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PureMathematicsnØêÆ,2022,12(1),1-13
PublishedOnlineJanuary2022inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2022.121001
ü>ÛÉÈ©†Lipschitz¼ê)¤
Cohen.†fCØª
@@@„„„!!!§§§ûûû•••²²²
∗
§ŒÆêƆÚOÆ§ìÀ§
ÂvFϵ2021c1128F¶¹^Fϵ2021c1231F¶uÙFϵ2022c17F
Á‡
©Ú\ü>ÛÉÈ©†Lipschitz¼ê)¤Cohen.†fCŽf.|^ü>í
{,ïáþãŽfl\ü>Triebel-Lizorkin˜m\Lebesgue˜mk.5.
'…c
ρ-C§Cohen.†f§ü>§\ü>Triebel-Lizorkin˜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
∗ÏÕŠö
©ÙÚ^:@„!,û•².ü>ÛÉÈ©†Lipschitz¼ê)¤Cohen.†fCØª[J].nØêÆ,2022,
12(1):1-13.DOI:10.12677/pm.2022.121001
@„!§û•²
Abstract
Inthispaper,weintroduceρ-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/
1.Úó†Ì‡(J
T={T
ε
}
ε
´˜Žf¿÷vlim
ε→0
T
ε
f(x)=Tf(x)A??¤á.²;•{´ÏLï
Ä(
P
∞
i=1
|T
ε
i
f−T
ε
i
+1
f|
2
)
1/2
a.¼ê5ÿþŽfx{T
ε
}Âñ„Ý.•˜„ó,½Âe¡
Žfµ
O(Tf)(x) =

∞
X
i=1
sup
t
i+1
≤ε
i+1
<ε
i
≤t
i
|T
ε
i
+1
f(x)−T
ε
i
f(x)|
2

1/2
,
Ù¥{t
i
}´˜‰½üNªu0ê.d,,˜«•{´•Äe¡ρCŽf:
V
ρ
(Tf)(x) =sup
ε
i
&0

∞
X
i=1
|T
ε
i
+1
f(x)−T
ε
i
f(x)|
ρ

1/ρ
,
Ù¥ρ>2,þ(.H¤k4~ªu0S{ε
i
}.
CŽfïÄ3VÇØ,H{nØ9NÚ©Û+•kX-‡A^.L´epingle[1]Äk
S† CØª.Bourgain[2]ïá?XÚH{²þCØª.k'†
CŽfïÄ9A^ë„©z[3][4][5].ÛÉÈ©Žf†fïÄ•´NÚ©Û¥-‡©|,
§3‡©•§¥kX4Ù-‡Š^.LiuÚWu[6]‰Ñ˜‡˜‘œ¹eäkIOØCalder´on-
ZygmundÛÉÈ©Žf†f†CŽf\L
p
k.5½½n.ü>ÛÉÈ©Ž
DOI:10.12677/pm.2022.1210012nØêÆ
@„!§û•²
f†BMO¼ê)¤†fCŽf\k.53©z[7]¥ïá.FuÚLin[8]ü
>Žf†Lipschitz¼ê)¤†f\k.5.ZhangÚWu[9]ïáÛÉÈ©†Lipschitz¼
ê)¤†f†CØª.ZhangÚHou[10]‰Ñü>ÛÉÈ©†Cohen.†f
LipschitzO. Cohen [11]Ú\ÛÉÈ©Cohen.†f¿ïÄÙk.5.ChenÚLu [12]ï
áÛÉÈ©Cohen.†flTriebel-Lizorkin˜mLebesguek.5.ɱþ(Jéu,
©òÚ\ü>ÛÉÈ©†Lipschitz¼ê)¤Cohen.†fCŽf¿ïÄÙ\k.5.
2.ý•£
3©z[13]¥,Aimar,Forzani†Mart´ın-ReyesÚ\ü>Calder´on-ZygmundÛÉÈ©:
T
+
f(x) =lim
ε→0
+
T
+
ε
f(x) =lim
ε→0
+
Z
∞
x+ε
K(x−y)f(y)dy,
Ù¥|3R
−
= (−∞,0)ؼêK¡•ü>Calder´on-ZygmundØ…÷v


