Bulletin of TICMI
Vol. 18, No. 1, 2014, 55–64
A Note on Vilenkin-Fejér Means on the
Martingale Hardy Spaces Hp
Lars-Erik Persson
a
and George Tephnadze
b ∗
a
Department of Engineering Sciences and Mathematics, Luleå University of Technology,
SE-971 87 Luleå, Sweden and Narvik University College, P.O. Box 385, N-8505, Narvik,
Norway, [email protected]
b
Department of Mathematics, Faculty of Exact and Natural Sciences, Tbilisi State
University, Chavchavadze str. 1, Tbilisi 0128, Georgia, [email protected]
(Received January 14, 2014; Revised April 21, 2014; Accepted May 30, 2014)
The main aim of this note is to derive necessary and sufficient conditions for the convergence
of Fejér means in terms of the modulus of continuity of the Hardy spaces Hp , (0 < p ≤ 1).
Keywords: Vilenkin system, Vilenkin-Fejér means, Martingale Hardy space.
AMS Subject Classification: 42C10.
1.
Introduction and preliminary results
Let P+ denote the set of the positive integers and P := P+ ∪ {0}.
Let m := (m0, m1 , . . . ) denote a sequence of the 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 group Zmj with the
product of the discrete topologies of Zmj ‘s.
The direct product µ of the measures
µk ({j}) := 1/mk , (j ∈ Zmk )
is the Haar measure on Gm with µ (Gm ) = 1.
In this paper we consider bounded Vilenkin groups only, which are
defined by the condition supn mn < ∞.
The elements of Gm are represented by sequences
x := (x0 , x1 , . . . , xk , . . . )
∗ Corresponding
author. Email: [email protected]
ISSN: 1512-0082 print
c 2014 Tbilisi University Press
⃝
( xk ∈ Zmk ) .
Bulletin of TICMI
56
It is easy to give a base for the neighbourhoods of Gm :
I0 (x) := Gm , In (x) := {y ∈ Gm | y0 = x0 , . . . , yn−1 = xn−1 } (x ∈ Gm , n ∈ P).
Denote In := In (0) and In := Gm \ In , for n ∈
(0, . . . , 0, xn = 1, 0, . . . ) ∈ Gm , (n ∈ P) .
The norm (or quasi-norm) of the space Lp (Gm ) is defined by
(∫
∥f ∥p :=
P. Let en
:=
)1/p
p
|f (x)| dµ(x)
(0 < p < ∞) .
Gm
The space weak − Lp (Gm ) consists of all measurable functions f, for which
∥f ∥pweak−Lp := supλp µ (f > λ) < +∞.
λ>0
If we define the so-called generalized number system based on m in the following
way:
M0 := 1,
(k ∈ P),
Mk+1 := mk Mk
∑
then every n ∈ P can be uniquely expressed as n = ∞
j=0 nj Mj , where nj ∈ Zmj
(j ∈ P) and only a finite number of nj ‘s differ from zero. Let |n| := max {j ∈ P;
nj ̸= 0}.
Next, we define the complex valued function rk (x) : Gm → C, called the generalized Rademacher functions in the following way:
(2
)
ı = −1, x ∈ Gm , k ∈ P .
rk (x) := exp (2πıxk /mk )
Moreover, the Vilenkin system ψ := (ψn : n ∈ P) on Gm is defined as follows:
ψn :=
∞
∏
(n ∈ P) .
rknk (x)
k=0
In particular, we call this system the Walsh-Paley one when m ≡ 2. It is known
that the Vilenkin system is orthonormal and complete in L2 (Gm ) (see e.g. [1, 15]).
Hence we can introduce analogues of the usual definitions in Fourier-analysis. If
f ∈ L1 (Gm ) we can define the Fourier coefficients, the partial sums of the Fourier
series, the Fejér means, the Dirichlet and Fejér kernels with respect to the Vilenkin
system in the usual manner:
fb(n) : =
∫
f ψ n dµ,
Sn f :=
Gm
Dn : =
n−1
∑
k=0
n−1
∑
fb(k) ψk ,
k=0
1∑
Dk
n
k=1
n
ψk ,
Kn :=
k=1
1∑
Sk f,
n
n
σn f :=
(n ∈ P+ ) .
Vol. 18, No. 1, 2014
57
Recall that
{
DMn (x) =
Mn , if x ∈ In ,
0, if x ∈
/ In .
