In this paper,we introduce Morrey–Herz spaces M K˙(·)q,p(·)(Rn) with variable exponents α(·) and p(·),and prove the boundedness of multilinear Caldern–Zygmund singular operators on the product of these spaces.
In this paper, we reintroduce the weighted multi-parameter Triebel-Lizorkin spaces F_p^(α,q) (ω; R^(n_1)× R^(n_2)) based on the Frazier and Jawerth' method in [11]. This space was′firstly introduced in [18]. Then we establish its dual space and get that(F_p^(α,q))*= CMO_p^(-α,q') for 0 < p ≤ 1.
Let α∈0,(n-1)/2 and T~α be the Bochner-Riesz operator of order α. In this paper, for n = 2 and n ≥ 3, the compactness on Lebesgue spaces and Morrey spaces are considered for the commutator of Bochner-Riesz operator generated by CMO(R^n) function and T~α.
Let TΩ be the singular integral operator with kernel Ω(x)/|x|~n,where Ω is homogeneous of degree zero,integrable and has mean value zero on the unit sphere S^(n-1).In this paper,by Fourier transform estimates,Littlewood-Paley theory and approximation,the authors prove that if Ω∈L(lnL)~2(S^(n-1)),then the commutator generated by T_Ω and CMO(R^n) function,and the corresponding discrete maximal operator,are compact on L^p(R^n,|x|^(γp)) for p∈(1,∞) and γ_p ∈(-1,p-1).
Let T_σ be the bilinear Fourier multiplier operator with associated multiplier σ satisfying the Sobolev regularity that sup κ∈Z∥σ_κ∥W^s(R^(2n))< ∞ for some s ∈ (n, 2n]. In this paper, it is proved that the commutator generated by T_σ and CMO(R^n) functions is a compact operator from L^(p1)(R^n, w_1) × L^(p2)(R^n, w_2) to L^p(R^n, ν_w) for appropriate indices p_1, p_2, p ∈ (1, ∞) with1 p=1/ p_1 +1/ p_2 and weights w_1, w_2 such that w = (w_1, w_2) ∈ A_(p/t)(R^(2n)).
In this paper, the behavior on the product of Lebesgue spaces is considered for the maximal operators associated with the bilinear singular integral operators whose kernels satisfy certain minimal regularity conditions.
