We give a necessary and sufficient conditions for that the weighted maximal function of the Walsh-Kaczmarz-Fejér kernels is inL1

ON THEL¹ NORM OF THE WEIGHTED MAXIMAL FUNCTION OF FEJÉR KERNELS WITH RESPECT TO THE WALSH-KACZMARZ SYSTEM

KÁROLY NAGY

INSTITUTE OFMATHEMATICS ANDCOMPUTERSCIENCE

COLLEGE OFNYÍREGYHÁZA

P.O. BOX166, NYÍREGYHÁZA

H-4400 HUNGARY

nkaroly@nyf.hu

Received 24 May, 2007; accepted 15 February, 2008 Communicated by Zs. Pales

ABSTRACT. 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 inL¹. 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. INTRODUCTION ANDPRELIMINARIES

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 the Walsh-Paley system. Namely, he showed in [9] that the inequalitylim sup^|D_log^n(x)|_n ≥C > 0 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 theL¹ norm ofsup_n|D_n|with respect to both systems is infinite. Gát in [3] raised the following problem: "What happens if we apply some weight functionα? That is, on what conditions do we find the inequality

sup

n

Dn

α(n) 1

<∞

170-07

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 functionαfor both rearrangements.

First we give a brief introduction to the theory of dyadic analysis [7, 1].

Denote byZ₂ the discrete cyclic group of order 2, that is Z₂ = {0,1}, the group operation is modulo2addition and every subset is open. The normalized Haar measure onZ₂ is given in the way that the measure of a singleton is1/2, that is,µ({0}) =µ({1}) = 1/2.Let

G:= ×^∞

k=0

Z₂,

G is called the Walsh group. The elements of G can be represented by a sequence x = (x₀, x₁, . . . , x_k, . . .),wherex_k∈ {0,1}(k ∈N) (N:={0,1, . . .},P:=N\{0}).

The group operation onGis coordinate-wise addition (denoted by+), the measure (denoted byµ) and the topology are the product measure and topology. Consequently, G is a compact Abelian group. Dyadic intervals are defined by

I₀(x) := G, I_n(x) := {y∈G:y = (x₀, . . . , xn−1, y_n, y_n+1. . .)}

forx∈ G, n∈P. They form a base for the neighborhoods ofG. Let0 = (0 :i∈N)∈Gand I_n:=I_n(0)forn ∈N.

Furthermore, letL^p(G)denote the usual Lebesgue spaces onG(with the corresponding norm k · k). The Rademacher functions are defined as

r_k(x) := (−1)^xk (x∈G, k ∈N).

Each natural numbern can be uniquely expressed asn = P∞

i=0n_i2ⁱ, n_i ∈ {0,1}(i ∈ N), where only a finite number of n_i’s are different from zero. Let the order of n > 0be denoted by|n|:= max{j ∈N:n_j 6= 0}.That is,|n|is the integer part of the binary logarithm ofn.

Define the Walsh-Paley functions by

ω_n(x) :=

∞

Y

k=0

(r_k(x))^nk = (−1)^P|n|k=0nkxk. Let the Walsh-Kaczmarz functions be defined byκ₀ = 1and forn ≥1

κ_n(x) := r|n|(x)

|n|−1

Y

k=0

(r|n|−1−k(x))^nk =r|n|(x)(−1)^{P|n|−1k=0 nkx|n|−1−k}.

The Walsh-Paley system isω := (ω_n :n∈N)and the Walsh-Kaczmarz system isκ := (κ_n: n∈N).It is well known that

{κ_n: 2^k≤n <2^k+1}={ω_n : 2^k≤n <2^k+1} for allk ∈Nandκ₀ =ω₀.

A relation between Walsh-Kaczmarz functions and Walsh-Paley functions was given by Skvorcov in the following way [8]. Let the transformationτ_A:G→Gbe defined by

τ_A(x) := (xA−1, xA−2, . . . , x₁, x₀, x_A, x_A+1, . . .) forA ∈N.We have that

κ_n(x) =r|n|(x)ω_n−2^|n|(τ|n|(x)) (n ∈N, x∈G).

