TWO-DIMENSIONAL SUNOUCHI OPERATOR WITH RESPECT TO VILENKIN-LIKE SYSTEMS

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Sunouchi Operator Chuanzhou Zhang and

Xueying Zhang vol. 9, iss. 4, art. 110, 2008

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TWO-DIMENSIONAL SUNOUCHI OPERATOR WITH RESPECT TO VILENKIN-LIKE SYSTEMS

CHUANZHOU ZHANG AND XUEYING ZHANG

College of Science

Wuhan University of Science and Technology Wuhan , 430065, China

EMail:zczwust@163.com zhxying315@sohu.com

Received: 05 May, 2007

Accepted: 19 October, 2008

Communicated by: S.S. Dragomir 2000 AMS Sub. Class.: 42C10.

Key words: Sunouchi operator, Vilenkin-like systems.

Abstract: In this paper two-dimensional Vilenkin-like systems will be investigated. We prove the Sunouchi operator is bounded fromH^qtoL^qfor(2/3< q≤1). As a consequence, we prove the Sunouchi operator isL^sbounded for1< s <∞ and of weak type(H^\, L¹).

Acknowledgements: Supported by Foundation of Hubei Scientific Committee under grant No.B20081102.

The author thanks the referees for their helpful advice.

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1. Introduction

The operatorU (called the Sunouchi operator) was first introduced and investigated by Sunouchi [1], [2] in Walsh-Fourier analysis.He showed a characterization for the L^pspaces forp >1by means ofU, since this characterization fails to hold forp= 1.

It was of interest to investigate the boundedness ofUon a Hardy space. In [3] Simon showed thatU is a sublinear bounded map from the dyadic Hardy spaceH¹intoL¹. The Vilenkin analogue of the Sunouchi operator was given by Gát [4], [5]. He investigated the boundedness of U from (Vilenkin) H¹ into L¹ and proved that if a Vilenkin group has an unbounded structure and H¹ is defined by means of the usual maximal function, thenU is not bounded. Furthermore, when they considered a modified H¹ space (introduced by Simon [6]), then a necessary and sufficient condition could be given for a Vilenkin group thatU : H¹ → L¹ be bounded. All Vilenkin groups with bounded structure and certain groups without this boundedness property satisfy the condition given by Gát. Thus, in the so-called bounded case, the (H¹, L¹)-boundedness of U remains true also for Vilenkin system. In [7] Simon extended this result, by showing the(H^q, L^q)-boundedness ofU for all0 < q ≤1.

Moreover, the equivalence

kfk_H^q ∼ kU fk_q 1

2 < q ≤1

was also obtained forf with mean value zero.

In this paper we consider a two-dimensional case with respect to generalized Vilenkin-like systems.

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2. Preliminaries and Notations

In this section, we introduce important definitions and notations. Furthermore, we formulate some known results with respect to Vilenkin-like systems, which play a basic role in further investigations. For details, see [8] by Vilenkin and [9] by Schipp, Wade, Simon and Pál.

Let m := (mk, k ∈ N) (N := {0,1, . . . ,}) be a sequence of integers, each of them not less than 2. Denote byZ_m_k them_k-th cyclic group(k ∈N). That is,Z_m_k can be represented by the set {0,1, . . . , m_k −1}, where the group operator is the modm_kaddition and every subset is open. The Harr measure onZ_m_k is given such thatµ({j}) = _m¹

k (j ∈Z_m_k, k ∈N).

LetG_mdenote the complete direct product ofZ_m_k’s equipped with product topology and product measure µ, then G_m forms a compact Abelian group with Haar measure 1. The elements of G_m are sequences of the form (x₀, x₁, . . . , x_k, . . .), wherex_k ∈ Z_m_k for every k ∈ Nand the topology of the groupG_m is completely determined by the sets

I_n(0) :={(x₀, x₁, . . . , x_k, . . .)∈G_m :x_k= 0 (k = 0, . . . , n−1)}

(I₀(0) := G_m).Let I_n(x) := I_n(0) +x (n ∈ N); M₀ := 1and M_k+1 := m_kM_k for k ∈ N, the so-called generalized powers. Then every n ∈ N can be uniquely expressed asn=P∞

k=0nkMk,0≤nk< mk, nk∈N.The sequence(n0, n1, . . .)is called the expansion ofn with respect tom. We often use the following notations:

|n|:= max{k ∈N:n_k 6= 0}(that is,M_|n| ≤n < M_|n|+1)andn^(k) =P∞

j=kn_jM_j. LetGˆ_m := {ψ_n : n ∈ N}denote the character group ofG_m. We enumerate the elements ofGˆ_mas follows. Fork ∈Nandx∈G_mdenote byr_kthek-th generalized Rademacher function:

r_k(x) := exp

2ψıx_k m_k

(x∈G_m, ı :√

−1, k ∈N).

