Some inequalities related to strong convergence of Riesz logarithmic means

In this paper we derive a new strong convergence theorem of Riesz logarithmic means of the one-dimensional Vilenkin–Fourier (Walsh–Fourier) series. The corresponding inequality is pointed out and it is also proved that the inequality is in a sense sharp, at least for the case with Walsh–Fourier series.


Introduction
Concerning definitions used in this introduction we refer to Sect. 2. Weisz [47] proved the boundedness of the maximal operator of Fejér means σ ψ, * with respect to bounded Vilenkin systems from the martingale Hardy space H p (G m ) to the space L p (G m ), for p > 1/2.
Theorem W The maximal operator of Fejér means σ ψ, * is bounded from the Hardy space H 1/2 (G m ) to the space weak-L 1/2 (G m ).
In [35] and [36] it was proved that the maximal operator σ For Walsh-Kaczmarzi system some analogical results were proved in [16] and [37]. Weisz [47] considered the norm convergence of the Fejér means of a Vilenkin-Fourier series and proved the following result.
Theorem W1 (Weisz) Let p > 1/2 and f ∈ H p (G m ). Then there exists an absolute constant c p , depending only on p, such that for all k = 1, 2, . . . and f ∈ H p (G m ) the following inequality holds: Moreover, in [34] it was proved that the assumption p > 1/2 in Theorem W1 is essential. In fact, the following is true.
Theorem W1 implies that For the Walsh system in [38] and for the bounded Vilenkin systems in [37] were proved that (1.2) holds, though Theorem T1 is not true for 0 < p < 1/2.
The Riesz logarithmic means with respect to the Walsh system was studied by Simon [31], Goginava [15], Gát, Nagy [13] and for Vilenkin systems by Gát [11] and Blahota, Gát [3], Persson, Ragusa, Samko, Wall [26]. Moreover, in [27] it was proved that the maximal operator of the Riesz logarithmic means of a Vilenkin-Fourier series is bounded from the martingale Hardy space H p (G m ) to the space L p (G m ) when p > 1/2 and is not bounded from the martingale Hardy space H p (G m ) to the space L p (G m ) when 0 < p ≤ 1/2.
In [35] and [36] it was proved that the Riesz logarithmic means has better properties than the Fejér means. In particular, one considered the maximal operator R there exists a martingale f ∈ H p (G m ), such that The main aim of this paper is to derive a new strong convergence theorem of the Riesz logarithmic means of one-dimensional Vilenkin-Fourier (Walsh-Fourier) series (see Theorem 1). The corresponding inequality is pointed out. The sharpness is proved in Theorem 2, at least for the case with Walsh-Fourier series.
The paper is organized as follows: In Sect. 2 some definitions and notations are presented. The main results are presented and proved in Sect. 3. Section 4 is reserved for some concluding remarks and open problems. Define the group G m as the complete direct product of the group Z m j with the product of the discrete topologies of the Z m j .

