1}}^{2^{s} - 1} {(n - 2^{j} + l)}^{α_{n} - 2} \\ \times \sum_{m = 0}^{i} 2^{m} (D_{2^{i}} (τ_{j - 1}) (x) + D_{2^{i}} (τ_{j - 1}) (x ∔ e_{m})) \\ = C n^{- α_{n}} \frac{1}{A_{n - 1}^{α_{n}} - 1} \sum_{j = 1}^{n_{1}} \sum_{i = 0}^{j - 2} γ_{i j} \sum_{m = 0}^{i} 2^{m} (D_{2^{i}} (τ_{j - 1}) (x) + D_{2^{i}} (τ_{j - 1}) (x ∔ e_{m})) \\ \leq C n^{- α_{n}} \frac{1}{A_{n - 1}^{α_{n}} - 1} \sum_{j = 1}^{n_{1}} \sum_{i = 0}^{j - 2} (γ_{i j} 2^{i} D_{2^{i}} (τ_{j - 1}) (x) + \sum_{m = 0}^{i} 2^{m} D_{2^{i}} (τ_{j - 1}) (x ∔ e_{m})), \end{array}$ \begin{align} & \left| {{\beta }_{3}} \right|\le C{{n}^{-{{\alpha }_{n}}}}\frac{1}{A_{n-1}^{{{\alpha }_{n}}}-1}\sum\limits_{j=1}^{{{n}_{1}}}{\sum\limits_{s=1}^{j-1}{\sum\limits_{l={{2}^{s-1}}}^{{{2}^{s}}-1}{\left| A_{n-{{2}^{j}}+l+1}^{{{\alpha }_{n}}-2} \right|}}} \\ & \,\,\,\,\,\,\,\,\,\,\,\,\times \sum\limits_{i=0}^{s-1}{\sum\limits_{m=0}^{i}{{{2}^{m}}\left( {{D}_{{{2}^{i}}}}\left( {{\tau }_{j-1}} \right)\left( x \right)+{{D}_{{{2}^{i}}}}\left( {{\tau }_{j-1}} \right)\left( x\dotplus {{e}_{m}} \right) \right)}} \\ & \,\,\,\,\,\,\,\,\le C{{n}^{-{{\alpha }_{n}}}}\frac{1}{A_{n-1}^{{{\alpha }_{n}}}-1}\sum\limits_{j=1}^{{{n}_{1}}}{\sum\limits_{i=0}^{j-2}{\sum\limits_{i=i+1}^{j-1}{\sum\limits_{l={{2}^{s-1}}}^{{{2}^{s}}-1}{{{\left( n-{{2}^{j}}+l \right)}^{{{\alpha }_{n}}-2}}}}}} \\ & \,\,\,\,\,\,\,\,\,\,\,\,\times \sum\limits_{m=0}^{i}{{{2}^{m}}\left( {{D}_{{{2}^{i}}}}\left( {{\tau }_{j-1}} \right)\left( x \right)+{{D}_{{{2}^{i}}}}\left( {{\tau }_{j-1}} \right)\left( x\dotplus {{e}_{m}} \right) \right)} \\ & =C{{n}^{-{{\alpha }_{n}}}}\frac{1}{A_{n-1}^{{{\alpha }_{n}}}-1}\sum\limits_{j=1}^{{{n}_{1}}}{\sum\limits_{i=0}^{j-2}{{{\gamma }_{ij}}}\sum\limits_{m=0}^{i}{{{2}^{m}}\left( {{D}_{{{2}^{i}}}}\left( {{\tau }_{j-1}} \right)\left( x \right)+{{D}_{{{2}^{i}}}}\left( {{\tau }_{j-1}} \right)\left( x\dotplus {{e}_{m}} \right) \right)}} \\ & \le C{{n}^{-{{\alpha }_{n}}}}\frac{1}{A_{n-1}^{{{\alpha }_{n}}}-1}\sum\limits_{j=1}^{{{n}_{1}}}{\sum\limits_{i=0}^{j-2}{\left( {{\gamma }_{ij}}{{2}^{i}}{{D}_{{{2}^{i}}}}\left( {{\tau }_{j-1}} \right)\left( x \right)+\sum\limits_{m=0}^{i}{{{2}^{m}}{{D}_{{{2}^{i}}}}\left( {{\tau }_{j-1}} \right)\left( x\dotplus {{e}_{m}} \right)} \right),}} \\ \end{align} where e_m := 2⁻^m⁻¹ = (0, . . . , 0, 1, 0, . . .) and $γ_{i j} = \sum_{i = i + 1}^{j = 1} \sum_{l = 2^{s - 1}}^{2^{s} - 1} {(n - 2^{j} + l)}^{α_{n} - 2} \leq C \int_{2^{i}}^{j - 1} {(n - 2^{j} + x)}^{α_{n} - 2} d μ (x) \leq C 2^{i (α_{n} - 1)} .