1 Introduction
Let \(N_{+}\) denote the set of positive integers, and let \(N:=N_{+} \cup \{0\}\). Let \(m:= ( m_{0},m_{1},\ldots ) \) be a sequence of positive integers not less than 2. Denote by \(Z_{m_{k}}:=\{0,1,\ldots,m _{k}-1\}\) the additive group of integers modulo \(m_{k}\). Define the group \(G_{m}\) as the complete direct product of the groups \(Z_{m_{j}}\) with the product of the discrete topologies of \(Z_{m_{j}}\).
The direct product of the measures
is the Haar measure on \(G_{m}\) with \(\mu ( G_{m} ) =1\). If the sequence m is bounded, then \(G_{m}\) is called a bounded Vilenkin group. In this paper, we consider only bounded Vilenkin groups. The elements of \(G_{m}\) can be represented by sequences \(x:= ( x_{0},x _{1},\ldots,x_{j},\ldots ) \) (\(x_{j}\in Z_{m_{j}} \)). The group operation + in \(G_{m}\) is given by
for \(x:= ( x_{0},\ldots,x_{k},\ldots ) \) and \(y:= ( y_{0},\ldots,y _{k},\ldots ) \in G_{m}\). The inverse of + will be denoted by −.
$$ \mu _{k} \bigl( \{j\} \bigr) :=\frac{1}{m_{k}} \quad ( j\in Z_{m_{k}} ) $$
$$ x+y= \bigl( ( x_{0}+y_{0} ) \operatorname{mod} m_{0},\ldots, ( x_{k}+y_{k} ) \operatorname{mod} m_{k},\ldots \bigr) $$
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It is easy to give a base for the neighborhoods of \(G_{m}\):
for \(x \in G_{m}\) and \(n \in N\). Define \(I_{n}:=I_{n} ( 0 ) \) for \(n\in N_{+}\). Set \(e_{n}:= ( 0,\ldots,0,1,0,\ldots ) \in G_{m}\), where the nth coordinate of which is 1, and the rest are zeros (\(n\in N \)).
$$ \begin{gathered} I_{0} ( x ) :=G_{m}, \\ I_{n} ( x ) :=\{y\in G_{m}|y_{0}=x_{0}, \ldots,y _{n-1}=x_{n-1}\} \end{gathered} $$
We define the so-called generalized number system based on m as follows: \(M_{0}:=1\), \(M_{k+1}:=m_{k}M_{k}\) (\(k\in N \)). Then every \(n \in N\) can be uniquely expressed as \(n=\sum_{j=0} ^{\infty }n_{j}M_{j}\), where \(n_{j} \in Z_{m_{j}}\) (\(j \in N_{+} \)), and only a finite number of \(n_{j}\) differ from zero. We also use the following notation: \(\vert n \vert :=\max \{k\in N:n_{k}\neq 0\}\) (i.e., \(M_{|n|}\leq n< M_{|n|+1}\), \(n\neq 0\)). For \(x \in G_{m}\), we denote \(\vert x \vert :=\sum_{j=0}^{ \infty }\frac{x_{j}}{M_{j+1}}\) (\(x_{j}\in Z_{m_{j}} \)).
Next, we introduce on \(G_{m}\) an orthonormal system, which is called the Vilenkin system. First, we define the complex-valued functions \(r_{k} ( x ) :G_{m}\rightarrow C\), the generalized Rademacher functions, as follows:
$$ r_{k}(x):=\exp \frac{2\pi ix_{k}}{m_{k}} \quad \bigl( i^{2}=-1,x\in G_{m},k\in N \bigr) . $$
Now we define the Vilenkin system \(\psi := ( \psi _{n}:n\in N ) \) on \(G_{m}\) as
$$ \psi _{n} ( x ) :=\prod_{k=0}^{\infty }r_{k}^{n_{k}} ( x ) \quad ( n\in N ) . $$
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In particular, if \(m=2\), then we call this system the Walsh–Paley system. Each \(\psi _{n}\) is a character of \(G_{m}\), and all characters of \(G_{m}\) are of this norm. Moreover, \(\psi _{n} ( -x ) =\bar{ \psi }_{n} ( x ) \).
The Dirichlet kernels are defined by
$$ D_{n}:=\sum_{k=0}^{n-1}\psi _{k} \quad ( n\in N_{+} ) . $$
Recall that (see [20] or [23])
$$ D_{M_{n}} ( x ) = \textstyle\begin{cases} M_{n}& \text{if }x \in I_{n}, \\ 0&\text{if }x\notin I_{n}. \end{cases} $$
(1)
The Vilenkin system is orthonormal and complete in \(L^{1} ( G _{m} ) \) (see [1]).
Next, we introduce some notation from the theory of two-dimensional Vilenkin system. Let m̃ be a sequence like m. The relation between the sequences \(( \tilde{m}_{n} ) \) and \(( \tilde{M} _{n}) \) is the same as between sequences \(( m_{n} ) \) and \((M_{n}) \). The group \(G_{m}\times G_{\tilde{m}}\) is called a two-dimensional Vilenkin group. The normalized Haar measure is denoted by μ as in the one-dimensional case. We also suppose that \(m=\tilde{m}\) and \(G_{m}\times G_{\tilde{m}}=G_{m}^{2}\).
The norm of the space \(L^{p} ( G_{m}^{2} ) \) is defined by
$$ \Vert f \Vert _{p}:= \biggl( \int _{G_{m}^{2}} \bigl\vert f ( x,y ) \bigr\vert ^{p}\,d \mu ( x,y ) \biggr) ^{1/p} \quad ( 1\leq p< \infty ) . $$
Denote by \(C ( G_{m}^{2} ) \) the class of continuous functions on the group \(G_{m}^{2}\) endowed with the supremum norm.
For brevity in notation, we write \(L^{\infty } ( G_{m}^{2} ) \) instead of \(C ( G_{m}^{2} ) \).
The two-dimensional Fourier coefficients, the rectangular partial sums of the Fourier series, and the Dirichlet kernels with respect to the two-dimensional Vilenkin system are defined as follows:
Denote
where
and
$$ \begin{gathered} \widehat{f} ( n_{1},n_{2} ) := \int _{G_{m}^{2}}f ( x,y ) \bar{\psi }_{n_{1}} ( x ) \bar{\psi }_{n_{2}} ( y ) \,d\mu ( x,y ) , \\ S_{n_{1},n_{2}} ( x,y,f ) :=\sum_{k_{1}=0}^{n_{1}-1} \sum_{k_{2}=0}^{n_{2}-1}\widehat{f} ( k_{1},k_{2} ) \psi _{k_{1}} ( x ) \psi _{k_{2}} ( y ) , \\ D_{n_{1},n_{2}} ( x,y ) :=D_{n_{1}} ( x ) D_{n_{2}} ( y ) , \end{gathered} $$
$$ \begin{gathered} S_{n}^{ ( 1 ) } ( x,y,f ) :=\sum _{l=0}^{n-1} \widehat{f} ( l,y ) \bar{\psi }_{l} ( x ) , \\ S_{m}^{ ( 2 ) } ( x,y,f ) :=\sum_{r=0}^{m-1} \widehat{f} ( x,r ) \bar{\psi }_{r} ( y ) , \end{gathered} $$
$$ \widehat{f} ( l,y ) = \int _{G_{m}}f ( x,y ) \psi _{l} ( x ) \,d\mu ( x ) $$
$$ \widehat{f} ( x,r ) = \int _{G_{m}}f ( x,y ) \psi _{r} ( y ) \,d\mu ( y ) . $$
The \((C,-\alpha )\) means of the double Vilenkin–Fourier series are defined as follows:
where
$$ \sigma _{n}^{-\alpha } ( f,x,y ) = \frac{1}{A_{n-1}^{-\alpha }}\sum _{j=1}^{n}A_{n-j}^{-\alpha -1}S _{j,j} ( f,x,y ) , $$
$$ A_{0}^{\alpha }=1, \qquad A_{n}^{\alpha }= \frac{ ( \alpha +1 ) \cdots ( \alpha +n ) }{n!}. $$
It is well known that (see [28])
and
where positive constants \(c_{1}\) and \(c_{2}\) depend on α.
