Fractional Cauchy transforms, multipliers and Cesàro operators

Let $B_n$ denote the unit ball in ${\mathbb{C}}^n$, $n\ge 1$. Given an $\alpha>0$, let ${\mathcal K}_\alpha (n)$ denote the class of functions defined for $z\in B_n$ by integrating the kernel $(1- \langle z, \zeta \rangle)^{-\alpha}$ against a complex Borel measure on the sphere $\{\zeta\in{\mathbb{C}}^n: \vert\zeta\vert=1\}$. We study properties of the holomorphic functions $g$ such that $fg\in{\mathcal K}_\alpha(n)$ for all $f\in{\mathcal K}_\alpha(n)$. Also, we investigate extended Cesàro operators on ${\mathcal K}_\alpha (n)$.