Uniformly perfect subsets of the real line and John domains (Q1884438)

A compact set \(E\subset{\mathbb C}\) is called uniformly perfect if there exists a constant \(0<c<1\) such that \[ E\cap\{\zeta:\,cr\leq| z-\zeta| \leq r\}\neq\emptyset, \quad 0<r<\text{diam}(E):= \sup_{z,\zeta\in E}| z-\zeta| , \] for all \(z\in E\). These sets arise in many areas of complex analysis. A simply connected domain \(\Omega\subset\overline{{\mathbb C}}\) such that \(\infty\in\Omega\) is called a John domain if there exists a positive constant \(c\) such that \[ \text{diam}(H)\leq c| a-b| \] holds for every rectilinear crosscut \([a,b]\) of \(\Omega\), where \(H\) is the bounded component of \(\Omega\setminus[a,b]\). The main purpose of the article is the description of a connection between these two notions in the case of \(E\) is a subset of the real line \({\mathbb R}\). The author supposes that \(E\subset[-1,1]\) is a compact set with positive (logarithmic) capacity \(\text{cap}(E)\), and \(\pm 1\in E\neq[-1,1]\). Let \(g_\omega(z):= g_\omega(z,\infty)\), \(z\in\Omega\), denote the Green function of \(\Omega\) with pole at \(\infty\). The following function defined in the upper half-plane \({\mathbb H}\) plays an important role in the author's investigations: \[ f(z):=\exp\left(\int_E\log(z-\zeta)\,d\mu_E(\zeta) -\log\text{cap}(E)\right). \] Here \(\mu_E\) is the equilibrium measure of the set \(E\). It is proven that \(f\) is univalent and maps \(\overline{{\mathbb C}}\setminus [-1,1]\) onto a starlike (with respect to \(\infty\)) domain \(\overline{{\mathbb C}}\setminus K_E\). The following theorem is the main result of the article. Theorem 1. A set \(E\) is uniformly perfect if and only if \(\overline{{\mathbb C}}\setminus K_E\) is a John domain. As an application, the author obtains a solution of inverse problem of the constructive approximation. Let \(E\subset{\mathbb R}\) be a regular compact set, and let \(E_n(f,E)\) denote the best uniform approximation of a function \(f\) on \(E\) by polynomials of degree at most \(n\) with real coefficients. For \(x\in E\) and \(\delta>0\) let denote \[ \Omega:=\overline{{\mathbb C}}\setminus E, \] \[ E_\delta:=\{z\in{\mathbb C}:\, g_\Omega(z)=\delta\}, \] \[ \rho_\delta(x):= \text{dist}(x,E_\delta)= \inf_{z\in E_\delta}| z-x| \] and define a function \(r(x,\delta)\) by the relation \( \rho_{r(x, \delta)}(x)=\delta. \) \textbf{Theorem~2.} Let \(E\subset{\mathbb R}\) be uniformly perfect. Suppose that \[ E_n(f,E)=O(n^{-\alpha})\quad\text{as }n\to\infty \] holds for a function \(f\in C(E)\) with some \(0<\alpha<1\). Then \[ | f(x_2)-f(x_1)| \leq cr(x_1,| x_2-x_1| )^\alpha \] for all \(x_1,x_2\in E\), where \(c>0\) is a constant independent of \(x_1\) and \(x_2\). In addition, an extended bibliography will help the reader to find necessary related information. The article is recommended for all researchers in complex analysis and approximation theory.