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PureMathematicsnØêÆ,2022,12(6),1011-1026
PublishedOnlineJune2022inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2022.126111
2ÂC•IMorrey˜mþMarcinkiewicz
È©õ‚5†f
¤¤¤+++•••
Ü“‰ŒÆ§êƆÚOÆ§[‹=²
ÂvFϵ2022c516F¶¹^Fϵ2022c621F¶uÙFϵ2022c628F
Á‡
/ÏC•ILebesgue˜mþk.5§|^¼ê©©)Ú¢CE|§Marcinkiewicz
È©ÚBMO ¼ê)¤õ‚5†f32ÂC•IMorrey˜mþk.5"
'…c
2ÂC•IMorrey˜m§MarcinkiewiczÈ©§õ‚5†f§BMO ¼ê
TheMultilinearCommutatorof
MarcinkiewiczIntegralonGeneralized
VariableExponentMorreySpaces
PengweiShi
CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu
Received:May16
th
,2022;accepted:Jun.21
st
,2022;published:Jun.28
th
,2022
Abstract
WiththehelpoftheboundednessoftheLebesguespacewithvariableexponent,
©ÙÚ^:¤+•.2ÂC•IMorrey˜mþMarcinkiewiczÈ©õ‚5†f[J].nØêÆ,2022,12(6):
1011-1026.DOI:10.12677/pm.2022.126111
¤+•
byapplyinghierarchicaldecompositionoffunctionandrealvariabletechniques,the
boundednessofMarcinkiewiczintegralanditsmultilinearcommutatorgeneratedby
BMOfunctionisobtainedongeneralizedvariableexponentMorreyspaces.
Keywords
GeneralizedVariableExponentMorreySpaces,MarcinkiewiczIntegral,Multilinear
Commutator,BMOFunction
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.Úó9̇(J
Ω´R
n
þ"gàg¼ê…÷vž”^‡
Z
S
n−1
Ω(x
0
)dσ(x
0
) = 0,(1.1)
Ù¥,S
n−1
L«R
n
(n≥2) ¥ü ¥¡,dσ(x
0
)•S
n−1
þLebesgueÿÝ,x
0
=
x
|x|
, x6= 0.
MarcinkiewiczÈ©Žfµ
Ω
½Â•
µ
Ω
(f)(x) =
Z
∞
0




Z
|x−y|≤t
Ω(x−y)
|x−y|
n−1
f(y)dy




2
dt
t
3
!
1
2
.(1.2)
1958c,Stein[1]ÄgÚ\/X(1.2)p‘MarcinkiewiczÈ©Žf,Óžy²Ω
ëY…Ω∈Lip
α
(S
n−1
)(0<α≤1)ž,µ
Ω
´(p,p).(1<p≤2)Úf(1,1)..ùp
Ω ∈Lip
α
(S
n−1
)(0 <α≤1)´••3˜‡~êC>0,¦
|Ω(x
1
)−Ω(x
2
)|≤C|x
1
−x
2
|
α
, ∀x
1
,x
2
∈S
n−1
.(1.3)
1962c,Benedek[2]y²Ω∈C
1
(S
n−1
),1<p<∞ž,µ
Ω
´(p,p)..2014c,
Wang[3]Marcinkiewicz È©Žf9Ù†f3C•IHerz ˜m
˙
K
α(·),p
q(·)
(R
n
)þk.5.
2018c,"Õ±Ú>V²[4]‘CþØMarcinkiewicz È©Žf3C•IHerz .Hardy ˜
mþk.5.Cc5,•õ'uMarcinkiewiczÈ©Žf9Ù†fk.5(J,Œë„©
z[5–7].
DOI:10.12677/pm.2022.1261111012nØêÆ
¤+•
b´Û܌ȼê,KBMO(k.²þÄ)¼ê˜m½ÂXe
BMO(R
n
) := {b∈L
1
loc
(R
n
) :kbk
∗
=sup
B⊂R
n
1
|B|
Z
B
|b(x)−b
B
|dx<∞},
Ù¥,b
B
=
1
|B|
R
B
b(y)dy.Kdµ
Ω
Úb)¤†f½Â•
[b,µ
Ω
](f)(x) := bµ
Ω
(f)(x)−µ
Ω
(bf)(x).(1.4)
m∈N,
~
b= (b
1
,b
2
,...,b
m
) ∈BMO
m
(R
n
),=b
i
∈BMO(R
n
),i= 1,2,...,m.ÉP´erezÚ
Trujillo-Gonz´alez[8]'uõ‚5†fïÄéu,·‚½ÂMarcinkiewicz È©Žfõ‚5
†f•
µ
Ω,
~
b
(f)(x) =


