Equivalence of the global and local Markov inequalities in the complex plane

Abstract

We prove that a compact subset of the complex plane satisfies the Global Markov Inequality if and only if it admits an extension property and a Kolmogorov type inequality in Jackson norms, generalizing a result established by Bos and Milman in the real case. We also show that the Global Markov Inequality is equivalent to the Local Markov Property for all compact subsets of the complex plane admitting the Jackson Property. The latter is a generalization of Jackson's approximation inequality, closely connected with the boundary behavior of the Green's function. Finally, we construct an example showing that there is no equivalence in general.