Some New Weak ( H p − L p ) -Type Inequality for Weighted Maximal Operators of Partial Sums of Walsh–Fourier Series

. In this paper, we introduce some new weighted maximal operators of the partial sums of the Walsh–Fourier series. We prove that for some “optimal” weights these new operators indeed are bounded from the martingale Hardy space H p ( G ) to the Lebesgue space weak − L p ( G ) , for 0 < p < 1 . Moreover, we also prove sharpness of this result. As a consequence we obtain some new and well-known results.


Introduction
All symbols used in this introduction can be found in Sect. 2.
It is well-known that the Walsh system does not form a basis in the space L 1 (G) (see e.g. [2,30]). Moreover, there exists a function in the dyadic Hardy space H 1 (G), such that the partial sums of f are not bounded in the L 1 -norm. Uniform and pointwise convergence and some approximation properties of partial sums in L 1 (G) norms were investigated by Avdispahić and Memić [1], Gát, Goginava and Tkebuchava [12,13], Nagy [17], Onneweer [18] and Persson, Schipp, Tephnadze and Weisz [21]. Fine [9] obtained sufficient conditions for the uniform convergence which are completely analogous to the Dini-Lipschits conditions. Gulicev [15] estimated the rate of uniform convergence of a Walsh-Fourier series2 by using Lebesgue constants and modulus of continuity. These problems for Vilenkin groups were investigated by Blatota, Nagy, Persson and Tephnadze [7] (see also [4][5][6]), Fridli [10] and Gát [11]. To study convergence of subsequences of Fejér means and their restricted maximal operators on the martingale Hardy spaces H p (G) for 0 < p ≤ 1/2, the central role is played by the fact that any natural number n ∈ N can be uniquely expressed as where only a finite numbers of n j differ from zero and their important characters [n] , |n| , ρ (n) and V (n) are defined by and In particular, (see [8,16,22]) from which it follows that, for any F ∈ L 1 (G), there exists an absolute constant c such that the following inequality holds: Moreover, for any f ∈ H 1 (G) (see [26]) For 0 < p < 1 in Refs. [24,25], the weighted maximal operator was investigated and it was proved that the following inequalities hold: Moreover, it was also proved that the rate of the sequence {(n + 1) given in the denominator of (3) can not be improved. In Refs. [26,27] (see also [3]), it was proved that if F ∈ H p (G), then there exists an absolute constant c p , depending only on p, such that In this paper, we prove that the weighted maximal operator of the partial sums of the Walsh-Fourier defined by is bounded from the martingale Hardy space H p (G) to the space weak − L p (G), for 0 < p < 1. We also prove the sharpness of this result (see Theorem 2). As a consequence, we obtain both some new and well-known results.
This paper is organized as follows: In order not to disturb our discussions later on some preliminaries are presented in Sect. 2. The main results and some of its consequences can be found in Sect. 3. The detailed proofs of the main results are given in Sect. 4.

Preliminaries
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 : Denote I n := I n (0) , I n := G\I n and e n := (0, . . . , 0, x n = 1, 0, . . .) ∈ G, for n ∈ N.
Then it is easy to prove that The norms (or quasi-norms) of the Lebesgue space L p (G) and the weak Lebesgue space L p,∞ (G) , (0 < p < ∞) are, respectively, defined by The k-th Rademacher function r k (x) is defined by Now, define the Walsh system w := (w n : n ∈ N) on G by The Walsh system is orthonormal and complete in L 2 (G) (see e.g. [14,22]).
If f ∈ L 1 (G) we can establish the Fourier coefficients, the partial sums of the Fourier series, the Dirichlet kernels with respect to the Walsh system in the usual manner: Recall that (see [19,20,22]) and Moreover, we have the following lower estimate (see [20]): The σ-algebra generated by the intervals {I n (x) : x ∈ G} will be denoted by ζ n (n ∈ N) .
Denote by F = (F n , n ∈ N) the martingale with respect to n (n ∈ N) (see e.g. [28]).
The maximal function F * of a martingale F is defined by In the case f ∈ L 1 (G) , the maximal function f * is given by For 0 < p < ∞, the Hardy martingale spaces H p (G) consists of all martingales for which It is easy to check that for every martingale F = (F n , n ∈ N) and every k ∈ N the limit exists and it is called the k-th Walsh-Fourier coefficients of F.
If F := (S 2 n f : n ∈ N) is a regular martingale, generated by f ∈ L 1 (G) , then (see e.g. [20,23,28]) A bounded measurable function a is called p-atom, if there exists a dyadic interval I, such that The dyadic Hardy martingale spaces H p (G) for 0 < p ≤ 1 have an atomic characterization. Namely, the following holds (see [20,28,29]): Lemma 2. A martingale F = (F n , n ∈ N) belongs to H p (G) (0 < p ≤ 1) if and only if there exists a sequence (a k , k ∈ N) of p-atoms and a sequence (μ k , k ∈ N) of real numbers such that for every n ∈ N, Moreover, F Hp inf ( ∞ k=0 |μ k | p ) 1/p , where the infimum is taken over all decomposition of F of the form (7).

