Bulletin of TICMIVol. 18, No. 1, 2014, 110–121

Approximation by Marcinkiewicz Means of

Walsh-Kaczmarz-Fourier Series in the Hardy Space H2/3

Karoly Nagya∗ and George Tephnadze b

aInstitute of Mathematics and Computer Sciences, College of Nyıregyhaza, P.O. Box 166,

Nyıregyhaza, H-4400 HungarybInstitute of Mathematics, Faculty of Exact and Natural Sciences, Tbilisi State

University, Chavchavadze str. 1, Tbilisi 0128, Georgia and Department of Engineering

Sciences and Mathematics, Lulea University of Technology, SE-971 87 Lulea, Sweden(Received January 24, 2014; Revised May 15, 2014; Accepted June 19, 2014)

Gat, Goginava and Nagy proved that the maximal operator σκ,∗ of Marcinkiewicz-Fejer meansof Walsh-Kaczmarz-Fourier series, is bounded from the dyadic Hardy space Hp into the spaceLp for p > 2/3 [4]. Moreover, Goginava and Nagy showed that σκ,∗ is not bounded from theHardy space H2/3 to the space L2/3 [8]. The main aim of this paper is to investigate the

endpoint case p = 2/3. In the present work we give necessary and sufficient conditions for theconvergence of Walsh-Kaczmarz-Marcinkiewicz means in terms of modulus of continuity onthe Hardy space H2/3.

Keywords: Walsh-Kaczmarz system, Marcinkiewicz means, maximal operator,two-dimensional system, modulus of continuity, Hardy space, convergence, Fourier series.

AMS Subject Classification: 42C10.

1. Introduction

In 1948, Sneider [17] introduced the Walsh-Kaczmarz system and showed that theinequality

lim supn→∞

Dκn(x)

log n≥ C > 0

holds a.e. In 1974, Schipp [14] and Young [21] proved that the Walsh-Kaczmarzsystem is a convergence system. In 1981, Skvortsov [16] showed the uniform con-vergence of the Fejer means with respect to the Walsh-Kaczmarz system for anycontinuous functions f . Gat [2] proved, that the Walsh-Kaczmarz-Fejer means ofan integrable function converge almost everywhere to the function. He showed thatthe maximal operator σκ,∗ of Walsh-Kaczmarz-Fejer means is weak type (1, 1) andof type (p, p) for all 1 < p ≤ ∞. Gat’s result was generalized by Simon [15], heshowed that the maximal operator σκ,∗ is of type (Hp, Lp) for p > 1/2. In theendpoint case p = 1/2 Goginava [5] proved that the maximal operator σκ,∗ is not

∗Corresponding author. Email: nkaroly@nyf.hu

Vol. 18, No. 1, 2014 111

bounded from the dyadic Hardy space H1/2 to the space L1/2. Moreover, Weisz[24] showed that the maximal operator is of weak type (H1/2, L1/2).

In [5, 18] it was proved that the maximal operators σκ,∗p defined by

σκ,∗p := supn∈N

|σκn|(n+ 1)1/p−2 log2[1/2+p] (n+ 1)

,

(where 0 < p ≤ 1/2 and [x] denotes the integer part of x) are bounded fromthe Hardy space Hp to the space Lp. Moreover, it was proved that the sequence

{(n+ 1)1/p−2 log2[1/2+p] (n+ 1)}∞n=1 can not be weakened.The second author [19] (see also [20]) found a necessary and sufficient condition

for the convergence σκnf → f in Hp norm, in terms of modulus of continuity of themartingale f ∈ Hp.

In particular, it was proved that if

ωHp

(1/2k, f

)= o

(2k(1/p−2) log2[1/2+p] k

), as k →∞,

then

‖σκnf − f‖Hp → 0, as n→∞, (0 < p ≤ 1/2) . (1)

Moreover, there exists a martingale f ∈ Hp, (0 < p ≤ 1/2) , for which

ω(

1/2k, f)Hp

= O(

2k(1/p−2) log2[1/2+p] k), as k →∞

and

‖σκnf − f‖p 9 0, as n→∞.

