Studia Scientiarum Mathematicarum Hungarica
DOI: 10.1556/012.2016.1342
A NOTE ON THE MAXIMAL OPERATORS OF
VILENKIN–NÖRLUND MEANS WITH NON-INCREASING
COEFFICIENTS∗
N. MEMIĆ1 , L. E. PERSSON2,3 and G. TEPHNADZE2,4
1
Department of Mathematics, University of Sarajevo, Zmaja od Bosne 33-35, Sarajevo,
Bosnia and Herzegovina
e-mail: [email protected]
2
Department of Engineering Sciences and Mathematics, Luleå University of Technology,
SE-971 87 Luleå, Sweden
e-mail: [email protected]
3
Narvik University College, P.O. Box 385, N-8505, Narvik, Norway
4
Department of Mathematics, Faculty of Exact and Natural Sciences,
Iv. Javakhishvili Tbilisi State University, Chavchavadze str. 1, Tbilisi 0128, Georgia
e-mail: [email protected]
Communicated by A. Kroó
(Received March 16, 2015; accepted June 19, 2015)
Abstract
In [14] we investigated some Vilenkin–Nörlund means with non-increasing coefficients.
In particular, it was proved that under some special conditions the maximal operators of
such summabily methods are bounded from the Hardy space H1/(1+α) to the space weak L1/(1+α) , (0 < α 5 1). In this paper we construct a martingale in the space H1/(1+α) ,
which satisfies the conditions considered in [14], and so that the maximal operators of these
Vilenkin–Nörlund means with non-increasing coefficients are not bounded from the Hardy
space H1/(1+α) to the space L1/(1+α) . In particular, this shows that the conditions under
which the result in [14] is proved are in a sense sharp. Moreover, as further applications,
some well-known and new results are pointed out.
1. Introduction and statement of the main result
Denote by N+ the set of the positive integers, N := N+ ∪ {0}. Let m :=
(m0, m1 , . . . ) be 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 groups Zmi ,
with the product of the discrete topologies of Zmj .
2010 Mathematics Subject Classification. Primary 42C10, 42B25.
Key words and phrases. Vilenkin system, Vilenkin group, Vilenkin–Fourier series,
Cesáro means, Fejér means, Nörlund means, martingale Hardy space, Lp spaces, weak Lp spaces, maximal operator.
∗
The research was supported by a Swedish Institute scholarship, provided within the
framework of the SI Baltic Sea Region Cooperation/Visby Programme.
c 2016 Akadémiai Kiadó, Budapest
⃝
2
N. MEMIĆ, L. E. PERSSON and G. TEPHNADZE
(
)
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 discuss bounded Vilenkin groups, i.e. the case when
supn mn < ∞.
The elements of Gm are 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 },
where x ∈ Gm , n ∈ N. Denote In := In (0) for n ∈ N+ , and In := Gm \In .
If we define the so-called generalized number system based on m in the
following way:
M0 := 1, Mk+1 := mk Mk (k ∈ N),
∑
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.
Next, we introduce on Gm an orthonormal system which is called the
Vilenkin system. At first, we define the complex-valued function rk (x) :
Gm → C, the generalized Rademacher functions, by
rk (x) := exp (2πixk /mk ) ,
(i2 = −1, x ∈ Gm , k ∈ N).
Now, define the Vilenkin system ψ := (ψn : n ∈ N) on Gm as:
ψn (x) :=
∞
∏
rknk (x),
(n ∈ N).
k=0
Specifically, we call this system the Walsh–Paley system when m ≡ 2.
The Vilenkin system is orthonormal and complete in L2 (Gm ) (see [17]).
The norm (or quasi-norm) of the space Lp (Gm ) and weak -Lp (Gm )
(0 < p < ∞) are respectively defined by
∫
p
∥f ∥p :=
|f |p dµ, ∥f ∥pweak-Lp := sup λp µ (f > λ) < +∞.
λ>0
Gm
If f ∈ L1 (Gm ) we can respectively define the Fourier coefficients, the
partial sums of the Fourier series, the Dirichlet kernels with respect to the
Vilenkin system in the usual manner:
∫
fb(n) :=
f ψ n dµ, (n ∈ N),
Gm
A NOTE ON THE MAXIMAL OPERATORS OF VILENKIN–NÖRLUND MEANS
Sn f :=
n−1
∑
fb(k)ψk ,
Dn :=
k=0
n−1
∑
ψk ,
3
(n ∈ N+ )
k=0
Recall that
{
(1)
DMn (x) =
Mn ,
if x ∈ In ,
0,
if x ∈
/ In .
{
}
The σ-algebra generated by the intervals In (x) : x ∈ Gm will be denoted by zn (n ∈ N). Denote by f = (f (n) , n ∈ N) a martingale with respect
to zn (n ∈ N). (for details see e.g. [18]).
