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PureMathematicsnØêÆ,2023,13(5),1157-1172
PublishedOnlineMay2023inHans.https://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2023.135121
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∈BMO(X)ÚT)¤†f[b
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'…c
V‚5Calder´on-ZygumdŽf§†f§¥Banach ¼ê˜m§2ÂMorrey˜m§k.5
BoundednessofBilinearC−ZOperators
andItsCommutatorGeneratedbyon
GeneralizedMorreySpacesAssociated
withBallBanachFunctionSpaces
XuemeiLi
CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu
©ÙÚ^:oÈr.†¥Banach¼ê˜mƒ'2ÂMorrey˜mþV‚5Calder´on-ZygumdŽf9Ù†f
k.5[J].nØêÆ,2023,13(5):1157-1172.DOI:10.12677/pm.2023.135121
oÈr
Received:Apr.2
nd
,2023;accepted:May4
th
,2023;published:May11
th
,2023
Abstract
In this paper, the authors mainly discussthe boundedness of bilinear C−Zoperator T
and itscommutator [b
1
,b
2
,T] on generalized Morreyspacesassociated with ballBanach
functionspacesM
u
(X).TheauthorsprovebilinearC−ZoperatorTisboundedfrom
product spaces M
u
1
(X
1
)×M
u
2
(X
2
)into spaces M
u
(Y).Further, theyalsoprove thatthe
commutator[b
1
,b
2
,T]generatedbyb
1
,b
2
∈BMO(X)andTareboundedfromproduct
spacesM
u
1
(X
1
)×M
u
2
(X
2
)intospacesM
u
(Y),whereu= u
1
u
2
.
Keywords
BilinearC−ZOperator,Commutator,BallBanachFunctionSpaces,Generalized
MorreySpace,Boundedness
Copyright
c
2023byauthor(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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DOI:10.12677/pm.2023.1351211158nØêÆ
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DOI:10.12677/pm.2023.1351211159nØêÆ
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DOI:10.12677/pm.2023.1351211160nØêÆ
oÈr
Ún2.2 [18]X´˜‡¥Banach¼ê˜m,K
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k
X
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X
0
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1
loc
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n
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t
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t
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1
loc
(R
n
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t>0
υ
2
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ω
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υ
1
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2
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R
∞
t
ω( s)ds
sup
s<τ<∞
υ
1
(τ)
<∞,A∼C.
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esssup
t>0
υ
2
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∗
ω
g(t) ≤Cesssup
t>0
υ
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t
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t
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(X)þV‚5ŽfT
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2
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B
k
Y
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k
X
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1
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Y
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X
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¤á.¼êu
1
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: R
n
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r
essinf
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u
1
(x,s)kχ
B
(x,s)k
X
kχ
B
(x,t)k
X
dt
t
≤Cu
2
(x,r),u= u
1
u
2
,(15)
K•3C>0,¦é¤kf
1
∈X
1
T
M
u
1
(X
1
)
Úf
2
∈X
2
T
M
u
2
(X
2
)
,k
kT(f
1
,f
2
)k
M
u
(Y)
≤Ckf
1
k
M
u
1
(X
1
)
kf
2
k
M
u
2
(X
2
)
.
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íØ3.2÷vÚn2.3^‡,…v
2
(t)=u
2
(z,t)
−1
,v
1
(t)=u
1
(z,t)
−1
kχ
B(z,t)
k
−1
X
,g(t)=
kχ
B(z,t)
fk
X
,ω(t) = t
−1
kχ
B(z,t)
k
−1
X
,k
sup
z∈X,r>0
u
2
(z,r)
−1
Z
∞
r
kfχ
B(z,t)
k
X
kχ
B(z,t)
k
−1
X
dt
t
≤Csup
z∈X,r>0
u
1
(z,r)
−1
kχ
B(z,r)
k
−1
X
kfχ
B(z,t)
k
X
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M
u
1
(X)
DOI:10.12677/pm.2023.1351211161nØêÆ
oÈr
½n3.1y²x∈B(z,r) ∈B,éf
i
?1Xe©):
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1
i
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i
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i
χ
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i
χ
X\2B
,i= 1,2(16)
K
kT(f
1
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2
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M
u
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1
1
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1
2
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M
u
(Y)
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1
1
,f
∞
2
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M
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∞
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1
2
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M
u
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∞
1
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∞
2
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M
u
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=: D
1
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2
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3
+D
4
k5OD
1
, dkT(f
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2
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Y
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X
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, u=u
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u
2
, kχ
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k
Y
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X
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X
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1
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B(z,r)
k
Y
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1
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1
2
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Y
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k
Y
kf
1
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B(z,2r)
k
X
1
kf
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χ
B(z,2r)
k
X