Z
∞
0





Z
|x−y|≤t
m
Y
i=1
(b
i
(x)−b
i
(y))
Ω(x−y)
|x−y|
n−1
f(y)dy





2
dt
t
3


1
2
.(1.5)
AO/,XJb
i
= b,m= 1,Kkµ
Ω,
~
b
(f)(x) = [b,µ
Ω
](f)(x).
2008c,Zhang[9]y²ω∈A
p
ž,µ
Ω,
~
b
\L
p
(1 <p<∞)k.5,¿˜«\
fL(logL) .O.2019c,WangÚShu[10]¼Marcinkiewicz È©Žfõ‚5†f3
C•ILebesgue˜mÚHerz.˜mþk.5.
C•I¼ê˜m36NÄåÆ,ã”?nÚäkšIOO•^‡‡©•§+•k2
•A^(ë„©z[11–13]).•ïÄý ‡©•§)ÛÜ1•9ÙA^,•@d
Morrey[13]JÑ ˜«¼ê˜mL
p,λ
(R
n
) (1<p<∞,0<λ<n),•Ò´y3²;Morrey ˜
m.CA›c5,N õŠöí2²;Morrey ˜m,ØÓ/ ªC•IMorrey ˜m¿…¼
Nõ²;Žf9Ù†fk.5(J.2008c,Almeida[14]y²4ŒŽfÚ ³Ž
f3C•IMorrey ˜mþk.5.2016c,>V²Úo ³³[15]ïáMarcinkiewiczÈ©9
Ù†f3C•IMorrey ˜mþk.5.Ho[16]‰Ñ©êgÈ©ÚÛÉÈ©Žf3C•I
Morrey ˜mþk.5¿©^‡,†dÓžŠö[17]©êgÈ©Žf3C•IMorrey ˜m
þf.O.•C,Guliyev[18]y²ω.Calder´on-ZygmundŽf9Ùõ‚5†f32
ÂC•IMorrey ˜mM
p(·),u
(R
n
)þk.5.ÉþãïÄ(Jéu,©̇8´ïÄ
Marcinkiewicz È©ÚBMO ¼ê)¤õ‚ 5†f32 C•IMorrey ˜mM
p(·),u
(R
n
)þ
k.5.•d,·‚Äk£e¡½Â.
é?¿x∈R
n
Úr>0,B(x,r)L«±x•¥%, r•Œ»¥N.B(x,r)
C
L«B(x,r) 
{8.^χ
B
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n
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1
,C
2
>0,¦
C
1
g≤f≤C
2
g.©¥C´˜‡~ê,3ØÓ/•ŒØÓŠ.
^P(R
n
)L«R
n
þ¤k÷ve^‡Œÿ¼êp(x) : R
n
→[1,∞)|¤8Ü
p
−
:= essinf{p(x) : x∈R
n
}>1,p
+
:= esssup{p(x) : x∈R
n
}<∞.
PP
0
(R
n
) ´d¤k÷v0<p
−
≤p
+
<∞Œÿ¼êp(x):R
n
→(0,∞)¤8Ü.P
1
(R
n
)
´d¤k÷v1≤p
−
≤p
+
<∞p(x): R
n
→(0,∞) ¤8Ü.1 <p(x)<∞,Pp
0
(x) •
DOI:10.12677/pm.2022.1261111013nØêÆ
¤+•
p(x)éó•I,=p
0
(x) =
p(x)
p(x)−1
.
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n
),÷v
|p(x)−p(y)|≤
−C
ln(|x−y|)
, |x−y|≤
1
2
,(1.6)
|p(x)−p(y)|≤
C
ln(|x|+e)
, |y|≥|x|,(1.7)
K¡p(·) ∈B(R
n
).ep(·) ∈P(R
n
),Kp(·) ∈B(R
n
)…=p
0
(·) ∈B(R
n
),„©z[10].
½½½ÂÂÂ1[3]p(x) ∈P(R
n
),C•ILebesgue ˜mL
p(·)
(R
n
)½Â•
L
p(·)
(R
n
) :=
n
f´Œÿ¼ê: é,‡~êλ>0,k
Z
R
n

|f(x) |
λ

p(x)
dx<∞
o
.
DƒXeLuxemburg-Nakano ‰êž,L
p(·)
(R
n
)´˜‡Banach¼ê˜m
kfk
L
p(·)
(R
n
)
= inf
n
λ>0 :
Z
R
n

