多重权理论与多线性算子的有界性

In this program, we will consider some important problems in multilinear harmonic analysis. Precisely, we have three main purpose. At first, we will consider the endpoint estimates for the multilinear singular integral operators(and its associated maxiaml operator) whose kernels satisfies minimum (or nearly minimum) regularity conditions; and consider the mapping properties for the multilinear singular integral operators with rough kernels; by establishing the weighted norm inequalities with general weights for the multilinear Calderon-Zygmund operator and the corresponding maximal multilinear singular integral operator, we will make clear the difference of the singularity of these operators, we also consider the same question for the multilinear singular integral operators with nonsmooth kernels. We will establish some endpoint estimates for the multilinear Calderon-Zygmund operator on nonhomogeneous spaces. Secondly, we will introduce some new class of multilinear Ap weights and give the characterizations of theses new class of weights, and we then introduce the multilinear A-\infty weights and establish some new characterization of this weight class, and study the weighted estimates for some multilinear operators with multiple weights, and generalize some of this result to the space of nonhomogeneous type. Also, we will consider the sharp weights bounds for the multilinear singular integral operators. Finally, we will consider the mapping properties of multilinear pseudodifferential operators whose symbols have minimum regularity on the spatial variable.

首先，研究核函数仅满足最弱（或接近最弱）光滑性条件时多线性奇异积分算子及其相应极大算子的弱端点估计；研究带粗糙核的多线性奇异积分算子的有界性和加权有界性；精确地确定多线性Calderon-Zygmund算子与相应的极大多线性算子的奇异程度的差别；建立带非光滑核的多线性奇异积分算子带一般权函数的加权估计并确定这类算子和相应的极大算子的奇异性的差别；建立非齐型空间上多线性Calderon-Zygmund算子的弱端点估计. 其次，从多(次)线性极大算子和多(次)线性分数次极大算子的变形出发，引进新的多重Ap权类并给出其特征刻画,再引进多重A无穷权类,利用新的多权理论研究一些多线性算子的带多重权的加权估计；在非齐型空间上引进新的多重Ap权类，对其做特征刻画并应用于多线性奇异积分算子；研究多线性奇异积分算子的多权估计的最佳常数.最后,研究象征关于空间变量的光滑性较弱时多线性拟微分算子的性质.

研究了核函数满足某些特殊条件的多线性奇异积分算子的加权估计， 并由此建立了乘子满足两类Sobolev正则性条件时双线性Fourier乘子算子在加权(包括带一般权函数也就是仅仅满足非负局部可积条件的权函数、多重Ap权函数)函数空间上的性质，证实了一类Sobolev正则性适合于研究双线性Fourier乘子算子在带单权的Lebesgue空间上的性质而另一类正则性条件适宜于研究双线性乘子算子在带多重权的Lebesgue空间上的性质；研究了多线性乘子算子与CMO函数的交换子的紧性.研究了向量值多线性奇异积分算子和多(次)线性极大算子在带多重权的函数空间上的性质；通过稀疏算子控制以及稀疏算子的加权估计, 建立了带非光滑核的多线性奇异积分算子、Cohen型多线性奇异积分算子在加权空间上的关于Ap常数之间的最佳界，由此得到了几类Calderon交换子在带多重权的加权空间上的界的精细估计, 还得到了带非光滑核的多线性奇异积分算子交换子的弱端点估计. 通过拓广多线性奇异积分算子理论, 建立了一类具有特殊象征的多线性拟微分算子在函数空间上的性质. 利用Fourier变换以及光滑算子逼近粗糙算子的方法, 建立了齐性奇异积分算子、Marcinkiewicz积分、变量核奇异积分算子等粗糙算子与CMO函数的交换子在函数空间上的紧性和全连续性.