2
=C
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u(z,r)
1
kχ
B(z,r)
k
Y
|B(z,r)|kf
1
χ
B(z,2r)
k
X
1
Z
∞
2r
dt
t
1+n
|B(z,r)|kf
2
χ
B(z,2r)
k
X
2
Z
∞
2r
dt
t
1+n
≤C
1
u(z,r)
1
kχ
B(z,r)
k
Y
|B(z,r)|
Z
∞
2r
kf
1
χ
B(z,t)
k
X
1
dt
t
1+n
|B(z,r)|
Z
∞
2r
kf
2
χ
B(z,t)
k
X
2
dt
t
1+n
≤C
1
u(z,r)
1
kχ
B(z,r)
k
Y
kχ
B(z,r)
k
X
1
kχ
B(z,r)
k
X
0
1
Z
∞
2r
kf
1
χ
B(z,t)
k
X
1
dt
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1+n
×kχ
B(z,r)
k
X
2
kχ
B(z,r)
k
X
0
2
Z
∞
2r
kf
2
χ
B(z,t)
k
X
2
dt
t
1+n
≤C
1
u(z,r)
1
kχ
B(z,r)
k
Y
kχ
B(z,r)
k
X
1
Z
∞
2r
kf
1
χ
B(z,t)
k
X
1
kχ
B(z,t)
k
X
0
1
dt
t
1+n
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k
X
2
Z
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2
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k
X
2
kχ
B(z,t)
k
X
0
2
dt
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1+n
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1
kχ
B(z,r)
k
Y
kχ
B(z,r)
k
X
1
Z
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kf
1
χ
B(z,t)
k
X
1
kχ
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k
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X
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k
X
2
Z
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kf
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χ
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k
X
2
kχ
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k
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X
2
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t
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1
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1
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1
kχ
B(z,r)
k
X
1
kχ
B(z,r)
k
X
1
Z
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kf
1
χ
B(z,t)
k
X
1
kχ
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k
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X
1
dt
t
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u
2
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kχ
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k
X
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X
2
Z
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k
X
2
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k
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X
2
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1
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M
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2
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M
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1
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k
M
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2
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2
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1
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2
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DOI:10.12677/pm.2023.1351211162nØêÆ
oÈr
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1
1
,f
∞
2
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Z
2B
Z
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1
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1
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2
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2
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1
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2
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1
dy
2
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Z
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1
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1
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1
Z
X\2B
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2
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2
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k
X
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k
X
0
1
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2
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2
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X
1
Z
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Z
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2
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2
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Z
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2
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Z
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1
χ
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X
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X
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2
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X
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X
0
1
Z
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χ
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X
1
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X
1
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2
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X
1
Z
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χ
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k
X
1
kχ
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k
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X
1
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t
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B(z,2r)
k
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X
1
×
Z
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kf
2
χ
B(z,t)
k
X
2
kχ
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k
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X
2
dt
t
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n
Z
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2r
kf
1
χ
B(z,t)
k
X
1
kχ
B(z,t)
k
−1
X
1
dt
t
Z
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2r
kf
2
χ
B(z,t)
k
X
2
kχ
B(z,t)
k
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X
2
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t
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B(z,r)
k
Y
