Boundary behavior of derivatives of analytic functions (Q1191907)

Denote by \(D\) the complex open unit disk and \(U\) the closed subalgebra of \(L^ \infty(D)\) generated by the bounded complex harmonic functions on \(D\). Define \(P\) to be the linear operator on \(L^ 1(D)\) \[ P(g,z):= {1\over\pi} \int_ D {{g(w)} \over {(1-z\overline{w})^ 2}}dw. \] The authors show that an analytic function \(f\) on \(D\) is in \(P(U)\) if and only if \((1-| z|^ 2)^ n f^{(n)}(z)\in U\) for some positive integer \(n\). Further they give several results that prove there is a closed connection between the behaviour of functions of the form \((1- | z|^ 2)^ n f^{(n)}(z)\), the Gleason parts, the Bloch space and the operator \(P\). Complete proofs are given.