In 1939, for the two-dimensional trigonometric Fourier partial sums Sj,jfMarcinkiewicz [9] has proved that the means

σnf =1

n

n∑j=1

Sj,jf

converge a.e. to f as n → ∞ for any f ∈ L logL([0, 2π]2). Zhizhiashvili [25] im-proved this result for f ∈ L1([0, 2π]2).

In 2006, the a.e. convergence of Walsh-Kaczmarz-Marcinkiewicz means wasproved by the first author [10]. He also proved that the maximal operator

σκ,∗f := supn∈P|σκnf | = sup

n∈P

1

n

∣∣∣∣∣∣n∑j=0

Sκj,jf

∣∣∣∣∣∣is of weak type (1, 1) and of type (p, p) for all 1 < p ≤ ∞. In [4] it was proved thatthe maximal operator σκ,∗ is bounded from the Hardy space Hp to the space Lpfor p > 2/3. In the paper [8] Goginava and the first author showed that, σκ,∗ is

112 Bulletin of TICMI

not bounded from the Hardy space H2/3 to the space L2/3. This means that, it isinteresting to discuss what does happen at the endpoint p = 2/3. Recently, it was

showed in [11] that the maximal operator σκ,∗f := supn∈P|σκnf |

log3/2(n+1), is bounded

from the Hardy space H2/3 into the space L2/3. Moreover, it was proved that the

sequence {log3/2(n + 1)}∞n=1 can not be weakened. That is, the order of deviant

behaviour of the n-th Walsh-Kaczmarz-Marcinkiewicz mean is exactly log3/2(n+1)in the endpoint case p = 2/3. As a corollary we get that

‖σκnf‖2/3 ≤ c log3/2(n+ 1)‖f‖H2/3(2)

for all f ∈ H2/3.The main aim of this paper is to continue the investigations at the endpoint

p = 2/3. We give necessary and sufficient conditions for the convergence of Walsh-Kaczmarz-Marcinkiewicz means in terms of modulus of continuity on the Hardyspace H2/3. That is, we give the two-dimensional version of the result of the secondauthor [19] (see equation (1)).

2. Definitions and notation

Now, we give a brief introduction to the theory of dyadic analysis [1, 13].Let P denote the set of positive integers, N := P∪{0}. Denote by Z2 the discrete

cyclic group of order 2, that is the elements of Z2 are 0,1, and the group operationis the modulo 2 addition. Let every subset be open. The Haar measure on Z2 isgiven such that µ({0}) = µ({1}) = 1/2. Let G be the complete direct productof the countable infinite copies of the compact groups Z2. The elements of G aresequences of the form x = (x0, x1, ..., xk, ...) with coordinates xk ∈ {0, 1} (k ∈ N) .The group operation on G is the coordinate-wise addition, the measure (denoted byµ) is the product measure and the topology is the product topology. The compactAbelian group G is called the Walsh group.

The dyadic intervals are defined by

I0 (x) := G, In (x) := In (x0, ..., xn−1) := {y ∈ G : y = (x0, ..., xn−1, yn, yn+1, ...)} ,

where (x ∈ G,n ∈ N) . They form a base for the neighbourhoods of G. Let 0 =(0 : i ∈ N) ∈ G denote the null element of G, and In := In (0) (n ∈ N) . Set en :=(0, ..., 0, 1, 0, ...) ∈ G, the n-th coordinate of which is 1 and the rest are zeros(n ∈ N) .

For k ∈ N the k-th Rademacher function is defined by

rk (x) := (−1)xk , (x = (x0, x1, ..., xk, ...) ∈ G)

If n ∈ N, then n can be expressed in the number system of base 2. That is,

n =∞∑i=0

ni2i can be written, where ni ∈ {0, 1} (i ∈ N). Denote the order of n by

|n| := max{j ∈ N :nj 6= 0}. We immediately have that 2|n| ≤ n < 2|n|+1.The Walsh-Paley system is defined as the product system of Rademacher func-

Vol. 18, No. 1, 2014 113

tions:

wn (x) :=

∞∏k=0

(rk (x))nk = r|n| (x) (−1)

|n|−1∑k=0

nkxk(x ∈ G,n ∈ P) .