$ {{\gamma }_{ij}}=\sum\limits_{i=i+1}^{j=1}{\sum\limits_{l={{2}^{s-1}}}^{{{2}^{s}}-1}{{{(n-{{2}^{j}}+l)}^{{{\alpha }_{n}}-2}}\le C}}\int_{{{2}^{i}}}^{j-1}{{{(n-{{2}^{j}}+x)}^{{{\alpha }_{n}}-2}}d\mu \left( x \right)\le C{{2}^{i\left( {{\alpha }_{n}}-1 \right)}}}. With a similar computation we show that the same estimation can be obtained for β₂. Thus, $Θ_{n_{1}}^{α_{n, 2}} (x)$ \Theta _{{{n}_{1}}}^{{{\alpha }_{n,2}}}\left( x \right) can be estimated as $\begin{array}{l} Θ_{n_{1}}^{α_{n, 2}} (x) = β_{2} + β_{3} \\ \leq C n^{- α_{n}} \sum_{j = 1}^{n_{1}} \sum_{i = 0}^{j - 2} 2^{i (α_{n} - 1)} (2^{i} D_{2^{i}} (τ_{j - 1}) (x) + \sum_{m = 0}^{i} 2^{m} D_{2^{i}} (τ_{j - 1}) (x ∔ e_{m})) . \end{array}$ \begin{align} & \Theta _{{{n}_{1}}}^{{{\alpha }_{n,2}}}\left( x \right)={{\beta }_{2}}+{{\beta }_{3}} \\ & \le C{{n}^{-{{\alpha }_{n}}}}\sum\limits_{j=1}^{{{n}_{1}}}{\sum\limits_{i=0}^{j-2}{{{2}^{i\left( {{\alpha }_{n}}-1 \right)}}\left( {{2}^{i}}{{D}_{{{2}^{i}}}}\left( {{\tau }_{j-1}} \right)\left( x \right)+\sum\limits_{m=0}^{i}{{{2}^{m}}{{D}_{{{2}^{i}}}}\left( {{\tau }_{j-1}} \right)\left( x\dotplus {{e}_{m}} \right)} \right).}} \\ \end{align} Applying (1.1), the previous estimation implies for $β_{2} + β_{3}‖$ \left\| {{\beta }_{2}}+{{\beta }_{3}} \right\| that ${β_{2} + β_{3}‖}_{1} \leq C n^{- α_{n}} \sum_{j = 1}^{n_{1}} \sum_{i = 0}^{j - 2} 2^{i (α_{n} - 1)} 2^{i} \leq C n^{- α_{n}} \sum_{j = 1}^{n_{1}} \sum_{i = 0}^{j - 2} 2^{i α_{n}} \leq C_{q} .$ {{\left\| {{\beta }_{2}}+{{\beta }_{3}} \right\|}_{1}}\le C{{n}^{-{{\alpha }_{n}}}}\sum\limits_{j=1}^{{{n}_{1}}}{\sum\limits_{i=0}^{j-2}{{{2}^{i\left( {{\alpha }_{n}}-1 \right)}}{{2}^{i}}\le C{{n}^{-{{\alpha }_{n}}}}}}\sum\limits_{j=1}^{{{n}_{1}}}{\sum\limits_{i=0}^{j-2}{{{2}^{i{{\alpha }_{n}}}}\le {{C}_{q}}}}. Analogically, it can also be obtained for the L¹-norm estimation of β₁. Consider that $w_{2^{j + 1} - 1} (τ_{j} (x)) = r_{j} (x)$ {{w}_{{{2}^{j+1}}-1}}\left( {{\tau }_{j}}\left( x \right) \right)={{r}_{j}}\left( x \right) when x₀ = . . . = x_j−₁ = 0. Then by (1.1) we get $w_{2^{j + 1} - 1} (τ_{j} (x)) D_{2^{j}} (x) = r_{j} (x) D_{2^{j}} (x) = D_{2^{j + 1}} (x) - D_{2^{j}} (x),$ {{w}_{{{2}^{j+1}}-1}}\left( {{\tau }_{j}}\left( x \right) \right){{D}_{{{2}^{j}}}}\left( x \right)={{r}_{j}}\left( x \right){{D}_{{{2}^{j}}}}\left( x \right)={{D}_{{{2}^{j+1}}}}\left( x \right)-{{D}_{{{2}^{j}}}}\left( x \right), that is $\begin{array}{l} β_{1} = 1 + \frac{1}{A_{n - 1}^{α_{n}}} \sum_{j = 0}^{n_{1} - 1} A_{n - 2^{j} - 1}^{α_{n}} (D_{2^{j + 1}} (x) - D_{2^{j}} (x)) \\ = 1 + \frac{1}{A_{n - 1}^{α_{n}}} (\sum_{j = 1}^{n_{1}} A_{n - 2^{j} - 1}^{α_{n}} D_{2^{j}} (x) - \sum_{j = 0}^{n_{1} - 1} A_{n - 2^{j} - 1}^{α_{n}} D_{2^{j}} (x)) \\ = 1 + \frac{1}{A_{n - 1}^{α_{n}}} A_{n - 2^{n_{1}} - 1}^{α_{n}} D_{2^{n_{1}}} (x) - \frac{1}{A_{n - 1}^{α_{n}}} A_{n - 