|f(x) |
λ

p(x)
dx≤1
o
.
w,,p(·)•~êž,KL
p(·)
(R
n
) = L
p
(R
n
)´²;Lebesgue ˜m.
ÛÜC•ILebesgue˜mL
p(·)
loc
(R
n
)½Â•
L
p(·)
loc
(R
n
) := {f: fχ
E
∈L
p(·)
(R
n
),E´R
n
?¿;f8}.
½½½ÂÂÂ2[6]λ(·):R
n
→(0,n)´˜‡Œÿ¼ê,p(·)∈P(R
n
).C•IMorrey˜m
L
p(·),λ(·)
(R
n
)½Â•
L
p(·),λ(·)
(R
n
) = {f: kfk
L
p(·),λ(·)
(R
n
)
=sup
x∈R
n
,r>0
r
−
λ(x)
p(x)
kχ
B(x,r)
fk
L
p(·)
(R
n
)
<∞}.
N´w,XJp(·),λ(·)þ•~ê,KL
p(·),λ(·)
(R
n
) = L
p,λ
(R
n
)´²;Morrey˜m,¿…d
žMorrey˜mŒwŠLebesgue˜mí2(ë„©z[13]).
3©¥,u(x,r),u
1
(x,r)Úu
2
(x,r)´R
n
×(0,∞)þšKŒÿ¼ê.¿…u(r),u
1
(r)Ú
u
2
(r)´(0,∞)þšKŒÿ¼ê.
½½½ÂÂÂ3[18,19]p(·)∈P
1
(R
n
),u(x,r):R
n
×(0,∞)→(0,∞).2ÂC•IMorrey˜m
M
p(·),u
(R
n
)½Â•
M
p(·),u
(R
n
) = {f∈L
p(·)
loc
(R
n
) : kfk
M
p(·),u
(R
n
)
=sup
x∈R
n
,r>0
1
u(x,r)r
θ
p
(x,r)
kχ
B(x,r)
fk
L
p(·)
(R
n
)
<∞},
Ù¥θ
p
(x,r) =
(
n
p(x)
,r≤1,
n
p(∞)
,r≥1,
p(∞) = lim
x→∞
p(x).
DOI:10.12677/pm.2022.1261111014nØêÆ
¤+•
5551Šâ½Â1.2,XJu(x,r)=r
−θ
p
(x,r)+
λ(x)
p(x)
,KkM
p(·),u
(R
n
)=L
p(·),λ(·)
(R
n
).XJ
u(x,r)=r
λ−n
p(x)
,…0<λ<n,KM
p(·),u
(R
n
)=L
p(·),λ
(R
n
)´dAlmeida[14]½ÂC•I
Morrey ˜m.eu(x,r)=r
−θ
p
(x,r)
,KM
p(·),u
(R
n
)=L
p(·)
(R
n
).d,·‚AT5¿,©‰Ñ
M
p(·),u
(R
n
)˜m†©z[14–17,20]¥¤½ÂC•IMorrey ˜m´k¤ØÓ.
•y²©̇(J,·‚I‡0e¡˜‡-‡Øª(„©z[18]).éu?¿
x∈R
n
,p(·) ∈B(R
n
),K•3˜‡~êC>0,¦
kχ
B(x,r)
(·)k
L
p(·)
(R
n
)
≤Cr
θ
p
(x,r)
.(1.8)
½½½ÂÂÂ4[21]C•IBMO
p(·)
(R
n
)˜m½Â•
BMO
p(·)
(R
n
) = {f∈L
1
loc
(R
n
) : kfk
BMO
p(·)
(R
n
)
=sup
x∈R
n
,r>0
k(f−f
B(x,r)
)χ
B(x,r)
k
L
p(·)
(R
n
)
kχ
B(x,r)
k
L
p(·)
(R
n
)
<∞}.
5552p(·) ∈B(R
n
),K‰êk·k
BMO
p(·)
Úk·k
∗
´ƒpd(„©z[21]).
©©©ÌÌ̇‡‡(((JJJXXXeee.
½½½nnn1Ω÷v(1.1)Ú(1.3), p(·) ∈B(R
n
).eu
1
Úu
2
÷v^‡
Z
∞
r
essinf
s<t<∞
u
1
(x,t)t
θ
p
(x,t)
s
θ
p
(x,s)
ds
s
≤C
0
u
2
(x,r),(1.9)
Ù¥C
0
´Ø•6uxÚr~ê.Kµ
Ω
´lM
p(·),u
1
(R
n
)M
p(·),u
2
(R
n
)þk..
½½½nnn2Ω÷v(1.1)Ú(1.3), p(·) ∈B(R
n
),
~
b∈BMO
m
(R
n
).eu
1
Úu
2
÷v^‡
Z
∞
r

1+ln
s
r

m
essinf
s<t<∞
u
1
(x,t)t
θ
p
(x,t)
s
θ
p
(x,s)
ds
s
≤C
0
u
2
(x,r),(1.10)
Ù¥C
0
´Ø•6uxÚr~ê.Kµ
Ω,
~
b
´lM
p(·),u
1
(R
n
)M
p(·),u
2
(R
n
)þk..
AO/,b
i
= b,m= 1ž,d½n2,·‚ke¡(J.
íííØØØ1[b,µ
Ω
] d(1.4) ¤½Â,Ω÷v(1.1)Ú(1.3), p(·)∈B(R
n
),b∈BMO.eu
1
Úu
2
÷v^‡
Z
∞
r

1+ln
s
r

essinf
s<t<∞
u
1
(x,t)t
θ
p
(x,t)
s
θ
p
(x,s)
ds
s
≤C
0
u
2
(x,r),(1.11)
Ù¥C
0
´Ø•6uxÚr~ê.K[b,µ
Ω
]´lM
p(·),u
1
(R
n
)M
p(·),u
2
(R
n
)þk..
2.½ny²
•y²Ì‡½n,·‚I‡±eÚn.
DOI:10.12677/pm.2022.1261111015nØêÆ
¤+•
ÚÚÚnnn2.1(222ÂÂÂH¨olderØØØªªª)[6]p(·)∈P(R
n
), ef∈L
p(·)
(R
n
),g∈L
p
0
(·)
(R
n
),Kfg
3R
n
þŒÈ,…k
Z
R
n
|f(x)g(x)|dx≤r
p
kfk
L
p(·)
(R
n
)
kgk
L
p
0
(·)
(R
n
)
,
Ù¥r
p
= 1+
1
p
−
−
1
p
+
.
ÚÚÚnnn2.2[7]p(·) ∈B(R
n
),K•3˜‡~êC>0,¦éR
n
¥¤k¥BÚ¤kŒÿ
f8S⊂B,Ñk
kχ
B
k
L
p(·)
(R
n
)
kχ
S
k
L
p(·)
(R
n
)
≤C
|B|
|S|
,
kχ
S
k
L
p(·)
(R
n
)
kχ
B
k
L
p(·)
(R
n
)
≤C

