EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 4, Article Number 6667
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Some New Weighted Results for the Hardy Operator on
Variable-Exponent λ-central Morrey Space

Muhammad Asim1, Khaled Suwais2, Nabil Mlaiki3,∗

1 Department of NUSASH, National University of Technology (NUTECH), Islamabad 44000,
Pakistan
2 Faculty of Computer Studies, Arab Open University, Riyadh 11681, Saudi Arabia
3 Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586,
Saudi Arabia

Abstract. The main purpose of this study is to demonstrate that the fractional Hardy operators
are bounded on the weighted variable central Morrey space. When the symbol functions belong
to the λ-central BMO space with a variable exponent, the estimates for their commutators are
similar.
2020 Mathematics Subject Classifications: 42B35, 26D10, 47B38, 47G10
Key Words and Phrases: Fractional operators, weighted Morrey space, variable exponent,
integral operators

1. Introduction

Let L1
loc(Rn) denote the set consisting of all complex-valued integrable functions on

Rn, In [1], the Hardy operator is defined as follows:

Hf(x) =
1

x

∫ x

0
f(t)dt, x > 0,

which was generalized by Faris [2] to n-dimensional Euclidean space and defined as:

Hf(x) =
1

vn|x|n

∫
|y|<|x|

f(y)dy,

where x ∈ Rn \ {0}, f ∈ L1
loc(Rn), and vn represents the volume of the unit ball. The

n-dimensional fractional Hardy operator Hα and its dual operator are defined as follows:

Hαf(x) = |x|α−n

∫
|y|<|x|

f(y)dy, H∗
αf(x) =

∫
|y|>|x|

|y|α−nf(y)dy.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6667

Email addresses: masim@math.qau.edu.pk (M. Asim),
khaled.suwais@arabou.edu.sa (K. Suwais), nmlaiki@psu.edu.sa (N. Mlaiki)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



M. Asim, K. Suwais, N. Mlaiki / Eur. J. Pure Appl. Math, 18 (4) (2025), 6667 2 of 20

Let b be a locally integrable function, and define the BMO norm as follows:

∥b∥BMO = sup
B

1

|B|

∫
B
|b(x)− bB|dx,

where the supremum is taken over all balls B in Rn, and bB is the average of b on B. A
function is called bounded mean oscillation if ∥b∥BMO < +∞. If b ∈ BMO(Rn), then the
commutator generated by the Hardy operator is defined as:

Hbf(x) = b(x)Hf(x)−H(bf)(x).

The commutator for the n-dimensional fractional Hardy operator Hα and their dual
operator H∗

α are defined in [3] as follows:

Hα,bf(x) = b(x)Hαf(x)−Hα(bf)(x)

H∗
α,bf(x) = b(x)H∗

αf(x)−H∗
α(bf)(x).

In real analysis, variable exponents function spaces are being observed with keen
interest. The theory of variable exponents has great applications in partial differential
equations and applied mathematics because they are used in image restoration and the
modeling of electrorheological fluids. Some basic properties for variable exponent function
spaces were defined by Kovacik and Rakosnik [4]. The boundedness of Hardy operators
on some function spaces is one of the main problems in this theory. The boundedness
of the Hardy operator on Lebesgue spaces and Sobolev spaces with variable exponents
can be verified in [5–7]. The boundedness of the commutator of the Hardy operator on
λ-central Morrey space and the variable exponent Herz space K̇α,p

q(·) is discussed in [8–13].
More generalized results, incorporating weights and variable exponents, can be found in
[14–29]. Some of these results include duality, boundedness of the Littlewood maximal
Hardy operator and sublinear operator, wavelet characterization, commutator of fractional
and singular integral, and more.

In this paper, our main focus is on the weighted λ-central Morrey space, which
holds importance in the theory of Harmonic analysis. The idea of λ-central Morrey space
and λ-central BMO space for variable exponents was initially discussed in [30]. Cen-
tral Morrey space, central BMO, and their corresponding function spaces have delightful
applications in exploring results for operators, including singular integral operators, as ex-
plained in [31–38]. In this article, we utilize the Reisz type potential operator to establish
the boundedness of the Hardy operator, defined as follows:

Iαf(x) =

∫
Rn

f(y)

|x− y|n−α
dy,

f is locally integrable function, 0 < α < n and x ∈ Rn \ {0}.
Permit me to elucidate the structural framework of the present manuscript. In

Section 3, a recapitulation shall be furnished of certain pivotal lemmas and propositions,
articulated within the analytical milieu of weighted Lebesgue space endowed with variable



M. Asim, K. Suwais, N. Mlaiki / Eur. J. Pure Appl. Math, 18 (4) (2025), 6667 3 of 20

exponents. Moving on to Section 4.1, we will establish the boundedness of fractional
Hardy operators in the weighted central Morrey space concerning variable exponents. In
Section 4.2, we will investigate the estimates of commutators induced by Hardy operators
and weighted λ-central BMO functions on the central Morrey space.

Additionally, |S| and χS represent the Lebesgue measure and characteristic function
of a measurable set S ⊂ Rn, respectively. When we write g ≈ h, it indicates the existence
of constants C1 and C2, both greater than zero, such that C1g ≤ h ≤ C2g. We define
Sj = S(0, 2j) = {x ∈ Rn : |x| ≤ 2j}, and for j ∈ Z, we set χj = χAj .

2. Preliminaries

According to some basic books and papers [4, 39–41], we have established the Lebesgue
space with a variable exponent. For a measurable function p(·) : Rn → [1,+∞) , the
Lebesgue space Lp(·)(Rn) with a variable exponent is a set containing complex valued
function g such that

Pp(g) =

∫
Rn

|g(x)|p(x)dx < +∞.

Lp(·) is a Banach function space with respect to the norm

∥g∥LP (·) = inf{ω > 0 : Pp(
g

ω
) ≤ 1}.

The set P represents a set consisting of all measurable function p(·) such that

p− = ess inf p(x) > 1 x ∈ Rn

+∞ > p+ = ess sup p(x) x ∈ Rn.