The Main Results with Applications
Our first main result reads: , n be defined by (1) and ρ (n) be defined by (2). Then, the weighted maximal operator S * ,∇ , defined by is bounded from the martingale Hardy space H p (G) to the space weak−L p (G).
Our second main result shows that Theorem 1 can not be improved in general, because it is sharp in some special senses: (b) Let 0 < p < 1, n be defined by (1) and ρ (n) be defined by (2). If ϕ : N → [1, ∞) is a nondecreasing function, satisfying the condition then there exists a sequence {f n , n ∈ N} of p-atoms, such that Theorem 1 implies the following result of Weisz [29] (see also [28]):

Proofs of the Theorems
Proof of Theorem 1. Since σ n is bounded from L ∞ to L ∞ , by Lemma 2, the proof of Theorem 1 will be complete, if we prove that for every p-atom a. In this paper, c p (or C p ) denotes a positive constant depending only on p but which can be different in different places. We may assume that a is an arbitrary p-atom, with support I, μ (I) = 2 −M and I = I M . It is easy to see that S n a (x) = 0, when n < 2 M . Therefore, we can suppose that n ≥ 2 M . Since a ∞ ≤ 2 M/p , we obtain that . Then, it is easy to see that x + t ∈ I s \I s+1 for t ∈ I M and if we combine (5) and (6) we get that D n (x + t) = 0, for t ∈ I M so that Let Then, it is easy to see that x + t ∈ I s \I s+1 for t ∈ I M and if we again combine (5) and (6), we find that By applying (11) and (12) for any x ∈ I s \I s+1 , 0 ≤ s < M, we find that It immediately follows that for s ≤ M , we have the following estimate S * ,∇ a (x) ≤ C p 2 M/p for any x ∈ I s \I s+1 , s = 0, 1, . . . , M and also that μ x ∈ I s \I s+1 : S * ,∇ a (x) > C p 2 k/p = 0, k = M, M + 1, . . . (14) By combining (4) and (13), we get that In view of (14) and (15), we can conclude that which shows that (10) holds and the proof of is complete. Proof of Theorem 2. a) Set It is evident that Then, we have that Since from (5), it follows that Let q s n k ∈ N be such that 2 n k < q s n k < 2 n k +1 , where [q s n k ] = s, s = 0, . . . , n k − 1. By combining (16) and (17), we can conclude that Let x ∈ I s+1 (e s ). By using Lemma 1, we find that so that 284 Page 10 of 13 D. Baramidze et al. MJOM Finally, by combining (18) and (20), we obtain that so the proof of part a) is complete.
(b) Under condition (9), we can choose q s k n k ∈ N for some 0 ≤ s k < n k such that 2 n k < q s k n k < 2 n k +1 , where [q n k ] = s k and lim k→∞ 2 ρ(q s k Let x ∈ I s k +1 (e s k ). By using (19) we get that .
Author Contributions DB, LEP and GT gave the idea and initiated the writing of this paper. SH followed up on this with some complementary ideas. All authors read and approved the final manuscript.
Funding Open access funding provided by UiT The Arctic University of Norway (incl University Hospital of North Norway).