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PureMathematicsnØêÆ,2023,13(4),917-934
PublishedOnlineApril2023inHans.https://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2023.134097
V‚5©êgÈ©Žf34ŒC•IHerz
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V‚5©êgÈ©Žf§4ŒC•IHerz˜m§k.5
BoundednessofBilinearFractional
IntegralOperatorsonGrand
VariableHerzSpaces
GuangjieFang
CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu
Received:Mar.18
th
,2023;accepted:Apr.19
th
,2023;published:Apr.26
th
,2023
Abstract
BasedontheboundednessofbilinearfractionalintegrationoperatorsonLebesgue
©ÙÚ^:•1#.V‚5©êgÈ©Žf34ŒC•IHerz˜mþk.5[J].nØêÆ,2023,13(4):917-934.
DOI:10.12677/pm.2023.134097
•1#
spaceswithvariableexponent,byusinghierarchicaldecompositionoffunctionand
realmethodsinharmonicanalysis,theboundednessofbilinearfractionalintegral
operatorsisobtainedongrandvariableHerzspaces.
Keywords
BilinearFractionalIntegralOperator,GrandVariableHerzSpace,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.2023.134097918nØêÆ
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DOI:10.12677/pm.2023.134097919nØêÆ
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DOI:10.12677/pm.2023.134097920nØêÆ
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DOI:10.12677/pm.2023.134097921nØêÆ
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ε
θ
∞
X
k=0
2
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X
l
1
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X
l
2
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l
1
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l
2
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k
k
L
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n
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.
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i
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i
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l
i
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l
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l
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n
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l
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l
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2
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1
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L
1
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l
2
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L
1
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1
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1
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1
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r
2
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r
i
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1
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X
l
1
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X
l
2
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l
1
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l
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k
k
L
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n
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X
l
1
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X
l
2
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l
1
k
L
p
1
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n
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1
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0
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l
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L
p
2
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n
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l
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L
p
0
2
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n
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k
L
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n
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X
l
1
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X
l
2
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0
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1
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L
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1
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2
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n
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k
L
r
1
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n
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k
L
r
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n
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X
l
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l
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r
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r
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L
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n
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l
2
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L
p
2
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n
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l
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1
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l
1
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L
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1
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n
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X
l
2
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p
2
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l
2
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L
p
2
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n
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.(5)
|^H¨olderØªÚ®•^‡
1
q
=
1
q
1
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1
q
2
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V
11
6Csup
ε>0
(
ε
θ
−1
X
k=−∞
k−2
X
l
1
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2
kα
1
(0)+(k−l
1
)(
n
p
1
(0)
−n)
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l
1
k
L
p
1
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(R
n
)
!
q
1
(1+ε)
)
1
q
1
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×sup
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(
ε
θ
−1
X
k=−∞
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X
l
2
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2
kα
2
(0)+(k−l
2
)(
n
p
2
(0)
−n)
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l
2
k
L
p
2
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n
)
!
q
2
(1+ε)
)
1
q
2
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:= V
111
V
112
.
DOI:10.12677/pm.2023.134097922nØêÆ
•1#
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1
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1
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n
p
1
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−n<0 …2
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1
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V
111
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ε>0
(
ε
θ
−1
X
k=−∞
k−2
X
l
1
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a
1
(k−l
1
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1
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2
2
l
1
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1
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1
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l
1
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q
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L
p
1
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n
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×
k−2
X
l
1
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1
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1
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1
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1
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1
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0
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q
1
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θ
−1
X
l
1
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l
1
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1
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1
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l
1
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q
1
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L
p
1
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n
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X
k=l
1
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2
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1
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1
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1
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2
!