2}^{α_{n}} \\ + \frac{1}{A_{n - 1}^{α_{n}}} \sum_{j = 1}^{n_{1} - 1} (A_{n - 2^{j - 1} - 1}^{α_{n}} - A_{n - 2^{j} - 1}^{α_{n}}) D_{2^{j}} (x) . \end{array}$ \begin{align} & {{\beta }_{1}}=1+\frac{1}{A_{n-1}^{{{\alpha }_{n}}}}\sum\limits_{j=0}^{{{n}_{1}}-1}{A_{n-{{2}^{j}}-1}^{{{\alpha }_{n}}}\left( {{D}_{{{2}^{j+1}}}}\left( x \right)-{{D}_{{{2}^{j}}}}\left( x \right) \right)} \\ & \,\,\,\,\,\,\,=1+\frac{1}{A_{n-1}^{{{\alpha }_{n}}}}\left( \sum\limits_{j=1}^{{{n}_{1}}}{A_{n-{{2}^{j}}-1}^{{{\alpha }_{n}}}{{D}_{{{2}^{j}}}}\left( x \right)}-\sum\limits_{j=0}^{{{n}_{1}}-1}{A_{n-{{2}^{j}}-1}^{{{\alpha }_{n}}}{{D}_{{{2}^{j}}}}\left( x \right)} \right) \\ & \,\,\,\,\,\,\,=1+\frac{1}{A_{n-1}^{{{\alpha }_{n}}}}A_{n-{{2}^{{{n}_{1}}}}-1}^{{{\alpha }_{n}}}{{D}_{{{2}^{{{n}_{1}}}}}}\left( x \right)-\frac{1}{A_{n-1}^{{{\alpha }_{n}}}}A_{n-2}^{{{\alpha }_{n}}} \\ & \,\,\,\,\,\,\,\,\,\,\,+\frac{1}{A_{n-1}^{{{\alpha }_{n}}}}\sum\limits_{j=1}^{{{n}_{1}}-1}{\left( A_{n-{{2}^{j-1}}-1}^{{{\alpha }_{n}}}-A_{n-{{2}^{j}}-1}^{{{\alpha }_{n}}} \right)}{{D}_{{{2}^{j}}}}\left( x \right). \\ \end{align} From this and (1.1) we get $\begin{array}{l} β_{1}‖ & \leq C_{q} + \frac{1}{A_{n - 1}^{α_{n}}} \sum_{j = 1}^{n_{1} - 1} (A_{n - 2^{j - 1} - 1}^{α_{n}} - A_{n - 2^{j} - 1}^{α_{n}}) \\ \leq C_{q} + \frac{1}{A_{n - 1}^{α_{n}}} (A_{n - 2^{j - 1} - 1}^{α_{n}} - A_{n - 2^{j} - 1}^{α_{n}}) \leq C_{q} . \end{array}$ \begin{align} {\left\| {{\beta }_{1}} \right\|} \ & {\le {{C}_{q}}+\frac{1}{A_{n-1}^{{{\alpha }_{n}}}}\sum\limits_{j=1}^{{{n}_{1}}-1}{\left( A_{n-{{2}^{j-1}}-1}^{{{\alpha }_{n}}}-A_{n-{{2}^{j}}-1}^{{{\alpha }_{n}}} \right)}} \\ {} & {\le {{C}_{q}}+\frac{1}{A_{n-1}^{{{\alpha }_{n}}}}\left( A_{n-{{2}^{j-1}}-1}^{{{\alpha }_{n}}}-A_{n-{{2}^{j}}-1}^{{{\alpha }_{n}}} \right)\le {{C}_{q}}.} \\ \end{align} Let us deal now with the situation ${β_{4}‖}_{1}$ {{\left\| {{\beta }_{4}} \right\|}_{1}} , ${β_{5}‖}_{1}$ {{\left\| {{\beta }_{5}} \right\|}_{1}} and ${β_{6}‖}_{1}$ {{\left\| {{\beta }_{6}} \right\|}_{1}} as follows: $\begin{array}{l} {β_{4}‖}_{1} & \leq C_{q} n^{- α_{n}} \sum_{k = 2}^{q} A_{n^{(k - 1)} - 1}^{α_{n}} \leq C_{q} n^{- α_{n}} \sum_{k = 2}^{q} {(n^{(k - 1)})}^{α_{n}} \\ \leq C_{q} n^{- α_{n}} \sum_{k = 2}^{q} 2^{n_{k} α_{n}} \leq C_{q} . \end{array}$ \begin{align} {{\left\| {{\beta }_{4}} \right\|}_{1}} \ & {\le {{C}_{q}}{{n}^{-{{\alpha }_{n}}}}\sum\limits_{k=2}^{q}{A_{{{n}^{\left( k-1 \right)}}-1}^{{{\alpha }_{n}}}\le {{C}_{q}}{{n}^{-{{\alpha }_{n}}}}}\sum\limits_{k=2}^{q}{{{({{n}^{\left( k-1 \right)}})}^{{{\alpha }_{n}}}}}} \\ {} & {\le {{C}_{q}}{{n}^{-{{\alpha }_{n}}}}\sum\limits_{k=2}^{q}{{{2}^{{{n}_{k}}{{\alpha }_{n}}}}\le {{C}_{q}}}.