$(H_p-L_p)$ type inequalities for subsequences of N\"orlund means of Walsh-Fourier series

We investigate the subsequence $\{t_{2^n}f \}$ of N\"{o}rlund means with respect to the Walsh system generated by non-increasing and convex sequences. In particular, we prove that a big class of such summability methods are not bounded from the martingale Hardy spaces $H_p$ to the space $weak-L_p $ for $0<p<1/(1+\alpha) $, where $0<\alpha<1$. Moreover, some new related inequalities are derived. As application, some well-known and new results are pointed out for well-known summability methods, especially for N\"{o}rlund logarithmic means and Ces\`aro means.


Introduction
The terminology and notations used in this introduction can be found in Section 2.
The fact that the Walsh system is the group of characters of a compact abelian group connects Walsh analysis with abstract harmonic analysis was discovered independently by Fine [6] and Vilenkin [36].For general references to the Haar measure and harmonic analysis on groups see Pontryagin [27], Rudin [28], and Hewitt and Ross [10].In particular, Fine investigated the group G, which is a direct product of the additive groups Z 2 =: {0, 1} and introduced the Walsh system {w j } ∞ j=0 .It is well-known that Walsh 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 [34]) there exists a martingale f ∈ H p (0 < p < 1) , such that On the other hand, (for details see e.g. the books [29] and [37] and especially the newest one [26]) 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: (1) S 2 n f Hp ≤ c p f Hp , n ∈ N, p > 0.
Weisz [38] proved that Fejér means of Vilenkin-Fourier series are bounded from the martingale Hardy space H p to the space H p , for p > 1/2.Goginava [12] (see also [25], [18,19,20,21]) proved that there exists a martingale f ∈ H 1/2 such that sup n∈N σ n f 1/2 = +∞.However, Weisz [38] (see also [22]) proved that for every f ∈ H p , there exists an absolute constant c p , such that the following inequality holds: (2) Móricz and Siddiqi [16] 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 [3,4].Fridli, Manchanda and Siddiqi [9] improved and extended the results of Móricz and Siddiqi [16] 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 [11] 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 f in the space In [5] (see also [17]) it was proved that there exists a martingale f ∈ H p , (0 < p < 1) such that A counterexample for p = 1 was proved in [24].However, Goginava [13] proved that for every f ∈ H 1 , there exists an absolute constant c, such that the following inequality holds: (3) In [2] it was proved that for any 0 < p < 1, there exists a martingale In [23] is was proved that for any non-decreasing sequence (q k , k ∈ N) satisfying the conditions (4) then, for every f ∈ H p , where p > 1/(1 + α), there exists an absolute constant c p , depending only on p, such that the following inequality holds: (5) Boundedness does not hold from H p to weak−L p , for 0 < p < 1/(1+α).As a consequence, (for details see [39]) we get that the Cesàro means σ α n is bounded from H p to L p , for p > 1/(1 + α), but they are not bounded from H p to weak − L p , for 0 < p < 1/(1 + α).In the endpoint case p = 1/(1 + α), Weisz and Simon [31] proved that the maximal operator σ α, * of Cesàro means define by is bounded from the Hardy space H 1/(1+α) to the space weak −L 1/(1+α) .
In this paper we develop some methods considered in [1,2,15] (see also the new book [26]) and prove that for any 0 < p < 1, there exists a martingale f ∈ H p such that Moreover, we prove that a big class of subsequence {t 2 n f } of Nörlund means with respect to the Walsh system generated by non-increasing and convex sequences are not bounded from the martingale Hardy spaces H p to the space weak − L p for 0 < p < 1/(1 + α), where 0 < α < 1.Moreover, some new related inequalities are derived.As application, some well-known and new results are pointed out for wellknown summability methods, especially for Nörlund logarithmic means and Cesàro means.
The main results in this paper are presented and proved in Section 4. Section 3 is used to present some auxiliary results, where, in particular, Lemma 2 is new and of independent interest.In order not to disturb our discussions later on some definitions and notations are given in Section 2.

