Abstract
Let E be a regular compact subset of the real line, let 
be the Green function of the complement of E with respect to the extended complex plane \({\overline {\rm C}}\) with pole at ∞. We construct two examples of sets E of the minimum Hausdorff dimension with 
satisfying the Hölder condition with p = 1/2 either uniformly or locally.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. V. Ahlfors, Lectures on Quasiconformal Mappings, Princeton, NJ, Van Nostrand, 1966.
V. V. Andrievskii, The highest smoothness of the Green function implies the highest density of a set, Arkiv för Matematik 42 (2004), 217–238.
V. V. Andrievskii and H.-P. Blatt, Discrepancy of Signed Measures and Polynomial Approximation, Springer-Verlag, Berlin/New York, 2002.
A. Baernstein,Integral means, univalent functions and circular symmetrization, Acta Math. 133 (1974), 139–169.
L. Bialas and A. Volberg, Markov’s property of the Cantor ternary set, Studia Math. 104 (1993), 259–268.
L. Carleson and V. Totik, Hölder continuity of Green’s functions, Acta Sci. Math. (Szeged) 70 (2004), 558–608.
J. A. Jenkins and K. Oikawa, On results of Ahlfors and Hayman, Illinois J. Math. 15 (1971), 664–671.
M. A. Lavrentiev, Sur la continuité des fonctions univalentes, C. R.: Akad. Sci. USSR 4 (1936), 215–217.
O. Lehto and K. I. Virtanen, Quasiconformal Mappings in the Plane, 2nd ed., Springer-Verlag, Berlin, 1973.
J. Lithner, Comparing two versions of Markov’s property, J. Approx. Theory 77 (1994), 202–211.
V. G. Maz’ja, On the modulus of continuity of a harmonic function at a boundary point, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 135 (1984), 87–95.
W. Pleśniak, A Cantor regular set which does not have Markov’s property, Ann. Pol. Math. LI (1990), 269–274.
Ch. Pommerenke, Boundary Behavior of Conformal Maps, Springer-Verlag, Berlin/New York, 1992.
T. Ransford, Potential Theory in the Complex plane, Cambridge University Press, Cambridge, 1995.
E. B. Saff and V. Totik, Logarithmic Potentials with External Fields, Springer-Verlag, New York/Berlin, 1997.
V. Totik, Markoff constants for Cantor sets, Acta Sci Math. (Szeged) 60 (1995), 715–734.
V. Totik, On Markoff’s inequality, Constr. Approx. 18 (2002), 427–441.
—, Metric properties of harmonic measure, manuscript.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Andrievskii, V.V. On Sparse Sets with the Green Function of the Highest Smoothness. Comput. Methods Funct. Theory 5, 301–322 (2006). https://doi.org/10.1007/BF03321100
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF03321100