Metrics associated with extremal plurisubharmonic functions

Klimek, Maciej

doi:10.1090/S0002-9939-1995-1307539-3

Metrics associated with extremal plurisubharmonic functions
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Proc. Amer. Math. Soc. 123 (1995), 2763-2770 Request permission

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

Richard Aron, Bernard Beauzamy, and Per Enflo, Polynomials in many variables: real vs complex norms, J. Approx. Theory 74 (1993), no. 2, 181–198. MR 1226355, DOI 10.1006/jath.1993.1060
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), no. 4, 598–600. MR 776186, DOI 10.1090/S0002-9939-1985-0776186-8

Supremum norms for polynomials

M. F. Barnsley and S. Demko, Iterated function systems and the global construction of fractals, Proc. Roy. Soc. London Ser. A 399 (1985), no. 1817, 243–275. MR 799111
Michael Barnsley, Fractals everywhere, Academic Press, Inc., Boston, MA, 1988. MR 977274
Eric Bedford and B. A. Taylor, A new capacity for plurisubharmonic functions, Acta Math. 149 (1982), no. 1-2, 1–40. MR 674165, DOI 10.1007/BF02392348
Hans Brolin, Invariant sets under iteration of rational functions, Ark. Mat. 6 (1965), 103–144 (1965). MR 194595, DOI 10.1007/BF02591353
Peter L. Duren, Univalent functions, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 259, Springer-Verlag, New York, 1983. MR 708494
John H. Hubbard and Peter Papadopol, Superattractive fixed points in $\textbf {C}^n$, Indiana Univ. Math. J. 43 (1994), no. 1, 321–365. MR 1275463, DOI 10.1512/iumj.1994.43.43014
John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713–747. MR 625600, DOI 10.1512/iumj.1981.30.30055
M. Klimek, Extremal plurisubharmonic functions and $L$-regular sets in $\textbf {C}^{n}$, Proc. Roy. Irish Acad. Sect. A 82 (1982), no. 2, 217–230. MR 701379

Pluripotential theory

Magnus Lundin, The extremal PSH for the complement of convex, symmetric subsets of $\textbf {R}^N$, Michigan Math. J. 32 (1985), no. 2, 197–201. MR 783573, DOI 10.1307/mmj/1029003186
M. J. Parker, Convex sets and subharmonicity of the distance function, Proc. Amer. Math. Soc. 103 (1988), no. 2, 503–506. MR 943074, DOI 10.1090/S0002-9939-1988-0943074-6
W. Pawłucki and W. Pleśniak, Markov’s inequality and $C^\infty$ functions on sets with polynomial cusps, Math. Ann. 275 (1986), no. 3, 467–480. MR 858290, DOI 10.1007/BF01458617
W. Pleśniak, Quasianalytic functions in the sense of Bernstein, Dissertationes Math. (Rozprawy Mat.) 147 (1977), 66. MR 427674
W. Pleśniak, Markov’s inequality and the existence of an extension operator for $C^\infty$ functions, J. Approx. Theory 61 (1990), no. 1, 106–117. MR 1047152, DOI 10.1016/0021-9045(90)90027-N
Józef 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 143946, DOI 10.1090/S0002-9947-1962-0143946-5

Extremal plurisubharmonic functions in

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Józef Siciak, Highly noncontinuable functions on polynomially convex sets, Univ. Iagel. Acta Math. 25 (1985), 95–107. MR 837828