Volume 9 (2008), Issue 1, Article 16, 9 pp.
ON THE L1 NORM OF THE WEIGHTED MAXIMAL FUNCTION OF FEJÉR
KERNELS WITH RESPECT TO THE WALSH-KACZMARZ SYSTEM
KÁROLY NAGY
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
nkaroly@nyf.hu
Received 24 May, 2007; accepted 15 February, 2008
Communicated by Zs. Pales
A BSTRACT. The main aim of this paper is to investigate the integral of the weighted maximal
function of the Walsh-Kaczmarz-Fejér kernels. We give a necessary and sufficient conditions
for that the weighted maximal function of the Walsh-Kaczmarz-Fejér kernels is in L1 . After this
we discuss the weighted maximal function of (C, α) kernels with respect to Walsh-Paley system
too.
Key words and phrases: Walsh-Kaczmarz system, Fejér kernels, Fejér means, Maximal operator.
2000 Mathematics Subject Classification. 42C10.
1. I NTRODUCTION AND P RELIMINARIES
The Walsh-Kaczmarz system was introduced in 1948 by Šneider [9]. He showed that the
behavior of the Dirichlet kernel of the Walsh-Kaczmarz system is worse than of the kernel of
n (x)|
the Walsh-Paley system. Namely, he showed in [9] that the inequality lim sup |Dlog
≥C>0
n
holds a.e. for the Dirichlet kernel with respect to the Walsh-Kaczmarz system. This allows us
to construct examples of divergent Fourier series [2].
On the other hand, Schipp [6] and Wo-Sang Young [10] proved that the Walsh-Kaczmarz
system is a convergence system. Skvorcov [8] verified the everywhere and uniform convergence
of the Fejér means for continous functions. Gát proved [4] that the Fejér-Lebesgue theorem
holds for the Walsh-Kaczmarz system.
It is easy to show that the L1 norm of supn |Dn | with respect to both systems is infinite. Gát
in [3] raised the following problem: "What happens if we apply some weight function α? That
is, on what conditions do we find the inequality
Dn sup n α(n) < ∞
1
170-07
2
K ÁROLY NAGY
to be valid?" He gave necessary and sufficient conditions for both rearrangements of the Walsh
system. The main aim of this paper to give necessary and sufficient conditions for the maximal
function of Fejér kernels with weight function α for both rearrangements.
First we give a brief introduction to the theory of dyadic analysis [7, 1].
Denote by Z2 the discrete cyclic group of order 2, that is Z2 = {0, 1}, the group operation
is 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, that is, µ({0}) = µ({1}) = 1/2. Let
∞
G := × Z2 ,
k=0
G is called the Walsh group. The elements of G can be represented by a sequence x =
(x0 , x1 , . . . , xk , . . . ), where xk ∈ {0, 1} (k ∈ N) (N := {0, 1, . . . }, P := N\{0}).
The group operation on G is coordinate-wise addition (denoted by +), the measure (denoted
by µ) and the topology are the product measure and topology. Consequently, G is a compact
Abelian group. Dyadic intervals are defined by
I0 (x) := G,
In (x) := {y ∈ G : y = (x0 , . . . , xn−1 , yn , yn+1 . . . )}
for x ∈ G, n ∈ P. They form a base for the neighborhoods of G. Let 0 = (0 : i ∈ N) ∈ G and
In := In (0) for n ∈ N.
Furthermore, let Lp (G) denote the usual Lebesgue spaces on G (with the corresponding norm
k · k). The Rademacher functions are defined as
rk (x) := (−1)xk
(x ∈ G, k ∈ N).
P
i
Each natural number n can be uniquely expressed as n = ∞
i=0 ni 2 , ni ∈ {0, 1} (i ∈ N),
where only a finite number of ni ’s are different from zero. Let the order of n > 0 be denoted
by |n| := max{j ∈ N : nj 6= 0}. That is, |n| is the integer part of the binary logarithm of n.
Define the Walsh-Paley functions by
∞
Y
P|n|
ωn (x) :=
(rk (x))nk = (−1) k=0 nk xk .
