In this dissertation, we consider various aspects of the boundary behavior of holomorphic
functions of several complex variables. In dimension one, a characterization
of the radial limit zero sets of nonconstant holomorphic functions on the disc has
been given by Lusin, Privalov, McMillan, and Berman. In higher dimensions, no such
characterization is known for holomorphic functions on the unit ball B. Rudin posed
the question as to the existence of nonconstant holomorphic functions on the ball
with radial limit zero almost everywhere. Hakim, Sibony, and Dupain showed that
such functions exist. Because the characterization in dimension one involves both
Lebesgue measure and Baire category, it is natural to also ask whether there exist
nonconstant holomorphic functions on the ball having residual radial limit zero sets.
We show here that such functions exist. We also prove a higher dimensional version
of the Lusin-Privalov Radial Uniqueness Theorem, but we show that, in contrast to
what is the case in dimension one, the converse does not hold. We show that any
characterization of radial limit zero sets on the ball must take into account the "complex structure" on the ball by giving an example that shows that the family of these sets is not closed under orthogonal transformations of the underlying real coordinates.
In dimension one, using the theorem of McMillan and Berman, it is easy to see that
radial limit zero sets are not closed under unions (even finite unions). Since there is
no analogous result in higher dimensions of the McMillan and Berman result, it is not obvious whether the radial limit zero sets in higher dimensions are closed under finite unions. However, we show that, as is the case in dimension one, these sets are
not closed under finite unions. Finally, we show that there are smooth curves of finite
length in S that are non-tangential limit uniqueness sets for holomorphic functions
on B. This strengthens a result of M. Tsuji.
| Identifer | oai:union.ndltd.org:tamu.edu/oai:repository.tamu.edu:1969.1/86034 |
| Date | 10 October 2008 |
| Creators | Fulkerson, Michael C |
| Contributors | Boas, Harold P. |
| Publisher | Texas A&M University |
| Source Sets | Texas A and M University |
| Language | en_US |
| Detected Language | English |
| Type | Book, Thesis, Electronic Dissertation, text |
| Format | electronic, born digital |
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