Markov's property for $k$th derivative
Volume 106 / 2012
Abstract
Consider the normed space $(\mathbb P(\mathbb C^N),\|\cdot \| )$ of all polynomials of $N$ complex variables, where $\|\,\|$ a norm is such that the mapping $L_g:(\mathbb P(\mathbb C^N),\|\cdot \| )\ni f\mapsto gf\in(\mathbb P(\mathbb C^N),\|\cdot \| )$ is continuous, with $g$ being a fixed polynomial. It is shown that the Markov type inequality $$ \left\| \frac{\partial}{\partial z_j}P\right\|\leq M(\deg P)^m\| P\|,\quad\ j=1, \dots, N, \, P\in \mathbb P(\mathbb C^N), $$ with positive constants $M$ and $m$ is equivalent to the inequality $$ \left\| \frac{\partial^N}{\partial z_1\dots \partial z_N}P\right\|\leq M'(\deg P)^{m'}\| P\|,\quad\ P\in \mathbb P(\mathbb C^N) , $$ with some positive constants $M'$ and $m'$. A similar equivalence result is obtained for derivatives of a fixed order $k\geq 2$, which can be more specifically formulated in the language of normed algebras. In addition, we give a nontrivial example of Markov's inequality in the Wiener algebra of absolutely convergent trigonometric series and show that the Banach algebra approach to Markov's property furnishes new tools in the study of polynomial inequalities.
