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Estimates of Variable Kernel Parameterized Littlewood–Paley Operators on Variable Herz

Spaces

Afif Abdalmonem1,∗ , Omer Khalil2, Omer Abdalrhman3

1Faculty of Science, Department of Mathematics, University of Dalanj, Dalanj, Sudan
afeefy86@gmail.com

2 College of Mathematics and Statistics, Northwest Normal University, China
us.omer2008@yahoo.com

3College of Education, Shendi University, Sudan
humoora@gmail.com

∗Correspondence: afeefy86@gmail.com

Abstract. In this article, we prove some boundedness results for variable kernel parameterizedLittlewood−Paley operators on the homogeneous Herz spaces K̇α(·),q(·)
p(·) (Rn). Several known resultsare extended.

1. Introduction
Suppose that Ψ(x1, z1) ∈ L∞(Rn)× Lb(Sn−1) (where b ≥ 1) satisfies:(1) Ψ(x1, αz1) = Ψ(x1, z1) and ∫

sn−1

Ψ(x1, z
′
1)dσ(z ′1) = 0, for all z1, x1 ∈ Rn, α > 0;

(2) ‖Ψ‖L∞(Rn)×Lb(Sn−1) := sup
x1∈Rn

(∫
sn−1 |Ψ(x1, z

′
1)|bdz ′1

) 1
b <∞,

where Sn−1 (for n ≥ 2) is the unit sphere in Rn equipped with Lebesgue measure dz ′1. Theparameterized Littlewood−Paley operators, denoted by µσΨ,s and µ∗,σΨ,λ, are closely associated withthe Lusin area integral and Littlewood−Paley g∗λ function. These operators are defined as follows
µσΨ,s(f )(x) =

(∫ ∫
Σ(x)

∣∣∣∣ 1

tσ

∫
|y1−z1|≤t

Ψ(y1, y1 − z1)

|y1 − z1|n−σ
f (z1)dz1

∣∣∣∣2 dy1dt

tn+1

) 1
2

and
µ∗,σΨ,λ(f )(x) =

(∫ ∫
Rn+1

+

(
t

t + |x1 − y1|

)λn ∣∣∣∣ 1

tσ

∫
|y1−z1|≤t

Ψ(y1, y1 − z1)

|y1 − z1|n−σ
f (z1)dz1

∣∣∣∣2 dy1dt

tn+1

) 1
2

,

where Σ(x) = {(y1, t) ∈ Rn+1
+ : |x1 − y1| < t and λ > 1}.

Received: 17 Jul 2025.
Key words and phrases. Littlewood−Paley operator; variable kernel; Herz space; variable exponent..1

https://adac.ee
https://doi.org/10.28924/ada/ma.5.22
https://orcid.org/0000-0002-6391-4243
https://orcid.org/0000-0003-0663-068X


Eur. J. Math. Anal. 10.28924/ada/ma.5.22 2The parameterized Littlewood−Paley operators µσΨ,s and µ∗,σΨ,λ were initially investigated bySakamoto and Yabuta in [1]. They proved that if Ψ ∈ Libβ(Sn−1) and 1 < p <∞, then µ∗,σΨ,λ and
µσΨ,s operators are bounded on Lp(Rn) space. Xue and Ding [2] established sharp Lp(w) boundedfor these operators (µ∗,σΨ,λ , µσΨ,s) in terms of the Aq characteristic of w , under the condition
Ψ ∈ Lb(Sn−1). Deringoz, Guliyev and Ragusa [3] obtained the boundedness of intrinsic squarefunctions and their commutators in the framework of Morrey-Orlicz spaces. The boundedness ofparametric Littlewood−Paley operators on Musielak−Orlicz Hardy spaces was further studiedin [4].As is well known, over the past thirty years, variable kernel integral operators have become anincreasingly active area of research. For example, Tao et al. [5] obtained the Lp(Rn) boundednessof variable kernel fractional integral operators TΨ,α, Chen and Ding [6] proved the Lp(Rn) bound-edness of variable kernel Littlewood−Paley operators, Shao [7] investigated the weighted estimatesfor variable kernel fractional integrals and their commutators on generalized Morrey spaces, in [8]the author proved the boundedness properties of Marcinkiewicz integral operator µΨ with variablekernel on the Hardy space Hp(Rn). Recently, Abdalmonem et. al. [9] obtained the boundedness oflittlewood−paley operators with variable kernel on the weighted variable herz-morrey spaces.Moreover, variable exponents Herz spaces have been extensively studied by many authors us-ing different methods( [14–20, 24, 25]). Izuki [23] defined the variable exponent homogeneous Herzspace K̇α,q