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It is known forx∈G_m, n∈Nthat (2.1)

mn−1

X

i=0

r_nⁱ(x) =

( 0, ifxn 6= 0;

mn, ifxn = 0.

Now we define theψ_nby

ψ_n:=

∞

Y

k=0

rⁿ_k^k (n ∈N).

Gˆ_mis a complete orthonormal system with respect toµ.

G. Gát introduced the so-called Vilenkin-like (or ψα) system. Let functions α_n, α_j^k:G_m → C (n, j, k∈N)satisfy:

i) α^k_j is measurable with respect toΣ_j (i.e.α_j^kdepends only onx₀, x₁, . . . , x_j−1 , j, k ∈N);

ii) |α^k_j|=α^k_j(0) =α^k₀ =α_j⁰ = 1 (j, k ∈N);

iii) αn :=Q∞

j=0αⁿ_j^(j) (n∈N).

Letχ_n :=ψ_nα_n (n ∈ N). The systemχ :={χ_n : n ∈ N}is called a Vilenkin- like(orψα)system (see [10] and [13] for examples).

1. Ifα^k_j = 1for eachk, j ∈N, then we have the "ordinary" Vilenkin systems.

2. Ifmj = 2for allj ∈Nandαⁿ_j^(j) = (βj)ⁿ^j,where βj(x) = exp

2πι

x_j−1

2² +· · ·+ x₀ 2^j+1

(n, j∈N, x∈Gm), then we have the character system of the group of 2-adic integers.

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3. If

χ_n(x) := exp 2πι

∞

X

j=0

n_j Mj+1

∞

X

j=0

x_jM_j

!!

(x∈G_m, n∈N), then we have a Vilenklin-like system which is useful in the approximation of limit periodic almost even arithmetical functions.

In [10] Gát proved that a Vilenkin-like system is orthonormal and complete in L¹(G_m). Define the Fourier coefficients, the Dirichlet kernels, and Fejér kernels with respect to the Vilenkin-like systemχas follows:

fˆ^χ(n) = ˆf(n) :=

Z

Gm

fχ¯n, fˆ^χ(0) :=

Z

Gm

f (f ∈L¹(Gm));

D_n^χ(y, x) = D_n(y, x) :=

n−1

X

k=0

χ_n(y) ¯χ_n(x);

K_n^χ(y, x) = K_n(y, x) := 1 n

n−1

X

k=0

D^χ_n(y, x);

K_h,H^χ (y, x) = K_h,H(y, x) :=

h+H−1

X

j=h

D_j^χ(y, x), where the bar means complex conjugation.

In [10] Gát also proved the following expression of the Dirichlet kernel functions.

(2.2) D^χ_M_n(y, x) = D_M^ψ_n(y−x) =

( M_n, ify−x∈I_n 0, ify−x∈G_m\I_n.

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Moreover,

D_n^χ(y, x) = α_n(y) ¯α_n(x)D_n^ψ(y−x)

=χ_n(y) ¯χ_n(x)





∞

X

j=0

D_M_j(y−x)

mj−1

X

k=mj−nj

r_j^k(y−x)





(n∈P:=N\{0}, y, x ∈Gm), where the systemψis the "ordinary" Vilenkin system.

Ifm˜ = ( ˜m_n, n∈N)is also a generating sequence then we consider the Vilenkin groupG_m_˜ as well. We writeM˜_ninstead ofM_n. LetG:=G_m×G_m_˜ and

χ_{k, l}(x, y) =χ_k(x)χ_l(y) (k, l∈N, x ∈G_m, y ∈G_m_˜) be the two-parameter Vilenkin groups and Vilenkin systems, respectively.

The symbolL^p(0 < p≤ ∞)will denote the usual Lebesgue space of complex- valued functionsf defined onGwith the norm (or quasinorm)

kfk_p :=

Z

G

|f|^p ¹_p

(0< p <∞), kfk∞ :=esssup|f|.