|S|
|B|

δ
1
,
kχ
S
k
L
p
0
(·)
(R
n
)
kχ
B
k
L
p
0
(·)
(R
n
)
≤C

|S|
|B|

δ
2
,
Ù¥δ
1
,δ
2
´~ê…k0 <δ
1
,δ
2
<1.
ÚÚÚnnn2.3[10]p(·),p
i
(·)∈P
0
(R
n
),i=1,2,...,m,÷v1/p(·)=1/p
1
(·)+···+ 1/p
m
(·).
@of
i
∈L
p
i
(·)
(R
n
),k





m
Y
i=1
f
i





L
p(·)
(R
n
)
≤C
m
Y
i=1
kf
i
k
L
p(·)
(R
n
)
.
ÚÚÚnnn2.4[22]f´Eþ¢ŠšKŒÿ¼ê,Kk

essinf
x∈E
f(x)

−1
= esssup
x∈E
1
f(x)
.
‰½˜‡¼ê$,·‚Äk0±eüa\HardyŽf.
H
$
g(t) :=
Z
∞
t
g(s)$(s)ds,t∈(0,∞),
H
?
$
g(t) :=
Z
∞
t

1+ln
s
t

m
g(s)$(s)ds,t∈(0,∞).
ÚÚÚnnn2.5[23]v
1
,v
2
Ú$´(0,∞)þ¼ê, v
1
(t)3:ƒ´k..éu,‡~ê
C>0Ú(0,∞)þ¤kšK,šO¼êg,eãØª
sup
t>0
v
2
(t)H
$
g(t) ≤Csup
t>0
v
1
(t)g(t)
DOI:10.12677/pm.2022.1261111016nØêÆ
¤+•
¤á…=
sup
t>0
v
2
(t)
Z
∞
t
$(s)ds
sup
s<σ<∞
v
1
(σ)
<∞.
ÚÚÚnnn2.6[19,24]v
1
,v
2
Ú$´(0,∞)þ¼ê, v
1
(t)3:ƒ´k..éu,‡
~êC>0Ú(0,∞)þ¤kšK,šO¼êg,eãØª
sup
t>0
v
2
(t)H
?
$
g(t) ≤Csup
t>0
v
1
(t)g(t)
¤á…=
sup
t>0
v
2
(t)
Z
∞
t

1+ln
s
t

m
$(s)ds
sup
s<σ<∞
v
1
(σ)
<∞.
ÚÚÚnnn2.7[24]b∈BMO(R
n
),K•3˜‡~êC>0,¦
|b
B(x,r)
−b
B(x,s)
|≤Ckbk
∗
ln
s
t
, 0 <2r<s,
Ù¥C´Ø•6ub,x,rÚs.
ÚÚÚnnn2.8[15]Ω÷v(1.1)Ú(1.3), p(·) ∈B(R
n
).K•3˜‡~êC>0,¦
kµ
Ω
(f)k
L
p(·)
(R
n
)
≤Ckfk
L
p(·)
(R
n
)
.
ÚÚÚnnn2.9[10]Ω∈Lip
α
(S
n−1
) (0<α≤1), p(·)∈B(R
n
),
~
b∈BMO
m
(R
n
).K•3˜‡~
êC>0,¦
kµ
Ω,
~
b
(f)k
L
p(·)
(R
n
)
≤Ck
~
bk
∗
kfk
L
p(·)
(R
n
)
,
Ù¥k
~
bk
∗
=
Q
m
j=1
kb
j
k
∗
.
½½½nnn1yyy²²²p(·) ∈B(R
n
).éu?¿x
0
∈R
n
Úr>0,B(x
0
,r)L«±x
0
•¥%,r•
Œ»¥N.é?¿f∈L
p(·)
loc
(R
n
),rf(x)©)•
f= f
1
+f
2
, f
1
= fχ
B(x
0
,2r)
, f
2
= fχ
B(x
0
,2r)
C
, r>0.(3.1)
?k,
kχ
B(x
0
,r)
µ
Ω
(f)k
L
p(·)
≤kχ
B(x
0
,r)
µ
Ω
(f
1
)k
L
p(·)
+kχ
B(x
0
,r)
µ
Ω
(f
2
)k
L
p(·)
=: E+F.
e¡©OOEÚF.éuE,|^µ
Ω
(L
p(·)
,L
p(·)
)k.5(Ún2.8),k
kχ
B(x
0
,r)
µ
Ω
(f
1
)k
L
p(·)
≤kµ
Ω
(f
1
)k
L
p(·)
≤Ckχ
B(x
0
,2r)
fk
L
p(·)
,
Ù¥~êC>0´Ø•6uf.
DOI:10.12677/pm.2022.1261111017nØêÆ
¤+•
,˜•¡,dÚn2.1,Ún2.2Ú(1.8),N´
kχ
B(x
0
,2r)
fk
L
p(·)
≤C|B(x
0
,r)|kχ
B(x
0
,2r)
fk
L
p(·)
Z
∞
2r
ds
s
n+1
≤C|B(x
0
,r)|
Z
∞
2r
kχ
B(x
0
,s)
fk
L
p(·)
ds
s
n+1
≤Cr
θ
p
(x
0
,r)
kχ
B(x
0
,r)
k
L
p
0
(·)
Z
∞
2r
kχ
B(x
0
,s)
fk
L
p(·)
ds
s
n+1
≤Cr
θ
p
(x
0
,r)
Z
∞
2r
kχ
B(x
0
,s)
fk
L
p(·)
kχ
B(x
0
,s)
k
L
p
0
(·)
ds
s
n+1
≤Cr
θ
p
(x
0
,r)
Z
∞
2r
kχ
B(x
0
,s)
fk
L
p(·)
t
−θ
p
(x
0
,s)
ds
s
.(3.2)
ÙgOF.5¿x∈B(x
0
,r),y∈B(x
0
,2r)
C
,k
1
2
|x
0
−y|≤|x−y|≤
3
2
|x
0
−y|.(3.3)
Ïd
|µ
Ω
(f
2
)(x)|≤
Z
|x
0
−y|
0




