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 εX on the open unit ball int B, with εX(z, w) tending to 0 as (z, w) tends to X. We give a growth condition on εX(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].
Citation
John Wermer. "Rationally convex sets on the unit sphere in ℂ2." Ark. Mat. 46 (1) 183 - 196, April 2008. https://doi.org/10.1007/s11512-007-0055-8
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