On the maximal operator of $(C, α)$-means of Walsh–Kaczmarz–Fourier series
Simon [J. Approxim. Theory, 127, 39–60 (2004)] proved that the maximal operator $σ^{α,κ,*}$ of the $(C, α)$-means of the Walsh–Kaczmarz–Fourier series is bounded from the martingale Hardy space $H_p$ to the space $L_p$ for $p > 1 / (1 + α), \;0 < α ≤ 1$. Recently, Gát and Goginava have...
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Institute of Mathematics, NAS of Ukraine
2010
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508839902707712 |
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| author | Goginava, U. Nagy, К. Гогінава, У. Надь, К. |
| author_facet | Goginava, U. Nagy, К. Гогінава, У. Надь, К. |
| author_sort | Goginava, U. |
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| description | Simon [J. Approxim. Theory, 127, 39–60 (2004)] proved that the maximal operator $σ^{α,κ,*}$ of the $(C, α)$-means of the Walsh–Kaczmarz–Fourier series is bounded from the martingale Hardy space $H_p$ to the space $L_p$ for $p > 1 / (1 + α), \;0 < α ≤ 1$. Recently, Gát and Goginava have proved that this boundedness result does not hold if $p ≤ 1 / (1 + α)$. However, in the endpoint case $p = 1 / (1 + α )$, the maximal operator $σ^{α,κ,*}$ is bounded from the martingale Hardy space $H_{1/(1+α)}$ to the space weak- $L_{1/(1+α)}$. The main aim of this paper is to prove a stronger result, namely, that, for any $0 < p ≤ 1 / (1 + α)$, there exists a martingale $f ∈ H_p$ such that the maximal operator $σ^{α,κ,*} f$ does not belong to the space $L_p$. |
| first_indexed | 2026-03-24T02:31:35Z |
| format | Article |
| fulltext |
UDC 517.9
U. Goginava* (Inst. Math., Tbilisi State Univ., Georgia),
K. Nagy (Inst. Math. and Comput. Sci., Hungary)
ON THE MAXIMAL OPERATOR OF (((( C, αααα ))))-MEANS
OF WALSH – KACZMARZ – FOURIER SERIES
PRO MAKSYMAL|NYJ OPERATOR
(((( C, αααα ))))-SEREDNIX RQDIV UOLÍA – KAÇMAÛA – FUR’{
Simon [J. Approxim. Theory. – 2004. – 127. – P. 39 – 60] proved that the maximal operator σα κ, ,* of
the ( C, α )-means of the Walsh – Kaczmarz – Fourier series is bounded from the martingale Hardy space
H p to the space L p for p > 1 / ( 1 + α ), 0 < α ≤ 1.
Recently, Gát and Goginava proved that this boundedness result does not hold if p ≤ 1 / ( 1 + α ).
However, in the endpoint case p = 1 / ( 1 + α ) the maximal operator σα κ, ,* is bounded from the
martingale Hardy space H1 1/( + )α to the space weak- L1 1/( + )α .
The main aim of this paper is to prove a stronger result, that is for any 0 < p ≤ 1 / ( 1 + α ) there
exists a martingale f ∈ H p such that the maximal operator σα κ, ,* f does not belong to the space L p .
Sajmon doviv [dyv. J. Approxim. Theory. – 2004. – 127. – P. 39 – 60], wo maksymal\nyj operator
σα κ, ,* ( C, α )-serednix rqdiv Uolßa – KaçmaΩa – Fur’[ [ obmeΩenym z martynhal\noho prosto-
ru Xardi H p do prostoru L p dlq p > 1 / ( 1 + α ), 0 < α ≤ 1.
Newodavno Hat i Hohinava dovely, wo cej rezul\tat pro obmeΩenist\ ne vykonu[t\sq, qkwo
p ≤ 1 / ( 1 + α ). Odnak u vypadku kincevo] toçky p = 1 / ( 1 + α ) maksymal\nyj operator σα κ, ,* [
obmeΩenym z martynhal\noho prostoru Xardi H1 1/( + )α do prostoru slabkoho- L1 1/( + )α .