A function p(·) is said to be globally log Holder continuous (LH(Rn)) if it fulfills the
following conditions:

|p(y)− p(x)| ≲ 1

log(|y − x|)
, x, y ∈ Rn : |y − x| ≤ 1

2
(1)

|p∞ − p(x)| ≲ 1

log(e+ |x|)
, (2)

for some real number p∞.
The Hardy maximal Littlewood operator denoted by M, is defined as:

Mg(x) = sup
1

|B|

∫
B
g(t)dt,

where B is a ball and x ∈ B. It is known that M is bounded on Lp(·)(Rn), for p(·) ∈
P(Rn)

⋂
LH(Rn) [42–44].



M. Asim, K. Suwais, N. Mlaiki / Eur. J. Pure Appl. Math, 18 (4) (2025), 6667 4 of 20

Let w be a weight, and q(·) ∈ P(Rn). Then Lp(·)(w) represents the weighted variable
exponent Lebesgue spaces variable exponent, which contains all complex valued measur-

able function f such that fw
1

p(·) ∈ Lp(·)(w). Lp(·)(w) forms a Banach function space with
respect to the given norm

∥f∥Lp(·)(w) = ∥fw
1

p(·) ∥Lp(·) .

Since, p
′
(·) is the conjugate of p(·) and 1

p(·) +
1

p′ (·) = 1, we have the conjugate exponent

relationship.
We introduce the concept of a A1 Muckenhoupt weight, defined as follows:

Definition 1. Weight w is called a A1 Muckenhoupt weight if it fulfills the condition
Mw(x) ≲ w(x), x ∈ Rn.

Definition 2. Let p(·) ∈ P(Rn). A weight is said to be Ap(·) if it satisfies the condition:

sup
B

1

|B|
∥w

1
p(·)χB∥Lp(·)∥w− 1

p(·)χB∥Lp
′
(·) < +∞.

and a weight is called ˜Ap(·) if the following condition holds:

sup
B

1

|B|pB
∥wχB∥L1∥w−1χB∥

L
p
′
(·)

p(·)
< +∞.

Where pB =
(

1
|B|

∫
B

1
p(x)dx

)−1
.

Based on this definition Izuki and Noi [16] have proved the following monotone prop-
erty.

Definition 3. Let α ∈ (0, n) and p2(·), p1(·) ∈ P(Rn), and 1
p1(·) = 1

p2(·) +
α
n . A weight w

is known as A(p2(·), p1(·)) if it satisfies the following inequality for all balls B ⊂ Rn

∥w−1χB∥
1

1−α
n

Lp
′
1(·)

∥wχB∥
1

1−α
n

Lp2(·)
≤ |B|.

Definition 4. If p(·) ∈ P and λ ∈ R, the weighted Morrey space with a variable exponent
Ḃp(·),λ(wp(·)) is defined as:

Ḃp(·),λ(w) = {f ∈ L
p(·)
loc (w) : ∥f∥Ḃp(·),λ(w) < +∞},

where

∥f∥Ḃp(·),λ(w) = sup
R>0

∥fχB(0,R)∥Lp(·)(w)

|B(0, R)|λ∥χB(0,R)∥Lp(·)(w)

.

Definition 5. If p(·) ∈ P and λ < 1
n the weighted λ− BMO space with variable exponent

CBMOp(·),λ(wp(·)) is defined as:

CBMOp(·),λ(w) = {f ∈ L
p(·)
loc (w) : ∥f∥CBMOp(·),λ(w) < +∞},

where

∥f∥CBMOp(·),λ(w) = sup
R>0

∥(f − fB(0,R))χB(0,R)∥Lp(·)(w)

|B(0, R)|λ∥χB(0,R)∥Lp(·)(w)

.



M. Asim, K. Suwais, N. Mlaiki / Eur. J. Pure Appl. Math, 18 (4) (2025), 6667 5 of 20

3. Important Lemmas

Lemma 1. [16] If p1(·), p2(·) ∈ P(Rn)
⋂
LH(Rn) and p1(·) < p2(·), then

A1 ⊂ Ap1(·) ⊂ Ap2(·).

Lemma 2. [16] Let 0 < α < n and p2(·), p1(·) ∈ P(Rn) where 1
p1(·) = 1

p2(·) +
α
n . Then

w ∈ A(p1(·), p2(·)) ⇔ wp2(·) ∈ A
1+

p2(·)
p
′
1(·)

.

Lemma 3. [45] Consider Y is a Banach function space. Then
1. Y

′
will also be a Banach function space. Specifically, ∥ · ∥(Y′ )′ and ∥ · ∥Y are both

equivalent norms and satisfy (Y
′
)
′
= Y.

2. If f ∈ Y and g ∈ Y
′
, then we have∫

Rn

|f(x)g(x)|dx ≤ ∥f∥Y∥g∥Y′ ,

which is known as Hölder inequality.

Lemma 4. [45] Let Y be a Banach function space. Then

1 ≤ 1

|B|
∥χB∥Y∥χB∥Y′ .

This holds for all balls B.

Lemma 5. [46] If M is weakly bounded on a Banach function space Y, i.e,

∥χ{Mf>λ}∥Y ≤ λ−1∥f∥Y, (3)

where λ > 0 and f ∈ Y, then

sup
1

|B|
∥χB∥Y∥χB∥Y′ < +∞. (4)

Now we define weighted Banach function space and explain some of their properties.
For a function W (x) ∈ (0,+∞), W (x) ∈ Yloc and W−1(x) ∈ Y

′
loc, where Yloc(Rn) comprises

of all measurable functions f such that fχB ∈ Y for any compact set B with |B| < +∞.
Then the weighted Banach function space is defined as

Y(Rn,W ) = {f ∈ M : fW ∈ Y}.

Then the accompanying lemma is true.

Lemma 6. [47]
1. Y(Rn,W ) is a Banach function space with the given norm

∥f∥Y(Rn,W ) = ∥fW∥Y.

