变量核奇异积分算子的有界性及交换子的紧性研究

Weakening the condition of kernels and obtaining the boundedness of singular integral operators are always the hot problems in harmonic analysis. The improvements play important roles in the development of Harmonic analysis. Applicant has obtained the sharp boundedness of fractional integral operator with variable kernel. Based on applicant's early works, by the three-line theorem and iterative methods, this project will focus on weakening the condition of kernels, establishing the boundedness of fractional integral operator with variable kernel and studying the necessary and sufficient conditions for the boundedness of fractional integral operator with variable kernel. Moreover, we will study the boundedness and compactness of the commutators of fractional integral operator with variable kernel and the commutators of fractional Marcinkiewicz integral operator with variable kernel . In order to obtain the main results, some methods, such as the spherical harmonic development, Fourier transform, approximation theory, may be used.

减弱奇异积分算子的核函数条件并保证算子在函数空间上有界一直是调和分析研究的热点问题, 这一条件减弱在调和分析的发展中也将具有实质性的意义。 申请人前期已获得变量核分数次积分算子的 sharp 有界性，基于申请人的前期工作，本项目将利用三线性插值、迭代等方法减弱变量核分数次积分算子的核函数条件并建立这个算子的有界性，进一步探讨变量核分数次积分算子有界性成立的充分必要条件。同时，利用球面调和函数展开、Fourier变换估计、逼近论等方法研究变量核分数次积分算子交换子、变量核分数次 Marcinkiewicz 积分算子交换子的有界性及紧性问题。

本项目在减弱核函数条件时，首先建立变量核分数次积分算子在Lebesgue 空间上的有界性，进一步建立变量核分数次积分算子交换子的加权有界性及加权紧性问题。其次研究变量核分数次Marcinkiewicz 积分算子与CMO中的函数生成的交换子在Lebesgue 空间上的紧性问题。再次研究Dini型多线性 Calderón-Zygmund 算子的多线性迭代交换子在非齐性空间上的有界性为奇异积分算子在偏微分方程中的应用和Banach 空间理论提供理论依据。