Some generalizations of Chirka’s extension theorem

In this paper, we generalize Chirka's theorem on the extension of functions holomorphic in a neighbourhood of S U (oD x D) where D is the open unit disc in C and S is the graph of a continuous D-valued function on D to higher dimensions, for certain classes of graphs S C D x Dn, n > 1. In particular, we show that Chirka's extension theorem generalizes to configurations in Cn+1, n > 1, involving graphs of (non-holomorphic) polynomial maps with small coefficients. 1. THE MAIN THEOREM This paper is motivated by an article by Chirka [1] (also see [2]) in which he proves the following result (in what follows, D will denote the open unit disc in C, while Dr will denote the open disc of radius r, centered at 0 C C): Theorem 1.1 (Chirka). Let 0: D -> C be a continuous function having sup cIq5(z)I 1. Rosay [3] showed that the theorem fails in general for higher dimensions. A natural question that arises is whether holomorphic extension to Dn+1r n > 1, occurs when the component functions of the Dn -valued map defining our graph are small in some appropriate sense (for instance, when the graph is a sufficiently small perturbation of a holomorphic graph). We are able to answer Chirka's question in the affirmative for the class of graphs described in Theorem 1.3 below. Before stating that theorem, however, we state the following proposition, which is a special case of Theorem 1.3. We highlight this as a separate proposition because of the clarity of its statement. Received by the editors May 1, 2000. 2000 Mathematics Subject Classification. Primary 32D15.