(1)
The σ-algebra generated( by the intervals
{In (x) : x ∈ Gm } will be denoted by zn
)
(n ∈ P) . Denote by f = f (n) , n ∈ P the martingale with respect to zn (n ∈ P)
(for details see e.g. [16]). The maximal function of the martingale f is defined by
f ∗ (x) = sup f (n) (x) .
n∈P
In the case f ∈ L1 (Gm ), the maximal functions can also be given by
∫
1
f (x) = sup
f (u) dµ (u)
|I
(x)|
n∈P n
In (x)
∗
For 0 < p < ∞ the Hardy martingale spaces Hp (Gm ) consist of all martingales
for which
∥f ∥Hp := ∥f ∗ ∥p < ∞.
(2)
If f ∈ L1 (Gm ), then (it is easy to
) show that the sequence (SMn (f ) : n ∈ P)
is a martingale. If f = f (n) , n ∈ P is a martingale, then the Vilenkin-Fourier
coefficients must be defined in a slightly different manner:
fb(i) := lim
∫
k→∞ Gm
f (k) (x) ψ i (x) dµ (x) .
The Vilenkin-Fourier coefficients of f ∈ L1 (Gm ) are the same as those of the
martingale (SMn (f ) : n ∈ P) obtained from f .
For the martingale f we consider the following maximal operators:
σ ∗ f := sup |σn f | ,
σ # f := sup |σMn f | ,
n∈P
n∈P
σ
ep∗ := sup
n∈P+
|σn |
2[1/2+p]
1/p−2
n
log
(n
+ 1)
,
where 0 < p ≤ 1/2 and [1/2 + p] denotes the integer part of 1/2 + p.
A weak type-(1, 1) inequality for the maximal operator of Fejér means σ ∗ can
be found in Schipp [8] for Walsh series and in Pl, Simon [7] for bounded Vilenkin
series. Fujji [3] and Simon [10] verified that σ ∗ is bounded from H1 to L1 .
Weisz [17] generalized this result and proved the following:
Theorem W1 (Weisz): The maximal operator σ ∗ is bounded from the martingale
space Hp to the space Lp for p > 1/2.
Simon [9] gave a counterexample, which shows that boundedness does not hold
for 0 < p < 1/2. The counterexample for p = 1/2 is due to Goginava [4], (see also
[2]). Weisz [18] proved that σ ∗ is bounded from the Hardy space H1/2 to the space
Lweak−1/2 .
Bulletin of TICMI
58
In [12] and [13] it was proved that the maximal operators σ
ep∗ with respect to
Vilenkin systems, where 0 < p ≤ 1/2 and [1/2 + p] denotes the integer part of
1/2+p, is bounded from the Hardy space Hp to the space Lp . Moreover, we showed
that the order of deviant behaviour of the n-th Fejér means was given exactly. As
a corollary it was pointed out that
∥σn f ∥p ≤ cp n1/p−2 log2[1/2+p] n ∥f ∥Hp ,
(n = 2, 3, ....) .
(3)
Weisz [19] also proved that the following is true:
Theorem W2 (Weisz): The maximal operator σ # f is bounded from the martingale Hardy space Hp (Gm ) to the space Lp (Gm ) for p > 0.
Moreover, he also considered the norm convergence of Fejér means of VilenkinFourier series and proved the following:
Theorem W3 (Weisz): Let k ∈ P. Then
∥σk f ∥Hp ≤ cp ∥f ∥Hp ,
( f ∈ Hp ,
p > 1/2)
∥σMk f ∥Hp ≤ cp ∥f ∥Hp ,
( f ∈ Hp ,
p > 1/2) .
and
For the Walsh system Goginava [6] proved a very unexpected fact:
Theorem G1 (Goginava): Let 0 < p ≤ 1. Then there exists a martingale f ∈ Hp ,
such that
sup ∥|σMk f |∥Hp = +∞,
(0 < p ≤ 1/2) .
n∈P
In [11] (see also [5]) it was proved that there exists a martingale f ∈ Hp , such
that
sup ∥σn f ∥Hp = +∞,
(0 < p ≤ 1/2) .
n∈P
In [14] it was proved that the following statements are true:
Theorem T2 (Tephnadze): a) Let 0 < p ≤ 1/2, f ∈ Hp , MN < n ≤ MN +1 and
(
)
1/p−2 2[1/2+p]
ωHp (1/MN , f ) = o 1/MN
N
, as N → ∞.
