几类具有非光滑核的多线性算子的有界性

The study of multilinear operators has been actively pursued in recent years due to their many applications in harmonic analysis and PDE. Boundedness of multilinear operators and its ramifications have proved to be a powerful tool in many aspects of harmonic analysis and PDE. Recently, most of the work aboout multilinear operators have been done by assuming sufficient smoothness on their symbols and kernels. With the development of multilinear operator theory, a lot of attentions have been attracted to multilinear operators with rough kernels. Multilinear operator has also attracted the application’s attentions recently. First we obtained a multilinear Mihlin-Hormander multilier theory by using a new Sobolev space. Second, when n=1, we established the boundedness of the bilinear singular operator with the rough kernel by using the Calderón--Zygmund method of rotations and the uniform boundedness of the bilinear Hilbert transforms. In this work the following important operators will be considered: multilinear Fourier multiplier operators, multilinear Calderón-Zygmund singular integral operators with non-smooth kernels, multilinear maximal Calderón-Zygmund singular integral operators with non-smooth kernels. In the mean while, we also study the vector valued commutators associated with these operators. The main purpose of this work is to establish the boundedness properties of these operators on some spaces and the vector valued inequality also will be considered. All of these problems are considered under the assumption that the kernels satisfy non-smooth conditions which extend the previous work assuming sufficient smoothness on their symbols and kernels.

多线性算子理论在调和分析和PDE中有着广泛的应用。调和分析和PDE中的很多问题都可以转化为多线性算子的有界性问题。目前对于多线性算子的研究主要是在核函数或符号的光滑性较好的前提下进行的。随着多线性理论的发展，对具有粗糙核的多线性算子的需求显得日益迫切。本项目将在核函数或符号的光滑性较弱的情况下来研究多线性算子的有界性。申请人在多线性算子领域已取得一系列的研究成果：通过引入一种新型的Sobolev空间建立了多线性 Mihlin-Hormander 乘子定理；使用旋转的方法解决了维数为1时具有粗糙核的双线性Calderón-Zygmund算子的有界性问题。本项目拟减弱多线性 Fourier 乘子符号的光滑性、建立具有非光滑核的多线性Calderón-Zygmund算子和多线性 Calderón-Zygmund 极大算子及相关向量值广义交换子等在各种空间中的有界性和一些向量值不等式。

多线性算子理论引起了很多专家学者的关注。但是, 目前对于多线性算子的研究主要是在核函数或符号的光滑性较好的前提下进行的。 随着多线性理论的发展，对具有粗糙核的多线性算子的需求显得日益迫切。本项目在核函数或符号的光滑性较弱的情况下来研究多线性算子的有界性。 本项目在乘子符号光滑性较弱的条件下建立了多线性 Fourier 乘子算子的加权有界性、建立具有非光滑核的多线性Calderón-Zygmund 算子和多线性 Calderón-Zygmund 极大算子及相关向量值广义交换子等在各种空间中的有界性和一些向量值不等式。 由于这些成果是在核函数光滑性较弱的情况下取得的，因此本项目改进了现有的研究成果。