Boundary behavior of singular integrals

Abstract

Let $\mu$ be a complex measure on the unit circle. In this thesis we study integral transforms of the measure defined on the unit disk. We investigate the behavior of its Cauchy, Poisson and conjugate Poisson transforms as the function argument approaches points lying in the support of the measure. We first show that if $\mu$ is absolutely continuous with respect to Lebesgue measure on the unit circle, the transforms tend to finite values almost everywhere on the unit circle. In contrast, if $\mu$ is singular with respect to Lebesgue measure, its Cauchy transform tends to infinity near the support of the measure. A key part of this thesis is dedicated to the normalized Cauchy transform associated with a measure $\mu$. This is a linear operator defined on the space of $\mu$-integrable functions. We discuss some properties of this operator and then turn to the main result concerning the boundary behavior of its image functions.<br />In the last chapter we present a classification of measures based on the relative speed of growth of their Poisson and conjugate Poisson transforms as their function argument approaches the support of $\mu$, and show how the results on the normalized Cauchy transform can be applied in the investigation of the boundary behavior of these transforms.