2. The associated space of the weighted Banach function space Y(Rn,W ) is also a
Banach function space .



M. Asim, K. Suwais, N. Mlaiki / Eur. J. Pure Appl. Math, 18 (4) (2025), 6667 6 of 20

Remark 1. Take q(·) ∈ P(Rn). Now, by the definition of the weighted Banach function

space Y(Rn,W ), Lq(·)(wq(·)) and Lq
′
(·)(w−q

′
(·)), we observe that

1. Let Y = Lq(·)(Rn) and W = w, then we have Lq(·)(Rn, w) = Lq(·)(wq(·)).

2. For Y = Lq
′
(·)(Rn) and W = w, we obtain Lq

′
(·)(Rn, w−1) = Lq

′
(·)(w−q

′
(·)). There-

fore, from lemma 6, we have

(Lq(·)(wq(·)))
′
= (Lq(·)(Rn, w−1))

′
= Lq

′
(·)(w−q

′
(·)).

Furthermore, define p1(·) so that 1
p1(·) =

α
n + 1

p2(·) . If p1(·) ∈ P ∩LH(Rn) and 0 < α < n
p1
,

now by applying the monotone property, we obtain wp1(·) ∈ A1 ⊂ A
1+

p2(·)
p
′
1(·)

. So, by lemma

2 we obtain w ∈ A(p1(·), p2(·)).

Lemma 7. [48] Consider a weight w on Rn. There exist p ∈ [1,+∞) such that w ∈ Ap,
then for any measurable set E subset of B, we have

w(B)

w(E)
≤ C

(
|B|
|E|

)p

w(E)

w(B)
≤ C

(
|E|
|B|

)δ

,

where 0 < δ < 1 represents a constant independent of E and B.

Lemma 8. [48] Let us consider p(·) ∈ P ∩ LH(Rn). If wp2(·) ∈ Ap2(·) and wp1(·) ∈
Ap1(·) imply w−p

′
2(·) ∈ A

p
′
2(·)

, w−p
′
1(·) ∈ A

p
′
1(·)

respectively . Thus, M is bounded on

Lp
′
2(·)(w−p

′
2(·)). There exists constants δ1, δ2 ∈ (0, 1) and E ⊂ B such that

∥χE∥Lp2(·)(wp2(·))

∥χB∥Lp2(·)(wp2(·))

=
∥χE∥

(Lp
′
2(·)w−p

′
2(·))′

∥χB∥
(Lp

′
2(·)w−p

′
2(·))′

≲

(
|E|
|B|

)δ1

, (5)

∥χE∥(Lp1(·)wp1(·))′

∥χB∥(Lp1(·)wp1(·))′
≲

(
|E|
|B|

)δ2

. (6)

Lemma 9. [16] Let p1(·) ∈ P ∩ LH(Rn) and 0 < α < n
p1(.)+

, and 1
p2(.)

= 1
p1(.)

− α
n . If

w ∈ A(p1(
.), p2(

.)), then Iα is bounded from Lp1(.)(wp1(.)) to Lp2(.)(wp2(.)).

Lemma 10. [49] If the Hardy Littlewood maximal operator is bounded on the Banach
function space Y, then for a measurable set E ⊂ B we have the following result

∥χB∥Y
∥χE∥Y

≲
|B|
|E|

.



M. Asim, K. Suwais, N. Mlaiki / Eur. J. Pure Appl. Math, 18 (4) (2025), 6667 7 of 20

4. Main Results and Their Proof

Lemma 11. If wq1(·) ∈ A1, where q1(·) ∈ P(Rn)
⋂
LH(Rn), define the variable exponent

q2(·) by
1

q2(x)
=

1

q1(x)
− α

n
,

then
∥χBk

∥Lq2(·)(wq2(·)) ≤ C2k(n−α)∥χBk
∥−1
(Lq1(·)(wq1(·)))′

.

Proof. Based on lemmas 9 and 5, we have

Iα(χBk
)(x) ≥ C2kαχBk

(x)

χBk
(x) ≤ C2−kαIα(χBk

)(x)

∥χBk
∥Lq2(·)(wq2(·)) ≤ C2−kα∥Iα(χBk

)∥Lq2(·)(wq2(·))

≤ C2−kα∥χBk
∥Lq1(·)(wq1(·))

≤ C2k(n−α)∥χBk
∥−1
(Lq1(·)(wq1(·)))′

.

(7)

4.1. Boundedness of Fractional Hardy Operators

Theorem 1. Let q1(·) ∈ P(Rn)
⋂
LH(Rn). Define the variable exponent q2(·) by

1

q2(x)
=

1

q1(x)
− α

n
.

If wq1(·) ∈ A1, λ2 = λ1 +
α
n , and δ2 + δλ2 + δ1 > 0, then

∥Hαf∥Ḃq2(·),λ2 (wq2(·)) ≤ C∥f∥Ḃq1(·),λ1 (wq1(·)).

Proof of Theorem 1
Using generalized Hölder inequality given in Lemma 3.

|Hαf(x) · χk(x)| ≤
1

|x|n−α

∫
Bk

|f(t)|dt · χk(x)

≤ C2−k(n−α)
k∑

j=−∞
∥χj∥(Lq1(·)(wq1(·)))′∥fj∥Lq1(·)(wq1(·)) · χk(x).

∥Hαf(x) · χk∥Lq2(·)(wq2(·))

≤ C2−k(n−α)
k∑

j=−∞
∥fj∥Lq1(·)(wq1(·))∥χj∥(Lq1(·)(wq1(·)))′∥χk∥Lq2(·)(wq2(·)).



M. Asim, K. Suwais, N. Mlaiki / Eur. J. Pure Appl. Math, 18 (4) (2025), 6667 8 of 20

By means of Lemma 5, we have

∥Hαf(x) · χk∥Lq2(·)(wq2(·))

≤ C2kα
k∑

j=−∞
∥χj∥(Lq1(·)(wq1(·)))′∥fj∥Lq1(·)(wq1(·))∥χk∥−1

(Lq2(·))(wq2(·)))′

≤ C2kα
k∑

j=−∞
∥fj∥Lq1(·)(wq1(·))

∥χj∥(Lq1(·)(wq1(·)))′

∥χk∥(Lq1(·)(wq1(·)))′
∥χk∥(Lq1(·)(wq1(·)))′∥χk∥−1

(Lq2(·)(wq2(·)))′
.