Holovna meta dano] statti — dovesty bil\ß vahomyj rezul\tat, tobto dovesty, wo dlq bud\-
qkoho 0 < p ≤ 1 / ( 1 + α ) isnu[ martynhal f ∈ H p takyj, wo maksymal\nyj operator σα κ, ,* f
ne naleΩyt\ prostoru L p .
1. Introduction. In 1948 Šneider [1] introduced the Walsh – Kaczmarz system and
showed that the inequality
lim sup
logn
nD x
n→∞
( )κ
≥ C > 0
holds a.e. In 1974 Schipp [2] and Young [3] proved that the Walsh – Kaczmarz system
is a convergence system. Skvortsov in 1981 [4] showed that the Fejér means with
respect to the Walsh – Kaczmarz system converge uniformly to f for any continuous
functions f. Gát [5] proved, for any integrable functions, that the Fejér means with
respect to the Walsh – Kaczmarz system converge almost everywhere to the function
and Gát proved that σκ* f C f
H1 1
≤ . Gát’s result was extended to the Hardy
space by Simon [6], who proved that σκ* is of type ( )H Lp p, for p > 1 / 2. Weisz
[7] showed that in endpoint case p = 1 / 2 the maximal operator is of weak type
( )H L1 2 1 2/ /, .
*
The first author is supported by the Georgian National Foundation for Scientific Research (grant no
GNSF/ST07/3-171).
© U. GOGINAVA, K. NAGY, 2010
158 ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 2
ON THE MAXIMAL OPERATOR OF ( C, α )-MEANS … 159
In paper [8] Simon proved the ( )H Lp p, -boundedness of the maximal operator of
( )C, α -means of Walsh – Kaczmarz – Fourier series, where 0 < α ≤ 1 and 1 / ( 1 +
+ α ) < p ≤ 1.
In the paper [9] Gát and Goginava proved that in theorem of Simon the assumption
p > 1 / ( 1 + α ) is essential, namely, this boundedness result does not hold if p ≤
≤ 1 / ( 1 + α ). However, in the endpoint case p = 1 / ( 1 + α ) the maximal operator
σα κ, ,* is bounded from the martingale Hardy space H1 1/( + )α to the space weak-
L1 1/( + )α .
The main aim of this paper is to prove a stronger result, for any 0 < p ≤ 1 / ( 1 + α )
there exists a martingale f H p∈ such that
σα κ, ,* f
p
= + ∞.
2. Dyadic Hardy space and (((( C, αααα ))))-means. Now, we give a brief introduction to
the theory of dyadic analysis [10]. Let denote by Z2 the discrete cyclic group of order
2, the group operation is the modulo 2 addition and every subset is open. The
normalized Haar measure on Z2 is given in the way that the measure of a singleton is
1 / 2. Let G : =
k=
∞×
0
Z2
, G be called the Walsh group. The elements of G are
sequences x = ( … …)x x xk0 1, , , , with xk ∈ { 0, 1 }, k ∈ N.
The group operation on G is the coordinate-wise addition (denoted by +), the
normalized Haar measure (denoted by µ ) and the topology are the product measure
and topology. Dyadic intervals are defined by
I x G0( ) =: , I x y G y x x y yn n n n( ) = ∈ = ( … …){ }− +: : , , , , ,0 1 1
for x ∈ G, n ∈ P. They form a base for the neighborhoods of G. Let 0 = ( 0 : i ∈
∈N ) ∈ G denote the null element of G and I In n: = ( )0 for n ∈ N.
Let Lp denote the usual Lebesgue spaces on G (with the corresponding norm or
quasinorm ⋅ p ).
The Rademacher functions are defined as
r xk
xk( ) = (− ): 1 , x ∈ G, k ∈ N.
Let the Walsh – Paley functions be the product functions of the Rademacher functions.
Namely, each natural number n can be uniquely expressed as
n = ni
i
i
2
0=
∞
∑ , ni ∈ { 0, 1 }, i ∈ N,
where only a finite number of ni’s different from zero. Let the order of n > 0 be
denoted by n j n j: max N := { ∈ ≠ }0 . Walsh – Paley functions are w0 = 1 and for
n ≥ 1
w x r x r xn k
n
k
n
n xk k kk
n
( ) = ( ) = ( )(− )∑( )
=
∞
∏ =
−
:
0
1 0
1
.
