On Estimates for Maximal Operators Associated with Tangential Regions

TL;DR

This thesis explores advanced convergence theorems for convolution operators, divergence properties along tangential curves, and the equivalence of differentiation bases in higher-dimensional spaces.

Contribution

It introduces $\lambda(r)$-convergence, generalizes theorems of Fatou and Littlewood, and studies differentiation basis equivalences in $\mathbb{R}^n$.

Findings

Established $\lambda(r)$-convergence as a generalization of non-tangential convergence.

Constructed analytic functions with divergence along tangential curves.

Proved the equivalence of certain differentiation bases in $\mathbb{R}^2$.

Abstract

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 $\lambda(r)$-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.