Z
|x−y|≤t
Ω(x−y)
|x−y|
n−1
f
2
(y)dy




2
dt
t
3
!
1
2
+
Z
∞
|x
0
−y|




Z
|x−y|≤t
Ω(x−y)
|x−y|
n−1
f
2
(y)dy




2
dt
t
3
!
1
2
=: I
1
+I
2
.
éuI
1
.d(3.3),Œ|x−y|≈|x
0
−y|.(Ü¥Š½n,k




1
|x−y|
2
−
1
|x
0
−y|
2




≤C
|x−x
0
|
|x−y|
3
.(3.4)
5¿|x−y|≈|x
0
−y|,dMinkowski’sØªÚ(3.4),Œ
I
1
≤C
Z
B(x
0
,2r)
C
|Ω(x−y)|
|x−y|
n−1
|f(y)|
Z
|x
0
−y|
|x−y|
dt
t
3
!
1
2
dy
≤C
Z
B(x
0
,2r)
C
|Ω(x−y)|
|x−y|
n−1
|f(y)|
|x−x
0
|
1/2
|x−y|
3/2
dy
≤C
1
|x
0
−y|
1/2
Z
B(x
0
,2r)
C
|Ω(x−y)|
|x
0
−y|
n
|f(y)|dy.
aq/,·‚•ÄI
2
,
I
2
≤C
Z
B(x
0
,2r)
C
|Ω(x−y)|
|x−y|
n−1
|f(y)|

Z
∞
|x
0
−y|
dt
t
3

1
2
dy
≤C
Z
B(x
0
,2r)
C
|Ω(x−y)|
|x
0
−y|
n
|f(y)|dy.
DOI:10.12677/pm.2022.1261111018nØêÆ
¤+•
Ï•Ω ∈Lip
α
(S
n−1
) ⊂L
∞
(S
n−1
),Ω´k..(ÜI
1
ÚI
2
O,k
|µ
Ω
(f
2
)(x)|≤C
Z
B(x
0
,2r)
C
|Ω(x−y)|
|x
0
−y|
n
|f(y)|dy
≤C
Z
B(x
0
,2r)
C
|f(y)|
|x
0
−y|
n
dy.
dFubini’s½n,Ún2.1Ú(1.8),N´
Z
B(x
0
,2r)
C
|f(y)|
|x
0
−y|
n
dy≈
Z
B(x
0
,2r)
C
|f(y)|

Z
∞
|x
0
−y|
ds
s
n+1

dy
≈
Z
∞
2r
Z
2r≤|x
0
−y|≤s
|f(y)|dy
ds
s
n+1
≤C
Z
∞
2r
Z
B(x
0
,s)
|f(y)|dy
ds
s
n+1
≤C
Z
∞
2r
kχ
B(x
0
,s)
fk
L
p(·)
t
−θ
p
(x
0
,s)
ds
s
.(3.5)
|^(1.8)
kχ
B(x
0
,r)
µ
Ω
(f
2
)k
L
p(·)
≤Ckχ
B(x
0
,r)
k
L
p(·)
Z
∞
2r
kχ
B(x
0
,s)
fk
L
p(·)
t
−θ
p
(x
0
,s)
ds
s
≤Cr
θ
p
(x
0
,r)
Z
∞
2r
kχ
B(x
0
,s)
fk
L
p(·)
t
−θ
p
(x
0
,s)
ds
s
.
l,
kχ
B(x
0
,r)
µ
Ω
(f)k
L
p(·)
≤Ckχ
B(x
0
,2r)
fk
L
p(·)
+Cr
θ
p
(x
0
,r)
Z
∞
2r
kχ
B(x
0
,s)
fk
L
p(·)
t
−θ
p
(x
0
,s)
ds
s
≤Cr
θ
p
(x
0
,r)
Z
∞
2r
kχ
B(x
0
,s)
fk
L
p(·)
t
−θ
p
(x
0
,s)
ds
s
.