} \\ \end{align} Similarly, $\begin{array}{l} β_{5}‖ & \leq C_{q} n^{- α_{n}} \sum_{k = 2}^{q} 2^{n_{k}} {(n^{(k - 1)})}^{α_{n} - 1} {K_{2^{n_{k}} - 1}‖}_{1} \\ \leq C_{q} n^{- α_{n}} \sum_{k = 2}^{q} 2^{n_{k}} (2^{(α_{n} - 1) n_{k}}) \leq C_{q} . \end{array}$ \begin{align} {\left\| {{\beta }_{5}} \right\|} \ & {\le {{C}_{q}}{{n}^{-{{\alpha }_{n}}}}\sum\limits_{k=2}^{q}{{{2}^{{{n}_{k}}}}{{({{n}^{\left( k-1 \right)}})}^{{{\alpha }_{n}}-1}}{{\left\| {{K}_{{{2}^{{{n}_{k}}}}-1}} \right\|}_{1}}}} \\{} & {\le {{C}_{q}}{{n}^{-{{\alpha }_{n}}}}\sum\limits_{k=2}^{q}{{{2}^{{{n}_{k}}}}\left( {{2}^{\left( {{\alpha }_{n}}-1 \right){{n}_{k}}}} \right)\le {{C}_{q}}}.} \\ \end{align} From (1.4), ${β_{6}‖}_{1}$ {{\left\| {{\beta }_{6}} \right\|}_{1}} can be estimated as follows: $\begin{matrix} {β_{6}‖}_{1} & \leq C n^{- α_{n}} \sum_{k = 2}^{r} \sum_{j = 1}^{2^{n_{k}} - 2} A_{n^{(k)} + j + 1}^{α_{n} - 1} j {K_{j}‖}_{1} \\ \leq C n^{- α_{n}} \frac{1}{A_{n - 1}^{α}} \sum_{k = 2}^{r} \sum_{l = 0}^{n_{k} - 1} \sum_{j = 2^{l}}^{2^{l + 1} - 1} {(n^{(k)} + j)}^{α_{n} - 2} j \\ \leq C n^{- α_{n}} \sum_{k = 2}^{r} \sum_{l = 0}^{n_{k} - 1} {(n^{(k)} + 2^{l})}^{α_{n} - 2} \sum_{j = 2^{l}}^{2^{l + 1} - 1} j \\ \leq C n^{- α_{n}} \sum_{k = 2}^{r} \sum_{l = 0}^{n_{k} - 1} {(n^{(k)} + 2^{l})}^{α_{n} - 2} 2^{2 l} \\ \leq C n^{- α_{n}} \sum_{k = 2}^{r} \sum_{l = 0}^{n_{k} - 1} 2^{l (α_{n} - 2)} 2^{2 l} \leq C n^{- α_{n}} \sum_{k = 2}^{r} 2^{α n_{k}} \leq C_{q} . \end{matrix}$ \begin{align} {{\left\| {{\beta }_{6}} \right\|}_{1}} & \le C{{n}^{-{{\alpha }_{n}}}}\sum\limits_{k=2}^{r}{\sum\limits_{j=1}^{{{2}^{{{n}_{k}}}}-2}{{}}}~A_{{{n}^{\left( k \right)}}+j+1}^{{{\alpha }_{n}}-1}j{{\left\| {{K}_{j}} \right\|}_{1}} \\ {} & \le C{{n}^{-{{\alpha }_{n}}}}\frac{1}{A_{n-1}^{\alpha }}\sum\limits_{k=2}^{r}{\sum\limits_{l=0}^{{{n}_{k}}-1}{\sum\limits_{j={{2}^{l}}}^{{{2}^{l+1}}-1}{{{({{n}^{\left( k \right)}}+j)}^{{{\alpha }_{n}}-2}}j}}} \\ {} & \le C{{n}^{-{{\alpha }_{n}}}}\sum\limits_{k=2}^{r}{\sum\limits_{l=0}^{{{n}_{k}}-1}{{{\left( {{n}^{\left( k \right)}}+{{2}^{l}} \right)}^{{{\alpha }_{n}}-2}}\sum\limits_{j={{2}^{l}}}^{{{2}^{l+1}}-1}{j}}} \\ {} & \le C{{n}^{-{{\alpha }_{n}}}}\sum\limits_{k=2}^{r}{\sum\limits_{l=0}^{{{n}_{k}}-1}{{{\left( {{n}^{\left( k \right)}}+{{2}^{l}} \right)}^{{{\alpha }_{n}}-2}}{{2}^{2l}}}} \\ {} & \le C{{n}^{-{{\alpha }_{n}}}}\sum\limits_{k=2}^{r}{\sum\limits_{l=0}^{{{n}_{k}}-1}{{{2}^{l\left( {{\alpha }_{n}}-2 \right)}}{{2}^{2l}}\le C{{n}^{-{{\alpha }_{n}}}}\sum\limits_{k=2}^{r}{{{2}^{\alpha {{n}_{k}}}}\le {{C}_{q}}}}}. \\\end{align} Thus, (I) follows. The results in (II) and (III) are a direct consequence of (I). Hence, the theorem follows.