The Walsh – Kaczmarz functions are defined by κ0 = 1 and for n ≥ 1
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 2
160 U. GOGINAVA, K. NAGY
κn n n k
n
k
n
n
n
x r x r x r xk( ) = ( ) ( ) = ( )(− )( )− −
=
−
∏: 1
0
1
1 kk n kk
n x − −=
−∑ 10
1
.
The set of Walsh – Kaczmarz functions and the set of Walsh – Paley functions is the
same in dyadic blocks. Namely,
{ ≤ < } = { ≤ < }+ +κn
k k
n
k kn w n: :2 2 2 21 1
for all k ∈ P and κ0 = w0
.
Skvortsov (see [4]) gave a relation between the Walsh – Kaczmarz functions and
the Walsh – Paley functions by the help of the transformation τA : G → G defined by
τA A A A Ax x x x x x x( ) = ( … …)− − +: , , , , , , ,1 2 1 0 1
for A ∈ N. By the definition of τA
, we have
κ τn n n nx r x w xn( ) = ( ) ( )− ( )
2
, n ∈ N, x ∈ G.
The Dirichlet kernels are defined by
Dn k
k
n
ψ ψ: =
=
−
∑
0
1
,
where ψn = wn or κn
, n ∈ P, D0
α : = 0. The 2n th Dirichlet kernels have a closed
form (see, e.g., [10])
D x D x D x
x I
x I
n n n
w n
n
n
2 2 2
0
2
( ) = ( ) = ( ) =
∉
∈
κ , if ,
, if .
If f L G∈ ( )1 , then the number
f̂ n f n
G
ψ ψ( ) = ∫
is said to the n th Walsh – (Kaczmarz) – Fourier coefficient.
Denote by Sn
ψ the n th partial sums of the Walsh – (Kaczmarz) – Fourier series of
a function f, namely
S f x f kn k
k
n
ψ ψ ψ( ) = ( )
=
−
∑; ˆ
0
1
.
The σ-algebra generated by the dyadic intervals of measure 2−k will be denoted
by Fk
, k ∈ N.
Denote by f f nn= ( ∈ )( ), N a martingale with respect to ( ∈ )F nn , N (for details
see, e. g., [11]). The maximal function of a martingale f is defined by
f f
n
n*
N
sup=
∈
( ) .
In case f L G∈ ( )1 , the maximal function can also be given by
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 2
ON THE MAXIMAL OPERATOR OF ( C, α )-MEANS … 161
f x
I x
f u d u
n n I xn
*
N
sup( ) =
( )
( ) ( )
∈ ( )( ) ∫
1
µ
µ , x ∈ G.
For 0 < p < ∞ the Hardy martingale space H Gp( ) consists of all martingales for
which
f f
H pp
: *= < ∞.
If f L G∈ ( )1 , then it is easy to show that the sequence ( ∈ )S f nn2
: N is a
martingale. If f is a martingale, that is f = ( …)f f( ) ( ), ,0 1 then the Walsh –
(Kaczmarz) – Fourier coefficients must be defined in a little bit different way:
ˆ limf i f x x d x
k
k
i
G
( ) = ( ) ( ) ( )
→∞
( )∫ ψ µ ( ψ = w or κ ).
The Walsh – (Kaczmarz) – Fourier coefficients of f L G∈ ( )1 are the same as the
ones of the martingale ( ∈ )S f nn2
: N obtained from f.
Set A
n
nn
α α α
:
!
= ( + )…( + )1
for any n ∈ N, α ∈ R, α ≠ – 1, – 2, … . It is known
that A nn
α α~ . For n = 1, 2, … and a martingale f the ( )C, α -means of the Walsh –
(Kaczmarz) – Fourier series of the function f is given by
σα ψ
α
α ψ
n
n
n j j
j
n
f x
A
A S f x, ;( ) = ( )
−
−
−
=
∑1
1
1
1
( ψ = w or κ ).