Define the Dirichlet and Fejér kernels by

D^φ_n :=

n−1

X

k=0

φ_k, K_n^φ:= 1 n

n

X

k=1

D_k^φ,

whereφ_n=ω_norκ_n(n∈P). D^φ₀, K₀^φ:= 0.

It is known [7] that

D₂ⁿ(x) =

(2ⁿ, x∈I_n,

0, otherwise(n∈N).

Letα, β : [0,∞)→[1,∞)be monotone increasing functions and define the weighted maxi- mal function of the Dirichlet kernelsD_α^φ,∗ and of the Fejér kernelsK_α^φ,∗:

D^φ,∗_α (x) := sup

n∈N

|D_n^φ(x)|

α([logn]), K_α^φ,∗(x) := sup

n∈N

|K_n^φ(x)|

α([logn]) (x∈G),

whereφis either the Walsh-Paley, or the Walsh-Kaczmarz system. For the the weighted maxi- mal function of the Dirichlet kernels with respect to the Walsh-Paley systemD_α^ω,∗Gát [3] proved thatD^ω,∗_α ∈L¹ if and only ifP∞

A=0 1

α(A) <∞.Moreover, he proved that 1

2

∞

X

A=0

1

α(A) ≤ kD^ω,∗_α k1 ≤2

∞

X

A=0

1 α(A).

For the Walsh-Kaczmarz system, he showed that the situation is changed, namelyD^κ,∗_α ∈L¹if and only ifP∞

A=1 A

α(A) < ∞.Moreover, he proved that there exists a positive constant Csuch that

kD^κ,∗_α k₁ ≥ 1 25

∞

X

A=1

A

α(A) −C.

The two conditions are quite different for the two rearrangements of the Walsh system.

2. THERESULTS

ForkK_α^ω,∗k1, we immediately obtain from Gát’s result the following lemma:

Lemma 2.1. K_α^ω,∗ ∈L¹if and only ifP∞ A=0

1

α(A) <∞.Moreover, 1

4

∞

X

A=0

1

α(A) ≤ kK_α^ω,∗k₁ ≤2

∞

X

A=0

1 α(A). Proof. The upper estimation follows trivially from

|K_n^ω(x)|

α(|n|) ≤ 1 n

n

X

j=1

|D_j^ω(x)|

α(|j|) ≤ 1 n

n

X

j=1

D_α^ω,∗(x)≤D_α^ω,∗(x), that is

K_α^ω,∗(x)≤D^ω,∗_α (x) (x∈G).

The lower estimation forφ=ωorκcomes from the following. On the setI_A\I_A+1 we have K₂^φA(x) = 1

2^A

2^A

X

k=1

k = 2^A+ 1 2 . Thus, we have

kK_α^φ,∗k₁ =

∞

X

A=0

Z

I_A\I_A+1

K_α^φ,∗(x)dµ(x)≥

∞

X

A=0

Z

I_A\I_A+1

K₂^φA(x) α(A) dµ(x)

=

∞

X

A=0

1 α(A)

Z

IA\IA+1

2^A+ 1

2 dµ(x)≥ 1 4

∞

X

A=0

1 α(A).

We will show that we can obtain as good an estimation for kK_α^κ,∗k₁ as for kK_α^ω,∗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 constantCsuch that 1

4

∞

X

A=0

1

α(A) ≤ kK_α^κ,∗k₁ ≤C

∞

X

A=0

1 α(A). Corollary 2.3. K_α^κ,∗ ∈L¹ if and only ifP∞

A=0 1

α(A) <∞.

Skvorcov in [8] proved that forn∈P, x∈G

nK_n^κ(x) = 1 +

|n|−1

X

i=0

2ⁱD₂ⁱ(x) +

|n|−1

X

i=0

2ⁱr_i(x)K₂^ωi(τ_i(x))

+ (n−2^|n|)(D₂|n|(x) +r_|n|(x)K_n−2^ω |n|(τ_|n|(x))).

To prove Theorem 2.2, we will use two lemmas by Gát [4].