The Walsh-Kaczmarz functions are defined by κ0 = 1 and for n ≥ 1

κn(x) := r|n|(x)

|n|−1∏k=0

(r|n|−1−k(x))nk = r|n|(x)(−1)∑|n|−1k=0 nkx|n|−1−k .

It is well-known that the set of Walsh-Kaczmarz functions and the set of Walsh-Paley functions are equal in dyadic blocks. Namely,

{κn : 2k ≤ n < 2k+1} = {wn : 2k ≤ n < 2k+1}

for all k ∈ P and κ0 = w0.Define the transformation τA : G→ G by

τA(x) := (xA−1, xA−2, ..., x1, x0, xA, xA+1, ...)

for A ∈ N. Skvortsov [16] gave a relation between the function κn and wn. Namely,

κn(x) = r|n|(x)wn−2|n|(τ|n|(x)) (n ∈ N, x ∈ G). (3)

The Dirichlet kernels are defined by

Dαn :=

n−1∑k=0

αk,

where αn = wn (for all n ∈ P) or κn (for all n ∈ P), Dα0 := 0. The 2n-th Dirichlet

kernels have a closed form (see e.g. [13])

Dw2n(x) = Dκ

2n(x) = D2n(x) =

{0, if x 6∈ In,2n, if x ∈ In.

(4)

The norm (or quasi-norm) of the space Lp(G2) (for the simplicity we write Lp)

is defined by

‖f‖p :=

(∫G2

|f(x1, x2)|p dµ(x1, x2)

)1/p

(0 < p <∞) .

The space weak − Lp consists of all measurable functions f for which

‖f‖weak−Lp := supλ>0

λµ (f > λ)1/p < +∞.

114 Bulletin of TICMI

The σ-algebra generated by the 2-dimensional cube of measure 2−2k will bedenoted by Fk (k ∈ N) . Denote by f =

(f (n), n ∈ N

)the one-parameter martingale

with respect to (Fn, n ∈ N) (for details see, e. g. [22, 23]). The maximal function ofa martingale f is defined by f∗ = sup

n∈N

∣∣f (n)∣∣ . For 0 < p <∞ the Hardy martingale

space Hp(G2) consists of all martingales for which ‖f‖Hp := ‖f∗‖p < ∞ (for

simplicity we use the notation Hp).The Kronecker product (αn,m : n,m ∈ N) of two Walsh-(Kaczmarz) system is

said to be the two-dimensional Walsh-(Kaczmarz) system. That is, αn,m(x1, x2

)=

αn(x1)αm(x2).

If f ∈ L1, then the number fα (n,m) :=∫G2

fαn,m (n,m ∈ N) is said to be the

(n,m)-th Walsh-(Kaczmarz)-Fourier coefficient of f. We can extend this definitionfor martingales in the usual way (see Weisz [22, 23]). Denote the (n,m)-th rect-angular partial sum of the Walsh-(Kaczmarz)-Fourier series of a martingale f bySαn,m. Namely,

Sαn,mf(x1, x2) :=

n−1∑k=0

m−1∑i=0

fα(k, i)αk,i(x1, x2).

The Marcinkiewicz-Fejer means of a martingale f are defined by

σαnf(x1, x2

):=

1

n

n∑k=0

Sαk,kf(x1, x2).

The two-dimensional Dirichlet kernels and Marcinkiewicz-Fejer kernels are de-fined by

Dαk,l(x

1, x2) := Dαk (x1)Dα

l (x2), Kαn (x1, x2) :=

1

n

n∑k=0

Dαk,k(x

1, x2).

