抛物型奇异积分算子在Triebel-Lizorkin空间的有界性

542011

6

11

ACTAMATHEMATICASINICA,ChineseSeries

Vol.54,No.6Nov.,2011

:0583-1431(2011)06-0893-08:A

Triebel–Lizorkin

014030

Email:nymmath@http://doc.guandang.net

730070

Email:taosp@http://doc.guandang.net

T

Triebel–Lizorkin

Tf(x)=p.v.Rn

Ω(y)

f(x y)dy,ρ(y)β

T

,

β≥n,ρ

.

MR(2010)

O174.2

;Triebel–Lizorkin42B20,42B25,42B35

;

BoundednessforParabolicSingularIntegral

onTriebel–LizorkinSpaces

YaoMingNIU

FacultyofMathematics,Baotouteacherscollege,Baotou014030,P.R.China

E-mail:nymmath@http://doc.guandang.net

ShuangPingTAO

CollegeofMathematicsandInformationScience,NorthwestNormalUniversity,

Lanzhou730070,P.R.ChinaE-mail:taosp@http://doc.guandang.net

AbstractInthispaper,theauthorsgivetheboundednessonTriebel–Lizorkinspaces

Ω(y)

forthesingularintegraloperatorwithroughkernelTf(x)=p.v.Rnρ(y)βf(x y)dy,whereβ≥nandρisanormfunctionwhichishomogeneouswithrespecttocertainnonistropicdialation.

Keywordsparabolicsingularintegral;Triebel–Lizorkinspace;roughkernelMR(2010)SubjectClassi cation42B20,42B25,42B35ChineseLibraryClassi cationO174.2

:2010-09-13;:

:2011-04-02

(11071250)

894

54

1

RnSn 1={x∈Rn:|x|=1}.β1,...,βnx=(x1,...,xn)∈Rn,

n x2i

F(x,ρ)=2ρβi

i=1

,

βi≥1.

ρ(ρ>0)ρ(0)=0,[1]

,

ρ=ρ(x)

ρ

Rn

.

Rn

F(x,ρ)=1.ρ(x)=t

ρ(δμx)=μρ(x),

μ>0,

.

(Rn,ρ)

δμ=exp(Alogμ)

,

A

x∈Rn,

x1=ρβ1cos 1···cos n 2cos n 1,x2=ρβ2cos 1···cos n 2sin n 1,...xn 1=ρβn 1cos 1sin 2,

xn=ρβnsin 1,

n

n 1

dx=ρβ 1J(x )dρdσ,β=,dσi=1βi,x∈S

Jacobian.FabesRivi`ere[2]ρβ 1J(x )

Sn 1

,

J(x )Rn

Sn 1.

C∞

,

1≤J(x )≤M.

,

βn≥βn 1≥···≥β1≥1.

Ω(x)Ω(x)

λ>0,

Bλ

Bλ=diag[λβ1,...,λβn],Ω(x),λ>0x∈Rn,Ω(x)

Ω(Bλx)=Ω(x).

(1.1)(1.2)

Sn 1

Ω(x )J(x )dσ(x )=0.

Grafakos

Stefanov

[2]

sup

ξ∈Sn 1

Sn 1

Ω(x)

|Ω(x)|ln

1|x ·ξ|

1+α

dσ(x )<∞, α>0.

(1.3)

[2]

(1.3)

Lr(Sn 1),r>1,

H1(Sn 1),Rivi`ere

[1]

H1(Sn 1)

Hardy

.

,Fabes

Tf(x)=p.v.

Rn

Ω(y)

f(x y)dy.ρ(y)β

(1.4)

Ω(x)Lp(Rn)

.

(1.1)(1.2),FabesNagelRivi`ere[3]

Rivi`ere[1]

,

Ω∈C1(Sn 1),T

Ω∈Llog+L(Sn 1)

,T

Lp(Rn)TLp(Rn)

.

,Chen,Ding

Fan

[4]

Ω∈H1(Sn 1)

,

.

