Date of Award

1-1-2011

Document Type

Campus Access Dissertation

Department

Mathematics

First Advisor

Manfred Stoll

Abstract

Let Bn be the unit ball in Cn. If f is a bounded holomorphic function, we say that f is inner provided that its modulus has a radial limit of 1 almost everywhere on Sn where Sn is the unit sphere.

If a > -1 and p > 0 then Apa denotes the weighted Bergman space of all holomorphic functions weighted by (1-|z|^2)a and for 0 < q < 1, set Bq := A1(n/q)-n-1. For p > 0 let Hp denote the usual Hardy space of holomorphic functions on the ball.

In this dissertation, we consider derivatives of inner functions in several spaces of holomorphic functions. If f is an inner function, membership of the radial derivative, Rf(z) = (d/dt)f(tz)|t=1 will be considered in the Bp spaces for p > n/(n+1) and will be related to

membership in weighted Dirichlet spaces, weighted Bergman spaces A2a for 0 < a < 1, and to the Ap spaces for 1 < p < 2. Moreover, it will be shown that if f is an inner function, n > 1, and either Rf belongs to B2n/(2n+1) , A3/2, or H1/2 then f must be constant.

In addition to these results, we will also provide similar results for higher order derivatives as well.

Finally, we will briefly consider derivatives of invariant generalized Green potentials in the unit ball and examine the spaces to which they belong.

Share

COinS