多线性Journe定理及Littlewood-Paley算子交换子的端点有界性

多线性Littlewood-Paley算子及其交换子的有界性研究是近年来调和分析的热点问题之一。本项目拟通过多线性复插值定理、Fourier乘子的衰减估计及局部光滑估计等几何与分析技巧，在申请人及其合作者已有关于多线性算子与交换子有界性研究的基础上，通过建立平面上双线性覆盖引理及插值定理，证明带有符号是一族互不相交且边与坐标轴平行的矩形的特征函数的双线性乘子算子的Littlewood-Paley平方函数的有界性；寻找BMO空间的最大子空间，使得多线性Marcinkiewicz积分、Lusin面积积分及g_\lambda^*函数与该子空间内的函数生成的交换子从乘积加权Hardy及Lebesgue空间到加权Lebesgue及弱型Lebesgue空间有界；减弱核的光滑条件，探索在该条件下前面提到的算子与BMO子空间内的函数生成的交换子在上述空间的有界性。这些研究将进一步丰富和发展多线性算子理论。

The study on the boundedness of multilinear Littlewood-Paley operator and its commutator has been one of the hot topics in harmonic analysis in recent years. This project, by making full use of the geometry and analysis techniques, such as, the multilinear complex interpolation, the decay estimates of the Fourier multipliers and the local smoothing estimates, and based on the existing works of the applicants and her collaborators on the study of the boundedness of multilinear operators and their commutators and so on, by establishing the covering lemma for bilinear operators on the plain and the bilinear interpolation theorem, we aim to prove the boundedness of the Littlewood-Paley square functions for bilinear multipliers with symbol of disjoint rectangles with sides parallel to the axes; find the biggest subspace of BMO space, such that the commutators generated by the multilinear Marcinkiewicz integral, Lusin area integral as well as the g_\lambda^* function and the functions in this subspace are bounded from product of weighted Hardy space and weighted Lebesgue space to weighted Lebesgue space and weighted weak Lebesgue space; try to weaken the smooth conditions of integral kernel, study the boundedness of the commutators generated by the operators above and the functions in the subspace of BMO in the spaces mentioned before. All of these will further enrich and develop the multilinear operator theory.