|
A short proof of Hara and Nakai's theorem
Author(s):
Byung-Geun
Oh
Journal:
Proc. Amer. Math. Soc.
136
(2008),
4385-4388.
MSC (2000):
Primary 30H05, 30D55
Posted:
July 23, 2008
MathSciNet review:
2431053
Retrieve article in:
PDF
Abstract |
References |
Similar articles |
Additional information
Abstract:
We give a short proof of the following theorem of Hara and Nakai: for a finitely bordered Riemann surface , one can find an upper bound of the corona constant of that depends only on the genus and the number of boundary components of .
References:
-
- 1.
- Lars L. Ahlfors, Open Riemann surfaces and extremal problems on compact subregions, Comment. Math. Helv. 24 (1950), 100-134. MR 0036318 (12:90b)
- 2.
- N. Alling, A proof of the corona conjecture for finite open Riemann surfaces, Bull. Amer. Math. Soc. 70 (1964), 110-112. MR 0156967 (28:209)
- 3.
- D. E. Barrett and J. Diller, A new construction of Riemann surfaces with corona, J. Geom. Anal. 8 (1998), 341-347. MR 1707732 (2000j:30076)
- 4.
- M. Behrens, The corona conjecture for a class of infinitely connected domains, Bull. Amer. Math. Soc. 76 (1970), 387-391. MR 0256166 (41:825)
- 5.
- M. Behrens, The maximal ideal space of algebras of bounded analytic functions on infinitely connected domains, Trans. Amer. Math. Soc. 161 (1971), 359-379. MR 0435420 (55:8380)
- 6.
- L. Carleson, Interpolations by bounded analytic functions and the corona problem, Ann. of Math. (2) 76 (1962), 547-559. MR 0141789 (25:5186)
- 7.
- Peter L. Duren, Theory of
spaces, Pure and Applied Mathematics 38, Academic Press, New York-London, 1970. MR 0268655 (42:3552) - 8.
- T. W. Gamelin, Localization of the corona problem, Pacific J. Math. 34 (1970), 73-81. MR 0276742 (43:2482)
- 9.
- T. W. Gamelin, Uniform algebras and Jensen measures, London Mathematical Society Lecture Note Series 32, Cambridge University Press, Cambridge-New York, 1978. MR 521440 (81a:46058)
- 10.
- J. B. Garnett, Bounded analytic functions, Pure and Applied Mathematics 96, Academic Press, Inc., New York-London, 1981. MR 628971 (83g:30037)
- 11.
- J. B. Garnett and P. W. Jones, The Corona theorem for Denjoy domains, Acta Math. 155 (1985), 27-40. MR 793236 (87e:30044)
- 12.
- Masaru Hara and Mitsuru Nakai, Corona theorem with bounds for finitely sheeted disks, Tohoku Math. J. (2) 37 (1985), no. 2, 225-240. MR 788130 (86h:30079)
- 13.
- M. Hayashi, Bounded analytic functions on Riemann surfaces, in: Aspects of complex analysis, differential geometry, mathematical physics and applications (St. Konstantin, 1998), World Sci. Publishing, River Edge, NJ, 1999, 45-59. MR 1731200 (2001i:30045)
- 14.
- P. Jones and D. Marshall, Critical points of Green's functions, harmonic measure and the corona problem, Ark. Mat. 23 (1985), 281-314. MR 827347 (87h:30101)
- 15.
- Mitsuru Nakai, The corona problem on finitely sheeted covering surfaces, Nagoya Math. J. 92 (1983), 163-173. MR 726148 (86b:30072)
- 16.
- Byung-Geun Oh, An explicit example of Riemann surfaces with large bounds on the corona solutions, Pacific J. Math. 228 (2006), no. 2, 297-304. MR 2274522 (2007h:30053)
- 17.
- E. L. Stout, Bounded holomorphic functions on finite Riemann surfaces, Trans. Amer. Math. Soc. 120 (1965), 255-285. MR 0183882 (32:1358)
Similar Articles:
Retrieve articles in Proceedings of the American Mathematical
Society
with
MSC (2000):
30H05, 30D55
Retrieve articles in all Journals with
MSC (2000):
30H05, 30D55
Additional Information:
Byung-Geun
Oh
Affiliation:
Department of Mathematics Education, Hanyang University, 17 Haengdang-dong, Seongdong-gu, Seoul 133-791, Korea
Email:
bgoh@hanyang.ac.kr
DOI:
10.1090/S0002-9939-08-09610-X
PII:
S 0002-9939(08)09610-X
Keywords:
Corona problem,
bounded analytic function
Received by editor(s):
November 14, 2007
Posted:
July 23, 2008
Additional Notes:
This work was supported by the research fund of Hanyang University (HY-2007-000-0000-4844).
Communicated by:
Mario Bonk
Copyright of article:
Copyright
2008,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|