The Properties Of Commutators Generated By Bochner-Riesz Operators And Besov Functions

For some integrable functions in L^s(Rⁿ) (s≥2), in this paper we research the almost everywhere convergence of the commutators T_λ;b^T. generated by Bochner-Riesz operator and Besov functions, as well as the boundedness of T_λ;b on L^s(Rⁿ) and on L^s(Rⁿ, r), where L^s(Rⁿ, r) denote the class of radical functions in L^s(Rⁿ).Firstly, when the summation index A is under the critical orderλ₀= (n-1)/2, and max(2, (?))≤s < 2n/(n-1-2λ), we study the almost everywhere convergence of the commutator T_λ;b^T on L^s(Rⁿ). To get this result, we research the (2, 2) boundedness and the two-weighted (2, 2) boundedness of the maximal operator of the commutator, which are generated by the multipliers of compactly supported smooth functions and Besov functions.Secondly, when the index 0 <λ< (n-1)/2, we studies the boundedness of the commutator T_λ;b^T on L^s(Rⁿ) and on L^s(Rⁿ, r), where index s always satisfies |(?)| < (?)+(?) Therefore, we make use of the decomposition of Bochner-Riesz operator to achieve some new results directly by duality and interpolation.The paper indicates the relationship among the summation index of the Bochner-Riesz operatorλ, the index of Besov spaceβ, p, q and the integrable spaces of order s and d profoundly.