Let E be a subvariety of the
unit polydisc
such that E is the zero set of a holomorphic function f on U^{N}, i.e., E = Z(f) where
Z(f) = {z ∈ U^{N} : f(z) = 0}. This amounts to saying that E is a subvariety of pure
dimension N − 1. In [2] Walter Rudin proved that if E is bounded away from the
torus T^{N} = {(z_{1},⋯,z_{N}) ∈ C^{N} : z_{i} = 1,1 ≦ i ≦ N}, then there is a bounded
holomorphic function F on U^{N} such that E = Z(F). Call such a subvariety E, that
is, a pure N − 1 dimensional subvariety of U^{N} bounded from T^{N}, a Rudin variety.
We are interested in the following question: When is it possible to extend every
bounded holomorphic function on a Rudin variety E to one on U^{N}? Examples
show this is not always possible. We will say that a pure N − 1 dimensional
subvariety E of U^{N} is a special Rudin variety if there exists an annular domain
Q^{N} = {(z_{1},⋯,z_{N}) ∈ C^{N} : r < z_{i} < 1,1 ≦ i ≦ N} for some r(0 < r < 1) and a δ > 0
such that
(i) E ∩ Q^{N} = ∅ and
(ii) if 1 ≦ k ≦ N and (z′,α,z′′) ∈ (Q^{k−1} × U × Q^{N−k}) ∩ E and
(z′,β,z′′) ∈ (Q^{k−1} ×U ×Q^{N−k}) ∩E and α≠β, then α−β≧ δ. Obviously (i) implies
that a special Rudin variety is a Rudin variety. We have the
Theorem. If E is a special Rudin variety in U^{N}, then there exists a bounded
linear transformation T : H^{∞}(E) → H^{∞}(U^{N}) (where H^{∞} is the corresponding
Banach space of bounded holomorphic functions under sup norm) which extends each
bounded holomorphic function on E to one on U^{N}.