During the proof of our main theorem we will use the following bellow estimationof Marcinkiewicz-Fejer kernels on the special indices, which was proved in [7]:

Lemma 2.1 (Goginava [7]) : Let

x1 ∈ I4A

(0, ..., 0, x1

4m = 1, 0, ..., 0, x14l = 1, x1

4l+1, ..., x14A−1

)and

x2 ∈ I4A

(0, ..., 0, x2

4l = 1, x14l+1..., x

14q−1, 1− x1

4q, x24q+1, ..., x

24A−1

).

Then

nA−1

∣∣KnA−1

(x1, x2

)∣∣ ≥ 24q+4l+4m−3,

Vol. 18, No. 1, 2014 115

where

nA = 24A + 24A−4 + ...+ 24 + 20.

The concept of modulus of continuity in Hp (0 < p ≤ 1) is given by

ωHp (1/2n, f) := ‖f − S2n,2nf‖Hp .

Let the maximal operators σκ,∗ and σκ,# be defined by

σκ,∗f = supn≥1|σκnf | , σκ,#f = sup

n∈N|σκ2nf | .

For the maximal operator σκ,# Gat, Goginava and Nagy [3] proved that thefollowing is true:

Theorem 2.2 (Gat, Goginava and Nagy [3]): The maximal operator σκ,# isbounded from the Hardy space Hp to the space Lp when p > 1/2.

Later, it was shown that the maximal operator σκ,# is not bounded from theHardy space Hp to the space Lp for 0 < p ≤ 1/2. Moreover, the maximal operatorσκ,# is bounded from the Hardy space H1/2 to the space weak-L1/2 (see [6]).

For the martingale

f =

∞∑n=0

(f (n) − f (n−1)

)the conjugate transforms are defined as

f (t) =

∞∑n=0

rn (t)(f (n) − f (n−1)

),

where t ∈ G is fixed. Note that f (0) = f. It is well-known (see [22]) that∥∥∥f (t)∥∥∥Hp

= ‖f‖Hp , ‖f‖pHp ∼∫

[0,1)

∥∥∥f (t)∥∥∥ppdt. (5)

As a consequence, we have that the conjugate transform of the n-th Marcinkieviczmeans of a function f coincides with the n-th Marcinkievicz means of the conjugatetransform of f .

3. Formulation of main results

Our main result reads as follows:

Theorem 3.1 : a) Let

ω

(1

2k, f

)H2/3

= o

(1

k3/2

), as k →∞. (6)

116 Bulletin of TICMI

Then

‖σκnf − f‖H2/3→ 0, when n→∞.

b) There exists a martingale f ∈ H2/3, for which

ω

(1

22k, f

)H2/3

= O

(1

23k/2

), as k →∞

and

‖σκnf − f‖2/3 9 0 as n→∞.

Proof : During the proof we follow the method of the second author in [19] (fordimension 1), but for the completeness we give the details. Moreover, the proofis based on the result of the first author [11] (see inequality (2)) and the methodof Weisz [22] improved for conjugate transforms (see equality (5)). Combining (2)and (5) we have

‖σκnf‖2/3H2/3

=

∫[0,1)

∥∥∥∥ ˜(σκnf)(t)

∥∥∥∥2/3

2/3

dt =

∫[0,1)

∥∥∥σκnf (t)∥∥∥2/3

2/3dt (7)

≤ c log (n+ 1)

∫[0,1)

∥∥∥f (t)∥∥∥2/3

H2/3

dt

= c log (n+ 1)

∫[0,1)‖f‖2/3H2/3

dt

= c log (n+ 1) ‖f‖2/3H2/3.

Let 2N < n ≤ 2N+1.