For a martingale f we consider the maximal operator
σ σα ψ α ψ, ,*
P
,supf f x
n
n= ( )
∈
( ψ = w or κ ).
The n th ( )C, α -kernel of the Walsh – (Kaczmarz) – Fourier series defined by
K x
A
A D xn
n
n k k
k
n
α ψ
α
α ψ, :( ) = ( )
−
−
−
=
∑1
1
1
1
( ψ = w or κ ).
A bounded measurable function a is a p-atom, if there exists a dyadic interval I,
such that
a) a d
I
µ∫ = 0;
b) a I p
∞
−≤ ( )µ 1/ ;
c) supp a ⊂ I.
The basic result of atomic decomposition is the following one.
Theorem A [11]. A martingale f f nn= ( ∈ )( ) : N is in Hp
, 0 < p ≤ 1, if and
only if there exists a sequence ( ∈ )a kk , N of p-atoms and a sequence ( ∈ )µk k, N
of real numbers such that for every n ∈ N,
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 2
162 U. GOGINAVA, K. NAGY
µk k
k
nS a fn2
0=
∞
( )∑ = ,
(1)
µk
p
k=
∞
∑
0
< ∞.
Moreover,
f
H k
p
k
p
p
∼
=
∞
∑inf
/
µ
0
1
,
where the infimum is taken over all decompositions of f of the form (1).
In the paper [8] Simon proved the following theorem.
Theorem B. Let 0 < α ≤ 1 and 1 / ( 1 + α ) < p ≤ 1. Then there exists a
constant C such that
σα κ, ,* f C f
p H p
≤
for all f H Gp∈ ( ) .
In this paper we prove that in theorem of Simon the assumption p > 1 / ( 1 + α ) is
essential. Moreover, we prove that the following is true.
Theorem 1. Let 0 < α ≤ 1 and 0 < p ≤ 1 / ( 1 + α ). Then there exists a
martingale f H Gp∈ ( ) such that
σα κ, ,* f
p
= + ∞.
3. Proof of main result. Proof. Let ( ∈ )m kk : N be an increasing sequence of
positive integers such that
1
0 mk
p
k=
∞
∑ < ∞, (2)
2 22
0
1 2m p
ll
k m p
k
l k
m m
/ /
=
−
∑ < , (3)
2 22
1
1m p
k
m
k
k k
m m
−
−
≤
/
. (4)
Let
f x aA
k k
k m Ak
( )
<
( ) = ∑:
,
λ
2
, where λk
km
: = 2
and
a x D x D xk
p mk
mk mk
( ) = ( ) − ( )( − ) − ( )+: /22 1 1 1
2 22 1 2 .
The martingale f : = ( … …)( )f f f A( ) ( ), , , ,0 1 is in H Gp( ) . Indeed, since
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 2
ON THE MAXIMAL OPERATOR OF ( C, α )-MEANS … 163
a ak
m p m
k
pk k
∞
( − )− + −= = ( )2 22 1 1 1 2 1 1/ /supp ,
S a x
A m
a x A m
A k
k
k k
2
0 2
2
( ) =
≤
( ) >
, if ,
, if ,
and
f x a x S a xA
k k
k m A
k k
kk
A
( )
< =
∞
( ) = ( ) = ( )∑ ∑λ λ
:2
2
0
by (2) and Theorem A we conclude that f H Gp∈ ( ) .
Now, we investigate the Fourier coefficients.
Let j m mk k∈{ … − }+2 2 12 2 1, , for some k = 0, 1, 2, … . Then it is evident that
ˆ : lim
/
f j f j
mA
A
m p
k
kκ κ
( ) = ( ) =
→∞
( )
( − )� 22 1 1
and f̂ jκ( ) = 0, if j m mk k∉{ … − }+2 2 12 2 1, , , k = 0, 1, 2, … .
Set qA s
A s
, : = +2 22 2 for any A > s. Now, we decompose the qm sk , th Walsh –
Kaczmarz ( )C, α -means as follows
σα κ α κ
q
q
q j j
j
m s
m s
m sk
k
k
f x
A
A S f x
,
,
,
, ( ) = ( )
−
−
−
=
1
1
1
11
2 12mk −
∑ +
+
1
1
1
22A
A S f x
q
q j j
j
q
m s
m sm
m s
k
k
k
k
,
,
,
−
−
−
=
( )∑ α κ = I + II.
