本学位论文主要研究平面上方向极大算子的几乎正交性准则. 由于极大算子在调和分析, 偏微分方程中扮演着非常基础的作用, 过去几十年来很多学者做了大量的研究工作. 在本学位论文中我们研究了当p是整数时平面上方向极大算子在弱L^p空间和L^p空间中的几乎正交性准则.

本学位论文的主要安排如下: 第一章主要介绍Hardy-Littlewood极大函数和平面上方向极大算子研究背景及本论文的主要结果; 第二章研究平面上方向极大算子在弱L^p空间中的几乎正交性估计; 第三章研究平面上方向极大算子在L^p空间中的几乎正交性估计. 最后, 我们在附录当中给出一个关键引理的详细证明.

This thesis mainly studies the almost regularity criteria for directional maximum operators on the plane. Because the maximum operator plays a very basic role in harmonic analysis and partial differential equations, many scholars have done a lot of research work in the past decades. In this thesis, we study the almost orthogonality criterion of the directional maximal operators on the plane in weak L^p spaces and L^p spaces when p is an integer.

The main arrangement of this thesis is as follows: Chapter 1 mainly introduces the research background of Hardy-Littlewood maximum functions and directional maximum operators in the plane, as well as the main results of this paper; In chapter 2, we study the almost orthogonality estimates of directional maximal operators in the plane in weak L^p spaces; In chapter 3, we study the almost orthogonality estimates of directional maximal operators in the plane in L^p space. Finally, we give a detailed proof of a key lemma in the appendix.