$$\begin{aligned}& A_{n}^{\alpha }=\sum_{k=0}^{n}A_{k}^{\alpha -1}, \end{aligned}$$
(2)
$$\begin{aligned}& A_{n}^{\alpha }-A_{n-1}^{\alpha }=A_{n}^{\alpha -1}, \end{aligned}$$
(3)
$$ c_{1}(\alpha )n^{\alpha }\leq A_{n}^{\alpha }\leq c_{2} (\alpha )n ^{\alpha }, $$
(4)
The dyadic partial moduli of continuity of a function \(f\in L^{p} ( G_{m}^{2} ) \) in the \(L^{p}\)-norm are defined by
and
whereas the dyadic mixed modulus of continuity is defined as follows:
It is clear that
$$ \omega _{1} \biggl( f,\frac{1}{M_{n}} \biggr) _{p}=\sup _{u\in I_{n}} \bigl\Vert f ( \cdot +u,\cdot ) -f ( \cdot ,\cdot ) \bigr\Vert _{p} $$
$$ \omega _{2} \biggl( f,\frac{1}{M_{n}} \biggr) _{p}=\sup _{v\in I_{n}} \bigl\Vert f ( \cdot ,\cdot +v ) -f ( \cdot ,\cdot ) \bigr\Vert _{p}, $$
$$ \begin{gathered} \omega _{1,2} \biggl( f,\frac{1}{M_{n}}, \frac{1}{M_{m}} \biggr) _{p} \\ \quad =\sup_{ ( u,v ) \in I_{n}\times I_{m}} \bigl\Vert f ( \cdot +u,\cdot +v ) -f ( \cdot +u,\cdot ) -f ( \cdot ,\cdot +v ) +f ( \cdot ,\cdot ) \bigr\Vert _{p}. \end{gathered} $$
$$ \omega _{1,2} \biggl( f,\frac{1}{M_{n}},\frac{1}{M_{m}} \biggr) _{p} \leq \omega _{1} \biggl( f,\frac{1}{M_{n}} \biggr) _{p}+\omega _{2} \biggl( f,\frac{1}{M _{m}} \biggr) _{p}. $$
The dyadic total modulus of continuity is defined by
The problems of summability of partial sums and Cesàro means for Walsh–Fourier series were studied in [2, 13‐19, 21, 22, 25, 26].
$$ \omega \biggl( f,\frac{1}{M_{n}} \biggr) _{p}= \sup _{ ( u,v ) \in I_{n}\times I_{n}} \bigl\Vert f ( \cdot +u,\cdot +v ) -f ( \cdot ,\cdot ) \bigr\Vert _{p}. $$
The convergence issue of Fejér (and Cesàro) means on the Walsh and Vilenkin groups for unbounded case were studied in [3‐11].
In his monograph [27], Zhizhinashvili investigated the behavior of Cesàro \((C, \alpha )\)-means for double trigonometric Fourier series in detail. Goginava [18] studied the analogous question in the case of the Walsh system. In particular, the following theorems were proved.
Theorem A
Let
f
belong to
\(L^{p} ( G_{2} ) \)
for some
\(p \in [ 1,\infty ] \)
and
\(\alpha \in ( 0,1 ) \). Then, for any
\(2^{k}\leq n<2^{k+1}\) (\(k, n\in N\)), we have the inequality
$$ \begin{aligned} \bigl\Vert \sigma _{2^{k}}^{-\alpha } ( f ) -f \bigr\Vert _{p}&\leq c ( \alpha ) \Biggl\{ 2^{k\alpha }\omega _{1} \bigl( f,1/2^{k-1} \bigr) _{p}+2^{k\alpha } \omega _{2} \bigl( f,1/2^{k-1} \bigr) _{p} \\ &\quad {} +\sum_{r=0}^{k-2}2^{r-k} \omega _{1} \bigl( f,1/2^{r} \bigr) _{p}+\sum _{s=0}^{k-2}2^{s-k}\omega _{2} \bigl( f,1/2^{s} \bigr) _{p} \Biggr\} . \end{aligned} $$
Theorem B
Let
f
belong to
\(L^{p} ( G_{2} ) \)
for some
\(p \in [ 1,\infty ] \)
and
\(\alpha \in ( 0,1 ) \). Then, for any
\(2^{k}\leq n<2^{k+1}\) (\(k, n\in N\)), we have the inequality
$$ \begin{aligned} \bigl\Vert \sigma _{n}^{-\alpha } ( f ) -f \bigr\Vert _{p} &\leq c ( \alpha ) \Biggl\{ 2^{k\alpha }k\omega _{1} \bigl( f,1/2^{k-1} \bigr) _{p}+2^{k\alpha }k \omega _{2} \bigl( f,1/2^{k-1} \bigr) _{p} \\ &\quad {} {}+\sum_{r=0}^{k-2}2^{r-k} \omega _{1} \bigl( f,1/2^{r} \bigr) _{p}+\sum _{s=0}^{k-2}2^{s-k}\omega _{2} \bigl( f,1/2^{s} \bigr) _{p} \Biggr\} . \end{aligned} $$
In this paper, we state and prove analogous results in the case of double Vilenkin–Fourier series. Our main results are the following theorems.
