Fractional derivatives of Bloch functions, growth rate, and interpolation

Abstract

Bloch functions on the ball are usually described by means of a restriction on the growth rate of ordinary derivatives of holomorphic functions. In this paper we first give a characterization of Bloch functions in terms of fractional derivatives. Then we show that the growth rate suggested by such a characterization is optimal in a certain sense. Also we prove a result concerning interpolating sequences for fractional derivatives of Bloch functions. 0. Introduction and Results Let B be the unit ball of the complex n-space C with norm |z| = 〈z, z〉 where 〈 , 〉 is the usual Hermitian inner product on C. A holomorphic function f on B is said to be a Bloch function if |∇f(z)|(1 − |z|2) is bounded on B where ∇f denotes the complex gradient of f . The space of Bloch functions endowed with norm ||f ||B = |f(0)|+ sup z∈B |∇f(z)|(1− |z|2) is called the Bloch space and denoted by B(B). If f ∈ B(B) satisfies the additional boundary vanishing condition |∇f(z)|(1 − |z|2) → 0 as |z| ↗ 1, we say f ∈ B0(B), the little Bloch space. In this paper we will investigate some properties of the Bloch space in terms of fractional derivatives. Let f be a function holomorphic on B with homogeneous expansion f = ∑∞ k=0 fk. Following [BB], we define the fractional derivative D f of order α > 0 as follows: D f(z) = ∞ ∑