1
q
1
(1+ε)
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θ
−1
X
l
1
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l
1
α
1
(0)q
1
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l
1
k
q
1
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L
p
1
(·)
(R
n
)
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1
q
1
(1+ε)
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k
˙
K
α
1
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1
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p
1
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n
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112
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k
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2
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p
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n
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k
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α
1
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1
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p
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n
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˙
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α
2
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2
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p
2
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12
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V
12
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ε
θ
∞
X
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2
kα(∞)q(1+ε)
−1
X
l
1
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X
l
2
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l
1
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l
2
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k
k
L
p(·)
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n
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q(1+ε)
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1
q(1+ε)
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(
ε
θ
∞
X
k=0
2
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X
l
1
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X
l
2
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α
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l
1
,f
l
2
)χ
k
k
L
p(·)
(R
n
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q(1+ε)
)
1
q(1+ε)
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121
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122
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11
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i
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L
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1
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n
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0
1
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L
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n
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0
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L
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1
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n
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l
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L
p
2
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n
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X
l
1
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1
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n
p
1
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l
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L
p
1
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l
2
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n
p
2
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l
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L
p
2
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n
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DOI:10.12677/pm.2023.134097923nØêÆ
•1#
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q
=
1
q
1
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1
q
2
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V
121
6Csup
ε>0
(
ε
θ
∞
X
k=0
−1
X
l
1
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2
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1
(∞)+(k−l
1
)(
n
p
1
(0)
−n)
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l
1
k
L
p
1
(·)
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n
)
!
q
1
(1+ε)
)
1
q
1
(1+ε)
×sup
ε>0
(
ε
θ
∞
X
k=0
−1
X
l
2
=−∞
2
kα
2
(∞)+(k−l
2
)(
n
p
2
(0)
−n)
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l
2
k
L
p
2
(·)
(R
n
)
!
q
2
(1+ε)
)
1
q
2
(1+ε)
:= D
1
D
2
.
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1
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1
(0)+
n
p
1
(0)
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D
1
6Csup
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(
ε
θ
∞
X
k=0
−1
X
l
1
=−∞
2
b
1
(k−l
1
)
q
1
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2
2
l
1
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1
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1
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l
1
k
q
1
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L
p
1
(·)
(R
n
)
!
×
−1
X
l
1
=−∞
2
b
1
(k−l
1
)
q
1
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0
2
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1
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(q
1
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0
)
1
q
1
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6Csup
ε>0
(
ε
θ
−1
X
k=−∞
2
−l
1
b
1
q
1
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2
2
l
1
α
1
(∞)q
1
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l
1
k
q
1
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L
p
1
(·)
(R
n
)
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∞
X
k=0
2
kb
1
q
1
(1+ε)
2
!)
1
q
1
(1+ε)
6Csup
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ε
θ
−1
X
l
1
=−∞
2
l
1
α
1
(∞)q
1
(1+ε)
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l
1
k
q
1
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L
p
1
(·)
(R
n
)
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1
q
1
(1+ε)
6Ckf
1
k
˙
K
α
1
(·),q
1
),θ
p
1
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(R
n
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1
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2
O.(ÜV
11
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12
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1
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p
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n
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p
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n
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2
.dMinkowskiØª, Œ
V
2
6Csup
ε>0
(
ε
θ
−1
X
k=−∞
2
kα(0)q(1+ε)
k−2
X
l
1
=−∞
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X
l
2
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l
1
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l
2
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k
k
L
p(·)
(R
n
)
!
q(1+ε)
)
1
q(1+ε)
+Csup
ε>0
(
ε
θ
∞
X
k=0
2
kα(∞)q(1+ε)
k−2
X
l
1
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X
l
2
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l
1
,f
l
2
)χ
k
k
L
p(·)
(R
n
)
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)
1
q(1+ε)
:= V
21
+V
22
.
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1
6k−2, k−1 6l
2
6k+1, x∈E
k
,y
1
∈E
l
1
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2
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l
2
ž,k
|x−y
1
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2
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1
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1
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k−2
,
DOI:10.12677/pm.2023.134097924nØêÆ
•1#
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β
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l
1
,f
l
2
)(x)|6C2
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l
1
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L
1
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n
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l
2
k
L
1
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n
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.
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p(x)
=
1
r
1
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r
2
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1
r
i
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=
1
p
i
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−
β
2n
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k−2
X
l
1
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k+1
X
l
2
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β
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l
1
,f
l
2
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k
k
L
p(·)
(R
n
)
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X
l
1
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2
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1
)(
n
p
1
(0)
−n)
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l
1
k
L
p
1
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(R
n
)
!