p(·)(Rn) and investigated the boundedness of some integral operators on these spaces.Wang [20] considered the boundedness results for certain rough kernel littlewood−paley opera-tors in homogeneous and homogeneous Herz spaces K̇α,q(·)
p(·) (Rn). In [21] the authors studied theboundedness of the vector-valued inequality for the intrinsic square function in variable exponentshomogeneous Herz spaces K̇α(·),q

p(·) (Rn). Izuki and Noi [12] considered the generalized Herz spaces
K̇
α(·)
q(·),p(·)(Rn) and obtained some boundedness results for integral operators and their commutatorson those spaces. In [13] the author established the boundedness properties of the rough kernelfractional integral operators in K̇α(·)

q(·),p(·)(Rn) spaces.Motivated by the work of [9, 13, 19],this paper discusses the boundedness of variable kernelparameterized Littlewood-Paley operators on homogeneous Herz spaces K̇α(·)
q(·),p(·)(Rn) with threevariable exponents. The results are also new for the case when α(·) is constant.

2. Mathematical background
Consider a Lebesgue measurable set E ⊂ Rn with positive measure |E| > 0. Denote by χE thecharacteristic function of E. In this paper, C represents a positive constant that may vary betweenoccurrences. We write g . f means g ≤ Cf , for some constant C > 0.

Definition 2.1 ( [22]). (variable Lebesgue space ) Suppose that p(·) : Γ→ [1,∞) is a measurablefunction. The Lp(·)(Γ) space is defined by

https://doi.org/10.28924/ada/ma.5.22


Eur. J. Math. Anal. 10.28924/ada/ma.5.22 3

Lp(·)(Γ) =

{
g is measurable :

∫
Γ

(
|g(x)|
β

)p(x)

dx <∞ for some constant β > 0

}
.

The local Lp(·)
loc (Γ) space is defined as
L
p(·)
loc (Γ) = {g is measurable : g ∈ Lp(·)(K) for any compact set K ⊂ Γ}.

With the given norm, the Lebesgue space Lp(·)
loc (Γ) is a Banach space

‖f ‖Lp(·)(Γ) = inf

{
η > 0 :

∫
E

(
|g(x)|
β

)p(x)

dx ≤ 1

}
.

Let p− = ess inf{p(x) : x ∈ Γ}, p+ = ess sup{p(x) : x ∈ Γ} denote the essential infimum andsupremum of p(Γ), respectively. P(Γ) represents the collection of all measurable functions p(·)with p− > 1. p+ < +∞. P0(Γ) consists of all measurable functions p(·) such that p− > 0and p+ < +∞. Furthermore, B(Rn) is defined as the subset of p(·) ∈ P(Rn) for which theHardy−Littlewood maximal operator M∗ is bounded in variable Lp(·) space.We know that, if p(·) ∈ P(Rn), then the operator M∗,
M∗g(x) = sup

B⊆Rn,B3x

1

|B|

∫
B

|g(y)|dy,

is bounded in variable Lp(·) space [24], where M∗ denotes the Hardy−Littlewood maximal operator.Let us now recall the definition of Herz space K̇α(·),q(·)
p(·) (Rn). Let Bk = {y ∈ Rn : |y | ≤ 2k},

k ∈ Z, Ck = Bk\Bk−1, χCk = χk .

Definition 2.2 ( [12]). Let α(·) : Rn −→ R, −∞ < α− ≤ α+ < ∞ and q(·), p(·) ∈ P(Rn). Thehomogeneous variable exponents Herz K̇α(·),q(·)
p(·) (Rn) space is defined by

K̇
α(·),q(·)
p(·) (Rn) = {f ∈ Lp(·)

loc (Rn\{0}) : ‖f ‖
K̇
α(·),q(·)
p(·) (Rn)

<∞},

where
‖f ‖

K̇
α(·),q(·)
p(·) (Rn)

:=
∥∥∥{2kα(·)|f χk |}∞k=−∞

∥∥∥
lq(·)(Lp(·))

= inf

{
β > 0 :

∞∑
k=−∞

∥∥∥∥∥
(

2kα(·)|f χk |
β

)q(·)∥∥∥∥∥
L
p(·)
q(·)

≤ 1

}
.