‖σκnf − f‖2/3H2/3≤∥∥σκnf − σκnS2N ,2Nf

∥∥2/3

H2/3+∥∥σκnS2N ,2Nf − S2N ,2Nf

∥∥2/3

H2/3

+∥∥S2N ,2Nf − f

∥∥2/3

H2/3

The inequality (7) gives immediately

∥∥σκnf − σκnS2N ,2Nf∥∥2/3

H2/3+∥∥S2N ,2Nf − f

∥∥2/3

H2/3≤∥∥σκn (S2N ,2Nf − f

)∥∥2/3

H2/3

+∥∥S2N ,2Nf − f

∥∥2/3

H2/3

≤ c (log (n+ 1) + 1)ω2/3

(1

2N, f

)H2/3

Vol. 18, No. 1, 2014 117

For∥∥σκnS2N ,2Nf − S2N ,2Nf

∥∥2/3

H2/3we will show that

∥∥σκnS2N ,2Nf − S2N ,2Nf∥∥2/3

H2/3≤(

2N

n

)2/3 ∥∥S2N ,2N (σκ2Nf − f)∥∥2/3

H2/3

≤ ‖σκ2Nf − f‖2/3H2/3→ 0, while n→∞. (8)

That is, we get that if

ω

(1

2n, f

)H2/3

= o

(1

n3/2

), as n→∞,

then

‖σκnf − f‖H2/3→ 0, while n→∞.

At last, we have to show inequality (8). We write

σκnS2N ,2Nf − S2N ,2Nf =1

n

2N∑k=0

Sκk,kS2N ,2Nf +1

n

n∑k=2N+1

Sκk,kS2N ,2Nf − S2N ,2Nf

=2N

n

(σκ2Nf − S2N ,2Nf

)=

2N

n

(S2N ,2Nσ

κ2Nf − S2N ,2Nf

)=

2N

nS2N ,2N (σκ2Nf − f) .

Combining (5), Theorem 2.2 and following the steps of estimation (7) we get in-equality (8). It completes the proof of the first part of our theorem.

Now, we prove the second part of Theorem 3.1. We use the martingale con-structed in [12]. We set

ai(x1, x2) = 22i

(D22i+1(x

1)−D22i (x1)) (D22i+1(x

2)−D22i (x2))

and

f (n)(x1, x2) =

n∑i=1

ai(x1, x2)

23i/2.

In the paper [12], it is shown that the martingale f satisfies the conditions of the sec-

ond part of Theorem 3.1. That is, f ∈ H2/3 and ω(

122k, f)H2/3

= O(

123k/2

), as k →

∞.Now, we show that

‖σκnf − f‖2/3 9 0 as n→∞.

118 Bulletin of TICMI

It is easy to calculate the Fourier coefficients

fκ(i, j) =

22k

23k/2 , if (i, j) ∈{

22k , ..., 22k+1 − 1}2, k = 0, 1, ...

0, if (i, j) /∈∞⋃k=0

{22k , ..., 22k+1 − 1

}2.

(9)

Set n2A−2 = 24·2A−2

+ 24·2A−2−4 + ...+ 24 + 20 = 22A + 22A−4 + ...+ 24 + 20 as inLemma 1.

σκn2k−2f − f =

22kσκ22kf

n2k−2

+1

n2k−2

n2k−2∑j=22k+1

Sκj,jf −22kf

n2k−2

− n2k−2−1f

n2k−2

(10)

Let 22k < j ≤ n2k−1 . Using equations (3) and (9) we have

Sκj,jf(x1, x2

)= S22k ,22kf

(x1, x2

)+

j−1∑v=22k

j−1∑s=22k

f(v, s)κv,s(x1, x2

)

= S22k ,22kf(x1, x2

)+

22k

23k/2

j−22k−1∑v=0

j−22k−1∑s=0

κv+22k

(x1)κs+22k

(x2)

= S22k ,22kf(x1, x2

)+

22kr2k(x1 + x2

)23k/2

j−22k−1∑v=0

j−22k−1∑s=0

wv(τ2k(x1))ws(τ2k(x2))

= S22k ,22kf(x1, x2

)+

22kr2k(x1 + x2

)Dwj−22k

(τ2k(x1))Dwj−22k

(τ2k(x2))

23k/2.