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B(z,r)
k
X
1
kχ
B(z,r)
k
X
2
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D
2
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1
u(z,r)
1
kχ
B(z,r)
k
Y
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1
1
,f
∞
2
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B(z,r)
k
Y
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z∈X,r>0
r
n
kχ
B(z,r)
k
X
1
kχ
B(z,r)
k
X
2
kχ
B(z,r)
k
Y
1
u
1
(z,r)
Z
∞
2r
kf
1
χ
B(z,t)
k
X
1
kχ
B(z,t)
k
−1
X
1
dt
t
×
1
u
2
(z,r)
Z
∞
2r
kf
2
χ
B(z,t)
k
X
2
kχ
B(z,t)
k
−1
X
2
dt
t
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1
k
M
u
1
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1
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kf
2
k
M
u
2
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2
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3
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M
u
1
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1
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M
u
2
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2
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DOI:10.12677/pm.2023.1351211163nØêÆ
oÈr
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∞
1
,f
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2
)|
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X\2B
Z
X\2B
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1
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2
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1
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1
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∞
2
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2
)|dy
1
dy
2
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Z
(X\2B)
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2
Y
i=1
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i
(y
i
)|
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i
|
n
dy
i
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Z
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2
Y
i=1
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i
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i
)|
|z−y
i
|
n
dy
i
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2
Y
i=1
Z
X\2B
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i
(y
i
)|
Z
∞
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i
|
dt
t
n+1
dy
i
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2
Y
i=1
Z
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Z
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i
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i
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i
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i
dt
t
n+1
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2
Y
i=1
Z
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Z
B(z,t)
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i
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i
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i
dt
t
n+1
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2
Y
i=1
Z
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2r
kf
i
χ
B(z,t)
k
X
i
kχ
B(z,t)
k
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X
i
dt
t
,
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D
4
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1
u(z,r)
1
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B(z,r)
k
Y
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∞
1
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∞
2
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B(z,r)
k
Y
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2
Y
i=1
sup
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1
u
i
(z,r)
1
kχ
B(z,r)
k
X
i
kχ
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k
X
i
Z
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kf
i
χ
B(z,t)
k
X
i
kχ
B(z,t)
k
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X
i
dt
t
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1
k
M
u
1
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1
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2
k
M
u
2
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2
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2
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3
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DOI:10.12677/pm.2023.1351211164nØêÆ
oÈr
kfk
BMO(X)
=sup
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n
kχ
B
(f−f
B
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X
kχ
B(z,r)
k
X
.(18)
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B(x,r)
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B(x,t)
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r
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k
Y
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k
X
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k
X
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X
1
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X
2
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1
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n
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t
r
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essinf
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1
(x,s)kχ
B
(x,s)k
X
kχ
B
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X
dt
t
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2
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1
u
2
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M
u
1
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1
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M
u
2
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2
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k[b
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2
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1
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2
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M
u
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1
k
∗
