On the Integrability of Derivatives of Holomorphic and M-Subharmonic Inner Functions In the Unit Ball of Cn

<p>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.</p> <p>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. </p> <p>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</p> <p> 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. </p> <p>In addition to these results, we will also provide similar results for higher order derivatives as well. </p> <p>Finally, we will briefly consider derivatives of invariant generalized Green potentials in the unit ball and examine the spaces to which they belong.</p>

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Campus Access Dissertation

Discipline thesis:degree_discipline

Mathematics

Year

2011

Author dc:creator

• Gamel, Matthew

Contributors dc:contributor

• Manfred Stoll

• Mathematics
• Physical Sciences and Mathematics
• Inner function
• M-Subharmonic function

Statement dc:rights

• © 2011, Matthew Gamel

Repository record dc:identifier

https://scholarcommons.sc.edu/etd/1595

OAI identifier oai:identifier

oai:scholarcommons.sc.edu:etd-2596