Fractional Cauchy Transforms
Fractional Cauchy transforms form a family of integral transformations depending on a parameter α generalizing the classical Cauchy transform. The concept was introduced by V. P. Havin in the 1960’s. The formula reduces to the well known Cauchy formula for α = 1. The present book studies the class Fα from several perspectives. The relation to Hardy and Besov spaces is treated in chapter 3. The behaviour near the boundary (radial limits) and the distribution of zeros are studied in chapters 4 and 5. Chapters 6 and 7 deal with multipliers, i.e. with functions g with the property that g maps (by multiplication) Fα to Fα. Compositions of Fα with analytic functions are studied in chapter 8. Chapter 9 is devoted to univalent functions. An analytic characterization of the class Fα is proved in chapter 10. However, in the general case, such a characterization remains an important open problem of the theory. The study of Fα goes back to the 60's but most of the results in the book are of recent origin. Though the exposition is to a large extent self-contained, the book is clearly aimed at specialists; good knowledge of complex and functional analysis is required for reading.