Journal of Inequalities in Pure and
Applied Mathematics
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
I NSTITUTE OF M ATHEMATICS AND C OMPUTER S CIENCE
C OLLEGE OF N YÍREGYHÁZA
P.O. B OX 166, N YÍREGYHÁZA
H-4400 H UNGARY
blahota@nyf.hu
I NSTITUTE OF M ATHEMATICS AND C OMPUTER S CIENCE
C OLLEGE OF N YÍREGYHÁZA
P.O. B OX 166, N YÍREGYHÁZA
H-4400 H UNGARY
gatgy@nyf.hu
D EPARTMENT OF M ECHANICS AND M ATHEMATICS
T BILISI S TATE U NIVERSITY
C HAVCHAVADZE STR . 1
T BILISI 0128, G EORGIA
z_goginava@hotmail.com
Received 12 June, 2006; accepted 22 November, 2006
Communicated by Zs. Páles
A BSTRACT. The main aim of this paper is to prove that the maximal operator σ ∗ := sup |σn | of
n
the Fejér means of the Vilenkin-Fourier series is not bounded from the Hardy space H1/2 to the
space L1/2 .
Key words and phrases: Vilenkin system, Hardy space, Maximal operator.
2000 Mathematics Subject Classification. 42C10.
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 by Zmk := {0, 1, . . . , mk − 1} the additive
group of integers modulo mk .
Define the group Gm as the complete direct product of the groups Zmj , with the product of
the discrete topologies of Zmj ’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
2
I STVÁN B LAHOTA , G YÖRGY G ÁT , AND U SHANGI G OGINAVA
The direct product µ of the measures
µk ({j}) :=
1
mk
(j ∈ Zmk )
is the Haar measure on Gm with µ(Gm ) = 1.
If the sequence m is bounded, then Gm is called a bounded Vilenkin group, else it is called an
unbounded one. The elements of Gm can be represented by sequences x := (x0 , x1 , . . . , xj , . . . )
(xj ∈ Zmj ). It is easy to give a base for the neighborhoods of Gm :
I0 (x) := Gm ,
In (x) := {y ∈ Gm |y0 = x0 , . . . , yn−1 = xn−1 }
for x ∈ Gm , n ∈ N. Define In := In (0) for n ∈ N+ .
If we define the so-called generalized number system based on m in the following way:
Mk+1 := mk Mk (k ∈ N),
P
then every n ∈ N can be uniquely expressed as n = ∞
j=0 nj Mj , where nj ∈ Zmj (j ∈ N+ )
and only a finite number of nj ’s differ from zero. We use the following notations.
Let (for
P∞
(k)
n > 0) |n| := max{k ∈ N : nk 6= 0} (that is, M|n| ≤ n < M|n|+1 ), n = j=k nj Mj and
n(k) := n − n(k) .
Denote by Lp (Gm ) the usual (one dimensional) Lebesgue spaces (k · kp the corresponding
norms) (1 ≤ p ≤ ∞).
Next, we introduce on Gm an orthonormal system which is called the Vilenkin system. At first
define the complex valued functions rk (x) : Gm → C, the generalized Rademacher functions
as
2πıxk
(ı2 = −1, x ∈ Gm , k ∈ N).
rk (x) := exp
mk
Now define the Vilenkin system ψ := (ψn : n ∈ N) on Gm as:
M0 := 1,
ψn (x) :=
∞
Y
rknk (x)
(n ∈ N).
k=0
Specifically, we call this system the Walsh-Paley one if m ≡ 2.
The Vilenkin system is orthonormal and complete in L1 (Gm ) [9].
Now, we introduce analogues of the usual definitions in Fourier-analysis. If f ∈ L1 (Gm ) we
can establish the following definitions in the usual manner:
Z
b
(Fourier coefficients)
f (k) :=
f ψ k dµ
(k ∈ N),
Gm
(Partial sums)
Sn f :=
n−1
X
(n ∈ N+ , S0 f := 0),
fb(k)ψk
k=0
n−1
(Fejér means)
1X
σn f :=
Sn f
n k=0
(Dirichlet kernels)
Dn :=
(n ∈ N+ ),
n−1
X
ψk
(n ∈ N+ ).
k=0
J. Inequal. Pure and Appl. Math., 7(4) Art. 149, 2006
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M AXIMAL O PERATORS OF F EJÉR M EANS ...