Lemma 2.4. LetA, t∈N, A > t.Suppose thatx∈I_t\I_t+1.Then

K₂^ωA(x) =







0 ifx−x_te_t6∈I_A, 2^t−1 ifx−x_te_t∈I_A. Ifx∈I_A,thenK₂^ωA(x) = ^2A₂⁺¹.

Set

K_a,b^ω :=

a+b−1

X

j=a

D_j^ω (a, b∈N), andn^(s):=P∞

i=sn_i2ⁱ(n, s∈N).Using simple calculations, we have nK_n^ω =

|n|

X

s=0

n_sK_n^ω(s+1),2^s +D_n^ω (n∈P).

Lemma 2.5. Lets, t, n ∈ N,andx ∈ I_t\I_t+1.Ifs ≤ t ≤ |n|,then|K_n^ω(s+1),2^s(x)| ≤ c2^s+t.If t < s≤ |n|,then we have

K_n^ω(s+1),2^s(x) =







0 ifx−x_te_t 6∈I_s, ω_n^(s+1)(x)2^s+t−1 ifx−xtet ∈Is.

Throughout the remainder of the paperCwill 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 1

nα(|n|)+ 1 nα(|n|)

|n|−1

X

i=0

2ⁱD₂ⁱ(x) + 1

nα(|n|)(n−2^|n|)D₂|n|(x)

≤ 1 α(1) + 1

n

|n|−1

X

i=0

2ⁱD₂ⁱ(x)

α(i) +D_α^ω,∗(x)≤ 1

α(1) +CD_α^ω,∗(x).

Now, we discuss

1 nα(|n|)

|n|−1

X

i=0

2ⁱr_i(x)K₂^ωi(τ_i(x)).

LetJ_tⁱ := {x ∈ G : xi−1 = · · ·= xi−t = 0, xi−t−1 = 1}andJ₀ⁱ :={x ∈ G : xi−1 = 1}.For every1≤i∈Nwe can decomposeGas the disjoint union:G:=Ii∪Si−1

t=0J_tⁱ.

By Gát’s Lemma 2.4, ifx ∈ J_tⁱ,thenK₂^ωi(τ_i(x)) 6= 0only in the case when xi−t−2 = · · · = x₀ = 0,and in this caseK₂^ωi(τ_i(x)) = 2^t−1.

Z

G

|r_i(x)K₂^ωi(τ_i(x))|dµ(x) = Z

Ii

K₂^ωi(τ_i(x))dµ(x) + Z

Ii

K₂^ωi(τ_i(x))dµ(x)

≤ 2ⁱ+ 1 2 · 1

2ⁱ +

i−1

X

t=0

Z

J_tⁱ

K₂^ωi(τ_i(x))dµ(x)

≤1 +

i−1

X

t=0

Z

{x∈G:xi−t−1=1,xj=0ifj<iandj6=i−t−1}

2^t−1dµ(x)

≤1 +

i−1

X

t=0

2^t−1 2ⁱ ≤2.

Thus, we have

sup

n

1 nα(|n|)

|n|−1

X

i=0

2ⁱr_i(x)K₂^ωi(τ_i(x)) 1

≤

∞

X

q=0

Z

G

sup

|n|=q

1 2^qα(q)

q−1

X

i=0

2ⁱ|r_i(x)K₂^ωi(τ_i(x))|dµ(x)

≤

∞

X

q=0

1 2^qα(q)

q−1

X

i=0

2ⁱ Z

G

|r_i(x)K₂^ωi(τ_i(x))|dµ(x)

≤

∞

X

q=0

1 2^qα(q)

q−1

X

i=0

2ⁱ⁺¹ ≤C

∞

X

q=0

1 α(q). We have to discuss

sup

n

n−2^|n|

nα(|n|)r|n|(x)K_n−2^ω |n|(τ|n|(x)) .

