Abstract
We deduce a polynomial estimate on a compact planar set from a polynomial estimate on its circular projection, which enables us to prove Markov and Bernstein-Walsh type inequalities for certain sets. We construct
-
–
totally disconnected Markov sets that are scattered around zero in different directions;
-
–
a Markov set \({E \subset \mathbb{R}}\) such that neither \({E \cap [0,+\infty)}\) nor \({E\cap (-\infty,0]}\) admit Markov’s inequality;
-
–
a Markov set that is not uniformly perfect.
Finally, we propose a construction based on a generalization of iterated function systems: a way of obtaining a big family of uniformly perfect sets.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Baribeau L., Brunet D., Ransford T., Rostand J.: Iterated function system, capacity and Green’s function. Comput. Methods Func. Theory 4(1), 47–58 (2004)
Białas L., Volberg A.: Markov’s property of the Cantor ternary set. St. Math. 104(3), 259–268 (1993)
Białas-Cież L.: Equivalence of Markov’s property and Hölder continuity of the Green function for Cantor-type sets. East J. Approx. 1(2), 249–253 (1995)
Białas-Cież L.: Markov sets in \({\mathbb{C}}\) are not polar. Bull. Pol. Ac. Math. 46(1), 83–89 (1998)
Białas-Cież, L., Eggink, R.: Equivalence of the local Markov inequality and a Sobolev type inequality in the complex plane. (in preparation)
Białas-Cież, L., Eggink, R.: Equivalence of the global and local Markov inequalities in the complex plane. (in preparation)
Bos, L.P., Milman, P.D.: On Markov and Sobolev type inequalities on sets in \({\mathbb{R}^n}\) . In: Topics in Polynomials in One and Several Variables and Their Applications, pp. 81–100. World Science Publishing, River Edge (1993)
Bos, L.P., Milman, P.D.: A Geometric Interpretation and the Equality of Exponents in Markov and Gagliardo-Nirenberg (Sobolev) Type Inequalities for Singular Compact Domains. (preprint)
Carleson L., Gamelin T.W.: Complex Dynamics. Universitext, Springer, New York (1993)
Carleson L., Totik V.: Hölder continuity of Green’s functions. Acta Sci. Math. (Szeged) 70, 557–608 (2004)
Goetgheluck P.: Inégalité de Markov dans les ensembles effilés. J. Approx. Theory 30, 149–154 (1980)
Goncharov, A., Uzun, H.B.: Markov’s property of compact sets in \({\mathbb{R}}\) . (preprint)
Hutchinson J.E.: Fractals and self similarity. Indiana Univ. Math. J. 30, 713–747 (1981)
Jonsson, A., Wallin, H.: Function Spaces on Subsets of \({\mathbb{R}^n}\) , Mathematical Reports, vol.2, Part 1. Harwood Academic, London (1984)
Klimek M.: Pluripotential Theory. Oxford Science Publications, Oxford (1991)
Klimek M., Kosek M.: Generalized iterated function systems, multifunctions and Cantor sets. Ann. Polon. Math. 96, 25–41 (2009)
Lithner J.: Comparing two versions of Markov’s inequality on compact sets. J. Approx. Theory 77, 202–211 (1994)
Pawłucki W., Pleśniak W.: Markov’s inequality and \({{\mathcal{C}}^\infty}\) functions on sets with polynomial cusps. Math. Ann. 275, 467–480 (1986)
Pleśniak W.: A Cantor regular set which does not have Markov’s property. Ann. Polon. Math. 51, 269–274 (1990)
Pleśniak W.: Markov’s inequality and the existence of an extension operator for C ∞ functions. J. Approx. Theory 61, 106–117 (1990)
Pleśniak, W.: Recent progress in multivariate Markov inequality. In: Approximation Theory, Monogr. Textbooks Pure Appl. Math., vol. 212, pp. 449–464. Dekker, New York (1998)
Pommerenke Ch.: On the derivative of a polynomial. Michigan Math. J. 6, 373–375 (1959)
Ransford T.: Potential Theory in the Complex Plane, London Math. Soc. Stud. Texts 28. Cambridge University Press, Cambridge (1995)
Totik V.: Markoff constants for Cantor sets. Acta Sci. Math. (Szeged) 60(3-4), 715–734 (1995)
Wallin H., Wingren P.: Dimensions and geometry of sets defined by polynomial inequalities. J. Approx. Theory 69, 231–249 (1992)
Author information
Authors and Affiliations
Corresponding author
Additional information
The research of Marta Kosek was supported in part by a grant of the Faculty of Mathematics and Computer Science of the Jagiellonian University.
Rights and permissions
About this article
Cite this article
Białas-Cież, L., Kosek, M. How to construct totally disconnected Markov sets?. Annali di Matematica 190, 209–224 (2011). https://doi.org/10.1007/s10231-010-0146-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10231-010-0146-1
Keywords
- Markov inequality
- Exceptional sets
- Leja-Siciak extremal function
- Green function
- Iterated function systems
- Attractors