Global and Local Markov Inequalities in the Complex Plane

We present the current state of the art concerning the global and local Markov inequalities in the complex plane. This paper is based on a talk given during the Workshop on Multivariate Approximation in honor of Prof. Len Bos 60th birthday, and rests on articles [4], [5] and [6]. Our research is inspired by papers by Bos and Milman where global and local Markov inequalities are compared in the real case (see [7], [8]). We are interested in obtaining analogous results in the complex plane in view of further investigation of properties of the Green’s function connected with Markov sets. We first recall two crucial results obtained by Bos and Milman (see [9]). Theorem A. Suppose that E ⊂ Rn is compact. Then a local Markov inequality with exponent r ≥ 1 is equivalent to a Geometric inequality with the same exponent r ≥ 1 and implies a Sobolev inequality (with Whitney norms), also with the same exponent r. A Sobolev inequality in Whitney norm implies a Sobolev inequality in the quotient norm. Conversely, if E admits a Sobolev inequality in the quotient norm with exponent r ≥ 1, then E admits a local Markov inequality with any exponent ρ > r. Moreover, in the regular case, r = 1, we may take ρ = r = 1. Theorem B. Suppose that E ⊂ Rn is compact and C∞-determining. Then the following are equivalent: 1. E admits a Sobolev type inequality in quotient norm, 2. E admits a bounded extension, 3. E admits a bounded linear extension, 4. E admits a Markov inequality. We were intrigued to obtain a corresponding result for sets in the complex plane because of the intricate interconnectedness of multiple distinct global and local properties: Markov inequalities, Kolmogorov (Sobolev) type inequalities, polynomial approximation, extension operators, geometric properties and, ultimately, the behavior of the Green’s function, i.e. L-regularity, Hölder continuity and the Łojasiewicz-Siciak inequality. However, a simple adaptation to the complex case of the proof given by Bos and Milman is not possible. Our goal is to prove or disprove the equivalence of GMI and LMP in the complex plane following the lead of Bos and Milman. Throughout this paper let E be a compact set in the complex plane. Definition 1. The set E admits the well known Global Markov Inequality GMI(k), where k ≥ 1, if there exists a constant M ≥ 1 such that for arbitrary n ∈ N and holomorphic polynomial p ∈ Pn of degree n we have ‖p‖E ≤ Mn‖p‖E where ‖ · ‖E is the supremum norm on E. If we take a compact set contained in Rn and we replace p′ by the gradient of p, then we obtain the Markov inequality considered by Bos and Milman. The next definition was inspired by the local inequality considered in theorems A and B. Definition 2. A compact set E ⊂ C admits the Local Markov Property LMP(m), where m≥ 1, if there exist constants c, k ≥ 1 such that ∀n ∈ N ∀z0 ∈ E ∀0< r ≤ 1 ∀p ∈ Pn ∀ j = 1, . . . , n : |p (z0)| ≤ cnk rm j ‖p‖E∩B(z0,r). It is evident that the Local Markov Property implies the Global Markov Inequality. aJagiellonian University, Faculty of Mathematics and Computer Science, Institute of Mathematics, 30-348 Kraków, Łojasiewicza 6, Poland, e-mail: [email protected] Bialas-Ciez · Eggink 35 In [3] we proved Theorem 3. LMP implies L-regularity, i.e. the continuity of the Green’s function of the unbounded component of the complement of E to the complex plane with logarithmic pole at infinity. We will use the following notations. Let • Eδ := {z ∈ C : dist(z, E)≤ δ}, • A∞(E) := ¦ f ∈ C∞(C) : ∂ f ∂ z̄ is flat on E © be the family of smooth functions that are ∂̄ -flat on the set E, • H∞(E) := ¦ f ∈ C∞(C) : ∂ f ∂ z̄ ≡ 0 in an open neighborhood of E © be the family of smooth functions that are holomorphic in some open neighborhood of the set E, • | f |E,` := ∑ α∈N0, |α|=` ‖D f ‖E , ‖ f ‖E,` := ‖ f ‖E + | f |E,` be regular supremum norms for f ∈ C∞(E) and ` ∈ N,