Hardy型算子与Hausdorff型算子的加权有界性

In this project we will study the weighted boundedness of Hardy type operators and Hausdorff type operators. Theses two types of operator are widely used in many various fields, such as Harmonic analysis and partial differential equations. Hausdorff type operators include Hardy type operators, but also include some other classical operators, such as Hilbert operators, Cesáro operators, and Riemann-Liouville integral. We aim to study weak type bounds for high dimensional Hardy operators and the conjugate operators on the Lebesgue spaces with power weight. Meanwhile, we work out the corresponding operator norms. We characterize weighted weak inequalites for the bilinear Hardy operators. As applications, we will yield the weighted estimates for the bilinear Hilbert transform and a class of bilinear singular integral operators. Finally we will focus on the corresponding weighted boundedness of Hausdorff type operators. As applications, we give the weak type estimates and operator norms for many operators such as Hilbert operators and Riemann-Liouville integral.

本项目研究Hardy型算子与Hausdorff型算子的加权有界性。这两种算子在调和分析及偏微分方程等许多数学分支中有广泛应用。Hausdorff型算子包含Hardy型算子，而且包含其它经典算子，比如Hilbert算子，Cesáro算子，Riemann-Liouville积分。本项目拟研究：高维Hardy算子与其共轭算子在带有幂权的Lebesgue空间上弱型有界的算子范数；给出双线性Hardy算子加权弱型有界的权的特征，作为应用研究双线性Hilbert变换和一类双线性奇异积分的加权有界性。 最后项目将研究Hausdorff型算子的相应有界性，作为应用给出Hilbert算子，Riemann-Liouville积分等算子的弱型有界及算子范数。

在本课题里，我们研究n维Hardy型算子在中心Morrey空间上的弱型有界，同时得到了精确的算子范数。此外，我们给出了共轭Hardy算子在加权Lebesgue空间上的弱型有界和相应的算子范数，作为应用得到Gamma函数的一个估计。. 我们研究了p进Hardy算子在p进Morrey-Herz空间的有界性。进一步确定了p进Hardy算子的交换子在p进函数空间如p进Lebesgue空间、p进Herz空间以及p进Morrey-Herz空间上的Lipschitz估计与CMO估计。这些结果对于分数次p进Hardy算子也是成立的。