Z
a<|x|<b
K(x)dx


≤C,0 <a<b,
|K(x)|≤C/|x|,x6= 0,(2.1)
|K(x−y)−K(x)|≤C|y|/|x|
2
,|x|>2|y|>0.(2.2)
ùaؼê˜‡~f•
K(x) =
sin(log|x|)
(xlog|x|)
χ
(−∞,0)
(x).
1986c,Sawyer[14]ÄkÚ\ü>MuckenhouptA
+
p
9A
−
p
5?ne¡ü>4ŒŽf:
M
+
f(x) := sup
h>0
1
h
Z
x+h
x
|f(y)|dy,M
−
f(x) := sup
h>0
1
h
Z
x
x−h
|f(y)|dy.
éu1 ≤p<∞,eω÷v
sup
a<b<c
1
(c−a)
p
Z
b
a
ω(x)dx

Z
c
b
ω(x)
1−p
0
dx

p−1
<∞,
½
sup
a<b<c
1
(c−a)
p
Z
c
b
ω(x)dx

Z
b
a
ω(x)
1−p
0
dx

p−1
<∞,
K¡ωáuA
+
p
½A
−
p
.p= 1ž,•3~êC¦ω÷v
M
−
ω(x) ≤Cω(x),M
+
ω(x) ≤Cω(x).
DOI:10.12677/pm.2022.1210013nØêÆ
@„!§û•²
K¡ωáuA
+
1
½A
−
1
.éu1≤p<∞,kA
p
$A
+
p
…A
p
$A
−
p
.5¿ω(x)=e
x
áuA
+
p
Ø
áuA
p
.aq,éu¤ka<b<c∈R,0 <α<1,1 <p<q…1/p−1/q= α,ω÷v
1
(c−a)
1−α

Z
b
a
ω
q

1/q

Z
c
b
ω
−p
0

1/p
0
<C,
½
1
(c−a)
1−α

Z
c
b
ω
q

1/q

Z
b
a
ω
−p
0

1/p
0
<C
K¡ωáuA
+
(p,q)½A
−
(p,q).p= 1,1−1/q= αž,•3~êC¦ω÷v
M
−
ω(x)
q
≤Cω(x)
q
,M
+
ω(x)
q
≤Cω(x)
q
K¡ωáuA
+
(1,q)½A
−
(1,q).éu0 <α<1,½Âü>©êgÈ©:
I
+
α
(f)(x) :=
Z
∞
x
1
(y−x)
1−α
f(y)dy.
Andersen†Sawyer[15]I
+
α
´lL
p
(ω
p
)L
q
(ω
q
)k.,Ù¥ω∈A
+
(p,q)…1/p−1/q= α.
-A(x)•RþÛ܌ȼê.éum≥1,R
m
(A;x,y)•¼êA(x)3y?V?êm{‘.
=
R
m
(A;x,y) := A(x)−
X
|k|≤m−1
1
k!
A
(k)
(y)(x−y)
k
.
½ÂXeü>ÛÉÈ©Cohen.†f:
T
ε,+
A,m
f(x) =
Z
∞
x+ε
K(x−y)
(y−x)
m−1
R
m
(A;x,y)f(y)dy.(2.3)
-Θ={β:β={ε
i
},i∈R,ε&0}.•Ä8ÜN×Θ…^F
ρ
L«¤k÷vkgk
F
ρ
<∞¼
êg(i,β)¤·Ü‰ê˜m,Ù¥
kgk
F
ρ
= sup
β

X
i
|g(i,β)|
ρ

1/ρ
.
‰½L
p
(R)þ˜ŽfT= {T
+
t
}
t>0
,•ÄF
ρ
-ŠŽfV(T) : f→V(T)f•:
V(T)f(x) =

T
ε
i
+1
f(x)−T
ε
i
f(x)

β={ε
i
}∈Θ
,
Ù¥

T
ε
i
+1
f(x)−T
ε
i
f(x)

β={ε
i
}∈Θ
´F
ρ
¥ƒ,…±e¡/ª‰Ñ
(i,β) = (i,{ε
i
}) →T
[ε
i
+1,ε
i
]
f(x).
DOI:10.12677/pm.2022.1210014nØêÆ
@„!§û•²
5¿
V
ρ
(Tf)(x) = kV(T)f(x)k
F
ρ
.
½Â2.1éu0 <α<1,e¼êf÷v
kfk
Lip
α
=sup
x,h∈R,h6=0
f(x+h)−f(x)
|h|
α
<∞,
K¡f∈Lip
α
.
½Â2.2éu0 <α<1,1 <p<∞9Ü·¼êω,e¼êf÷v
kfk
˙
F
α,∞
p,+
≈



sup
h>0
1
h
1+α
Z
x+h
x


f(y)−f
[x,x+h]