By Lemma 8 and condition (6), we acquire

∥Hαf(x) · χk∥Lq2(·)(wq2(·))

≤ C2kα
k∑

j=−∞
2nδ2(j−k)∥χk∥(Lq1(·)(wq1(·)))′∥fj∥Lq1(·)(wq1(·))∥χk∥−1

(Lq2(·)(wq2(·)))′
. (8)

Using inequality (7) in (8) and by the result of Lemma 5

∥Hαf(x) · χk∥Lq2(·)(wq2(·))

≤ C2kα
k∑

j=−∞
2nδ2(j−k)2k(n−α)∥fj∥Lq1(·)(wq1(·))∥χk∥−1

Lq2(·)(wq2(·))
∥χk∥−1

(Lq2(·)(wq2(·)))′

≤ C
k∑

j=−∞
2nδ2(j−k)∥fj∥Lq1(·)(wq1(·))

(
2−kn∥χk∥Lq2(·)(wq2(·))∥χk∥(Lq2(·)(wq2(·)))′

)−1

≤ C
k∑

j=−∞
2nδ2(j−k)∥fj∥Lq1(·)(wq1(·))

≤ C∥f∥Ḃq1(·),λ1 (wq1(·))

k∑
j=−∞

2nδ2(j−k)w(Bj)
λ1∥χj∥Lq1(·)(wq1(·)).

∥χj∥Lq1(·)(wq1(·)) ≈ w(B)
1

q1(·) ≈ w(B)
1

q2(·)
+α

n ≈ w(B)
α
n ∥χj∥Lq2(·)(wq2(·)).

∥Hαf(x) · χk∥Lq2(·)(wq2(·))

≤ C∥f∥Ḃq1(·),λ1 (wq1(·))

k∑
j=−∞

2nδ2(j−k)w(Bj)
λ1+

α
n ∥χj∥Lq2(·)(wq2(·))

= C∥f∥Ḃq1(·),λ1 (wq1(·))

k∑
j=−∞

2nδ2(j−k)w(Bk)
λ2

w(Bj)
λ2

w(Bk)λ2
∥χk∥Lq2(·)(wq2(·))

∥χj∥Lq2(·)(wq2(·))

∥χk∥Lq2(·)(wq2(·))

.



M. Asim, K. Suwais, N. Mlaiki / Eur. J. Pure Appl. Math, 18 (4) (2025), 6667 9 of 20

Applying Lemmas 8 and 7

∥Hαf(x) · χk∥Lq2(·)(wq2(·))

≤ C∥f∥Ḃq1(·),λ1 (wq1(·))

k∑
j=−∞

2nδ2(j−k)|Bk|λ2

(
|Bj |
|Bk|

)δλ2

∥χk∥Lq2(·)(wq2(·))

∥χj∥Lq2(·)(wq2(·))

∥χk∥Lq2(·)(wq2(·))

≤ C∥f∥Ḃq1(·),λ1 (wq1(·))|Bk|λ2∥χk∥Lq2(·)(wq2(·))

k∑
j=−∞

2(j−k)(nδ3+nδ1+nδλ2),

∥Hαf(x) · χk∥Ḃq2(·),λ2 (wq2(·)) ≤ C∥f∥Ḃq1(·),λ1 (wq1(·))

k∑
j=−∞

2n(j−k)(δ2+δ1+δλ2).

Since it is given that δ2 + δ1 + δλ2 > 0, which gives the required result:

∥Hαf(x) · χk∥Ḃq2(·),λ2 (wq2(·)) ≤ C∥f∥Ḃq1(·),λ1 (wq1(·)).□

Theorem 2. Let q1(·), q2(·) and α be the same as in theorem 1 . If λ2 = λ1 + α
n ,

wq1(·) ∈ A1 and α < −n(1 + λ2), then

∥H∗
αf∥Ḃq2(·),λ2 (wq2(·)) ≤ C∥f∥Ḃq1(·),λ1 (wq1(·)).

Proof of Theorem 2

|H∗
αf(x) · χk(x)| ≤

∫
Rn\Bk

|f(t)||t|α−ndt · χk(x)

≤ C
∞∑

j=k+1

2j(α−n)∥χj∥(Lq1(·)(wq1(·)))′∥fj∥Lq1(·)(wq1(·))χk(x).

∥H∗
αf(x) · χk∥Lq2(·)(wq2(·))

≤ C

∞∑
j=k+1

2j(α−n)∥χj∥(Lq1(·)(wq1(·)))′∥fj∥Lq1(·)(wq1(·))∥χk∥Lq2(·)(wq2(·)).

≤ C
∞∑

j=k+1

2j(α−n)
∥χj∥(Lq1(·)(wq1(·)))′

∥χk∥(Lq1(·)(wq1(·)))′
∥χk∥(Lq1(·)(wq1(·)))′∥χk∥Lq2(·)(wq2(·))∥fj∥Lq1(·)(wq1(·)).

By virtue of Lemmas 2, 10, 6 and by the definition of A(q2(·), q1(·)) we obtain the
following inequalities:

∥H∗
αf(x) · χk∥Lq2(·)(wq2(·))

≤ C
∞∑

j=k+1

2j(α−n)2n(j−k)∥χk∥(Lq1(·)(wq1(·)))′∥fj∥Lq1(·)(wq1(·))∥χk∥Lq2(·)(wq2(·)).

≤ C

∞∑
j=k+1

2(j−k)(α−n)2n(j−k)∥fj∥Lq1(·)(wq1(·)).