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2
k
∗
kf
1
k
M
u
1
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1
)
kf
2
k
M
u
2
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2
)
.
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k
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X
,g(t)=
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fk
X
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k
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X
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u
2
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Z
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r
(1+ln
t
r
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B(z,t)
k
X
kχ
B(z,t)
k
−1
X
dt
t
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z∈X,r>0
u
1
(z,r)
−1
kχ
B(z,r)
k
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X
kfχ
B(z,t)
k
X
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M
u
1
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i
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i
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i
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i
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i
χ
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i
χ
X\2B
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2
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2
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M
u
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1
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2
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1
1
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1
2
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M
u
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1
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2
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1
1
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2
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M
u
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1
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2
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1
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1
2
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M
u
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1
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2
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1
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2
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M
u
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1
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2
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3
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4
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1
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2
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1
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2
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Y
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k
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2
k
∗
kf
1
k
X
1
kf
2
k
X
2
,u= u
1
u
2
,
kχ
B(z,r)
k
Y
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B(z,r)
k
X
1
kχ
B(z,r)
k
X
2
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DOI:10.12677/pm.2023.1351211165nØêÆ
oÈr
1
u(z,r)
1
kχ
B(z,r)
k
Y
kχ
B(z,r)
[b
1
,b
2
,T](f
1
1
,f
1
2
)k
Y
≤C
1
u(z,r)
1
kχ
B(z,r)
k
Y
k[b
1
,b
2
,T](f
1
1
,f
1
2
)k
Y
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1
u(z,r)
1
kχ
B(z,r)
k
Y
kb
1
k
∗
kb
2
k
∗
kf
1
χ
B(z,2r)
k
X
1
kf
2
χ
B(z,2r)
k
X
2
=Ckb
1
k
∗
kb
2
k
∗
1
u(z,r)
1
kχ
B(z,r)
k
Y
2
Y
i=1
|B(z,r)|kf
i
χ
B(z,2r)
k
X
i
Z
∞
2r
dt
t
n+1
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1
k
∗
kb
2
k
∗
1
u(z,r)
1
kχ
B(z,r)
k
Y
2
Y
i=1
kχ
B(z,r)
k
X
i
kχ
B(z,r)
k
X
0
i
Z
∞
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kf
i
χ
B(z,t)
k
X
i
dt
t
n+1
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1
k
∗
kb
2
k
∗
1
u(z,r)
1
kχ
B(z,r)
k
Y
2
Y
i=1
kχ
B(z,r)
k
X
i
Z
∞
2r
kf
i
χ
B(z,t)
k
X
i
kχ
B(z,t)
k
−1
X
i
dt
t
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1
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∗
kb
2
k
∗
1
u
1
(z,r)
kχ
B(z,r)
k
X
1
kχ
B(z,r)
k
X
1
Z
∞
2r
kf
1
χ
B(z,t)
k
X
1
kχ
B(z,t)
k
−1
X
1
dt
t
×
1
u
2
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kχ
B(z,r)
k
X
2
kχ
B(z,r)
k
X
2
Z
∞
2r
kf
2
χ
B(z,t)
k
X
2
kχ
B(z,t)
k
−1
X
2
dt
t
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1
k
∗
kb
2
k
∗
kf
1
k
M
u
1
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1
)
kf
2
k
M
u
2
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2
)
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1
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1
k
∗
kb
2
k
∗
kf
1
k
M
u
1
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1
)
kf
2
k
M
u
2
(X
2
)
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2
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|[b
1
,b
1
,T](f
1
1
,f
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2
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c
(z,2r),k
1
2
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3
2
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|[b
1
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1
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1
1
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2
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X\2B
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1
,y
2
)||b
1
(x)−b
1
(y
1
)||b
2
(x)−b
1
(y
2
)||f
1
1
(y
1
)||f
∞
2
(y
2
)|dy
1
dy
2
≤C
Z
B(z,2r)
Z
B
c
(z,2r)
|b
1
(x)−b
1
(y
1
)||b
2
(x)−b
1
(y
2
)||f
1
(y
1
)||f
2
(y
2
)|
(|x−y
1
|+|x−y
2
|)
2n
dy
1
dy
2
≤C
Z
B(z,2r)
|b
1
(x)−b
1
(y
1
)||f
1
(y
1
)|dy
1
Z
B
c
(z,2r)
|b
2
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1
(y
2
)||f
2
(y
2
)|
|x−y
2
|
2n
dy
2
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
|b
1
(x)−(b
1
)
B(z,2r)
|
Z
B(z,2r)
|f
1
(y
1
)|dy
1
+
Z
B(z,2r)
|b
1
(y
1
)−(b
1
)
B(z,2r)
||f
1
(y
1
)|dy
1

