带变量核分数次极大算子在λ-中心Morrey 空间上的加权估计 Weighted Estimates of Fractional Maximal Operator with Variable Kernel on λ-Central Morrey Spaces

doi:10.12677/PM.2023.133068

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PureMathematicsnØêÆ,2023,13(3),636-643

PublishedOnlineMarch2023inHans.https://www.hanspub.org/journal/pm

https://doi.org/10.12677/pm.2023.133068

‘CþØ©êg4ŒŽf3λ-¥%

Morrey˜mþ\O

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'…c

\λ-¥%Morrey˜m§©êg4ŒŽf§CþØ

WeightedEstimatesofFractional

MaximalOperatorwithVariable

Kernelonλ-Central

MorreySpaces

YuheYang

1,2

,ZhenXin

1,2

,QiaoxiaLi

1,2

,SusuXu

1,2

SchoolofMathematicsandStatistics,YiliNormalUniversity, YiningXinjiang

InstituteofAppliedMathematics,YiliNormalUniversity, YiningXinjiang

Received:Feb.22

,2023;accepted:Mar.23

,2023;published:Mar.30

,2023

©ÙÚ^:…Ö,"û,o|_,M€€.‘CþØ©êg4ŒŽf3λ-¥%Morrey˜mþ\O[J].nØê

Æ,2023,13(3):636-643.DOI:10.12677/pm.2023.133068

…Ö

Abstract

Byapplyingtheweightedinequalitiesandtherealvariablemethods,theboundedness

ofthefractionalmaximaloperatorwithvariablekernelisobtainedintheweighted

λ-centralMorreyspaceswiththehelpofthecorrespondingboundednessontheL

spaces.

Keywords

Weightedλ-CentralMorreySpace,FractionalMaximalOperator,VariableKernel

2023byauthor(s)andHansPublishersInc.

This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).

http://creativecommons.org/licenses/by/4.0/

1.Úó9Ì‡(J

Morrey˜m3NÚ©Û9Ù ‡©•§+•kX2•A^,Óž,•Š•Lebesgue˜

m˜«g,í2[1].2000c,Alvarez,Lakey,ÚGuzman-Partide3©z[2]¥ïÄ¥%BMO

˜mÚMorrey˜m'Xž,Ú\λ-¥%Morrey˜m.2019c,>V²<3©z[3]¥

Marcinkiewicz È©9Ù†f3λ-¥%Morrey˜mþ\O.2021c§Mj<3©

z[4]¥ïÄäko÷ØV‚5HardyŽf†f 3λ-¥%Morrey ˜m¥k.5.k'ù

a˜mþÈ©Žfk.5?˜ÚïÄŒë„[5–9].Óž§‘CþØŽf3NÚ©ÛïÄ¥

Ók-‡/ .2011c,M[u3©z[10]¥y²‘ CþØ©êgÈ©Žf3Hardy ˜m

þk.5.2013c,ëp3©z[11]¥ïÄ‘CþØMarcinkiewiczÈ©\k.5,?

NõÆöé‘CþØŽf‰2•ïÄ[12–16].Éþ¡ïÄéu,©òÏLïÄ‘Cþ

Ø©êg4ŒŽf3L

˜mO§l‘CþØ©êg4ŒŽf3λ-¥%Morrey˜m

þ\O.3Qã©Ì‡(Jƒc,I‡Ú\e¡VgÚPÒ.

n−1

•R

(n≥2)¥ü ¥¡,ÙþCLebesgueÿÝdσ=dσ(z

).½Â3

×R

þ¼êΩ ∈L

∞

)×L

n−1

)(t≥1) ÷v

kΩk

∞

)×L

n−1

)

=sup

x∈R

(

n−1

|Ω(x,z

dσ(z

))

<∞,(1)

Ù¥,z

|z|

,∀z∈R

\{0}.

DOI:10.12677/pm.2023.133068637nØêÆ

…Ö

¿Ω÷v^‡Ω(x,λz) = Ω(x,z),∀x,z∈R

,∀λ>0 †ž”^‡

n−1

Ω(x,z

)d(z

) = 0,∀x∈R

.(2)

éu0 <β<n,‘CþØ©êg4ŒŽfM

Ω,β

½Â•

Ω,β

(f)(x) = sup

|Q|

n−β

|Ω(x,x−y)||f(y)|dy.(3)

©òy²CþØ©êg4ŒŽf3\λ-¥%Morrey˜mþk.5.