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.
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 numbers of n j differ from zero.Let |n| := max{k ∈ N : n k = 0}.The norms (or quasi-norms) of the spaces L p (G) and weak − L p (G) , (0 < p < ∞) are, respectively, defined by The k-th 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.[29]) the Walsh system is orthonormal and complete in L 2 (G) .Moreover, for any n ∈ N, we define the Fourier coefficients, partial sums and Dirichlet kernel by Recall that (for details see e.g.[29]): Let {q k , k ≥ 0} be a sequence of nonnegative numbers.The Nörlund means for the Fourier series of f are defined by In this paper we consider convex {q k , k ≥ 0} sequences, that is If the function ψ(x) is any real valued and convex function (for example and we also get that (9) q n−2 + q n+2 − 2q n ≥ 0, for all n ∈ N.
In the special case when {q k = 1, k ∈ N}, we have the Fejér means Moreover, if q k = 1/(k + 1), then we get the Nörlund logarithmic means: (10) The Cesàro means σ α n (sometimes also denoted (C, α)) is also wellknown example of Nörlund means defined by where We also define U α n means as Let us also define V α n means as .
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.[37]).
We say that this martingale belongs to the Hardy martingale spaces H p (G) , where 0 < p < ∞, if In the case f ∈ L 1 (G) , the maximal functions are also given by , then it is easy to show that the sequence F = (S 2 n f : n ∈ N) is a martingale and F * = M(f ).
, n ∈ N is a martingale, then the Walsh-Fourier coefficients must be defined in a slightly different manner: A bounded measurable function a is p-atom, if there exists an interval I, such that supp (a) ⊂ I,

Auxiliary Results
The Hardy martingale space H p (G) has an atomic characterization (see Weisz [37], [38]): if and only if there exist 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 : where Moreover, the following two-sided inequality holds , where the infimum is taken over all decompositions of f of the form (12).
We also state and prove the following new lemma of independent interest: Lemma 2. Let k ∈ N, {q k : k ∈ N} be any convex and non-increasing sequence and x ∈ I 2 (e 0 + e 1 ) ∈ I 0 \I 1 .Then, for any {α k }, the following inequality holds: Proof.Let x ∈ I 2 (e 0 + e 1 ) ∈ I 0 \I 1 .According to (7) and ( 8) we get that D j (x) = w j , if j is odd number, 0, if j is even number, and By using ( 9) we find that Hence, if we apply we find that The proof is complete.

The Main result
In previous Sections we have discussed a number of inequalities and sometimes their sharpness.Our main result is the following new sharpness result: Theorem 1.Let 0 ≤ α ≤ 1, β be any non-negative real number and t n be Nörlund means with convex and non-increasing sequence {q k : k ∈ N} satisfying the condition for some positive constant C.Then, for any 0 < p < 1/(1 + α) there exists a martingale f ∈ H p such that Proof.Let 0 < p < 1/(1 + α).Under condition ( 13) there exists a sequence {n k : k ∈ N} such that Let {α k : k ∈ N} ⊂ {n k : k ∈ N} be an increasing sequence of positive integers such that ( 14) (2 2αη ) where From ( 14) and Lemma 1 we find that f ∈ H p .
It is easy to prove that (17) Moreover, 15), ( 16) and ( 17) we can conclude that Hence, for II we can conclude that √ α k Let x ∈ I 2 (e 0 + e 1 ) ∈ I 0 \I 1 .According to that α 0 ≥ 1 we get that 2α k ≥ 2, for all k ∈ N and if use (7) we get that D 2 2α k = 0 and if we use Lemma 2 we can also conclude that By combining ( 16), ( 18)-( 21) for x ∈ I 2 (e 0 + e 1 ) we have that Hence, we can conclude that The proof is complete.
In a concrete case we get a result for Nörlund logarithmic means {L n } proved in [2]: Proof.It is easy to show that and condition (13) holds true for α = β = 0.
We also get similar new result for the V n means: Proof.It is easy to show that We also get a corresponding new result for the Cesàro means σ α 2 n .Corollary 3. Let 0 < p < 1/(1 + α), for some 0 < α ≤ 0.56.Then there exists a martingale f ∈ H p such that Proof.By a routine calculation we find that It is easy to show that when 0 < α < 0.56 this expression is positive.Hence, condition (13) holds true for β = 0 and 0 < α < 1.

Open questions and final remarks
Remark 1.This article can be regarded as a complement of the new book [26].In this book also a number of open problems are raised.Also this new investigation implies some corresponding open questions.