(4)
Then
∥σn f − f ∥p → 0, when n → ∞.
b) Let 0 < p < 1/2 and MN < n ≤ MN +1 . Then there exists a martingale
Vol. 18, No. 1, 2014
59
f ∈ Hp (Gm ), for which
(
)
1/p−2
ωHp (1/MN , f ) = O 1/MN
, as N → ∞
(5)
and
∥σn f − f ∥Lp,∞ 9 0, as n → ∞.
c) Let MN < n ≤ MN +1 . Then there exists a martingale f ∈ H1/2 (Gm ), for which
(
)
ωH1/2 (1/MN , f ) = O 1/N 2 , as N → ∞
(6)
and
∥σn f − f ∥1/2 9 0, as n → ∞.
In this paper we will show that Theorem W3 of Weisz are simple corollary of
Theorems W1 and W2. It is very important, because we do not have definition
of conjugate transform of martingales, with the same properties as Walsh series.
Moreover we will improve inequality (3) and show that
∥σn f ∥Hp ≤ cp n1/p−2 log2[1/2+p] n ∥f ∥Hp ,
(n = 2, 3, ....) .
On the other hand, it gives chance to generalize Theorem T2 and derive necessary
and sufficient conditions for the convergence of Fejér means in terms of the modulus
of continuity of the Hardy spaces Hp , (0 < p ≤ 1). We will also generalize Theorem
G1 for the bounded Vilenkin system.
2.
The main result
Theorem 2.1 : a) Let f ∈ Hp , where 1/2 < p ≤ 1. Then
∥σn f ∥Hp ≤ cp ∥f ∥Hp , (n ∈ P) .
b) Let f ∈ Hp , where 0 < p ≤ 1/2. Then
∥σn f ∥Hp ≤ cp n1/p−2 log2[1/2+p] n ∥f ∥Hp ,
(n ∈ P) .
c) Let f ∈ Hp , where p > 0. Then
∥σMn f ∥Hp ≤ cp ∥f ∥Hp ,
(n ∈ P) .
d) Let p > 1/2 and f ∈ Hp . Then
∥σn f − f ∥Hp → 0, when n → ∞.
Bulletin of TICMI
60
e) Let 0 < p ≤ 1/2, f ∈ Hp , MN < n ≤ MN +1 and
(
)
1/p−2 2[1/2+p]
ωHp (1/MN , f ) = o 1/MN
N
, as N → ∞.
(7)
Then
∥σn f − f ∥Hp → 0, when n → ∞.
Proof : Let f ∈ Hp , p > 1 and MN ≤ n < MN +1 . Then
(
Eσn f := (SMk σn f : k ≥ 0) =
MN
M0
σM0 f, ...,
σMN f, σn f
n
n
)
(8)
and
Mk
(Eσn f ) ≤ sup σMk f + |σn f | ≤ σ # f + |σn f | .
n
0≤k≤N
∗
By combining (2) and (3) we get
∥σn f ∥Hp : = ∥(Eσn f )∗ ∥p ≤ σ # f + ∥σn f ∥p
(9)
p
≤ cp ∥f ∥Hp ,
(1/2 < p ≤ 1)
and
∥σn f ∥Hp
: = ∥(σn f )∗ ∥p ≤ sup |σMk f | + ∥σn f ∥p
k∈+
p
(
)
≤ cp n1/p−2 log2[1/2+p] n ∥f ∥Hp , (0 < p ≤ 1/2) .
(10)
On the other hand, if n = MN , for some n ∈ P, by using (8), we obtain that
Mk
σMk f ≤ sup |σMk f | =: σ # f
(EσMN f ) ≤ sup n
k∈N+
0≤k≤N
∗
and
∥σMN f ∥Hp : = ∥(EσMN f )∗ ∥p ≤ σ # f ≤ cp ∥f ∥Hp ,
p
(11)
(p > 0) .
It is easy to show that (see [14])
σn SMN f − SMN f =
MN
SMN (σMN f − f ) .
n
(12)
Vol. 18, No. 1, 2014
61
Hence, according to (12), we have
∥σn f − f ∥Hp
≤ cp ∥σn f − σn SMN f ∥Hp + cp ∥σn SMN f − SMN f ∥Hp + cp ∥SMN f − f ∥Hp
= cp ∥σn (SMN f − f )∥Hp + cp ∥SMN f − f ∥Hp +
cp MN
∥SMN σMN f − SMN f ∥Hp
n
: = III + IV + V.
