EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 4, Article Number 6519
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Boundedness of the Sublinear Operators on Anisotropic
Herz-Slice Spaces

Babar Sultan1, Amjad Hussain1,∗, Mehvish Sultan2, Ioan-Lucian Popa3,4,∗

1 Department of Mathematics, Quaid-I-Azam University, Islamabad 45320, Pakistan
2 Department of Mathematics, Capital University of Science and Technology, Islamabad,
Pakistan
3 Department of Computing, Mathematics and Electronics, “1 Decembrie 1918” University
of Alba Iulia, 510009 Alba Iulia, Romania
4 Faculty of Mathematics and Computer Science, Transilvania University of Brasov,
Iuliu Maniu Street 50, 500091 Brasov, Romania

Abstract. We will define the idea of anisotropic Herz-slice spaces and prove some properties
of these spaces. As an application we obtain the bounds for sublinear operator on anisotropic
Herz-slice spaces.
2020 Mathematics Subject Classifications: 42B35, 47B38
Key Words and Phrases: Herz spaces, slice spaces, Herz-slice spaces, integral operators, atomic
decomposition, boundedness

1. Introduction

Let p ∈ (0,∞), the Lebesgue space is defined as

Lp(E) :=

{
f is measurable: Ip

(
f

γ

)
< ∞ for some constant γ > 0

}
where

Ip(f) :=

∫
E
|g(x)|pdx

and
‖f‖Lp(E) := inf

{
γ > 0 : Ip

(
f

γ

)
6 1

}
.

∗Corresponding author.
∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6519

Email addresses: babarsultan40@yahoo.com (B. Sultan),
a.hussain@qau.edu.pk (A. Hussain), mehvishsultanbaz@gmail.com (M. Sultan),
lucian.popa@uab.ro (I.-L. Popa)

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



B. Sultan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6519 2 of 11

Afterward, Lp(E) is a Banach space, and ‖ · ‖Lp(E) is its norm.
Slice spaces were initially presented in [1] to study weak solutions of boundary value

problems for time-independent elliptic systems in the upper half-plane. This concept was
later expanded upon in [2], where generalized slice spaces were used to study the mapping
properties of sublinear operators on tent spaces, the Hardy-Littlewood maximal operator
M , and the Calderón-Zygmund operators. Let u ∈ (0,∞), p ∈ [1,∞) and r ∈ (1,∞), the
slice space (Ep

r )u is defined as the set of all measurable functions g such that

‖f‖(Ep
r

)
u
:=

∥∥∥∥∥∥∥
 1

|B(·, u)|

∫
B(·,u)

|f(y)|r dy


∥∥∥∥∥∥∥
Lp

< ∞.

More recently, great attention was paid to the study on the Herz spaces since there
are several remarkable works to push forward the study on the Herz spaces. Conversely,
Herz spaces, a class of function spaces, have been pivotal in real analysis due to their
norms, which explicitly incorporate both local and global information of functions. For
more results generalized versions of Herz space like boundedness of sublinear operators,
fractional integrals, the commutator of singular integrals with BMO functions, and the
commutator of fractional integrals with BMO functions see [3–25].

Let α ∈ R, q ∈ (0,∞], and 0 < p ≤ ∞, then the homogeneous version of Herz spaces
K̇α,q

p are defined by

K̇α,q
p =

{
g ∈ Lp

loc(R
n \ {0}) : ‖g‖K̇α,q

p
< ∞

}
,

where

‖g‖K̇α,q
p

=

( ∞∑
`=−∞

2`αq‖gχ`‖qLp

) 1
q

.

Firstly, we define the idea of anisotropic Herz-slice spaces by using anisotropic Herz
spaces and slice spaces. We will establish the atomic decomposition in these spaces. Then,
we obtain boundedness for sublinear in these spaces.

