Markov-type inequalities on certain irrational arcs and domains (Q1764873)

Let \({\mathcal P}^d_n\) denote the set of real algebraic polynomials of \(d\) variables of total degree at most \(n\). For a compact set \(K \subset {\mathbb R}^d\), the Markov factor is \[ M_n(K) = \max{\{ \| D_{\omega}P\| _{C(K)} : P \in {\mathcal P}^d_n,\;\| P\| _{C(K)} \leq 1,\;\omega \in S^{d-1} \} }. \] Here \(D_{\omega}\) denotes the directional derivative, and \( S^{d-1}\) stands for the Euclidean unit sphere in \({\mathbb R}^d\). Furthermore, given a smooth curve \(\Gamma \subset {\mathbb R}^d\), the tangential Markov factor of \(\Gamma\) is defined by \[ M_n^T(\Gamma) = \max{\{ \| D_{T}P\| _{C(\Gamma)} : P \in {\mathcal P}^d_n,\;\| P\| _{C(\Gamma)}\} }, \] where \(D_{T}\) denotes the tangential derivative along \(\Gamma\). Let \(d=2\), \[ \Gamma_{\alpha} = \{ (x,x^{\alpha}) : 0 \leq x \leq 1 \}, \] and \[ \Omega_{\alpha} = \left\{ (x,y) : 0 \leq x \leq 1,\;\frac{1}{2}x^{\alpha} \leq y \leq 2x^{\alpha} \right\}. \] The following statements are proved: 1) For every irrational number \(\alpha > 0\) there are constants \(A,B>1\) depending only on \(\alpha\) such that \[ A^n \leq M_n^T(\Gamma_{\alpha}) \leq B^n \] for every sufficiently large \(n\). 2) For every \(\alpha > 1\) there exists a constant \(c>0\) depending only on \(\alpha\) such that \[ M_n(\Omega_{\alpha}) \leq n^{c \log{n}}. \] 3) Let \((\beta_n)\) be an arbitrary increasing sequence of positive numbers tending to \(\infty\). Then there exists an irrational number \(\alpha>1\) so that \[ \liminf_{n\to\infty}{M_n(\Omega_{\alpha})\,n^{-\beta_n}} < \infty. \]