Let j < 22mk . Then (3) gives that
S f x fj
l
k m
m
m
l
l
l
κ κ( ) ≤ ( ) ≤
=
−
=
− (+
∑∑ ˆ v
v 2
2 1
0
1 2 1
2
2 1
2 // /p
l
m
l
k m p
km m
l
k− )
=
−
−
∑ <
−1
2
0
1 2
1
2 2
2 1
and
I c
A
A S f x c
q
q j j
jm s
m s
m
k
k
k
≤ ( ) ≤ (
−
−
−
=
−
∑1
1
1
1
2 12
,
,
α κ αα)
−
−
22
1
1m p
k
k
m
/
. (5)
Now, we discuss II.
For 22m
m s
k
k
j q≤ < , we have the following:
S f x f x fj
mk
κ κ κκ κ( ) = ( ) ( ) + ( ) (
=
−− +
∑ ˆ ˆv vv
v
v
0
2 12 11
xx
mk
j
)
=
−
∑
v 2
1
2
=
= ˆ ˆf x f
m
m
l
l
l
k
κ κκ κ( ) ( ) + ( ) (
=
−
=
− +
∑∑ v vv
v
v
2
2 1
0
1
2
2 1
xx
mk
j
)
=
−
∑
v 2
1
2
=
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 2
164 U. GOGINAVA, K. NAGY
=
2 22 1 1
2
2 1
0
1 2
2
2 1 m p
ll
k l
m
m
m
x
l
l ( − )
=
−
=
−
( ) +
+
∑∑
/
κ v
v
mm p
k
jk
mm
x
k
( − )
=
−
( )∑
1 1
2
1
2
/
κ v
v
=
=
2 22 1 1
0
1
2 22 1 2
m p
ll
k l
ml mlm
D x D x
( − )
=
−
∑ ( ) ++ ( ) − ( )
/ 22 1 1
22
m p
k
j
k
mkm
D x D x
( − )
( )( ) − ( )
/
κ .
This gives that
II =
1 2
1
1
2
2 1 1
2A
A
q
q j
j
q m p
m s
m sm
m s l
k
k
k
k
,
,
, /
−
−
−
=
( −
∑α
α
))
=
−
∑ ( )+ ( ) − ( )
m
D x D x
ll
k
ml ml
0
1
2 22 1 2 +
+
22 1 1
1
1
22
m p
q k
q j
j
q
k
m s
m sm
m
A m
A
k
k
k
k( − )
−
−
−
=
/
,
,
,
α
α
ss
mk
D x D xj∑ ( )( ) − ( )κ
22 = : II1 + II2 .
To discuss II1 , we use (3) and D xn
n
2
2( ) ≤ . Thus, we can write
II c
m
c
m p
ll
k
m
ml
l
l
1
2 1 1
0
1
2 1
22
2
2≤ ( ) ≤ ( )
( − )
=
−
+∑α α
/ // /p
ll
k m p
km
c
m
k
=
−
−
∑ < ( )
−
0
1 2
1
2 1
α . (6)
From σα κ
qm sk
f x
,
, ( ) = I + II1 + II2 and (5), (6) we have
σα κ
q
m p
km s
k
k
f x II I II II c
m,
,
/
( ) ≥ − − ≥ −
−
−
2 1 2
2
1
2 1
. (7)
Now, we discuss II2 . We can write the n th Dirichlet kernel with respect to the Walsh
– Kaczmarz system in the following form:
D x D x r x w xn k k k
k
n
n n
n
κ τ( ) = ( ) + ( ) ( )−
=
−
( )∑2 2
2
1
=
= D x r x D xn nn n
w
n2 2
( ) + ( ) ( )
−
( )τ .