2. Preliminaries

Now we give some notations.
Let l ∈ Z, define Bl :=

{
z ∈ Rn : |z| 6 2l

}
, Rl := Bl\Bl−1, and χl := χRl

. A n × n
real matrix O is called dilation or expansive matrix if |γ| > 1, where γ is the eigenvalue.
Let γ1, · · · , γn are eigenvalues of O such that 1 < |γ1| ≤ · · · ≤ |γn| and γ−, γ+ are two
numbers such that 1 < γ− < |γ1| ≤ |γn| < γ+. Let | · | is the Euclidean norm, P is
nondegenerate matrix of order n× n and r > 1, then the set ∆ ⊂ Rn is called ellipsoid if
∆ = {z ∈ Rn : |Pz| < 1}. consider an ellipsoid ∆ and r > 1, dilation O, ∆ ⊂ r∆ ⊂ A∆
and |∆| = 1, |∆| is the Lebesgue measure of ∆. If B` = O`∆ for ` ∈ Z, then we get
B` ⊂ rB` ⊂ B`+1, and |B`| = b` such that b =

∏n
i=1 |γi| = |detO| > 1. Assume that w is



B. Sultan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6519 3 of 11

the smallest integer such that 2B0 ⊂ OwB0 = Bw. A quaisi-norm with expansive matrix
O is a measurable mapping τO : Rn → [0,∞) such that

τO(y) > 0 for y 6= 0 ,

τO(Ay) = |detA|τ(y) for y ∈ Rn ,

τO(x+ y) ≤ C (τO(x) + τO(y)) for x , y ∈ Rn ,

where C ≥ 1. The step homogeneous quasi-norm τ on Rn defined by dilation O is given
as (

τ(x) = bj for x ∈ Bj+1 \ Bj

)
and (0 for x = 0 ) .

If x, y ∈ Rn, then we have

τ(x+ y) ≤ bw (τ(x) + τ(y)) . (2.1)

Definition 1. If α ∈ R, u ∈ (0,∞), p, q ∈ [1,∞) and r ∈ (1,∞), then the homogeneous
version of anisotropic Herz-slice spaces

(
K̇Eα,p

q,r

)
u
(O;Rn) are defined by

(
K̇Eα,p

q,r

)
u
(O;Rn) =

{
g ∈ (Ep

r )u : ‖g‖(
K̇Eα,p

q,r

)
u
(O;Rn)

< ∞
}
,

where

‖g‖(
K̇Eα,p

q,r

)
u
(O;Rn)

=

( ∞∑
`=−∞

b`αq‖gχ`‖q(Ep
r

)
u

) 1
q

.

Now we state the Hölder’s inequality for slice space.

Lemma 2. [26] Let 1 ≤ p ≤ ∞, 0 < u < ∞ and 1 < r < ∞,

‖fg‖L1(Rn) ≤ ‖f‖(Ep
r

)
u
‖g‖(

Ep′
r′

)
u

where 1
p + 1

p′ =
1
r +

1
r′ = 1.

Lemma 3. [27] Let u ∈ (0,∞), p, r ∈ (1,∞), if x0 ∈ Rn, and 1 < r0 < ∞, then the
characteristic function on B (x0, r0) fulfills∥∥χB(x0,r0)

∥∥(
Ep

r

)
u

≤ Cr
n/p
0 .

Remark 4. [27] Let u ∈ (0,∞), p.r ∈ (1,∞), ` ∈ Z, then the characteristic function on
R` fulfills

‖χR`
‖(Ep

r

)
u
≤ ‖χB`

‖ (
Ep

r

)
u

) ≤ Cb`n/p.



B. Sultan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6519 4 of 11

3. Atomic Decomposition of anisotropic Herz-slice spaces

Definition 5. Let α ∈ R, and p, q, r, u ∈ (0,∞]. Then

(i) A measurable function a(y) is called central (α, p, r, u)-block if supp a ⊂ Bl and
‖a‖(Ep

r

)
u
≤ b−lα.

(ii) A measurable function a(y) is called central (α, p, r, u)-block if supp a ⊂ Bl.

Theorem 3.1. Let α ∈ R, and p, q, r, u ∈ (0,∞]. Let b` be a central (α, p, r, u)-block with
support d in B` and

∑∞
`=−∞ |γ`|q < ∞. Then the following two statements are equivalent:

(i) g ∈
(
K̇Eα,p

q,r

)
u
(O;Rn).