The maximal function of a martingale f is defined by
f ∗ = sup |f (n) |.
n∈N
For 0 < p < ∞ the Hardy martingale spaces Hp (Gm ) consist of all martingales for which
∥f ∥Hp := ∥f ∗ ∥p < ∞.
If f = (f (n) , n ∈ N) is a martingale, then the Vilenkin–Fourier coefficients
must be defined in a slightly different manner:
∫
b
f (i) := lim
f (k) ψ i dµ.
k→∞
Gm
A bounded measurable function a is a p-atom (p > 0), if there exists an
interval I, such that
∫
adµ = 0, ∥a∥∞ 5 µ(I)−1/p , supp(a) ⊂ I.
I
We also need the following auxiliary result (see [19]):
Lemma 1. A martingale f = (f (n) , n ∈ N) is in Hp (0 < p 5 1) if
and only if there exists a sequence (ak , k ∈ N) of p-atoms and a sequence
(µk , k ∈ N) of real numbers, such that, for every n ∈ N,
(2)
∞
∑
k=0
µk SMn ak = f (n) ,
∞
∑
k=0
|µk |p < ∞.
4
N. MEMIĆ, L. E. PERSSON and G. TEPHNADZE
∑
p 1/p
Moreover, ∥f ∥Hp v inf ( ∞
, where the infimum is taken over
k=0 |µk | )
all decompositions of f of the form (2).
Let {qk : k = 0} be a sequence of nonnegative numbers. The n-th Nörlund mean for a Fourier series of f is defined by
1 ∑
qn−k Sk f,
tn f =
k=1
Qn
n
(3)
∑
where Qn := n−1
k=0 qk .
We always assume that q0 > 0 and limn→∞ Qn = ∞. In this case it is
well-known that the summability method generated by {qk : k = 0} is regular if and only if
qn−1
= 0.
lim
n→∞ Qn
Concerning this fact and related basic results, we refer to [6].
The (C, α)-means (Cesáro means) of the Vilenkin–Fourier series are defined by
n
1 ∑
α
Aα−1 Sk f,
σn f = α
k=1 n−k
An
where
Aα0 = 0,
Aαn =
(α + 1) . . . (α + n)
,
n!
α ̸= −1, −2, . . .
When α = 1 the Cesáro means coincide with the Fejér means
1∑
σn f =
Sk f.
k=1
n
n
For the martingale f we consider the following maximal operators:
t∗ f := sup |tn f |, σ ∗ f := sup |σn f |, σ α,∗ f := sup σnα f .
n∈N
n∈N
n∈N
In the one-dimensional case the result with respect to the a.e. convergence of Fejér is due to Pál and Simon [11] (c.f. also [2]) for bounded Vilenkin
series. Weisz [20] proved that the maximal operator of the Fejér means σ ∗
is bounded from the Hardy space H1/2 to the space weak-L1/2 . Simon [12]
gave a counterexample, which shows that boundedness does not hold for
0 < p < 1/2. A counterexample for p = 1/2 was given in [16].
In [4] Goginava investigated the behaviour of Cesáro means of Walsh–
Fourier series in detail. The a.e. convergence of Cesáro means of f ∈ L1 was
proved in [5]. Furthermore, Simon and Weisz [13] showed that the maximal
A NOTE ON THE MAXIMAL OPERATORS OF VILENKIN–NÖRLUND MEANS
5
operator σ α,∗ (0 < α < 1) of the (C, α) means is bounded from the Hardy
space H1/(1+α) to the space weak -L1/(1+α) . Moreover, Goginava [3] gave
a counterexample, which shows that boundedness does not hold for 0 < p 5
1/(1 + α).
Móricz and Siddiqi [7] investigated the approximation properties of some
special Nörlund means of Walsh–Fourier series of Lp functions in norm. In
the two-dimensional case approximation properties of Nörlund was considered by Nagy (see [8]–[10]). In [1] and [15] it was proved strong convergence
theorems for Nörlund means of Vilenkin–Fourier series with monotone coefficients. Moreover, there was also shown boundedness of weighted maximal
operators of such Nörlund means on martingale Hardy spaces. Recently,
in [14] it was proved that the following is true:
Theorem A. a) Let 0 < α 5 1. Then the maximal operator t∗ of
summability method (3) with non-increasing sequence {qk : k = 0}, satisfying the conditions
(4)
|qn − qn+1 |
= O(1),
nα−2
n α q0
= O(1),
Qn
as n → ∞,
is bounded from the Hardy space H1/(1+α) to the space weak-L1/(1+α) .
b) Let 0 < α 5 1, 0 5 p < 1/(1 + α) and {qk : k = 0} be a non-increasing
sequence, satisfying the condition
q0
c
= α,
Qn
n
(5)
(c > 0).
