A compact set with noncompact disc-hull

Abstract

The disc-hull of a set is the union of the set arid all HiX discs whose boundaries lie in the set. We give an example of a, compact set in C2 whose disc-hull is not compact, answering a question posed by P. Ahern and W. Rudin. The polynomial hull of a compact set X c is the set X of all points x E C' at which the inequality IP(x)l : U --* C,7 is a non-constant map whose components are in H' (U), its range d>(U) is called an H?-disc, parametrized by d>. If lim.,/1(4>(rei0)) E X for almost all e'O on the unit circle T, then 4)(U) is an H'-disc whose boundary lies in They further define the disc-hull D(X) to be the union of X and all Hc-discs whose boundaries lie in X. Because of the maximum principle, D(X) C X. One of the questions posed in [1] (see p. 25) is whether the disc-hull D(X) is always compact for a compact set X C C. Below we answer this question negatively by constructing a counter-exaimple in c2 1. Define w {z E U: Rez > 4 Let Ro: U -> W be the Riemann map satisfying o(+i) = 1 + 23i and ,o(1) = 1. Therefore Re,o(e&0) = 2 for Ree0o ,cn(0) = 0. 2. Let X {((, ?7) E C2 ( E T, q E F(}, where the fiber F, is defined as follows. 1( = T for Re( > 0, 1T( U for ( ?i, and 1( {f01(() 'n E N} U {0} for ReX D? (0) (0, 0); therefore (0, 0) E D(X). Received by the editors August 31, 1999. 2000 Mathematics Subject Classification. Primary 32E20.