非齐型空间上齐次~Morrey-Herz 空间中某些交换子的有界性

Morrey-Herz∗( 730070): Morrey-Herz Calder´o n-Zygmund RBMO(µ) .: ; Morrey-Herz ;Calder´o n-Zygmund ; ;RBMO(µ):O.177.6MR :42B25BOUNDEDNESS OF SOME COMMUTATORS ON HOMOGENEOUS MORREY-HERZ SPACES WITH NON DOUBLING MEASURESWU Jiang-long(College of Mathematics and Information Science,Northwest Normal University,Lanzhou730070,Gansu,China)Abstract:The boundedness of some commutators,which are generated by Calder´o n-Zygmund operators and RBMO(µ)functions,are established on homogeneous Morrey-Herz spaces with non doubling measures.Keywords:commutator;homogeneous Morrey-Herz space;Calder´o n-Zygmund operator;non-doubling measure;RBMO(µ)1Calder´o n-Zygmund .. µ Cx∈suppµ r>0, µ(B(x,2r))≤Cµ(B(x,r)), B(x,r)={y∈R d:|y−x|<r}., [1]∼[3] , R d Radon µ , , µ ,C0>0, x∈R d r>0,µ(B(x,r))≤C0r n,(1)∗ : R n .E–mail:wuhjl@n 0<n≤d . R d Radon(1), .[4]∼[7].1997 ,Lu Yang[4] µ d– Lebesgue , Calder´o n-Zygmund BMO(R d) ;2004 , [8] , Calder´o n-Zygmund RBMO(µ)Herz . Herz Morrey-Herz, .,C , . 1≤s≤∞,s s , s =ss−1.2Calder´o n-Zygmund RBMO(µ) Morrey-Herz. , .k∈Z, B k={x∈R d:|x|≤2k},A k=B k\B k−1 χk =χA k, χA kA k .2.1[9] α∈R,0<p≤∞,0<q<∞ λ≥0, Morrey-Herz M˙Kα,λp,q(µ)M˙Kα,λp,q (µ)={f∈L qloc(R d\{0},µ): fM˙Kα,λp,q(µ)<∞},fM˙Kα,λp,q(µ)=supk0∈Z2−k0λ(k0k=−∞2kαp fχkpL q(µ))1p.p=∞ , .2.2[8] α∈R,0<p≤∞ 0<q<∞. Herz ˙Kα,pq(µ)˙Kα,p q (µ)={f∈L qloc(R d\{0},µ): f ˙Kα,pq(µ)<∞},f ˙Kα,pq(µ)=(∞k=−∞2kαp fχkpL q(µ))1p.p=∞ , .Morrey-Herz M˙Kα,λp,q(µ) Herz ˙Kα,pq(µ), M˙Kα,0p,q(µ)=˙Kα,pq(µ).K(·,·)∈L1loc(R d×R d\{(x,y):x=y}) Calder´o n-Zygmund , ,(a)|K(x,y)|≤C|x−y|;(2)(b) 0<δ≤1, |x−x |≤|x−y|2,|K(x,y)−K(x ,y)|+|K(y,x)−K(y,x )|≤|x−x |δ|x−y|.K(·,·) µ Calder´o n-ZygmundT f(x)=R dK(x,y)f(y)dµ(y).(3) K(x,y) x=y , f , Tε(ε> 0)Tεf(x)=|x−y|>εK(x,y)f(y)dµ(y).(4) : ε>0, Tε L p(µ) , T L p(µ) , 1<p<∞., ε>0, Tε M˙Kα,λp,q(µ) , T M ˙Kα,λp,q(µ) , α∈R,0<p≤∞,0<q<∞, λ≥0.γ>1,β>γn βd=2infβ, Q⊂R d (γ,β) Q µ(γQ)≤βµ(Q), γQ Q , γl(Q) . R d Q1⊂Q2,K Q1,Q2=1+N Q1,Q2k=1µ(2k Q1)l(2Q1),N Q1,Q2 l(2k Q1)≥l(Q2)( Q2=R d=Q1 , N Q1,Q2=∞)k. K Q1,Q2[2].2.3[2]ρ>1 , b∈L1loc(µ) RBMO(µ) bB>0, supp(µ) Q,sup Q1µ(ρQ)Q|b(x)−me Q(b)|dµ(x)≤B<∞,|mQ1(b)−mQ2(b)|≤BK Q1,Q2Q1⊂Q2,supp(µ) , Q 2k Q(k∈N) (γ,βd) . me Q(b) b Q ,me Q (b)=1µ( Q)e Qb(x)dµ(x).B b RBMO(µ) , b ∗.Tolsa[2] RBMO(µ) ρ>1 . , ρ=γ=2,βd=2d+1. , (2,2d+1) .,Calder´o n-Zygmund T RBMO(µ) b [b,T](f)=bT(f)−T(bf), T b f(x)=b(x)T f(x)−T(bf)(x)=T((b(x)−b(·))f(·))(x), TL2(µ) {Tε}ε>0 ε→0 . T L2(µ), L2(µ) fT f(x)=R d K(x,y)f(y)dµ(y)µ−a.e.x∈R d\supp(f),(5)K(x,y) (3) .Tolsa[2] T L2(µ) , L q(µ), 1<q<∞.:2.1 (5) T L2(µ) b∈RBMO(µ), T b M˙Kα,λp,q(µ), −nq +λ<α<n(1−1q)+λ,λ≥0,0<p≤1 1<q<∞.2.1, .2.1 b∈RBMO(µ),M b f(x)=supx∈Q1l(Q)nQ|b(x)−b(y)||f(y)|dµ(y),M b M˙Kα,λp,q(µ) , −n q+λ<α<n(1−1q)+λ,λ≥0,0<p≤∞ 1<q<∞.p=∞ , M b L q(µ) [10], M b M˙Kα,λp,q(µ) . 