Rationally convex sets on the unit sphere in ℂ2

Abstract

Let X be a rationally convex compact subset of the unit sphere S in ℂ², 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 $\bar{\delta}_{b}$ 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 ℂ², we denote the rationally convex hull of Y by $\widehat{Y}$. A general reference is Rudin [8] or Aleksandrov [1].