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k+1
X
l
2
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2
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n
p
2
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−n)
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l
2
k
L
p
2
(·)
(R
n
)
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.
|^H¨older ØªÚ
1
q
=
1
q
1
+
1
q
2
,
V
21
6Csup
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(
ε
θ
−1
X
k=−∞
k−2
X
l
1
=−∞
2
kα
1
(0)+(k−l
1
)(
n
p
1
(0)
−n)
kf
l
1
k
L
p
1
(·)
(R
n
)
!
q
1
(1+ε)
)
1
q
1
(1+ε)
×sup
ε>0
(
ε
θ
−1
X
k=−∞
k+1
X
l
2
=k−1
2
kα
2
(0)+(k−l
2
)(
n
p
2
(0)
−n)
kf
l
2
k
L
p
2
(·)
(R
n
)
!
q
2
(1+ε)
)
1
q
2
(1+ε)
:= V
211
V
212
.
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211
= V
111
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212
.
V
212
6Csup
ε>0
(
ε
θ
−1
X
k=−∞
2
kα
2
(0)q
2
(1+ε)
k+1
X
l
2
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2
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2
)(
n
p
2
(0)
−n)
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l
2
k
L
p
2
(·)
(R
n
)
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q
2
(1+ε)
)
1
q
2
(1+ε)
6Csup
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θ
−1
X
k=−∞
2
kα
2
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2
(1+ε)
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l
2
k
q
2
(1+ε)
L
p
2
(·)
(R
n
)
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1
q
2
(1+ε)
6Ckf
2
k
˙
K
α
2
(·),q
2
),θ
p
2
(·)
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n
)
.
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22
OXe.dMinkowskiØª, Œ
V
22
6Csup
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(
ε
θ
∞
X
k=0
2
kα(∞)q(1+ε)
−1
X
l
1
=−∞
k+1
X
l
2
=k−1
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β
(f
l
1
,f
l
2
)χ
k
k
L
p(·)
(R
n
)
!
q(1+ε)
)
1
q(1+ε)
+Csup
ε>0
(
ε
θ
∞
X
k=0
2
kα(∞)q(1+ε)
k−2
X
l
1
=0
k+1
X
l
2
=k−1
kBI
β
(f
l
1
,f
l
2
)χ
k
k
L
p(·)
(R
n
)
!
q(1+ε)
)
1
q(1+ε)
:= V
221
+V
222
.
DOI:10.12677/pm.2023.134097925nØêÆ
•1#
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2
(0) 6p
2
(∞)9Ún1-3,Œ
k−2
X
l
1
=−∞
k+1
X
l
2
=k−1
kBI
β
(f
l
1
,f
l
2
)χ
k
k
L
p(·)
(R
n
)
6C
−1
X
l
1
=−∞
2
(k−l
1
)(
n
p
1
(0)
−n)
kf
l
1
k
L
p
1
(·)
(R
n
)
!
×
k+1
X
l
2
=k−1
2
(k−l
2
)(
n
p
2
(∞)
−n)
kf
l
2
k
L
p
2
(·)
(R
n
)
!
.
5¿
1
q
=
1
q
1
+
1
q
2
.|^H¨older Øª,
V
221
6Csup
ε>0
(
ε
θ
∞
X
k=0
−1
X
l
1
=−∞
2
kα
1
(∞)+(k−l
1
)(
n
p
1
(0)
−n)
kf
l
1
k
L
p
1
(·)
(R
n
)
!
q
1
(1+ε)
)
1
q
1
(1+ε)
×sup
ε>0
(
ε
θ
∞
X
k=0
k+1
X
l
2
=k−1
2
kα
2
(∞)+(k−l
2
)(
n
p
2
(∞)
−n)
kf
l
2
k
L
p
2
(·)
(R
n
)
!
q
2
(1+ε)
)
1
q
2
(1+ε)
:= E
1
E
2
.
duE
1
duD
1
, •I^α
1
(∞) “Oα
1
(0); E
2
O†V
212
Oƒq, •I^α
2
(∞) Ú
p
2
(∞)©O“Oα
2
(0)Úp
2
(0).
