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PureMathematics
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,2022,12(1),1-13
PublishedOnlineJanuary2022inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2022.121001
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Triebel-Lizorkin
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m
VariationInequalitiesforCohen
TypeCommutatorofOne-Sided
SingularIntegralwithLipschitz
Function
JiaxinJin,XianmingHou
∗
SchoolofMathematicsandStatistics,LinyiUniversity,LinyiShandong
Received:Nov.28
th
,2021;accepted:Dec.31
st
,2021;published:Jan.7
th
,2022
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12(1):1-13.DOI:10.12677/pm.2022.121001
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Abstract
Inthispaper,weintroduce
ρ
-variationoperatorofCohentypecommutatorofone-
sidedsingularintegral.Bytheextrapolationofone-sidedweights,weestablishthe
boundednessoftheaboveoperatorfromweightedLebesguespacestoweighedone-
sidedTriebel-Lizorkinspaces.
Keywords
ρ
-Variation,CohenTypeCommutator,One-SidedWeights,WeighedOne-Sided
Triebel-LizorkinSpaces
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
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k
A
(
m
−
1)
k
Lip
α
|
x
−
y
|
m
−
1+
α
;(3.1)
DOI:10.12677/pm.2022.1210015
n
Ø
ê
Æ
@
„
!
§
û
•
²
|
R
m
(
A
;
x,y
)
−
R
m
(
A
;
z,y
)
|
≤
C
k
A
(
m
−
1)
k
Lip
α
m
−
1
X
l
=1
|
x
−
z
|
l
|
z
−
y
|
m
−
1
−
l
+
α
+
|
x
−
z
|
m
−
1+
α
.
(3.2)
Ú
n
3.2
[17]
-
T
•
½
Â
3
C
∞
c
(
R
)
þ
g
‚
5
Ž
f
¿
÷
v
k
ωTf
k
∞
≤
C
k
fω
k
∞
,
Ù
¥
ω
−
1
∈
A
−
1
;
@
o
,
é
u
1
<p<
∞
,
ω
∈
A
+
p
k
k
Tf
k
L
p
(
ω
)
≤
C
k
f
k
L
p
(
ω
)
.
Ú
n
3.3
[18]
ω
∈
A
−
1
,
•
3
ε
1
>
0
¦
,
é
?
¿
1
<r
≤
1+
ε
1
,
k
ω
r
∈
A
−
1
.
y
[
½
n
2.1
y
²
]
*
χ
(
y
+
ε
i
+1
,y
+
ε
i
]
(
z
)
i
∈
N
,γ
=
{
ε
i
}∈
Θ
F
ρ
≤
1
,
∀
y
∈
R
.
(3.3)
|
^
(3.1),
Œ
•
|V
ρ
(
T
+
A,m
)
f
(
x
)
|≤
V
(
T
+
A,m
)
f
(
x
)
F
ρ
≤
Z
∞
x
χ
(
x
+
ε
i
+1
,x
+
ε
i
]
(
y
)
i
∈
N
,γ
=
{
ε
i
}∈
Θ
F
ρ
|
K
(
x
−
y
)
|
|
y
−
x
|
m
−
1
|
R
m
(
A
;
x,y
)
f
(
y
)
|
dy
.
k
A
(
m
−
1)
k
Lip
α
Z
∞
x
|
f
(
y
)
|
|
x
−
y
|
1
−
α
dy
=
k
A
(
m
−
1)
k
Lip
α
I
+
α
(
|
f
|
)(
x
)
.
|
^
I
+
α
L
p
(
ω
p
)
L
q
(
ω
q
)
k
.
5
,
Œ
kV
ρ
(
T
+
A,m
)
f
k
L
q
(
w
q
)
.
k
A
(
m
−
1)
k
Lip
α
k
I
+
α
(
|
f
|
)
k
L
q
(
w
q
)
.
k
A
(
m
−
1)
k
Lip
α
k
f
k
L
p
(
w
p
)
.
ù
Ò
¤
½
n
2.1
y
²
.
y
[
½
n
2.2
y
²
]
-
x
∈
R
,h>
0
…
I
= [
x,x
+8
h
].
P
f
=
f
1
+
f
2
,
Ù
¥
f
1
(
x
) :=
fχ
I
(
x
).