M. Asim, K. Suwais, N. Mlaiki / Eur. J. Pure Appl. Math, 18 (4) (2025), 6667 10 of 20

∥H∗
αf(x) · χk∥Ḃq2(·),λ2 (wq2(·)) ≤ C∥f∥Ḃq1(·),λ1 (wq1(·))

∞∑
j=k+1

2(j−k)(α+n+nλ2).

By using α < −n(1 + λ2), we get the final result:

∥H∗
αf∥Ḃq2(·),λ2 (wq2(·)) ≤ C∥f∥Ḃq1(·),λ1 (wq1(·)).□

4.2. Commutators of Fractional Hardy Operators

Theorem 3. Let 0 < α < n, and p1(·), p(·) ∈ P(Rn)
⋂
LH(Rn) . Define the variable

exponent p2(·) by
1

p2(x)
=

1

p(x)
+

1

p1(x)
− α

n
.

If wP1(·) ∈ A1, b ∈ ∥b∥CBMOp(·),λ(wp(·)) , µ = λ1 +
α
n and λ2 = λ+ λ1 +

α
n , then

∥[b,Hα]f∥Ḃp2(·),λ2 (wp2(·)) ≤ C∥b∥CBMOp(·),λ(wp(·))∥f∥Ḃp1(·),λ1 (wp1(·)).

Proof of Theorem 3

|[b,Hα]f(x) · χB(x)| ≤
1

|x|n−α

∫
B(0,|x|)

|(b(x)− b(v))f(v)|dv · χB(x)

≤ 1

|x|n−α

∫
B(0,|x|)

|(b(x)− bB)f(v)|dv · χB(x)

+
1

|x|n−α

∫
B(0,|x|)

|(b(v)− bB)f(v)|dv · χB(x)

= A1 +A2.

First, we estimate A1. Denote 1
S(x) =

1
p1(x)

− α
n , then

1
p2(x)

= 1
S(x) +

1
p(x) .

A1 =
1

|x|n−α

∫
B(0,|x|)

|(b(x)− bB)f(v)|dv · χB(x)

= |(b(x)− bB)χB(x)||Hαf(x)|,

∥A1∥Lp2(·)(wp2(·)) = ∥(b(x)− bB)χB(x)Hαf(x)∥Lp2(·)(wp2(·)).

Using Hölder inequality ( 1
p2(·) =

1
S(·) +

1
p(·))

∥A1∥Lp2(·)(wp2(·)) ≤ ∥(b(x)− bB)χB(x)∥Lp(·)(wp(·))∥Hαf(x)χB∥LS(·)(wS(·))

= C∥b∥CBMOp(·),λ(wp(·))|B|λ∥χB∥Lp(·)(wp(·))|B|µ∥χB∥LS(·)(wS(·))∥Hαf∥Ḃµ,S(·)(wS(·)),

∥A1∥Lp2(·)(wp2(·)) ≤ C∥b∥CBMOp(·),λ(wp(·))|B|λ∥χB∥Lp(·)(wp(·))|B|µ∥χB∥LS(·)(wS(·))∥Hαf∥Ḃµ,S(·)(wS(·)).



M. Asim, K. Suwais, N. Mlaiki / Eur. J. Pure Appl. Math, 18 (4) (2025), 6667 11 of 20

∥χB∥Lp2(·)(wp2(·)) ≈ w(B)
1

p2(·) ≈ w(B)
1

S(·)+
1

p(·) ≈ ∥χB∥Lp(·)(wp(·))∥χB∥LS(·)(wS(·))

Given that µ = λ1 +
α
n , using the result of theorem 1

∥A1∥Lp2(·)(wp2(·)) ≤ C∥b∥CBMOp(·),λ(wp(·))|B|λ2∥χB∥Lp2(·)(wp2(·))∥f∥Ḃp1(·),λ1 (wp1(·)),

A2 =
1

|x|n−α

∫
B(0,|x|)

|(b(v)− bB)f(v)|dv · χB(x).

A2 =
0∑

k=−∞

1

|x|n−α

k∑
l=−∞

∫
2lB\2l−1B

|(b(v)− bB)f(v)|dv · χ2kB\2k−1B(x)

≤
0∑

k=−∞

1

|x|n−α

k∑
l=−∞

∫
2lB\2l−1B

|(b(v)− b2lB)f(v)|dv · χ2kB\2k−1B(x)

+
0∑

k=−∞

1

|x|n−α

k∑
l=−∞

∫
2lB\2l−1B

|(bB − b2lB)f(v)|dv · χ2kB\2k−1B(x)

= A21 +A22

A21 =

0∑
k=−∞

|2kB|
α
n
−1

k∑
l=−∞

∫
2lB\2l−1B

|(b(v)− b2lB)f(v)|dv · χ2kB\2k−1B(x).

Using Hölder inequality ( 1
p1(·) +

1
t(·) +

1
p(·) = 1).

A21 ≤ C
0∑

k=−∞
|2kB|

α
n
−1χ2kB\2k−1B(x)

k∑
l=−∞

∥(b(v)− b2lB)χ2lB∥Lp(·)(wp(·))∥fχ2lB∥Lp1(·)(wp1(·))∥χ2lB∥Lt(·)(wt(·))

= C

0∑
k=−∞

|2kB|
α
n
−1χ2kB\2k−1B(x)

k∑
l=−∞

∥b∥CBMOp(·),λ(wp(·))|2
lB|λ∥χ2lB∥Lp(·)(wp(·))

∥f∥Ḃp1(·),λ1 (wp1(·))|2
lB|λ1∥χ2lB∥Lp1(·)(wp1(·))∥χ2lB∥Lt(·)(wt(·))

∥χ2lB∥
(Lp

′
1(·)(wp1(·)))′

≈ w(2lB)
1

p
′
1(·) ≈ w(2lB)

1
p(·)+

1
t(·) ≈ ∥χ2lB∥Lp(·)(wp(·))∥χ2lB∥Lt(·)(wt(·))

A21 = C

0∑
k=−∞

|2kB|
α
n
−1χ2kB\2k−1B(x)∥b∥CBMOp(·),λ(wp(·))∥f∥Ḃp1(·),λ1 (wp1(·))

k∑
l=−∞

|2lB|λ+λ1∥χ2lB∥
(Lp

′
1(·)(wp1(·)))′

∥χ2lB∥Lp1(·)(wp1(·)).