éV
222
.|^H¨older Øª,Ún1-3 Úp
i
(0) 6p
i
(∞),Œ
k−2
X
l
1
=0
k+1
X
l
2
=k−1
kBI
β
(f
l
1
,f
l
2
)χ
k
k
L
p(·)
(R
n
)
6C
k−2
X
l
1
=0
2
(k−l
1
)(
n
p
1
(∞)
−n)
kf
l
1
k
L
p
1
(·)
(R
n
)
!
×
k+1
X
l
2
=k−1
2
(k−l
2
)(
n
p
2
(∞)
−n)
kf
l
2
k
L
p
2
(·)
(R
n
)
!
.
dH¨older ØªÚ
1
q
=
1
q
1
+
1
q
2
,
V
222
6Csup
ε>0
(
ε
θ
∞
X
k=0
k−2
X
l
1
=0
2
kα
1
(∞)+(k−l
1
)(
n
p
1
(∞)
−n)
kf
l
1
k
L
p
1
(·)
(R
n
)
!
q
1
(1+ε)
)
1
q
1
(1+ε)
×sup
ε>0
(
ε
θ
∞
X
k=0
k+1
X
l
2
=k−1
2
kα
2
(∞)+(k−l
2
)(
n
p
2
(∞)
−n)
kf
l
2
k
L
p
2
(·)
(R
n
)
!
q
2
(1+ε)
)
1
q
2
(1+ε)
:= F
1
F
2
.
duF
2
= E
2
,F
1
O†V
212
Oƒq,ÏdV
22
O..(ÜV
21
ÚV
22
O,
V
2
6Ckf
1
k
˙
K
α
1
(·),q
1
),θ
p
1
(·)
(R
n
)
kf
2
k
˙
K
α
2
(·),q
2
),θ
p
2
(·)
(R
n
)
.
DOI:10.12677/pm.2023.134097926nØêÆ
•1#
e¡OV
3
.
V
3
6Csup
ε>0
(
ε
θ
−1
X
k=−∞
2
kα(0)q(1+ε)
k−2
X
l
1
=−∞
∞
X
l
2
=k+2
kBI
β
(f
l
1
,f
l
2
)χ
k
k
L
p(·)
(R
n
)
!
q(1+ε)
)
1
q(1+ε)
+Csup
ε>0
(
ε
θ
∞
X
k=0
2
kα(∞)q(1+ε)
k−2
X
l
1
=−∞
∞
X
l
2
=k+2
kBI
β
(f
l
1
,f
l
2
)χ
k
k
L
p(·)
(R
n
)
!
q(1+ε)
)
1
q(1+ε)
:= V
31
+V
32
.
l
1
6k−2, l
2
>k+2, x∈E
k
,y
1
∈E
l
1
9y
2
∈E
l
2
ž,k
|x−y
1
|>|x|−|y
1
|>2
k−2
,
Ú
|x−y
2
|>|y
2
|−|x|>2
l
2
−1
−2
k
>2
l
2
−1
−2
l
2
−2
>2
l
2
−2
,
¤±k
|BI
β
(f
l
1
,f
l
2
)(x)|6C2
−k(n−
β
2
)
kf
l
1
k
L
1
(R
n
)
2
−l
2
(n−
β
2
)
kf
l
2
k
L
1
(R
n
)
.
