与Hardy算子相关的权函数的特征及其应用

The theory of Muchenhoupt weight plays a key role in Harmonic Analysis. Althogh Mp weights which are closed to Muchenhoupt weights attracted many mathematicians from the date of the occurrence, they have not been researched systematacially except for few results. Comparing with Muchenhoupt weights,Mp weights consist of a larger class of function spaces. Besides their close relation to Hardy operators,Mp weights are also a very convenient tool for two-weight estimates of many operators of harmonic analysis-Hilbert transform and maximal operators. We will study the characterization of the weighted BMO spaces by the means of the Hardy operators, the properties of Mp weights and the relationship between the bilinear Hardy operators and the bilinear Mp weights. As applications, we will yield the two-weight estimates for many operators such as the bilinear Hilbert transform, bilinear maximal operators, bilinear Hilbert operators and bilinear Riemann-Liouville operators. We will adopt the idea of the Muchenhoupt weight and the rotation method. We need to point out that the above research work is based on the n-dimensional Euclidean space.

以Muchenhoupt权函数为代表的权函数理论是调和分析研究中的核心问题之一.与 Muchenhoupt权密切相关的Mp权自出现之日起，虽备受关注,但只有零星的结果,至今未得到系统的研究.同Muchenhoupt权相比较： 一方面,Mp权的应用范围更广；另一方面,Mp权同Hardy算子密切相关可以应用Hardy算子的Mp双权模不等式研究Hilbert变换、极大算子的加双权模估计.本项目拟研究：借助高维Hardy算子刻画加权BMO空间、Mp权函数的性质、双线性Mp 权与双线性Hardy算子之间的联系.作为应用，研究双线性Hilbert变换及双线性极大算子、（双线性）Hilbert算子、（双线性）Riemann-Liouville算子等的加双权模估计.我们将借鉴研究Muchenhoupt权函数的方法并结合"旋转的方法"来研究上述问题.需要指出的是，上述研究基于n维欧氏空间中.

Hardy算子作为调和分析中的一类重要的平均算子，与其相关的权函数尤其是与高维Hardy算子相关的权函数并未得到系统的研究。本项目研究了高维Hardy算子的权函数的性质、带幂权的Hardy型算子的最佳常数、利用Hardy型算子在函数空间的有界性刻画函数空间。作为应用，研究了与Hardy算子密切相关的Hausdorff算子在欧氏空间、海森堡群、与量子积分有关问题以及多线性算子的问题。这些结果的取得对于研究平均算子的性质提供了重要的理论依据和方法。