On inner functions with [formula omitted] and [formula omitted] derivatives in the unit ball of [formula omitted

Abstract

Let B n be the unit ball in C n . If f is a bounded holomorphic function, we say that f is inner provided that lim r → 1 − | f ( r ζ ) | = 1 σ -a.e. , ζ ∈ S n where S n is the unit sphere and σ is normalized surface measure on S n . If β > − 1 and p > 0 then A β p denotes the weighted Bergman space of all holomorphic functions weighted by ( 1 − | z | 2 ) β . For 0 < q < 1 , set B q : = A n q − n − 1 1 and if p > 0 let H p denote the usual Hardy space of holomorphic functions on the ball. In this paper, we consider derivatives of inner functions in several spaces of holomorphic functions. If f is an inner function, membership of the radial derivative, R f = ∑ j = 1 n z j ∂ f ∂ z j , will be considered in the B p spaces for p > n n + 1 and will be related to membership in weighted Dirichlet spaces, weighted Bergman spaces A α 2 for 0 < α < 1 , and to the A p spaces for 1 < p < 2 . Moreover, it will be shown that if f is an inner function, n > 1 , and either R f ∈ B 2 n 2 n + 1 , R f ∈ A 3 / 2 , or R f ∈ H 1 / 2 then f must be constant.