The nonhomogeneous variable exponents Herz K̇α(·),q(·)
p(·) (Rn) space is defined by

K
α(·),q(·)
p(·) (Rn) = {f ∈ Lp(·)

loc (Rn\{0}) : ‖f ‖
K
α(·),q(·)
p(·) (Rn)

<∞},

where
‖f ‖

K
α(·),q(·)
p(·) (Rn)

:=
∥∥∥{2kα(·)|f χk |}∞k=0

∥∥∥
lq(·)(Lp(·))

= inf

β > 0 :

∞∑
k=0

∥∥∥∥∥∥
(

2kα(·)|f χk |
β

)q(·)
∥∥∥∥∥∥
L
p(·)
q(·)

≤ 1

 .

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Eur. J. Math. Anal. 10.28924/ada/ma.5.22 4

Remark. (1) If
K̇
α(·),q(·)
p(·) (Rn) = K̇

α(·),q
p(·) (Rn),

then, q(·) is a constant.(2) If
K̇
α(·),q(·)
p(·) (Rn) = K̇α,q

p(·)(Rn),

then, both α(·), q(·) are constants.(3) If
K̇
α(·),q(·)
p(·) (Rn) = K̇α,qp (Rn),

then, α(·), p(·), q(·) are all constants.(4) Moreover, if p(·) = q(·) and α(·) = 0 , then
K̇
α(·),q(·)
p(·) (Rn) = Lp(·)(Rn).

Next, we present some key lemmas needed to prove our main theorems.
Lemma 2.3 ( [22]). (Generalized Hölder’s inequality) Let f ∈ Lp1(·)(Rn), g ∈ Lp

′
1(·)(Rn), and

p(·) ∈ P(Rn). Then, the following inequality is satisfied:∫
Rn
|g(x)f (x)|dx ≤ C‖g‖

Lp
′
1(·)(Rn)

‖f ‖Lp1(·)(Rn),

here C = 1− 1
p+

+ 1
p−

.

Lemma 2.4 ( [23]). Suppose p(·) ∈ B(Rn). For a given C > 0, the following inequality is satisfied:

C ≥
1

|B|‖χB‖Lp1(·)(Rn)‖χB‖Lp′1(·)(Rn)
,

here B ⊂ Rn.

Lemma 2.5 ( [23]). Suppose p(·) ∈ B(Rn). For n = 1, 2, there are constants δn1, δn2 > 0 for which
the following inequalities hold:

‖χB‖Lp(·)(Rn)

‖χS‖Lp(·)(Rn)

.
|B|
|S| ,

‖χS‖Lp′1(·)(Rn)

‖χB‖Lp′1(·)(Rn)

.

(
|S|
|B|

)δn1

,
‖χS‖Lp1(·)(Rn)

‖χB‖Lp1(·)(Rn)

.

(
|S|
|B|

)δn2

,

here B ⊂ Rn, S ⊂ B.

Lemma 2.6 ( [20]). Let p1(·), q1(·) ∈ P0(Rn), g ∈ Lp1(·)q1(·)(Rn), and 0 < q− ≤ p1(·) ≤ q+. Then,
we have

min(‖g‖q+

Lp1(·)q1(·) , ‖g‖
q−
Lp1(·)q1(·) ) ≤ ‖|g|q1(·)‖Lp1(·) ≤ max(‖g‖q+

Lp1(·)q1(·) , ‖g‖
q−
Lp1(·)q1(·) ).

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Eur. J. Math. Anal. 10.28924/ada/ma.5.22 5

Lemma 2.7 ( [10]). Suppose that α(·) ∈ L∞(Rn) and r0 > 0. If α(·) be a function that is log-
Hölder continuous both both at the origin and at infinity, then for any x ∈ B(0, r0) \ B(0, r0/2),
x ′ ∈ B(0, r1) \ B(0, r1/2), we have

r
α(x)
0 . rα(x ′)

1 ×


[ r0r1 ]α+ , 0 < r1 ≤ r0/2,

1, r0/2 < r1 ≤ 2r0,

[ r0r1 ]α− , r1 > 2r0.

3. Boundedness of the parameterized Littlewood-Paley operators
In this section, we discuss the boundedness of variable kernel parameterized Littlewood-Paleyoperators on homogeneous Herz spaces K̇α(·)

q(·),p(·)(Rn). The results are also new for the case when
α(·) is constant.Let 1 < q <∞, q′ = q

q−1 and w be a weight. For every cube Q ⊆ Rn, we say w ∈ Aq if thereexists C > 0, the following inequality is satisfied:(
1

|Q|

∫
Q

w(x)dx

)(
1

|Q|

∫
Q

w(x)1−q′dx

)q−1

≤ C <∞.