5¿
1
p(x)
=
1
r
1
(x)
+
1
r
2
(x)
,
1
r
i
(x)
=
1
p
i
(x)
−
β
2n
Úp
2
(0) 6p
2
(∞).dH¨older Øª9Ún1-3, 
k−2
X
l
1
=−∞
∞
X
l
2
=k+2
kBI
β
(f
l
1
,f
l
2
)χ
k
k
L
p(·)
(R
n
)
6C
k−2
X
l
1
=−∞
∞
X
l
2
=k+2
2
−k(n−
β
2
)
kf
l
1
k
L
1
(R
n
)
2
−l
2
(n−
β
2
)
kf
l
2
k
L
1
(R
n
)
kχ
k
k
L
r
1
(·)
(R
n
)
kχ
k
k
L
r
2
(·)
(R
n
)
6C
k−2
X
l
1
=−∞
∞
X
l
2
=k+2
2
−k(n−
β
2
)
2
l
1
n
p
0
1
(0)
2
−l
2
(n−
β
2
)
2
l
2
n
p
0
2
(∞)
2
kn
r
1
(0)
2
kn
r
2
(0)
kf
l
1
k
L
p
1
(·)
(R
n
)
kf
l
2
k
L
p
2
(·)
(R
n
)
6C
k−2
X
l
1
=−∞
2
(k−l
1
)(
n
p
1
(0)
−n)
kf
l
1
k
L
p
1
(·)
(R
n
)
!
∞
X
l
2
=k+2
2
(k−l
2
)(
n
p
2
(∞)
−
β
2
)
kf
l
2
k
L
p
2
(·)
(R
n
)
!
.(7)
kwV
31
.|^H¨older ØªÚ
1
q
=
1
q
1
+
1
q
2
,
V
31
6Csup
ε>0
(
ε
θ
−1
X
k=−∞
k−2
X
l
1
=−∞
2
kα
1
(0)+(k−l
1
)(
n
p
1
(0)
−n)
kf
l
1
k
L
p
1
(·)
(R
n
)
!
q
1
(1+ε)
)
1
q
1
(1+ε)
×sup
ε>0
(
ε
θ
−1
X
k=−∞
∞
X
l
2
=k+2
2
kα
2
(0)+(k−l
2
)(
n
p
2
(∞)
−
β
2
)
kf
l
2
k
L
p
2
(·)
(R
n
)
!
q
2
(1+ε)
)
1
q
2
(1+ε)
:= V
311
V
312
.
DOI:10.12677/pm.2023.134097927nØêÆ
•1#
duV
311
= V
111
,•IOV
312
.
V
312
6Csup
ε>0
(
ε
θ
−1
X
k=−∞
−1
X
l
2
=k+2
2
(k−l
2
)(α
2
(0)+
n
p
2
(∞)
−
β
2
)
2
l
2
α
2
(0)
kf
l
2
k
L
p
2
(·)
(R
n
)
!
q
2
(1+ε)
)
1
q
2
(1+ε)
+Csup
ε>0
(
ε
θ
−1
X
k=−∞
∞
X
l
2
=0
2
(k−l
2
)(α
2
(0)+
n
p
2
(∞)
−
β
2
)
2
l
2
α
2
(0)
kf
l
2
k
L
p
2
(·)
(R
n
)
!
q
2
(1+ε)
)
1
q
2
(1+ε)
:= G
1
+G
2
.
d®•^‡d
2
:= α
2
(0)+
n
p
2
(∞)
−
β
2
>0.|^H¨older ØªÚFubini½n, Œ
G
1
6Csup
ε>0
(
ε
θ
−1
X
k=−∞
−1
X
l
2
=k+2
2
l
2
α
2
(0)q
2
(1+ε)
kf
l
2
k
q
2
(1+ε)
L
p
2
(·)
(R
n
)
2
(k−l
2
)d
2
q
2
(1+ε)
2
!
×
−1
X
l
2
=k+2
2
(k−l
2
)d
2
(q
2
(1+ε))
0
2
!
q
2
(1+ε)
(q
2
(1+ε))
0
)
1
q
2
(1+ε)
6Csup
ε>0
ε
θ
−1
X
k=−∞
−1
X
l
2
=k+2
2
l
2
α
2
(0)
2
(k−l
2
)d
2
q
2
(1+ε)
2
kf
l
2
k
q
2
(1+ε)
L
p
2
(·)
(R
n
)
!
