Abstract
UDC 517.9
Let
Ω
be homogeneous of degree zero, have mean value zero, and integrable on the unit sphere. For
m
∈
ℕ
,
let and let the higher-order commutator of the Marcinkiewicz integral
μ
Ω
,
b
m
be defined by
μ
Ω
,
b
m
(
f
)
(
x
)
=
(
∫
0
∞
|
∫
|
x
-
y
|
≤
t
Ω
(
x
-
y
)
|
x
-
y
|
n
-
1
[
b
(
x
)
-
b
(
y
)
]
m
f
(
y
)
ⅆ
y
|
2
ⅆ
t
t
3
)
1
2
.
We establish a sparse domination of
μ
Ω
,
b
m
for
Ω
∈
L
i
p
(
𝕊
n
-
1
)
. Moreover, we also give Bloom-type characterizations of the two-weighted boundedness of the higher-order commutators
μ
Ω
,
b
m
,
μ
Ω
,
α
,
b
*
,
m
, and
μ
Ω
,
S
,
b
m
, where the higher-order commutators
μ
Ω
,
α
,
b
*
,
m
and
μ
Ω
,
S
,
b
m
are defined, respectively, by
μ
Ω
,
α
,
b
*
,
m
(
f
)
(
x
)
=
(
∫
∫
ℝ
+
n
+
1
(
t
t
+
|
x
-
y
|
)
n
α
|
∫
|
y
-
z
|
≤
t
Ω
(
y
-
z
)
|
y
-
z
|
n
-
1
[
b
(
x
)
-
b
(
z
)
]
m
f
(
z
)
ⅆ
z
|
2
ⅆ
y
ⅆ
t
t
n
+
3
)
1
2
,
α
>
1
,
and
μ
Ω
,
S
,
b
m
(
f
)
(
x
)
=
(
∫
∫
|
x
-
y
|
<
t
|
∫
|
y
-
z
|
≤
t
Ω
(
y
-
z
)
|
y
-
z
|
n
-
1
[
b
(
x
)
-
b
(
z
)
]
m
f
(
z
)
ⅆ
z
|
2
ⅆ
y
ⅆ
t
t
n
+
3
)
1
2
.
Publisher
SIGMA (Symmetry, Integrability and Geometry: Methods and Application)