v
2
(r) = u
−1
2
(x
0
,r), v
1
(t) = u
−1
1
(x
0
,t)t
−θ
p
(x
0
,t)
,
g(s) = kχ
B(x,s)
fk
L
p(·)
, $(s) = s
−1
t
−θ
p
(x
0
,s)
.
dÚn2.4,Œ
1
sup
s<t<∞
v
1
(t)
= essinf
s<t<∞
u
1
(x,t)t
θ
p
(x,t)
.(3.6)
(Ü(1.9)Ú(3.6),N´e¡Øª
sup
r>0
v
2
(r)
Z
∞
r
$(s)ds
sup
s<t<∞
v
1
(t)
<∞
DOI:10.12677/pm.2022.1261111019nØêÆ
¤+•
´¤á.¤±,dÚn2.5Œ
kµ
Ω
(f)k
M
p(·),u
2
≤Csup
x∈R
n
,r>0
1
u
2
(x,r)
Z
∞
r
kχ
B(x
0
,s)
fk
L
p(·)
t
−θ
p
(x
0
,s)
ds
s
≤Csup
x∈R
n
,r>0
1
u
1
(x,r)r
θ
p
(x
0
,r)
kχ
B(x,r)
fk
L
p(·)
=Ckfk
M
p(·),u
1
.
½½½nnn2yyy²²²Ø”˜„5,·‚=•Äm=2œ¹.é?¿f∈L
p(·)
loc
(R
n
),rf(x)©)
•f= f
1
+f
2
, f
1
= fχ
B(x
0
,2r)
, f
2
= fχ
B(x
0
,2r)
C
.?k
kχ
B(x
0
,r)
µ
Ω,
~
b
(f)k
L
p(·)
≤kχ
B(x
0
,r)
µ
Ω,
~
b
(f
1
)k
L
p(·)
+kχ
B(x
0
,r)
µ
Ω,
~
b
(f
2
)k
L
p(·)
.=: G+H.
e¡©OOGÚH.éuG,|^µ
Ω,
~
b
(L
p(·)
,L
p(·)
)k.5(Ún2.9),k
kχ
B(x
0
,r)
µ
Ω,
~
b
(f
1
)k
L
p(·)
≤kµ
Ω,
~
b
(f
1
)k
L
p(·)
≤Ck
~
bk
∗
kχ
B(x
0
,2r)
fk
L
p(·)
,
Ù¥~êC>0´Ø•6uf.
éuH.••BOŽ,^(b
i
)
B
L«¼êb
i
,i= 1,23¥NB(x
0
,r)þ²þ.Ïdk
µ
Ω,
~
b
(f
2
)(x)=(b
1
(x)−(b
1
)
B
)(b
2
(x)−(b
2
)
B
)µ
Ω
(f
2
)(x)
−(b
1
(x)−(b
1
)
B
)µ
Ω
((b
2
(·)−(b
2
)
B
)(f
2
))(x)
+(b
2
(x)−(b
2
)
B
)µ
Ω
((b
1
(·)−(b
1
)
B
)(f
2
))(x)
−µ
Ω
((b
1
(·)−(b
1
)
B
)(b
2
(·)−(b
2
)
B
)(f
2
))(x).
5¿x∈B(x
0
,r),y∈B(x
0
,2r)
C
ž,k
1
2
|x
0
−y|≤|x−y|≤
3
2
|x
0
−y|.d½n1¥'u
|µ
Ω
(f
2
)(x)|O,Œ
|µ
Ω,
~
b
(f
2
)(x)|≤C|b
1
(x)−(b
1
)
B
||b
2
(x)−(b
2
)
B
|
Z
B(x
0
,2r)
C
|f(y)|
|x
0
−y|
n
dy
+C|b
1
(x)−(b
1
)
B
|
Z
B(x
0
,2r)
C
|b
2
(y)−(b
2
)
B
|
|f(y)|
|x
0
−y|
n
dy
+C|b
2
(x)−(b
2
)
B
|
Z
B(x
0
,2r)
C
|b
1
(y)−(b
1
)
B
|
|f(y)|
|x
0
−y|
n
dy
+C
Z
B(x
0
,2r)
C
|b
1
(y)−(b
1
)
B
||b
2
(y)−(b
2
)
B
|
|f(y)|
|x
0
−y|
n
dy.
DOI:10.12677/pm.2022.1261111020nØêÆ
¤+•
u´,·‚k
kχ
B(x
0
,r)
µ
Ω,
~
b
(f
2
)k
L
p(·)
≤C





2
Y
i=1
b
i
(·)−(b
i
)
B
χ
B(x
0
,r)





L
p(·)
Z
B(x
0
,2r)
C
|f(y)|
|x
0
−y|
n
dy
+Ckb
1
(·)−(b
1
)
B
χ
B(x
0
,r)
k
L
p(·)
Z
B(x
0
,2r)
C
|b
2
(y)−(b
2
)
B
|
|f(y)|
|x
0
−y|
n
dy
+Ckb
2
(·)−(b
2
)
B
χ
B(x
0
,r)
k
L
p(·)
Z
B(x
0
,2r)
C
|b
1
(y)−(b
1
)
B
|
|f(y)|
|x
0
−y|
n
dy
+Ckχ
B(x
0
,r)
k
L
p(·)
Z
B(x
0
,2r)
C
2
Y
i=1
|b
i
(y)−(b
i
)
B
|
|f(y)|
|x
0
−y|
n
dy
:=H
1
+H
2
+H
3
+H
4
.
éuH
1
,5¿p(·) ∈B(R
n
),|^Ún2.3,k
H
1
≤C
2
Y
i=1


b
i
(·)−(b
i
)
B
χ
B(x
0
,r)