Theorem 1
Let
f
belong to
\(L^{p} ( G_{m}^{2} ) \)
for some
\(p \in [ 1,\infty ] \)
and
\(\alpha \in ( 0,1 ) \). Then, for any
\(M_{k}\leq n< M_{k+1}\) (\(k, n\in N\)), we have the inequality
$$ \begin{aligned} \bigl\Vert \sigma _{M_{k}}^{-\alpha } ( f ) -f \bigr\Vert _{p}&\leq c ( \alpha ) \Biggl( \omega _{1} ( f,1/M_{k-1} ) _{p}M_{k}^{\alpha }+\omega _{2} ( f,1/M_{l-1} ) _{p}M_{k}^{\alpha }\\ &\quad {} +\sum_{r=0}^{k-2} \frac{M_{r}}{M_{k}}\omega _{1} ( f,1/M _{r} ) _{p}+ \sum_{s=0}^{k-2}\frac{M_{s}}{M_{k}}\omega _{2} ( f,1/M_{s} ) _{p} \Biggr). \end{aligned} $$
Theorem 2
Let
f
belong to
\(L^{p} ( G_{m}^{2} ) \)
for some
\(p \in [ 1,\infty ] \)
and
\(\alpha \in ( 0,1 ) \). Then, for any
\(M_{k}\leq n< M_{k+1}\) (\(k, n\in N\)), we have the inequality
$$ \begin{gathered} \bigl\Vert \sigma _{n}^{-\alpha } ( f ) -f \bigr\Vert _{p} \\ \quad \leq c ( \alpha ) \Biggl( \omega _{1} ( f,1/M_{k-1} ) _{p}M_{k}^{\alpha }\log n+\omega _{2} ( f,1/M_{l-1} ) _{p}M _{k}^{\alpha }\log n \\ \qquad {} +\sum_{r=0}^{k-2} \frac{M_{r}}{M_{k}}\omega _{1} ( f,1/M _{r} ) _{p}+ \sum_{s=0}^{k-2}\frac{M_{s}}{M_{k}}\omega _{2} ( f,1/M_{s} ) _{p} \Biggr). \end{gathered} $$
2 Auxiliary lemmas
Lemma 1
Let
\(\alpha _{1}, \alpha _{2},\ldots,\alpha _{n}\)
be real numbers. Then
$$ \frac{1}{n} \int _{G} \Biggl\vert \sum_{k=1}^{n} \alpha _{k}D _{k}(x) \Biggr\vert \,d\mu (x)\leq \frac{c}{\sqrt{n}} \Biggl( \sum_{k=1}^{n} \alpha _{k}^{2} \Biggr) ^{1/2}. $$
Lemma 2
Let
\(\alpha _{1}, \alpha _{2},\ldots,\alpha _{n}\)
be real numbers. Then
$$ \frac{1}{n} \int _{G_{m}^{2}} \Biggl\vert \sum_{k=1}^{n} \alpha _{k}D_{k} ( x ) D_{k} ( y ) \Biggr\vert \,d \mu ( x,y ) \leq \frac{c}{\sqrt{n}} \Biggl( \sum_{k=1} ^{n}\alpha _{k}^{2} \Biggr) ^{1/2}. $$
Lemma 3
Let
\(0\leq j< n_{s}M_{s}\)
and
\(0\leq n_{s}< m_{s}\). Then
$$ D_{n_{s}M_{s}-j}=D_{n_{s}M_{s}}-\psi _{n_{s}M_{s}-1}\bar{D}_{j}. $$
We also need the following new nemmas of independent interest.
Lemma 4
Let
f
belong to
\(L^{p} ( G_{m}^{2} ) \)
for some
\(p \in [ 1,\infty ] \). Then, for every
\(\alpha \in ( 0,1 ) \), we have the inequality
where
\(M_{k}\leq n< M_{k+1}\).
$$ \begin{aligned} I&:=\frac{1}{A_{n}^{-\alpha }} \Biggl\Vert \int _{G_{m}^{2}} \sum_{i=1}^{M_{k-1}}A_{n-i}^{-\alpha -1}D_{i} ( u ) D _{i} ( v ) \bigl[ f ( \cdot -u,\cdot - ) -f ( \cdot ,\cdot ) \bigr] \,d\mu (u,v) \Biggr\Vert _{p} \\ & \leq \sum_{r=0}^{k-2} \frac{M_{r}}{M_{k}}\omega _{1} ( f,1/M_{r} ) _{p}+ \sum_{s=0}^{k-2}\frac{M_{s}}{M _{k}}\omega _{2} ( f,1/M_{s} ) _{p} , \end{aligned} $$
Lemma 5
Let
\(\alpha \in ( 0,1 ) \)
and
\(p=M_{k},M_{k}+1,\ldots \) . Then
$$ \mathit{II}:= \int _{G_{m}^{2}} \Biggl\vert \sum_{i=1}^{M_{k}}A_{p-i} ^{-\alpha -1}D_{i} ( u ) D_{i} ( v ) \Biggr\vert \,d \mu (u,v)\leq c ( \alpha ) < \infty ,\quad k=1,2,\ldots . $$
Lemma 6
We have the inequality
$$ \mathit{III}:= \int _{G_{m}^{2}} \Biggl\vert \sum_{i=1}^{n}A_{n-i} ^{-\alpha -1}D_{i} ( u ) D_{i} ( v ) \Biggr\vert \,d \mu (u,v)\leq c ( \alpha ) \log n $$
3 The detailed proofs
Proof of Lemma 3
Applying Abel’s transformation, from (2) we get
where the first and second terms on the right side of inequality (5) are denoted by \(I_{1}\) and \(I_{2}\), respectively.
$$ \begin{aligned}[b] I&\leq \frac{1}{A_{n}^{-\alpha }} \Biggl\Vert \int _{G_{m}^{2}} \sum_{i=1}^{M_{k-1}-1}A_{n-i}^{-\alpha -2} \sum_{l=1} ^{i}D_{i} ( u ) D_{i} ( v ) \bigl[ f ( \cdot -u, \cdot -v ) -f ( \cdot ,\cdot ) \bigr] \,d\mu (u,v) \Biggr\Vert _{p} \\ &\quad {}+\frac{1}{A_{n}^{-\alpha }} \Biggl\Vert \int _{G_{m}^{2}}A_{n-M _{k-1}}^{-\alpha -1}\sum _{i=1}^{M_{k-1}}D_{i} ( u ) D _{i} ( v ) \bigl[ f ( \cdot -u,\cdot -v ) -f ( \cdot ,\cdot ) \bigr] \,d\mu (u,v) \Biggr\Vert _{p} \\ &:=I_{1}+I_{2}, \end{aligned} $$
(5)
For \(I_{2}\), we have the estimate
where the first, second, and third terms on the right side of inequality (6) are denoted by \(I_{21}\), \(I_{22}\), and \(I_{23}\), respectively.
