sup n |σn|of the Fejér means of the Vilenkin-Fourier series is not bounded from the Hardy spaceH1/2to the spaceL1/2

http://jipam.vu.edu.au/

Volume 7, Issue 4, Article 149, 2006

MAXIMAL OPERATORS OF FEJÉR MEANS OF VILENKIN-FOURIER SERIES

ISTVÁN BLAHOTA, GYÖRGY GÁT, AND USHANGI GOGINAVA INSTITUTE OFMATHEMATICS ANDCOMPUTERSCIENCE

COLLEGE OFNYÍREGYHÁZA

P.O. BOX166, NYÍREGYHÁZA

H-4400 HUNGARY

blahota@nyf.hu

INSTITUTE OFMATHEMATICS ANDCOMPUTERSCIENCE

COLLEGE OFNYÍREGYHÁZA

P.O. BOX166, NYÍREGYHÁZA

H-4400 HUNGARY

gatgy@nyf.hu

DEPARTMENT OFMECHANICS ANDMATHEMATICS

TBILISISTATEUNIVERSITY

CHAVCHAVADZE STR. 1 TBILISI0128, GEORGIA

z_goginava@hotmail.com

Received 12 June, 2006; accepted 22 November, 2006 Communicated by Zs. Páles

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

n

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

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

2000 Mathematics Subject Classification. 42C10.

LetN⁺ denote the set of positive integers,N:=N⁺∪ {0}.Letm := (m0, m1, . . .)denote a sequence of positive integers not less than2.Denote byZm_k :={0,1, . . . , mk−1}the additive group of integers modulom_k.

Define the groupG_m as the complete direct product of the groupsZ_mj,with the product of the discrete topologies ofZ_mj’s.

ISSN (electronic): 1443-5756 c

2006 Victoria University. All rights reserved.

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.

276-06

The direct productµof the measures

µ_k({j}) := 1

m_k (j ∈Z_mk) is the Haar measure onG_mwithµ(G_m) = 1.

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

I₀(x) :=G_m,

In(x) := {y∈Gm|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 onmin the following way:

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

j=0njMj, wherenj ∈ Zmj (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 and n_(k) :=n−n^(k).

Denote by L_p(G_m)the usual (one dimensional) Lebesgue spaces (k · k_p 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) := exp2πıxk

m_k (ı² =−1, x∈G_m, k ∈N).

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

ψ_n(x) :=

∞

Y

k=0

rⁿ_k^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. Iff ∈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

f(k)ψb _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+).

Recall that

(1) D_Mn(x) =

( Mn, ifx∈In, 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)^1p <+∞.

Theσ-algebra generated by the intervals{I_n(x) : (x)∈G_m}will be denoted byF_n (n∈N). Denote byf = f⁽ⁿ⁾, n∈N

a martingale with respect to(Fn, 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) .

For0< p <∞the Hardy martingale spacesH_p(G_m)consist of all martingales for which kfk_H

p :=kf^∗k_p <∞.

Iff ∈L1(Gm),then it is easy to show that the sequence(SMn(f) :n ∈N)is a martingale.

Iff is a martingale, that isf = (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_Mn(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 spaceH_pto 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 thatσ^∗ is bounded from the Hardy spaceH_1/2to the space weak-L_1/2. By interpolation it follows that σ^∗ is not bounded fromH_p to the space weak-L_pfor 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_1/2to 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)+Ms)K_n^(s+1)_+Ms(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]). Let 2 < 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_2k6= 0,0, . . . ,0, x_2s6= 0, x_2s+1, . . . , x2A−1), k = 0,1, . . . , A−3, s=k+ 2, k+ 3, . . . , A−1.

Proof of Theorem 1. LetA∈N+and

f_A(x) := D_M2A+1(x)−D_M2A(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,

where q = sup{m₀, m₁, . . .} and constant c depends only on q. 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_M2A(x), if i=M_2A+ 1, . . . , M_2A+1−1, f_A(x), if i≥M_2A+1,

0, otherwise.

Since

f_A^∗(x) = sup

n∈N

|S_Mn(fA;x)|=|f_A(x)|,

from (1) we get

kfAk_H

1/2 =kf_A^∗k_1/2 =

DM_2A+1−DM_2A

1/2

(3)

= Z

I2A\I_2A+1

M

1 2

2A+ Z

I2A+1

|M_2A+1−M_2A|¹²

!2

= m_2A−1 M2A+1

M

1 2

2A+(m_2A−1)¹² M2A+1

M

1 2

2A

!2

≤2²m2AM_2A⁻¹

≤cM_2A⁻¹. Since

D_k+M2A −D_M2A =ψ_M2AD_k, k= 1,2, . . . , M_2A, from (2) we obtain

σ^∗f_A(x) = sup

n∈N

|σ_n(f_A;x)| ≥ σ_n^∗

A(f_A;x) (4)

= 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_M2A(x))

= 1 n^∗_A

n^∗_A−1−1

X

i=1

(D_i+M2A(x)−D_M2A(x))

= n^∗_A−1 n^∗_A

K_n^∗_A−1(x) . Letq := sup{m_i : i ∈}. For everyl = 1, . . . ,h

1

4log_q√ Ai

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

M2A

√ A 1

q^4l ≤M_2k²_l < M2A

√ A 1

q^4l−4 hold.

Denote

I_2A^k,s(x) := I_2A(0, . . . ,0, x_2k 6= 0,0, . . . ,0, x_2s 6= 0, x_2s+1, . . . , x2A−1) and let

x∈I_2A^kl,kl+1(z) Then from Lemma 3 and (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

m2kl+3−1

X

x2kl+3=0

· · ·

m2A−1−1

X

x2A−1=0

√4

A q^2l µ

I_2A^kl,kl+1(x)

≥c√⁴ A

[¹4log_q√ A] X

l=1

m_2kl+3· · ·m2A−1

q^2lM_2A

=c√⁴ A

[¹4log_q√ A] X

l=1

1 q^2lM_2kl+2

≥c√⁴ A

[¹₄^logq

√ A] X

l=1

1 q^2lM_2kl

≥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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