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volume 7, issue 4, article 149, 2006.

Received 12 June, 2006;

accepted 22 November, 2006.

Communicated by:Zs. Páles

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Journal of Inequalities in Pure and Applied Mathematics

MAXIMAL OPERATORS OF FEJÉR MEANS OF VILENKIN-FOURIER SERIES

ISTVÁN BLAHOTA, GYÖRGY GÁT AND USHANGI GOGINAVA

Institute of Mathematics and Computer Science College of Nyíregyháza

P.O. Box 166, Nyíregyháza H-4400 Hungary

EMail:blahota@nyf.hu EMail:gatgy@nyf.hu

Department of Mechanics and Mathematics Tbilisi State University

Chavchavadze str. 1 Tbilisi 0128, Georgia

EMail:z_goginava@hotmail.com

c

2000Victoria University ISSN (electronic): 1443-5756 276-06

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Maximal Operators of Fejér Means of Vilenkin-Fourier

Series

István Blahota, György Gát and Ushangi Goginava

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J. Ineq. Pure and Appl. Math. 7(4) Art. 149, 2006

Abstract

The main aim of this paper is to prove that the maximal operatorσ^∗:= sup

n

|σ_n| of the Fejér means of the Vilenkin-Fourier series is not bounded from the Hardy spaceH_1/2to the spaceL_1/2.

2000 Mathematics Subject Classification:42C10.

Key words: Vilenkin system, Hardy space, Maximal operator.

The first author is supported by the Békésy Postdoctoral fellowship of the Hungarian Ministry of Education Bö 91/2003, the second author is supported by the Hungarian National Foundation for Scientific Research (OTKA), grant no. M 36511/2001., T 048780 and by the Széchenyi fellowship of the Hungarian Ministry of Education Szö 184/2003.

Let N+ denote the set of positive integers, N := N+ ∪ {0}. Let m :=

(m₀, m₁, . . .) denote a sequence of positive integers not less than 2. Denote byZ_m_k :={0,1, . . . , m_k−1}the additive group of integers modulom_k.

Define the groupGm as the complete direct product of the groupsZmj,with the product of the discrete topologies ofZ_m_j’s.

The direct productµof the measures µ_k({j}) := 1

mk

(j ∈Z_m_k)

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http://jipam.vu.edu.au

is the Haar measure onG_m withµ(G_m) = 1.

If the sequencemis bounded, thenG_m is called a bounded Vilenkin group, else it is called an unbounded one. The elements ofG_m can be represented by sequencesx:= (x₀, x₁, . . . , x_j, . . .) (x_j ∈Z_m_j).It is easy to give a base for the neighborhoods ofG_m :

I₀(x) :=G_m,

I_n(x) := {y ∈G_m|y₀ =x₀, . . . , yn−1 =xn−1} forx∈G_m, n∈N. DefineI_n :=I_n(0)forn∈N⁺.

If we define the so-called generalized number system based on m in the following way:

M₀ := 1, M_k+1 :=m_kM_k(k ∈N), then every n ∈ Ncan be uniquely expressed as n = P∞

j=0n_jM_j,wheren_j ∈ Z_m_j (j ∈ N+) and only a finite number of n_j’s differ from zero. We use the following notations. Let (for n > 0) |n| := max{k ∈ N : n_k 6= 0}(that is, M_|n|≤n < M_|n|+1),n^(k) =P∞

j=kn_jM_j andn_(k) :=n−n^(k).

Denote byLp(Gm)the usual (one dimensional) Lebesgue spaces (k · kp the corresponding norms)(1≤p≤ ∞).

Next, we introduce onG_man orthonormal system which is called the Vilenkin system. At first define the complex valued functionsr_k(x) :G_m →C, the generalized Rademacher functions as

r_k(x) := exp2πıx_k

m_k (ı² =−1, x∈G_m, k ∈N).

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Now define the Vilenkin systemψ := (ψ_n :n∈N)onG_m as:

ψ_n(x) :=

∞

Y

k=0

r_kⁿ^k(x) (n∈N).

Specifically, we call this system the Walsh-Paley one ifm≡2.

The Vilenkin system is orthonormal and complete inL₁(G_m)[9].

Now, we introduce analogues of the usual definitions in Fourier-analysis. If f ∈L₁(G_m)we can establish the following definitions in the usual manner:

(Fourier coefficients) fb(k) :=

Z

Gm

f ψ_kdµ (k ∈N),

(Partial sums) S_nf :=

n−1

X

k=0

fb(k)ψ_k (n ∈N⁺, S₀f := 0),

(Fejér means) σ_nf := 1

n

n−1

X

k=0

S_nf (n ∈N+),

(Dirichlet kernels) D_n :=

n−1

X

k=0

ψ_k (n ∈N+).

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Recall that

(1) DMn(x) =

( M_n, ifx∈I_n, 0, ifx∈G_m\I_n. The norm (or quasinorm) of the spaceL_p(G_m)is defined by

kfk_p :=

Z

Gm

|f(x)|^pµ(x) ¹_p

(0< p <+∞).

