Some new ( H p − L p ) type inequalities 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 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
It is well-known that the Walsh system does not form a basis in the space L 1 (see e.g. [2,27]). Moreover, there exists a function f in the dyadic Hardy space H 1 , such that Davit Baramidze and George Tephnadze have contributed equally to this work.
the partial sums of f are not bounded in L 1 -norm, but the partial sums S n of the Walsh-Fourier series of a function f ∈ L 1 convergence in measure (see [9,12]). Uniform and pointwise convergence and some approximation properties of partial sums in L 1 norm were investigated by Onneweer [16], Goginava [10], Goginava and Tkebuchava [11], Nagy [15], Avdispahić and Memić [1], Persson et al. [18]. Fine [6] obtained sufficient conditions for the uniform convergence, which are complete analogies with the Dini-Lipschits conditions. Guličev [13] estimated the rate of uniform convergence of a Walsh-Fourier series by using Lebesgue constants L(n) := D n 1 and modulus of continuity. These problems for Vilenkin groups were considered by Blahota [3], Fridli [7] and Gát [8].
Above, and in the sequel, all used notations can be found in Sect. 2. For example, the notations D n and S n are given in (6).
To study convergence of subsequences of partial sums in the martingale Hardy spaces H p (G) for 0 < p ≤ 1, the central role plays uniquely expression of any 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 In particular, (see [5,14,19]) from which it follows that, for any F ∈ L 1 (G), there exists an absolute constant c such that Moreover, for any f ∈ H 1 , In [23] and [24] it was proved that if 0 < p < 1 and F ∈ H p , then there exists an absolute constant c p , depending only on p, such that Moreover, if 0 < p < 1, {n k : k ≥ 0} is any increasing sequence of positive integers such that sup k∈N ρ (n k ) = ∞ and : N + → [1, ∞) is any nondecreasing function, satisfying the condition For 0 < p < 1 in [21,22] was investigated and it was proved that the following estimate holds: Moreover, it was also proved that the rate of the sequence (n + 1) 1/ p−1 given in denominator of (3) can not be improved, but it was proved only for the special subsequences.
For p = 1 analogical results for the maximal operator was proved in [22]. One main aim of this paper is to generalize the estimate (4) for f ∈ H p (G), 0 < p < 1. Our main idea is to investigate much more general maximal operators by replacing the weights (n + 1) 1/ p−1 in (3) by more general "optimal" weights where ϕ : R + → R + is any nonnegative and nondecreasing function satisfying the condition ∞ n=1 1/ϕ p (n) < c < ∞ and prove that it is bounded from the martingale Hardy space H p (G) to the Lebesgue space L p (G), for 0 < p < 1. As a consequence we obtain some new and well-known results. In particular, we prove that the maximal operator S * ,∇ , defined by is bounded from the Hardy space H p (G) to the Lebesgue space L p (G) for any ε > 0 and is not bounded from the Hardy space H p (G) to the Lebesgue space L p (G) when ε = 0. This paper is organized as follows: In order not to disturb our discussions later on some definitions and notations are presented in Sect. 2. The main results and some of its consequences can be found in Sect. 3. The detailed proofs 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 Then it is easy to show that The norms (or quasi-norm) of the spaces L p (G) and weak− 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. [19]). 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 [17,19]) and Moreover, we have the following lower estimate (see [17]): The σ -algebra generated by the intervals {I n (x) : x ∈ G} will be denoted by ζ n (n ∈ N) . It is easy to see that I n+1 (x) ⊂ I n (x) and I n (x) = m n −1 x n =0 I n+1 (x) , for any x ∈ G and n ∈ N.
It follows that Denote by F = (F n , n ∈ N) the martingale with respect to n (n ∈ N) (for details see e.g. [25]).
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 (for details see e.g. [17,20] and [25]) 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 for 0 < p ≤ 1 have an atomic characterization. Namely, the following theorem holds (see [17,25,26]): (μ k , k ∈ N) of real numbers such that for every n ∈ N ∞ k=0 μ k S 2 n a k = F n ,

Lemma 2 A martingale F = (F n , n ∈ N) belongs to H p (0 < p ≤ 1) if and only if there exists a sequence (a k , k ∈ N) of p-atoms and a sequence
where ∞ k=0 |μ k | p < ∞.

Moreover, F H p inf
∞ k=0 |μ k | p 1/ p , where the infimum is taken over all decomposition of F of the form (9).

The main results
Our first main result reads: Theorem 1 Let 0 < p < 1, f ∈ H p (G) and ϕ : N + → R + be any nonnegative and nondecreasing function satisfying the condition Then the weighted maximal operator S * ,∇ , defined by

is bounded from the Hardy space H p (G) to the Lebesgue space L p (G).
Theorem 1 can be of special interest even if we restrict to subsequences.
Corollary 1 Let 0 < p < 1, f ∈ H p (G), ϕ : N + → R + be any nonnegative and nondecreasing function satisfying the condition (10) and {n k : k ≥ 0} be any sequence of positive numbers. Then the weighted maximal operator S * ,∇ , defined by is bounded from the Hardy space H p (G) to the Lebesgue space L p (G).
We also prove sharpness of Theorem 1: Theorem 2 Let 0 < p < 1, {n k : k ≥ 0} be a sequence of positive numbers and ϕ : N + → R + be any nonnegative and nondecreasing function satisfying the condition Then there exists p-atoms a k , such that Then, the maximal operator (11) can not be estimated by Hence, Theorem 1 and Remark 1 and Theorem proved in [21,22] follows that if 0 < p < 1, f ∈ H p (G) and ϕ : N + → R + be any nonnegative and nondecreasing function satisfying the condition (10), then the weighted maximal operator S * ,∇ , defined by is bounded from the Hardy space H p (G) to the Lebesgue space L p (G).
Now, we formulate a result proved in [22], which follows from Theorems 1 and 2:

Proofs of the Theorems
Proof of Theorem 1. By using Lemma 2 the proof of Theorem 1 will be complete, if we prove that for every p-atom a, with support I and μ (I ) = 2 −M . We may assume that this arbitrary p-atom a has support 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 find that Let x ∈ I M . Since V (n) ≤ ρ (n) + 2, by applying (2) we get that Let t ∈ I M and x ∈ I s \I s+1 , 0 ≤ s ≤ M − 1 < [n] or 0 ≤ s < [n] ≤ M − 1. Then x + t ∈ I s \I s+1 . By using (8) we get that D n (x + t) = 0 and S n a(x) Let x ∈ I s \I s+1 , [n] ≤ s ≤ M − 1. Then x + t ∈ I s \I s+1 , for t ∈ I M . By using (8) we find that Hence, by applying (15) we get that S n a(x) 2 (1/ p−1)ρ(n) ϕ(ρ (n)) By now using (18) for 0 < [n] < s/2 we can conclude that S n a(x) Moreover, according to (18) for s/2 ≤ [n] ≤ s we have that By combining (17), (19) and (20), for all x ∈ I s \I s+1 , 0 ≤ s ≤ M − 1 we get that . (21) By now combining (5) and (21) By combining (16) and (22) we obtain that (14) holds and the proof is complete.

Proof of Theorem 2.
In view of the condition (12) we have that Set f n k (x) = D 2 n k +1 (x) − D 2 n k (x) , n k ≥ 3. Finally, by combining (23) and (25)  The proof is complete.