$$ \begin{aligned}[b] I_{2}&\leq \frac{1}{A_{n}^{-\alpha }} \Biggl\Vert \int _{G_{m} ^{2}}A_{n-M_{k-1}}^{-\alpha -1}\sum _{r=1}^{k-2}\sum_{i=M_{r}}^{M_{r+1}-1}D_{i} (u ) D_{i} ( v ) \\ &\quad {}\times \bigl[ f ( \cdot -u,\cdot -v ) -f ( \cdot ,\cdot ) \bigr] \Biggr\Vert _{p}\,d\mu (u,v) \\ &\leq \frac{1}{A_{n}^{-\alpha }} \Biggl\Vert \int _{G_{m}^{2}}A _{n-M_{k-1}}^{-\alpha -1}\sum _{r=1}^{k-2}\sum_{i=M_{r}} ^{M_{r+1}-1}D_{i} (u ) D_{i} ( v ) \\ &\quad {}\times \bigl[ f ( \cdot -u,\cdot -v ) -S_{M_{r},M_{r}} ( \cdot -u,\cdot -v,f ) \bigr] \,d\mu (u,v) \Biggr\Vert _{p} \\ &\quad {}+\frac{1}{A_{n}^{-\alpha }} \Biggl\Vert \int _{G_{m}^{2}}A_{n-M _{k-1}}^{-\alpha -1}\sum _{r=1}^{k-2}\sum_{i=M_{r}}^{M _{r+1}-1}D_{i} ( u ) D_{i} ( v ) \\ &\quad {}\times \bigl[ S_{M_{r},M_{r}} ( \cdot -u,\cdot -v,f ) -S_{M_{r},M_{r}} ( \cdot ,\cdot ,f ) \bigr] \,d\mu (u,v) \Biggr\Vert _{p} \\ &\quad {}+\frac{1}{A_{n}^{-\alpha }} \Biggl\Vert \int _{G_{m}^{2}}A_{n-M _{k-1}}^{-\alpha -1}\sum _{r=1}^{k-2}\sum_{i=M_{r}}^{M _{r+1}-1}D_{i} ( u ) D_{i} ( v ) \\ &\quad {}\times \bigl[ S_{M_{r},M_{r}} ( \cdot ,\cdot ,f ) -f ( \cdot ,\cdot ) \bigr] \,d\mu (u,v) \Biggr\Vert _{p}\\ &:=I_{21}+I_{22}+I_{23}, \end{aligned} $$
(6)
It is evident that
$$ \begin{aligned}[b] & \int _{G_{m}^{2}}\sum_{i=M_{r}}^{M_{r+1}-1}D_{i} ( u ) D_{i} ( v ) \bigl[ S_{M_{r},M_{r}} ( \cdot -u, \cdot -v,f ) -S_{M_{r},M_{r}} ( \cdot ,\cdot ,f ) \bigr] \,d\mu (u,v) \\ &\quad =\sum_{i=M_{r}}^{M_{r+1}-1} \biggl( \int _{G_{m}^{2}}D_{i} ( u ) D_{i} ( v ) S_{M_{r},M_{r}} ( \cdot -u, \cdot -v,f ) \,d\mu (u,v)-S_{M_{r},M_{r}} ( \cdot , \cdot ,f ) \biggr) \\ &\quad =\sum_{i=M_{r}}^{M_{r+1}-1} \bigl( S_{i} \bigl( \cdot ,\cdot ,S _{M_{r},M_{r}} ( f ) \bigr) -S_{M_{r},M_{r}} ( \cdot , \cdot ,f ) \bigr) \\ &\quad =\sum_{i=M_{r}}^{M_{r+1}-1} \bigl( S_{M_{r},M_{r}} ( \cdot ,\cdot ,f ) -S_{M_{r},M_{r}} ( \cdot ,\cdot ,f ) \bigr) =0. \end{aligned} $$
Hence
$$ I_{22}=0. $$
(7)
Moreover, by the generalized Minkowski inequality, Lemma 2, and by (1) and (4) we obtain
$$ \begin{aligned}[b] I_{21}&\leq \frac{1}{A_{n}^{-\alpha }} \bigl\vert A_{n-M_{k-1}}^{- \alpha -1} \bigr\vert \sum _{r=1}^{k-2} \int _{G_{m}^{2}} \Biggl\vert \sum_{i=M_{r}}^{M_{r+1}-1}D_{i} ( u ) D _{i} ( v ) \Biggr\vert \\ &\quad {}\times \bigl\Vert f ( \cdot -u,\cdot -v ) -S_{M_{r},M_{r}} ( \cdot -u,\cdot -v,f ) \bigr\Vert _{p}\,d\mu (u,v) \\ &\leq \frac{c ( \alpha ) }{M_{k}}\sum_{r=1}^{k-2} \bigl( \omega _{1} ( f,1/M_{r} ) _{p}+\omega _{2} ( f,1/M _{r} ) _{p} \bigr) \\ &\quad {}\times \int _{G_{m}^{2}} \Biggl\vert \sum_{i=M_{r}}^{M_{r+1}-1}D _{i} ( x ) D_{i} ( y ) \Biggr\vert \,d\mu (u,v) \\ &\leq c ( \alpha ) \sum_{r=1}^{k-2} \frac{M_{r}}{M_{k}} \bigl( \omega _{1} ( f,1/M_{r} ) _{p}+ \omega _{2} ( f,1/M_{r} ) _{p} \bigr) . \end{aligned} $$
(8)
The estimation of \(I_{23}\) is analogous to that of \(I_{21}\):
$$ I_{23}\leq c ( \alpha ) \sum_{r=1}^{k-2} \frac{M_{r}}{M _{k}} \bigl( \omega _{1} ( f,1/M_{r} ) _{p}+\omega _{2} ( f,1/M _{r} ) _{p} \bigr) . $$
(9)
Analogously, we can estimate \(I_{1}\) as follows:
$$\begin{aligned} I_{1}&\leq \frac{1}{A_{n}^{-\alpha }}\sum _{r=1}^{k-2} \Biggl\Vert \int _{G_{m}^{2}}\sum_{i=M_{r}}^{M_{r+1}-1}A_{n-i} ^{-\alpha -2}\sum_{l=1}^{i}D_{l} ( u ) D_{l} ( v ) \\ &\quad {}\times \bigl[ f ( \cdot -u,\cdot -v ) -S_{M_{r},M_{r}} ( \cdot -u,\cdot -v,f ) \bigr] \,d\mu (u,v) \Biggr\Vert _{p} \\ &\quad {}+\frac{1}{A_{n}^{-\alpha }}\sum_{r=1}^{k-2} \Biggl\Vert \int _{G_{m}^{2}}\sum_{i=M_{r}}^{M_{r+1}-1}A_{n-i}^{- \alpha -2} \sum_{l=1}^{i}D_{l} ( u ) D_{l} ( v ) \\ &\quad {}\times \bigl[ S_{M_{r},M_{r}} ( \cdot -u,\cdot -v,f ) -S_{M _{r},M_{r}} ( \cdot ,\cdot ,f ) \bigr] \Biggr\Vert _{p}\,d \mu (u,v) \\ &\quad {}+\frac{1}{A_{n}^{-\alpha }}\sum_{r=1}^{k-2} \Biggl\Vert \int _{G_{m}^{2}}\sum_{i=M_{r}}^{M_{r+1}-1}A_{n-i}^{- \alpha -2} \sum_{l=1}^{i}D_{l} ( u ) D_{l} ( v ) \\ &\quad {}\times \bigl[ S_{M_{r},M_{r}} ( \cdot ,\cdot ,f ) -f ( \cdot ,\cdot ) \bigr] \,d\mu (u,v) \Biggr\Vert _{p} \\ &\leq \frac{1}{A_{n}^{-\alpha }}\sum_{r=1}^{k-2} \int _{G_{m}^{2}} \Biggl\vert \sum_{i=M_{r}}^{M_{r+1}-1}A_{n-i} ^{-\alpha -2}\sum_{l=1}^{i}D_{l} ( u ) D_{l} ( v ) \Biggr\vert \\ &\quad {}\times \bigl\Vert f ( \cdot -u,\cdot -v ) -S_{M_{r},M_{r}} ( \cdot -u,\cdot -v,f ) \bigr\Vert _{p}\,d\mu (u,v) \\ &\quad {}+\frac{1}{A_{n}^{-\alpha }}\sum_{r=1}^{k-2} \int _{G_{m} ^{2}} \Biggl\vert \sum_{i=M_{r}}^{M_{r+1}-1}A_{n-i}^{-\alpha -2} \sum_{l=1}^{i}D_{l} ( u ) D_{l} ( v ) \Biggr\vert \\ &\quad {}\times \bigl\Vert S_{M_{r},M_{r}} ( \cdot ,\cdot ,f ) -f ( \cdot ,\cdot ) \bigr\Vert _{p} \,d\mu (u,v) \\ &\leq c ( \alpha ) M_{k}^{\alpha }\sum _{r=1}^{k-2} \sum_{i=M_{r}}^{M_{r+1}-1} ( n-i ) ^{-\alpha -2}i \bigl( \omega _{1} ( f,1/M_{r} ) _{p}+\omega _{2} ( f,1/M _{r} ) _{p} \bigr) \\ &\leq c ( \alpha ) M_{k}^{\alpha }\sum _{r=1}^{k-2} \sum_{i=M_{r}}^{M_{r+1}-1} ( n-M_{r+1}-1 ) ^{-\alpha -2}i \bigl( \omega _{1} ( f,1/M_{r} ) _{p}+\omega _{2} ( f,1/M _{r} ) _{p} \bigr) \\ &\leq c ( \alpha ) \sum_{r=0}^{k-2} \frac{M_{r}}{M_{k}} \bigl( \omega _{1} ( f,1/M_{r} ) _{p}+ \omega _{2} ( f,1/M_{r} ) _{p} \bigr) . \end{aligned}$$
(10)
Proof of Lemma 4
It is evident that
where the first and second terms on the right side of inequality (12) are denoted by \(\mathit{II}_{1}\) and \(\mathit{II}_{2}\), respectively.