3
Recall that
(
(1)
DMn (x) =
Mn , if x ∈ In ,
0,
if x ∈ Gm \In .
The norm (or quasinorm) of the space Lp (Gm ) is defined by
Z
p1
p
kf kp :=
|f (x)| µ (x)
(0 < p < +∞) .
Gm
The space weak-Lp (Gm ) consists of all measurable functions f for which
1
kf kweak−Lp (Gm ) := sup λµ (|f | > λ) p < +∞.
λ>0
The σ-algebra generated by the
intervals {In (x) : (x) ∈ Gm } will be denoted by Fn (n ∈ N) .
(n)
Denote by f = f , n ∈ N a martingale with respect to (Fn , n ∈ N) (for details see, e. g.
[10, 14]).
The maximal function of a martingale f is defined by
f ∗ = sup f (n) ,
n∈N
respectively.
In case f ∈ L1 (Gm ), the maximal functions are also be given by
Z
1
∗
f (u) µ (u) .
f (x) = sup
n∈N µ (In (x))
In (x)
For 0 < p < ∞ the Hardy martingale spaces Hp (Gm ) consist of all martingales for which
kf kHp := kf ∗ kp < ∞.
If f ∈ L1 (Gm ) , then it is easy to show that the sequence (SMn (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:
Z
b
f (i) = lim
f (k) (x) ψ i (x) µ (x) .
k→∞
Gm
The Vilenkin-Fourier coefficients of f ∈ L1 (Gm ) are the same as those of the martingale
(SMn (f ) : n ∈ N) obtained from f .
For a martingale f the maximal operators of the Fejér means are defined by
σ ∗ f (x) = sup |σn (f ; x)|.
n∈N
In this one-dimensional case the weak type inequality
c
µ (σ ∗ f > λ) ≤ kf k1 (λ > 0)
λ
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 that σ ∗ is bounded from H1 to L1 . Weisz [11, 13] generalized this result and proved the
boundedness of σ ∗ from the martingale Hardy space Hp to the space Lp for p > 1/2. Simon [7]
gave a counterexample, which shows that this boundedness does not hold for 0 < p < 1/2. In
the endpoint case p = 1/2 Weisz [15] proved that σ ∗ is bounded from the Hardy space H1/2 to
the space weak-L1/2 . By interpolation it follows that σ ∗ is not bounded from Hp to the space
weak-Lp for all 0 < p < 1/2.
J. Inequal. Pure and Appl. Math., 7(4) Art. 149, 2006
http://jipam.vu.edu.au/
4
I STVÁN B LAHOTA , G YÖRGY G ÁT , AND U SHANGI G OGINAVA
Theorem 1. For any bounded Vilenkin system the maximal operator σ ∗ of the Fejér means is
not bounded from the Hardy space H1/2 to the space L1/2 .
The Fejér kernel of order n of the Vilenkin-Fourier series is defined by
n−1
Kn (x) :=
1X
Dk (x) .
n k=0
In order to prove the theorem we need the following lemmas.
Lemma 2 ([4]). Suppose that s, t, n ∈ N and x ∈ It \It+1 . If t ≤ s ≤ |n|, then
(n(s+1) + Ms )Kn(s+1) +Ms (x) − n(s+1) Kn(s+1)
(
Mt Ms ψn(s+1) (x) 1−r1t (x) ,
=
0,
if x − xt et ∈ Is ,
otherwise.
Lemma 3 ([2]). Let 2 < A ∈ N+ , k ≤ s < A and n∗A := M2A + M2A−2 + · · · + M2 + M0 .
Then
M M
2k
2s
n∗A−1 Kn∗A−1 (x) ≥
4
for
x ∈ I2A (0, . . . , 0, x2k 6= 0, 0, . . . , 0, x2s 6= 0, x2s+1 , . . . , x2A−1 ),
k = 0, 1, . . . , A − 3,
s = k + 2, k + 3, . . . , A − 1.
Proof of Theorem 1. Let A ∈ N+ and
fA (x) := DM2A+1 (x) − DM2A (x) .
