Hausdorff算子的有界性及相关应用

算子在各类空间中的有界性研究是近代调和分析理论研究中最为活跃的课题之一。本项目将致力于研究Hausdorff算子的有界性。该算子在Fourier分析，几何分析和偏微分方程等各个领域均有广泛应用。值得一提的是，Hausdorff算子涵盖了调和分析领域中很多重要算子，如Hardy算子、Cesaro算子等平均算子。因此，Hausdorff算子的研究是一项有重大理论意义和应用价值的工作。首先，本项目拟研究带矩阵的Hausdorff算子在Lp空间上的最佳常数和分数次Hausdorff算子在加幂权Lp空间上的有界性及其最佳常数。其次，项目将研究高维Hausdorff算子在Hp空间上的有界性。最后项目将致力于建立齐次群上Hausdorff算子及其交换子在中心Morrey空间上的有界性理论。在整个项目的研究过程中，我们也将关注它在其它算子理论研究中的应用。

Researching on the boundedness of operators is one of the most active subjects in harmonic analysis. This project will focus on the boundedness of the Hausdorff operator. This operator is widely used in various fields, such as Fourier analysis, geometric analysis and partial differential equations. It is worth to point out that the Hausdorff operator includes many important operators in harmonic analysis, for example, the Hardy operator, the Cesaro operator. Firstly, the project will aim to study the best constant of the Hausdorff operator with a matric on Lp spaces and the fractional Hausdorff operator on Lp spaces with power weights. Secondly, we will study the boundedness of the high dimensional Hausdorff operator on Hp spaces. Finally, we shall work for the boundedness of the Hausdorff operators and their commutators in the setting of homogeneous groups. In the whole process, we will also focus on the applications of our theorems in other fields.