(ii) g are given as

g(y) =

∞∑
`=−∞

γ`b`(y). (2.1)

Proof.
We first prove (i) implies (ii). For every g ∈

(
K̇Eα,p

q,r

)
u
(O;Rn), write

g(y) =

∞∑
`=−∞

g(y)χ`(y)

=

∞∑
`=−∞

b`α ‖gχ`‖(Ep
r

)
u

g(y)χ`(y)

b`α ‖χ`‖(Ep
r

)
u

=

∞∑
`=−∞

γ`b`(x),

where γ` = b`α ‖gχ`‖(Ep
r

)
u

and b`(x) =
g(y)χ`(y)

b`α‖χ`‖(Ep
r

)
u

.

It is easy to note that supp b` ⊂ B` and ‖b`‖(Ep
r

)
u
= |B`|−α/n. So every b` is a central

(α, p, r, u)-block with support contained in B`

∞∑
`=−∞

|γ`|q =
∞∑

`=−∞
b`α ‖gχ`‖q(Ep

r

)
u

= ‖g‖q(
K̇Eα,p

q,r

)
u
(O;Rn)

< ∞.

Now we prove (ii) implies (i). Let g(y) =
∑∞

`=−∞ γ`b`(y), we get

‖gχj‖(Ep
r

)
u
≤

∞∑
`=j

|γ`| ‖b`‖(Ep
r

)
u
. (3.1)

If 0 < q ≤ 1. From (3.1) it follows that

‖g‖q(
K̇Eα,p

q,r

)
u
(O;Rn)

=

∞∑
`=−∞

b`α ‖gχ`‖q(Ep
r

)
u



B. Sultan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6519 5 of 11

≤
∞∑

`=−∞
b`αq ‖gχ`‖q(Ep

r

)
u

≤
∞∑

`=−∞
b`αq

 ∞∑
j=`

|γj |q ‖bj‖q(Ep
r

)
u

 .

Let I =
∞∑

`=−∞
b`αq

(∑∞
j=` |γj |

q ‖bj‖q(Ep
r

)
u

)
. By using the fact 0 < α < ∞, we get

I =

∞∑
`=−∞

b`αq

 ∞∑
j=`

|γj |q ‖bj‖q(Ep
r

)
u


.

∞∑
`=−∞

b`αq
∞∑
j=`

|γj |q b−jαq

.
∞∑

`=−∞

∞∑
j=`

|γj |q b(`−j)αq

.
∞∑

j=−∞

j∑
`=−∞

|γj |q b(`−j)αq

.
∞∑

j=−∞
|γj |q .

If 1 < q < ∞, we have
For I, if 0 < α < ∞, then (3.1) and Hölder’s inequality yields

I .
∞∑

`=−∞
b`αq

 ∞∑
j=`

|γj | ‖bj‖(Ep
r

)
u


.

∞∑
`=−∞

 ∞∑
j=`

|γj | b(`−j)α

q

.
∞∑

`=−∞

 ∞∑
j=`

|γj |q b(`−j)αq/2

 ∞∑
j=`

b(`−j)αq′/2

q/q′

.
∞∑

`=−∞

 ∞∑
j=`

|γj |q b(`−j)αq/2


.

∞∑
j=−∞

j∑
`=−∞

|γj |q b(`−j)αq/2



B. Sultan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6519 6 of 11

.
∞∑

j=−∞
|γj |q .

Thus the proof of the Theorem is completed. Non-homogeneous version of the Theorem
is obtained similarly.

Remark 6. By using the Theorem, note that if g ∈
(
K̇Eα,p

q,r

)
u
(O;Rn) and g(y) =∑∞

`=−∞ γ`b`(x) is a central (α, p, r, u)-block decomposition, then

‖g‖(
K̇Eα,p

q,r

)
u
(O;Rn)

≈

( ∞∑
`=−∞

|γ`|q
)1/q

.

4. Boundedness of sublinear operators on anisotropic Herz-slice spaces

As applications of the decomposition theorem, we will prove the boundedness of sub-
linear operators on anisotropic Herz-slice spaces.

Theorem 4.1. Let α ∈ R, with 0 < α < 1/p′ and p, q, r, u ∈ (0,∞]. Let the sublinear T
fulfills the size condition

|Tg(y)| .
∫
Rn

|g(y)|
τ(x− y)

dy, x /∈ supp g, (4.1)

for any g ∈ (Ep
r )u with a compact support and T is bounded on (Ep

r )u. Then T is
bounded in

(
K̇Eα,p

q,r

)
u
(O;Rn).