k=0
Let the Walsh-Kaczmarz functions be defined by κ0 = 1 and for n ≥ 1
|n|−1
κn (x) := r|n| (x)
Y
P|n|−1
(r|n|−1−k (x))nk = r|n| (x)(−1)
k=0
nk x|n|−1−k
.
k=0
The Walsh-Paley system is ω := (ωn : n ∈ N) and the Walsh-Kaczmarz system is κ := (κn :
n ∈ N). It is well known that
{κn : 2k ≤ n < 2k+1 } = {ωn : 2k ≤ n < 2k+1 }
for all k ∈ N and κ0 = ω0 .
A relation between Walsh-Kaczmarz functions and Walsh-Paley functions was given by
Skvorcov in the following way [8]. Let the transformation τA : G → G be defined by
τA (x) := (xA−1 , xA−2 , . . . , x1 , x0 , xA , xA+1 , . . . )
for A ∈ N. We have that
κn (x) = r|n| (x)ωn−2|n| (τ|n| (x))
(n ∈ N, x ∈ G).
Define the Dirichlet and Fejér kernels by
Dnφ
:=
n−1
X
n
φk ,
k=0
J. Inequal. Pure and Appl. Math., 9(1) (2008), Art. 16, 9 pp.
Knφ
1X φ
:=
D ,
n k=1 k
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L1 N ORM
OF THE
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3
where φn = ωn or κn (n ∈ P). D0φ , K0φ := 0.
It is known [7] that
(
2n , x ∈ In ,
D2n (x) =
0,
otherwise (n ∈ N).
Let α, β : [0, ∞) → [1, ∞) be monotone increasing functions and define the weighted maximal function of the Dirichlet kernels Dαφ,∗ and of the Fejér kernels Kαφ,∗ :
|Knφ (x)|
|Dnφ (x)|
,
Kαφ,∗ (x) := sup
(x ∈ G),
n∈N α([log n])
n∈N α([log n])
where φ is either the Walsh-Paley, or the Walsh-Kaczmarz system. For the the weighted maximal function of the Dirichlet kernels
respect to the Walsh-Paley system Dαω,∗ Gát [3] proved
P∞ with
1
ω,∗
1
that Dα ∈ L if and only if A=0 α(A) < ∞. Moreover, he proved that
Dαφ,∗ (x) := sup
∞
∞
X
1X 1
1
ω,∗
≤ kDα k1 ≤ 2
.
2 A=0 α(A)
α(A)
A=0
For the Walsh-Kaczmarz
system, he showed that the situation is changed, namely Dακ,∗ ∈ L1 if
P∞
A
and only if A=1 α(A) < ∞. Moreover, he proved that there exists a positive constant C such
that
∞
1 X A
kDακ,∗ k1 ≥
− C.
25 A=1 α(A)
The two conditions are quite different for the two rearrangements of the Walsh system.
2. T HE R ESULTS
For kKαω,∗ k1 , we immediately obtain from Gát’s result the following lemma:
P
1
Lemma 2.1. Kαω,∗ ∈ L1 if and only if ∞
A=0 α(A) < ∞. Moreover,
∞
∞
X
1X 1
1
ω,∗
≤ kKα k1 ≤ 2
.
4 A=0 α(A)
α(A)
A=0
Proof. The upper estimation follows trivially from
n
n
|Knω (x)|
1 X |Djω (x)|
1 X ω,∗
≤
≤
D (x) ≤ Dαω,∗ (x),
α(|n|)
n j=1 α(|j|)
n j=1 α
that is
Kαω,∗ (x) ≤ Dαω,∗ (x) (x ∈ G).
The lower estimation for φ = ω or κ comes from the following. On the set IA \IA+1 we have
A
K2φA (x)
2
1 X
2A + 1
= A
k=
.
2 k=1
2
Thus, we have
kKαφ,∗ k1
=
=
∞ Z
X
A=0
∞
X
Kαφ,∗ (x)dµ(x)
IA \IA+1
1
α(A)
A=0
Z
IA \IA+1
≥
∞ Z
X
A=0
IA \IA+1
K2φA (x)
dµ(x)
α(A)
∞
2A + 1
1X 1
dµ(x) ≥
.
2
4 A=0 α(A)
J. Inequal. Pure and Appl. Math., 9(1) (2008), Art. 16, 9 pp.
http://jipam.vu.edu.au/
4
K ÁROLY NAGY
We will show that we can obtain as good an estimation for kKακ,∗ k1 as for kKαω,∗ k1 . This
means that the behavior of the Walsh-Kaczmarz-Fejér kernels is better than the behavior of the
Walsh-Kaczmarz-Dirichlet kernels. This is the main reason, why we have so many convergence
theorems for Walsh-Kaczmarz-Fejér means [4, 8]. Namely,
Theorem 2.2. There is positive absolute constant C such that
∞
∞
X
1X 1
1
κ,∗
≤ kKα k1 ≤ C
.
4 A=0 α(A)
α(A)
A=0
P
1
Corollary 2.3. Kακ,∗ ∈ L1 if and only if ∞
A=0 α(A) < ∞.
Skvorcov in [8] proved that for n ∈ P, x ∈ G
|n|−1
nKnκ (x)
=1+
X
|n|−1
i
2 D2i (x) +
X
2i ri (x)K2ωi (τi (x))
i=0
i=0
ω
+ (n − 2|n| )(D2|n| (x) + r|n| (x)Kn−2
|n| (τ|n| (x))).
To prove Theorem 2.2, we will use two lemmas by Gát [4].
Lemma 2.4. Let A, t ∈ N, A > t. Suppose that x ∈ It \It+1 . Then

