Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Metrics associated with extremal plurisubharmonic functions


Author: Maciej Klimek
Journal: Proc. Amer. Math. Soc. 123 (1995), 2763-2770
MSC: Primary 32F05; Secondary 31C10, 32H50
DOI: https://doi.org/10.1090/S0002-9939-1995-1307539-3
MathSciNet review: 1307539
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A natural metric is introduced on the family of all polynomially convex compact L-regular sets in $ {\mathbb{C}^n}$, thus turning this family into a complete metric space. An application in complex dynamics is described.


References [Enhancements On Off] (What's this?)

  • [ABE] R. Aron, B. Beauzamy, and P. Enflo, Polynomials in many variables. Real vs complex norms, J. Approx. Theory 74 (1993), 181-198. MR 1226355 (94h:32003)
  • [AK1] D. H. Armitage and Ü. Kuran, The convexity of a domain and the superharmonicity of the signed distance function, Proc. Amer. Math. Soc. 93 (1985), 598-600. MR 776186 (86k:31005)
  • [AK2] R. Aron and M. Klimek, Supremum norms for polynomials, preprint.
  • [B-D] M. Barnsley and S. Demko, Iterated function systems and the global construction of fractals, Proc. Roy. Soc. London Ser. A 399 (1985), 243-275. MR 799111 (87c:58051)
  • [BAR] M. Barnsley, Fractals everywhere, Academic Press, Boston, 1988. MR 977274 (90e:58080)
  • [B-T] E. Bedford and B. A. Taylor, A new capacity for plurisubharmonic functions, Acta Math. 149 (1982), 1-40. MR 674165 (84d:32024)
  • [BRO] H. Brolin, Invariant sets under iteration of rational functions, Ark. Mat. 6 (1965), 103-144. MR 0194595 (33:2805)
  • [DUR] P. L. Duren, Univalent functions, Springer-Verlag, New York, 1983. MR 708494 (85j:30034)
  • [H-P] J. H. Hubbard and P. Papadopol, Superattractive fixed points in $ {\mathbb{C}^n}$, Indiana Univ. Math. J. 43 (1994), 321-365. MR 1275463 (95e:32025)
  • [HUT] J. E. Hutchinson, Fractals and self similarity, Indiana Univ. Math. J. 30 (1981), 713-747. MR 625600 (82h:49026)
  • [KL1] M. Klimek, Extremal plurisubharmonic functions and L-regular sets in $ {\mathbb{C}^n}$, Proc. Roy. Irish Acad. Sect. A 82 (1982), 217-230. MR 701379 (84f:32019)
  • [KL2] -, Pluripotential theory, Oxford Univ. Press, Oxford, 1991.
  • [LUN] M. Lundin, The extremal plurisubharmonic function for convex symmetric subsets of $ {\mathbb{R}^n}$, Michigan Math. J. 32 (1985), 197-201. MR 783573 (86h:32030)
  • [PAR] M. J. Parker, Convex sets and subharmonicity of the distance function, Proc. Amer. Math. Soc. 103 (1988), 503-506. MR 943074 (89e:31005)
  • [P-P] W. Pawlucki and W. Pleśniak, Markov's inequality and $ {\mathcal{C}^\infty }$ functions on sets with polynomial cusps, Math. Ann. 275 (1986), 467-480. MR 858290 (87k:32031)
  • [PL1] W. Pleśniak, Quasianalytic functions in the sense of Bernstein, Dissertationes Math. (Rozprawy Mat.) 147 (1977), 1-69. MR 0427674 (55:705)
  • [PL2] -, Markov's inequality and the existence of an extension operator for $ {\mathcal{C}^\infty }$ functions, J. Approx. Theory 61 (1990), 106-117. MR 1047152 (91h:46065)
  • [SI1] J. Siciak, On some extremal functions and their applications in the theory of analytic functions of several complex variables, Trans. Amer. Math. Soc. 105 (1962), 322-357. MR 0143946 (26:1495)
  • [SI2] -, Extremal plurisubharmonic functions in $ {\mathbb{C}^n}$, Ann. Polon. Math. 29 (1981), 175-211.
  • [SI3] -, Highly noncontinuable functions on polynomially convex sets, Uni. Iagel. Acta Math. 25 (1985), 95-107. MR 837828 (87i:32020)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 32F05, 31C10, 32H50

Retrieve articles in all journals with MSC: 32F05, 31C10, 32H50


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1995-1307539-3
Keywords: Extremal plurisubharmonic functions, pluricomplex Green functions, complex dynamics
Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society