In the sequel we are going to prove for the function fA that
kσ ∗ fA k1/2
≥ c log2q A,
kfA kH1/2
where q = sup {m0 , m1 , . . . } and constant c depends only on q. This inequality obviously
would show the unboundedness of σ ∗ .
It is evident that
(
1, if i = M2A , . . . , M2A+1 − 1,
fbA (i) =
0, otherwise.
Then we can write
(2)

Di (x) − DM2A (x) , if i = M2A + 1, . . . , M2A+1 − 1,



fA (x) ,
if i ≥ M2A+1 ,
Si (fA ; x) =



0,
otherwise.
Since
fA∗ (x) = sup |SMn (fA ; x)| = |fA (x)| ,
n∈N
J. Inequal. Pure and Appl. Math., 7(4) Art. 149, 2006
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M AXIMAL O PERATORS OF F EJÉR M EANS ...
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from (1) we get
(3)
kfA kH1/2 = kfA∗ k1/2 = DM2A+1 − DM2A 1/2
!2
Z
Z
1
1
2
=
M2A
+
|M2A+1 − M2A | 2
I2A \I2A+1
I2A+1
1
=
(m2A − 1) 2 12
m2A − 1 12
M2A
M2A +
M2A+1
M2A+1
!2
−1
≤ 22 m2A M2A
−1
≤ cM2A
.
Since
Dk+M2A − DM2A = ψM2A Dk ,
k = 1, 2, . . . , M2A ,
from (2) we obtain
σ ∗ fA (x) = sup |σn (fA ; x)| ≥ σn∗A (fA ; x)
n∈N
∗
nA −1
X
1 = ∗ Si (fA ; x)
nA i=0
∗
nA −1
X
1 = ∗ (Di (x) − DM2A (x))
nA i=M +1
2A
∗
nA−1 −1
X
1 (Di+M2A (x) − DM2A (x))
= ∗ nA i=1
∗
n
∗
= A−1
K
(x) .
∗ nA−1
nA
h
√ i
Let q := sup{mi : i ∈}. For every l = 1, . . . , 14 logq A − 1 (A is supposed to be large
enough) let kl be the smallest natural numbers, for which
(4)
√ 1
√
1
2
M2A A 4l ≤ M2k
<
M
A
2A
l
q
q 4l−4
hold.
Denote
k,s
I2A
(x) := I2A (0, . . . , 0, x2k 6= 0, 0, . . . , 0, x2s 6= 0, x2s+1 , . . . , x2A−1 )
and let
kl ,kl +1
x ∈ I2A
(z)
Then from Lemma 3 and (4) we obtain that
√
2
M2k
A
l
σ fA (x) ≥ c
≥ c 4l
M2A
q
∗
J. Inequal. Pure and Appl. Math., 7(4) Art. 149, 2006
http://jipam.vu.edu.au/
6
I STVÁN B LAHOTA , G YÖRGY G ÁT , AND U SHANGI G OGINAVA
On the other hand,
√
A] m2kl +3 −1
[ 14 log
m2A−1 −1 √
q
q
X
X
X 4 A k ,k +1
l l
∗
kσ fA k1/2 ≥ c
···
µ
I
(x)
2A
q 2l
x
=0
x
=0
l=1
2A−1
2kl +3
√
1
4
A]
[ log
q
X
m2kl +3 · · · m2A−1
≥c A
q 2l M2A
l=1
√
4
√
√
4
=c A
√
4
≥c A
√
4
≥c A
A]
[ 14 log
q
X
l=1
√
logq A]
1
q 2l M
[ 14 X
1
4
l=1
√
logq A]
2kl +2
1
q 2l M
2kl
[ X
l=1
q 2l
q
1
√
M2A Aq −4l+4
logq A
≥ c√
.
M2A
Combining this with (3) we obtain
c log2q A
kσ ∗ fA k1/2
≥
M2A = c log2q A → ∞
kfA kH1/2
M2A
as A → ∞.
Thus, the theorem is proved.
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M AXIMAL O PERATORS OF F EJÉR M EANS ...
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J. Inequal. Pure and Appl. Math., 7(4) Art. 149, 2006
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