Families of fractional Cauchy transforms in the ball

Abstract:

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.