Iff ∈ L¹, thenfˆ(k, l) :=R

Gf χ_k,l (k, L∈N)is the usual Fourier coefficient of f. LetS_n,lf (n, l∈N)be the(n, l)-th rectangular partial sum off:

S_n,lf :=

n−1

X

k=0 l−1

X

j=0

f(k, j)χˆ _k,j.

The so-called (martingale) maximal function off is given by f^∗(x, y) = sup

n, l

M_nM˜_l Z

In(x)

Z

Il(y)

f

(x∈G_m, y ∈G_m_˜).

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Furthermore, letf^\ be the hybrid maximal function off defined by f^\(x, y) := sup

n

M_n Z

In(x)

f(t, y)dt

(x∈G_m, y ∈G_m_˜).

Define the Hardy spaceH^p(G_m×G_m_˜)for0< p <∞as the space of functions f for which

kfkH^p :=kf^∗kp <∞.

ThenkfkH^p is equivalent tokQfkp, whereQf is the quadratic variation off:

Qf :=

∞

X

n=0

∞

X

l=0

|∆_n,lf|²

!¹₂

:=

∞

X

n=0

∞

X

l=0

S_M_n_,M˜lf−S_M_n_,M˜l−1f−S_M_n−1_,M˜lf+S_M_n−1_,M˜l−1f

2!¹₂

S_M

n,M˜−1f :=S_M

−1,M˜_lf :=S_M

−1,M˜−1f := 0 (n, l∈N).

LetH^\be the set of functionsf such that

kfk_H\ :=kf^\k₁ <∞.

In [11] Weisz defined the two-dimensional Sunouchi operator as follows:

U f :=

∞

X

n=0

∞

X

m=0

|S₂ⁿ_,2^mf−S₂ⁿσ₂^mf −σ₂ⁿS₂^mf +σ₂ⁿσ₂^mf|²

!¹₂

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whereσf is the Cesàro means of the Walsh Fourier series off ∈L¹. Now we extend the definition to the two-dimensional Vilenkin-like systems as follows:

U f :=





∞

X

n=0

∞

X

s=0

Mn+1−1

X

j=1

M˜s+1−1

X

k=1

jk M_n+1M˜_s+1

fˆ(j, k)χ_j,k

2



1 2

(f ∈L¹).

If α = (α_n, n ∈ N), β = (β_n, n ∈ N) are bounded sequences of complex numbers, then let

T_α,βf := sup

n,l Mn−1

X

i=0 M˜l−1

X

j=0

α_nβ_kf(n, k)χˆ _n,k be defined at least onL².

Moreover, let α_j := jM_l⁻¹ (l ∈ N, j = M_l, . . . , M_l+1 −1)and β_k := kM˜_t⁻¹ (t∈N, k= ˜M_t, . . . ,M˜_t+1−1)then

U f =





∞

X

n=0

∞

X

s=0

n

X

l=0 s

X

t=0

M_lM˜_t∆_l+1,t+1(T_α,βf)

2



1 2

.

In this paper we assume the sequencesm, m˜ are bounded. In the investigations of some operators defined on Hardy spaces, the concept of aq-atom is very useful.

The function a is called aq-atom if eithera is identically equal to 1 or there exist intervalsI_n(τ)⊂G_m, I_L(γ)⊂G_m_˜ (N, L∈N, τ ∈G_m, γ ∈G_m_˜)such that

i) a(x, y) = 0if(x, y)∈G\(IN(τ)×IL(γ)), ii) kak₂ ≤µ(I_N(τ)×I_L(γ))¹²⁻¹^q,

iii) Z

Gm

a(t, y)dt= Z

Gm˜

a(x, u)du= 0ifx∈G_m, y∈G_m_˜.

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Lemma 2.1 ([1]). Let T be an operator defined at least on L₂ and assume that T is L₂ bounded. If there exists δ > 0 such that for all q-atoms a with support I_N(τ)×I_L(γ)and for allr ∈N, we have

Z

G\I_N−r(τ)×I_L−r(γ)

|T a|^q ≤C_q2^−δr, thenT is bounded fromH_qtoL_qfor all0< q ≤1.

Lemma 2.2. Let ²₃ < q ≤ 1. Then there exist δ > 0and a constantC_q depending only onqsuch that forN, L, r∈N

M¹⁻

q 2

N

∞

X

n=N+1

Z

Gm\IN−r



 Z

IN

Mn+1−1

X

k=Mn

kχ_k(x)χ_k(t) Mn

2

dt





q 2

dx≤C_q2^−δr. Proof. Forn∈N, n≥N, we have

M_nK_M_n(x, t) =

Mn−2

X

i=0

χ_i(x) ¯χ_i(t)

Mn−1

X

k=i+1

1

=

Mn−2

X

i=0

(M_n−i−1)χ_i(x) ¯χ_i(t)

= (M_n−1)D_M_n−1(x, t)−

Mn−1

X

i=0

iχ_i(x) ¯χ_i(t).