×

|b
2
(x)−(b
2
)
B(z,2r)
|
Z
B
c
(z,2r)
|f
2
(y
2
)|
|z−y
2
|
2n
dy
2
+
Z
B
c
(z,2r)
|b
2
(y
2
)−(b
2
)
B(z,2r)
||f
2
(y
2
)|
|z−y
2
|
2n
dy
2

≤C

|b
1
(x)−(b
1
)
B(z,2r)
|kf
1
χ
B(z,2r)
k
X
1
kχ
B(z,2r)
k
X
0
1
+kf
1
χ
B(z,2r)
k
X
1
kχ
B(z,2r)
(b
1
(·)−(b
1
)
B(z,2r)
)k
X
0
1

×

|b
2
(x)−(b
2
)
B(z,2r)
|
Z
B
c
(z,2r)
|f
2
(y
2
)|
Z
∞
|z−y|
dt
t
n+1
dy
2
+
Z
B
c
(z,2r)
|b
2
(y
2
)−(b
2
)
B(z,2r)
||f
2
(y
2
)|
Z
∞
|z−y|
dt
t
n+1
dy
2

DOI:10.12677/pm.2023.1351211166nØêÆ
oÈr
=C

|b
1
(x)−(b
1
)
B(z,2r)
|kf
1
χ
B(z,2r)
k
X
1
kχ
B(z,2r)
k
X
0
1
+kf
1
χ
B(z,2r)
k
X
1
kχ
B(z,2r)
(b
1
(·)−(b
1
)
B(z,2r)
)k
X
0
1
kχ
B(z,2r)
k
X
0
1
kχ
B(z,2r)
k
X
0
1

×

|b
2
(x)−(b
2
)
B(z,2r)
|
Z
∞
2r
Z
2r≤|z−y|<t
|f
2
(y
2
)|dy
2
dt
t
n+1
+
Z
∞
2r
Z
2r≤|z−y|<t
|b
2
(y
2
)−(b
2
)
B(z,2r)
||f
2
(y
2
)dy
2
|
dt
t
n+1

≤C

|b
1
(x)−(b
1
)
B(z,2r)
|kf
1
χ
B(z,2r)
k
X
1
kχ
B(z,2r)
k
X
0
1
+kf
1
χ
B(z,2r)
k
X
1
kb
1
k
∗
kχ
B(z,2r)
k
X
0
1

×

|b
2
(x)−(b
2
)
B(z,2r)
|
Z
∞
2r
Z
B(z,t)
|f
2
(y
2
)|dy
2
dt
t
n+1
+
Z
∞
2r
Z
B(z,t)
|b
2
(y
2
)−(b
2
)
B(z,2r)
||f
2
(y
2
)dy
2
|
dt
t
n+1

≤Ckf
1
χ
B(z,2r)
k
X
1
kχ
B(z,2r)
k
X
0
1

kb
1
k
∗
+|b
1
(x)−(b
1
)
B(z,2r)
|

×

|b
2
(x)−(b
2
)
B(z,2r)
|
Z
∞
2r
kf
2
χ
B(z,t)
k
X
2
kχ
B(z,t)
k
X
0
2
dt
t
n+1
+
Z
∞
2r
Z
B(z,t)
|b
2
(y
2
)−(b
2
)
B(z,t)
||f
2
(y
2
)dy
2
|
dt
t
n+1
+
Z
∞
2r
Z
B(z,t)
|(b
2
)
B(z,2r)
−(b
2
)
B(z,t)
||f
2
(y
2
)dy
2
|
dt
t
n+1

≤C|B(z,r)|kf
1
χ
B(z,2r)
k
X
1
Z
∞
2r
dt
t
n+1
|B(z,2r)|kχ
B(z,2r)
k
−1
X
1

kb
1
k
∗
+|b
1
(x)−(b
1
)
B(z,2r)
|

×

|b
2
(x)−(b
2
)
B(z,2r)
|
Z
∞
2r
kf
2
χ
B(z,t)
k
X
2
kχ
B(z,t)
k
−1
X
2
dt
t
+
Z
∞
2r
k(b
2
(·)−(b
2
)
B(z,t)
)χ
B(z,t)
k
X
0
2
kf
2
χ
B(z,t)
k
X
2
dt
t
n+1
+
Z
∞
2r
|(b
2
)
B(z,2r)
−(b
2
)
B(z,t)
|kf
2
χ
B(z,t)
k
X
2
kχ
B(z,t)
k
X
0
2
dt
t
n+1