By the help of this, we immediately get
II
A m
A
m p
q k
q j
j
k
m s
m s
mk
k
k
2
2 1 1
1
2
12
2=
( − )
−
− −
−
/
,
,
α
α
==
+∑ ( )( ) − ( )
1
2
2 2
2
2 2
s
m mk
D x D x
j k
κ =
=
22 1 1
1
2 2
1
2
m p
q k
m j j
w
j
k
m s
k s
A m
r x A D
k
( − )
−
−
−
=
( )
/
,
α
α
11
2
2
2s
km x∑ ( )( )τ =
=
22 1 1
2 1
1
2 2
2
2
m p
k q
w
m
k s
m s
s km
A
A
K x
k
( − )
−
−
( (
/
,
,
α
α
α τ ))) ≥
≥ c
A
m
K x
m p m
k
w
m
k k
s
s k
( ) (
( − )−
− (α τ
α α
α
22 1 1 2
2 1
2 2
2
2
/
, ))) .
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 2
ON THE MAXIMAL OPERATOR OF ( C, α )-MEANS … 165
Thus, from (7) and (4) we have
σα κ
α α
q
m p m
km s
k k
s
k
f x c
A
m
K
,
,
/
( ) ≥
( − )−
−
22 1 1 2
2 1
2
2
2ss k
k
w
m
m
k
x c
m
α τ, ( )( ) −2
2
.
On the set I s2
A K A l Cs s s
s
w
l
l
s
2 1 2 2
1
0
2 1
2 1
2 2 2
2
2
− −
−
=
−
( += ≥∑α α α α, ))
and
σα κ
α α
q
m p s
km s
k
k
f x C
m
c
,
,
/
( ) ≥ −
( −( + )) ( + )2 2 22 1 1 2 1 mm
k
k
m
.
We decompose the set G as the following disjoint union
G = I JA t
A
t
A
∪ ∪
=
−
0
1
,
where A > t ≥ 1 and J x G x x xt
A
A A t A t: : ,= { ∈ = … = = = }− − − −1 10 1 , J A
0 : =
: = { ∈ = }−x G xA: 1 1 . Notice that, by the definition of τA we have τA t
A
t tJ I I( ) = +\ 1 .
Therefore, we can write
σ µ σ µα κ α κ, ,* , ,*f d f d
p
G
p
Jt
m
t
m
k
k
∫ ∫∑≥
=
−
21
2 1
≥
≥ σ µ σα κ α κ, ,*
/
,
,
f d f
p
Js m
m
q
s
mk
k
m s
k
k
2
22 1
1
∫∑
=[ ]+
−
≥
pp
Js m
m
d
s
mk
k
k
µ
2
22 1
1
∫∑
=[ ]+
−
/
≥
≥ c
A
m
K c
m
m p
k
w
m
mk
s
s k
k2 2
2 1 1
2 1
2 2
2
2
( −( + ))
− −
/
,
α α
α τ�
kk
p
Js m
m
d
s
mk
k
k
∫∑
=[ ]+
−
µ
2
22 1
1
/
≥
≥ c
A
m
K c
m
m p
k
w
m
k
k
s
s
k2 2
2 1 1
2 1
2
2
2
( −( + ))
− −
/
,
α α
α
+
∫∑
=[ ]+
−
p
I Is m
m
d
s sk
k
µ
2 2 1
2 1
1
\/
≥
≥ C
m
c
m
d
m p s
k
m
k
p
I
k k2 2 22 1 1 2 1( −( + )) ( + )
−
/ α α
µ
22 2 1
2 1
1
s sk
k
Is m
m
\/ +
∫∑
=[ ]+
−
and
σ µ µα κ
α α
, ,*
/
f d c
m
d
p
G
m p s
k
p
I
k
∫ ≥
( −( + )) ( + )2 22 1 1 2 1
22 2 1
2 1
1
s sk
k
Is m
m
\/ +
∫∑
=[ ]+
−
≥
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 2
166 U. GOGINAVA, K. NAGY
≥ c
m
s p m p
k
p
s m
m k
k
k 2 22 1 1 2 1 1
2 1
(( + ) − ) ( − ( + ))
=[ ]+
α α
/
−−
∑
1
≥
≥
cm p
c
m
p
k
p
m p
k
p
k
1
1 1
1
1
2
0
1
1
−
( − ( + ))
=
+
< <
+
, ,
, .