dy



L
p
(ω)
<∞,
K¡fáu\ü>Tribel-Lizorkin˜m
˙
F
α,∞
p,+
.
3©¥,·‚òïáXe(J:
½n2.1-m≥1,ρ>2,0<α<1…A
(k)
∈Lip
α
,k=0,1,...,m−1.T
+
A,m
={T
ε,+
A,m
}
ε>0
.
e1 <p<q<∞,1/p−1/q= α,Kéω∈A
+
(p,q),k
kV
ρ
(T
+
A,m
)fk
L
q
(w
q
)
.kA
(m−1)
k
Lip
α
kfk
L
p
(w
p
)
.
½n2.2-m≥1,0<α<1…A
(k)
∈Lip
α
,k= 0,1,...,m−1.T
+
A,m
= {T
ε,+
A,m
}
ε>0
.e1 <p<
∞,ρ>max{2,1/(1−α)},Kéω∈A
+
p
,k
kV
ρ
(T
+
A,m
)fk
˙
F
α,∞
p,+
.kfk
L
p
(w)
.
••Bå„,3©¥·‚^i1CL«˜‡†ŸCþ Ã'~ê,3ØÓ ˜ÙŠ
Œ±ØÓ.ef≤CgÚf.g.f,©OP•f.g,f∼g.
3.½n2.1Ú½n2.2y²
3y²½n2.1ƒc,·‚k‰Ñe¡Ún.
Ún3.1[16]eA
(m−1)
∈Lip
α
(0 <α≤1),•3~êC¦
|R
m
(A;x,y)|≤CkA
(m−1)
k
Lip
α
|x−y|
m−1+α
;(3.1)
DOI:10.12677/pm.2022.1210015nØêÆ
@„!§û•²
|R
m
(A;x,y)−R
m
(A;z,y)|
≤CkA
(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)
Ún3.2[17]-T•½Â3C
∞
c
(R)þg‚5Žf¿÷v
kωTfk
∞
≤Ckfωk
∞
,
Ù¥ω
−1
∈A
−
1
;@o,éu1 <p<∞,ω∈A
+
p
k
kTfk
L
p
(ω)
≤Ckfk
L
p
(ω)
.
Ún3.3[18]ω∈A
−
1
,•3ε
1
>0¦,é?¿1 <r≤1+ε
1
,kω
r
∈A
−
1
.
y[½n2.1y²]*



χ
(y+ε
i+1
,y+ε
i
]
(z)

i∈N,γ={ε
i
}∈Θ


F
ρ
≤1,∀y∈R.(3.3)
|^(3.1),Υ
|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
.kA
(m−1)
k
Lip
α
Z
∞
x
|f(y)|
|x−y|
1−α
dy
= kA
(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
)fk
L
q
(w
q
)
.kA
(m−1)
k
Lip
α
kI
+
α
(|f|)k
L
q
(w
q
)
.kA
(m−1)
k
Lip
α
kfk
L
p
(w
p
)
.
ùÒ¤½n2.1y².
y[½n2.2y²]-x∈R,h>0…I= [x,x+8h].Pf=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+2h
x


V
ρ
(T
+
A,m
)f(y)−V
ρ
(T
+
A,m
)f
2
(x)


dy
≤
2
h
1+α
Z
x+2h
x


V
ρ
(T
+
A,m
)f
1
(y)


dy+
2
h
1+α
Z
x+2h
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.2022.1210016nØêÆ
@„!§û•²
éuI
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).
ddŒ


V
ρ
(T
+
A,m
)f
2
(y)−V
ρ
(T
+
A,m
)f
2
(x)


≤kV(T
+
A,m
)f
2
(y)dy−V(T
+
A,m
)f
2
(x)


F
ρ
≤kJ
1
(x,y)


F
ρ
+kJ
2
(x,y)


F
ρ
+kJ
3
(x,y)


F
ρ
.
éukJ
1
(x,y)


F
ρ
,5¿z∈(x+8h,∞),y∈(x,x+2h).Œ|x−z|∼|z−y|,|x−y|≤|z−y|.
d(3.2),Υ
|R
m
(A;x,z)−R
m
(A;y,z)|.kA
(m−1)
k
Lip
α