Xue et al. [2] proved the following Lp−boundedness of µσΨ,s and µ∗,σΨ,λ.
Lemma 3.1 ( [2]). Let 1 < p < ∞ and Ψ ∈ L∞(Rn) × L2(Sn−1) satisfies (1) and (2). Then, we
have

‖µσΨ,s f ‖Lp(w) . ‖f ‖Lp(w)

and
‖µ∗,σΨ,λf ‖Lp(w) . ‖f ‖Lp(w).

Lemma 3.2 ( [21]). Given a family of functions F , if for some p1, 1 < p1 < ∞, p1 ≤ p− and
(p(·)
p1

)′ ∈ B(E) and every w1 ∈ Ap1 ,∫
Rn

f1(x)p1w1(x)dx .
∫
Rn

g1(x)p1w1(x)dx, (f , g) ∈ F .

If p(·) ∈ P(E) and f1 ∈ Lp(·)(E), then for all (f1, g1) ∈ F ,

‖f1‖Lp(·)(E) . ‖g1‖Lp(·)(E).

Since Aq/s ′ ⊂ A∞, using Lemma 3.1 and Lemma 3.2, its simple to obtain the Lp(·)-boundednessof µσΨ,s and µ∗,σΨ,λ.
Theorem 3.3. Assume that p1(·) ∈ B(Rn), q1(·), q2(·) ∈ P(Rn), λ > 2, 2σ − n > 0, and Ψ ∈
L∞(Rn) × L2(Sn−1) satisfies (1) and (2). Let α(·) ∈ L∞(Rn) be a function that is log-Hölder
continuous both at the origin and at infinity, such that

−nδ11 < α− ≤ α+ < nδ12,

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Eur. J. Math. Anal. 10.28924/ada/ma.5.22 6

where δn1, δn2 (n = 1, 2) are the same as in Lemma 2.4. Then, µσΨ,s operator is bounded from
K̇
α(·)
p1(·),q1(·)(Rn) to K̇α(·)

p1(·),q2(·)(Rn) for all f ∈ K̇α(·)
p1(·),q1(·)(Rn).

Theorem 3.4. Assume that p1(·) ∈ B(Rn), q1(·), q2(·) ∈ P(Rn), λ > 2, 2σ − n > 0, and Ψ ∈
L∞(Rn) × L2(Sn−1) satisfies (1) and (2). Let α(·) ∈ L∞(Rn) be a function that is log-Hölder
continuous both at the origin and at infinity, such that

−nδ11 < α− ≤ α+ < nδ12,

where δn1, δn2 (n = 1, 2) are the same as in Lemma 2.4. Then, µ∗,σΨ,λ operator is bounded from
K̇
α(·)
p1(·),q1(·)(Rn) to K̇α(·)

p1(·),q2(·)(Rn) for all f ∈ K̇α(·)
p1(·),q1(·)(Rn).

Before proving Theorems, we first establish a necessary inequality.
Remark. Let 1 ≤ pm <∞, am ≥ 0, m ∈ N. We have

∞∑
m=0

apmm ≤

( ∞∑
m=0

am

)p•
,

here
p• =


min
m∈N

pm if ∞∑
m=0

am ≤ 1,

max
m∈N

pm if ∞∑
m=0

am > 1.

Remark. From ( [11], p.89]), we recall the estimate µσΨ,s f (x) ≤ 2nλµ∗,σσ,λf (x). Therefore, we presentonly the proof of Theorem 3.4.
Proof. We present the proof of K̇α(·),q(·)

p(·) (Rn) (homogeneous case). The same argument holds truefor Kα(·),q(·)
p(·) (Rn) (nonhomogeneous case).Let f ∈ K̇α(·),q1(·)

p1(·) (Rn). Decomposet f as:
f (x) =

∞∑
j=−∞

f (x)χj(x) =

∞∑
j=−∞

fj(x).

From homogeneous K̇α(·),q(·)
p(·) (Rn) (Definition 2.2), we have

‖µ∗,σΨ,λ(f )‖
K̇
α(·),q2(·)
p1(·) (Rn)

= inf

η > 0 :

∞∑
k=−∞

∥∥∥∥∥∥
(

2kα(·)|µ∗,σΨ,λ(f )χk |
β

)q2(·)
∥∥∥∥∥∥
L

p1(·)
q2(·)

≤ 1

 .
We have ∥∥∥∥∥

(
2kα(·)|µ∗,σΨ,λ(f )χk |

β

)q2(·)
∥∥∥∥∥
L

p1(·)
q2(·)