M. Asim, K. Suwais, N. Mlaiki / Eur. J. Pure Appl. Math, 18 (4) (2025), 6667 12 of 20

By using the result of lemma 5, we will have

A21 = C
0∑

k=−∞
|2kB|

α
n
−1χ2kB\2k−1B(x)∥b∥CBMOp(·),λ(wp(·))∥f∥Ḃp1(·),λ1 (wp1(·))

k∑
l=−∞

|2l|λ+λ1+1|B|λ+λ1+1

= C∥b∥CBMOp(·),λ(wp(·))∥f∥Ḃp1(·),λ1 (wp1(·))

0∑
k=−∞

|2k|
α
n
+λ+λ1χ2kB\2k−1B(x)|B|λ+λ1+

α
n

∥A21∥Lp2(·)(wp2(·)) ≤ C∥b∥CBMOp(·),λ(wp(·))∥f∥Ḃp1(·),λ1 (wp1(·))

0∑
k=−∞

|2k|
α
n
+λ+λ1∥χ2kB∥Lp2(·)(wp2(·))|B|λ+λ1+

α
n

= C∥b∥CBMOp(·),λ(wp(·))∥f∥Ḃp1(·),λ1 (wp1(·))

0∑
k=−∞

|2k|
α
n
+λ+λ1w(2kB)

1
p2(·) |B|λ+λ1+

α
n

= C∥b∥CBMOp(·),λ(wp(·))∥f∥Ḃp1(·),λ1 (wp1(·))w(B)
1

p2(·) |B|λ2

0∑
k=−∞

|2|k(λ2+
1

p2(·)
)

∥A21∥Lp2(·)(wp2(·)) ≤ C∥b∥CBMOp(·),λ(wp(·))∥f∥Ḃp1(·),λ1 (wp1(·))∥χB∥Lp2(·)(wp2(·))|B|λ2

A22 =
0∑

k=−∞
|2kB|

α
n
−1

k∑
l=−∞

∫
2lB\2l−1B

|(bB − b2lB)f(y)|dy · χ2kB\2k−1B(x)

|(bB − b2lB)| =
−1∑
i=l

|(b2i+1B − b2iB)

=

−1∑
i=l

1

|2iB|

∫
2iB

|b(y)− b2i+1B|dy

≤ C
−1∑
i=l

1

|2iB|
∥(b− b2i+1b)χ2i+1B∥Lp(·)(wp(·))∥χ2i+1B∥(Lp(·)(wp1(·)))′

By virtue of Lemma 5, we have

|(bB − b2lB)| ≤ C

−1∑
i=l

1

|2iB|
∥(b− b2i+1b)χ2i+1B∥Lp(·)(wp(·))

|2i+1B|
∥χ2i+1B∥Lp(·)(wp(·))

≤ C
−1∑
i=l

∥b∥CBMOp(·),λ(wp(·))|2
i+1B|λ

≤ C∥b∥CBMOp(·),λ(wp(·))

−1∑
i=l

|2i+1B|λ

≤ C∥b∥CBMOp(·),λ(wp(·))|2
l+1B|λ|l| (9)



M. Asim, K. Suwais, N. Mlaiki / Eur. J. Pure Appl. Math, 18 (4) (2025), 6667 13 of 20

A22 ≤ C

0∑
k=−∞

χ2kB\2k−1B(x)|2kB|
α
n
−1

k∑
l=−∞

∥b∥CBMOp(·),λ(wp(·))|2
l+1B|λ|l|

∥fχ2lB∥Lp1(·)(wp1(·))∥χ2lB∥(Lp1(·)(wp1(·)))′

≤ C∥b∥CBMOp(·),λ(wp(·))

0∑
k=−∞

χ2kB\2k−1B(x)|2kB|
α
n
−1

k∑
l=−∞

|2l+1B|λ|l|

∥fχ2lB∥Lp1(·)(wp1(·))

|2lB|
∥χ2lB∥Lp1(·)(wp1(·))

≤ C∥b∥CBMOp(·),λ(wp(·))

0∑
k=−∞

χ2kB\2k−1B(x)|2kB|
α
n
−1

k∑
l=−∞

|2lB|λ+1|l|∥f∥Ḃp1(·),λ1 (wp1(·))|2
lB|λ1

≤ C∥b∥CBMOp(·),λ(wp(·))∥f∥Ḃp1(·),λ1 (wp1(·))

0∑
k=−∞

χ2kB\2k−1B(x)|2kB|
α
n
−1

k∑
l=−∞

|2lB|λ+λ1+1|l|

≤ C∥b∥CBMOp(·),λ(wp(·))∥f∥Ḃp1(·),λ1 (wp1(·))

0∑
k=−∞

χ2kB\2k−1B(x)|2kB|
α
n
−1|2kB|λ+λ1+1|k|

≤ C∥b∥CBMOp(·),λ(wp(·))∥f∥Ḃp1(·),λ1 (wp1(·))

0∑
k=−∞

|k||2kB|λ+λ1+
α
nχ2kB\2k−1B(x)

∥A22∥Lp2(·)(wp2(·)) ≤ C∥b∥CBMOp(·),λ(wp(·))∥f∥Ḃp1(·),λ1 (wp1(·))

0∑
k=−∞

|k||2kB|λ+λ1+
α
n ∥χ2kB∥Lp2(·)

≤ C∥b∥CBMOp(·),λ(wp(·))∥f∥Ḃp1(·),λ1 (wp1(·))

0∑
k=−∞

|k||2kB|λ+λ1+
α
nw(2kB)

1
p2(·)

≤ C∥b∥CBMOp(·),λ(wp(·))∥f∥Ḃp1(·),λ1 (wp1(·))

0∑
k=−∞

|k||2k|λ2+
1

p2(·) |B|λ2w(B)
1

p2(·)

