T
ẠP CHÍ KHOA HỌC
TRƯ
ỜNG ĐẠI HỌC SƯ PHẠM TP HỒ CHÍ MINH
Tập 22, Số 3 (2025): 414-423
HO CHI MINH CITY UNIVERSITY OF EDUCATION
JOURNAL OF SCIENCE
Vol. 22, No. 3 (2025): 414-423
ISSN:
2734-9918
Websit
e: https://journal.hcmue.edu.vn https://doi.org/10.54607/hcmue.js.22.3.4266(2025)
414
Research Article1
CHARACTERIZING THE COMPACTNESS
OF CALDERÓN-ZYGMUND COMMUTATORS OF TYPE THETA
ON GENERALIZED MORREY-LORENTZ SPACES
Pham Ngoc Xuan Vy, Phan Thanh Phat, Le Minh Thuc, Du Kim Thanh, Tran Tri Dung*
Ho Chi Minh City University of Education, Vietnam
*Corresponding author: Tran Tri Dung– Email: dungtt@hcmue.edu.vn
Received: May 08, 2024; Revised: June 19, 2024; Accepted: September 18, 2024
ABSTRACT
In this paper, we characterize the compactness of the Calderón-Zygmund commutator
[ ]
,bT
of type
θ
on generalized Morrey-Lorentz spaces
( )
,npr
ϕ
M
. More precisely, we prove that if
( )
n
b CMO∈
, which is the
( )
n
BMO
closure of
, then
[ ]
,bT
is a compact operator
on
()
,n
pr
ϕ
M
for all
1p< <∞
and
1r≤ <∞
.
Keywords: Calderón-Zygmund commutator of type
θ
; generalized Morrey-Lorentz space;
compactness
1. Introduction
It is known that the theory of commutators has many important applications to some
nonlinear partial differential equations. When
T
is a Calderón-Zygmund operator and
( )
n
b BMO∈
, the
p
L
- compactness of [b, T] was first obtained by Uchiyama (1978). Since
then, the compactness of commutators of classical operators on various function spaces has
been considered. Beatrous and Li (1993) proved the boundedness and compactness
characterization of
[ ]
,bT
on
( )
p
LX
, where
X
is a space of homogeneous type. After that,
Chen (2011) obtained the compactness of commutators for singular integrals on Morrey
spaces. Later, the Lorentz boundedness and compactness of integral commutators on spaces
of homogeneous type were proved by Dao and Krantz (2021). More recently, Tran et al.
(2024) proved a compactness characterization of commutators of Calderón-Zygmund type
in generalized Morrey-Lorentz spaces.
Cite this article as: Pham, N. X. V., Phan, T. P., Le, M. T., Du, K. T., & Tran, T. D. (2025). Characterizing the
compactness of Calderón-Zygmund commutators of type theta on generalized Morrey-Lorentz spaces. Ho Chi
Minh City University of Education Journal of Science, 22(3), 414-423.
https://doi.org/10.54607/hcmue.js.22.3.4266(2025)
HCMUE Journal of Science
Vol. 22, No. 3 (2025): 414-423
415
On the other hand, Yabuta (1985) first introduced Calderón-Zygmund operators of
type theta to facilitate his study of certain classes of pseudodifferential operators. Since then,
many researchers have further studied the properties of these operators and their
commutators. Liu et al. (2002) showed that if
( )
n
b BMO∈
and
T
is a Calderón-Zygmund
operator of type theta, then
[ ]
,bT
is bounded from
( )
1n
H
to weak
( )
1n
L
. Later, Thai et
al. (2022) obtained the boundedness of Calderón-Zygmund operators and commutators of
type theta on generalized weighted Lorentz spaces
( )
p
u
wΛ
. Recently, Le et al. (2024)
proved that the commutators
[ ]
,bT
of type theta are also bounded on generalized Morrey-
Lorentz spaces.
For the reader’s convenience, we recall below the definition of a generalized Morrey-
Lorentz space and Calderón-Zygmund operator
T
of type theta.
Definition 1.1. Let
0p< <∞
,
0r< ≤∞
and
ϕ
be functions satisfying the following
conditions:
()()
: 0; 0;
) s,
) () s , ,
) (2 ) ( ), , .0 01
p
tt
i is nonincrea ing
ii B t is nondecrea ing for any ball B X
iii sfor ome c ttD ntt ons a Dt
ϕ
ϕ
ϕϕ
∞→ ∞
⊂
≤ ∀> < <
(1.1)
Then the generalized Morrey-Lorentz space
( )
,npr
ϕ
M
is defined as a set of all real-valued
functions
f
with finite norm:
()
( )
( )
,
,
,
1/
( ,)
: sup ,
, ()
p
r
r
p
L Bxt
p
B xt
f
fB xt t
ϕ
ϕ
=
M
(1.2)
where the supremum is taken over all balls
( )
,B xt
in
n
, and
( )
()
,
,
pr
L Bxt
f
denotes the
Lorentz norm of
f
on
( )
,B xt
(see Grafakos, 2008, Definition 1.4.6).
Definition 1.2. (Yabuta, 1985) Let
θ
be a nonnegative, nondecreasing function on
(0, )∞
with
()
11
0
t t dt
θ
−<∞
∫
. (1.3)
A continuous function
(,)Kxy
on
{( , ): }\
nn n
xx x×∈
is said to be a standard kernel
of type
θ
if it satisfies the following conditions.
(i)
| ( , )| .
||
n
C
Kxy xy
≤−
(1.4)
HCMUE Journal of Science
Pham Ngoc Xuan Vy et al.
416
(ii)
0
0 00 0
||
| ( , ) ( , )| | ( , ) ( , )| | | ,
||
nxx
Kxy Kx y Kyx Kyx C x y yx
θ
−
−
− + − ≤− 
−

