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1997 | 66 | 1 | 203-221
Tytuł artykułu

Wiener's type regularity criteria on the complex plane

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Abstrakty
EN
We present a number of Wiener's type necessary and sufficient conditions (in terms of divergence of integrals or series involving a condenser capacity) for a compact set E ⊂ ℂ to be regular with respect to the Dirichlet problem. The same capacity is used to give a simple proof of the following known theorem [2, 6]: If E is a compact subset of ℂ such that $d(t^{-1}E ∩ {|z-a| ≤ 1}) ≥ const > 0$ for 0 < t ≤ 1 and a ∈ E, where d(F) is the logarithmic capacity of F, then the Green function of ℂ \ E with pole at infinity is Hölder continuous.
Twórcy
  • Institute of Mathematics Jagiellonian University Reymonta 4 30-059 Kraków, Poland
Bibliografia
  • [1] L. Białas and A. Volberg, Markov's property of the Cantor ternary set, Studia Math. 104 (1993), 259-268.
  • [2] L. Carleson and T. W. Gamelin, Complex Dynamics, Springer, 1993.
  • [3] W. Hayman and Ch. Pommereneke, On analytic functions of bounded mean oscillation, Bull. London Math. Soc. 10 (1978), 219-224.
  • [4] O. D. Kellogg and F. Vasilesco, A contribution to the theory of capacity, Amer. J. Math. 51 (1929), 515-526.
  • [5] N. S. Landkof, Foundations of Modern Potential Theory, Nauka, Moscow, 1966.
  • [6] J. Lithner, Comparing two versions of Markov's inequality on compact sets, J. Approx. Theory 77 (1994), 202-211.
  • [7] W. Pleśniak, A Cantor regular set which does not have Markov's property, Ann. Polon. Math. 51 (1990), 269-274.
  • [8] W. Pleśniak, Markov's inequality and the existence of an extension operator for $C^∞$ functions, J. Approx. Theory 61 (1990), 106-117.
  • [9] Ch. Pommerenke, Uniformly perfect sets and the Poincaré metric, Arch. Math. (Basel) 32 (1980), 192-199.
  • [10] J. Siciak, Extremal Plurisubharmonic Functions and Capacities in $ℂ^n$, Sophia Kokyuroku in Math. 14, Sophia University, Tokyo, 1982.
  • [11] J. Siciak, Rapid polynomial approximation on compact sets in $ℂ^N$, Univ. Iagel. Acta Math. 30 (1993), 145-154.
  • [12] J. Siciak, Compact sets in $ℝ^n$ admitting polynomial inequalities, Trudy Mat. Inst. Steklov. 203 (1994), 441-448.
  • [13] J. Siciak, Wiener's type sufficient conditions in $ℂ^N$, to appear.
  • [14] V. Totik, Markoff constants for Cantor sets, to appear.
  • [15] M. Tsuji, Potential Theory in Modern Function Theory, Maruzen, Tokyo, 1959.
  • [16] C. de la Vallée-Poussin, Points irréguliers. Détermination des masses par les potentiels, Bull. Cl. Sci. Bruxelles (5) 24 (1938), 672-689.
  • [17] H. Wallin and P. Wingren, Dimension and geometry of sets defined by polynomial inequalities, J. Approx. Theory 69 (1992), 231-249.
  • [18] J. L. Walsh, Interpolation and Approximation by Rational Functions in the Complex Domain, Amer. Math. Soc. Colloq. Publ. 20, Amer. Math. Soc., Providence, R.I., 1935. Third edition, 1960.
  • [19] N. Wiener, The Dirichlet problem, J. Math. Phys. Mass. Inst. Techn. 3 (1924), 127-146.
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