≤

∥∥∥∥∥∥∥
2kα(·)|

∞∑
j=−∞

µ∗,σΨ,λ(fj )χk |

β01+β02+β03

q2(·)∥∥∥∥∥∥∥
L

p1(·)
q2(·)

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Eur. J. Math. Anal. 10.28924/ada/ma.5.22 7

≤ C

∥∥∥∥∥∥∥∥
2kα(·)|

k−2∑
j=−∞

µ∗,σΨ,λ(fj )χk |

β01


q2(·)

∥∥∥∥∥∥∥∥
L

p1(·)
q2(·)

+ C

∥∥∥∥∥∥∥∥
2kα(·)|

k+1∑
j=k−1

µ∗,σΨ,λ(fj )χk |

β02


q2(·)

∥∥∥∥∥∥∥∥
L

p1(·)
q2(·)

+C

∥∥∥∥∥∥∥
2kα(·)|

∞∑
j=k+2

µ∗,σΨ,λ(fj )χk |

β03

q2(·)∥∥∥∥∥∥∥
L

p1(·)
q2(·)

,

where
β01 =

∥∥∥∥∥
{

2kα(·)|
k−2∑
j=−∞

µ∗,σΨ,λ(fj)χk |

}∞
k=−∞

∥∥∥∥∥
lq2(·)(Lp1(·))

,

β02 =

∥∥∥∥∥
{

2kα(·)|
k+1∑
j=k−1

µ∗,σΨ,λ(fj)χk |

}∞
k=−∞

∥∥∥∥∥
lq2(·)(Lp1(·))

,

β03 =

∥∥∥∥∥
{

2kα(·)|
∞∑

j=k+2

µ∗,σΨ,λ(fj)χk |

}∞
k=−∞

∥∥∥∥∥
lq2(·)(Lp1(·))

.

If β0 = β01 + β02 + β03 thus
∞∑

k=−∞

∥∥∥∥∥∥
(

2kα(·)|µ∗,σΨ,λ(fj)χk |
β0

)q2(·)
∥∥∥∥∥∥
L

p1(·)
q2(·)

. 1.

Then
‖µ∗,σΨ,λ(f )χk‖K̇α(·),q1(·)

p1(·) (Rn)
. β0 . [β01 + β02 + β03].Therefore, if we can conclude that

β01 ≤ C‖f ‖K̇α(·),q1(·)
p1(·) (Rn)

, β02 ≤ C‖f ‖K̇α(·),q1(·)
p1(·) (Rn)

, β03 ≤ C‖f ‖K̇α(·),q1(·)
p1(·) (Rn)

,

we are finished. Let us set β0 = ‖f ‖
K̇
α(·),q1(·)
p1(·) (Rn)

.First, we estimate β02. From Lemma 2.6 and Lemma 2.7, we have
∞∑

k=−∞

∥∥∥∥∥∥∥∥∥∥


2kα(·)|
k+1∑
j=k−1

µ∗,σΨ,λ(fj)χk |

β0


q2(·)∥∥∥∥∥∥∥∥∥∥

L

p1(·)
q2(·)

.
∞∑

k=−∞

∥∥∥∥∥∥∥∥∥
2kα(·)|

k+1∑
j=k−1

µ∗,σΨ,λ(fj)χk |

β0

∥∥∥∥∥∥∥∥∥
(q0

2 )k

Lp1(·)

.
∞∑

k=−∞

∥∥∥∥∥∥∥∥∥
|
k+1∑
j=k−1

µ∗,σΨ,λ(2αj fj)χk |

β0

∥∥∥∥∥∥∥∥∥
(q0

2 )k

Lp1(·)

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Eur. J. Math. Anal. 10.28924/ada/ma.5.22 8

.
∞∑

k=−∞

 k+1∑
j=k−1

∥∥∥∥∥ |µ∗,σΨ,λ(2αj fj)χk |
β0

∥∥∥∥∥
Lp1(·)

(q0
2 )k

,

where
(q0

2)k =


(q2)+

∥∥∥∥∥∥∥∥
2kα(·)|

k+1∑
j=k−1

µ∗,σΨ,λ(fj )χk |

β0


q2(·)

∥∥∥∥∥∥∥∥
L
p1(·)
q2(·)

≥ 1,

(q2)− otherwise.

