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The corona problem with two pieces of data
Author(s):
Steven
G.
Krantz
Journal:
Proc. Amer. Math. Soc.
138
(2010),
3651-3655.
MSC (2010):
Primary 30H80, 32A38, 32A65
Posted:
May 10, 2010
MathSciNet review:
2661563
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Abstract:
We study the corona problem on the unit ball in , and more generally on strongly pseudoconvex domains in . When the corona problem has just two pieces of data, and an extra geometric hypothesis is satisfied, then we are able to solve it.
References:
-
- 1.
- B. Berndtsson, Integral formulas for the
-equation and zeros of bounded holomorphic functions in the unit ball, Math. Ann. 249(1980), 163-176. MR 578723 (81m:32012) - 2.
- B. Berndtsson, S. Y. Chang, and K. C. Lin, Interpolating sequences in the polydisc, Trans. Amer. Math. Soc. 302(1987), 161-169. MR 887503 (88i:32011)
- 3.
- L. Carleson, Interpolation by bounded analytic functions and the corona problem, Ann. of Math. (2) 76(1962), 547-559. MR 0141789 (25:5186)
- 4.
- U. Cegrell and A. Fällström, Spectrum of certain Banach algebras and
-problems, Ann. Polon. Math. 90(2007), 51-58. MR 2283112 (2008d:32009) - 5.
- J. E. Fornæss and N. Sibony, Smooth pseudoconvex domains in
for which the corona theorem and estimates for fail, Complex Analysis and Geometry, 209-222, Univ. Ser. Math., Plenum, New York, 1993. MR 1211882 (94a:32028) - 6.
- T. W. Gamelin, Uniform Algebras, Prentice-Hall, Englewood Cliffs, NJ, 1969. MR 0410387 (53:14137)
- 7.
- J. B. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981. MR 628971 (83g:30037)
- 8.
- J. B. Garnett and P. W. Jones, The corona theorem for Denjoy domains, Acta Math. 155(1985), 27-40. MR 793236 (87e:30044)
- 9.
- I. M. Gelfand, To the theory of normed rings. II. On absolutely convergent trigonometrical series and integrals, C. R. (Doklady) Acad. Sci. URSS (N.S.) 25(1939), 570-572. MR 0001984 (1:330f)
- 10.
- I. M. Gelfand, To the theory of normed rings. III. On the ring of almost periodic functions, C. R. (Doklady) Acad. Sci. URSS (N.S.) 25(1939), 573-574. MR 0001985 (1:331a)
- 11.
- M. Hakim and N. Sibony, Fonctions holomorphes bornées et limites tangentielles, Duke Math. Journal 50(1983), 133-141. MR 700133 (84m:32011)
- 12.
- S. G. Krantz, Optimal Lipschitz and
regularity for the equation on strongly pseudo-convex domains, Math. Annalen 219(1976), 233-260. MR 0397020 (53:880) - 13.
- S. G. Krantz, Function Theory of Several Complex Variables,
ed., American Mathematical Society, Providence, RI, 2001. MR 1846625 (2002e:32001) - 14.
- S. G. Krantz, Cornerstones of Geometric Function Theory: Explorations in Complex Analysis, Birkhäuser Publishing, Boston, 2006. MR 2167675 (2006e:30001)
- 15.
- S. G. Krantz and S. Y. Li, Some remarks on the corona problem on strongly pseudoconvex domains in
, Illinois J. Math. 39(1995), 323-349. MR 1316541 (96g:32014) - 16.
- S. G. Krantz and S. Y. Li, Explicit solutions for the corona problem with Lipschitz data in the polydisc, Pacific J. Math. 174(1996), 443-458. MR 1405596 (97h:46088)
- 17.
- S. G. Krantz and S. Y. Li, Factorization of functions in subspaces of
and applications to the corona problem, Indiana Univ. Math. J. 45(1996), 83-102. MR 1406685 (97h:46089) - 18.
- N. Sibony, Problème de la couronne pour des domaines pseudoconvexes à bord lisse, Ann. of Math. (2) 126(1987), 675-682. MR 916722 (88k:32012)
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Additional Information:
Steven
G.
Krantz
Affiliation:
Department of Mathematics, Washington University in St. Louis, St. Louis, Missouri 63130
Email:
sk@math.wustl.edu
DOI:
10.1090/S0002-9939-10-10462-6
PII:
S 0002-9939(10)10462-6
Keywords:
Corona problem,
bounded holomorphic functions,
domain of holomorphy,
ball
Received by editor(s):
January 12, 2010
Posted:
May 10, 2010
Additional Notes:
The author was supported in part by the National Science Foundation and by the Dean of the Graduate School at Washington University
Communicated by:
Mei-Chi Shaw
Copyright of article:
Copyright
2010,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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