This follows

Mn+1−1

X

i=Mn

iχ_i(x) ¯χ_i(t)

M_n =m_n(D_M_n+1(x, t)−K_M_n+1(x, t))

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−(D_M_n(x, t)−K_M_n(x, t))−D_M_n+1(x, t)−D_M_n(x, t)

M_n .

If x ∈ G_m\I_N−r, t ∈ I_N,then there exists u (0 ≤ u ≤ N −r−1)such that x ∈ I_u\I_u+1. Since x−t ∈ I_u\I_u+1, we have D_M_k(x, t) = 0for all (k ≥ u+ 1).

Suppose thats > u. From the definitions of the functionα_nand the Fejér kernel, we have, ifx∈I_u(t)\I_u+1(t),

K_n^(s)_{, M}_s(x, t) =

n^(s)+Ms−1

X

k=n^(s)

u−1

X

j=0

k_jM_j

!

χ_k(x) ¯χ_k(t)

+

n^(s)+Ms−1

X

k=n^(s)

M_u

mu−1

X

p=mu−k_u

r_t^p(x−t)χ_k(x) ¯χ_k(t)

=:

1

X+

2

X,

where

1

X=

ms−1−1

X

ks−1=0

· · ·

mu+1−1

X

ku+1=0

mu−1−1

X

ku−1=0

· · ·

m0−1

X

k0=0 t−1

X

j=0

k_jM_j

!

·

∞

Y

l=u+1

r^k_l^l(x−t)α^k_l^(l)(x) ¯α^k_l^(l)(t)

mu−1

X

ku=0

r_u^k^u(x−t)

=

mu−1

X

ku=0

r_u^k^u(x−t)φ(x, t),

and the functionφdoes not depend onk_t. Consequently,P1

= 0(see [12]).

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Since the sequencemis bounded, we have Z

IN

2

X

2

dt ≤CM_u²

mu−1

X

p=0

Z

IN

Ms−1

X

k, l=0;ku=mu=p

χ_n^(s)_+k(t) ¯χ_n^(s)_+l(t) ¯χ_n^(s)_+k(x)χ_n^(s)_+l(x)dt

≤CM_u² 1

M_NMsMu. Recall thatk^(u+1)6=l^(u+1)implies

Z

IN

χ_n^s_+k(x) ¯χ_n(s)+l(x)dx= 0.

Ifs≤u, then|K_n(s), Ms(x, t)| ≤CM_uM_s.Then

M_N^1−q/2

∞

X

n=N+1

Z

Gm\IN−r



 Z

IN

Mn+1−1

X

k=Mn

kχ_k(x) ¯χ_i(t) M_n

2

dt





q 2

dx

≤M_N^1−q/2

∞

X

n=N+1

Z

Gm\I_N−r

Z

IN

C(|DMn+1(x, t)−KMn+1(x, t)|²

+

|D_M_n(x, t)−K_M_n(x, t)|+

D_M_n+1(x, t)−D_M_n(x, t) M_n

2

dt

!^q₂ dx

=M_N^1−q/2

∞

X

n=N+1

Z

Gm\I_N−r

Z

IN

C(|K_M_n+1(x, t)|²+|K_M_n(x, t)|²)dt ^q₂

dx

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≤C_qM_N^1−q/2

∞

X

n=N+1 N−r−1

X

u=0

1 M_n+1

n+1

X

s=0 ns−1

X

j=0

Z

Iu\I_u+1

Z

IN

|K_n(s+1)+jMs, Ms(x, t)|²dt ^q₂

dx

+C_qM_N^1−q/2

∞

X

n=N+1 N−r−1

X

u=0

1 M_n

n

X

s=0 ns−1

X

j=0

Z

Iu\Iu+1

Z

IN

|K_n(s+1)+jMs, Ms(x, t)|²dt ^q₂

dx

≤C_qM_N^1−q/2

∞

X

n=N+1 N−r−1

X

u=0

1 Mn+1

n+1

X

s=0 ns−1

X

j=0

Z

Iu\Iu+1

M_u³M_s MN

^q₂ dx

+CqM_N^1−q/2

∞

X

n=N+1 N−r−1

X

u=0

1 M_n

n

X

s=0 ns−1

X

j=0

Z

Iu\I_u+1

M_u³M_s M_N

^q₂ dx

≤C_qM_N^1−q/2

∞

X

n=N+1 N−r−1

X

u=0

M_u^3q/2−1M_n^−q/2M_N^−q/2

≤C_qM_N^1−q/2M_N−r−1^3q/2−1M_N^−q =C_q(mN−r· · ·m_N−1)^{−(3q/2−1)} ≤C_q2^−δr (δ = 3q/2−1>0).