5
¿
1
h
1+
α
Z
x
+
h
x
V
ρ
(
T
+
A,m
)
f
(
y
)
−
V
ρ
(
T
+
A,m
)
f
[
x,x
+
h
]
dy
≤
2
h
1+
α
Z
x
+2
h
x
V
ρ
(
T
+
A,m
)
f
(
y
)
−V
ρ
(
T
+
A,m
)
f
2
(
x
)
dy
≤
2
h
1+
α
Z
x
+2
h
x
V
ρ
(
T
+
A,m
)
f
1
(
y
)
dy
+
2
h
1+
α
Z
x
+2
h
x
V
ρ
(
T
+
A,m
)
f
2
(
y
)
dy
−V
ρ
(
T
+
A,m
)
f
2
(
x
)
dy
=:
I
1
(
x
)+
I
2
(
x
)
.
DOI:10.12677/pm.2022.1210016
n
Ø
ê
Æ
@
„
!
§
û
•
²
é
u
I
2
(
x
),
k
V
(
T
+
A,m
)
f
2
(
y
)
−
V
(
T
+
A,m
)
f
2
(
x
)
≤
Z
R
|
k
(
y
−
z
)
|
|
y
−
z
|
m
−
1
|
R
m
(
A
;
x,z
)
−
R
m
(
A
;
y,z
)
||
f
2
(
z
)
|
χ
(
y
+
ε
i
+1
,y
+
ε
i
)
(
z
)
dz
+
Z
R
k
(
y
−
z
)
(
z
−
y
)
m
−
1
−
k
(
x
−
z
)
(
z
−
x
)
m
−
1
|
R
m
(
A
;
x,z
)
f
2
(
z
)
|
χ
(
y
+
ε
i
+1
,y
+
ε
i
)
(
z
)
dz
+
Z
R
|
k
(
x
−
z
)
|
|
x
−
z
|
m
−
1
χ
(
y
+
ε
i
+1
,y
+
ε
i
)
(
z
)
−
χ
(
x
+
ε
i
+1
,x
+
ε
i
)
(
z
)
|
R
m
(
A
;
x,z
)
f
2
(
z
)
|
dz
=:
J
1
(
x,y
)+
J
2
(
x,y
)+
J
3
(
x,y
)
.
d
d
Œ
V
ρ
(
T
+
A,m
)
f
2
(
y
)
−V
ρ
(
T
+
A,m
)
f
2
(
x
)
≤k
V
(
T
+
A,m
)
f
2
(
y
)
dy
−
V
(
T
+
A,m
)
f
2
(
x
)
F
ρ
≤k
J
1
(
x,y
)
F
ρ
+
k
J
2
(
x,y
)
F
ρ
+
k
J
3
(
x,y
)
F
ρ
.
é
u
k
J
1
(
x,y
)
F
ρ
,
5
¿
z
∈
(
x
+8
h,
∞
),
y
∈
(
x,x
+2
h
).
Œ
|
x
−
z
|∼|
z
−
y
|
,
|
x
−
y
|≤|
z
−
y
|
.
d
(3.2),
Œ
•
|
R
m
(
A
;
x,z
)
−
R
m
(
A
;
y,z
)
|
.
k
A
(
m
−
1)
k
Lip
α
m
−
1
X
l
=1
|
x
−
y
|
l
|
z
−
y
|
m
−
1
−
l
+
α
+
|
x
−
y
|
m
−
1+
α
.
k
A
(
m
−
1)
k
Lip
α
|
x
−
y
||
z
−
y
|
m
−
2+
α
.
2
(
Ü
(2.1)
Ú
(3.3),
Œ
k
J
1
(
x,y
)
k
F
ρ
≤
Z
R
χ
(
y
+
ε
i
+1
,y
+
ε
i
]
(
z
)
i
∈
N
,γ
=
{
ε
i
}∈
Θ
F
ρ
|
k
(
y
−
z
)
|
|
y
−
z
|
m
−
1
×
|
R
m
(
A
;
x,z
)
−
R
m
(
A
;
y,z
)
||
f
2
(
z
)
|
dz
.
Z
R
|
x
−
y
|
|
x
−
z
|
2
−
α
|
f
2
(
z
)
|
dz
≤
h
Z
R
1
|
x
−
z
|
2
−
α
|
f
2
(
z
)
|
dz.
