Multipliers of Fractional Cauchy Transforms

摘要

Let B-n denote the unit ball of C-n, n >= 2. Given an alpha > 0, let K-alpha(n) denote the class of functions defined for z is an element of B-n by integrating the kernel (1 - < z, zeta >)(-alpha) against a complex-valued measure on the sphere {zeta is an element of C-n : vertical bar zeta vertical bar = 1}. Let Hol(B-n) denote the space of holomorphic functions in the ball. A function g is an element of Hol(B-n) is called a multiplier of K-alpha(n) provided that fg is an element of K-alpha(n) for every f is an element of K-alpha(n). In the present paper, we obtain explicit analytic conditions on g is an element of Hol(B-n) which imply that g is a multiplier of K-alpha(n). Also, we discuss the sharpness of the results obtained.