$$ \begin{aligned}[b] \mathit{II}&\leq \int _{G_{m}^{2}} \Biggl\vert \sum_{i=1}^{M_{k}-1}A _{p-M_{k}+i}^{-\alpha -1}D_{M_{k}-i} ( u ) D_{M_{k}-i} ( v ) \Biggr\vert \,d\mu (u,v) \\ &\quad {}+ \bigl\vert A_{p-M_{k}}^{-\alpha -1} \bigr\vert \int _{G_{m} ^{2}}D_{M_{k}} ( u ) D_{M_{k}} ( v ) \,d\mu (u,v)\\ &:=\mathit{II} _{1}+\mathit{II}_{2}, \end{aligned} $$
(12)
From (1) by \(\vert A_{p-M_{k}}^{-\alpha -1} \vert \leq 1\) we get that
Moreover, by Lemma 3 we have that
where the first, second, third, and fourth terms on the right side of inequality (14) are denoted by \(\mathit{II}_{11}\), \(\mathit{II}_{12}\), \(\mathit{II}_{13}\), and \(\mathit{II}_{14}\) respectively.
$$ \mathit{II}_{2}\leq 1. $$
(13)
$$\begin{aligned} \mathit{II}_{1}&\leq \int _{G_{m}^{2}} \Biggl\vert \sum_{i=1}^{M_{k}-1}A _{p-M_{k}+i}^{-\alpha -1}\bar{D}_{i} ( u ) \bar{D}_{i} ( v ) \Biggr\vert \,d\mu (u,v) \\ &\quad {}+ \int _{G_{m}^{2}}D_{M_{k}} ( u ) \Biggl\vert \sum _{i=1}^{M_{k}-1}A_{p-M_{k}+i}^{-\alpha -1} \bar{D}_{i} ( v ) \Biggr\vert \,d\mu (u,v) \\ &\quad {}+ \int _{G_{m}^{2}}D_{M_{k}} ( v ) \Biggl\vert \sum _{i=1}^{M_{k}-1}A_{p-M_{k}+i}^{-\alpha -1} \bar{D}_{i} ( u ) \Biggr\vert \,d\mu (u,v) \\ &\quad {}+ \Biggl\vert \sum_{i=1}^{M_{k}-1}A_{p-M_{k}+i}^{-\alpha -1} \Biggr\vert \int _{G_{m}^{2}}D_{M_{k}} ( u ) D_{M_{k}} ( v ) \,d\mu (u,v) \\ &:=\mathit{II}_{11}+\mathit{II}_{12}+\mathit{II}_{13}+ \mathit{II}_{14}, \end{aligned}$$
(14)
From (1) and (4) it follows that
$$ \mathit{II}_{14}\leq c ( \alpha ) \sum_{v=1}^{\infty }v^{- \alpha -1}< \infty . $$
(15)
By Applying Abel’s transformation, in view of Lemma 2, we have that
$$ \begin{aligned}[b] \mathit{II}_{11}&\leq \int _{G_{m}^{2}} \Biggl\vert \sum_{i=1}^{M _{k}-2}A_{p-M_{k}+i}^{-\alpha -2} \sum_{l=1}^{i}\bar{D}_{l} ( u ) \bar{D}_{l} ( v ) \Biggr\vert \,d\mu (u,v) \\ &\quad {}+ \int _{G_{m}^{2}} \Biggl\vert A_{p-1}^{-\alpha -1}\sum _{i=1}^{M_{k}-1}\bar{D}_{i} ( u ) \bar{D}_{i} ( v ) \Biggr\vert \,d\mu (u,v) \\ &\leq c ( \alpha ) \Biggl\{ \sum_{v=1}^{M_{k}-2} ( p-M_{k}+i ) ^{-\alpha -2}i+ ( p-1 ) ^{-\alpha -1}M_{k} \Biggr\} \\ &\leq c ( \alpha ) \Biggl\{ \sum_{i=1}^{\infty }i^{- \alpha -1}+M_{k}^{-\alpha } \Biggr\} < \infty . \end{aligned} $$
(16)
The estimation of \(\mathit{II}_{12}\) and \(\mathit{II}_{13}\) are analogous to the estimation of \(\mathit{II}_{11}\). Applying Abel’s transformation, in view of Lemma 1, we find that
and
$$ \begin{aligned}[b] \mathit{II}_{12}&\leq \int _{G_{m}^{2}}D_{M_{k}} ( u ) \Biggl\vert \sum _{i=1}^{M_{k}-2}A_{p-M_{k}+i}^{-\alpha -2}\sum _{l=1} ^{i}\bar{D}_{l} ( v ) \Biggr\vert \,d\mu (u,v) \\ &\quad {}+ \int _{G_{m}^{2}}D_{M_{k}} ( u ) \Biggl\vert A_{p-1} ^{-\alpha -1}\sum_{i=1}^{M_{k}-1} \bar{D}_{i} ( v ) \Biggr\vert \,d \mu (u,v) \\ &\leq c ( \alpha ) \Biggl\{ \sum_{v=1}^{M_{k}-2} ( p-M_{k}+i ) ^{-\alpha -2}i+ ( p-1 ) ^{-\alpha -1}M_{k} \Biggr\} \\ &\leq c ( \alpha ) \Biggl\{ \sum_{i=1}^{\infty }i^{- \alpha -1}+M_{k}^{-\alpha } \Biggr\} < \infty \end{aligned} $$
(17)
$$\begin{aligned} \mathit{III}_{12}&\leq \int _{G_{m}^{2}}D_{M_{k}} ( v ) \Biggl\vert \sum _{i=1}^{M_{k}-2}A_{p-M_{k}+i}^{-\alpha -2}\sum _{l=1} ^{i}\bar{D}_{l} ( u ) \Biggr\vert \,d\mu (u,v) \\ &\quad {}+ \int _{G_{m}^{2}}D_{M_{k}} ( v ) \Biggl\vert A_{p-1} ^{-\alpha -1}\sum_{i=1}^{M_{k}-1} \bar{D}_{i} ( u ) \Biggr\vert \,d \mu (u,v) \\ &\leq c ( \alpha ) \Biggl\{ \sum_{v=1}^{M_{k}-2} ( p-M_{k}+i ) ^{-\alpha -2}i+ ( p-1 ) ^{-\alpha -1}M_{k} \Biggr\} \\ &\leq c ( \alpha ) \Biggl\{ \sum_{i=1}^{\infty }i^{- \alpha -1}+M_{k}^{-\alpha } \Biggr\} < \infty . \end{aligned}$$
(18)
Proof of Lemma 5
Let
$$ n=n_{k_{1}}M_{k_{1}}+\cdots+n_{k_{s}}M_{k_{s}}, \quad k_{1}>\cdots >k_{s} \geq 0. $$
Denote
$$ n^{ ( i ) }=n_{k_{i}}M_{k_{i}}+\cdots +n_{k_{s}}M_{k_{s}}, \quad i=1,2,\ldots,s. $$
Since (see [20])
we find that
where the first, second, third, fourth, and fifth terms on the right side of inequality (20) are denoted by \(\mathit{III}_{1}\), \(\mathit{III}_{2}\), \(\mathit{III}_{3}\), \(\mathit{III}_{4}\), and \(\mathit{III}_{5}\), respectively.
