on the boundedness of the maximal operators of double walsh-logarithmic means of marcinkiewicz type

��

DOI: 10.2478/s12175-013-0138-x

Math. Slovaca 63 (2013), No. 4, 839–848

ON THE BOUNDEDNESS

OF THE MAXIMAL OPERATORS

OF DOUBLE WALSH-LOGARITHMIC MEANS

OF MARCINKIEWICZ TYPE

Ushangi Goginava* — Karoly Nagy**

(Communicated by Jan Borsık )

ABSTRACT. The main aim of this paper is to investigate the (Hp, Lp)-typeinequality for the maximal operators of Riesz and Norlund logarithmic means ofthe quadratical partial sums of Walsh-Fourier series. Moreover, we show thatthe behavior of Norlund logarithmic means is worse than the behavior of Rieszlogarithmic means in our special sense.

c©2013Mathematical Institute

Slovak Academy of Sciences

1. Introduction

Let us denote by N the set of natural numbers and by P the set of positiveintegers. Let Z2 be the discrete cyclic group of order 2, the group operation is themodulo 2 addition and every subset is open. The normalized Haar measure on Z2

is given in the way that the measure of a singleton is 1/2. Let G :=∞×k=0

Z2, G is

called the Walsh group. The elements ofG are sequences x = (x0, x1, . . . , xk, . . . )with xk ∈ {0, 1} (k ∈ N).

The group operation on G is the coordinate-wise addition (denoted by +),the normalized Haar measure (denoted by µ) is the product measure and thetopology is the product topology. Dyadic intervalls are defined by

I0(x) := G, In(x) :={y ∈ G : y = (x0, . . . , xn−1, yn, yn+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 In := In(0) for n ∈ N.

2010 Mathemat i c s Sub j e c t C l a s s i f i c a t i on: Primary 42C10.Keywords: Walsh system, Hardy space, logarithmic means, Norlund means, double Walsh-Fourier series.The research of U. Goginava was supported by Shota Rustaveli National Science Foundationgrant no. 31/48 (Operators in some function spaces and their applications in Fourier analysis).

U. GOGINAVA — K. NAGY

Let Lp denote the usual Lebesgue spaces on G (with the corresponding normor quasinorm ‖ · ‖p). The space weak-Lp consists of all measurable functions ffor which

‖f‖weak-Lp:= sup

λ>0λµ (|f | > λ)

1/p< +∞.

The Rademacher functions are defined as

rk(x) := (−1)xk (x ∈ G, k ∈ N).

Let the Walsh-Paley functions be the product functions of the Rademacher func-tions. Namely, each natural number n can be uniquely expressed as

n =∞∑i=0

ni2i, ni ∈ {0, 1} (i ∈ N),

where only a finite number of ni’s different from zero. Let the order of n > 0 bedenoted by |n| := max{j ∈ N : nj �= 0}. Walsh-Paley functions are w0 = 1 andfor n ≥ 1

wn(x) :=

∞∏k=0

(rk(x))nk = r|n|(x)(−1)

|n|−1∑

k=0

nkxk

.

The Dirichlet kernels are defined by

Dn :=

n−1∑k=0

wk,

where D0 := 0. The 2nth Dirichlet kernels have a closed form (see e.g. [6])

D2n(x) =

{0, if x �∈ In2n, if x ∈ In.

(1)

The σ-algebra generated by the dyadic cubes Ik,k(x, y) := Ik(x) × Ik(y) ofmeasure 2−2k will be denoted by Fk (k ∈ N).

Denote by f =(f (n,n), n ∈ N

)a martingale with respect to (Fn, n ∈ N) (for

details see, e.g. [9]). The maximal function of a martingale f is defined by

f∗ = supn∈N

∣∣f (n,n)∣∣.

In the case f ∈ L1

(G2), the maximal function can also be given by

f∗ (x, y) = supn∈N

1

µ (In,n(x, y))

∣∣∣∣∫In,n(x,y)

f (u, v) dµ (u, v)

∣∣∣∣, (x, y) ∈ G2.

For 0 < p < ∞ the Hardy martingale space Hp(G2) consists of all martingales

for which‖f‖Hp

:= ‖f∗‖p < ∞.