(1.5)
for every
0
,,xx y
with
00
2| | | |xx yx− <−
.
Definition 1.3. (Yabuta, 1985) Let
θ
be a function as in Definition 1.3. A linear operator
T
from
()
n
to
()
n
′
is said to be a Calderón-Zygmund operator of type
θ
if it satisfies
the following conditions.
(i)
T
is bounded on
2
()
n
L
, which means
22
LL
Tf C f≤
for every
( )
0n
fC
∞
∈
. (1.6)
(ii) There exists a standard kernel
K
of type
θ
such that for every function
0
()
n
fC
∞
∈
and
()
x supp f∉
() (,) () .
n
Tf x K x y f y dy=
∫
(1.7)
Definition 1.4. Let
T
be a Calderón-Zygmund operator of type
θ
with strong conditions in
(1.7). Suppose that
( )
n
b BMO∈
, then the commutator
[ ]
,bT
of type
θ
is defined by
[ ]
( ) ( ) ( )( ) ( )( )
,:bTfx bxTfxTbfx= −
(1.8)
for measurable functions
.f
Definition 1.5. (i) A function
()
1n
loc
bL∈
is said to belong to
( )
n
BMO
if
( )
1
: sup ( ) ,
nB
BMO B
B
b b x b dx
B
= − <∞
∫
where
1()
BB
b b x dx
B
=
∫
, and the supremum is taken over all balls
n
B⊂
(ii) We denote by
( )
n
CMO
, the
BMO
closure of
( )
n
c
C
∞
, where
is the set
of all functions in
( )
n
C
∞
with compact support.
Inspired by the above works, this study aims to study the compactness of Calderón–
Zygmund commutators of type theta on generalized Morrey – Lorentz spaces in this paper.
More specifically, in Section 2, we first recall a characterization of a precompact subset in
( )
,npr
ϕ
M
for
1p< <∞
and
1r≤ <∞
(see Lemma 2.1). Then we prove that if
( )
n
b CMO∈
then
[ ]
,bT
is a compact operator on
( )
,npr
ϕ
M
(see Theorem 2.1).
As usual, for any
1q≤ <∞
, we denote by
'q
the conjugate exponent of
q
, that is,
111
'qq
+=
. We also denote a constant by
C
, which only depends on
,,,,pqrn
ϕ
and may
HCMUE Journal of Science
Vol. 22, No. 3 (2025): 414-423
417
change on different lines. In addition, we write
A B
if there exists a constant
0C>
such
that
A CB≤
and
AB
if
A B
and
B A
. Finally, we denote by
t
B
a ball in
n
with
radius
0
t>
and by
1A
the characteristic function of a subset
n
A⊂
.
2. Main results
The following lemmas are needed for proving this study’s main result.
Lemma 2.1. (Tran et al., 2024, Lemma 5.1) Let
1p< <∞
,
1r≤ ≤∞
and
ϕ
satisfy (1.1).
Assume that the set
in
( )
,npr
ϕ
M
satisfies the following conditions:
(i)
,
sup
pr
M
f
f
ϕ
∈
<∞
,
(ii)
( ) ( )
,
0
lim 0
pr
M
y
f yf
ϕ
→
⋅+−⋅ =
uniformly in
f∈
,
(iii)
,
lim 1 0
cpr
R
B
RM
f
ϕ
→∞
=
uniformly in
f∈
.
Then
is precompact in
( )
,n
pr
ϕ
M
.
Lemma 2.2. (Le et al., 2024, Theorem 2.2) Let
1p< <∞
,
1r≤ ≤∞
and
ϕ
satisfy (1.1). Let
T
be a Calderón – Zygmund operator of type
θ
with
( )
11
0
logt t t dt
θ
−
<∞
∫
. If
( )
n
b BMO
∈
then
[
]
,
bT
maps
( ) ( )
,,pr prnn
ϕϕ
→MM
. Moreover, we have
( )
,
,
,
pr
pr
BMO
bT f fb
ϕ
ϕ


