Multipliers of Fractional Cauchy Transforms

Abstract.

Let B _n denote the unit ball of $\mathbb{C}^n$, n ≥ 2. Given an α > 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 measure on the sphere $\{\zeta \in \mathbb{C}^n : |\zeta| = 1\}$. Let $\mathcal{H}ol(B_n)$ denote the space of holomorphic functions in the ball. A function $g \in \mathcal{H}ol(B_n)$ is called a multiplier of $\mathcal{K}_\alpha (n)$ provided that $fg \in \mathcal{K}_\alpha (n)$ for every $f \in \mathcal{K}_\alpha (n)$. In the present paper, we obtain explicit analytic conditions on $g \in \mathcal{H}ol(B_n)$ which imply that g is a multiplier of $\mathcal{K}_\alpha(n)$. Also, we discuss the sharpness of the results obtained.