α
α
α
That is σα κ, ,* f
p
= + ∞ for 0 < p ≤ 1 / ( 1 + α ). The proof is complete.
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176.
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2004. – 127. – P. 39 – 60.
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Received 21.05.09
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 2
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| id | umjimathkievua-article-2852 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:31:35Z |
| publishDate | 2010 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/e7/1f573d17f1f71e2285aca613dfaa49e7.pdf |
| spelling | umjimathkievua-article-28522020-03-18T19:39:03Z On the maximal operator of $(C, α)$-means of Walsh–Kaczmarz–Fourier series Про максимальний оператор $(C, α)$-середніх рядів Уолша - Качмажа - Фур'є Goginava, U. Nagy, К. Гогінава, У. Надь, К. Simon [J. Approxim. Theory, 127, 39–60 (2004)] proved that the maximal operator $σ^{α,κ,*}$ of the $(C, α)$-means of the Walsh–Kaczmarz–Fourier series is bounded from the martingale Hardy space $H_p$ to the space $L_p$ for $p > 1 / (1 + α), \;0 < α ≤ 1$. Recently, Gát and Goginava have proved that this boundedness result does not hold if $p ≤ 1 / (1 + α)$. However, in the endpoint case $p = 1 / (1 + α )$, the maximal operator $σ^{α,κ,*}$ is bounded from the martingale Hardy space $H_{1/(1+α)}$ to the space weak- $L_{1/(1+α)}$. The main aim of this paper is to prove a stronger result, namely, that, for any $0 < p ≤ 1 / (1 + α)$, there exists a martingale $f ∈ H_p$ such that the maximal operator $σ^{α,κ,*} f$ does not belong to the space $L_p$. Саймон довів [див. J. Approxim. Theory. - 2004. - 127. - P. 39 - 60], що максимальний оператор $σ^{α,κ,*}$ $(C, α)$-середніх рядів Уолша - Качмажа - Фур'є є обмеженим з маргингального простору Харді $H_p$ до простору $L_p$ для $p > 1 / (1 + α), \;0 < α ≤ 1$. Нещодавно Гат і Гогінава довели, що цей результат про обмеженість не виконується, якщо $p ≤ 1 / (1 + α)$. Однак у випадку кінцевої точки $p = 1 / (1 + α )$ максимальний оператор $σ^{α,κ,*}$ к обмеженим з мартипгального простору Харді $H_{1/(1+α)}$ до простору слабкого $L_{1/(1+α)}$. Головна ме та даної статіі —довести більш вагомий результат, тоб то довес ти, що для будь-якого $0 < p ≤ 1 / (1 + α)$ існує мартингал $f ∈ H_p$ такий, що максимальний оператор $σ^{α,κ,*} f$ не належить простору $L_p$. Institute of Mathematics, NAS of Ukraine 2010-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2852 Ukrains’kyi Matematychnyi Zhurnal; Vol. 62 No. 2 (2010); 158–166 Український математичний журнал; Том 62 № 2 (2010); 158–166 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2852/2456 https://umj.imath.kiev.ua/index.php/umj/article/view/2852/2457 Copyright (c) 2010 Goginava U.; Nagy К. |
| spellingShingle | Goginava, U. Nagy, К. Гогінава, У. Надь, К. On the maximal operator of $(C, α)$-means of Walsh–Kaczmarz–Fourier series |
| title | On the maximal operator of $(C, α)$-means of Walsh–Kaczmarz–Fourier series |
| title_alt | Про максимальний оператор $(C, α)$-середніх рядів Уолша - Качмажа - Фур'є |
| title_full | On the maximal operator of $(C, α)$-means of Walsh–Kaczmarz–Fourier series |
| title_fullStr | On the maximal operator of $(C, α)$-means of Walsh–Kaczmarz–Fourier series |
| title_full_unstemmed | On the maximal operator of $(C, α)$-means of Walsh–Kaczmarz–Fourier series |
| title_short | On the maximal operator of $(C, α)$-means of Walsh–Kaczmarz–Fourier series |
| title_sort | on the maximal operator of $(c, α)$-means of walsh–kaczmarz–fourier series |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2852 |
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