≤C|B(z,r)|
Z
∞
2r
kf
1
χ
B(z,t)
k
X
1
dt
t
n+1
|B(z,2r)|kχ
B(z,2r)
k
−1
X
1

kb
1
k
∗
+|b
1
(x)−(b
1
)
B(z,2r)
|

×

|b
2
(x)−(b
2
)
B(z,2r)
|
Z
∞
2r
kf
2
χ
B(z,t)
k
X
2
kχ
B(z,t)
k
−1
X
2
dt
t
+
Z
∞
2r
k(b
2
(·)−(b
2
)
B(z,t)
)χ
B(z,t)
k
X
0
2
kχ
B(z,t)
k
X
0
2
kf
2
χ
B(z,t)
k
X
2
kχ
B(z,t)
k
−1
X
2
dt
t
+
Z
∞
2r
|(b
2
)
B(z,2r)
−(b
2
)
B(z,t)
|kf
2
χ
B(z,t)
k
X
2
kχ
B(z,t)
k
−1
X
2
dt
t

DOI:10.12677/pm.2023.1351211167nØêÆ
oÈr
≤Ckχ
B(z,r)
k
X
1
Z
∞
2r
kf
1
χ
B(z,t)
k
X
1
kχ
B(z,t)
k
X
0
1
dt
t
n+1
|B(z,2r)|kχ
B(z,2r)
k
−1
X
1
×

kb
1
k
∗
+|b
1
(x)−(b
1
)
B(z,2r)
|

×

|b
2
(x)−(b
2
)
B(z,2r)
|
Z
∞
2r
kf
2
χ
B(z,t)
k
X
2
kχ
B(z,t)
k
−1
X
2
dt
t
+kb
2
k
∗
Z
∞
2r
(1+ln
t
r
)kf
2
χ
B(z,t)
k
X
2
kχ
B(z,t)
k
−1
X
2
dt
t
≤Cr
n
Z
∞
2r
kf
1
χ
B(z,t)
k
X
1
kχ
B(z,t)
k
−1
X
1
dt
t

kb
1
k
∗
+|b
1
(x)−(b
1
)
B(z,2r)
|

×

|b
2
(x)−(b
2
)
B(z,2r)
|
Z
∞
2r
kf
2
χ
B(z,t)
k
X
2
kχ
B(z,t)
k
−1
X
2
dt
t
+kb
2
k
∗
Z
∞
2r
(1+ln
t
r
)kf
2
χ
B(z,t)
k
X
2
kχ
B(z,t)
k
−1
X
2
dt
t
,
?,dª(4),(18)ÚíØ4.5,±9kχ
B(z,r)
k
Y
= kχ
B(z,r)
k
X
1
kχ
B(z,r)
k
X
2
,
E
2
=sup
z∈X,r>0
1
u(z,r)
1
kχ
B(z,r)
k
Y
k[b
1
,b
2
,T](f
1
1
,f
∞
2
)χ
B(z,r)
k
Y
≤Csup
z∈X,r>0
r
n
kχ
B(z,r)
k
X
1
kχ
B(z,r)
k
X
2
kχ
B(z,r)
k
Y
kb
1
k
∗
1
u
1
(z,r)
Z
∞
2r
kf
1
χ
B(z,t)
k
X
1
kχ
B(z,t)
k
−1
X
1
dt
t
×

kb
2
k
∗
1
u
2
(z,r)
Z
∞
2r
kf
2
χ
B(z,t)
k
X
2
kχ
B(z,t)
k
−1
X
2
dt
t
+kb
2
k
∗
1
u
2
(z,r)
Z
∞
2r
(1+ln
t
r
)kf
2
χ
B(z,t)
k
X
2
kχ
B(z,t)
k
−1
X
2
dt
t

