## Arkiv för Matematik

• Ark. Mat.
• Volume 46, Number 1 (2008), 183-196.

### Rationally convex sets on the unit sphere in ℂ2

John Wermer

#### 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].

#### Article information

Source
Ark. Mat., Volume 46, Number 1 (2008), 183-196.

Dates
Revised: 1 May 2007
First available in Project Euclid: 31 January 2017

https://projecteuclid.org/euclid.afm/1485907020

Digital Object Identifier
doi:10.1007/s11512-007-0055-8

Mathematical Reviews number (MathSciNet)
MR2379690

Zentralblatt MATH identifier
1145.32005

Rights

#### Citation

Wermer, John. Rationally convex sets on the unit sphere in ℂ 2. Ark. Mat. 46 (2008), no. 1, 183--196. doi:10.1007/s11512-007-0055-8. https://projecteuclid.org/euclid.afm/1485907020

#### References

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