Theorem 2.3. Let ²₃ < q ≤1. Then there exists a constantC_q such that kU fk_q ≤C_qkfk_H^q (∀f ∈H^q(G_m×G_m_˜)).

Proof. Letabe aq-atom. It can be assumed that the support ofaisI_N×I_Lfor some N, L∈N, that is

kak₂ ≤(M_NP_L⁰)¹^q⁻¹² and Z

IL

a(x, t)dt = Z

IN

a(u, y)du= 0 for all x∈G_m, y ∈G_m_˜.

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This last property implies that ˆ

a(i, j) = 0ifi= 0, . . . , M_N −1orj = 0, . . . ,M˜_L−1.

Letαandβ as above. Then from the Cauchy inequality we have

Tα,βa(x, y)

≤

∞

X

n=N+1

∞

X

j=L+1

Z

I_N

Z

J_L

|a(t, u)|

M_n+1−1

X

k=M_n

k Mn

χk(x) ¯χk(t)

Mj+1−1

X

l=M_j

l Mj

χl(y) ¯χl(u)|dtdu

≤ kak2

∞

X

n=N+1

∞

X

j=L+1



 Z

I_N

Z

J_L

M_n+1−1

X

k=M_n

k Mn

χk(x) ¯χi(t)

M_j+1−1

X

l=M_j

l Mj

χl(y) ¯χl(u)

2

dtdu





1 2

. (2.3)

First we will show T_α,β is q-quasi local. Let r ∈ N and define the setsX_i (i = 1,2,3,4)as follows:

X₁ := (G_m\IN−r)×I_L, X₂ := (G_m\IN−r)×(G_m_˜\I_L), X₃ :=I_N ×(G_m_˜\IL−r), X₄ := (G_m\I_N)×(G_m_˜\IL−r).

It is clear that

Z

(G\I_N−r×I_L−r)

(T_α,βa)^q ≤

4

X

i=1

Z

Xi

(T_α,βa)^q. To estimate the integral overX₁, we have

Z

X1

(T_α,βa)^q(x, y)dxdy

≤ |I_L|¹⁻^q²

∞

X

n=N+1

Z

Gm\IN−r

Z

IL

Z

In

Mn+1−1

X

k=Mn

k Mn

χ_k(x) ¯χ_k(t)

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× sup Z

IL

a(t, u)

l

X

j=L+1

Mj+1−1

X

l=Mj

l

M_jχ_l(y) ¯χ_l(u)|du|dt





2

dy





q 2

dx

≤ |I_L|^1−q/2

∞

X

n=N+1

Z

Gm\IN−r



 Z

IN

Mn+1−1

X

k=Mn

k

M_nχ_k(x) ¯χ_k(t)

2

dt





q 2

dx

× Z

IN

Z

JL

|a(t, y)|²dydt ^q₂

.

From the definition ofq-atoms and Lemma2.2, we have Z

X1

(Tα,βa)^q(x, y)dxdy

≤ kak^q₂|I_L|¹⁻^q²

∞

X

n=N+1

Z

Gm\IN−r



 Z

IN

Mn+1−1

X

k=Mn

kχ_k(x) ¯χ_k(t) M_n

2

dt





q 2

dx

≤C_qM¹⁻

q 2

N

∞

X

n=N+1

Z

Gm\IN−r



 Z

I_N

Mn+1−1

X

k=Mn

2

dt





q 2

dx

≤Cq2^−δr. (2.4)

In a similar way, we have (2.5)

Z

X3

(T_α,βa)^q(x, y)dxdy≤C_q2^−δr.