Hence, we write the following

1

n2k−2

n2k−2∑j=22k+1

Sκj,jf(x1, x2

)=n2k−2−1S22k ,22kf

(x1, x2

)n2k−2

+22kr2k

(x1 + x2

)n2k−223k/2

n2k−2−1∑j=1

Dwj

(τ2k(x1))Dwj

(τ2k(x2))

=n2k−2−1S22k ,22kf

(x1, x2

)n2k−2

+22kr2k

(x1 + x2

)n2k−2−1K

wn2k−2−1

(τ2k(x1), τ2k

(x2))

n2k−223k/2.

Vol. 18, No. 1, 2014 119

Equality (10) yields

‖σκn2k−2f − f‖2/32/3 ≥

c

2k‖n2k−2−1K

wn2k−2−1

◦ (τ2k × τ2k) ‖2/32/3

−

(22k

n2k−2

)2/3

‖σκ22kf − f‖

2/32/3

−(n2k−2−1

n2k−2

)2/3

‖S22k ,22kf − f‖2/32/3.

For a fixed 2k we give a subset of G2 as the following disjoint union:

G×G ⊇2k−3⋃m=1

2k−2⋃l=m+1

2k−1⋃q=l+1

(Jm,l2k × L

l,q2k

),

where

Jm,l2k :={x1 ∈ G : x1

2k−1 = ... = x12k−4m = 0,

x12k−4m−1 = 1, x1

2k−4m−2 = ... = x12k−4m−4l = 0, x1

2k−4l−1 = 1}

and

Ll,q2k :={x2 ∈ G : x2

2k−1 = ... = x22k−4l = 0,

x22k−4l−1 = 1, x1

2k−4l−2, ..., x12k−4q, x

22k−4q−1 = 1− x1

2k−4q−1

}.

Notice that, for any(x1, x2

)∈ Jm,l2k ×L

l,q2k , by the definition of τ2k and Lemma 1

we have

n2k−2−1

∣∣∣Kwn2k−2−1

(τ2k(x1), τ2k

(x2))∣∣∣ ≥ 24q+4l+4m−3.

This immediately yields

∫G2

(n2k−2−1

|Kwn2k−2−1

(τ2k(x1), τ2k(x

2))|)2/3dµ(x1, x2)

≥ c2k−2−3∑m=1

2k−2−2∑l=m+1

2k−2−1∑q=l+1

∫Jm,l2k×Ll,q

2k

(n2k−2−1|Kwn2k−2−1

(τ2k(x1), τ2k(x

2))|)2/3dµ(x1, x2)

≥ c2k−2−3∑m=1

2k−2−2∑l=m+1

2k−2−1∑q=l+1

µ(Jm,l2k × L

l,q2k

)2(8q+8l+8m)/3

120 Bulletin of TICMI

≥ c2k−2−3∑m=1

2k−2−2∑l=m+1

2k−2−1∑q=l+1

2(8q+8l+8m)/32−4l2−4q

≥ c2k−2−3∑m=1

28m/32k−2−2∑l=m+1

2−4l/32k−2−1∑q=l+1

2−4q/3

≥ c2k−2−3∑m=1

1 ≥ c2k.

By inequality (11) we have

lim supk→∞

‖σκn2k−2f − f‖2/3 ≥ c > 0.

The proof of Theorem 3.1 is complete. �

Acknowledgements.

The research was supported by project TAMOP-4.2.2.A-11/1/KONV-2012-0051and Shota Rustaveli National Science Foundation grant no.13/06 (Geometry offunction spaces, interpolation and embedding theorems).

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[2] G. Gat, On (C, 1) summability of integrable functions with respect to the Walsh-Kaczmarz system,Studia Math., 130, 2 (1998), 135-148

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[4] G. Gat, U. Goginava, K. Nagy, On the Marcinkiewicz-Fejer means of double Fourier series withrespect to the Walsh-Kaczmarz system, Studia Sci. Math. Hungar., 46, 3 (2009), 399-421

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