840

ON THE BOUNDEDNESS OF THE MAXIMAL OPERATORS

If f ∈ L1

(G2), then it is easy to show that the sequence (S2n,2nf : n ∈ N)

is a martingale. If f is a martingale, that is f = (f (0,0), f (1,1), . . . ) then theWalsh-Fourier coefficients must be defined in a little bit different way [9]:

f (i, j) = limk→∞

∫G2

f (k,k) (x, y)wi(x)wj (y) dµ (x, y) .

The Walsh-Fourier coefficients of f ∈ L1

(G2)are the same as the ones of the

martingale (S2n,2nf : n ∈ N) obtained from f .

The (n,m)th rectangular partial sum of the Walsh-Fourier series are definedas follows:

Sn,mf(x, y) =

n−1∑i=0

m−1∑j=0

f (i, j)wi(x)wj(y).

The Marcinkiewicz-Fejer means and the maximal function of Marcinkiewicz-Fejer means are given by

Mnf(x, y) :=1

n

n∑k=1

Sk,kf(x, y), M∗f := supn∈P

|Mnf |.

The nth Riesz’s logarithmic mean of quadratical partial sums is defined by

Rnf(x, y) :=1

ln

n∑k=1

Sk,kf(x, y)

k,

where ln :=n∑

k=1

1k . Sometimes it is called Riesz’s logarithmic mean of Marcinki-

ewicz type. The nth Norlund logarithmic mean of quadratical partial sums isdefined by

Lnf(x, y) :=1

ln

n−1∑k=0

Sk,kf(x, y)

n− k,

it is a kind of “reverse” Riesz’s logarithmic mean.For martingale f we consider the maximal operators R∗ and L∗, which are

defined byR∗f := sup

n∈P|Rnf |, L∗f := sup

n∈P|Lnf |.

A bounded measurable function a is a p-atom, if there exists a dyadic cubeI2, such that

a)∫I2

a dµ = 0;

b) ‖a‖∞ ≤ µ(I2)−1/p;

c) supp a ⊂ I2.

The basic result of atomic decomposition is the following one.

841


�� (Weisz)� ([9]) A martingale f =(f (n,n) : n ∈ N

)is in Hp (0 <

p ≤ 1) if and only if there exists a sequence (ak, k ∈ N) of p-atoms and a sequence(µk, k ∈ N) of real numbers such that for every n ∈ N,

∞∑k=0

µkS2n,2nak = f (n,n), (2)

∞∑k=0

|µk|p < ∞.

Moreover,

‖f‖Hp∼ inf

( ∞∑k=0

|µk|p)1/p

,

where the infimum is taken over all decompositions of f of the form (2).

2. The main results

For Riesz logarithmic means of Marcinkiewicz type the Abel’s transformationimmediately gives

Rnf =1

ln

n−1∑j=1

Mjf

j + 1+

Mnf

ln.

This implies thatR∗f ≤ cM∗f.

From this inequality, we conclude that the maximal operator R∗ of Riesz log-arithmic means of quadratical partial sums has got so nice properties as themaximal operator of Marcinkiewicz means M∗ has. The results of Weisz [11]yield that the maximal operator R∗ is of weak type (1, 1) and of type (Hp, Lp)for p > 2/3. Moreover, R∗ is of weak type (H2/3, L2/3) which follows from thework of Goginava [4]. In one-dimension the readers are referred to [5,7,8,10].

�� 1�

a) Let p > 2/3. Then the maximal operator R∗ is bounded from the Hardyspace Hp(G

2) to the space Lp(G2).

b) Let 0 < p ≤ 2/3. Then there exists a martingale f ∈ Hp

(G2)such that

‖R∗f‖p = +∞.

�� 1� Let 0 < p ≤ 2/3. Then there exists a martingale f ∈ Hp

(G2)

such that‖M∗f‖p = +∞.

842


On the other hand, we show in Theorem 2 that the behavior of the maximaloperator of Walsh-Norlund logarithmic means of quadratical partial sums isworse than the behavior of the maximal operator of Walsh-Riesz logarithmicmeans of Marcinkiewicz type in our special sense.

We note that the behavior of Walsh-Norlund logarithmic means was discussedin [1–3]. In 2006 the first author, Gat and Tkebuchava [1] proved that the Walsh-Norlund logarithmic means of Marcinkiewicz type does not improve the conver-gence in measure. That is, they proved that for any Orlicz space, which is nota subspace of L lnL(I2), the set of the functions having this means convergentin measure is of first Baire category.

