Positive harmonic functions of finite order in a Denjoy type domain
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- by Hiroaki Aikawa PDF
- Proc. Amer. Math. Soc. 131 (2003), 3873-3881 Request permission
Abstract:
We introduce a Denjoy type domain and prove that the dimension of the cone of positive harmonic functions of finite order in the domain with vanishing boundary values is one or two, whenever the boundary is included in a certain set.References
- Hiroaki Aikawa, Norm estimate of Green operator, perturbation of Green function and integrability of superharmonic functions, Math. Ann. 312 (1998), no. 2, 289–318. MR 1671780, DOI 10.1007/s002080050223
- Alano Ancona, Une propriété de la compactification de Martin d’un domaine euclidien, Ann. Inst. Fourier (Grenoble) 29 (1979), no. 4, ix, 71–90 (French, with English summary). MR 558589
- Alano Ancona, Sur la frontière de Martin des domaines de Denjoy, Ann. Acad. Sci. Fenn. Ser. A I Math. 15 (1990), no. 2, 259–271 (French). MR 1087334, DOI 10.5186/aasfm.1990.1502
- David H. Armitage and Stephen J. Gardiner, Classical potential theory, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 2001. MR 1801253, DOI 10.1007/978-1-4471-0233-5
- Michael Benedicks, Positive harmonic functions vanishing on the boundary of certain domains in $\textbf {R}^{n}$, Ark. Mat. 18 (1980), no. 1, 53–72. MR 608327, DOI 10.1007/BF02384681
- Nicolas Chevallier, Frontière de Martin d’un domaine de $\textbf {R}^n$ dont le bord est inclus dans une hypersurface lipschitzienne, Ark. Mat. 27 (1989), no. 1, 29–48 (French). MR 1004720, DOI 10.1007/BF02386358
- Michael C. Cranston and Thomas S. Salisbury, Martin boundaries of sectorial domains, Ark. Mat. 31 (1993), no. 1, 27–49. MR 1230263, DOI 10.1007/BF02559496
- S. Friedland and W. K. Hayman, Eigenvalue inequalities for the Dirichlet problem on spheres and the growth of subharmonic functions, Comment. Math. Helv. 51 (1976), no. 2, 133–161. MR 412442, DOI 10.1007/BF02568147
- Stephen J. Gardiner, Minimal harmonic functions on Denjoy domains, Proc. Amer. Math. Soc. 107 (1989), no. 4, 963–970. MR 991695, DOI 10.1090/S0002-9939-1989-0991695-8
- W. K. Hayman and P. B. Kennedy, Subharmonic functions. Vol. I, London Mathematical Society Monographs, No. 9, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1976. MR 0460672
- Axel Lömker, Martin boundaries of quasi-sectorial domains, Potential Anal. 13 (2000), no. 1, 11–67. MR 1776044, DOI 10.1023/A:1008774010423
- Pietro Poggi-Corradini, On the failure of a generalized Denjoy-Wolff theorem, Conform. Geom. Dyn. 6 (2002), 13–32. MR 1882086, DOI 10.1090/S1088-4173-02-00075-9
- Shigeo Segawa, Martin boundaries of Denjoy domains, Proc. Amer. Math. Soc. 103 (1988), no. 1, 177–183. MR 938665, DOI 10.1090/S0002-9939-1988-0938665-2
- Shigeo Segawa, Martin boundaries of Denjoy domains and quasiconformal mappings, J. Math. Kyoto Univ. 30 (1990), no. 2, 297–316. MR 1068793, DOI 10.1215/kjm/1250520073
- Emanuel Sperner Jr., Zur Symmetrisierung von Funktionen auf Sphären, Math. Z. 134 (1973), 317–327 (German). MR 340558, DOI 10.1007/BF01214695
Additional Information
- Hiroaki Aikawa
- Affiliation: Department of Mathematics, Shimane University, Matsue 690-8504, Japan
- Email: haikawa@math.shimane-u.ac.jp
- Received by editor(s): May 31, 2002
- Received by editor(s) in revised form: August 6, 2002
- Published electronically: April 24, 2003
- Additional Notes: This work was supported in part by Grant-in-Aid for Scientific Research (A) (No. 11304008), (B) (No. 12440040) and Exploratory Research (No. 13874023) Japan Society for the Promotion of Science.
- Communicated by: Juha M. Heinonen
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 131 (2003), 3873-3881
- MSC (2000): Primary 31A05, 31B05, 31B25
- DOI: https://doi.org/10.1090/S0002-9939-03-06977-6
- MathSciNet review: 1999936
Dedicated: Dedicated to Professor Kaoru Hatano on the occasion of his 60th birthday