摘要
本文研究由Bochner-Ricsz算子与Besov函数生成的交换子T_(λ;b)~T在某些可积函数空间L~s(R~n)(s≥2)中的几乎处处收敛性,同时讨论T_(λ;b)在L~s(R~n)和L~s(R~n)中径向函数类上的有界性问题.
我们首先研究求和指标λ小于临界阶λ_0=(n-1)/2,且max{2,(?)}≤s<2n/(n-1-2λ)时,交换子T_(λ;b)~T在L~s(R~n)上的几乎处处收敛性.为了得到这一结果,我们利用紧支光滑函数的乘子算子与Besov函数生成交换子的相应的极大算子,分别进行(2,2)的有界性估计和权函数为幂权的双权(2,2)有界性估计.
其次,我们研究了当指标0<λ<(n-1)/2时,交换子T_(λ;b)在L~s(R~n)和L~s(R~n)中径向函数类上的有界性,其中s均满足|(?)|<(?)+(?).为此,我们利用Bochner-Riesz算子的分解形式T_(λ)f(x)=(?)*f(x),由对偶性及插值定理得到了新的结论.
文中深刻阐明了Bochner-Riesz算子的求和指标λ,Besov空间中的相关指数β,p,q与可积空间的指标s和d之间的相互关系.
For some integrable functions in L~s(R~n) (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(R~n) and on L~s(R~n, r), where L~s(R~n, r) denote the class of radical functions in L~s(R~n).
Firstly, when the summation index A is under the critical orderλ_0= (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(R~n). 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(R~n) and on L~s(R~n, 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.
引文
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