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On the setX₂, by inequality (2.3) we have Z

X3

(T_α,βa)^q(x, y)dxdy

≤ kak^q₂

∞

X

n=N+1

∞

X

j=L+1

Z

Gm\I_N−r

Z

Gm˜\I_l



 Z

IN

Z

JL

Mn+1−1

X

k=Mn

kχ_k(x) ¯χ_k(t) Mn

Mj−1

X

l=Mj−1

l Mj

χ_l(y) ¯χ_l(u)

2

dtdu





q 2

dxdy

≤(M_NP_L)¹⁻^q²

∞

X

n=N+1

∞

X

j=L+1

Z

Gm\IN−r

Z

Gm˜\I_l



 Z

IN

Z

JL

Mn+1−1

X

k=Mn

Mj+1−1

X

l=Mj

l

M_jχ_l(y) ¯χ_l(u)

2

dtdu





q 2

dxdy

≤M¹⁻

q 2

N

∞

X

n=N+1

Z

Gm\I_N−r



 Z

IN

Mn+1−1

X

k=Mn

2

dt





q 2

dx

≤C_q2^−δr( ˜M_L)¹⁻^q²

∞

X

j=L+1

Z

Gm˜\JL



 Z

IL

Mj+1−1

X

l=Mj

l

M_jχ_l(y) ¯χ_l(u)|²du)^q²dy





≤C_q2^−δr.

An analogous estimate withX4 instead ofX2 can be obtained using a similar ar-

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gument and these prove that the operatorT_α,βisq-quasi local. By Parseval’s equality, it is clear that the operatorT_α,β isL² bounded. Since

U f =





∞

X

n=0

∞

X

s=0

Mn+1−1

X

j=1

M˜s+1−1

X

k=1

jk M_n+1M˜_s+1

fˆ(j, k)χ_j,k

2



1 2

≤CQ(T_α,βf), where the operatorQis a two-dimensional quadratic variation off.By Lemma2.1, we have

kU fk_q ≤C_qkQ(T_α,βf)k_q ≤C_qkT_α,βfk_H_q ≤C_qkfk_H_q.

Applying known theorems on the interpolation of operators and a duality argu- ment gives the following:

Theorem 2.4. The operatorU isL^s →L^sbounded and of weak type(H^\, L¹), i.e., there exists a constantCsuch that for allδ >0andf ∈H^\we have

µ{(x, y)∈G:|U f(x, y)|> δ} ≤Ckfk_H^\ δ .

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References

[1] G.-I. SUNOUCHI, On the Walsh-Kaczmarz series, Proc. Amer. Math. Soc., 2 (1951), 5–11.

[2] G.-I. SUNOUCHI, Strong summability of Walsh-Fourier series, Tohoku Math.

J., 16 (1969), 228–237.

[3] P. SIMON, (L¹, H)-type estimations for some operators with respect to the Walsh-Paley system, Acta Math. Hungar., 46 (1985), 307–310.

[4] G. GÁT, Investigation of some operators with respect to Vilenkin systems, Acta Math. Hungar., 61 (1993), 131–144.

[5] G. GÁT, On the lower bound of Sunouchi’s operator with respect to Vilenkin system, Analysis Math., 23 (1997), 259–272.

[6] P. SIMON, Investigation with respect to Vilenkin systems, Ann. Univ. Sci. Bu- dapest. Sect. Math., 27 (1982), 87–101.

[7] P. SIMON, A note on the Sunouchi operator with respect to the Vilenkin sys- tem, Ann. Univ. Sci. Budapest. Sect. Math., 43 (2000), 101–116.

[8] N.Ya. VILENKIN, On a class of complete orthonormal systems, Izd. Akad.

Nauk SSSR., 11 (1947), 363–400 (in Russian).

[9] F. SCHIPP, W.R. WADE, P. SIMON,ANDJ.PÁL, Walsh series, An Introduction to Dyadic Harmonic Analysis, Adam Hilger. Bristol-new York ,1990.

[10] G. GÁT, Orthonormal systems on Vilenkin groups, Acta Mathematica Hungar- ica, 58(1-2) (1991), 193–198.

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[11] F. WEISZ, The boundedness of the two-parameter Sunouchi operators on Hardy spaces, Acta Math. Hungar., 72 (1996), 121–152.

[12] G. GÁT, Convergence and Summation With Respect to Vilenkin-like Systems in: Recent Developments in Abstract Harmonic Analysis with Applications in Signal Processing, Nauka, Belgrade and Elektronsik Fakultet, Nis, 1996, 137–

146.

[13] G. GÁT, On(C,1)summability for Vilenkin-like systems, Studia Math., 144(2) (2001), 101–120.