0
if x − xt et 6∈ IA ,
K2ωA (x) =
2t−1 if x − x e ∈ I .
t t
A
If x ∈ IA , then K2ωA (x) =
2A +1
.
2
Set
ω
Ka,b
:=
a+b−1
X
Djω
(a, b ∈ N),
j=a
and n
(s)
:=
P∞
i=s
i
ni 2 (n, s ∈ N). Using simple calculations, we have
nKnω
=
|n|
X
ns Knω(s+1) ,2s + Dnω
(n ∈ P).
s=0
Lemma 2.5. Let s, t, n ∈ N, and x ∈ It \It+1 . If s ≤ t ≤ |n|, then |Knω(s+1) ,2s (x)| ≤ c2s+t . If
t < s ≤ |n|, then we have

0
if x − xt et 6∈ Is ,
Knω(s+1) ,2s (x) =
ω
s+t−1
if x − xt et ∈ Is .
n(s+1) (x)2
Throughout the remainder of the paper C will denote a positive absolute constant, though not
always the same at different occurences.
Proof of the Theorem 2.2. We will use Skvorcov’s result and
|n|−1
X
1
1
1
+
2i D2i (x) +
(n − 2|n| )D2|n| (x)
nα(|n|) nα(|n|) i=0
nα(|n|)
|n|−1
1
1 X i D2i (x)
1
≤
+
2
+ Dαω,∗ (x) ≤
+ CDαω,∗ (x).
α(1) n i=0
α(i)
α(1)
J. Inequal. Pure and Appl. Math., 9(1) (2008), Art. 16, 9 pp.
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L1 N ORM
OF THE
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5
Now, we discuss
|n|−1
X
1
2i ri (x)K2ωi (τi (x)).
nα(|n|) i=0
Let Jti := {x ∈ G : xi−1 = · · · = xi−t = 0, xi−t−1 = 1} and J0i := {x ∈ G : xi−1 = 1}. For
S
i
every 1 ≤ i ∈ N we can decompose G as the disjoint union: G := Ii ∪ i−1
t=0 Jt .
i
ω
By Gát’s Lemma 2.4, if x ∈ Jt , then K2i (τi (x)) 6= 0 only in the case when xi−t−2 = · · · =
x0 = 0, and in this case K2ωi (τi (x)) = 2t−1 .
Z
Z
Z
ω
ω
K2ωi (τi (x))dµ(x)
|ri (x)K2i (τi (x))|dµ(x) =
K2i (τi (x))dµ(x) +
G
Ii
Ii
2i + 1 1
· i+
2
2
i−1 Z
X
≤1+
≤
t=0
≤1+
t=0
Jti
K2ωi (τi (x))dµ(x)
2t−1 dµ(x)
{x∈G:xi−t−1 =1,xj =0 if j<i and j6=i−t−1}
i−1 t−1
X
2
t=0
i−1 Z
X
2i
≤ 2.
Thus, we have
|n|−1
q−1
∞ Z
X
X
1
1 X i
i
ω
sup 2 ri (x)K2i (τi (x)) ≤
2 |ri (x)K2ωi (τi (x))|dµ(x)
sup q
n nα(|n|)
2
α(q)
|n|=q
q=0 G
i=0
i=0
1
≤
∞
X
q=0
≤
∞
X
q=0
q−1
1 X i
2
2q α(q) i=0
Z
|ri (x)K2ωi (τi (x))|dµ(x)
G
q−1
∞
X 1
1 X i+1
2
≤
C
.
2q α(q) i=0
α(q)
q=0
We have to discuss
n − 2|n|
ω
sup r|n| (x)Kn−2|n| (τ|n| (x)) .
nα(|n|)
n
n − 2|n|
ω
sup r|n| (x)Kn−2|n| (τ|n| (x)) dµ(x)
nα(|n|)
G n
Z
∞
X
1
n − 2|n| ω
sup
≤
Kn−2|n| (τ|n| (x)) dµ(x)
α(l) G |n|=l
n
l=1
Z
∞
X
1
n − 2|n| ω
=
sup
Kn−2|n| (τ|n| (x)) dµ(x)
α(l) Il |n|=l
n
l=1
Z
∞
X
1
n − 2|n| ω
+
sup
Kn−2|n| (τ|n| (x)) dµ(x)
α(l) Il |n|=l
n
l=1
Z
=: S 1 + S 2 .
J. Inequal. Pure and Appl. Math., 9(1) (2008), Art. 16, 9 pp.