�� 2� Let 0 < p ≤ 1. Then there exists a martingale f ∈ Hp

(G2)such

that‖L∗f‖p = +∞.

P r o o f o f T h e o r e m 1. Let {mk : k ∈ N} be a monotone increasing se-quence of positive integers such that

∞∑k=0

1

mp/3k

< ∞, (3)

k−1∑l=0

24ml/p

3√ml

<24mk/p

3√mk

, (4)

24mk−1/p

3√mk−1

<2mk

mk. (5)

We note that, we could construct such a sequence which satisfies conditions(3)–(5).

Let

f (A,A) (x, y) :=∑

{k:2mk<A}λkak(x, y), where λk :=

43√mk

and

ak (x, y) := 24mk(1/p−1)−2 (D22mk+1 (x)−D22mk (x)) (D22mk+1 (y)−D22mk (y)) .

It is easy to see that the martingale f :=(f (0,0), f (1,1), . . . , f (A,A), . . .

)is in

Hp

(G2)(0 < p ≤ 1). Indeed, since

S2A,2Aak (x, y) =

{0, if A ≤ 2mk,

ak (x, y) , if A > 2mk,

and

f (A,A) (x, y) =∑

{k: 2mk<A}λkak(x, y) =

∞∑k=0

λkS2A,2Aak (x, y)

by (3) and Theorem W we conclude that f ∈ Hp

(G2).

843


Now, we give the Fourier coefficients.

f (i, j) :=

{24mk(1/p−1)

3√mk

, if i, j ∈ {22mk , . . . , 22mk+1 − 1} for some k

0, otherwise.

SetqA,s := 22A + 22s for s < A.

Now, we decompose the qmk,sth Riesz logarithmic means as follows.

Rqmk,sf (x, y) =

1

lqmk,s

22mk−1∑j=1

Sj,jf (x, y)

j+

1

lqmk,s

qmk,s∑j=22mk

Sj,jf (x, y)

j

=: I + II.

(6)

First, we discuss I. Thus, let j < 22mk . Then (4) yields that

|Sj,jf (x, y)| ≤k−1∑l=0

22ml+1−1∑ν=22ml

22ml+1−1∑µ=22ml

|f (ν, µ) |

≤k−1∑l=0

24ml(1/p−1)

3√ml

24ml < 224mk−1/p

3√mk−1

(7)

and

|I| ≤ 1

lqmk,s

22mk−1∑j=1

|Sj,jf (x, y)|j

≤ c

mk

24mk−1/p

3√mk−1

22mk−1∑j=1

1

j< c

24mk−1/p

3√mk−1

. (8)

Now, we discuss II. For 22mk ≤ j ≤ qmk,s we get that

Sj,jf (x, y) =

k−1∑l=0

22ml+1−1∑ν=22ml

22ml+1−1∑µ=22ml

f (ν, µ)wν (x)wµ(y)

+

j−1∑ν=22mk

j−1∑µ=22mk

f (ν, µ)wν (x)wµ(y)

=

k−1∑l=0

22ml+1−1∑ν=22ml

22ml+1−1∑µ=22ml

24ml(1/p−1)

3√ml

wν (x)wµ(y)

+24mk(1/p−1)

3√mk

j−1∑ν=22mk

j−1∑µ=22mk

wν (x)wµ(y)

=

k−1∑l=0

24ml(1/p−1)

3√ml

(D22ml+1(x)−D22ml (x)) (D22ml+1(y)−D22ml (y))

+24mk(1/p−1)

3√mk

(Dj (x)−D22mk (x)) (Dj (y)−D22mk (y)) . (9)

844


This yields that

II =1

lqmk,s

qmk,s∑j=22mk

1

j

k−1∑l=0

24ml(1/p−1)

3√ml

(D22ml+1 (x)−D22ml (x))×

× (D22ml+1 (y)−D22ml (y))

+1

lqmk,s

24mk(1/p−1)

3√mk

qmk,s∑j=22mk

(Dj (x)−D22mk (x)) (Dj (y)−D22mk (y))

j

=: II1 + II2.