≤ C∥b∥CBMOp(·),λ(wp(·))∥f∥Ḃp1(·),λ1 (wp1(·))|B|λ2∥χB∥Lp2(·)(wp2(·))

Combine all the results of A1, A2, A21, A22, we obtain the required result

∥[b,Hα]f∥Lp2(·)(wp2(·)) ≤ C∥b∥CBMOp(·),λ(wp(·))∥f∥Ḃp1(·),λ1 (wp1(·))|B|λ2∥χB∥Lp2(·)(wp2(·))

∥[b,Hα]f∥Ḃp2(·),λ2 (wp2(·)) ≤ C∥b∥CBMOp(·),λ(wp(·))∥f∥Ḃp1(·),λ1 (wp1(·)).□

Theorem 4. Let p1(·), p2(·), p(·), and α be defined the same way as in Theorem 3.
If b ∈ ∥b∥CBMOp(·),λ(wp(·)) , µ = λ1 +

α
n and λ2 = λ+ λ1 +

α
n , then

∥[b,H∗
α]f∥Ḃp2(·),λ2 (wp2(·)) ≤ C∥b∥CBMOp(·),λ(wp(·))∥f∥Ḃp1(·),λ1 (wp1(·)).



M. Asim, K. Suwais, N. Mlaiki / Eur. J. Pure Appl. Math, 18 (4) (2025), 6667 14 of 20

Proof of Theorem 4

|[b,H∗
α]f(x) · χB(x)| ≤

∫
B(0,|x|)c

|(b(x)− b(v))f(v)|
|v|n−α

dv · χB(x)

≤
∫
B(0,|x|)c

|(b(x)− bB)f(v)|
|v|n−α

dv · χB(x)

+

∫
B(0,|x|)c

|(b(v)− bB)f(v)|
|v|n−α

dv · χB(x)

= D1 +D2.

D1 =

∫
B(0,|x|)c

|(b(x)− bB)f(v)|
|v|n−α

dv · χB(x)

= |(b(x)− bB)χB(x)||H∗
αf(x)|,

∥D1∥Lp2(·)(wp2(·)) = ∥(b(x)− bB)χBH
∗
αf∥Lp2(·)(wp2(·)).

Using Hölder inequality ( 1
p2(·) =

1
S(·) +

1
p(·))

∥D1∥Lp2(·)(wp2(·)) ≤ C∥(b(x)− bB)χB∥Lp(·)(wp(·))∥H
∗
αfχB∥LS(·)(wS(·))

= C∥b∥CBMOp(·),λ(wp(·))|B|λ∥χB∥Lp(·)(wp(·))|B|µ∥χB∥LS(·)(wS(·))∥H
∗
αf∥Ḃµ,S(·)(wS(·)),

Given that µ = λ1 +
α
n , using the result of Theorem 2

∥D1∥Lp2(·)(wp2(·)) ≤ C∥b∥CBMOp(·),λ(wp(·))|B|λ2∥χB∥Lp(·)(wp(·))∥χB∥LS(·)(wS(·))∥f∥Ḃp1(·),λ1 (wp1(·)),

≤ C∥b∥CBMOp(·),λ(wp(·))|B|λ2∥χB∥Lp2(·)(wp2(·))∥f∥Ḃp1(·),λ1 (wp1(·)),

D2 =

∫
B(0,|x|)c

|(b(v)− bB)f(v)|
|v|n−α

dv · χB(x).

D2 =

0∑
k=−∞

∫
2lB\2l−1B

|(b(v)− bB)f(v)|
|v|n−α

dv · χ2kB\2k−1B(x)

≤
0∑

k=−∞
χ2kB\2k−1B(x)

∞∑
l=k+1

|2lB|
α
n
−1

∫
2lB\2l−1B

|(b(v)− b2lB)f(v)|dv

+
0∑

k=−∞
χ2kB\2k−1B(x)

∞∑
l=k+1

|2lB|
α
n
−1

∫
2lB\2l−1B

|(bB − b2lB)f(v)|dv

= D21 +D22

D21 =

0∑
k=−∞

χ2kB\2k−1B(x)

∞∑
l=k+1

|2lB|
α
n
−1

∫
2lB\2l−1B

|(b(v)− b2lB)f(v)|dv.



M. Asim, K. Suwais, N. Mlaiki / Eur. J. Pure Appl. Math, 18 (4) (2025), 6667 15 of 20

Using Hölder inequality ( 1
t(·) +

1
p1(·) +

1
p(·) = 1).

D21 ≤ C
0∑

k=−∞
χ2kB\2k−1B(x)

∞∑
l=k+1

|2lB|
α
n
−1∥(b(v)− b2lB)χ2lB∥Lp(·)(wp(·))∥fχ2lB∥Lp1(·)(wp1(·))∥χ2lB∥Lt(·)(wt(·))

= C
0∑

k=−∞
χ2kB\2k−1B(x)

∞∑
l=k+1

|2lB|
α
n
−1∥b∥CBMOp(·),λ(wp(·))|2

lB|λ∥χ2lB∥Lp(·)(wp(·))

∥f∥Ḃp1(·),λ1 (wp1(·))|2
lB|λ1∥χ2lB∥Lp1(·)(wp1(·))∥χ2lB∥Lt(·)(wt(·))

≤ C

0∑
k=−∞

χ2kB\2k−1B(x)

∞∑
l=k+1

|2lB|
α
n
−1∥b∥CBMOp(·),λ(wp(·))|2

lB|λ

∥f∥Ḃp1(·),λ1 (wp1(·))|2
lB|λ1∥χ2lB∥Lp1(·)(wp1(·))∥χ2lB∥(Lp(·)(wp(·)))′

= C

0∑
k=−∞

χ2kB\2k−1B(x)∥b∥CBMOp(·),λ(wp(·))∥f∥Ḃp1(·),λ1 (wp1(·))