By the boundedness of µ∗,σΨ,λ on Lp(·), we have
∞∑

k=−∞

∥∥∥∥∥∥∥∥∥∥


2kα(·)|
k+1∑
j=k−1

µ∗,σΨ,λ(fj)χk |

β0


q2(·)∥∥∥∥∥∥∥∥∥∥

L

p1(·)
q2(·)

.
∞∑

k=−∞

 k+1∑
j=k−1

∥∥∥∥∥ |(2jα(·)fj)|
β0

∥∥∥∥∥
Lp1(·)

(q0
2 )k

,

.
∞∑

k=−∞

∥∥∥∥∥∥
(

2kα(·)|fk |
β0

)q1(·)
∥∥∥∥∥∥

(q1
2 )k

(q1)+

Lp1(·)

Lq1(·)

.

 ∞∑
k=−∞

∥∥∥∥∥∥
(

2kα(·)|fk |
β0

)q1(·)
∥∥∥∥∥∥
Lp1(·)

Lq1(·)


q•

. 1,

Here q• = min
k∈N

(q0
2 )k

(q1)+
≥ 1.The previous calculations imply that

β02 . β0 . ‖f ‖K̇α(·),q1(·)
p1(·) (Rn)

.

We must examine µ∗,σΨ,λfj . By applying the Minkowski inequality, we have
|µ∗,σΨ,λ(fj)(x)|

=

(∫ ∞
0

∫
Rn

(
(t)(t + |x1 − y1|)−1

)λn ∣∣∣∣ 1

tσ

∫
|y1−z1|≤t

Ψ(y1, y1 − z1)

|y1 − z1|n−σ
fj(z1)dz1

∣∣∣∣2 dy1dt

tn+1

) 1
2

≤
∫
Rn
fj(z1)

(∫ ∞
0

∫
|y1−z1|≤t

(
(t)(t + |x1 − y1|)−1

)λn |Ψ(y1, y1 − z1)|2

|y1 − z1|2n−2σ

dy1dt

t2σ+n+1

) 1
2

dz1

≤
∫
Rn
fj(z1)

(∫ |x1−z1|

0

∫
|y1−z1|≤t

(
(t)(t + |x1 − y1|)−1

)λn |Ψ(y1, y1 − z1)|2

|y1 − z1|2n−2σ

dy1dt

t2σ+n+1

) 1
2

dz1

https://doi.org/10.28924/ada/ma.5.22


Eur. J. Math. Anal. 10.28924/ada/ma.5.22 9

+

∫
Rn
fj(z1)

(∫ ∞
|x1−z1|

∫
|y1−z1|≤t

(
(t)(t + |x1 − y1|)−1

)λn |Ψ(y1, y1 − z1)|2

|y1 − z1|2n−2σ

dy1dt

t2σ+n+1

) 1
2

dz1.

Let 2ρ− n > 0 and Ψ ∈ L∞(Rn)× L2(Sn−1). Then, the following inequality is satisfied∫
|y1−z1|≤t

|Ψ(y1, y1 − z1)|2

|y1 − z1|2n−2σ
dy1 ≤

∫
sn−1

∫ t

0

|Ψ(sy ′1 + z1, y
′
1)|2

s2n−2σ
sn−1dsdσ(y ′1)

. ‖Ψ‖2
L∞(Rn)×L2(Sn−1)t

2σ−n.

Because |x1− z1| ≤ |y1− z1|+ |x1− y1| ≤ |x1− y1|+ t , for λ > 2 and 0 < ε < (λ− 2)n, we obtain∫ |x1−z1|

0

∫
|y1−z1|≤t

(
(t)(t + |x1 − y1|)−1

)λn |Ψ(y1, y1 − z1)|2

|y1 − z1|2n−2σ

dy1dt

t2σ+n+1

.
∫ |x1−z1|

0

∫
|y1−z1|≤t

(
(t)(t + |x1 − y1|)−1

)λn−2n−ε 1

|x1 − z1|2n+ε

|Ψ(y1, y1 − z1)|2

|y1 − z1|2n−2σ

dy1dt

t2σ−n−ε+1

.
1

|x1 − z1|2n+ε

∫ |x1−z1|

0

∫
|y1−z1|≤t

|Ψ(y1, y1 − z1)|2

|y1 − z1|2n−2σ

dy1dt

t2σ−n−ε+1

.
‖Ψ‖2

L∞(Rn)×L2(Sn−1)

|x1 − z1|2n+ε

∫ |x1−z1|

0

tε−1dt

. |x1 − z1|−2n.

Let λ0n − 2n < 0, λ0n − n > 0 and 1 < λ0 < 2. Then, we get∫ ∞
|x1−z1|

∫
|y1−z1|≤t

(
(t)(t + |x1 − y1|)−1

)λn |Ψ(y1, y1 − z1)|2

|y1 − z1|2n−2σ

dy1dt

t2σ+n+1

.
∫ ∞
|x1−z1|

∫
|y1−z1|≤t

|x1 − z1|−λ0n
|Ψ(y1, y1 − z1)|2

|y1 − z1|2n−2σ

dy1dt

t2σ−λ0n+n+1

.

