Fractional derivatives of Bloch functions, growth rate, and interpolation (Q1921961)

Let \(C\) be the unit ball in \(\mathbb{C}^n\). Let \(B(C)\) be the set of holomorphic functions \(f = \sum^\infty_{k = 0} f_k\) on \(C\) such that \(|\nabla f(z) |(1 - |z |^2)\) are bounded on \(C\), where \(\nabla f\) denotes the complex gradient of \(f\). This set is endowed with the following norm \[ |f |= \bigl |f(0) \bigr |+ \sup_{z \in C} \bigl |\nabla f(z) \bigr |\bigl( 1 - |z |^2 \bigr) \] and is called the Bloch space. The authors investigate some properties of the Bloch space in terms of fractional derivatives \(D^\alpha f(z) = \sum^\infty_{k = 0} (k + 1)^\alpha f_k (z)\) (Theorem A) and the behavior of these derivatives (Theorems B,C).