m−1
X
l=1
|x−y|
l
|z−y|
m−1−l+α
+|x−y|
m−1+α

.kA
(m−1)
k
Lip
α
|x−y||z−y|
m−2+α
.
2(Ü(2.1)Ú(3.3),Œ
kJ
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.
éuJ
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.2022.1210017nØêÆ
@„!§û•²
éuz∈(x+8h,∞),y∈(x,x+2h),´y|y−z|≥2|x−y|.d(2.2),(3.3)Ú(3.1),Œ
kJ
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.
|^aqukJ
21
(x,y)k
F
ρ
?n,Œ
kJ
22
(x,y)k
F
ρ
.h
Z
R
1
|x−z|
2−α
|f
2
(z)|dz.
nþŒ
kJ
2
(x,y)k
F
ρ
.h
Z
R
1
|x−z|
2−α
|f
2
(z)|dz.
éu{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
kJ
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
=: kJ
31
(x,y)k
F
ρ
+kJ
32
(x,y)k
F
ρ
.
éui∈N
1
,´y
kJ
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
=: kL
1
(x,y)k
F
ρ
+kL
2
(x,y)k
F
ρ
.
DOI:10.12677/pm.2022.1210018nØêÆ
@„!§û•²
1/(1−α) <r<ρ,dH¨olderØª9(2.1),(3.1),Œ
kL
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
.
aqŒ
kL
2
(x,y)k
F
ρ
.h
1/r
0

Z
R
|f
2
(z)|
r
|z−x|
(1−α)r
dz

1/r
.
éui∈N
2
,5¿ε
i
−ε
i+1
≥y−x<2h.dH¨olderØª9(2.1),(3.1),ÏLaqé
ukL
1
(x,y)k
F
ρ
?n,k
kJ
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
.
nþŒ•,
kJ
3
(x,y)k
F
ρ
.h
1/r
0

Z
R
|f
2
(z)|
r
|z−x|
(1−α)r
dz

1/r
.
•Ä½Â3C
∞
c
(R)þn‡g‚5Žf:
M
+
1
f(x) = sup
h>0
1
h
1+α
Z
x+2h
x


V
ρ
(T
+
A,m
)f
2
(y)


dy,
M
+
2
f(x) = sup
h>0
1
h
1+α
Z
x+2h
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+2h
x
h
1/r
0

Z
R
|f
2
(z)|
r
|z−x|
(1−α)r
dz

1/r
dy.
DOI:10.12677/pm.2022.1210019nØêÆ
@„!§û•²
éuω
−1
∈A
−
1
,|^Ún3.3,•3s>1¦ω
−s
∈A
−
1
.éuI(x),dH¨olderØªÚ½n2.1,
Œ
1
h
1+α
Z
x+2h
x


V
ρ
(T
+
A,m
)f
2
(y)


dy≤
1
h
α+1/t

Z
x+2h
x


V
ρ
(T
+
A,m
)f
1
(y)


t
dy

1/t
.
1
h
α+1/t

Z
x+2h
x
|f
1
(y)|
s
dy

1/s
.
h
1/s
h
α+1/t

1
8h
Z
x+8h
x
|f(y)|
s
ω(y)
s
ω(y)
−s
dy

1/s
.kfωk
∞
ω(x)
−1
,
Ù¥1/s−1/t= α,ω
−s
∈A
−
1
éuω
−1
∈A
−
1
.@o,
kωM
+
1
fk
∞
.kfωk
∞
.
2(ÜÚn3.2,Œ•
kM
+
1
fk
L
p
(ω)
.kfk
L
p
(ω)
.(3.4)
éuM
+
2
f,ÏLH¨olderØªŒ
1
h
1+α
Z
x+2h
x
Z
R
h
|x−z|
2−α
|f
2
(z)|dzdy
≤
1
h
α
Z
x+2h
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
≤kfωk
∞
∞
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
.kfωk
∞
ω(x)
−1
,
Ù¥0 <α<1,ω
−s
∈A
−
1
éuω
−1
∈A
−
1
.K
kωM
+
2
fk
∞
.kfωk
∞
.
2(ÜÚn3.2í
kM
+
2
fk
L
p
(ω)
.kfk
L
p
(ω)
.(3.5)
DOI:10.12677/pm.2022.12100110nØêÆ
@„!§û•²
éuM
+
3
f,k
1
h
1+α
Z
x+2h
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+2h
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
≤kfωk
∞
∞
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
.kfωk
∞
ω(x)
−1
,
Ù¥0 <α<1,ω
−r
∈A
−
1
éuω
−1
∈A
−
1
.Ïd,
kωM
+
3
fk
∞
.kfωk
∞
.
|^Ún3.2,
kM
+
3
fk
L
p
(ω)
.kfk
L
p
(ω)
.
2(Ü(3.4)9(3.5),¤½n2.2y².
Ä7‘8
ìÀŽg,‰ÆÄ7]Ï‘8(No:ZR2020QA006,GrantNos:ZR2019YQ04,2020KJI002,
ZR2021MA079)"
ë•©z
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