To discuss II1, we use (4) and (1). Thus, we have that

|II1| ≤ c

k−1∑l=0

24ml(1/p−1)

3√ml

24ml ≤ c24mk−1/p

3√mk−1

. (10)

By Rqmk,sf(x) = I + II1 + II2, and inequalities (8), (10) we have

|Rqmk,sf(x)| ≥ |II2| − |I| − |II1| ≥ |II2| − c

24mk−1/p

3√mk−1

. (11)

Now, we discuss II2. The nth Dirichlet kernel can be written in the followingform [6]:

Dn(x) = D2|n|(x) + r|n|(x)Dn−2|n|(x). (12)

By the help of this, we immediately get

II2 =24mk(1/p−1)

lqmk,s3√mk

r2mk(x) r2mk

(y)

22s∑j=0

Dj(x)Dj(y)

22mk + j.

Thus, by (11) and (5) we write

|Rqmk,sf(x, y)| ≥ c

24mk(1/p−1)

m4/3k

∣∣∣∣ 22s∑j=0

Dj(x)Dj(y)

22mk + j

∣∣∣∣− c2mk

mk.

Let (x, y) ∈ (I2s\I2s+1)×(I2s\I2s+1), for some s = [mk/2], [mk/2]+1, . . . ,mk.Then we have that∣∣∣∣ 22s∑

j=0

Dj(x)Dj(y)

22mk + j

∣∣∣∣ = 22s∑j=0

j2

22mk + j≥ c

22mk

22s∑j=0

j2 ≥ c26s

22mk

and ∣∣∣Rqmk,sf (x, y)

∣∣∣ ≥ c24mk(1/p−1)

m4/3k

26s

22mk− c

2mk

mk≥ c

22mk(2/p−3)

m4/3k

26s

845


for k big enough. Hence, for 0 < p ≤ 2/3∫G2

|R∗f (x, y) |p dµ (x, y) ≥mk∑

s=[mk/2]

∫(I2s\I2s+1)2

|R∗f (x, y) |p dµ (x, y)

≥mk∑

s=[mk/2]

∫(I2s\I2s+1)2

|Rqmk,sf (x, y) |p dµ (x, y)

≥ c

mk∑s=[mk/2]

∫(I2s\I2s+1)2

(22mk(2/p−3)

m4/3k

26s

)p

dµ (x, y)

≥ c

mk∑s=[mk/2]

22s(3p−2) 22mk(2−3p)

m4p/3k

≥{cm

1/9k , for p = 2/3,

c2mk(2−3p)

m4p/3k

, for p < 2/3.

k → ∞ yields that ‖R∗f‖p = +∞. The proof is complete. �

Now, we prove Theorem 2.

P r o o f o f T h e o r e m 2. To prove Theorem 2, we use the counterexamplemartingale given in the previous proof.

Now, we decompose the qmk,sth Norlund logarithmic means as follows (qmk,s

is the same number as above, that is, qmk,s := 22mk + 22s):

Lqmk,sf (x, y) =

1

lqmk,s

22mk−1∑j=0

Sj,jf (x, y)

qmk,s − j+

1

lqmk,s

qmk,s−1∑j=22mk

Sj,jf (x, y)

qmk,s − j

=: III + IV.

Since, for j < 22mk the inequality |Sj,jf (x, y)| < c24mk−1/p

3√mk−1

holds (see (7)), we

have that

|III| ≤ c

mk

22mk−1∑j=1

24mk−1/p

3√mk−1

1

qmk,s − j≤ c

24mk−1/p

3√mk−1

.

The inequality (9) gives that

IV =1

lqmk,s

qmk,s−1∑j=22mk

1

qmk,s − j

k−1∑l=0

24ml(1/p−1)

3√ml

(D22ml+1 (x)−D22ml (x))×

× (D22ml+1 (y)−D22ml (y))

846


+24mk(1/p−1)

lqmk,s3√mk

qmk,s−1∑j=22mk

(Dj (x)−D22mk (x)) (Dj (y)−D22mk (y))

qmk,s − j

=: IV1 + IV2.