0< p<∞ .f∈M˙Kα,λp,q(µ), ff(x)=∞j=−∞f(x)χj(x)≡∞j=−∞f j(x),M b(f)M˙Kα,λp,q(µ)=supk0∈Z2−k0λ(k0k=−∞2kαp M b(f)χkpL q(µ))1p≤C supk0∈Z2−k0λ(k0k=−∞2kαp(k−2j=−∞M b(f j)χkL q(µ))p)1p+C supk0∈Z2−k0λ(k0k=−∞2kαp(k+1j=k−1M b(f j)χkL q(µ))p)1p+C supk0∈Z2−k0λ(k0k=−∞2kαp(∞j=k+2M b(f j)χkL q(µ))p)1p ≡E1+E2+E3.E2, M b L q(µ) E2≤C fM˙Kα,λp,q(µ).E1, , x∈A k,j≤k−2 y∈A j , |x−y|∼|x|, 2|y|≤|x|. , x∈A k, Q j A j , b j=m˜Qj(b)( ),M b(f j)(x)≤C2−knA j|b(x)−b(y)||f(y)|dµ(y)≤C2−knA j |b(x)−b j||f(y)|dy+C2−knA j|b(y)−b j||f(y)|dµ(y),, j≤k−2, RBMO(µ) ,H¨o lder ,K˜Qj,˜Qk≤C(k−j−1),(1) [2] 3.5,χk M b(f j) L q(µ)≤C2−knA j|f(y)|dµ(y)(A k|b(x)−b j|q dµ(x))1q+C2−knA j|b(y)−b j||f(y)|dµ(y)(A kdµ(x))1q≤C2−kn+kn q(k−j−1) b ∗A j|f(y)|dµ(y)+C2−kn+kn q f j L q(µ)(A j|b(y)−b j|q dµ(y))q≤C2(j−k)n q (k−j) b ∗ f j L q(µ),α<n(1−1q)+λ,E1≤C supk0∈Z 2−k0λ b ∗(k0k=−∞2kαp(k−2j=−∞2(j−k)n q (k−j) f j L q(µ))p)1p≤C supk0∈Z 2−k0λ(k0k=−∞(k−2j=−∞2(j−k)(n q −α)(k−j)(jl=−∞2lαp f l pL q(µ))1p)p)1p≤C supk0∈Z 2−k0λ(k0k=−∞2kλp(k−2j=−∞(k−j)2(j−k)(n q −α+λ))p)1p fM˙Kα,λp,q(µ)≤C supk0∈Z 2−k0λ(k0k=−∞2kλp)1p fM˙Kα,λp,q(µ)=C fM˙Kα,λp,q(µ).E3. E1 ,E3≤C b ∗supk0∈Z 2−k0λ(k0k=−∞2kαp(∞j=k+22(k−j)n q(j−k) f j L q(R n))p)1p≤C fM˙Kα,λp,q(µ)., 2.1 .22.1.2.1 f∈M˙Kα,λp,q(µ), ff(x)=∞j=−∞f(x)χj(x)≡∞j=−∞f j(x),T b(f)M˙Kα,λp,q(µ)=supk0∈Z2−k0λ(k0k=−∞2kαp T b(f)χkpL q(µ))1p≤C supk0∈Z2−k0λ(k0k=−∞2kαp χkT b(k−2j=−∞f j) pL q(µ))1p+C supk0∈Z2−k0λ(k0k=−∞2kαp(k+1j=k−1χkT b(f j) L q(µ))p)1p+C supk0∈Z2−k0λ(k0k=−∞2kαp(∞j=k+2χkT b(f j) L q(µ))p)1p ≡E1+E2+E3.T b L q(µ) E2≤C fM˙Kα,λp,q(µ).E1, , x∈A k,j≤k−2 y∈A j , |x−y|∼|x|. (2) (3) x∈A k,|T b(k−2j=−∞f j)(x)|≤C|x|−nR d|b(x)−b(y)||k−2j=−∞f j(y)|dµ(y)≤CM b(k−2j=−∞f j)(x)≤CM b(f)(x),2.1E1≤C M b(f)M˙Kα,λp,q(µ)≤C fM˙Kα,λp,q(µ).E3, , x∈A k,j≥k+2 y∈A j , |x−y|∼|y|, , j≥k+2,RBMO(µ) ,H¨o lder ,K˜Qk,˜Qj≤C(j−k−1), (2), (1) [2]3.5,χk T b(f j) L q(µ)≤C2−jn(A k(A j|b(x)−b(y)||f(y)|dµ(y))q dµ(x))1q ≤C2−jn(A k|b(x)−b j|q(A j|f(y)|dµ(y))q dµ(x))1q+C2−jn(A k(A j|b(y)−b j||f(y)|dµ(y))q dµ(x))1q ≤C2(k−j)n q(j−k) b ∗ f j L q(µ).2.1 E3 , E3≤C fM˙Kα,λp,q(µ)., .2λ=0 , [8] .[1]Orobitg J,P´e rez C.A p weights for non doubling measures in R n and applications[J].TransAmer Math Soc,2002,354:2013-2033.[2]Tolsa X.BMO,H1and Calder´o n-Zygmund operators for non doubling measures[J],Math.Ann.319(2001):89—149.[3]Tolsa X.Littlewood-Paley theory and the T(1)theorem with non-doubling measures[J],Adv.Math.164(2001):57—116.[4]Lu Shanzhen,Yang Dachun.The continucity of commutators on Herz-type spaces[J],MichiganMath J,1997,44:255[5] , , . 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