Fractional Cauchy transforms, multipliers and Cesàro operators

Abstract

Let B n denote the unit ball in ℂ n , n > 1. Given an α > 0, let K α (n) denote the class of functions defined for z ∈ B n by integrating the kernel (1 - (z,ζ〉) -α against a complex Borel measure on the sphere {ζ ∈ ℂ n : |ζ| = 1}. We study properties of the holomorphic functions g such that fg ∈ K α (n) for all f ∈ K α (n). Also, we investigate extended Cesaro operators on K α (n).