é
u
J
2
(
x,y
),
5
¿
J
2
(
x,y
) =
Z
R
k
(
y
−
z
)
(
z
−
y
)
m
−
1
−
k
(
x
−
z
)
(
z
−
x
)
m
−
1
|
R
m
(
A
;
x,z
)
f
2
(
z
)
|
χ
(
y
+
ε
i
+1
,y
+
ε
i
)
(
z
)
dz
≤
Z
R
|
k
(
y
−
z
)
−
k
(
x
−
z
)
|
|
z
−
y
|
m
−
1
|
R
m
(
A
;
x,z
)
f
2
(
z
)
|
χ
(
y
+
ε
i
+1
,y
+
ε
i
)
(
z
)
dz
+
Z
R
1
(
z
−
y
)
m
−
1
−
1
(
z
−
x
)
m
−
1
|
k
(
y
−
z
)
R
m
(
A
;
x,z
)
f
2
(
z
)
|
χ
(
y
+
ε
i
+1
,y
+
ε
i
)
(
z
)
dz
=:
J
21
(
x,y
)+
J
22
(
x,y
)
.
DOI:10.12677/pm.2022.1210017
n
Ø
ê
Æ
@
„
!
§
û
•
²
é
u
z
∈
(
x
+8
h,
∞
),
y
∈
(
x,x
+2
h
),
´
y
|
y
−
z
|≥
2
|
x
−
y
|
.
d
(2.2),(3.3)
Ú
(3
.
1),
Œ
k
J
21
(
x,y
)
k
F
ρ
≤
Z
R
χ
(
y
+
ε
i
+1
,y
+
ε
i
]
(
z
)
i
∈
N
,γ
=
{
ε
i
}∈
Θ
F
ρ
×
|
k
(
y
−
z
)
−
k
(
x
−
z
)
|
|
z
−
y
|
m
−
1
|
R
m
(
A
;
x,z
)
f
2
(
z
)
|
dz
.
Z
R
|
x
−
y
|
|
x
−
z
|
2
−
α
|
f
2
(
z
)
|
dz
≤
h
Z
R
1
|
x
−
z
|
2
−
α
|
f
2
(
z
)
|
dz.
|
^
a
q
u
k
J
21
(
x,y
)
k
F
ρ
?
n
,
Œ
k
J
22
(
x,y
)
k
F
ρ
.
h
Z
R
1
|
x
−
z
|
2
−
α
|
f
2
(
z
)
|
dz.
n
þ
Œ
k
J
2
(
x,y
)
k
F
ρ
.
h
Z
R
1
|
x
−
z
|
2
−
α
|
f
2
(
z
)
|
dz.
é
u
{
ε
i
}∈
Θ,
-
N
1
=
{
i
∈
Z
:
ε
i
−
ε
i
+1
≥
y
−
x
}
,
N
2
=
{
i
∈
Z
:
ε
i
−
ε
i
+1
<y
−
x
}
.
K
k
J
3
(
x,y
)
k
F
ρ
≤
n
Z
R
χ
(
y
+
ε
i
+1
,y
+
ε
i
)
(
z
)
−
χ
(
x
+
ε
i
+1
,x
+
ε
i
)
(
z
)
×
k
(
x
−
z
)
(
z
−
x
)
m
−
1
R
m
(
A
;
x,z
)
f
2
(
z
)
dz
o
i
∈
N
1
,β
=
{
ε
i
}∈
Θ
F
ρ
+
n
Z
R
χ
(
y
+
ε
i
+1
,y
+
ε
i
)
(
z
)
−
χ
(
x
+
ε
i
+1
,x
+
ε
i
)
(
z
)
×
k
(
x
−
z
)
(
z
−
x
)
m
−
1
R
m
(
A
;
x,z
)
f
2
(
z
)
dz
o
i
∈
N
2
,β
=
{
ε
i
}∈
Θ
F
ρ
=:
k
J
31
(
x,y
)
k
F
ρ
+
k
J
32
(
x,y
)
k
F
ρ
.
é
u
i
∈
N
1
,
´
y
k
J
31
(
x,y
)
k
F
ρ
≤
n
Z
R
χ
(
x
+
ε
i
+1
,y
+
ε
i
+1
)
(
z
)
k
(
x
−
z
)
(
z
−
x
)
m
−
1
R
m
(
A
;
x,z
)
f
2
(
z
)
dz
o
i
∈
N
1
,β
=
{
ε
i
}∈
Θ
F
ρ
+
n
Z
R
χ
(
x
+
ε
i
,y
+
ε
i
)
(
z
)
k
(
x
−
z
)
(
z
−
x
)
m
−
1
R
m
(
A
;
x,z
)
f
2
(
z
)
dz
o
i
∈
N
1
,β
=
{
ε
i
}∈
Θ
F
ρ
=:
k
L
1
(
x,y
)
k
F
ρ
+
k
L
2
(
x,y
)
k
F
ρ
.