1<p,q<∞,R

þšKÛÜŒÈ¼êω(x) ¡•A(p,q) ,XJ•3~êC>0,¦

eª¤á

|Q|

ω(x)

|Q|

ω(x)

−p

≤C,(4)

þª¥•~êC^[ω]

A(p,q)

L«.

½½½ÂÂÂ[17,18]λ∈R, 1 <q<∞, ω

Úω

•ÛÜŒÈšKŒÿ¼ê, \λ-¥%Morrey

˜m½Â•

q,λ

,ω

) = {f: kfk

q,λ

,ω

= sup

r>0

(B(0,r))

1+λq

B(0,r)

|f(x)|

(x)dx

<∞},(5)

Ù¥,B(0,r) L«R

¥±:•¥%, r•Œ»¥,¿…ω

= ω

:= ωž,{P

q,λ

,ω

) =

q,λ

©Ì‡(JXe.

½½½nnn1<t<∞,0≤β<n, Ω∈L

∞

)×L

n−1

),‘CþØ©êg4ŒŽfM

Ω,β

dª(3) ¤½Â,@o

−

,1 <p

<t,t

<p<q<∞,1 <t

<p<

,λ

= λ

<0,¿

…ω(x)

∈A

(

)

ž,•3˜‡†fÃ'~êC,¦

Ω,β

(f)k

q,λ

≤Ckfk

p,λ

,ω

©¥,p

L«péó•I,=1/p+1/p

=1.C´Ø•6uÌ‡¼ê½ëþ~ê,3ØÓ1

¥$–3Ó˜1¥Œ±ØÓ.^ω∈∆

L«÷vV^‡¼êω¤8Ü,=•3~ê

C>0,¦é?¿•NQ⊂R

,¤áω(2Q) ≤Cω(Q).

2.½ny²

3y²½nƒc,k‰Ñe¡Ún.

DOI:10.12677/pm.2023.133068638nØêÆ

…Ö

ÚÚÚnnn1[13]XJ0 <α<n, 1 <p<

,±9

−

,eω∈A

(p,q)

(

f(x)ω(x)]

dx)

≤C(

[f(x)ω(x)]

dx)

Ù¥C´Ø•6uf~ê.

ÚÚÚnnn2[18]XJω∈A

(1 ≤q<∞),Ké?¿k∈Z

,l<0,±9•NB⊂R

ω(2

≤D

ω(B)

Ù¥,1 <D

<2.

ÚÚÚnnn3[19]XJω∈A

,1 ≤p<∞,Kω∈∆

.é¤kα>1,þk

ω(αQ) ≤α

[ω]

ω(Q).

ÚÚÚnnn41<t<∞, Ω∈L

∞

)×L

n−1

),e0<β<n, 1≤t

<p<

−

ω(x)

∈A

(

)

.K•3Ø•6uf~êC,¦

(

Ω,β

f(x)ω(x)]

dx)

≤C(

[f(x)ω(x)]

dx)

Ún4y²1 <t<∞,Ω ∈L

∞

)×L

n−1

).dH¨olderØªk

Ω,β

f(x) =sup

r>0

n−β

|y|≤r

|Ω(x,y)||f(x−y)|dy

≤sup

r>0

n−β

|y|≤r

|Ω(x,y)|

|y|≤r

|f(x−y)|

Ù¥,

(

|y|≤r

|Ω(x,y)|

dy)

≤Cr

kΩk

∞

)×L

n−1

)

Ïd,

Ω,β

f(x) ≤Cr

kΩk

∞

)×L

n−1

)

sup

r>0

n−β

(

|y|≤r

|f(x−y)|

dy)

≤Csup

r>0

n−βt

(

|y|≤r

|f(x−y)|

dy)

≤CM

βt

.(6)

5¿1 <t

<p<

−

Ú1 <

βt

(

)

(

)

−

βt

.dÚn1 9ª(6) k

(

Ω,β

f(x)ω(x)]

dx)

≤C(

βt

f(x)ω(x)]

dx)

≤C(

[f(x)ω(x)]

dx)

DOI:10.12677/pm.2023.133068639nØêÆ

…Ö

Ún4y..