$$ D_{j+n_{A}M_{A}}=D_{n_{A}M_{A}}+\psi _{n_{A}M_{A}}D_{j}, $$
(19)
$$ \begin{aligned}[b] \mathit{III}&\leq \int _{G_{m}^{2}} \Biggl\vert \sum_{i=1}^{n_{k_{1}}M _{k_{1}}}A_{n-i}^{-\alpha -1}D_{i} ( u ) D_{i} ( v ) \Biggr\vert \,d\mu (u,v) \\ &\quad {}+ \int _{G_{m}^{2}} \Biggl\vert \sum_{i=1}^{n^{ ( 2 ) }}A_{n^{ ( 2 ) }-i}^{-\alpha -1}D_{i} ( u ) D_{i} ( v ) \Biggr\vert \,d\mu (u,v) \\ &\quad {}+ \int _{G_{m}^{2}}D_{n_{k_{1}}M_{k_{1}}} ( u ) D_{n _{k_{1}}M_{k_{1}}} ( v ) \Biggl\vert \sum_{i=1}^{n^{ ( 2 ) }}A_{n^{ ( 2 ) }-i}^{-\alpha -1} \Biggr\vert \,d \mu (u,v) \\ &\quad {}+ \int _{G_{m}^{2}}D_{n_{k_{1}}M_{k_{1}}} ( u ) \Biggl\vert \sum _{i=1}^{n^{ ( 2 ) }}A_{n^{ ( 2 ) }-i}^{- \alpha -1}D_{i} ( v ) \Biggr\vert \,d\mu (u,v) \\ &\quad {}+ \int _{G_{m}^{2}}D_{n_{k_{1}}M_{k_{1}}} ( v ) \Biggl\vert \sum _{i=1}^{n^{ ( 2 ) }}A_{n^{ ( 2 ) }-i}^{- \alpha -1}D_{i} ( u ) \Biggr\vert \,d\mu (u,v) \\ &:=\mathit{III}_{1}+\mathit{III}_{2}+\mathit{III}_{3}+ \mathit{III}_{4}+\mathit{III}_{5}, \end{aligned} $$
(20)
By (1) we have that
$$ \mathit{III}_{3}\leq c ( \alpha ) . $$
(21)
Moreover, since (see [24])
for \(\mathit{III}_{4}\), we get that
$$ \Biggl\vert \sum_{i=1}^{n}A_{n-i}^{-\alpha -1}D_{i} ( u ) \Biggr\vert =O \bigl( \vert u \vert ^{\alpha -1} \bigr) , $$
(22)
$$ \begin{aligned}[b] \mathit{III}_{4}&\leq \int _{G_{m}^{2}}D_{n_{k_{1}}M_{k_{1}}} ( u ) \vert v \vert ^{\alpha -1}\,d\mu (u,v) \\ &\leq \int _{G_{m}} \vert v \vert ^{\alpha -1}\,d\mu ( v ) = \frac{1}{ \alpha }< \infty . \end{aligned} $$
(23)
Analogously, we find that
$$ \begin{aligned}[b] \mathit{III}_{5}&\leq \int _{G_{m}^{2}}D_{n_{k_{1}}M_{k_{1}}} ( v ) \vert u \vert ^{\alpha -1}\,d\mu (u,v) \\ &\leq \int _{G_{m}} \vert u \vert ^{\alpha -1}\,d\mu ( v ) = \frac{1}{ \alpha }< \infty . \end{aligned} $$
(24)
For \(r\in \{0,\ldots m_{A}-1\}\) and \(0\leq j< M_{A}\) (see [20]), this yields that
$$ D_{j+rM_{A}}= \Biggl( \sum_{q=0}^{r-1} \psi _{M_{A}}^{q} \Biggr) D _{M_{A}}+\psi _{M_{A}}^{r}D_{j}. $$
Thus we have
$$ \begin{gathered} \int _{G_{m}^{2}}\sum_{i=1}^{n_{k_{1}}M_{k_{1}}-1}A_{n-i} ^{-\alpha -1}D_{i} ( u ) D_{i} ( v ) \,d\mu (u,v) \\ \quad \leq \int _{G_{m}^{2}}\sum_{r=0}^{n_{k_{1}}-1} \sum_{i=0}^{M_{k_{1}}-1}A_{n-i-rM_{k_{1}}}^{-\alpha -1}D_{i+rM_{k _{1}}} ( u ) D_{i+rM_{k_{1}}} ( v ) \,d\mu (u,v) \\ \quad \leq \int _{G_{m}^{2}}\sum_{r=0}^{n_{k_{1}}-1} \sum_{i=0}^{M_{k_{1}}-1}A_{n-i-rM_{k_{1}}}^{-\alpha -1} \Biggl( \sum_{q=0}^{r-1}\psi _{M_{k_{1}}}^{q} \Biggr) D_{M_{k_{1}}} ( u ) \\ \qquad {}\times \Biggl( \sum_{q=0}^{r-1} \psi _{M_{k_{1}}}^{q} \Biggr) D _{M_{k_{1}}} ( v ) \,d\mu (u,v) \\ \qquad {}+ \int _{G_{m}^{2}}\sum_{r=0}^{n_{k_{1}}-1} \sum_{i=0}^{M_{k_{1}}-1}A_{n-i-rM_{k_{1}}}^{-\alpha -1} \Biggl( \sum_{q=0}^{r-1}\psi _{M_{k_{1}}}^{q} \Biggr) D_{M_{k_{1}}} ( u ) \psi _{M_{A}}^{r}D_{i} ( v ) \,d\mu (u,v) \\ \qquad {}+ \int _{G_{m}^{2}}\sum_{r=0}^{n_{k_{1}}-1} \sum_{i=0}^{M_{k_{1}}-1}A_{n-i-rM_{k_{1}}}^{-\alpha -1} \psi _{M_{A}}^{r}D _{i} ( u ) \Biggl( \sum _{q=0}^{r-1}\psi _{M_{k_{1}}} ^{q} \Biggr) D_{M_{k_{1}}} ( v ) \,d\mu (u,v) \\ \qquad {}+ \int _{G_{m}^{2}}\sum_{r=0}^{n_{k_{1}}-1} \sum_{i=0}^{M_{k_{1}}-1}A_{n-i-rM_{k_{1}}}^{-\alpha -1} \psi _{M_{A}}^{r}D _{i} ( u ) \psi _{M_{A}}^{r}D_{i} ( v ) \,d\mu (u,v). \end{gathered} $$
On the other hand, by (1) and (4) we obtain that
$$ \int _{G_{m}^{2}}A_{n-n_{k_{1}}M_{k_{1}}}^{-\alpha -1}D_{n_{k _{1}}M_{k_{1}}} ( u ) D_{n_{k_{1}}M_{k_{1}}} ( v ) \,d \mu (u,v)\leq c ( \alpha ). $$
Consequently, for \(\mathit{III}_{1}\), we have the estimate
where the first, second, third, and fourth terms on the right side of inequality (25) are denoted by \(\mathit{III}_{11}\), \(\mathit{III}_{12} \), \(\mathit{III}_{13}\), and \(\mathit{III}_{14}\), respectively.