http://jipam.vu.edu.au/
6
K ÁROLY NAGY
ω
If x ∈ I|n| , then τ|n| (x) ∈ I|n| and Kn−2
(τ
(x))
≤ C(n − 2|n| ) and
|n|
|n|
Z
∞
X
(n − 2|n| )2
1
S ≤C
sup
dµ(x)
α(l) Il |n|=l
n
l=1
Z
∞
X
1
≤C
sup (n − 2|n| )dµ(x)
α(l) Il |n|=l
l=1
Z
∞
∞
X
X
1
1
≤C
2l dµ(x) ≤ C
.
α(l)
α(l)
I
l
l=1
l=1
1
Now, we investigate S 2 .
∞
l−1 Z
X
1 X
n − 2|n| ω
sup sup
Kn−2|n| (τ|n| (x)) dµ(x)
S ≤
α(l) t=0 Jtl |n|=l |n−2|n| |=q
n
l=1
2
q<l
≤
∞
X
l=1
1
α(l)
+
t=0
∞
X
l=1
=:
X
K
+
q
1 X ω
sup sup
ns Kn(s+1) ,2s (τ|n| (x)) dµ(x)
l
n
|n|
Jt |n|=l |n−2 |=q
s=0
l−1 Z
X
X
q<l
l−1
1 X
α(l) t=0
Z
1 ω
Dn−2|n| (τ|n| (x)) dµ(x)
Jtl |n|=l |n−2|n| |=q n
sup
sup
q<l
.
D
Let x ∈ Jtl . By Lemma 2.5
t, then Knω(s+1) ,2s (τ|n| (x)) 6=
ω
of Gát, if s ≤ t, then Kn(s+1) ,2s (τ|n| (x)) ≤ 2s+t , if q ≥ s >
0 if and only if xl−t−2 = · · · = xl−s = 0, and in this case
ω
Kn(s+1) ,2s (τ|n| (x)) = 2s+t .
X
K
q
l−1 Z
l−1
∞
X
X
1 X
1 X ω
sup
≤C
Kn(s+1) ,2s (τ|n| (x)) dµ(x)
l
q
α(l) t=0 Jtl |n|=l q=0 2 + 2 s=0
l=1
q Z
∞
l−1 t
X
1 XX 1 X
≤C
2s+t dµ(x)
l + 2q
l
α(l)
2
t=0 q=0
s=0 Jt
l=1
∞
l−1 l−1
t Z
X
1 X X
1 X
+C
2s+t dµ(x)
l + 2q
l
α(l)
2
t=0 q=t+1
s=0 Jt
l=1
Z
q
∞
l−1 l−1
X
X
1 X X
1
+C
2s+t dµ(x)
l
q
α(l) t=0 q=t+1 2 + 2 s=t+1 {x∈Jtl :xl−t−2 =···=xl−s =0}
l=1
q
∞
l−1 t
∞
l−1
X
X
1 XX 1 X s
1 X 2t (l − t)
≤C
2 +C
l + 2q
α(l)
2
α(l) t=0
2l
t=0 q=0
s=0
l=1
l=1
∞
l−1
X
1 X 2t (l − t)2
+C
α(l) t=0
2l
l=1
J. Inequal. Pure and Appl. Math., 9(1) (2008), Art. 16, 9 pp.
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L1 N ORM
OF THE
W EIGHTED M AXIMAL
7
∞
X
1
≤C
.
α(l)
l=1
ω
The inequality Dn−2
(τ
(x))
≤ n − 2|n| gives
|n|
|n|
X
D
l−1 Z
∞
X
n − 2|n|
1 X
sup sup
dµ(x)
≤
l |n|=l
α(l)
n
|n| |=q
J
|n−2
t
t=0
l=1
q<l
l−1
∞
∞
X
X
1 X −t
1
2 ≤C
.
α(l)
α(l)
t=0
l=1
l=1
≤C
The lower estimation comes from Lemma 2.1.
This completes the proof of Theorem 2.2.
Let α ∈ R, and define the nth (C, α) Fejér kernel Knφ,α and the weighted maximal function
of the (C, α) Fejér kernels Kβφ,α,∗ by
Knφ,α :=
n
|Knφ,α |
1 X α−1 φ
φ,α,∗
A
D
,
K
:=
sup
,
β
Aαn k=0 n−k k
n∈N β([log n])
where φ = ω or κ and Aαn := (1+α)...(n+α)
for any n ∈ N, α ∈ R(α 6= −1, −2, . . . ). It is known
n!
that Aαn ∼ nα .
To investigate Kβω,α,∗ , we have to use the following lemma of Gát and Goginava [5]:
Lemma 2.6 (G. Gát, U. Goginava). Let α ∈ (0, 1) and n := n(A) = nA 2A + · · · + n0 20 , then


