Abstract:
The classical boundary theorem of F. and M. Riesz asserts that, if the radial limits of a bounded holomorphic function f(z)f(z) in the disk |z|<1|z|<1 lie in a set of capacity zero for a set of positive measure on the circle |z|=1|z|=1, then f(z)≡constantf(z)≡constant. The main result of this paper is the proof of an analogous theorem for maps F:D→CnF:D→Cn, where DD is a domain in CnCn. We take as uniqueness set on the boundary any set of positive Lebesgue measure on a generating submanifold.
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