$$ \begin{aligned}[b] \mathit{III}_{1}&\leq \int _{G_{m}^{2}}D_{M_{k_{1}}} ( u ) D_{M _{k_{1}}} ( v ) \Biggl\vert \sum_{r=0}^{n_{k_{1}}-1} \sum _{i=1}^{M_{k_{1}}}A_{n-i-rM_{k_{1}}}^{-\alpha -1} \Biggr\vert \,d \mu (u,v) \\ &\quad {}+ \int _{G_{m}^{2}}D_{M_{k_{1}}} ( u ) \Biggl\vert \sum _{r=0}^{n_{k_{1}}-1}\sum_{i=1}^{M_{k_{1}}}A_{n-i-rM_{k _{1}}}^{-\alpha -1}D_{i} ( v ) \Biggr\vert \,d\mu (u,v) \\ &\quad {}+ \int _{G_{m}^{2}}D_{M_{k_{1}}} ( v ) \Biggl\vert \sum _{r=0}^{n_{k_{1}}-1}\sum_{i=1}^{M_{k_{1}}}A_{n-i-rM_{k _{1}}}^{-\alpha -1}D_{i} ( u ) \Biggr\vert \,d\mu (u,v) \\ &\quad {}+ \int _{G_{m}^{2}} \Biggl\vert \sum_{r=0}^{n_{k_{1}}-1} \sum_{i=1}^{M_{k_{1}}}A_{n-i-rM_{k_{1}}}^{-\alpha -1}D_{i} ( u ) D_{i} ( v ) \Biggr\vert \,d\mu (u,v)+c ( \alpha ) \\ &:=\mathit{III}_{11}+\mathit{III}_{12}+ \mathit{III}_{13}+\mathit{III}_{14}+c ( \alpha ) , \end{aligned} $$
(25)
From Lemma 4 we have that
$$ \mathit{III}_{14}\leq c ( \alpha ) . $$
(26)
The estimation of \(\mathit{III}_{11}\) is analogous to that of \(\mathit{III}_{3}\), and we find that
$$ \mathit{III}_{11}\leq c ( \alpha ) . $$
(27)
The estimation of \(\mathit{III}_{12}\) and \(\mathit{III}_{13}\) is analogous to that of \(\mathit{III}_{4}\), and we obtain that
and
$$ \mathit{III}_{12}< \infty $$
(28)
$$ \mathit{III}_{13}< \infty . $$
(29)
After substituting (21) and (23)–(29) into (20), we conclude that
$$ \begin{gathered} \int _{G_{m}^{2}} \Biggl\vert \sum_{i=1}^{n}A_{n-i}^{- \alpha -1}D_{i} ( u ) D_{i} ( v ) \Biggr\vert \,d \mu (u,v) \\ \quad \leq \int _{G_{m}^{2}} \Biggl\vert \sum_{i=1}^{n^{ ( 2 ) }}A_{n^{ ( 2 ) }-i}^{-\alpha -1}D_{i} ( u ) D_{i} ( v ) \Biggr\vert \,d\mu (u,v)+c ( \alpha ) \\ \quad \leq \cdots\leq \int _{G_{m}^{2}} \Biggl\vert \sum_{i=1}^{n ^{ ( s ) }}A_{n^{ ( s ) }-i}^{-\alpha -1}D_{i} ( u ) D_{i} ( v ) \Biggr\vert \,d\mu (u,v)+c ( \alpha ) s \\ \quad \leq c ( \alpha ) +c ( \alpha ) s\leq c ( \alpha ) \log n. \end{gathered} $$
The proof is complete. □
Now we are ready to prove the main results.
Proof of Theorem 1
It is evident that
$$ \begin{gathered}[b] \bigl\Vert \sigma _{M_{k}}^{-\alpha } ( f ) -f \bigr\Vert _{p} \\ \quad \leq \frac{1}{A_{M_{k}-1}^{-\alpha }} \Biggl\Vert \int _{G_{m} ^{2}}\sum_{i=1}^{M_{k-1}}A_{M_{k}-i}^{-\alpha -1}D_{i} ( u ) D_{i} ( v ) \bigl[ f ( \cdot -u,\cdot -v ) -f ( \cdot ,\cdot ) \bigr] \,d\mu (u,v) \Biggr\Vert _{p} \\ \qquad {}+\frac{1}{A_{M_{k}-1}^{-\alpha }} \Biggl\Vert \int _{G_{m}^{2}} \sum_{i=M_{k-1}+1}^{M_{k}}A_{M_{k}-i}^{-\alpha -1}D_{i} ( u ) D_{i} ( v ) \bigl[ f ( \cdot -u,\cdot -v ) -f ( \cdot ,\cdot ) \bigr] \,d\mu (u,v) \Biggr\Vert _{p} \\ \quad :=I+\mathit{II}. \end{gathered} $$
(30)
From Lemma 5 it follows that
$$ I\leq c ( \alpha ) \sum_{r=0}^{k-2} \frac{M_{r}}{M _{k}} \bigl( \omega _{1} ( f,1/M_{r} ) _{p}+\omega _{2} ( f,1/M _{r} ) _{p} \bigr) . $$
(31)
Moreover, for II, we have the estimate
where the first and second terms on the right side of inequality (32) are denoted by \(\mathit{II}_{1}\) and \(\mathit{II}_{2}\), respectively.
$$ \begin{aligned}[b] \mathit{II}&\leq \frac{1}{A_{M_{k}-1}^{-\alpha }} \Biggl\Vert \int _{G_{m}^{2}}\sum_{i=M_{k-1}+1}^{M_{k}}A_{M_{k}-i}^{- \alpha -1}D_{i} ( u ) D_{i} ( v ) \\ &\quad {}\times \bigl[ f ( \cdot -u,\cdot -v ) -S_{M_{k-1}} ^{ ( 1 ) } ( \cdot -u,\cdot -v,f ) \bigr] \,d\mu (u,v) \Biggr\Vert _{p} \\ &\quad {}+\frac{1}{A_{M_{k}-1}^{-\alpha }} \Biggl\Vert \int _{G_{m}^{2}} \sum_{i=M_{k-1}+1}^{M_{k}}A_{M_{k}-i}^{-\alpha -1}D_{i} ( u ) D_{i} ( v ) \\ &\quad {}\times \bigl[ S_{M_{k-1}}^{ ( 1 ) } ( \cdot -u, \cdot -v,f ) -f ( \cdot ,\cdot ) \bigr] \,d\mu (u,v) \Biggr\Vert _{p}\\ &:=\mathit{II}_{1}+\mathit{II}_{2}, \end{aligned} $$
(32)
In view of the generalized Minkowski inequality, by (4) and Lemma 5 we get that
$$ \begin{aligned}[b] \mathit{II}_{1}&\leq \frac{1}{A_{M_{k}-1}^{-\alpha }} \int _{G_{m}^{2}} \Biggl\vert \sum_{i=M_{k-1}+1}^{M_{k}}A_{M_{k}-i}^{-\alpha -1}D _{i} ( u ) D_{i} ( v ) \Biggr\vert \\ &\quad {}\times \bigl\Vert f ( \cdot -u,\cdot -v ) -S_{M_{k-1}}^{ ( 1 ) } ( \cdot -u,\cdot -v,f ) \bigr\Vert _{p}\,d \mu (u,v) \\ &\leq c ( \alpha ) M_{k}^{\alpha }\omega _{1} ( f,1/M_{k-1} ) _{p}. \end{aligned} $$
(33)
The estimation of \(\mathit{II}_{2}\) is analogous to that of \(\mathit{II}_{1}\), and we find that
$$ \mathit{II}_{2}\leq c ( \alpha ) M_{k}^{\alpha }\omega _{2} ( f,1/M _{k-1} ) _{p}. $$
(34)
Proof of Theorem 2
It is evident that
where the first, second, and third terms on the right side of inequality (35) are denoted by I, II, and III, respectively.