p −1
A X
i
2X

X
c(α)
|Knω,α | ≤ α
2p(α−1)
|Kjω | + 2iα |K2ωi −1 | + 2iα D2i .


n
p−1
i=0
p=1
j=2
Theorem 2.7. Let 0 < α ≤ 1, then there are positive absolute constants c, C (c, C depend only
on α) such that
∞
∞
X
X
1
1
c
≤ kKβω,α,∗ k1 ≤ C
.
β(A)
β(A)
A=0
A=0
This means that the behavior of the weighted maximal function of the (C, α) kernels is the
same as the behavior of the weighted maximal function of the (C, 1) kernels with respect to this
issue.
P
1
Corollary 2.8. Kβω,α,∗ ∈ L1 if and only if ∞
A=0 β(A) < ∞.
Proof. α = 1 is given by Lemma 2.1.
Let |n| = A. Then by Lemma 2.6 of Gát and Goginava we have


p −1
A
i
2


ω,α
|Kn |
C(α) X X p(α−1) X
iα
ω
iα
ω
i
≤ Aα
2
|Kj | + 2 |K2i −1 | + 2 D2

β(A)
2 β(A) i=0  p=1
j=2p−1


p −1
A
i
2

ω
ω
|K i |
|Kj |
C(α) X X p(α−1) X
Di
≤ Aα
2
+ 2iα 2 −1 + 2iα 2
 p=1
2
β(p − 1)
β(i − 1)
β(i) 
p−1
i=0
j=2
≤
C(α)(Kβω,∗
+
Dβω,∗ ).
J. Inequal. Pure and Appl. Math., 9(1) (2008), Art. 16, 9 pp.
http://jipam.vu.edu.au/
8
K ÁROLY NAGY
This, Lemma 2.1 and [3] of Gát gives that the upper estimation holds for Kβω,α,∗ .
To make the lower estimation we need to investigate K2φ,α
A , where φ = ω or κ.
On the set IA \IA+1 we have
A
2
X
A
Aα−1
Dφ (x)
2A −j j
2
X
=
A
Aα−1
j
2A −j
=
j=0
j=0
2
X
Aα−1
(2A − l).
l
l=0
α−1 α+l
Therefore by an Abel transformation and Aα−1
< Aα−1
it follows that
l+1 = Al
l
l+1
A
2
X
Alα−1 (2A − l) =
l=0
A −2
2X
(Aα−1
− Aα−1
l
l+1 )
l
X
(2A − j) + Aα−1
2A −1
j=1
l=0
2A −1
≥ Aα−1
2A −1
X
(2A − l) = Aα−1
2A −1
l=1
A −1
2X
(2A − l)
l=1
2A (2A − 1)
>0
2
and
A
K2φ,α
A (x)
2
1 X α−1 φ
1
2A (2A − 1)
= α
A2A −j Dj (x) ≥ α Aα−1
.
A
A2A j=0
A2A 2 −1
2
Thus,
kKβφ,α,∗ k1
=
∞ Z
X
A=0
≥
IA \IA+1
φ,α K2A (x)
∞ Z
X
A=0
∞
X
IA \IA+1
1
≥
β(A)
A=0
≥c
Kβφ,α,∗ (x)dµ(x)
β(A)
Z
IA \IA+1
dµ(x)
1 α−1 2A (2A − 1)
AA
dµ(x)
Aα2A 2 −1
2
∞
X
1
.
β(A)
A=0
This completes the proof of Theorem 2.7.
R EFERENCES
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