Proof. Let g ∈
(
K̇Eα,p

q,r

)
u
(O;Rn)/ such that g(y) =

∑∞
a0=−∞ γa0ba0(x) where ba0 is

a central (α, p.r, u)-block with support contained in Ba0 , By applying the decomposition
theorem, we get

‖g‖(
K̇Eα,p

q,r

)
u
(O;Rn)

≈

( ∞∑
a0=−∞

|γa0 |
q

)1/q

.

Therefore, we get

‖Tg‖q(
K̇Eα,p

q,r

)
u
(O;Rn)

=
∞∑

`=−∞
b`αq ‖(Tg)χ`‖q(Ep

r

)
u

.
∞∑

`=−∞
b`αq ‖(Tg)χ`‖q(Ep

r

)
u

.
∞∑

`=−∞
b`αq

(
`−w−1∑
a0=−∞

|γa0 | ‖(Tba0)χ`‖(Ep
r

)
u

)q



B. Sultan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6519 7 of 11

+
∞∑

`=−∞
b`αq

 ∞∑
a0=`−w

|γa0 | ‖(Tba0)χ`‖(Ep
r

)
u

q

= I1 + I2.

To find the estimate of I1. Let x ∈ C`, y ∈ Ba0 such that a0 ≤ ` − w − 1,, then (2.1)
yields

b−w(1− 1/b)τ(x) = b−wτ(x)− b−w−1τ(x) ≤ b−wτ(x)− τ(y) ≤ τ(x− y).

Hence (4.1) and Hölder’s inequality yields

|Tba0(x)| ≤ Cτ(x)−1

∫
Ba0

|ba0(y)| dy

≤ Cb−` ‖ba0‖(Ep
r

)
u

∥∥χBa0

∥∥(
Ep′

r′

)
u

.

Applying Lemmas 2 and 4, we get

‖(Tba0)χ`‖(Ep
r

)
u
. b−` ‖ba0‖(Ep

r

)
u

∥∥χBa0

∥∥(
Ep′

r′

)
u

‖χB`
‖(Ep

r

)
u

. b−` ‖ba0‖(Ep
r

)
u

(
|B`| ‖χB`

‖−1(
Ep′

r′

)
u

)∥∥χBa0

∥∥(
Ep′

r′

)
u

. ‖ba0‖(Ep
r

)
u

∥∥χBa0

∥∥(
Ep′

r′

)
u

‖χB`
‖(

Ep′
r′

)
u

. b(a0−`)/p′ ‖ba0‖(Ep
r

)
u
.

Therefore, for 0 < q ≤ 1, and 0 < α < 1/p′, we have

I1 =
∞∑

`=−∞
b`αq

(
`−w−1∑
a0=−∞

|γa0 | ‖(Tba0)χ`‖(Ep
r

)
u

)q

.
∞∑

`=−∞
b`αq

(
`−w−1∑
a0=−∞

|γa0 |
q b[(a0−`)/p′−a0α]q

)

.
−w−2∑
a0=−∞

|γa0 |
q

∞∑
`=a0+w+1

b(a0−`)[1/p′−α]q

.
−w−2∑
a0=−∞

|γa0 |
q



B. Sultan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6519 8 of 11

. ‖g‖q(
K̇Eα,p

q,r

)
u
(O;Rn)

.

When 1 < q < ∞, 0 < α < 1/p′, and the Hölder inequality yields

I1 .
∞∑

`=−∞
b`αq

(
`−w−1∑
a0=−∞

|γa0 | b(a0−`)/p′−a0α

)q

.
∞∑

`=−∞

(
`−w−1∑
a0=−∞

|γa0 |
q b(a0−`)[1/p′−α]q/2

)(
`−w−1∑
a0=−∞

b(a0−`)[1/p′−α]q′/2

)q/q′

.
∞∑

`=−∞

(
`−w−1∑
a0=−∞

|γa0 |
q b(a0−`)[1/p′−α]q/2

)

.
−w−2∑
a0=−∞

|γa0 |
q

−1∑
`=a0+w+1

b(a0−`)[1/p′−α]q/2

.
−w−2∑
a0=−∞

|γa0 |
q

. ‖g‖q(
K̇Eα,p

q,r

)
u
(O;Rn)

.

