Rationally convex sets on the unit sphere in ℂ2

• Published in: Arkiv för matematik 2008-04, Vol.46 (1), p.183-196
• Main Author: Wermer, John
• Format: Article
• Language: English
• Subjects: Mathematics; Mathematics and Statistics
• ISSN: 0004-2080; 1871-2487
• DOI: 10.1007/s11512-007-0055-8

Let X be a rationally convex compact subset of the unit sphere S in ℂ 2 , of three-dimensional measure zero. Denote by R ( X ) the uniform closure on X of the space of functions P / Q , where P and Q are polynomials and Q ≠0 on X . When does R ( X )= C ( X )? Our work makes use of the kernel function for the operator on S , introduced by Henkin in [5] and builds on results obtained in Anderson–Izzo–Wermer [3]. We define a real-valued function ε X on the open unit ball int B , with ε X ( z , w ) tending to 0 as ( z , w ) tends to X . We give a growth condition on ε X ( z , w ) as ( z , w ) approaches X , and show that this condition is sufficient for R ( X )= C ( X ) (Theorem 1.1). In Section 4, we consider a class of sets X which are limits of a family of Levi-flat hypersurfaces in int B . For each compact set Y in ℂ 2 , we denote the rationally convex hull of Y by . A general reference is Rudin [8] or Aleksandrov [1].