Definitions and notations
The direct product μ of the measures is a Haar measure on G m with μ(G m ) = 1. If sup n∈N m n < ∞, then we call G m a bounded Vilenkin group. If the generating sequence m is not bounded, then G m is said to be an unbounded Vilenkin group. In this paper we discuss only bounded Vilenkin groups. The elements of G m are represented by the sequences It is easy to give a base for the neighborhood of G m , namely Denote I n := I n (0) for n ∈ N and I n := G m \I n . Let It is evident that If we define the so-called generalized number system based on m in the following way: then every n ∈ N can be uniquely expressed as n = ∞ k=0 n j M j , where n j ∈ Z m j (j ∈ N) and only a finite number of the n j differ from zero. Let |n| := max{j ∈ N; n j = 0}.
The norm (or quasi-norm when p < 1) of the space L p (G m ) is defined by The space weak-L p (G m ) consists of all measurable functions f for which Next, we introduce on G m an orthonormal system which is called the Vilenkin system. Let us define complex valued function r k (x) : G m → C, the generalized Rademacher functions, as Now, define the Vilenkin system ψ := (ψ n : n ∈ N) on G m as The Vilenkin systems are orthonormal and complete in L 2 (G m ) (for details see e.g. [1]). Specifically, we call this system Walsh-Paley if m k = 2, for all k ∈ N. In this case we have the dyadic group G 2 = ∞ j=0 Z 2 , which is called the Walsh group and the Vilenkin system coincides with the Walsh functions defined by (for details see e.g. [17] and [29]) where n k = 0 ∨ 1 and x k = 0 ∨ 1. Now, we introduce analogues of the usual definitions in Fourier analysis.
If f ∈ L 1 (G m ), then we can establish the Fourier coefficients, the partial sums of the Fourier series, the Fejér means, the Dirichlet and Fejér kernels with respect to the Vilenkin system ψ (Walsh system w) in the usual manner: It is well known that (see e.g. [1]) The σ -algebra generated by the intervals {I n (x) : x ∈ G m } will be denoted by n (n ∈ N). Denote by f = (f (n) , n ∈ N) a martingale with respect to n (n ∈ N) (for details see e.g. [5,23,46]). The maximal function of a martingale f is defend by In the case f ∈ L 1 (G m ), the maximal functions are also given by For 0 < p < ∞ the Hardy martingale spaces H p (G m ) consist of all martingales for which If f ∈ L 1 (G m ), then it is easy to show that S M n f is n measurable and the sequence (S M n f : n ∈ N) is a martingale. If f = (f (n) , n ∈ N) is a martingale, then the Vilenkin-Fourier (Walsh-Fourier) coefficients must be defined in a slightly different manner, namely The Vilenkin-Fourier coefficients of f ∈ L 1 (G m ) are the same as those of the martingale (S M n f : n ∈ N) obtained from f . For the martingale f we consider the following maximal operators: , R α, * p f := sup n∈N log(n + 1)|R α n f | (n + 1) 1/p-2 (α = w or ψ).
A bounded measurable function a is a p-atom, if there exists an interval I, such that I a dμ = 0, a ∞ ≤ μ(I) -1/p , supp(a) ⊂ I.
In order to prove our main results we need the following lemma of Weisz (for details see e.g. Weisz [49]). By using atomic characterization (see Proposition 1) it can be easily proved that the following statement holds (see e.g. Weisz [50]).

Proposition 2 Suppose that an operator T is sub-linear and for some
for every p 0 -atom a, where I denotes the support of the atom. If T is bounded from L p 1 to L p 1 (1 < p 1 ≤ ∞), then  [6] (see also Later [19], Torchinsky [44], Wilson [51]).

Main results
Our first main result reads as follows. For the proof of Theorem 1 we will use the following lemmas.
Lemma 1 (see [38]) Let x ∈ I N (x k e k +x l e l ), 1 ≤ x k ≤ m k -1, 1 ≤ x l ≤ m l -1, k = 0, . . . , N -2, l = k + 1, . . . , N -1. Then Lemma 2 (see [39]) Let x ∈ I N (x k e k +x l e l ), 1 ≤ x k ≤ m k -1, 1 ≤ x l ≤ m l -1, k = 0, . . . , N -2, l = k + 1, . . . , N -1. Then Proof By using an Abel transformation, the kernels of the Riesz logarithmic means can be rewritten as (see also [39]) Hence, according to (2.2) we get sup n∈N G m L α n dμ ≤ c < ∞, where α = w or ψ and it follows that R ψ n is bounded from L ∞ to L ∞ . By Proposition 2, the proof of Theorem 1 will be complete, if we show that ∞ n=1 log p n Ī |R ψ n a| p dμ n 2-2p ≤ c p < ∞, for 0<p < 1/2, (3.3) for every p-atom a, where I denotes the support of the atom. Let a be an arbitrary p-atom with support I and μ(I) = M -1 N . We may assume that I = I N . It is easy to see that R ψ n a = σ ψ n (a) = 0, when n ≤ M N . Therefore we suppose that n > M N . Since a ∞ ≤ cM 2 N if we apply (3.2), then we can conclude that It is easy to see that It means that (3.3) holds true and the proof is complete.
Our next main result shows in particular that the inequality in Theorem 1 is in a special sense sharp at least in the case of Walsh-Fourier series (cf. also Problem 2 in the next section). Then there exists a martingale f ∈ H p (G 2 ) such that where R w n f denotes the nth Riesz logarithmic means with respect to Walsh-Fourier series of f .
Proof It is evident that if we assume that Φ(n) ≥ cn, where c is some positive constant then log p nΦ(n) n 2-2p ≥ n 1-2p log p n → ∞, as n → ∞, and also (3.10) holds. So, without loss of generality we may assume that there exists an increasing sequence of positive integers {α k : k ∈ N} such that Let {α k : k ∈ N} ⊆ {α k : k ∈ N} be an increasing sequence of positive integers such that α 0 ≥ 2 and . (3.14) We note that under condition (3.11) we can conclude that and it immediately follows that such an increasing sequence {α k : k ∈ N}, which satisfies conditions (3.12)-(3.14), can be constructed. Let and a k = 2 2α k (1/p-1) (D 2 2α k +1 -D 2 2α k ).