M
M
,
for any
( )
,npr
f
ϕ
∈M
.
It is now ready to prove the following main theorem.
Theorem 2.1. Let
( )
1,p∈∞
,
[ ]
1,r∈∞
and
ϕ
satisfy (1.1). If
()
∈
n
b CMO
, and
T
is a
Calderón – Zygmund operator of type
θ
with
( )
11
0
logt t t dt
θ
−
<∞
∫
, then
[ ]
,bT
is a compact
operator on
( )
,npr
ϕ
M
.
Proof. Assume that
()
∈
n
b CMO
. Let
be a bounded set in
( )
,npr
ϕ
M
. We need to show
that
[ ]
( )
,bT
is precompact in
( )
,npr
ϕ
M
.
Indeed, since
( )
∈
n
b CMO
, then for every
0
ε
>
, there exists a function
( )
n
c
bC
ε
∞
∈
such that
BMO
bb
ε
ε
−<
.
By the triangle inequality and Lemma 2.1, we have for every
f∈
HCMUE Journal of Science
Pham Ngoc Xuan Vy et al.
418
[ ]
( )
[ ]
( )
[ ]
( )
[ ]
( )
[ ]
( )
, ,,
,,
,
, ,,
,
,.
pr pr pr
pr pr
pr
M MM
BMO M M
M
bTf bbTf bTf
b b f bT f
C bT f
ϕ ϕϕ
ϕϕ
ϕ
εε
εε
ε
ε
≤− +
−+
+
With this inequality in mind, it suffices to show that
[ ]
( )
,bT
ε
is precompact in
( )
,npr
ϕ
M
for a given
0
ε
>
small enough.
Since
is a bounded set in
( )
,npr
ϕ
M
, it follows from Lemma 2.2 that
[ ]
( )
,bT
ε
satisfies
(i) of Lemma 2.1.
Next, we show that
[ ]
( )
,bT
ε
also satisfies (iii) of Lemma 2.1. Indeed, suppose that
( )
supp
R
bB
ε
ε
⊂
, for some
10
R
ε
>
. Then, for any
f∈
and for
c
R
xB∈
with
10RR
ε
>
, we
observed that
xy x−≈
for any
R
yB
ε
∈
.
Thus, for any
c
R
xB∈
we have
[ ]
( )( ) ( )( ) ( ) ( )
( )
( )
( )
( )
( )
0,
0,
0,
,,
LBR
n
LBR
n
LBR
bTfx Tbfx b Kxyfydy
C
b f y dy
xy
C
b f y dy
RR
ε
ε
ε
ε εε
ε
ε
ε
∞
∞
∞
= ≤
≤−
≤−
∫
∫
∫
( )
( )
,
.
1
2
LnBxR
bf x w dw
R
ε
ε
∞
≤−



∫
(2.10)
For every ball
( )
0
,
t
B Bxt=
in
n
, by (2.10), and Minkowski’s inequality, we obtain:
[ ]
( )
( )
( ) ( )
( )
( )
( )
( )
( )
( ) ( )
( ) ( )
,
,
,0,
1,
1
,
1
,,
0
,1
1
2
11
,
22
cpr
Rt
pr t
pr
pr
BLB LnLB
BxR
pp
tt
L B x wt
LL
nn
M
BxR BxR
p
bT f bf w dw
Bt B tR
f
bb
dw f dw
B x wt t
RR
ε
ϕ
εε
εε
εε
ϕϕ
ϕ
∞
∞∞
−
⋅−



 
−
 
 
∫
∫∫