∫ ∞
|x1−z1|

|x1 − z1|−λ0n

∫
|y1−z1|≤t

|Ψ(y1, y1 − z1)|2

|y1 − z1|2n−λ0n

dy1dt

tn+1

.
∫ ∞
|x1−z1|

|x1 − z1|−λ0n

∫
sn−1

∫ t

0

|Ψ(y ′1, (y1 − z1)′)|2

s2n−λ0n
sn−1dsdσ(y ′1)

dt

tn+1

. ‖Ψ‖2
L∞(Rn)×L2(Sn−1)|x1 − z1|−λ0n

∫ ∞
|x1−z1|

tλ0n−2n−1dt

. |x1 − z1|−2n.

By combining these estimates, we find
µ∗,σΨ,λ(f )(x) .

∫
Rn

|fj(z1)|
|x1 − z1|n

dz1. (∗)

Next, we consider β01. Since j ≤ k − 2, by applying Hölder’s inequality (Lemma 2.3), we obtain
µ∗,σΨ,λ(f )(x) .

∫
Rn

|fj(z1)|
|x1 − z1|n

dz

. 2−kn‖χj‖Lp′1(·)(Rn)
‖fj‖Lp1(·)(Rn).

https://doi.org/10.28924/ada/ma.5.22


Eur. J. Math. Anal. 10.28924/ada/ma.5.22 10Then, by Lemmas 2.5 - 2.7, we deduce that
∞∑

k=−∞

∥∥∥∥∥∥∥∥∥∥


2kα(·)|
k−2∑
j=−∞

µ∗,σΨ,λ(fj)χk |

β0


q2(·)∥∥∥∥∥∥∥∥∥∥

L

p1(·)
q2(·)

.
∞∑

k=−∞

∥∥∥∥∥∥∥∥∥
2kα(·)|

k−2∑
j=−∞

µ∗,σΨ,λ(fj)χk |

β0

∥∥∥∥∥∥∥∥∥
(q01

2 )k

Lp1(·)

.
∞∑

k=−∞

2kα(·)
k−2∑
j=−∞

∥∥∥∥ fjβ0

∥∥∥∥
Lp1(·)(Rn)

‖χBj‖Lp′1(·)‖χBk‖Lp1(·) 2−kn

(q01
2 )k

.
∞∑

k=−∞

2kα(·)
k−2∑
j=−∞

∥∥∥∥ fjβ0

∥∥∥∥
Lp1(·)(Rn)

‖χBj‖Lp′1(·)

‖χBk‖Lp′1(·)

(q01
2 )k

.
∞∑

k=−∞

2kα(·)
k−2∑
j=−∞

2(j−k)nδ11

∥∥∥∥ fjβ0

∥∥∥∥
Lp1(·)(Rn)

(q01
2 )k

.
∞∑

k=−∞

 k−2∑
j=−∞

2(k−j)(−nδ11+α+)

∥∥∥∥∥∥
(
|2jα(·)f χj |

β0

)q1(·)
∥∥∥∥∥∥

1
(q1)+

Lp1(·)q1(·)(Rn)


(q01

2 )k

,

where
(q01

2 )k =


(q2)−

∥∥∥∥∥∥∥∥
2kα(·)|

k−2∑
j=−∞

µ∗,σΨ,λ(fj )χk |

β0


q2(·)

∥∥∥∥∥∥∥∥
L

p1(·)
q2(·)

≥ 1,

(q2)+ otherwise.

If (q1)+ 6 1, then by Lemma 2.6, we have
∞∑

k=−∞

∥∥∥∥∥∥∥∥∥∥


2kα(·)|
k−2∑
j=−∞

µ∗,σΨ,λ(fj)χk |

β0


q2(·)∥∥∥∥∥∥∥∥∥∥

L

p1(·)
q2(·)

.
∞∑

k=−∞

 k−2∑
j=−∞

2(k−j)(q1)+(α+−nδ11)

∥∥∥∥∥∥
(
|2jα(·)f χj |

β0

)q1(·)
∥∥∥∥∥∥
Lp1(·)q1(·)(Rn)


(q01

2 )k
(q1)+

https://doi.org/10.28924/ada/ma.5.22


Eur. J. Math. Anal. 10.28924/ada/ma.5.22 11

.

