论文标题
关于与切向区域相关的最大运算符的估计值
On Estimates for Maximal Operators Associated with Tangential Regions
论文作者
论文摘要
论文包括三章。 第1章调查了Fatou定理对具有一般近似身份的卷积类型积分运算符的概括。它引入了$λ（r）$ - 收敛性，这是单位光盘中非区域收敛的概括。在不同的功能空间中描述了此类运算符的一般近似身份与最佳收敛区域之间的连接。 第2章研究了Littlewood定理的一些概括，这是对Fatou定理的重要补充，构建了分析功能，几乎在给定的切向曲线上具有不同的特性。相同的卷积类型积分运算符被视为比近似身份更一般的内核。获得了利特伍德定理的两种概括，在任何地方都拥有不同的财产。 第3章专门提及$ \ mathbb {r}^n $中分化基础的等效性问题。研究了稀有二元矩形的基础的完全等效性，以及$ \ mathbb {r}^2 $中完全二元矩形的基础。此外，它在$ \ mathbb {r}^n $中的两个分化库之间引入了准等效性，并被视为差异化的一组函数。
The thesis comprises three chapters. Chapter 1 investigates generalizations of the theorem of Fatou for convolution type integral operators with general approximate identities. It is introduced $λ(r)$-convergence, which is a generalization of non-tangential convergence in the unit disc. The connections between general approximate identities and optimal convergence regions for such operators are described in different functional spaces. Chapter 2 studies some generalizations of the theorem of Littlewood, which makes an important complement to the theorem of Fatou, constructing analytic function possessing almost everywhere divergent property along a given tangential curve. The same convolution type integral operators are considered with more general kernels than approximate identities. Two kinds of generalizations of the theorem of Littlewood are obtained, possessing everywhere divergent property. Chapter 3 is devoted to some questions of equivalency of differentiation bases in $\mathbb{R}^n$. The complete equivalence of basis of rare dyadic rectangles and the basis of complete dyadic rectangles in $\mathbb{R}^2$ is investigated. Besides, it is introduced quasi-equivalence between two differentiation bases in $\mathbb{R}^n$ and is considered the set of functions that such bases differentiate.