$$ \begin{gathered}[b] \bigl\Vert \sigma _{n}^{-\alpha } ( f ) -f \bigr\Vert _{p} \\ \quad \leq \frac{1}{A_{n-1}^{-\alpha }} \Biggl\Vert \int _{G_{m}^{2}} \sum_{i=1}^{M_{k-1}}A_{n-i}^{-\alpha -1}D_{i} ( u ) D _{i} ( v ) \bigl[ f ( \cdot -u,\cdot -v ) -f ( \cdot ,\cdot ) \bigr] \,d\mu (u,v) \Biggr\Vert _{p} \\ \qquad {}+\frac{1}{A_{n-1}^{-\alpha }} \Biggl\Vert \int _{G_{m}^{2}} \sum_{i=M_{k-1}+1}^{M_{k}}A_{n-i}^{-\alpha -1}D_{i} ( u ) D_{i} ( v ) \bigl[ f ( \cdot -u,\cdot -v ) -f ( \cdot ,\cdot ) \bigr] \,d\mu (u,v) \Biggr\Vert _{p} \\ \qquad {}+\frac{1}{A_{n-1}^{-\alpha }} \Biggl\Vert \int _{G_{m}^{2}} \sum_{i=M_{k}+1}^{n}A_{n-i}^{-\alpha -1}D_{i} ( u ) D _{i} ( v ) \bigl[ f ( \cdot -u,\cdot -v ) -f ( \cdot ,\cdot ) \bigr] \,d\mu (u,v) \Biggr\Vert _{p} \\ \quad :=I+\mathit{II}+\mathit{III}, \end{gathered} $$
(35)
From Lemma 4 it follows that
$$ I\leq c ( \alpha ) \sum_{r=0}^{k-2} \frac{M_{r}}{M _{k}} \bigl( \omega _{1} ( f,1/M_{r} ) _{p}+\omega _{2} ( f,1/M _{r} ) _{p} \bigr) . $$
(36)
Next, we repeat the arguments just in the same way as in the proof of Theorem 1 and find that
$$ \mathit{II}\leq c ( \alpha ) M_{k}^{\alpha } \bigl( \omega _{1} ( f,1/M_{k-1} ) _{p}+\omega _{2} ( f,1/M_{k-1} ) _{p} \bigr) . $$
(37)
On the other hand, for III, we have
where the first, second, and third terms on the right side of inequality (38) are denoted by \(\mathit{III}_{1}\), \(\mathit{III}_{2}\), and \(\mathit{III}_{3}\), respectively.
$$\begin{aligned} \mathit{III}&\leq \frac{1}{A_{n-1}^{-\alpha }} \Biggl\Vert \int _{G_{m} ^{2}}\sum_{i=M_{k}+1}^{n}A_{n-i}^{-\alpha -1}D_{i} ( u ) D_{i} ( v ) \\ &\quad {}\times \bigl[ f ( \cdot -u,\cdot -v ) -f ( \cdot ,\cdot ) \bigr] \Biggr\Vert _{p}\,d\mu (u,v) \\ &\leq \frac{1}{A_{n}^{-\alpha }} \Biggl\Vert \int _{G_{m}^{2}} \sum_{i=M_{k}+1}^{n}A_{n-i}^{-\alpha -1}D_{i} ( u ) D _{i} ( v ) \\ &\quad {}\times \bigl[ f ( \cdot -u,\cdot -v ) -S_{M_{k},M_{k}} ( \cdot -u,\cdot -v,f ) \bigr] \,d\mu (u,v) \Biggr\Vert _{p} \\ &\leq \frac{1}{A_{n}^{-\alpha }} \Biggl\Vert \int _{G_{m}^{2}} \sum_{i=M_{k}+1}^{n}A_{n-i}^{-\alpha -1}D_{i} ( u ) D _{i} ( v ) \\ &\quad {}\times \bigl[ S_{M_{k},M_{k}} ( \cdot -u,\cdot -v,f ) -S_{M_{k},M_{k}} ( \cdot ,\cdot ,f ) \bigr] \,d\mu (u,v) \Biggr\Vert _{p} \\ &\leq \frac{1}{A_{n}^{-\alpha }} \Biggl\Vert \int _{G_{m}^{2}} \sum_{i=M_{k}+1}^{n}A_{n-i}^{-\alpha -1}D_{i} ( u ) D _{i} ( v ) \\ &\quad {}\times \bigl[ S_{M_{k},M_{k}} ( \cdot ,\cdot ,f ) -f ( \cdot ,\cdot ) \bigr] \,d\mu (u,v) \Biggr\Vert _{p} \\ &:= \mathit{III}_{1}+\mathit{III}_{2}+\mathit{III}_{3}, \end{aligned}$$
(38)
It is easy to show that
$$ \mathit{III}_{2}=0. $$
(39)
By the generalized Minkowski inequality and Lemma 5, for \(\mathit{III}_{1}\), we obtain that
$$ \begin{aligned}[b] \mathit{III}_{1}&\leq \frac{1}{A_{n}^{-\alpha }} \int _{G_{m}^{2}} \Biggl\vert \sum_{i=M_{k}+1}^{n}A_{n-i}^{-\alpha -1}D_{i} ( u ) D_{i} ( v ) \Biggr\vert \\ &\quad {}\times \bigl\Vert f ( \cdot -u,\cdot -v ) -S_{M_{r},M_{r}} ( \cdot -u,\cdot -v,f ) \bigr\Vert _{p}\,d\mu (u,v) \\ &\leq c ( \alpha ) M_{k}^{\alpha } \bigl( \omega _{1} ( f,1/M _{k-1} ) _{p}+\omega _{2} ( f,1/M_{k-1} ) _{p} \bigr) \\ &\quad {}\times \int _{G_{m}^{2}} \Biggl\vert \sum_{v=M_{k}+1}^{n}A _{n-v}^{-\alpha -1}D_{v} ( u ) D_{v} ( v ) \Biggr\vert \,d \mu (u,v) \\ &\leq c ( \alpha ) M_{k}^{\alpha }\log n \bigl( \omega _{1} ( f,1/M_{k-1} ) _{p}+\omega _{2} ( f,1/M_{k-1} ) _{p} \bigr) . \end{aligned} $$
(40)
The estimation of \(\mathit{III}_{3}\) is analogous to that of \(\mathit{III}_{2}\), and we find that
$$ \mathit{III}_{3}\leq c ( \alpha ) M_{k}^{\alpha } \log n \bigl( \omega _{1} ( f,1/M_{k-1} ) _{p}+\omega _{2} ( f,1/M_{k-1} ) _{p} \bigr) . $$
(41)
Acknowledgements
The authors would like to thank the referees for helpful suggestions.
Competing interests
The authors declare that they have no competing interests.
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