PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 129, Number 5, Pages 1473–1475 S 0002-9939(00)05704-X Article electronically published on October 25, 2000

A COMPACT SET WITH NONCOMPACT DISC-HULL

BUMA FRIDMAN, LOP-HING HO, AND DAOWEI MA

(Communicated by Steven R. Bell)

Abstract. The disc-hull of a set is the union of the set and all H^{∞} discs whose boundaries lie in the set. We give an example of a compact set in ℂ^{2} whose disc-hull is not compact, answering a question posed by P. Ahern and W. Rudin.

The polynomial hull of a compact set X ⊂ ℂ^{n} is the set X̂ of all points x ∈ ℂ^{n} at which the inequality |P (x)| ≤ max{|P (z)| : z ∈ X} holds for every polynomial P. Let U denote the unit disc in ℂ. In [1] P. Ahern and W. Rudin introduced the following definition.

“If Φ : U → ℂ^{n} is a non-constant map whose components are in H^{∞}(U), its range Φ(U) is called an H^{∞}-disc, parametrized by Φ. If lim_{r↗1}(Φ(re^{iθ})) ∈ X for almost all e^{iθ} on the unit circle T, then Φ(U) is an H^{∞}-disc whose boundary lies in X.”

They further define the disc-hull D(X) to be the union of X and all H^{∞}-discs whose boundaries lie in X. Because of the maximum principle, D(X) ⊂ X̂. One of the questions posed in [1] (see p. 25) is whether the disc-hull D(X) is always compact for a compact set X ⊂ ℂ^{n}.

Below we answer this question negatively by constructing a counter-example in ℂ^{2}.

1. Define ω = \{z ∈ U : Re z > \frac{1}{2}\}. Let φ : \overline{U} → \overline{ω} be the Riemann map satisfying φ(±i) = \frac{1}{2} ± \frac{\sqrt{3}}{2}i and φ(1) = 1. Therefore Re φ(e^{iθ}) = \frac{1}{2} for Re e^{iθ} ≤ 0. Also, 0 ∉ φ(\overline{U}), | φ(0) |< 1, and hence lim_{n→∞} φ^{n}(0) = 0.

2. Let X = {(ζ, η) ∈ ℂ^{2} : ζ ∈ T, η ∈ Γ_{ζ}}, where the fiber Γ_{ζ} is defined as follows. Γ_{ζ} = T for Re ζ > 0, Γ_{ζ} = \overline{U} for ζ = ±i, and Γ_{ζ} = \{φ^{n}(ζ) : n ∈ N\} ∪ \{0\} for Re ζ < 0.

One can check that the complement ℂ^{2}\X of X is an open set and, since X is also bounded, it is compact. One can also notice that X is connected.

3. For each n consider

Φ_{n}(z) = (z, φ^{n}(z)) : U → ℂ^{2}.

By construction, Φ_{n}(T) ⊂ X, so, Φ_{n}(U) ∈ D(X). One can see that lim_{n→∞} Φ_{n}(0) = (0, 0); therefore (0, 0) ∈ \overline{D(X)}.

\begin{matrix}c \\ ◯ \end{matrix}2000 American Mathematical Society

Received by the editors August 31, 1999.

2000 Mathematics Subject Classification. Primary 32E20.

Key words and phrases. Polynomial convexity, disc-hull.

4. To complete the example we now need to show that (0, 0) ∉ D(X). If not, then there is an H^{∞}-disc Φ(U), Φ (z) = (α(z), β(z)) : U → ℂ^{2}, such that lim_{r↗1}(Φ(rt)) ∈ X for almost all t ∈ T and (0, 0) ∈ Φ(U). Without loss of generality we may assume that Φ(0) = (α(0), β(0)) = (0, 0) (one can consider Φ ∘ ψ in place of Φ for a suitable Möbius transformation ψ). Since by construction lim_{r↗1}(α(re^{iθ})) ∈ T for almost all e^{iθ}, α(z) is a nonconstant inner function.

For any function u(z) ∈ H^{∞}(U) the radial lim_{r↗1}(u(re^{iθ})) exists for almost all t = e^{iθ} ∈ T. Let u(t) denote the corresponding limit. The following property of an H^{∞} function u(z) (see [2], p. 339) will be used:

If u(t) = 0 for t ∈ T′ ⊂ T, and T′ has positive measure,
then u(z) = 0 for all z ∈ U.

(1)

For the function α(z) we introduce the set

T_{0} = {t ∈ T : α(t) exists and α(t) ∈ T },

so T_{0} is almost all of T. Consider the following sets:

\begin{aligned}S^{−} & = α^{−1}\{e^{iθ} : Re e^{iθ} < 0\} ∩ T_{0}, \\ S^{+} & = α^{−1}\{e^{iθ} : Re e^{iθ} > 0\} ∩ T_{0}, \\ S^{0} & = α^{−1}\{e^{iθ} : Re e^{iθ} = 0\} ∩ T_{0}. \end{aligned}

Our main goal now is to prove that S^{−} has positive measure. Notice that S^{−} ∪ S^{+} ∪ S^{0} = T_{0} which has full circle measure.

The set α(S^{0}) consists of two points and if S^{0} had positive measure, then according to (1), α(z) would be constant. Therefore, S^{0} has measure 0. If S^{+} had the full measure, then Re α(0) would be positive since it is the average of its values on T, but α(0) = 0. Therefore, the measure of S^{−} is positive.

Introduce now the following functions: u_{p}(z) = β(z) − φ^{p}(α(z)) for p = 1, 2, ...; u_{0}(z) = β(z). All of them are in H^{∞}(U). Define S_{p} = {t ∈ T : u_{p}(t) = 0}. One can see that by construction almost all points of S^{−} lie in ⋃ S_{p}. Therefore there exists a q such that S_{q} has positive measure.

If q = 0, then by (1), β(z) = u_{0}(z) = 0 on U. This implies that X (containing almost all of Φ(T)) contains almost all points (e^{iθ}, 0). This is impossible since for all Re e^{iθ} > 0, the point (e^{iθ}, 0) ∉ X.

Therefore q > 0. So, by (1), u_{q}(z) = β(z) − φ^{q}(α(z)) = 0 for all of U. Now 0 = β(0) = φ^{q}(α(0)), and φ(0) = 0, contradicting 0 ∉ φ(\overline{U}).

Remark. One can see that the entire disc (U, 0) ⊂ X̂ and all the points of this disc belong to \overline{D(X)} but not to D(X).

Added after posting

After the galley proofs were returned, the authors were informed by J. Globevnik that a different counterexample was published by Herb Alexander in “A disc-hull in ℂ^{2}”, Proc. Amer. Math. Soc. 120 (1994), 1207–1209.

References

[1] Patrick Ahern and Walter Rudin. Hulls of 3-spheres in ℂ^{3}. Contemporary Math., v 137, Amer. Math. Soc., Providence, RI, 1992, 1-27. MR 93k:32020

[2] Walter Rudin. Real and Complex Analysis, 2nd ed. McGraw-Hill, New York, 1974. MR 49:8783

Department of Mathematics and Statistics, Wichita State University, Wichita, Kansas 67260-0033

E-mail address: fridman@math.twsu.edu

Department of Mathematics and Statistics, Wichita State University, Wichita, Kansas 67260-0033

E-mail address: lho@twsuvm.uc.twsu.edu

Department of Mathematics and Statistics, Wichita State University, Wichita, Kansas 67260-0033

E-mail address: dma@math.twsu.edu