Theorem 2.4.

Let α = (α_n, n ∈ ℕ), where 0 < α_n < 1 and f ∈ L¹[0, 1). Then:

(I)
the maximal operator ${sup}_{n \in ℕ_{α_{n}, q}} σ_{n}^{α_{n}} f|$ \text{su}{{\text{p}}_{n\in {{\mathbb{N}}_{{{\alpha }_{n}},q}}}}\left| \sigma _{n}^{{{\alpha }_{n}}}f \right| is of weak type (L¹, L¹),
(II)
$μ {σ_{n}^{α_{n}} f - f| > 0} = 0$ \mu \{\left| \sigma _{n}^{{{\alpha }_{n}}}f-f \right|>0\}=0 as n → ∞ where n ∈ ℕ_{α_n,q}, where constant C_q depends on q indicated in equation (1.5) above.

Proof

To prove (I) of this theorem, we apply the Calderon–Zygmund decomposition Lemma [11]. That is, let f ∈ L¹[0, 1) and ${f‖}_{1} < δ$ {{\left\| f \right\|}_{1}}<\delta . Then there is a decomposition: $f = f_{0} + \sum_{j = 1}^{\infty} f_{j}$ f={{f}_{0}}+\sum\limits_{j=1}^{\infty }{{{f}_{j}}} such that ${f_{0}‖}_{\infty} \leq C δ, {f_{0}‖}_{1} \leq C {f‖}_{1}$ {{\left\| {{f}_{0}} \right\|}_{\infty }}\le C\delta ,\,\,\,\,\,\,~{{\left\| {{f}_{0}} \right\|}_{1}}\le C{{\left\| f \right\|}_{1}} and ${[0, 1)}^{j} = I_{k_{j}} (u^{j})$ {{[0,1)}^{j}}={{I}_{{{k}_{j}}}}\left( {{u}^{j}} \right) are disjoint intervals for which $supp f_{j} \subset I_{k_{j}} (u^{j}), \int_{I_{k_{j}} (u^{j})} f_{j} d μ = 0, u^{j} \in 0, 1), k_{j} \in ℕ, j \in ℕ_{+},$ \text{supp}\,{{f}_{j}}\subset {{I}_{{{k}_{j}}}}\left( {{u}^{j}} \right),\,\,\,\,\,\,\,\,\int_{{{I}_{{{k}_{j}}}}\left( {{u}^{j}} \right)}{{{f}_{j}}d\mu =0},\,\,\,\,\,{{u}^{j}}\in \left[ 0,1 \right),\,\,{{k}_{j}}\in \mathbb{N},\,\,j\in {{\mathbb{N}}_{+}}, and $F| \leq \frac{C {f‖}_{1}}{δ}, where F = \cup_{i = 1}^{\infty} I_{k_{j}} (u^{j}) .$ \left| F \right|\le \frac{C\|f{{\|}_{1}}}{\delta },\,\,\,\,\,\,\text{where}\,\,F=\bigcup\limits_{i=1}^{\infty }{{{I}_{{{k}_{j}}}}\left( {{u}^{j}} \right)}.

