In this paper, the authors consider the weighted estimates for the commutators of multilinear Calderón-Zygmund operators.By introducing an operator which shifts the commutation, and establishing the weighted esti...In this paper, the authors consider the weighted estimates for the commutators of multilinear Calderón-Zygmund operators.By introducing an operator which shifts the commutation, and establishing the weighted estimates for this new operator, the authors prove that, if p_1 ∈ （1,∞）, p_2,…,p_m ∈（1,∞], p ∈ （0,∞） with 1/p =Σ1≤k≤ m 1/pk, then for any weight w, the commutators of m-linear Galderón-Zygmund operator are bounded from L P1（R n,M_l（logL） σw）× p2（Rn,M~w）×...×Lpm（Rn,Mw） to Lp（Rn,w）with σ to be a constant depending only on p_1 and the order of commutator展开更多
The commutators of oscillatory singular integral operators with homogeneous kernel $\frac{{\Omega (x)}}{{\left| x \right|^n }}$ are studied, where Ω is homogeneous of degree zero, has mean value zero on the unit sphe...The commutators of oscillatory singular integral operators with homogeneous kernel $\frac{{\Omega (x)}}{{\left| x \right|^n }}$ are studied, where Ω is homogeneous of degree zero, has mean value zero on the unit sphere. It is proved that Ω∈L (logL)K+1(Sn-1) is a sufficient condition under which the k-th order commutator is bounded on L2(Rn).展开更多
The paper is to establish a boundedness criterion for some commutators of linear operators when these linear operators don't satisfy the general Ap weight estimates but satisfy some radial weight estimates.
The authors consider the multilinear oscillatory singular integral operator defined by T A 1,A 2,…,A k f(x)=∫ R n e iP(x,y) ∏k j=1 R m j (A j;x,y)Ω(x-y)|x-y| n+M f(y)dy,where P(x,y) ...The authors consider the multilinear oscillatory singular integral operator defined by T A 1,A 2,…,A k f(x)=∫ R n e iP(x,y) ∏k j=1 R m j (A j;x,y)Ω(x-y)|x-y| n+M f(y)dy,where P(x,y) is a real-valued polynomial on R n× R n , Ω is homogeneous of degree zero, R m j (A j;x,y) denotes the m j -th order Taylor series remainder of A j at x expanded about y , M=∑kj=1 m j . It is shown that if Ω belongs to the space L log +L(S n-1 ) and has vanishing moment up to order M , then‖T A 1,A 2,…,A k f‖ q C ∏kj=1∑|α|=mj‖D αA j‖ r j ‖f‖ p, provided that 1<p,q<∞ , 1<r j ∞ (j=1,2,...,k) and 1/q=1/p+∑kj=1 1/r j . The corresponding maximal operator is also considered.展开更多
Let TΩ be the singular integral operator with kernel Ω（x）/｜x｜n where is homogeneous of degree zero, integrable and has mean value zero on the unit sphere Sn-1. In this paper, by Fourier transform estimates, L...Let TΩ be the singular integral operator with kernel Ω（x）/｜x｜n where is homogeneous of degree zero, integrable and has mean value zero on the unit sphere Sn-1. In this paper, by Fourier transform estimates, Littlewood-Paley theory and approximation, the authors prove that if Ω∈（lnL）2 （Sn- 1）, then the commutator generated by TΩ and CMO（Rn） function, and the corresponding discrete maximal operator, are compact on LP（Rn, ｜s｜γp） for p∈ （1, ∞） and γp ∈ （-1, p-l）展开更多
基金This research was supported by the NSFC （10971228）.
文摘In this paper, the authors consider the weighted estimates for the commutators of multilinear Calderón-Zygmund operators.By introducing an operator which shifts the commutation, and establishing the weighted estimates for this new operator, the authors prove that, if p_1 ∈ （1,∞）, p_2,…,p_m ∈（1,∞], p ∈ （0,∞） with 1/p =Σ1≤k≤ m 1/pk, then for any weight w, the commutators of m-linear Galderón-Zygmund operator are bounded from L P1（R n,M_l（logL） σw）× p2（Rn,M~w）×...×Lpm（Rn,Mw） to Lp（Rn,w）with σ to be a constant depending only on p_1 and the order of commutator
文摘The commutators of oscillatory singular integral operators with homogeneous kernel $\frac{{\Omega (x)}}{{\left| x \right|^n }}$ are studied, where Ω is homogeneous of degree zero, has mean value zero on the unit sphere. It is proved that Ω∈L (logL)K+1(Sn-1) is a sufficient condition under which the k-th order commutator is bounded on L2(Rn).
基金National Natural Science Foundation of China(No.1 990 1 0 2 1 ) and Beijing Ed-ucation Commission FoundationNatural Science Foundation of Beijing (1 0 1 3 0 0 6)
文摘The paper is to establish a boundedness criterion for some commutators of linear operators when these linear operators don't satisfy the general Ap weight estimates but satisfy some radial weight estimates.
文摘The authors consider the multilinear oscillatory singular integral operator defined by T A 1,A 2,…,A k f(x)=∫ R n e iP(x,y) ∏k j=1 R m j (A j;x,y)Ω(x-y)|x-y| n+M f(y)dy,where P(x,y) is a real-valued polynomial on R n× R n , Ω is homogeneous of degree zero, R m j (A j;x,y) denotes the m j -th order Taylor series remainder of A j at x expanded about y , M=∑kj=1 m j . It is shown that if Ω belongs to the space L log +L(S n-1 ) and has vanishing moment up to order M , then‖T A 1,A 2,…,A k f‖ q C ∏kj=1∑|α|=mj‖D αA j‖ r j ‖f‖ p, provided that 1<p,q<∞ , 1<r j ∞ (j=1,2,...,k) and 1/q=1/p+∑kj=1 1/r j . The corresponding maximal operator is also considered.
基金supported by National Natural Science Foundation of China(Grant No.11371370)
文摘Let TΩ be the singular integral operator with kernel Ω（x）/｜x｜n where is homogeneous of degree zero, integrable and has mean value zero on the unit sphere Sn-1. In this paper, by Fourier transform estimates, Littlewood-Paley theory and approximation, the authors prove that if Ω∈（lnL）2 （Sn- 1）, then the commutator generated by TΩ and CMO（Rn） function, and the corresponding discrete maximal operator, are compact on LP（Rn, ｜s｜γp） for p∈ （1, ∞） and γp ∈ （-1, p-l）