Next we find the estimate of I2. If 0 < q ≤ 1, by (Ep
r )u boundedness of T , we get

I2 =

∞∑
`=−∞

b`αq

 ∞∑
a0=`−w

|γa0 | ‖(Tba0)χ`‖(Ep
r

)
u

q

.
∞∑

`=−∞
b`αq

 ∞∑
a0=`−w

|γa0 |
q ‖ ba0 |

q(
Ep

r

)
u


.

∞∑
`=−∞

b`αq

 ∞∑
a0=`−w

|γa0 |
q b−a0αq


.

∞∑
`=−∞

∞∑
a0=`−w

|γa0 |
q b(`−a0)αq

.
∞∑

a0=−∞
|γa0 |

q
a0+w∑
`=−∞

b(`−a0)αq

.
∞∑

a0=−∞
|γa0 |

q

. ‖g‖q(
K̇Eα,p

q,r

)
u
(O;Rn)

.



B. Sultan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6519 9 of 11

If 1 < q < ∞, by using (Ep
r )u boundedness of T and again Hölder inequality to obtain

I2 .
∞∑

`=−∞
b`αq

 ∞∑
a0=`−w

|γa0 | ‖ba0‖(Ep
r

)
u

q

.
∞∑

`=−∞

 ∞∑
a0=`−w

|γa0 | b(`−a0)α

q

.
∞∑

`=−∞

 ∞∑
a0=`−w

|γa0 |
q b(`−a0)αq/2

 ∞∑
a0=`−w

b(`−a0)α(q)
′/2

q/q′

.
∞∑

a0=−∞
|γa0 |

q
a0+w∑
`=−∞

b(`−a0)αq/2

.
∞∑

a0=−∞
|γa0 |

q

. ‖g‖q(
K̇Eα,p

q,r

)
u
(O;Rn)

.

Combining these estimates we get

‖Tg‖(
K̇Eα,p

q,r

)
u
(O;Rn)

. ‖g‖(
K̇Eα,p

q,r

)
u
(O;Rn)

.

Thus, the proof of the Theorem 4.1 is completed.

5. Ethics declarations

Conflict of interest

The authors declare that they have no known competing financial interests or personal
relationships that could have appeared to influence the work reported in this paper.

Ethics approval and consent to participate

This manuscript has not and will not be submitted to more than one journal for
simultaneous consideration. The submitted work is original and will not be published
elsewhere.

Funding

Authors state no funding involved.



B. Sultan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6519 10 of 11

Availability of data

No data is available for this study.

References

[1] P. Auscher and M. Mourgoglou. Representation and uniqueness for boundary
value elliptic problems via first order systems. Revista Matemática Iberoamericana,
35:241–315, 2019.

[2] P. Auscher and C. Prisuelos-Arribas. Tent space boundedness via extrapolation.
Mathematische Zeitschrift, 286:1575–1604, 2017.

[3] M. Sultan and B. Sultan. Λ-Central Musielak–Orlicz–Morrey spaces. Arabian Journal
of Mathematics, 14:357–363, 2025.

[4] M. Sultan and B. Sultan. Boundedness of sublinear operators on grand central Orlicz-
Morrey spaces. Bulletin des Sciences Mathématiques, 205:103704, 2025.

[5] B. Sultan and M. Sultan. Boundedness of higher order commutators of Hardy op-
erators on grand Herz-Morrey spaces. Bulletin des Sciences Mathématiques, 190,
2024.

[6] M. Sultan, B. Sultan, A. Khan, and T. Abdeljawad. Boundedness of Marcinkiewicz
integral operator of variable order in grand Herz-Morrey spaces. AIMS Mathematics,
8(9):22338–22353, 2023.

[7] B. Sultan, M. Sultan, and I. Khan. On Sobolev theorem for higher commutators
of fractional integrals in grand variable Herz spaces. Communications in Nonlinear
Science and Numerical Simulation, 126, 2023.

[8] B. Sultan and M. Sultan. Boundedness of commutators of rough Hardy operators on
grand variable Herz spaces. Forum Mathematicum, 2023.