By the σ-sublinearity of the maximal operator with an appropriate constant C_q we have $μ (\sup_{n \in ℕ_{α, q}} σ_{n}^{α_{n}} f| > 2 C_{q} δ) \leq μ (\sup_{n \in ℕ_{α_{n}, q}} σ_{n}^{α_{n}} f_{0}| > C_{q} δ) + μ (\sup_{n \in ℕ_{α_{n}, q}} σ_{n}^{α_{n}} f_{j}| > C_{q} δ) = : A + B .$ \mu \left( \underset{n\in {{\mathbb{N}}_{\alpha ,\,q}}}{\mathop{\sup }}\,\left| \sigma _{n}^{{{\alpha }_{n}}}f \right|>2{{C}_{q}}\delta \right)\le \mu \text{ }\!\!~\!\!\text{ }\left( \underset{n\in {{\mathbb{N}}_{{{\alpha }_{n}},\,q}}}{\mathop{\sup }}\,\left| \sigma _{n}^{{{\alpha }_{n}}}{{f}_{0}} \right|>{{C}_{q}}\delta \right)\text{ }\!\!~\!\!\text{ }+\mu \left( \underset{n\in {{\mathbb{N}}_{{{\alpha }_{n}},\,q}}}{\mathop{\sup }}\,\left| \sigma _{n}^{{{\alpha }_{n}}}{{f}_{j}} \right|>{{C}_{q}}\delta \right)=:A+B. Since ${sup}_{n \in ℕ_{α_{n}, q}} σ_{n}^{α_{n}}|$ \text{su}{{\text{p}}_{n\in {{\mathbb{N}}_{{{\alpha }_{n}},q}}}}\left| \sigma _{n}^{{{\alpha }_{n}}} \right| is of type (L^∞, L^∞), we have $‖ \sup_{n \in ℕ_{α_{n}, q}} σ_{n}^{α_{n}} f_{0}| ‖_{\infty} \leq C_{q} {f_{0}‖}_{\infty} \leq C_{q} δ .$ \|\underset{n\in {{\mathbb{N}}_{{{\alpha }_{n}},q}}}{\mathop{\sup }}\,\left| \sigma _{n}^{{{\alpha }_{n}}}{{f}_{0}} \right|{{\|}_{\infty }}\le {{C}_{q}}{{\left\| {{f}_{0}} \right\|}_{\infty }}\le {{C}_{q}}\delta . Then we have A = 0. The case for B becomes, $\begin{array}{l} B = μ (\sup_{n \in ℕ_{α_{n}, q}} σ_{n}^{α_{n}} f_{j}| > C_{q} δ) \leq F| + μ (\bar{F} \cap [\sup_{n \in ℕ_{α_{n}, q}} σ_{n}^{α_{n}} \sum_{j = 1}^{\infty} f_{j}| > C_{q} δ]) \\ \leq \frac{C {f‖}_{1}}{δ} + \frac{C_{q}}{δ} \sum_{j = 1}^{\infty} \int_{0, 1) \ I_{k_{j}} (u^{j})} \sup_{n \in ℕ_{α_{n}, q}} σ_{n}^{α_{n}} f_{j}| d μ = : \frac{C {f‖}_{1}}{δ} + \frac{C_{q}}{δ} \sum_{j = 1}^{\infty} N_{j}, \end{array}$ \begin{align} & B=\mu (\underset{n\in {{\mathbb{N}}_{{{\alpha }_{n}},q}}}{\mathop{\sup }}\,\left| \sigma _{n}^{{{\alpha }_{n}}}{{f}_{j}} \right|>{{C}_{q}}\delta )\le \left| F \right|+\mu (\bar{F}\cap [\underset{n\in {{\mathbb{N}}_{{{\alpha }_{n}},q}}}{\mathop{\sup }}\,\left| \sigma _{n}^{{{\alpha }_{n}}}\sum\limits_{j=1}^{\infty }{{{f}_{j}}} \right|>{{C}_{q}}\delta ]) \\ & \le \frac{C{{\left\| f \right\|}_{1}}}{\delta }+\frac{{{C}_{q}}}{\delta }\sum\limits_{j=1}^{\infty }{\int_{\left[ 0,1 \right)\backslash {{I}_{{{k}_{j}}}}\left( {{u}^{j}} \right)}{\underset{n\in {{\mathbb{N}}_{{{\alpha }_{n}},q}}}{\mathop{\sup }}\,}\left| \sigma _{n}^{{{\alpha }_{n}}}{{f}_{j}} \right|d\mu =:\frac{C{{\left\| f \right\|}_{1}}}{\delta }+\frac{{{C}_{q}}}{\delta }\sum\limits_{j=1}^{\infty }{{{N}_{j}}}}, \\ \end{align} where $N_{j} = \int_{0, 1) \ I_{k_{j}} (u^{j})} \sup_{n \in ℕ_{α_{n}, q}} σ_{n}^{α_{n}} f_{j}| d μ .$ {{N}_{j}}=\int_{\left[ 0,1 \right)\backslash {{I}_{{{k}_{j}}}}\left( {{u}^{j}} \right)}{\underset{n\in {{\mathbb{N}}_{{{\alpha }_{n}},q}}}{\mathop{\sup }}\,}\left| \sigma _{n}^{{{\alpha }_{n}}}{{f}_{j}} \right|d\mu . From Lemma 2.2 we get $\begin{array}{l} N_{j} & \leq \int_{0, 1) \ I_{k_{j}} (u^{j})} \sup_{n \in ℕ_{α_{n}, q}} \int_{I_{k_{j}} (u^{j})} f_{j} (x) Θ_{n}^{α_{n}} (y + x) d μ (x)| d μ (y) \\ \leq C_{q} {f_{j}‖}_{1} . \end{array}$ \begin{align} {{N}_{j}} & {\le \int_{\left[ 0,1 \right)\backslash {{I}_{{{k}_{j}}}}\left( {{u}^{j}} \right)}{\underset{n\in {{\mathbb{N}}_{{{\alpha }_{n}},q}}}{\mathop{\sup }}\,\left| \int_{{{I}_{{{k}_{j}}}}\left( {{u}^{j}} \right)}{{{f}_{j}}\left( x \right)\Theta _{n}^{{{\alpha }_{n}}}\left( y+x \right)d\mu \left( x \right)} \right|d\mu \left( y \right)}} \\ {} & {\le {{C}_{q}}{{\left\| {{f}_{j}} \right\|}_{1}}.} \\ \end{align} Finally, we have $μ (\sup_{n \in ℕ_{α_{n}, q}} σ_{n}^{α_{n}} f| > 2 C_{q} δ) \leq C_{q} \frac{{f‖}_{1}}{δ} .$ \mu (\underset{n\in {{\mathbb{N}}_{{{\alpha }_{n}},q}}}{\mathop{\sup }}\,\left| \sigma _{n}^{{{\alpha }_{n}}}f \right|>2{{C}_{q}}\delta )\le {{C}_{q}}\frac{{{\left\| f \right\|}_{1}}}{\delta }. This shows that the maximal operator ${sup}_{n \in ℕ_{α_{n}, q}} σ_{n}^{α_{n}}|$ \text{su}{{\text{p}}_{n\in {{\mathbb{N}}_{{{\alpha }_{n}},q}}}}\left| \sigma _{n}^{{{\alpha }_{n}}} \right| is of weak type (L¹, L¹).

