Fractional Cauchy Transforms

Rita A. Hibschweiler

• Publisher: Chapman and Hall/CRC
• Year: 2020
• ISBN:
  9780429525063
• Edition: 1
• Topic: Matemáticas
• Subtopic: Matemáticas general

Presenting new results along with research spanning five decades. Fractional Cauchy Transforms provides a full treatment of the topic, from its roots in classical complex analysis to its current state. Self-contained, it includes introductory material and classical results, such as those associated with complex-valued measures on the unit circle, that form the basis of the developments that follow. The authors focus on concrete analytic questions, with functional analysis providing the general framework., After examining basic properties, the authors study integral means and relationships between the fractional Cauchy transforms and the Hardy and Dirichlet spaces. They then study radial and nontangential limits, followed by chapters devoted to multipliers, composition operators, and univalent functions. The final chapter gives an analytic characterization of the family of Cauchy transforms when considered as functions defined in the complement of the unit circle.

CHAPTER 1

Introduction

Preamble. This chapter introduces the topic of this book, the families of fractional Cauchy transforms. For each α ≥ 0 a family ${\mathcal{F}}_{\alpha }$ is defined as the collection of functions that can be expressed as the Cauchy-Stieltjes integral of a suitable kernel. The case α = 1 corresponds to the set of Cauchy transforms of measures on the unit circle $\text{T}=\{\text{z}\in ℂ:|\text{z}|=1\}$. Each function in ${\mathcal{F}}_{\alpha }$ is analytic in $\mathcal{D}=\{\text{z}\in ℂ:|\text{z}|<1\}$.

Some facts are recalled about the Hardy spaces H^p and the harmonic classes h^p, and it is noted that ${\text{H}}^1\subset {\mathcal{F}}_1$. Other connections between H^p and ${\mathcal{F}}_{\alpha }$ are given in Chapter 3 and in later chapters. Properties of complex-valued measures on T are obtained. Subsequently, these properties are shown to be related to properties of functions in ${\mathcal{F}}_{\alpha }$.

The Riesz-Herglotz formula is quoted and the correspondence between measures and functions given by this formula is shown to be one-to-one. As a consequence of this formula, any function which is analytic in $\mathcal{D}$ and has a range contained in a half-plane belongs to the family ${\mathcal{F}}_1$.

The F. and M. Ri...