For IV we have that
IV = cp ωHp (1/Mn , f ) → 0,
as
n → ∞,
(p > 0) .
Since
∥SMn f ∥Hp ≤ cp ∥f ∥Hp , p > 0
(13)
we obtain
V ≤ ∥SMN (σMN f − f )∥Hp ≤ ∥σMN f − f ∥Hp → 0,
as
n → ∞.
Let 1/2 < p ≤ 1. Then, by using (9) we obtain
III ≤ cp ∥SMN f − f ∥Hp ≤ cp ωHp (1/MN , f ) → ∞,
as
n → ∞.
On the other hand, for 0 < p ≤ 1/2 we can apply (10) and under condition (7)
we get
(
)
III ≤ cp n1/p−2 log2[1/2+p] n ωHp (1/MN , f ) → 0,
as
n → ∞.
The proof is complete.
Theorem 2.2 : Let 0 < p ≤ 1. Then the operator |σMn f | is not bounded from the
martingale Hardy space Hp (Gm ) to the martingale Hardy space Hp (Gm ) .
Proof : Let
fA = DMA+1 − DMA .
It is evident that
fbA (i) =
{
1, if i = MA , ..., MA+1 − 1,
0, otherwise.
Then we can write

 Di − DMA ,
Si fA =
f ,
 A
0,
if i = MA , ..., MA+1 − 1,
if i ≥ MA+1 ,
otherwise.
(14)
Bulletin of TICMI
62
From (1) we get (c.f. [12] and [13])
1−1/p
= sup SMn (fA )
.
= ∥fA ∥p ≤ MA
∥fA ∥Hp
n∈P
(15)
p
Let x ∈ IA+1 . Applying (14), we obtain that
σMA+1 fA (x) =
=
=
1
MA+1
1
MA+1
∑
MA+1
1
MA+1
Sj fA (x) =
j=0
∑ (
)
Dj (x) − DMA (x) =
MA+1
j=MA
∑
MA+1
1
MA+1
1
MA+1
Sj fA (x)
(16)
j=MA +1
∑
MA+1
(j − MA )
j=MA
(mA −1)MA
∑
j ≥ cMA .
j=0
By using (16), we find that
)
(
SMN σMA+1 fA ; x =
∫
σMA+1 fA (t) DMN (x − t) dµ (t)
(17)
Gm
∫
≥
IA+1
σMA+1 fA (x) DMN (x − t) dµ (t)
∫
≥ cMA
DMN (x − t) dµ (t) .
IA+1
According to (17), we have that
)
(
SMN σMA+1 fA ; x ≥ cDMN (x) , N = 0, 1, ..., A,
and
)
)
(
(
sup SMN σMA+1 fA ; x ≥ sup SMN σMA+1 fA ; x ≥ c sup DMN (x) .
N
1≤N <A
1≤N <A
Let x ∈ IN \IN +1 , for some s = 0, 1, ..., A. Then, from (1) it follows that
)
(
sup SMN σMA+1 fA ; x ≥ cMN .
N ∈P
Vol. 18, No. 1, 2014
63
Let 0 < p < 1. Then
σMA+1 fA p
Hp
) p
(
= sup SMN σMA+1 fA ; x 1≤N <A−1
≥
sup
(
A ∫
∑
s=1
SMN
)
) p
(
σMA+1 fA ; x
dµ (x)
1≤N <A−1
Gm
≥
p
(
∫
(18)
sup
IN \IN +1
SMN
)
(
σMA+1 fA ; x
)p
dµ (x)
1≤N <A−1
A
∑
Msp
= cp > 0.
≥c
Ms
s=1
Let p = 1. Then we obtain
σMA+1 fA H1
≥ cA.
(19)
By combining (15), (18) and (19) we can conclude that
σMA+1 fA Hp
∥fA ∥Hp
≥
cp
1−1/p
MA
→ ∞,
as
A → ∞,
0<p<1
and
σMA+1 fA ∥fA ∥H1
H1
≥ cA → ∞,
as
A → ∞.
The proof is complete.
As the consequence of our result we have the following negative result:
Corollary 2.3: Let 0 < p ≤ 1. Then the maximal operator σ # f is not bounded
from the martingale Hardy space Hp (Gm ) to the martingale Hardy space Hp (Gm ) .
Acknowledgment.
The research was supported by Shota Rustaveli National Science Foundation grant
no.52/54 (Bounded operators on the martingale Hardy spaces).
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