Vol. 29, No. 3, 1969

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Extending bounded holomorphic functions from certain subvarieties of a polydisc

Herbert James Alexander

Vol. 29 (1969), No. 3, 485–490

Let E be a subvariety of the unit polydisc

UN = {(z1,⋅⋅⋅ ,zN ) ∈ CN : |zi| < 1,1 ≦ i ≦ N }

such that E is the zero set of a holomorphic function f on UN, i.e., E = Z(f) where Z(f) = {z UN : 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 TN = {(z1,,zN) CN : |zi| = 1,1 i N}, then there is a bounded holomorphic function F on UN such that E = Z(F). Call such a subvariety E, that is, a pure N 1 dimensional subvariety of UN bounded from TN, 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 UN? Examples show this is not always possible. We will say that a pure N 1 dimensional subvariety E of UN is a special Rudin variety if there exists an annular domain QN = {(z1,,zN) CN : r < |zi| < 1,1 i N} for some r(0 < r < 1) and a δ > 0 such that

(i) E QN = and

(ii) if 1 k N and (z,α,z′′) (Qk1 × U × QNk) E and (z,β,z′′) (Qk1 ×U ×QNk) E and αβ, then |αβ|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 UN, then there exists a bounded linear transformation T : H(E) H(UN) (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 UN.

Mathematical Subject Classification
Primary: 32.20
Received: 8 January 1968
Published: 1 June 1969
Herbert James Alexander