[9] B. Sultan, M. Sultan, Q. Q. Zhang, and N. Mlaiki. Boundedness of Hardy operators
on grand variable weighted Herz spaces. AIMS Mathematics, 8(10):24515–24527,
2023.

[10] B. Sultan, F. Azmi, M. Sultan, T. Mahmood, N. Mlaiki, and N. Souayah. Bound-
edness of Fractional Integrals on Grand Weighted Herz-Morrey Spaces with Variable
Exponent. Fractal and Fractional, 6(11):660–670, 2022.

[11] B. Sultan, M. Sultan, M. Mehmood, F. Azmi, M. A. Alghafli, and N. Mlaiki. Bound-
edness of fractional integrals on grand weighted Herz spaces with variable exponent.
AIMS Mathematics, 8(1):752–764, 2023.

[12] B. Sultan, F. Azmi, M. Sultan, M. Mehmood, and N. Mlaiki. Boundedness of Riesz
Potential Operator on Grand Herz-Morrey Spaces. Axioms, 11(11):583, 2022.

[13] M. Sultan and B. Sultan. A note on the boundedness of Marcinkiewicz integral
operator on continual Herz-Morrey spaces. Filomat, 39(6):2017–2027, 2025.

[14] J. Younas, A. Hussain, H. Alhazmi, A. F. Aljohani, and I. Khan. BMO estimates
for commutators of the rough fractional Hausdorff operator on grand-variable-Herz-
Morrey spaces. AIMS Mathematics, 9(9):23434–23448, 2024.



B. Sultan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6519 11 of 11

[15] A. Hussain, I. Khan, and A. Mohamed. Variable Her-Morrey estimates for rough
fractional Hausdorff operator. Journal of Inequalities and Applications, 33, 2024.

[16] A. Hussain and G. Gao. Multilinear singular integrals and commutators on Herz
space with variable exponent. ISRN Mathematical Analysis, pages 1–10, 2014.

[17] M. Sultan, B. Sultan, and A. Hussain. Grand Herz–Morrey Spaces with variable
exponent. Mathematical Notes, 114(5):957–977, 2023.

[18] M. Sultan and B. Sultan. A note on the boundedness of higher order commutators
on fractional integrals in grand variable Herz-Morrey spaces. Kragujevac Journal of
Mathematics, 50(7):1063–1080, 2026.

[19] M. Sultan, B. Sultan, and R. E. Castillo. Weighted composition operator on Gamma
spaces with variable exponent. Journal of Pseudo-Differential Operators and Appli-
cations, 15:46, 2024.

[20] B. Sultan and M. Sultan. Sobolev-type Theorem for commutators of Hardy operators
in grand Herz spaces. Ukrainian Mathematical Journal, 76:1196–1213, 2024.

[21] B. Sultan, M. Sultan, A. Khan, and T. Abdeljawad. Boundedness of commutators
of variable Marcinkiewicz fractional integral operator in grand variable Herz spaces.
Journal of Inequalities and Applications, page 93, 2024.

[22] B. Sultan, A. Hussain, and M. Sultan. Characterization of generalized Campanato
spaces with variable exponents via fractional integrals. Journal of Pseudo-Differential
Operators and Applications, 16:22, 2025.

[23] B. Sultan, M. Sultan, and A. Hussain. Boundedness of the Bochner–Riesz operators
on the weighted Herz–Morrey type Hardy spaces. Complex Analysis and Operator
Theory, 19:49, 2025.

[24] A. Hussain and G. Gao. Some new estimates for the commutators of n-dimensional
Hausdorff operator. Applied Mathematics. A Journal of Chinese Universities,
29(2):139–150, 2014.

[25] A. Hussain and G. Gao. Multidimensional Hausdorff operators and commutators on
Herz-type spaces. Journal of Inequalities and Applications, page 594, 2013.

[26] C. Heil. Wiener amalgam spaces in generalized harmonic analysis and wavelet theory.
PhD thesis, University of Maryland, College Park, MD, 1990.

[27] Y. Lu, J. Zhou, and S. Wang. Herz-slice spaces and applications, 2022.
arXiv:2204.08635.


	Introduction
	Preliminaries
	Atomic Decomposition of anisotropic Herz-slice spaces
	 Boundedness of sublinear operators on anisotropic Herz-slice spaces
	Ethics declarations