L
2p(·)
Z
B(x
0
,2r)
C
|f(y)|
|x
0
−y|
n
dy.
d52,(1.8)Ú(3.5),Œ
H
1
≤Ck
~
bk
∗
kχ
B(x
0
,r)
k
2
L
2p(·)
Z
B(x
0
,2r)
C
|f(y)|
|x
0
−y|
n
dy
≤Ck
~
bk
∗
r
θ
p
(x
0
,r)
Z
B(x
0
,2r)
C
|f(y)|
|x
0
−y|
n
dy
≤Ck
~
bk
∗
r
θ
p
(x
0
,r)
Z
∞
2r
kχ
B(x
0
,s)
fk
L
p(·)
t
−θ
p
(x
0
,s)
ds
s
éuH
2
,aqu(3.5),k
H
2
≤Ckb
1
k
∗
r
θ
p
(x
0
,r)
Z
B(x
0
,2r)
C
|b
2
(y)−(b
2
)
B
|
|f(y)|
|x
0
−y|
n
dy
≤Ckb
1
k
∗
r
θ
p
(x
0
,r)
Z
B(x
0
,2r)
C
|b
2
(y)−(b
2
)
B
||f(y)|

Z
∞
|x
0
−y|
ds
s
n+1

dy
≤Ckb
1
k
∗
r
θ
p
(x
0
,r)
Z
∞
2r
Z
B(x
0
,s)
|b
2
(y)−(b
2
)
B
||f(y)|dy
ds
s
n+1
.
DOI:10.12677/pm.2022.1261111021nØêÆ
¤+•
¯¢þ,d52,Ún2.7,Œ
kb
2
(·)−(b
2
)
B
χ
B(x
0
,s)
k
L
p
0
(·)
≤Ckb
2
(·)−(b
2
)
B(x
0
,s)
k
L
p
0
(·)
(B(x
0
,s))
+Ck(b
2
)
B(x
0
,s)
−(b
2
)
B
k
L
p
0
(·)
(B(x
0
,s))
≤Ckb
2
k
∗
kχ
B(x
0
,s)
k
L
p
0
(·)

1+ln
s
r

.(3.7)
dÚn2.1,52Ú(3.7),N´
H
2
≤Ckb
1
k
∗
r
θ
p
(x
0
,r)
Z
∞
2r
kχ
B(x
0
,s)
fk
L
p(·)
kb
2
(·)−(b
2
)
B
χ
B(x
0
,s)
k
L
p
0
(·)
ds
s
n+1
≤C
2
Y
i=1
kb
i
k
∗
r
θ
p
(x
0
,r)
Z
∞
2r
kχ
B(x
0
,s)
fk
L
p(·)
kχ
B(x
0
,s)
k
L
p
0
(·)

1+ln
s
r

ds
s
n+1
≤Ck
~
bk
∗
r
θ
p
(x
0
,r)
Z
∞
2r

1+ln
s
r

kχ
B(x
0
,s)
fk
L
p(·)
t
−θ
p
(x
0
,s)
ds
s
n+1
.
éuH
3
,A^†þãOƒÓ?Ø•{,Œ
H
3
≤Ckb
2
k
∗
r
θ
p
(x
0
,r)
Z
B(x
0
,2r)
C
|b
1
(y)−(b
1
)
B
|
|f(y)|
|x
0
−y|
n
dy
≤Ckb
2
k
∗
r
θ
p
(x
0
,r)
Z
∞
2r
Z
B(x
0
,s)
|b
1
(y)−(b
1
)
B
||f(y)|dy
ds
s
n+1
≤Ckb
2
k
∗
r
θ
p
(x
0
,r)
Z
∞
2r
kχ
B(x
0
,s)
fk
L
p(·)
kb
1
(·)−(b
1
)
B
χ
B(x
0
,s)
k
L
p
0
(·)
ds
s
n+1
≤Ck
~
bk
∗
r
θ
p
(x
0
,r)
Z
∞
2r

1+ln
s
r

kχ
B(x
0
,s)
fk
L
p(·)
t
−θ
p
(x
0
,s)
ds
s
n+1
.
•·‚OH
4
.d(1.8)ÚÚn2.1,†(3.5)aq,Œ
H
4
≤Cr
θ
p
(x
0
,r)
Z
B(x
0
,2r)
C
2
Y
i=1
|b
i
(y)−(b
i
)
B
|
|f(y)|
|x
0
−y|
n
dy
≤Cr
θ
p
(x
0
,r)
Z
B(x
0
,2r)
C
2
Y
i=1
|b
i
(y)−(b
i
)
B
||f(y)|

