Arkiv för Matematik

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

Rationally convex sets on the unit sphere in ℂ2

John Wermer

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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

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

Received: 21 February 2006
Revised: 1 May 2007
First available in Project Euclid: 31 January 2017

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2007 © Institut Mittag-Leffler


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.

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  • Aleksandrov, A., Function theory in the ball, in Current Problems in Mathematics. Fundamental Directions, vol. 8, pp. 115–190, 274, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz Inst. Nauchn. i Tekhn. Inform., Moscow, 1985 (Russian). English transl.: in Several Complex Variables II, Encyclopaedia of Math. Sci. 8, pp. 107–178, Springer, Berlin–Heidelberg, 1994.
  • Alexander, H. and Wermer, J., Several Complex Variables and Banach Algebras, Grad. Texts Math. 35, Springer, New York, 1998.
  • Anderson, J. T., Izzo, A. J. and Wermer, J., Rational approximation on the unit sphere in ℂ2, Michigan Math. J. 52 (2004), 105–117.
  • Duchamp, T. and Stout, E. L., Maximum modulus sets, Ann. Inst. Fourier (Grenoble) 31:3 (1981), 37–69.
  • Henkin, G. M., H. Lewy’s equation and analysis on pseudoconvex manifolds, Uspekhi Mat. Nauk 32:3 (1977), 57–118, 247 (Russian). English transl.: Russian Math. Surveys 32 (1977), 59–130.
  • Kytmanov, A. M., On removable singularities of integrable CR-functions, Mat. Sb. 136(178) (1988), 178–186, 301 (Russian). English transl.: Math. USSR-Sb. 64 (1989), 177–185.
  • Lee, H. P. and Wermer, J., Orthogonal measures for subsets of the boundary of the ball in ℂ2, in Recent Developments in Several Complex Variables (Princeton Univ., Princeton, NJ, 1979), Ann. of Math. Stud. 100, pp. 277–289, Princeton Univ. Press, Princeton, NJ, 1981.
  • Rudin, W., Function Theory in the Unit Ball ofn, Grundlehren Math. Wiss. 241, Springer, New York, 1980.
  • Słodkowski, Z., Analytic set-valued functions and spectra, Math. Ann. 256 (1981), 363–386.
  • Stout, E. L., The Theory of Uniform Algebras, Bogden & Quigley, New York, 1971.
  • Wermer, J., Maximum modulus algebras and singularity sets, Proc. Roy. Soc. Edinburgh Sect. A 86 (1980), 327–331.