Then there exists a martingale f ∈ Hp , such that
sup ∥tn f ∥weak-Lp = ∞.
n∈N
c) Let {qk : k = 0} be a non-increasing sequence, satisfying the condition
(6)
q0 n α
= ∞,
n→∞ Qn
lim
(0 < α 5 1).
Then there exists an martingale f ∈ H1/(1+α) , such that
sup ∥tn f ∥weak-L1/(1+α) = ∞.
n∈N
In this paper we complement this result by proving sharpness of both
conditions of (4). Our main result reads:
6
N. MEMIĆ, L. E. PERSSON and G. TEPHNADZE
Theorem 1. Let 0 < α 5 1 and {qk : k = 0} be a non-increasing sequence, satisfying the conditions
(7)
lim
n→∞
|qn − qn+1 |
= c,
nα−2
(c > 0),
and
nα q0
= c,
Qn
(8)
(c > 0, n ∈ N).
Then there exists a martingale f ∈ H1/(1+α) , such that
sup ∥tn f ∥1/(1+α) = ∞.
n∈N
The proof can be found in the Section 2 and some applications and final
remark in the Section 3.
2. Proof of Theorem 1
Proof. Under the condition (7), there exists an increasing sequence
{nk : k ∈ N} of positive integers such that
(9)
α
M2n
k +1
> cα > 0,
QM2nk +1
k ∈ N.
Let {αk : k ∈ N} ∈ {nk : k ∈ N} be an increasing sequence of positive
integers such that:
(10)
∞
∑
1/(1+α)
1/αk
< ∞,
k=0
(11)
λ
k−1
∑
Mα1+α
η
η=0
αη
<
Mα1+α
k
αk
and
(12)
32λMα1+α
k−1
αk−1
<
α+1
M[α
k /2]
αk
,
A NOTE ON THE MAXIMAL OPERATORS OF VILENKIN–NÖRLUND MEANS
7
where λ = supn mn and [αk /2] denotes the integer part of αk /2.
We note that such increasing sequence {αk : k ∈ N} which satisfies conditions (10)–(12) can be constructed.
Let the martingale f := (f (n) : n ∈ N) be defined by
∑
f (n) =
(13)
λk θαk ,
{k: αk <n}
where
λ
λk =
αk
(14)
and θαk
)
Mααk (
=
DMαk +1 − DMαk .
λ
Since
{
SMA θk =
θk ,
if αk < A,
0,
if αk = A,
∫
θk dµ = 0,
supp(θk ) = Iαk ,
∥θk ∥∞ 5 Mα1+α
= (supp θk )1+α ,
k
Iαk
if we apply Lemma 1 and (10) we can conclude that f ∈ H1/(1+α) .
Moreover, it is easy to see that
 Mα
{
}
αk

,
if
j
∈
M
,
.
.
.
,
M
−
1
, k = 0, 1, 2 . . . ,

α
α
+1
k
k
 αk
∞
b
∪{
(15)
f (j) =
}

if j ∈
/
Mαk , . . . , Mαk +1 − 1 .

0,
k=1
Let s = 0, . . . , k − 1. We can write that
tMαk +Ms f
=
1
QMαk +Ms
M αk
∑
j=0
qj S j f +
1
QMαk +Ms
Mαk +Ms
∑
qj S j f
j=Mαk +1
:= I + II.
Let Mαs 5 j 5 Mαs +1 , where s = 0, . . . , k − 1. Moreover,
Dj − DMα 5 2j 5 λMαs , (s ∈ N)
s
8
N. MEMIĆ, L. E. PERSSON and G. TEPHNADZE
so that, according to (1) and (15), we have that
|Sj f |
(16)
−1
Mαs−1
j−1
∑
∑+1
b
b
f (v)ψv f (v)ψv +
=
v=0
s−1
∑
5 η=0
+
v=Mαs
Mαη +1 −1
∑
v=Mαη
ψv αη
Mααη
)
Mααs (
Dj − DMαs |
|
αs
∑
s−1 Mααη (
)
=
DMαη +1 − DMαη αη
η=0
+
5λ
)
Mααs (
Dj − DMαs |
|
αs
s−1
∑
Mαα+1
η
αη
η=0
5
+
λMαα+1
s
αs
2λMαα+1
λMαα+1
λMαα+1
k−1
s
s
+
5
.
αs
αs
αk−1
Let Mαs−1 +1 + 1 5 j 5 Mαs , where s = 1, . . . , k. Analogously to (16) we
find that
Mα +1 −1
s−1
s−1
∑
∑
b
|Sj f | = f (v)ψv = v=0
Mαη +1 −1
η=0
∑
v=Mαη
∑
s−1 Mααη (
) 2λMαα+1
k−1
=
DMαη +1 − DMαη 5
.