Z

G

sup

n

n−2^|n|

nα(|n|)r|n|(x)K_n−2^ω |n|(τ|n|(x))

dµ(x)

≤

∞

X

l=1

1 α(l)

Z

G

sup

|n|=l

n−2^|n|

n

K_n−2^ω |n|(τ|n|(x)) dµ(x)

=

∞

X

l=1

1 α(l)

Z

Il

sup

|n|=l

n−2^|n|

n

K_n−2^ω |n|(τ|n|(x)) dµ(x)

+

∞

X

l=1

1 α(l)

Z

Il

sup

|n|=l

n−2^|n|

n

K_n−2^ω |n|(τ|n|(x)) dµ(x)

=:S¹+S².

Ifx∈I|n|,thenτ|n|(x)∈I|n|and

K_n−2^ω |n|(τ|n|(x))

≤C(n−2^|n|)and S¹ ≤C

∞

X

l=1

1 α(l)

Z

Il

sup

|n|=l

(n−2^|n|)² n dµ(x)

≤C

∞

X

l=1

1 α(l)

Z

Il

sup

|n|=l

(n−2^|n|)dµ(x)

≤C

∞

X

l=1

1 α(l)

Z

Il

2^ldµ(x)≤C

∞

X

l=1

1 α(l). Now, we investigateS².

S² ≤

∞

X

l=1

1 α(l)

l−1

X

t=0

Z

J_t^l

sup

|n|=l q<l

sup

|n−2^|n||=q

n−2^|n|

n

K_n−2^ω |n|(τ|n|(x)) dµ(x)

≤

∞

X

l=1

1 α(l)

l−1

X

t=0

Z

J_t^l

sup

|n|=l q<l

sup

|n−2^|n||=q

1 n

q

X

s=0

n_s

K_n^ω(s+1),2^s(τ_|n|(x)) dµ(x)

+

∞

X

l=1

1 α(l)

l−1

X

t=0

Z

J_t^l

sup

|n|=l q<l

sup

|n−2^|n||=q

1 n

D^ω_n−2|n|(τ|n|(x)) dµ(x)

=:X

K

+X

D

.

Let x ∈ J_t^l. By Lemma 2.5 of Gát, if s ≤ t, then

K_n^ω_(s+1),2s(τ_|n|(x))

≤ 2^s+t, if q ≥ s >

t, then K_n^ω(s+1),2^s(τ|n|(x)) 6= 0 if and only if xl−t−2 = · · · = xl−s = 0, and in this case

K_n^ω(s+1),2^s(τ|n|(x))

= 2^s+t. X

K

≤C

∞

X

l=1

1 α(l)

l−1

X

t=0

Z

J_t^l

sup

|n|=l l−1

X

q=0

1 2^l+ 2^q

q

X

s=0

K_n^ω(s+1),2^s(τ|n|(x)) dµ(x)

≤C

∞

X

l=1

1 α(l)

l−1

X

t=0 t

X

q=0

1 2^l+ 2^q

q

X

s=0

Z

J_t^l

2^s+tdµ(x)

+C

∞

X

l=1

1 α(l)

l−1

X

t=0 l−1

X

q=t+1

1 2^l+ 2^q

t

X

s=0

Z

J_t^l

2^s+tdµ(x)

+C

∞

X

l=1

1 α(l)

l−1

X

t=0 l−1

X

q=t+1

1 2^l+ 2^q

q

X

s=t+1

Z

{x∈J_t^l:xl−t−2=···=xl−s=0}

2^s+tdµ(x)

≤C

∞

X

l=1

1 α(l)

l−1

X

t=0 t

X

q=0

1 2^l+ 2^q

q

X

s=0

2^s+C

∞

X

l=1

1 α(l)

l−1

X

t=0

2^t(l−t) 2^l +C

∞

X

l=1

1 α(l)

l−1

X

t=0

2^t(l−t)² 2^l

≤C

∞

X

l=1

1 α(l). The inequality

D_n−2^ω |n|(τ_|n|(x))

≤n−2^|n|gives X

D

≤

∞

X

l=1

1 α(l)

l−1

X

t=0

Z

J_t^l

sup

|n|=l q<l

sup

|n−2^|n||=q

n−2^|n|

n dµ(x)

≤C

∞

X

l=1

1 α(l)

l−1

X

t=0

2^−t≤C

∞

X

l=1

1 α(l). The lower estimation comes from Lemma 2.1.