1
q
2
(1+ε)
6Csup
ε>0
ε
θ
−1
X
l
2
=−∞
2
l
2
α
2
(0)q
2
(1+ε)
kf
l
2
k
q
2
(1+ε)
L
p
2
(·)
(R
n
)
l
2
−2
X
k=−∞
2
(k−l
2
)d
2
2
!
1
q
2
(1+ε)
6Csup
ε>0
ε
θ
−1
X
l
2
=−∞
2
l
2
α
2
(0)q
2
(1+ε)
kf
l
2
k
q
2
(1+ε)
L
p
2
(·)
(R
n
)
!
1
q
2
(1+ε)
6Ckf
2
k
˙
K
α
2
(·),q
2
),θ
p
2
(·)
(R
n
)
.
éuG
2
.5¿d
2
>0,α
2
(0) 6α
2
(∞).dH¨older ØªÚFubini ½n,Œ
G
2
6Csup
ε>0
(
ε
θ
−1
X
k=−∞
∞
X
l
2
=0
2
l
2
α
2
(0)q
2
(1+ε)
kf
l
2
k
q
2
(1+ε)
L
p
2
(·)
(R
n
)
2
(k−l
2
)d
2
q
2
(1+ε)
2
!
×
∞
X
l
2
=0
2
(k−l
2
)d
2
(q
2
(1+ε))
0
2
!
q
2
(1+ε)
(q
2
(1+ε))
0
)
1
q
2
(1+ε)
6Csup
ε>0
ε
θ
−1
X
k=−∞
∞
X
l
2
=0
2
l
2
α
2
(0)q
2
(1+ε)
kf
l
2
k
q
2
(1+ε)
L
p
2
(·)
(R
n
)
2
(k−l
2
)d
2
q
2
(1+ε)
2
!
1
q
2
(1+ε)
6Csup
ε>0
ε
θ
−1
X
k=−∞
∞
X
l
2
=0
2
(k−l
2
)d
2
q
2
(1+ε)
2
∞
X
j=l
2
2
jα
2
(∞)q
2
(1+ε)
kf
2
χ
j
k
q
2
(1+ε)
L
p
2
(·)
(R
n
)
!
1
q
2
(1+ε)
6Csup
ε>0
ε
θ
−1
X
k=−∞
∞
X
l
2
=0
2
(k−l
2
)d
2
q
2
(1+ε)
2
kf
2
k
q
2
(1+ε)
˙
K
α
2
(·),q
2
),θ
p
2
(·)
(R
n
)
!
1
q
2
(1+ε)
6Ckf
2
k
˙
K
α
2
(·),q
2
),θ
p
2
(·)
(R
n
)
.
DOI:10.12677/pm.2023.134097928nØêÆ
•1#
ÏdV
312
O.(ÜV
311
ÚV
312
O,Œ
V
31
6Ckf
1
k
˙
K
α
1
(·),q
1
),θ
p
1
(·)
(R
n
)
kf
2
k
˙
K
α
2
(·),q
2
),θ
p
2
(·)
(R
n
)
.
2OV
32
.
V
32
6Csup
ε>0
(
ε
θ
∞
X
k=0
2
kα(∞)q(1+ε)
−1
X
l
1
=−∞
∞
X
l
2
=k+2
kBI
β
(f
l
1
,f
l
2
)χ
k
k
L
p(·)
(R
n
)
!
q(1+ε)
)
1
q(1+ε)
+Csup
ε>0
(
ε
θ
∞
X
k=0
2
kα(∞)q(1+ε)
k−2
X
l
1
=0
∞
X
l
2
=k+2
kBI
β
(f
l
1
,f
l
2
)χ
k
k
L
p(·)
(R
n
)
!
q(1+ε)
)
1
q(1+ε)
:= V
321
+V
322
.
5¿
1
p(x)
=
1
r
1
(x)
+
1
r
2
(x)
,
1
r
i
(x)
=
1
p
i
(x)
−
β
2n
Úp
1
(0) 6p
1
(∞).dH¨older Øª9Ún1-3, 
k−2
X
l
1
=−∞
∞
X
l
2
=k+2
kBI
β
(f
l
1
,f
l
2
)χ
k
k
L
p(·)
(R
n
)
6C
k−2
X
l
1
=−∞
2
(k−l
1
)(
n
p
1
(0)
−n)
kf
l
1
k
L
p
1
(·)
(R
n
)
!