αη
αk−1
η=0
Hence,
(17)
|I| 5
1
QMαk +Ms
Mαk
∑
j=0
qj |Sj f |
ψv αη
Mααη
A NOTE ON THE MAXIMAL OPERATORS OF VILENKIN–NÖRLUND MEANS
5
5
2λMαα+1
k−1
1
αk−1
QMαk +Ms
2λMαα+1
k−1
αk−1
9
Mαk
∑
qj
j=0
.
Let x ∈ Is /Is+1 . Since
(18)
Dj+Mn = DMn + ψMn Dj = DMn + rn Dj ,
when j < Mn ,
if we now apply Abel transformation, (15) and inequalities of (8) and
(9) we get that
Mα
αk
|II| =
QMαk +Ms αk
1
Mαk +Ms
∑
qMαk +Ms −j
j=Mαk +1
(
) Dj − DMαk Ms
Mα ∑
)
(
αk
qMs −j Dj+Mαk − DMαk =
QMαk +Ms αk
1
j=1
Ms
ψM M α ∑
αk
αk
qMs −j Dj =
QMαk +Ms αk
1
j=1
M
s
∑
Mααk
=
q
j
Ms −j αk QMαk +Ms j=1
M
s
(
) 2 c ∑
qMs −j − qMs −j−1 j =
αk j=1
cMs2
=
αk
Ms
∑
qMs −j − qMs −j−1 j=[Ms /2]
[Ms /2]
cMs2 ∑
=
|qj − qj+1 |
αk
j=0
[Ms /2]
cMs2 ∑ α−2
=
j
αk
j=0
10
N. MEMIĆ, L. E. PERSSON and G. TEPHNADZE
=
cMsα+1
cMsα−1 Ms2
=
.
αk
αk
Let [αk /2] < s 5 αk . Therefore, it yields that
∫
tMα
(19)
+Ms f (x)
k
1/(1+α)
dµ(x)
Gm
α+1
cMs1+α 4λMαk−1
cMs1+α
= |II| − |I| =
−
=
.
αk
αk−1
αk
By combining (17) and (19) we get that
∫
|t∗ f |1/(1+α) dµ
Gm
αk
∑
=
∫
tMα
+Ms f
k
1/(1+α)
dµ
s=[αk /2]+1I /I
s s+1
=c
=
αk
∑
Ms
1/(1+α)
s=[αk /2] Ms αk
c
αk
∑
1/(1+α)
αk
s=[αk /2]
α/(1+α)
= cαk
→ ∞,
1=
=c
α∑
k −3
1
1/(1+α)
s=[αk /2] αk
cαk
1/(1+α)
αk
as k → ∞.
The proof is complete.
3. Applications and final remark
Remark 1. We note that under the both conditions of (7) in Theorem 1
the conditions (4) in Theorem A can still be fulfilled. So our main result
shows that under the both conditions of (7) in part a) of Theorem A are in
a sence sharp and the point p = 1/(1 + α) is the smallest number for which
we have boundedness from the Hardy space H1/(1+α) to the space weak L1/(1+α) .
A NOTE ON THE MAXIMAL OPERATORS OF VILENKIN–NÖRLUND MEANS
11
Our main result Theorem 1 immediately implies the following results of
Goginava [3] and Tephnadze [16]:
Corollary 1 (Goginava). The maximal operator of the (C, α)-means
σ α,∗ is not bounded from the Hardy space H1/(1+α) to the space L1/(1+α) ,
where 0 < α 5 1.
Corollary 2 (Tephnadze). The maximal operator of the Fejér means
σ ∗ is not bounded from the Hardy space H1/2 to the space L1/2 .
Let θnα denote the Nörlund mean, where {q0 = 0, qk = k α−1 : k = 1}, that
is
1 ∑
(n − k)α−1 Sk f.
k=1
Qn
n
θnα f =
It is easy to see that
(20)
( α
)
|qn − qn+1 |
n
(n + 1)α
1
−
= α−2
nα−2
n
n
n+1
)
( α
nα
1
1
n
nα
−
= α−2
5 α−2
n
n
n+1
n
n(n + 1)
1
2
5 α−2 2−α = O(1), as n → ∞.
n
n
Since
Qn :=
n−1
∑
n−1
∫
k
α−1
=
k=0
xα−1 dx = cnα
1
we obtain that
(21)
n α q0
= O(1),
Qn
as n → ∞.
By combining inequalities (20) and (21) we get the following new result:
Corollary 3. The maximal operator of the θnα -means
θα,∗ := sup |θn f |
n∈N
is not bounded from the martingale Hardy space H1/(1+α) to the Lebesgue
space L1/(1+α) , where 0 < α 5 1.
12
N. MEMIĆ, L. E. PERSSON and G. TEPHNADZE: A NOTE ON . . .
Acknowledgment. We thank the referee for some valuable remarks, which
have improved the final version of this paper.
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