½½½nnnyyy²²²f∈

p,λ

,ω

,‰½?¿¥B=B(0,r),r=

√

nl(Q),©)f•f= f

,Ù¥

= fχ

(B)

1+λ

Ω,β

(f)(x)|

(x)dx≤

(B)

1+λ

Ω,β

)(x)|

(x)dx

(B)

1+λ

Ω,β

)(x)|

(x)dx

:=I+II.

éI,dÚn4,¿5¿ω

∈A

(

)

Ω,β

)(x)|

(x)dx≤C

|f(x)|

(x)dx

≤

(2B)

1+λ

|f(x)|

(x)dx

(2B)

(1+λ

≤Ckfk

p,λ

,ω

(2B)

+λ

1+λ

q≥0 ž,dÚn3,¿5¿1 <t

<p<q<∞,λ

−λ

−

,·‚k

(B)

1+λ

Ω,β

)(x)|

(x)dx

≤Ckfk

p,λ

,ω

(2B)

1+λ

(B)

1+λ

≤Ckfk

p,λ

,ω

.(7)

1+λ

q<0ž,dÚn2ÓnŒƒÓO.

e¡OII,x∈B, y∈2

j+1

B,2

j−1

≤|y−x|<2

j+2

(

j+1

|Ω(x,x−y)

dy)

≤CkΩk

∞

)×L

n−1

)

·|2

j+1

.(8)

5¿ω(x)

∈A

(

)

,dª(4),é?¿•NQ⊂R

|Q|

[−(

)

]

(

)

≤C.

DOI:10.12677/pm.2023.133068640nØêÆ

…Ö

Ïd,

|Q|

t−1

(−(

p(t−1)

pt−p−t

))

pt−p−t

p(t−1)

≤C.



−

pt−p−t

dx≤C

|Q|

(pt−t−p)q

|Q|

(Q)

(pt−p−t)q

¤±,

−t

(

)

dx=

−

pt−p−t

dx≤

|Q|

(pt−t−p)q

|Q|

(Q)

(pt−p−t)q

Ïd,k

−

pt−p−t

1−

−

|Q|

(pt−t−p)q

|Q|

(Q)

(pt−p−t)q

pt−p−t

≤

|Q|

1−

−

(Q)

.(9)

Šâþ¡O,dH¨olderØªÚλ

−λ

9ª(8),(9),k

Ω,β

(x) =

|Q|

1−

(0,2r)

|Ω(x,x−y)||f(y)|dy

≤

|B|

1−

∞

j=1

r<|y|≤2

j+1

|Ω(x,x−y)||f(y)|dy

≤C

|B|

1−

∞

j=1

j+1

|f(y)|

j+1

|Ω(x,x−y)|

j+1

ω(y)

−

pt−p−t

1−

−

≤C

|B|

1−

∞

j=1

j+1

|f(y)|

j+1

1−

−

j+1

kΩk

∞

)×L

n−1

)

·|2

j+1

≤CkΩk

∞

)×L

n−1

)

∞

j=1

j+1

1−

|B|

1−

j+1

|f(y)|

≤C

∞

j=1

j+1

−

j+1

1+λ

j+1

|f(y)|

j+1

(1+λ

j+1

≤Ckfk

p,λ

,ω

∞

j=1

j+1

−

j+1

(1+λ

j+1

≤Ckfk

p,λ

,ω

∞

j=1

j+1

DOI:10.12677/pm.2023.133068641nØêÆ

…Ö

II=

(B)

1+λ

Ω,β

)(x)|

(x)dx

≤C

(B)

1+λ

kfk

p,λ

,ω

∞

j=1

j+1

≤Ckfk

p,λ

,ω

∞

j=1

j+1

(B)

≤Ckfk

p,λ

,ω

∞

j=1

[

j+1

(B)

]

Ón,ω

∈A

(

)

,1 <t

<p<q<∞ž,kω

∈A

,Ù¥s= 1+

.λ

<0,dÚn2 

II≤Ckfk

p,λ

,ω

∞

j=1

[

j+1

(B)

]

≤Ckfk

p,λ

,ω

.(10)

(ÜIÚIIO,½ny..

Ä7‘8

žh“‰ŒÆ?‘8(2021YSYB073).

ë•©z

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