A Compact Set with Noncompact Disc-Hull
Author(s): Buma Fridman, Lop-Hing Ho, Daowei Ma
Source: Proceedings of the American Mathematical Society, Vol. 129, No. 5 (May, 2001), pp.
1473-1475
Published by: American Mathematical Society
Stable URL: http://www.jstor.org/stable/2668758 .
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PROCEEDINGS
OF THE
AMERICAN
MATHEMATICAL
SOCIETY
Voltuml-e 129, Nuimber 5, Pages 1473-1475
S 0002-9939(00)05704-X
Article electroniica.lly piublislhed oin October
A COMPACT
25, 200(
SET WITH NONCOMPACT
DISC-HULL
BUMA FRIDMAN, LOP-HING HO, AND DAOWEI MA
(Communicated by Steven R. Bell)
The disc-hull of a set is the union of the set arid all HiX discs
ABSTRACT.
whose boundaries lie in the set. We give an example of a, compact set in C2
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 C'" is the set X of all points x E C"'
at which the inequality IP(x)l < max{ P(z)I: z E 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 d>: U --* C,7 is a non-constant map whose components are in H' (U), its
range d>(U) is called an H?-disc, parametrized by d>.If lim.,/1(4>(rei0)) E X for
almost all e'Oon the unit circle T, then 4)(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 Hc-discs
whose boundaries lie in X. Because of the maximum principle, D(X) C X. 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 C".
Below we answer this question negatively by constructing a counter-exaimple in
c2
1. Define w
{z
4
Let Ro: U -> W be the Riemann map
U: Rez >
satisfying o(+i) = 1 + 23i and ,o(1) = 1. Therefore Re,o(e&0)= 2 for Ree0o < 0.
o I< 1, and hence lim,l,> ,cn(0) = 0.
Also, 0 , ,o(U), ,(0)
{((, ?7) E C2 ( E T, q E F(}, where the fiber F, is defined as follows.
2. Let X
?i, and 1(
{f01(() 'n E N} U {0} for
1( = T for Re( > 0, 1T( U for (
E
ReX < 0.
One can check that the complement C2\X of X is an open set anid, since X is
also bounded, it is compact. One can also notice that X is connected.
3. For each n consider
(Z)
=(z,
By construction, 1?l(T) c X, so, d1(U)
(0, 0); therefore (0, 0) E D(X).
phn(z))
E
U
-*
C2
D(X). One can see that lihm.,7> D? (0)
Received by the editors August 31, 1999.
2000 Mathematics Subject Classification. Primary 32E20.
Key words and phrases. Polynomial convexity, disc-hull.
(C)2000 Americani
1473
IVlathleiiiaticail SocietY
1474
BUMA FRIDMAN,
LOP-HING
HO, AND DAOWEI
MA
4. To complete the example we now need to show that (0,0) , D(X). If
not, then there is an H?-disc d>(U), >(z) = (a(z),,3(z)):
U -* C2, such that
limr /7(1(rt)) E X for almost all t E T and (0,0) E d>(U). Without loss of generality we mav assume that d>(0) = (a(0),,3(0)) = (0,0) (one can consider d) o V
in place of d) for a suitable Mobius transformation ). Since by construction
limr,z(a(rei0)) E T for almost all e&o,a(z) is a nonconstant inner function.
For any function u(z) E H?(U) the radial limr,/i(u(r&e0)) exists for almost all
t = eio E 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' C T, and T' has positive measure,
then u(z) = 0 for all z E U.
E
For the function a(z) we introduce the set
To = {t
E
T aa(t) exists and a(t)
E
T},
so To is almost all of T. Consider the following sets:
S- = a-l {ei0: Re e'o < O} n To,
S+ = a'l{ei0
= crS?
i:
Re eo > 0}onTo,
Ree2O= 0}
nTo.
Our main goal now is to prove that S- has positive measure. Notice that S- U
S+ U So = To which has full circle measure.
The set a(SO) consists of two points and if So had positive measure, then according to (1), a(z) would be constant. Therefore, So has measure 0. If S+ had
the full measure, then Re a(0) would be positive since it is the average of its values
on T, but a(0) = 0. Therefore, the measure of S- is positive.
Introduce now the following functions: u?(z) = /3(z) - SOP(a(z)) for p = 1, 2, ...
Uo(Z) =3p(z). All of them are in H'(U). Define Sp {t E T: up(t) = 0}. One can
see that by construction almost all points of S- lie in U Sp. Therefore there exists
a q such that Sq has positive measure.
If q = 0, then by (1), p3(z) = u0(z) = 0 on U. This implies that X (containing
almost all of >(T)) contains almost all points (ei, 0). This is impossible since for
all Re eio > 0, the point (eio,0) , X.
Therefore q > 0. So, by (1), uq(z) = 3(z) - q(a(z)) = 0 for all of U. Now
0 = ,3(0) = q(C(0)), and ,o(0) = 0, contradicting 0 , po(U).
Remark. One can see that the entire disc (U, 0)
belong to D(X) but not to D(X).
c
X and all the points of this disc
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 C27, Proc. Amer. Math. Soc. 120 (1994), 1207-1209.
NONCOMPACT
DISC-HULL
1475
REFERENCES
v 137, Amer.
[1] Patrick Ahern and Walter Rudin. Hulls of 3-spheres in C3. Contemporary JAVath.,
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
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UNIVERSITY,
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WICHITA
STATE
UNIVERSITY,
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67260-0033
E-mail address: fridxmanKmath.twsu.edu
DEPARTMENT
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E-mail address: lho@twsuvm
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