Now, we prove (II). Let t ≥ 2^k. Then we have S_tp ≡ p, where p is a Walsh–Kaczmarz polynomial which can be given by $p (x) = \sum_{i = 0}^{2^{k} - 1} C_{i} ψ_{i} (x) .$ p\left( x \right)=\sum\limits_{i=0}^{{{2}^{k}}-1}{{{C}_{i}}{{\psi }_{i}}\left( x \right)}. This implies the statement $σ_{n}^{α_{n}} p \to p$ \sigma _{n}^{{{\alpha }_{n}}}p\to p holds everywhere not only for $n \in ℕ_{α_{n}, q}$ n\in {{\mathbb{N}}_{{{\alpha }_{n}},q}} .

Now, fix η, ϵ > 0, f ∈ L¹[0, 1). Let p be a one dimensional Walsh–Kaczmarz polynomial such that ${f - p‖}_{1} < η .$ {{\left\| f-p \right\|}_{1}}<\eta .

Since from (I) the maximal operator ${sup}_{n \in ℕ_{α, q}} σ_{n}^{α_{n}}|$ \text{su}{{\text{p}}_{n\in {{\mathbb{N}}_{\alpha ,q}}}}\left| \sigma _{n}^{{{\alpha }_{n}}} \right| is of weak type (L¹, L¹), we get $\begin{array}{l} μ ({\lim^{¯}}_{n \in ℕ_{α_{n}, q}} σ_{n}^{α_{n}} f - f| > ϵ) \leq μ ({\lim^{¯}}_{n \in ℕ_{α_{n}, q}} σ_{n}^{α_{n}} (f - p)| > \frac{ϵ}{3}) \\ + μ ({\lim^{¯}}_{n \in ℕ_{α_{n}, q}} σ_{n}^{α_{n}} p - p| > \frac{ϵ}{3}) + μ ({\lim^{¯}}_{n \in ℕ_{α_{n}, q}} p - f| > \frac{ϵ}{3}) \\ \leq μ (\sup_{n \in ℕ_{α_{n}, q}} σ_{n}^{α_{n}} (f - p)| > \frac{ϵ}{3}) + 0 + \frac{3}{ϵ} {p - f‖}_{1} \leq C_{q} {p - f‖}_{1} \frac{3}{ϵ} \leq \frac{C_{q}}{ϵ} η . \end{array}$ \begin{align} & \mu \left( {{\overline{\lim }}_{n\in {{\mathbb{N}}_{{{\alpha }_{n}},\,q}}}}\left| \sigma _{n}^{{{\alpha }_{n}}}f-f \right|>\epsilon \right)\le \mu \left( {{\overline{\lim }}_{n\in {{\mathbb{N}}_{{{\alpha }_{n}},\,q}}}}\left| \sigma _{n}^{{{\alpha }_{n}}}\left( f-p \right) \right|>\frac{\epsilon }{3} \right) \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,+\mu \left( {{\overline{\lim }}_{n\in {{\mathbb{N}}_{{{\alpha }_{n}},\,q}}}}\left| \sigma _{n}^{{{\alpha }_{n}}}p-p \right|>\frac{\epsilon }{3} \right)+\mu \left( {{\overline{\lim }}_{n\in {{\mathbb{N}}_{{{\alpha }_{n}},\,q}}}}\left| p-f \right|>\frac{\epsilon }{3} \right) \\ & \le \mu \left( \underset{n\in {{\mathbb{N}}_{{{\alpha }_{n}},q}}}{\mathop{\sup }}\,\left| \sigma _{n}^{{{\alpha }_{n}}}\left( f-p \right) \right|>\frac{\epsilon }{3} \right)+0+\frac{3}{\epsilon }{{\left\| p-f \right\|}_{1}}\le {{C}_{q}}{{\left\| p-f \right\|}_{1}}\frac{3}{\epsilon }\le \frac{{{C}_{q}}}{\epsilon }\eta . \\ \end{align} This is true for all η > 0.

Thus, we get $μ ({\lim^{¯}}_{n \in ℕ_{α_{n}, q}} σ_{n}^{α_{n}} f - f| > ϵ) = 0,$ \mu \left( {{\overline{\lim }}_{n\in {{\mathbb{N}}_{{{\alpha }_{n}},\,q}}}}\left| \sigma _{n}^{{{\alpha }_{n}}}f-f \right|>\epsilon \right)=0, for an arbitrary ϵ > 0. As a result, we have $μ ({\lim^{¯}}_{n \in ℕ_{α_{n}, q}} σ_{n}^{α_{n}} f - f| > 0) = 0.$ \mu \left( {{\overline{\lim }}_{n\in {{\mathbb{N}}_{{{\alpha }_{n}},\,q}}}}\left| \sigma _{n}^{{{\alpha }_{n}}}f-f \right|>0 \right)=0. Finally, for all $n \in ℕ_{α_{n}, q}$ n\in {{\mathbb{N}}_{{{\alpha }_{n}},q}} , $μ σ_{n}^{α_{n}} f - f| > 0\} = 0.$ \mu \left\{ \left| \sigma _{n}^{{{\alpha }_{n}}}f-f \right|>0 \right\}=0. Hence, the theorem follows.

Theorem 2.5.

The maximal operator ${sup}_{n \in ℕ_{α_{n}, q}} σ_{n}^{α_{n}} f|$ \text{su}{{\text{p}}_{n\in {{\mathbb{N}}_{{{\alpha }_{n}},q}}}}\left| \sigma _{n}^{{{\alpha }_{n}}}f \right| is of strong type (H¹, L¹) and (L^p, L^p), for all 1 < p ≤ ∞.

Proof

By combining Lemma 2.4 and Marcinkiewicz interpolation theorem of [13], it is possible to get that operator ${sup}_{n \in ℕ_{α_{n}, q}} σ_{n}^{α_{n}} f|$ \text{su}{{\text{p}}_{n\in {{\mathbb{N}}_{{{\alpha }_{n}},q}}}}\left| \sigma _{n}^{{{\alpha }_{n}}}f \right| is of type (L^p, L^p) for all 1 < p ≤ ∞. Moreover, by the σ-sublinearity of ${sup}_{n \in ℕ_{α_{n}, q}} σ_{n}^{α_{n}} f|$ \text{su}{{\text{p}}_{n\in {{\mathbb{N}}_{{{\alpha }_{n}},q}}}}\left| \sigma _{n}^{{{\alpha }_{n}}}f \right| and since $σ_{n}^{α_{n}}$ \sigma _n^{{\alpha _n}} is A_k measurable for n < 2^k, we prove that it is of type (H¹, L¹).

Almost Everywhere Convergence of Varying Parameter Setting Cesàro Means of Fourier Series With Respect to Walsh–Kaczmarz System

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