Some inequalities for Cesàro means of double Vilenkin–Fourier series

In this paper, we state and prove some new inequalities related to the rate of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{p}$\end{document}Lp approximation by Cesàro means of the quadratic partial sums of double Vilenkin–Fourier series of functions from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{p}$\end{document}Lp.


Introduction
Let N + denote the set of positive integers, and let N := N + ∪ {0}. Let m := (m 0 , m 1 , . . .) be a sequence of positive integers not less than 2. Denote by Z m k := {0, 1, . . . , 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 μ(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 , . . . , x j , . . .) (x j ∈ Z m j ). The group operation + in G m is given by x + y = (x 0 + y 0 ) mod m 0 , . . . , (x k + y k ) mod m k , . . . We define the so-called generalized number system based on m as follows: M 0 := 1, M k+1 := m k M k (k ∈ N ). Then every n ∈ N can be uniquely expressed as n = ∞ j=0 n j M j , where n j ∈ Z m j (j ∈ N + ), and only a finite number of n j differ from zero. We also use the following notation: |n| := max{k ∈ N : n k = 0} (i.e., M |n| ≤ n < M |n|+1 , n = 0). For x ∈ G m , we denote |x| := ∞ j=0 x j M j+1 (x j ∈ 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 → C, the generalized Rademacher functions, as follows: Now we define the Vilenkin system ψ := (ψ n : n ∈ N) on G m as In particular, if m = 2, then we call this system the Walsh-Paley system. Each ψ n is a character of G m , and all characters of G m are of this norm. Moreover, ψ n (-x) =ψ n (x).
The Dirichlet kernels are defined by Recall that (see [20] or [23]) 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. Letm be a sequence like m. The relation between the sequences (m n ) and (M n ) is the same as between sequences (m n ) and (M n ). The group G m × Gm 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 =m and G m × Gm = G 2 m . The norm of the space L p (G 2 m ) is defined by Denote by C(G 2 m ) the class of continuous functions on the group G 2 m endowed with the supremum norm.
For brevity in notation, we write L ∞ (G 2 m ) instead of C(G 2 m ). 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: The (C, -α) means of the double Vilenkin-Fourier series are defined as follows: It is well known that (see [28]) and where positive constants c 1 and c 2 depend on α.
The dyadic partial moduli of continuity of a function f ∈ L p (G 2 m ) in the L p -norm are defined by and whereas the dyadic mixed modulus of continuity is defined as follows: It is clear that 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].
In his monograph [27], Zhizhinashvili investigated the behavior of Cesàro (C, α)-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 ∈ [1, ∞] and α ∈ (0, 1). Then, for any 2 k ≤ n < 2 k+1 (k, n ∈ N ), we have the inequality Theorem B Let f belong to L p (G 2 ) for some p ∈ [1, ∞] and α ∈ (0, 1). Then, for any 2 k ≤ n < 2 k+1 (k, n ∈ N ), we have the inequality In this paper, we state and prove analogous results in the case of double Vilenkin-Fourier series. Our main results are the following theorems.
To make the proofs of these theorems clearer, we formulate some auxiliary lemmas in Sect. 2. Some of these lemmas are new and of independent interest. Detailed proofs can be found in Sect. 3.

Lemma 6
We have the inequality

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.
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.

It is evident that
Hence Moreover, by the generalized Minkowski inequality, Lemma 2, and by (1) and (4) we The estimation of I 23 is analogous to that of I 21 : Analogously, we can estimate I 1 as follows: By combining (7)-(9) with (10) for I we find that The proof of Lemma 3 is complete.

Proof of Lemma 4 It is evident that
where the first and second terms on the right side of inequality (12) are denoted by II 1 and II 2 , respectively. From (1) by |A -α-1 p-M k | ≤ 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 II 11 , II 12 , II 13 , and II 14 respectively.
From (1) and (4) it follows that By Applying Abel's transformation, in view of Lemma 2, we have that The estimation of II 12 and II 13 are analogous to the estimation of II 11 . Applying Abel's transformation, in view of Lemma 1, we find that and The proof is complete by combining (12)- (18).
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 III 1 , III 2 , III 3 , III 4 , and III 5 , respectively. By (1) we have that Moreover, since (see [24]) for III 4 , we get that Analogously, we find that For r ∈ {0, . . . m A -1} and 0 ≤ j < M A (see [20]), this yields that Thus we have On the other hand, by (1) and (4)