Families of fractional Cauchy transforms in the ball

Abstract

Let Bn denote the unit ball in C, n > 1. Given an α > 0, let Kα(n) denote the class of functions defined for z ∈ Bn by integrating the kernel (1−⟨z, ζ⟩)−α against a complex-valued Borel measure on the sphere {ζ ∈ C : |ζ| = 1}. The families of fractional Cauchy transforms Kα(1) have been intensively investigated by several authors. Various properties of Kα(n), n > 2, are studied in this paper. In particular, we obtain relations between Kα(n) and other spaces of holomorphic functions in the ball. Also, we investigate pointwise multipliers for the spaces Kα(n).