≤Ckb
1
k
∗
kb
2
k
∗
kf
1
k
M
u
1
(X
1
)
kf
2
k
M
u
2
(X
2
)
.
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2
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E
3
≤Ckb
1
k
∗
kb
2
k
∗
kf
1
k
M
u
1
(X
1
)
kf
2
k
M
u
2
(X
2
)
.
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4
,éu∀x∈B(z,r),y∈B
c
(z,2r),Œ±
1
2
|z−y|≤|x−y|≤
3
2
|z−y|.$^ª
(5),(10),(11),(12),(18),(19),(20)ÚíØ4.5±9Fuibin’s ½n,k
|[b
1
,b
1
,T](f
∞
1
,f
∞
2
)(x)|
≤C
Z
(X\2B)
2
|b
1
(x)−b
1
(y
1
)||b
2
(x)−b
2
(y
2
)||f
1
(y
1
)||f
2
(y
2
)|
(|x−y
1
|+|x−y
2
|)
2n
dy
1
dy
2
≤C
2
Y
i=1
Z
X\2B
|b
i
(x)−b
i
(y
i
)||f
i
(y
i
)|
|x−y
i
|
n
dy
i
≤C
2
Y
i=1
|b
i
(x)−(b
i
)
B(z,r)
|
Z
X\2B
|f
i
(y
i
)|
|z−y
i
|
n
dy
i
+C
2
Y
i=1
Z
X\2B
|b
i
(y
i
)−(b
i
)
B(z,r)
||f
i
(y
i
)|
|z−y
i
|
n
dy
i
DOI:10.12677/pm.2023.1351211168nØêÆ
oÈr
≤C
2
Y
i=1
|b
i
(x)−(b
i
)
B(z,r)
|
Z
X\2B
|f
i
(y
i
)|
Z
∞
|z−y|
dt
t
n+1
dy
i
+C
2
Y
i=1
Z
X\2B
|b
i
(y
i
)−(b
i
)
B(z,r)
||f
i
(y
i
)|
Z
∞
|z−y|
dt
t
n+1
dy
i
=C
2
Y
i=1
|b
i
(x)−(b
i
)
B(z,r)
|
Z
∞
2r
Z
2r≤|z−y|<t
|f
i
(y
i
)|dy
i
dt
t
n+1
dy
i
+C
2
Y
i=1
Z
∞
2r
Z
2r≤|z−y|<t
|b
i
(y
i
)−(b
i
)
B(z,r)
||f
i
(y
i
)|dy
i
dt
t
n+1
≤C
2
Y
i=1
|b
i
(x)−(b
i
)
B(z,r)
|
Z
∞
2r
Z
B(z,t)
|f
i
(y
i
)|dy
i
dt
t
n+1
dy
i
+C
2
Y
i=1
Z
∞
2r
Z
B(z,t)
|b
i
(y
i
)−(b
i
)
B(z,r)
||f
i
(y
i
)|dy
i
dt
t
n+1
≤C
2
Y
i=1
|b
i
(x)−(b
i
)
B(z,r)
|
Z
∞
2r
kf
i
χ
B(z,t)
k
X
i
kχ
B(z,t)
k
X
0
i
dt
t
n+1
+C
2
Y
i=1

Z
∞
2r
Z
B(z,t)
|b
i
(y
i
)−(b
i
)
B(z,t)
||f
i
(y
i
)|dy
i
dt
t
n+1
+
Z
∞
2r
Z
B(z,t)
|(b
i
)
B(z,r)
−(b
i
)
B(z,t)
||f
i
(y
i
)|dy
i
dt
t
n+1

≤C
2
Y
i=1
|b
i
(x)−(b
i
)
B(z,r)
|
Z
∞
2r
kf
i
χ
B(z,t)
k
X
i
kχ
B(z,t)
k
−1
X
i
dt
t
+C
2
Y
i=1

Z
∞
2r
k

b
i
(·)−(b
i
)
B(z,t)

χ
B(z,t)
k
X
0
i
kf
i
χ
B(z,t)
k
X
i
dt
t
n+1
+
Z
∞
2r
|(b
i
)
B(z,r)
−(b
i
)
B(z,t)
|kf
i
χ
B(z,t)
k
X
i
kχ
B(z,t)
k
X
0
i
dt
t
n+1

≤C
2
Y
i=1
|b
i
(x)−(b
i
)
B(z,r)
|
Z
∞
2r
kf
i
χ
B(z,t)
k
X
i
kχ
B(z,t)
k
−1
X
i
dt
t
+C
2
Y
i=1

Z
∞
2r
k

b
i
(·)−(b
i
)
B(z,t)

χ
B(z,t)
k
X
0
i
kχ
B(z,t)
k
X
0
i
kf
i
χ
B(z,t)
k
X
i
kχ
B(z,t)
k
−1
X
i
dt
t
+
Z
∞
2r
|(b
i
)
B(z,r)
−(b
i
)
B(z,t)
|kf
i
χ
B(z,t)
k
X
i
kχ
B(z,t)
k
−1
X
i
dt
t