DOI:10.12677/pm.2022.1210018
n
Ø
ê
Æ
@
„
!
§
û
•
²
1
/
(1
−
α
)
<r<ρ
,
d
H¨older
Ø
ª
9
(2.1),(3.1),
Œ
k
L
1
(
x,y
)
k
F
ρ
.
n
Z
R
χ
(
x
+
ε
i
+1
,y
+
ε
i
+1
)
(
z
)
|
f
2
(
z
)
|
|
z
−
x
|
1
−
α
dz
o
i
∈
N
1
,β
=
{
ε
i
}∈
Θ
F
ρ
≤
h
1
/r
0
sup
β
X
i
∈
N
1
Z
R
χ
(
x
+
ε
i
+1
,x
+
ε
i
)
(
z
)
|
f
2
(
z
)
|
r
|
z
−
x
|
(1
−
α
)
r
dz
ρ/r
1
/ρ
≤
h
1
/r
0
Z
R
|
f
2
(
z
)
|
r
|
z
−
x
|
(1
−
α
)
r
dz
1
/r
.
a
q
Œ
k
L
2
(
x,y
)
k
F
ρ
.
h
1
/r
0
Z
R
|
f
2
(
z
)
|
r
|
z
−
x
|
(1
−
α
)
r
dz
1
/r
.
é
u
i
∈
N
2
,
5
¿
ε
i
−
ε
i
+1
≥
y
−
x<
2
h
.
d
H¨older
Ø
ª
9
(2.1),(3.1),
Ï
L
a
q
é
u
k
L
1
(
x,y
)
k
F
ρ
?
n
,
k
k
J
32
(
x,y
)
k
F
ρ
≤
n
Z
R
χ
(
x
+
ε
i
+1
,x
+
ε
i
)
(
z
)
k
(
x
−
z
)
(
z
−
x
)
m
−
1
R
m
(
A
;
x,z
)
f
2
(
z
)
dz
o
i
∈
N
2
,β
=
{
ε
i
}∈
Θ
F
ρ
+
n
Z
R
χ
(
y
+
ε
i
+1
,y
+
ε
i
)
(
z
)
k
(
x
−
z
)
(
z
−
x
)
m
−
1
R
m
(
A
;
x,z
)
f
2
(
z
)
dz
o
i
∈
N
2
,β
=
{
ε
i
}∈
Θ
F
ρ
.
h
1
/r
0
sup
β
X
i
∈
N
2
Z
R
χ
(
x
+
ε
i
+1
,x
+
ε
i
)
(
z
)
|
f
2
(
z
)
|
r
|
z
−
x
|
(1
−
α
)
r
dz
ρ/r
1
/ρ
.
h
1
/r
0
Z
R
|
f
2
(
z
)
|
r
|
z
−
x
|
(1
−
α
)
r
dz
1
/r
.
n
þ
Œ
•
,
k
J
3
(
x,y
)
k
F
ρ
.
h
1
/r
0
Z
R
|
f
2
(
z
)
|
r
|
z
−
x
|
(1
−
α
)
r
dz
1
/r
.
•
Ä
½
Â
3
C
∞
c
(
R
)
þ
n
‡
g
‚
5
Ž
f
:
M
+
1
f
(
x
) = sup
h>
0
1
h
1+
α
Z
x
+2
h
x
V
ρ
(
T
+
A,m
)
f
2
(
y
)
dy,
M
+
2
f
(
x
) = sup
h>
0
1
h
1+
α
Z
x
+2
h
x
Z
R
h
|
x
−
z
|
2
−
α
|
f
2
(
z
)
|
dzdy,
M
+
3
f
(
x
) = sup
h>
0
1
h
1+
α
Z
x
+2
h
x
h
1
/r
0
Z
R
|
f
2
(
z
)
|
r
|
z
−
x
|
(1
−
α
)
r
dz
1
/r
dy.
DOI:10.12677/pm.2022.1210019
n
Ø
ê
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@
„
!
§
û
•
²
é
u
ω
−
1
∈
A
−
1
,
|
^
Ú
n
3.3,
•
3
s>
1
¦
ω
−
s
∈
A
−
1
.