 ∞∑
j=−∞

∥∥∥∥∥∥
(
|2jα(·)f χj |

β0

)q1(·)
∥∥∥∥∥∥
Lp1(·)q1(·)(Rn)

∞∑
k=j+2

2(k−j)(q1)+(α+−nδ11)


q•

. 1,

here q• = min
k∈N

(q01
2 )k

(q1)+
.If (q1)+ > 1, using Hölder’s inequality, we deduce that

∞∑
k=−∞

∥∥∥∥∥∥∥∥∥∥


2kα(·)|
k−2∑
j=−∞

µ∗,σΨ,λ(fj)χk |

β0


q2(·)∥∥∥∥∥∥∥∥∥∥

L

p1(·)
q2(·)

.
∞∑

k=−∞

 k−2∑
j=−∞

2(k−j)(α+−nδ11)
(q1)+

2

∥∥∥∥∥∥
(
|2jα(·)f χj |

β0

)q1(·)
∥∥∥∥∥∥
Lp1(·)q1(·)(Rn)


(q01

2 )k

(q1)+

×

 k−2∑
j=−∞

2(k−j)(α+−nδ11)
((q1)+)′

2


(q01

2 )+

((q1)+)′

.

 ∞∑
j=−∞

∥∥∥∥∥∥
(
|2jα(·)f χj |

β0

)q1(·)
∥∥∥∥∥∥
Lp1(·)q1(·)(Rn)

∞∑
k=j+2

2(k−j)(α+−nδ11)
(q1 )+

2


q•

. 1.

Therefore, we conclude the estimate
β01 . β0 . ‖f ‖K̇α,q1(·)

p1(·) (Rn)
.

Finally, we estimate β03. Since j ≥ k + 2, by (∗) and applying Hölder’s inequality (Lemma 2.3), we have
µ∗,σΨ,λ(f )(x) .

∫
Rn

|fj(z1)|
|x1 − z1|n

dz1

. 2−jn‖χj‖Lp′1(·)(Rn)
‖fj‖Lp1(·)(Rn).

Then, by Lemma2.5, Lemma2.6 and Lemma2.7, we get
∞∑

k=−∞

∥∥∥∥∥∥∥∥∥
2kα(·)|

∞∑
k+2

µ∗,σΨ,λ(fj)χk |

β0


q2(·)∥∥∥∥∥∥∥∥∥

L

p1(·)
q2(·)

.
∞∑

k=−∞

∥∥∥∥∥∥∥∥∥
2kα(·)|

∞∑
j=k+2

µ∗,σΨ,λ(fj)χk |

β0

∥∥∥∥∥∥∥∥∥
(q02

2 )k

Lp1(·)

https://doi.org/10.28924/ada/ma.5.22


Eur. J. Math. Anal. 10.28924/ada/ma.5.22 12

.
∞∑

k=−∞

2kα(·)
∞∑

j=k+2

2−jn‖χBj‖Lp′1(·)‖χBk‖Lp1(·)

∥∥∥∥ fjβ0

∥∥∥∥
Lp1(·)(Rn)

(q02
2 )k

.
∞∑

k=−∞

2kα(·)
∞∑

j=k+2

2−jn
‖χBk‖Lp1(·)

‖χBj‖Lp1(·)
|Bj |

∥∥∥∥ fjβ0

∥∥∥∥
Lp1(·)(Rn)

(q02
2 )k

.
∞∑

k=−∞

 ∞∑
j=k+2

2(k−j)(nδ12+α−)

∥∥∥∥∥∥
(

2jα(·)f χj
β0

)q1(·)
∥∥∥∥∥∥

1
(q1)+

Lp1(·)q1(·)


(q02

2 )k

,

where
(q02

2 )k =


(q2)−

∥∥∥∥∥∥∥
2kα(·)|

∞∑
j=k+2

µ∗,σΨ,λ(fj )χk |

β0

q2(·)∥∥∥∥∥∥∥
L

p1(·)
q2(·)

≥ 1,

(q2)+, otherwise.Notice that α− > −nδ12, and applying the same estimation technique used for β01, we get that
β03 . β0 . ‖f ‖K̇α,q1(·)

p1(·) (Rn)
.

Thus, the theorem 3.4 is proven. �

Author Contributions: All authors contributed equally to the writing of this paper. All authors readand approved the final manuscript.
Conflicts of Interest: The authors declare that there is no conflict of interest regarding the publi-cation of this article.

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http://eudml.org/doc/126704

	1. Introduction
	2. Mathematical background
	3. Boundedness of the parameterized Littlewood-Paley operators 
	Author Contributions:
	Conflicts of Interest:

	References

