Maximal operators of Fejér means of Walsh-Kaczmarz-Fourier series

The main aim of this paper is to prove that there exists a martingale f ∈ H1/2 such that the maximal Fejér operator with respect to WalshKaczmarz system does not belong to the space L1/2 . For the two-dimensional case, we prove that there exists a martingale f ∈ H 1/2(f ∈ H1/2) such that the restricted (unrestricted) maximal operator of Fejér means of two-dimensional Walsh-Kaczmarz-Fourier series does not belong to the space weak-L1/2 .


Introduction
The first result with respect to the a.e.convergence of the Walsh-Fejér means σ n f is due to Fine [1].Later, Schipp [9] showed that the maximal operator σ * f := sup n |σ n f | is of weak type (1,1), from which the a. e. convergence follows by standard argument.Schipp's result implies by interpolation also the boundedness of σ * : L p → L p (1 < p ≤ ∞) .This fails to hold for p = 1 , but Fujii [2] proved that σ * is bounded from the dyadic Hardy space H 1 to the space L 1 .Fujii's theorem was extended by Weisz [19].Namely, he proved that the maximal operator of the Fejér means of the one-dimensional Walsh-Fourier series is bounded from the martingale Hardy space H p (G) to the space L p (G) for p > 1/2.Simon [13] gave a counterexample, which shows that this boundedness does not hold for 0 < p < 1/2.In the endpoint case p = 1/2 Weisz [21] proved that σ * is bounded from the Hardy space H 1/2 (G) to the space weak-L 1/2 (G) (see also [14]).In [5] the first author proved that the maximal operator σ * is not bounded from the Hardy space H 1/2 (G) to the space L 1/2 (G).
In 1948 S neider [16] introduced the Walsh-Kaczmarz system and showed that the inequality lim sup n→∞ D κ n (x) log n ≥ C > 0 holds a.e.In 1974 Schipp [10] and Young [17] proved that the Walsh-Kaczmarz system is a convergence system.Skvortsov in 1981 [15] showed that the Fejér means with respect to the Walsh-Kaczmarz system converge uniformly to f for any continuous functions f .Gát [3] proved, for any integrable functions, that the Fejér means with respect to the Walsh-Kaczmarz system converge almost everywhere to the function and Gát proved that σ κ * 1 ≤ C |f | H1 .The result of Gát was extended to the Hardy space by Simon [11], who proved that σ κ * is of type (H p , L p ) for p > 1/2 .Weisz [21] showed that in endpoint case p = 1/2 the maximal operator is of weak type In this paper we will prove a stronger result than the unboundedness of the maximal operator from the Hardy space H 1/2 to the space L 1/2 , in particular, we prove that there exists a martingale f ∈ H 1/2 such that For the two-dimensional Walsh-Kaczmarz-Fourier series Simon proved [12] that the restricted maximal operator σ κ * λ is bounded from the Hardy space H p to the space L p for all p > 1/2.
In the paper [7] it was proved that the assumption p > 1/2 is essential.Namely, the maximal operator σ κ * := sup n |σ κ n,n | of the Fejér means of double Fourier series with respect to the Walsh-Kaczmarz system is not bounded from the Hardy space H 1/2 to the space weak-L 1/2 .In this paper we will prove a stronger result than in the paper [7], in particular, we prove that there exists a martingale f Thus, as regards boundedness of σ κ, * the case of two-dimensional Walsh-Kaczmarz series differs from the case of one-dimensional Walsh-Kaczmarz series.
Let denote by Z 2 the discrete cyclic group of order 2, the group operation is the modulo 2 addition and every subset is open.The normalized Haar measure on Z 2 is given in the way that the measure of a singleton is 1/2.
The group operation on G is the coordinate-wise addition (denoted by + ), the normalized Haar measure (denoted by μ) and the topology are the product measure and topology.Dyadic intervalls are defined by I 0 (x) := G, I n (x) := {y ∈ G : y = (x 0 , ..., x n−1 , y n , y n+1 ...)} for x ∈ G, n ∈ P.They form a base for the neighborhoods of G. Let 0 = (0 : i ∈ N) ∈ G denote the null element of G and I n := I n (0) for n ∈ N.
Let L p denote the usual Lebesgue spaces on G (with the corresponding norm or quasinorm .p ).The space weak-L p consists of all measurable functions f for which The Rademacher functions are defined as Let the Walsh-Paley functions be the product functions of the Rademacher functions.Namely, each natural number n can be uniquely expressed as where only a finite number of n i 's different from zero.Let the order of n > 0 be denoted by |n| := max{j ∈ N : n j = 0}.Walsh-Paley functions are w 0 = 1 and for n ≥ 1 The Walsh-Kaczmarz functions are defined by κ 0 = 1 and for n ≥ 1 The set of Walsh-Kaczmarz functions and the set of Walsh-Paley functions is the same in dyadic blocks.Namely, for all k ∈ P and κ 0 = w 0 .V. A. Skvortsov (see [15]) gave a relation between the Walsh-Kaczmarz functions and the Walsh-Paley functions by the help of the transformation τ A : G → G defined by for A ∈ N. By the definition of τ A , we have The Dirichlet kernels are defined by where α n = w n or κ n (n ∈ P), D α 0 := 0. The 2 n th Dirichlet kernels have a closed form (see e.g.[8]) The σ -algebra generated by the dyadic intervals of measure 2 −k will be denoted by F k (k ∈ N) .
Denote by f = f (n) , n ∈ N a martingale with respect to (F n , n ∈ N) (for details see, e. g. [20]).The maximal function of a martingale f is defined by In case f ∈ L 1 (G) , the maximal function can also be given by For 0 < p < ∞ the Hardy martingale space H p (G) consists of all martingales for which (1) , ...) then the Walsh-(Kaczmarz)-Fourier coefficients must be defined in a little bit different way: The Walsh-(Kaczmarz)-Fourier coefficients of f ∈ L 1 (G) are the same as the ones of the martingale (S 2 n f : n ∈ N) obtained from f .
The two-dimensional dyadic cubes are of the form By F n,n , we denote the σ -algebra generated by the dyadic rectangles Denote by f = f (n,n) , n ∈ N a martingale with respect to (F n,n , n ∈ N) (for details see, e. g. [20]).The maximal function of a martingale f is defined by In case f ∈ L 1 (G × G) , the maximal function can also be given by We denote by F n,m (n, m ∈ N), the σ -algebra generated by the dyadic rectangles {I n,m (x, y) : (x, y) ∈ G × G} .
Denote by f = f (n,m) , n, m ∈ N a martingale with respect to (F n,m , n, m ∈ N) (for details see, e. g. [20]).
The maximal function of a martingale f is defined by For 0 < p < ∞ the Hardy martingale space H p (G × G) consists of all martingales for which , maximal functions can also be given by