This completes the proof of Theorem 2.2.

Letα ∈R, and define thenth(C, α)Fejér kernel K_n^φ,α and the weighted maximal function of the(C, α)Fejér kernelsK_β^φ,α,∗ by

K_n^φ,α := 1 A^α_n

n

X

k=0

A^α−1_n−kD^φ_k, K_β^φ,α,∗ := sup

n∈N

|K_n^φ,α| β([logn]), whereφ=ωorκandA^α_n := (1+α)...(n+α)

n! for anyn ∈N, α ∈R(α 6=−1,−2, . . .).It is known thatA^α_n∼n^α.

To investigateK_β^ω,α,∗, we have to use the following lemma of Gát and Goginava [5]:

Lemma 2.6 (G. Gát, U. Goginava). Letα∈(0,1)andn :=n^(A) =n_A2^A+· · ·+n₀2⁰,then

|K_n^ω,α| ≤ c(α) n^α

A

X

i=0







i

X

p=1

2^p(α−1)

2^p−1

X

j=2^p−1

|K_j^ω|+ 2^iα|K₂^ωi−1|+ 2^iαD₂ⁱ





 .

Theorem 2.7. Let0< α≤1, then there are positive absolute constantsc, C (c, Cdepend only onα) such that

c

∞

X

A=0

1

β(A) ≤ kK_β^ω,α,∗k₁ ≤C

∞

X

A=0

1 β(A).

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.

Corollary 2.8. K_β^ω,α,∗ ∈L¹ if and only ifP∞ A=0

1

β(A) <∞.

Proof. α= 1is given by Lemma 2.1.

Let|n|=A.Then by Lemma 2.6 of Gát and Goginava we have

|K_n^ω,α|

β(A) ≤ C(α) 2^Aαβ(A)

A

X

i=0







i

X

p=1

2^p(α−1)

2^p−1

X

j=2^p−1

|K_j^ω|+ 2^iα|K₂^ωi−1|+ 2^iαD₂ⁱ







≤ C(α) 2^Aα

A

X

i=0







i

X

p=1

2^p(α−1)

2^p−1

X

j=2^p−1

|K_j^ω|

β(p−1)+ 2^iα |K₂^ωi−1|

β(i−1) + 2^iαD₂ⁱ β(i)







≤C(α)(K_β^ω,∗+D_β^ω,∗).

This, Lemma 2.1 and [3] of Gát gives that the upper estimation holds forK_β^ω,α,∗. To make the lower estimation we need to investigateK₂^φ,αA ,whereφ=ωorκ.

On the setI_A\I_A+1 we have

2^A

X

j=0

A^α−1₂A−jD_j^φ(x) =

2^A

X

j=0

A^α−1₂A−jj =

2^A

X

l=0

A^α−1_l (2^A−l).

Therefore by an Abel transformation andA^α−1_l+1 =A^α−1_l ^α+l_l+1 < A^α−1_l it follows that

2^A

X

l=0

A^α−1_l (2^A−l) =

2^A−2

X

l=0

(A^α−1_l −A^α−1_l+1)

l

X

j=1

(2^A−j) +A^α−1₂A−1 2^A−1

X

l=1

(2^A−l)

≥A^α−1₂A−1 2^A−1

X

l=1

(2^A−l) =A^α−1₂A−1

2^A(2^A−1) 2 >0 and

K₂^φ,αA (x) = 1 A^α₂A

2^A

X

j=0

A^α−1₂A−jD_j^φ(x)≥ 1 A^α₂A

A^α−1₂A−1

2^A(2^A−1)

2 .

Thus,

kK_β^φ,α,∗k₁ =

∞

X

A=0

Z

IA\I_A+1

K_β^φ,α,∗(x)dµ(x)

≥

∞

X

A=0

Z

IA\I_A+1

K₂^φ,αA (x) β(A) dµ(x)

≥

∞

X

A=0

1 β(A)

Z

IA\I_A+1

1

A^α_2AA^α−1₂A−1

2^A(2^A−1) 2 dµ(x)

≥c

∞

X

A=0

1 β(A).

This completes the proof of Theorem 2.7.

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