é
u
I
(
x
),
d
H¨older
Ø
ª
Ú
½
n
2.1,
Œ
1
h
1+
α
Z
x
+2
h
x
V
ρ
(
T
+
A,m
)
f
2
(
y
)
dy
≤
1
h
α
+1
/t
Z
x
+2
h
x
V
ρ
(
T
+
A,m
)
f
1
(
y
)
t
dy
1
/t
.
1
h
α
+1
/t
Z
x
+2
h
x
|
f
1
(
y
)
|
s
dy
1
/s
.
h
1
/s
h
α
+1
/t
1
8
h
Z
x
+8
h
x
|
f
(
y
)
|
s
ω
(
y
)
s
ω
(
y
)
−
s
dy
1
/s
.
k
fω
k
∞
ω
(
x
)
−
1
,
Ù
¥
1
/s
−
1
/t
=
α
,
ω
−
s
∈
A
−
1
é
u
ω
−
1
∈
A
−
1
.
@
o
,
k
ωM
+
1
f
k
∞
.
k
fω
k
∞
.
2
(
Ü
Ú
n
3.2,
Œ
•
k
M
+
1
f
k
L
p
(
ω
)
.
k
f
k
L
p
(
ω
)
.
(3.4)
é
u
M
+
2
f
,
Ï
L
H¨older
Ø
ª
Œ
1
h
1+
α
Z
x
+2
h
x
Z
R
h
|
x
−
z
|
2
−
α
|
f
2
(
z
)
|
dzdy
≤
1
h
α
Z
x
+2
h
x
∞
X
k
=3
1
(2
k
h
)
2
−
α
Z
x
+2
k
+1
h
x
+2
k
h
|
f
(
z
)
|
dzdy
≤
∞
X
k
=3
1
2
k
(1
−
α
)
1
2
k
+1
h
Z
x
+2
k
+1
h
x
|
f
(
z
)
|
s
ω
(
z
)
s
ω
(
z
)
−
s
dz
1
/s
≤k
fω
k
∞
∞
X
k
=3
1
2
k
(1
−
α
)
1
2
k
+1
h
Z
x
+2
k
+1
h
x
ω
(
z
)
−
s
dz
1
/s
.
k
fω
k
∞
ω
(
x
)
−
1
,
Ù
¥
0
<α<
1,
ω
−
s
∈
A
−
1
é
u
ω
−
1
∈
A
−
1
.
K
k
ωM
+
2
f
k
∞
.
k
fω
k
∞
.
2
(
Ü
Ú
n
3.2
í
k
M
+
2
f
k
L
p
(
ω
)
.
k
f
k
L
p
(
ω
)
.
(3.5)
DOI:10.12677/pm.2022.12100110
n
Ø
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@
„
!
§
û
•
²
é
u
M
+
3
f
,
k
1
h
1+
α
Z
x
+2
h
x
h
1
/r
0
Z
R
|
f
2
(
z
)
|
r
|
z
−
x
|
(1
−
α
)
r
dz
1
/r
dy
≤
1
h
α
+1
/r
Z
x
+2
h
x
∞
X
k
=3
1
(2
k
h
)
(1
−
α
)
r
Z
x
+2
k
+1
h
x
+2
k
h
|
f
(
z
)
|
dz
1
/r
dy
.
∞
X
k
=3
1
2
k
(1
−
α
−
1
/r
)
1
2
k
+1
h
Z
x
+2
k
+1
h
x
|
f
(
z
)
|
r
dz
1
/r
≤
∞
X
k
=3
1
2
k
(1
−
α
−
1
/r
)
1
2
k
+1
h
Z
x
+2
k
+1
h
x
|
f
(
z
)
|
r
ω
(
z
)
r
ω
(
z
)
−
r
dz
1
/r
≤k
fω
k
∞
∞
X
k
=3
1
2
k
(1
−
α
−
1
/r
)
1
2
k
+1
h
Z
x
+2
k
+1
h
x
ω
(
z
)
−
r
dz
1
/r
.
k
fω
k
∞
ω
(
x
)
−
1
,
Ù
¥
0
<α<
1,
ω
−
r
∈
A
−
1
é
u
ω
−
1
∈
A
−
1
.
Ï
d
,
k
ωM
+
3
f
k
∞
.
k
fω
k
∞
.
|
^
Ú
n
3.2,
k
M
+
3
f
k
L
p
(
ω
)
.
k
f
k
L
p
(
ω
)
.
2
(
Ü
(3.4)
9
(3.5),
¤
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p
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Ø
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Æ