The space weak-L_p(G_m)consists of all measurable functionsf for which kfk_weak−L

p(Gm) := sup

λ>0

λµ(|f|> λ)^p¹ <+∞.

The σ-algebra generated by the intervals {I_n(x) : (x)∈G_m} will be de- noted byFn (n∈N).

Denote byf = f⁽ⁿ⁾, n ∈N

a martingale with respect to(F_n, n∈N)(for details see, e. g. [10,14]).

The maximal function of a martingalef is defined by f^∗ = sup

n∈N

f⁽ⁿ⁾ , respectively.

In casef ∈L₁(G_m), the maximal functions are also be given by f^∗(x) = sup

n∈N

1 µ(I_n(x))

Z

In(x)

f(u)µ(u) .

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For0 < p <∞the Hardy martingale spacesH_p(G_m)consist of all martin- gales for which

kfk_H

p :=kf^∗k_p <∞.

Iff ∈L₁(G_m),then it is easy to show that the sequence(S_M_n(f) :n ∈N) is a martingale. If f is a martingale, that is f = (f⁽ⁿ⁾ : n ∈ N), then the Vilenkin-Fourier coefficients must be defined in a slightly different manner:

fb(i) = lim

k→∞

Z

Gm

f^(k)(x)ψ_i(x)µ(x).

The Vilenkin-Fourier coefficients of f ∈ L₁(G_m)are the same as those of the martingale(S_M_n(f) :n ∈N)obtained fromf.

For a martingalef the maximal operators of the Fejér means are defined by σ^∗f(x) = sup

n∈N

|σ_n(f;x)|.

In this one-dimensional case the weak type inequality µ(σ^∗f > λ)≤ c

λkfk₁ (λ >0)

can be found in Zygmund [16] for the trigonometric series, in Schipp [6] for Walsh series and in Pál, Simon [5] for bounded Vilenkin series. Again in one- dimension, Fujji [3] and Simon [8] verified thatσ^∗ is bounded fromH₁ toL₁. Weisz [11, 13] generalized this result and proved the boundedness ofσ^∗ from the martingale Hardy space H_p to the spaceL_p forp > 1/2. Simon [7] gave a counterexample, which shows that this boundedness does not hold for0< p <

1/2. In the endpoint casep = 1/2Weisz [15] proved that σ^∗ is bounded from

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the Hardy space H_1/2 to the space weak-L_1/2. By interpolation it follows that σ^∗ is not bounded fromHpto the space weak-Lp for all0< p <1/2.

Theorem 1. For any bounded Vilenkin system the maximal operatorσ^∗ of the Fejér means is not bounded from the Hardy spaceH_1/2 to the spaceL_1/2.

The Fejér kernel of ordernof the Vilenkin-Fourier series is defined by K_n(x) := 1

n

n−1

X

k=0

D_k(x).

In order to prove the theorem we need the following lemmas.

Lemma 2 ([4]). Suppose thats, t, n∈Nandx∈I_t\I_t+1.Ift≤s ≤ |n|,then (n^(s+1)+M_s)K_n^(s+1)_+M_s(x)−n^(s+1)K_n^(s+1)

=

( M_tM_sψ_n(s+1)(x)_1−r¹

t(x), ifx−x_te_t∈I_s,

0, otherwise.

Lemma 3 ([2]). Let2< A∈N+,k ≤s < Aandn^∗_A :=M_2A+M2A−2+· · ·+ M₂+M₀. Then

n^∗_A−1 K_n^∗

A−1(x)

≥ M_2kM_2s 4 for

x∈I_2A(0, . . . ,0, x_2k 6= 0,0, . . . ,0, x_2s 6= 0, x_2s+1, . . . , x2A−1), k = 0,1, . . . , A−3, s=k+ 2, k+ 3, . . . , A−1.

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Proof of Theorem1. LetA∈N+and

f_A(x) :=D_M_2A+1(x)−D_M_2A(x). In the sequel we are going to prove for the functionf_Athat

kσ^∗f_Ak_1/2 kf_Ak_H

1/2

≥clog²_qA,

whereq = sup{m₀, m₁, . . .}and constantcdepends only onq. This inequality obviously would show the unboundedness ofσ^∗.

It is evident that fb_A(i) =

( 1, ifi=M_2A, . . . , M_2A+1−1, 0, otherwise.

Then we can write

(2) S_i(f_A;x) =











D_i(x)−D_M_2A(x), if i=M_2A+ 1, . . . , M_2A+1−1, fA(x), if i≥M2A+1,

0, otherwise.