∞∑
l=k+1

|2lB|
α
n
−1|2lB|λ+λ1+1

= C

0∑
k=−∞

χ2kB\2k−1B(x)∥b∥CBMOp(·),λ(wp(·))∥f∥Ḃp1(·),λ1 (wp1(·))

∞∑
l=k+1

|2lB|λ2

= C∥b∥CBMOp(·),λ(wp(·))∥f∥Ḃp1(·),λ1 (wp1(·))

0∑
k=−∞

|2(k+1)B|λ2χ2kB\2k−1B(x)

∥D21∥Lp2(·)(wp2(·)) ≤ C∥b∥CBMOp(·),λ(wp(·))∥f∥Ḃp1(·),λ1 (wp1(·))

0∑
k=−∞

|2(k+1)B|λ2∥χ2kB∥Lp2(·)(wp2(·))

≤ C∥b∥CBMOp(·),λ(wp(·))∥f∥Ḃp1(·),λ1 (wp1(·))

0∑
k=−∞

|2(k+1)B|λ2w(2kB)
1

p2(·)

≤ C∥b∥CBMOp(·),λ(wp(·))∥f∥Ḃp1(·),λ1 (wp1(·))|B|λ2w(B)
1

p2(·)

0∑
k=−∞

|2(k+1)|λ2+
1

p2(·)

≤ C∥b∥CBMOp(·),λ(wp(·))∥f∥Ḃp1(·),λ1 (wp1(·))|B|λ2∥χB∥Lp2(·)(wp2(·))

D22 =
0∑

k=−∞
χ2kB\2k−1B(x)

k∑
l=−∞

|2lB|
α
n
−1

∫
2lB\2l−1B

|(bB − b2lB)f(v)|dv.



M. Asim, K. Suwais, N. Mlaiki / Eur. J. Pure Appl. Math, 18 (4) (2025), 6667 16 of 20

Here we use inequality (9)

D22 ≤ C
0∑

k=−∞
χ2kB\2k−1B(x)

∞∑
l=k+1

|2lB|
α
n
−1∥b∥CBMOp(·),λ(wp(·))|2

l+1B|λ|l|

∥fχ2lB∥Lp1(·)(wp1(·))∥χ2lB∥(Lp1(·)(wp1(·)))′

≤ C∥b∥CBMOp(·),λ(wp(·))

0∑
k=−∞

χ2kB\2k−1B(x)
∞∑

l=k+1

|2lB|
α
n
−1|2l+1B|λ|l|

∥fχ2lB∥Lp1(·)(wp1(·))

|2lB|
∥χ2lB∥Lp1(·)(wp1(·))

≤ C∥b∥CBMOp(·),λ(wp(·))

0∑
k=−∞

χ2kB\2k−1B(x)
∞∑

l=k+1

|2lB|
α
n
−1|2lB|λ+1|l|∥f∥Ḃp1(·),λ1 (wp1(·))|2

lB|λ1

≤ C∥b∥CBMOp(·),λ(wp(·))∥f∥Ḃp1(·),λ1 (wp1(·))

0∑
k=−∞

χ2kB\2k−1B(x)
∞∑

l=k+1

|2lB|λ+λ1+
α
n |l|

≤ C∥b∥CBMOp(·),λ(wp(·))∥f∥Ḃp1(·),λ1 (wp1(·))

0∑
k=−∞

χ2kB\2k−1B(x)|2k+1B|λ2 |k + 1|

≤ C∥b∥CBMOp(·),λ(wp(·))∥f∥Ḃp1(·),λ1 (wp1(·))

0∑
k=−∞

|k + 1||2k+1B|λ2χ2kB\2k−1B(x)

∥D22∥Lp2(·)(wp2(·)) ≤ C∥b∥CBMOp(·),λ(wp(·))∥f∥Ḃp1(·),λ1 (wp1(·))

0∑
k=−∞

|k + 1||2k+1B|λ2∥χ2kB∥Lp2(·)(wp2(·))

≤ C∥b∥CBMOp(·),λ(wp(·))∥f∥Ḃp1(·),λ1 (wp1(·))

0∑
k=−∞

|k + 1||2k+1B|λ2w(2kB)
1

p2(·)

≤ C∥b∥CBMOp(·),λ(wp(·))∥f∥Ḃp1(·),λ1 (wp1(·))

0∑
k=−∞

|k + 1||2k+1|λ2+
1

p2(·) |B|λ2w(B)
1

p2(·)

≤ C∥b∥CBMOp(·),λ(wp(·))∥f∥Ḃp1(·),λ1 (wp1(·))|B|λ2∥χB∥Lp2(·)(wp2(·))

Combine all results of D1, D2, D21, D22, we obtain the required result

∥[b,H∗
α]fχB∥Lp2(·)(wp2(·)) ≤ C∥b∥CBMOp(·),λ(wp(·))∥f∥Ḃp1(·),λ1 (wp1(·))|B|λ2∥χB∥Lp2(·)(wp2(·))

∥[b,H∗
α]f∥Ḃp2(·),λ2 (wp2(·)) ≤ C∥b∥CBMOp(·),λ(wp(·))∥f∥Ḃp1(·),λ1 (wp1(·)).□

5. Conclusion

This scholarly treatise vouchsafes momentous progressions in the analytical dissec-
tion of fractional Hardy operators, meticulously ensconced within the structural fabric of



M. Asim, K. Suwais, N. Mlaiki / Eur. J. Pure Appl. Math, 18 (4) (2025), 6667 17 of 20

weighted central Morrey spaces imbued with variable exponents. The present intellectual
enterprise is poised to inaugurate uncharted pathways for erudition in the mathematical
and physical sciences—most conspicuously within the abstruse territories of Quantum Me-
chanics and Mathematical Modeling—thereby engendering novel prospects for exploratory
inquiry and theoretical augmentation.

Acknowledgements

The author Khaled Suwais would like to thank Arab Open University for supporting
this work.
The authors N. Mlaiki would like to thank Prince Sultan University for the support through
the TAS research lab.

Authors’ contributions

M.A., S.H. and N.M. wrote the main manuscript text. All authors reviewed the
manuscript.

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