The one-dimensional maximal operator
For n = 1, 2, ... and a martingale f the Fejér means of the Walsh-(Kaczmarz)-Fourier series of the function f is given by For a martingale f we consider the maximal operator The nth Fejér kernel of the Walsh-(Kaczmarz)-Fourier series defined by A bounded measurable function a is a p-atom, if there exists a dyadic interval I, such that a) The basic result of atomic decomposition is the following one.
, where the infimum is taken over all decompositions of f of the form (1).
We will use the following lemma of Goginava: We will prove the following theorem.
We can write the nth Dirichlet kernel with respect to the Walsh-Kaczmarz system in the following form: By the help of this, we immediately get Thus, from (9) we have Define the set J l,s 2A (x) for l < s < A by .., m k − 3, and s = l + 2, l + 3, ..., m k − 1 , then from Lemma 1 and (4) we have That is σ κ, * f 1/2 = +∞.The proof is complete.

The two-dimensional restricted maximal operator
For α = w or κ the rectangular partial sums of the double Walsh-(Kaczmarz)-Fourier series are defined as follows: where the number is said to be the (i, j)th Walsh-(Kaczmarz)-Fourier coefficient of the function f. n) : n ∈ N) then the Walsh-(Kaczmarz)-Fourier coefficients must be defined in a little bit different way: The Walsh-(Kaczmarz)-Fourier coefficients of f ∈ L 1 (G × G) are the same as the ones of the martingale (S 2 n ,2 n (f ) : n ∈ N) obtained from f .
For n, m ∈ P and a martingale f the (n, m)th Fejér mean of the double Walsh-(Kaczmarz)-Fourier series is given by For the martingale f the restricted maximal operator is defined by A bounded measurable function a is a p-atom, if there exists a dyadic 2-dimensional cube I × I, such that a) The basic result of atomic decomposition is the following one.[20]).A martingale f = f (n,n) : n ∈ N is in H p (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 ,

Theorem B. (Weisz
We will prove the following theorem.

Theorem 2. There exists a martingale
Proof.To prove Theorem 2 we modify the sequence {m k : k ∈ P} and atoms a k given in the previous section in the following way.
Let {m k : k ∈ N} be an increasing sequence of positive integers such that ( 12) The martingale f := f (0,0) , f (1,1) , ..., f (A,A) , ... 12) and Theorem B we conclude that f ∈ H 1/2 (G × G) .Now, we investigate the Fourier coefficients.Since we can write (15) We decompose the (q m k , q m k )th Fejér means as follows for some k .Then from ( 15) and ( 13) it is easy to show that Consequently, we have Combining ( 16)-( 19) we obtain that (20) Now, we discuss IV.Let (i, j) ∈ 2 2m k , ..., q m k − 1 × 2 2m k , ..., q m k − 1 .Then from (15) we have and By ( 13), ( 14) and |D 2 n (x)| ≤ 2 n we get that By the help of the equation (10) we immediatelly have for IV 2 Then from Lemma 1 we can write and This completes the proof of this theorem.m) : n, m ∈ N) then the Walsh-(Kaczmarz)-Fourier coefficients must be defined in a little bit different way:

The two-dimensional unrestricted maximal operator
The Walsh-(Kaczmarz)-Fourier coefficients of f ∈ L 1 (G × G) are the same as the ones of the martingale (S 2 n ,2 m (f ) : n, m ∈ N) obtained from f .
For the martingale f the unrestricted maximal operator of the Fejér mean is defined by The basic result of atomic decomposition is the following one.
Theorem C. (Weisz [20]).A martingale f = f (n,m) : n, m ∈ N is in H p (0 < p ≤ 1) if there exists a sequence (a k , k ∈ N) of rectangle p-atoms and a sequence (μ k , k ∈ N) of real numbers such that for every n, m ∈ N , Theorem 3 is proved.