≤C
2
Y
i=1
|b
i
(x)−(b
i
)
B(z,r)
|
Z
∞
2r
kf
i
χ
B(z,t)
k
X
i
kχ
B(z,t)
k
−1
X
i
dt
t
+C
2
Y
i=1
kb
i
k
∗
Z
∞
2r
(1+ln
t
r
)kf
i
χ
B(z,t)
k
X
i
kχ
B(z,t)
k
−1
X
i
dt
t
DOI:10.12677/pm.2023.1351211169nØêÆ
oÈr
?˜Ú,dª(4),(18)ÚíØ4.5,
E
4
=sup
z∈X,r>0
1
u(z,r)
1
kχ
B(z,r)
k
Y
k[b
1
,b
2
,T](f
∞
1
,f
∞
2
)χ
B(z,r)
k
Y
≤C
2
Y
i=1
sup
z∈X,r>0
1
u(z,r)
1
kχ
B(z,r)
k
Y
k

b
i
(x)−(b
i
)
B(z,r)

χ
B(z,r)
k
Y
×
Z
∞
2r
kf
i
χ
B(z,t)
k
X
i
kχ
B(z,t)
k
−1
X
i
dt
t
+C
2
Y
i=1
sup
z∈X,r>0
1
u(z,r)
1
kχ
B(z,r)
k
Y
kχ
B(z,r)
k
Y
kb
i
k
∗
×
Z
∞
2r
(1+ln
t
r
)kf
i
χ
B(z,t)
k
X
i
kχ
B(z,t)
k
−1
X
i
dt
t
≤C
2
Y
i=1
sup
z∈X,r>0
1
u(z,r)
1
kχ
B(z,r)
k
Y
k

b
i
(x)−(b
i
)
B(z,r)

χ
B(z,r)
k
Y
kχ
B(z,r)
k
Y
kχ
B(z,r)
k
Y
×
Z
∞
2r
kf
i
χ
B(z,t)
k
X
i
kχ
B(z,t)
k
−1
X
i
dt
t
+C
2
Y
i=1
kb
i
k
∗
sup
z∈X,r>0
1
u(z,r)
Z
∞
2r
(1+ln
t
r
)kf
i
χ
B(z,t)
k
X
i
kχ
B(z,t)
k
−1
X
i
dt
t
≤C
2
Y
i=1
kb
i
k
∗
kf
i
k
M
u
i
(X
i
)
.
(ÜE
1
,E
2
,E
3
O,½n4.4y..
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Ü“‰ŒÆ2022cÝïÄ)‰ï]Ï‘8(2022KYZZ-S121)"
ë•©z
[1]Coifman,R.R.andMeyer,Y.(1975)OnCommutatorsofSingularIntegralsandBilinear
SingularIntegrals.TransactionsoftheAMS,212,315-331.
https://doi.org/10.1090/S0002-9947-1975-0380244-8
[2]Hu, G.E.and Meng,Y. (2012) Multilinear Calder´on-Zygmund Operator on Products of Hardy
Spaces.ActaMathematicaSinica,EnglishSeries,28,281-294.
https://doi.org/10.1007/s10114-012-0240-y
[3]Lu,Y.andZhu,Y.P.(2014)BoundednessofMultilinearCalder´on-ZygmundSingularOpera-
torsonMorrey-HerzSpaceswithVariableExponents.ActaMathematica Sinica,30,1180-1194.
https://doi.org/10.1007/s10114-014-3410-2
DOI:10.12677/pm.2023.1351211170nØêÆ
oÈr
[4]Wang,P.W.andLiu,Z.G.(2017)WeightedNormInequalitiesforMultilinearCalder´on-
ZygmundOperatorsinGeneralizedMorreySpaces.JournalofInequalitiesandApplications,
2017,ArticleNo.48.https://doi.org/10.1186/s13660-017-1325-z
[5]Sawano, Y.,Ho, K.-P.,Yang,D.andYang, S.(2017)HardySpacesforBallQuasi-BanachFunc-
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