For IV1 we immediately get

|IV1| < c24mk−1/p

3√mk−1

(see condition (4) and equation (1)). That is, we have

|Lqmk,sf(x)| ≥ |IV2| − |III| − |IV1| ≥ |IV2| − c

24mk−1/p

3√mk−1

. (13)

The equation (12) yields

IV2 = c24mk(1/p−1)

m4/3k

r2mk(x) r2mk

(y)

22s−1∑j=0

Dj(x)Dj(y)

22s − j.

Set (x, y) ∈ (I2s\I2s+1)2, then Dj(x) = Dj(y) = j for j < 22s. Consequently,

22s−1∑j=0

Dj(x)Dj(y)

22s − j=

22s−1∑j=0

j2

22s − j=

22s−1∑j=0

(24s

22s − j− (j + 22s)

)≥ cs24s.

By this and inequalities (13), (5)

|IV2| ≥ c24mk(1/p−1)

m4/3k

s24s

and

|Lqmk,sf(x, y)| ≥ c

24mk(1/p−1)

m4/3k

s24s − c2mk

mk≥ c

24mk(1/p−1)

m4/3k

s24s

for (x, y) ∈ (I2s\I2s+1)2, s = [mk/2], . . . ,mk and k big enough. For 0 < p ≤ 1

we have∫G2

|L∗f (x, y) |p dµ (x, y) ≥mk∑

s=[mk/2]

∫(I2s\I2s+1)2

|L∗f (x, y) |p dµ (x, y)

≥mk∑

s=[mk/2]

∫(I2s\I2s+1)2

|Lqmk,sf (x, y) |p dµ (x, y)

≥ c

mk∑s=[mk/2]

∫(I2s\I2s+1)2

(24mk(1/p−1)

m4/3k

s24s

)p

dµ (x, y)

≥ c

mk∑s=[mk/2]

24s(p−1) 24mk(1−p)sp

m4p/3k

847


≥{cm

2/3k , for p = 1,

c 22mk(1−p)

mp/3k

, for 0 < p < 1.

k → ∞ yields that ‖L∗f‖p = +∞. The proof of this theorem is complete. �

REFERENCES

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[2] GAT, G.—GOGINAVA, U.: Uniform and L-convergence of logarithmic means of Walsh-Fourier series, Acta Math. Sin. (Engl. Ser.) 22 (2006), 497–506.

[3] GAT, G.—GOGINAVA, U.: Uniform and L-convergence of logarithmic means of cubicalpartial sums of double Walsh-Fourier series, East J. Approx. 10 (2004), 391–412.

[4] GOGINAVA, U.: The weak type inequality for the maximal operator of the Marcinkiewicz-Fejer means of the two-dimensional Walsh-Fourier series, J. Approx. Theory 154 (2008),161–180.

[5] GOGINAVA, U.: The maximal operator of the Fejer means of the character system of thep-series field in the Kaczmarz rearrangement, Publ. Math. Debrecen 71 (2007), 43–55.

[6] SCHIPP, F.—WADE, W.R.—SIMON, P.—PAL, J.: Walsh Series. An Introduction toDyadic Harmonic Analysis, Adam Hilger, Bristol-New York, 1990.

[7] SIMON, P.—WEISZ, F.: Weak inequalities for Cesaro and Riesz summability of Walsh-Fourier series, J. Approx. Theory. 151 (2008), 1–19.

[8] WEISZ, F.: Cesaro summability of one and two-dimensional Walsh-Fourier series, Anal.Math. 22 (1996), 229–242.

[9] WEISZ, F.: Summability of Multi-dimensional Fourier Series and Hardy Space, KluwerAcademic, Dordrecht, 2002.

[10] WEISZ, F.: θ-summability of Fourier series, Acta Math. Hungar. 103 (2004), 139–176.[11] WEISZ, F.: Convergence of double Walsh-Fourier series and Hardy spaces, Approx.

Theory Appl. (N.S.) 17 (2001), 32–44.

Received 26. 1. 2011Accepted 23. 5. 2011

* Institute of MathematicsFaculty of Exact and Natural SciencesChavchavadze str. 1Tbilisi 0128GEORGIA

E-mail : z [email protected]

** Institute of Mathematicsand Computer SciencesCollege of NyıregyhazaP.O. Box 166H–4400 NyıregyhazaHUNGARY

E-mail : [email protected]

848

on the boundedness of the maximal operators of double walsh-logarithmic means of marcinkiewicz type

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