Some new results and inequalities for subsequences of Nörlund logarithmic means of Walsh–Fourier series

We prove that there exists a martingale f∈Hp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f\in H_{p} $\end{document} such that the subsequence {L2nf}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\{L_{2^{n}}f \}$\end{document} of Nörlund logarithmic means with respect to the Walsh system are not bounded from the martingale Hardy spaces Hp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H_{p}$\end{document} to the space weak−Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$weak-L_{p} $\end{document} for 0<p<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0< p<1 $\end{document}. We also prove that for any f∈Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f\in L_{p}$\end{document}, p≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p\geq 1 $\end{document}, L2nf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L_{2^{n}}f$\end{document} converge to f at any Lebesgue point x. Moreover, some new related inequalities are derived.


Introduction
The terminology and notations used in this introduction can be found in Sect. 2.
It is well known that Vilenkin systems do not form bases in the space L 1 . Moreover, there is a function in the Hardy space H 1 , such that the partial sums of f are not bounded in the L 1 -norm. Moreover, (see Tephnadze [22]) there exists a martingale f ∈ H p (0 < p < 1), such that sup n∈N S 2 n +1 f weak-L p = ∞.
On the other hand, (for details see, e.g., the books [20] and [25]) the subsequence {S 2 n } of partial sums is bounded from the martingale Hardy space H p to the space H p , for all p > 0, that is, the following inequality holds: It is also well known that (see [20] and [16]) , for all Lebesgue points of f ∈ L p , where p ≥ 1.
Weisz [26] considered the norm convergence of Fejér means of Vilenkin-Fourier series and proved that the inequality σ k f p ≤ c p f H p , p > 1/2 and f ∈ H p , holds. Moreover, Goginava [8] (see also [12][13][14][15]18]) proved that the assumption p > 1/2 in (3) is essential. In particular, he showed that there exists a martingale f ∈ H 1/2 such that sup n∈N σ n f 1/2 = +∞. However, Weisz [26] (see also [17]) proved that for every f ∈ H p , there exists an absolute constant c p , such that the following inequality holds: Móricz and Siddiqi [11] investigated the approximation properties of some special Nörlund means of Walsh-Fourier series of L p functions in norm. Approximation properties for general summability methods can be found in [2,3]. Fridli, Manchanda and Siddiqi [5] improved and extended the results of Móricz and Siddiqi [11] to martingale Hardy spaces. The case when {q k = 1/k : k ∈ N} was excluded, since the methods are not applicable to Nörlund logarithmic means. In [6] Gát and Goginava proved some convergence and divergence properties of the Nörlund logarithmic means of functions in the Lebesgue space L 1 . In particular, they proved that there exists a function in the space L 1 , such that sup n∈N L n f 1 = ∞.
In [4] (see also [10]) it was proved that there exists a martingale f ∈ H p , (0 < p < 1) such that sup n∈N L n f p = ∞.
In [19] (see also [24]) it was proved that there exists a martingale f ∈ H 1 such that However, Goginava [7] proved that From this result it immediately follows that for every f ∈ H 1 , there exists an absolute constant c, such that the inequality holds for all n ∈ N. Goginava [7] also proved that for any f ∈ L 1 (G), a.e., as n → ∞.
Question 1 Is the subsequence {L 2 n } also bounded on the martingale Hardy spaces H p (G) when 0 < p < 1?
In Theorem 2 of this paper we give a negative answer to this question. In particular, we further develop some methods considered in [1,9] and prove that for any 0 < p < 1, there exists a martingale f ∈ H p such that sup n∈N L 2 n f weak-L p = ∞. Moreover, in our Theorem 1 we generalize the result of Goginava [7] and prove that for any f ∈ L 1 (G) and for any Lebesgue point x, The main results in this paper are presented and proved in Sect. 4. Section 3 is used to present some auxiliary lemmas, where, in particular, Lemma 2 is new and of independent interest. In order not to disturb our discussions later some definitions and notations are given in Sect. 4. Finally, Sect. 5 is reserved for some open questions we hope can be a source of inspiration for further research in this interesting area.