×
∞
X
l
2
=k+2
2
(k−l
2
)(
n
p
2
(∞)
−
β
2
)
kf
l
2
k
L
p
2
(·)
(R
n
)
!
.
dH¨older ØªÚ
1
q
=
1
q
1
+
1
q
2
,Œ
V
321
6Csup
ε>0
(
ε
θ
∞
X
k=0
−1
X
l
1
=−∞
2
(k−l
1
)(α
1
(∞)−
n
p
0
1
(0)
)
2
l
1
α
1
(∞)
kf
l
1
k
L
p
1
(·)
(R
n
)
!
q
1
(1+ε)
)
1
q
1
(1+ε)
×sup
ε>0
(
ε
θ
∞
X
k=0
∞
X
l
2
=k+2
2
(k−l
2
)(α
2
(∞)+
n
p
2
(∞)
−
β
2
)
2
l
2
α
2
(∞)
kf
l
2
k
L
p
2
(·)
(R
n
)
!
q
2
(1+ε)
)
1
q
2
(1+ε)
:= H
1
H
2
.
Ï•H
1
= D
1
,H
2
O†V
312
Oƒq,•I^α
2
(∞)“Oα
2
(0).
éV
322
,5¿
1
p(x)
=
1
r
1
(x)
+
1
r
2
(x)
,
1
r
i
(x)
=
1
p
i
(x)
−
β
2n
.dH¨older Øª9Ún1-3, 
k−2
X
l
1
=−∞
∞
X
l
2
=k+2
kBI
β
(f
l
1
,f
l
2
)χ
k
k
L
p(·)
(R
n
)
6C
k−2
X
l
1
=−∞
2
(k−l
1
)(
n
p
1
(∞)
−n)
kf
l
1
k
L
p
1
(·)
(R
n
)
!
×
∞
X
l
2
=k+2
2
(k−l
2
)(
n
p
2
(∞)
−
β
2
)
kf
l
2
k
L
p
2
(·)
(R
n
)
!
.
DOI:10.12677/pm.2023.134097929nØêÆ
•1#
dH¨older Øª,
1
q
=
1
q
1
+
1
q
2
,Œ
V
322
6Csup
ε>0
(
ε
θ
∞
X
k=0
k−2
X
l
1
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l
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n
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l
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l
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k
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k
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l
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k
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l
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n
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1
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L
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n
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l
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θ
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l
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p
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l
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L
p
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1
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q
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X
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X
l
2
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n
p
2
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β
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l
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2
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l
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L
p
2
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2
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q
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V
612
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l
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l
2
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2
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n
p
2
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β
2
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l
2
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2
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l
2
k
L
p
2
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n
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2
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2
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621
V
622
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DOI:10.12677/pm.2023.134097931nØêÆ
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l
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l
2
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91
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i
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L
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n
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l
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p
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θ
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X
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X
l
1
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2
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1
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1
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n
p
1
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β
2
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1
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k
L
p
1
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n
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1
(1+ε)
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1
q
1
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(
ε
θ
−1
X
k=−∞
∞
X
l
2
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2
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2
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2
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n
p
2
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−
β
2
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l
2
α
2
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l
2
k
L
p
2
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(R
n
)
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2
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)
1
q
2
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:= V
911
V
912
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aquV
312
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911
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912
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.
V
92
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X
l
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1
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1
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n
p
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β
2
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l
1
α
1
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l
1
k
L
p
1
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n
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q
1
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1
q
1
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(
ε
θ
∞
X
k=0
∞
X
l
2
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2
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2
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2
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n
p
2
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−
β
2
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2
l
2
α
2
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l
2
k
L
p
2
(·)
(R
n
)
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q
2
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)
1
q
2
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921
V
922
.
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322
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922
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k
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p
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½n1y..
ë•©z
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