Families of fractional Cauchy transforms in the ball

Let $B_n$ denote the unit ball in ${\mathbb{C}}^n$, $n\ge 1$. Given $\alpha>0$, 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-valued Borel measure on the sphere $\{\zeta\in{\mathbb{C}}^n:|\zeta|=1\}$. The families ${\mathcal K}_\alpha(1)$ of fractional Cauchy transforms have been investigated intensively by several authors. In the paper, various properties of $\mathcal K_\alpha(n)$, $n\ge 2$, are studied. In particular, relations between ${\mathcal K}_\alpha(n)$ and other spaces of holomorphic functions in the ball are obtained. Also, pointwise multipliers for the spaces ${\mathcal K}_\alpha (n)$ are investigated.