Definitions and notations
Let N + denote the set of the positive integers, N := N + ∪ {0}. Denote by Z 2 the discrete cyclic group of order 2, that is Z 2 := {0, 1}, where the group operation is the modulo 2 addition and every subset is open. The Haar measure on Z 2 is given so that the measure of a singleton is 1/2.
Define the group G as the complete direct product of the group Z 2 , with the product of the discrete topologies of Z 2 s. The elements of G are represented by sequences It is easy to give a base for the neighborhood of x ∈ G, namely: Denote I n := I n (0), I n := G\I n and e n := (0, . . . , 0, x n = 1, 0, . . .) ∈ G, for n ∈ N. It is easy to show that I M = M-1 s=0 I s \I s+1 . If n ∈ N, then every n can be uniquely expressed as n = ∞ k=0 n j 2 j , where n j ∈ Z 2 (j ∈ N) and only a finite number of n j differ from zero. Let |n| := max{k ∈ N : n k = 0}.
The norms (or quasinorms) of the spaces L p (G) and weak -L p (G), (0 < p < ∞) are, respectively, defined by The kth Rademacher function is defined by Now, define the Walsh system w := (w n : n ∈ N) on G as: It is well known that (see, e.g., [20]) The Walsh system is orthonormal and complete in L 2 (G) (see, e.g., [20]). If f ∈ L 1 (G) let us define Fourier coefficients, partial sums and the Dirichlet kernel by Recall that (for details see, e.g., [20]): and Let {q k : k ≥ 0} be a sequence of nonnegative numbers. The Nörlund means for the Fourier series of f are defined by The Riesz logarithmic means are defined by We note that this is an inverse of the Nörlund logarithmic means. The convolution of two functions f , g ∈ L 1 (G) is defined by It is well known that if f ∈ L p (G), g ∈ L 1 (G) and 1 ≤ p < ∞. Then, f * g ∈ L p (G) and the corresponding inequality holds: The representations for n ∈ N play a central role in the following, where P n := 1 Q n n k=1 q n-k D k and Y n := 1 Q n n k=1 q k D k are called the kernels of the Nörlund logaritmic and the Reisz means, respectively. It is well known that (see, e.g., Goginava [7] and Tephnadze [23]): Moreover, for all n ∈ N, In the case f ∈ L 1 (G) the maximal functions are given by It is well known (for details see, e.g., [20]) that if f ∈ L 1 (G), then According to a density argument of Calderon-Zygmund (see [20]) we obtain that if f ∈ L 1 (G), then 2 n I n (x) f (u) dμ(u) → 0, as n → ∞.
A point x on the Walsh group is called a Lebesgue point of f ∈ L 1 (G), if According to (2) we find that if f ∈ L 1 (G), then a.e. point is a Lebesgue point. Let f := (f (n) , n ∈ N) be a martingale with respect to n (n ∈ N), which are generated by the intervals {I n (x) : x ∈ G} (for details see, e.g., [25]).
We say that a martingale belongs to Hardy martingale spaces n ∈ N) is a martingale, then the Walsh-Fourier coefficients must be defined in a slightly different manner:

Auxiliary results
The Hardy martingale space H p (G) has an atomic characterization (see Weisz [25,26]). (a k , k ∈ N) of p-atoms, which means that they satisfy the conditions I a k dμ = 0, a k ∞ ≤ μ(I) -1/p , supp(a k ) ⊂ I, and a sequence (μ k , k ∈ N) of real numbers such that for every n ∈ N:

Lemma 1 A martingale f = (f (n) , n ∈ N) is in H p (0 < p ≤ 1) if and only if there exist a sequence
Moreover, f H p inf( ∞ k=0 |μ k | p ) 1/p , where the infimum is taken over all decompositions of f of the form (14).
We also state and prove a new lemma of independent interest: Lemma 2 Let n ∈ N and x ∈ I 2 (e 0 + e 1 ) ∈ I 0 \I 1 . Then, Proof Let x ∈ I 2 (e 0 + e 1 ) ∈ I 0 \I 1 . According to (8) and (9) we obtain that if j is an odd number, 0, if j is an even number, if we apply w 4k+2 = w 2 w 4k = -w 4k , for x ∈ I 2 (e 0 + e 1 ), we find that The proof is complete.

Main results
Our first main result reads: Theorem 1 Let p ≥ 1 and f ∈ L p (G). Then, Moreover, for all Lebesgue points of f , Proof Let n ∈ N. By combining (11) and (13) we immediately obtain L 2 n f p ≤ c p f p for all n ∈ N, which immediately implies (15).
To prove a.e. convergence we use identity (12) to obtain that By applying (2) we can conclude that I = S 2 n f (x) → f (x) for all Lebesgue points of f ∈ L p . Moreover, by using (7) we find that In view of (13) we see that and also note that II describes the Fourier coefficients of an integrable function. Hence, according to the Riemann-Lebesgue Lemma it vanishes as n → ∞, i.e., II → 0 for any x ∈ G, n → ∞.
The proof is complete.
Our next main result is the following answer of Question 1. Proof Let {α k : k ∈ N} be an increasing sequence of the positive integers such that and Let where λ k = 1 √ α k and a k = 2 2α k (1/p-1) (D 2 2α k +1 -D 2 2α k ).
From (16) and Lemma 1 we find that f ∈ H p . It is easy to show that Moreover, := I + II.