6

:

Triebel–Lizorkin

895

β1=β2=···=βn=1

,ρ(x)=|x|,

(Rn,ρ)=(Rn,|·|),(1.4)

T

,

Triebel–Lizorkin

[5–8].

,,

Ω(x)Chen

(1.1),(1.2)(1.3)

˙α,q(Rn)FZhang[5]p

,Grafakos

Stefanov

T

[2]

.

TΩ(x)

Lp(Rn)

(1.1),(1.2)

.

(1.3)

,

Triebel–Lizorkin

T.

?

0≤φ≤1,supp(φ) {x∈Rn:1/2≤|x|≤2},3/5≤|x|≤5/3

2

,φ(x)>C>0.φk(x)=φ(2kx),Skf=Φk f,k∈Zφk(x)=1,x=0. k(ξ)=φk(ξ),Ψ∈S(Rn),supp(Ψ) {ξ:|ξ|≤2}, |≥c>0,P(Rn)Φ|ξ|≤5,|ΨRn3

∞

φ∈C0(Rn)

.

Triebel–Lizorkin

1q s,q ˙ps,qF(Rn)=f∈S (Rn)/P(Rn): f F2 ksq|Skf|q ˙p(Rn)= <∞.

k∈Z

p

Triebel–Lizorkin

1q s,q s,qFp(Rn)=f∈S (Rn)/P(Rn): f Fp2 ksq|Skf|q (Rn)= + Ψ f p<∞.

k≥1

p

Triebel–Lizorkin

.

1<p,q<∞,

(1)(2)(3)(4)

1111

p+p =1,q+q =1.0,2˙0,2(Rn);Lp(Rn)=Fp(Rn)=Fp

s,q˙s,q(Rn)∩Lp(Rn)s,qs,qFp(Rn)～F f Fp˙p(Rn)～ f Lp(Rn)+ f Fp(Rn),

s,q s,q s,q˙s,q(Rn)) =F˙ (Fp(Rn)) =Fp(Rn)(F(Rn); pp

s,q1s,q2˙s,q1(Rn) F˙s,q2(Rn)FFp(Rn) Fp(Rn),q1≤q2.pp

pn

s>0;

T

L(R)

.

1.1

T

(1.4)T

.

Ω∈L1(Sn 1)

(1.1),(1.2),

α>0,

Ω

(1.3).

T

Lp(Rn)

,

(2+2α)/(1+2α)<p<2+2α.

Triebel–Lizorkin

.

Ω∈L1(Sn 1)

,

1.2

(1.2),

s∈R,1<p,q<∞,Tα>0,Ω(1.3)T

(1.4)

˙s,q(Rn)Fp

.

(1.1),

s,qs,q Tf F˙p˙p(Rn)≤C f F(Rn).

1.1

1.2,

(2)

(3),1.2

.

1.1

s∈R,1<p,q<∞,T,Ω

,

T

s,q

Fp(Rn)

.

C

.

2

.

E(x,r)

jE(x,r)

x

r

,

x

j

,

E(x,r)={y:ρ(x y)<r}.j>0,jE(x,r)={y:ρ(x y)<jr},|E(x,r)|

896

54

E(x,r)

Lebesgue

,

|E(x,r)|,

rβA

.

E

x

r

.

· AA

Hardy–Littlewood

Mf(x)=supr β

r>0

x y A≤r

|f(y)|dy,

Mf(x)=sup

x∈E

1|E|

E

|f(y)|dy.

[9]

(Mf)ν∈A1,

[10]

0≤ ≤1

Mf

.

2.11≤p≤∞,

E

ω(x)∈Ap,

E,

1|E|

p 1

11

ω(x)dxω(x) p 1dx≤C,1<p<∞,

|E|E

supE

1|E|

E

,

A∞=2.1

p≥1

y∈E

ω(x)dx≤Cω(y),p=1

Ap,

RnL

.

[10]

0≤δ≤1

y

C>0

x

{βi},

2

dλ

≤C| x,y | δ/L,e i Bλx,y λ1

.

x,y∈Rn,

Tf(x)

Tf(x)=p.v.

Rn

Ω(y)

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