Since

f_A^∗ (x) = sup

n∈N

|S_M_n(f_A;x)|=|f_A(x)|,

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from (1) we get kf_Ak_H

1/2 =kf_A^∗k_1/2 =

D_M_2A+1−D_M_2A 1/2

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= Z

I2A\I2A+1

M

1 2

2A+ Z

I2A+1

|M_2A+1−M_2A|¹²

!2

= m2A−1 M_2A+1 M

1 2

2A+(m2A−1)¹² M_2A+1 M

1 2

2A

!2

≤2²m_2AM_2A⁻¹

≤cM_2A⁻¹. Since

Dk+M_2A −DM_2A =ψM_2ADk, k = 1,2, . . . , M2A, from (2) we obtain

σ^∗fA(x) = sup

n∈N

|σn(fA;x)|

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≥ σ_n^∗

A(f_A;x)

= 1 n^∗_A

n^∗_A−1

X

i=0

S_i(f_A;x)

= 1 n^∗_A

n^∗_A−1

X

i=M2A+1

(D_i(x)−D_M_2A(x))

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= 1 n^∗_A

n^∗_A−1−1

X

i=1

(D_i+M_2A(x)−D_M_2A(x))

= n^∗_A−1 n^∗_A

K_n^∗

A−1(x) . Letq := sup{mi :i∈}.For everyl= 1, . . . ,

h1

4log_q√ A

i

−1(Ais supposed to be large enough) letk_lbe the smallest natural numbers, for which

M_2A√ A 1

q^4l ≤M_2k²

l < M_2A√ A 1

q^4l−4 hold.

Denote

I_2A^k,s(x) :=I_2A(0, . . . ,0, x_2k6= 0,0, . . . ,0, x_2s6= 0, x_2s+1, . . . , x_2A−1) and let

x∈I_2A^k^l^,k^l⁺¹(z) Then from Lemma3and (4) we obtain that

σ^∗f_A(x)≥cM_2k²

l

M_2A ≥c

√ A q^4l On the other hand,

q

kσ^∗f_Ak_1/2 ≥c

[¹4log_q√ A] X

l=1

m₂_kl₊₃−1

X

x2kl+3=0

· · ·

m2A−1−1

X

x2A−1=0

√4

A q^2l µ

I_2A^k^l^,k^l⁺¹(x)

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≥c√⁴ A

[¹₄^logq

√ A] X

l=1

m_2k_l₊₃· · ·m_2A−1 q^2lM_2A

=c√⁴ A

[¹4log_q√ A] X

l=1

1 q^2lM_2k_l₊₂

≥c√⁴ A

[¹₄^logq

√ A] X

l=1

1 q^2lM_2k_l

≥c√⁴ A

[¹4log_q√ A] X

l=1

1 q^2l

q M_2A√

Aq^−4l+4

≥clog_qA

√M_2A.

Combining this with (3) we obtain kσ^∗f_Ak_1/2

kf_Ak_H

1/2

≥ clog²_qA

M_2A M_2A=clog²_qA→ ∞ as A→ ∞.

Thus, the theorem is proved.

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References

[1] G.N. AGAEV, N.Ya. VILENKIN, G.M. DZHAFARLI ANDA.I. RUBIN- SHTEJN, Multiplicative systems of functions and harmonic analysis on zero-dimensional groups, Baku, Ehlm, 1981 (in Russian).

[2] I. BLAHOTA, G. GÁTAND U. GOGINAVA, Maximal operators of Fejér means of double Vilenkin-Fourier series, Colloq. Math., to appear.

[3] N.J. FUJII, Cesàro summability of Walsh-Fourier series, Proc. Amer.

Math. Soc., 77 (1979), 111–116.

[4] G. GÁT, Pointwise convergence of the Fejér means of functions on un- bounded Vilenkin groups, J. of Approximation Theory, 101(1) (1999), 1–

36.

[5] J. PÁL AND P. SIMON, On a generalization of the concept of derivate, Acta Math. Hung., 29 (1977), 155–164.

[6] F. SCHIPP, Certain rearrangements of series in the Walsh series, Mat. Za- metki, 18 (1975), 193–201.

[7] P. SIMON, Cesaro summability with respect to two-parameter Walsh sys- tem, Monatsh. Math., 131 (2000), 321–334.

[8] P. SIMON, Investigations with respect to the Vilenkin system, Annales Univ. Sci. Budapest Eötv., Sect. Math., 28 (1985), 87–101.

[9] N. Ya. VILENKIN, A class of complete orthonormal systems, Izv. Akad.

Nauk. U.S.S.R., Ser. Mat., 11 (1947), 363–400

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[10] F. WEISZ, Martingale Hardy Spaces and their Applications in Fourier Analysis, Springer, Berlin - Heidelberg - New York, 1994.

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

[12] F. WEISZ, Hardy spaces and Cesàro means of two-dimensional Fourier series, Bolyai Soc. Math. Studies, 5 (1996), 353–367.

[13] F. WEISZ, Bounded operators on Weak Hardy spaces and applications, Acta Math. Hungar., 80 (1998), 249–264.

[14] F. WEISZ, Summability of Multi-dimensional Fourier Series and Hardy Space, Kluwer Academic, Dordrecht, 2002.

[15] F. WEISZ,ϑ-summability of Fourier series, Acta Math. Hungar., 103(1-2) (2004), 139–176.

[16] A. ZYGMUND, Trigonometric Series, Vol. 1, Cambridge Univ. Press, 1959.