Z
∞
|x
0
−y|
ds
s
n+1

dy
≤Cr
θ
p
(x
0
,r)
Z
∞
2r
Z
B(x
0
,s)
2
Y
i=1
|b
i
(y)−(b
i
)
B
||f(y)|dy
ds
s
n+1
≤Cr
θ
p
(x
0
,r)
Z
∞
2r
kχ
B(x
0
,s)
fk
L
p(·)





2
Y
i=1
|b
i
(·)−(b
i
)
B
|χ
B(x
0
,s)





L
p
0
(·)
ds
s
n+1
.
DOI:10.12677/pm.2022.1261111022nØêÆ
¤+•
dÚn2.3,Ún2.7Ú52,k
Z
∞
2r
kχ
B(x
0
,s)
fk
L
p(·)





2
Y
i=1
|b
i
(·)−(b
i
)
B
|χ
B(x
0
,s)





L
p
0
(·)
ds
s
n+1
≤C
Z
∞
2r
kχ
B(x
0
,s)
fk
L
p(·)





2
Y
i=1
|b
i
(·)−(b
i
)
B(x
0
,s)
|χ
B(x
0
,s)





L
p
0
(·)
ds
s
n+1
+C
Z
∞
2r
kχ
B(x
0
,s)
fk
L
p(·)
|(b
1
)
B(x
0
,s)
−(b
1
)
B
|kb
2
(·)−(b
2
)
B(x
0
,s)
χ
B(x
0
,s)
k
L
p
0
(·)
ds
s
n+1
+C
Z
∞
2r
kχ
B(x
0
,s)
fk
L
p(·)
|(b
2
)
B(x
0
,s)
−(b
2
)
B
|kb
1
(·)−(b
1
)
B(x
0
,s)
χ
B(x
0
,s)
k
L
p
0
(·)
ds
s
n+1
+C
Z
∞
2r
kχ
B(x
0
,s)
fk
L
p(·)





2
Y
i=1
|(b
i
)
B(x
0
,s)
−(b
i
)
B
|χ
B(x
0
,s)





L
p
0
(·)
ds
s
n+1
≤C
Z
∞
2r
kχ
B(x
0
,s)
fk
L
p(·)
2
Y
i=1


b
i
(·)−(b
i
)
B(x
0
,s)
χ
B(x
0
,s)


L
2p
0
(·)
ds
s
n+1
+Ckb
1
k
∗
kb
2
k
∗
Z
∞
2r
ln
s
r
kχ
B(x
0
,s)
fk
L
p(·)
kχ
B(x
0
,s)
k
L
p
0
(·)
ds
s
n+1
+C
Z
∞
2r
2
Y
i=1
|(b
i
)
B(x
0
,s)
−(b
i
)
B
|kχ
B(x
0
,s)
fk
L
p(·)
kχ
B(x
0
,s)
k
L
p
0
(·)
ds
s
n+1
≤Ck
~
bk
∗
Z
∞
2r

1+ln
s
r

2
kχ
B(x
0
,s)
fk
L
p(·)
kχ
B(x
0
,s)
k
L
p
0
(·)
ds
s
n+1
≤Ck
~
bk
∗
Z
∞
2r

1+ln
s
r

2
kχ
B(x
0
,s)
fk
L
p(·)
t
−θ
p
(x
0
,s)
ds
s
.
u´
H
4
≤Ck
~
bk
∗
r
θ
p
(x
0
,r)
Z
∞
2r

1+ln
s
r

2
kχ
B(x
0
,s)
fk
L
p(·)
t
−θ
p
(x
0
,s)
ds
s
.
(ÜH
1
–H
4
O,Œ
kχ
B(x
0
,r)
µ
Ω,
~
b
(f
2
)k
L
p(·)
≤Ck
~
bk
∗
r
θ
p
(x
0
,r)
Z
∞
2r

1+ln
s
r

2
kχ
B(x
0
,s)
fk
L
p(·)
t
−θ
p
(x
0
,s)
ds
s
.
Ïd,d(3.2)Œ
kχ
B(x
0
,r)
µ
Ω,
~
b
(f)k
L
p(·)
≤Ck
~
bk
∗
r
θ
p
(x
0
,r)
Z
∞
2r

1+ln
s
r

2
kχ
B(x
0
,s)
fk
L
p(·)
t
−θ
p
(x
0
,s)
ds
s
.
Ón
v
2
(r) = u
−1
2
(x
0
,r), v
1
(t) = u
−1
1
(x
0
,t)t
−θ
p
(x
0
,t)
,
g(s) = kχ
B(x,s)
fk
L
p(·)
, $(s) = s
−1
s
−θ
p
(x
0
,s)
.
DOI:10.12677/pm.2022.1261111023nØêÆ
¤+•
Ïd,dÚn2.6,Œ
kµ
Ω,
~
b
(f)k
M
p(·),u
2
≤Ck
~
bk
∗
sup
x∈R
n
,r>0
1
u
2
(x,r)
Z
∞
r

1+ln
s
r

2
kχ
B(x
0
,s)
fk
L
p(·)
t
−θ
p
(x
0
,s)
ds
s
≤Ck
~
bk
∗
sup
x∈R
n
,r>0
1
u
1
(x,r)r
θ
p
(x
0
,r)
kχ
B(x,r)
fk
L
p(·)
=Ck
~
bk
∗
kfk
M
p(·),u
1
.
ë•©z
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