Kolmogorov and Markov type inequalities on certain algebraic varieties (Q6057450)

Let \(\mathcal{P(\mathbb{R^N})}\) be the space of all polynomials of \(N\) real variables with real coefficients of degree at most \(n\). For a function \(f\), defined on a compact set \(K\in \mathbb{R^N}\), let us consider the norm \(\vert \vert f\vert \vert_K=\sup_{x\in K}\vert f(x)\vert \). The classical Markov inequality for derivatives \(D^\alpha P,\ P\in \mathcal{P(\mathbb{R^N})},\) gives the following sharp upper bounds: \(\vert \vert D^\alpha P\vert \vert_K\leq C^{\vert \alpha\vert}n^{k}\vert \vert P\vert \vert_K\), where \(D^\alpha=\frac{\partial^{\vert \alpha\vert}}{\partial x_1^{\alpha_1}\cdots \partial x_N^{\alpha_N}},\ \alpha =(\alpha_1,\cdots,\alpha_N)\in Z_+^N,\ \vert \alpha\vert =\alpha_1 +\cdots +\alpha_N\). The local Markov inequality is also well known, in which the quantity \(\vert D^\alpha\vert \) is estimated in above by \(\vert \vert P\vert \vert_{\widetilde{K}}\), where \(\widetilde{K}\) is some compact neighborhood of \(x_0\in \mathbb{R^N}\). It is proved that the local Markov inequality is equivalent to Markov classical ones. A generalization of the classical Markov and of its local form to compact subsets of certain algebraic varieties are introduced. Besides, the Kolmogorov-type inequality is proposed. It is proved that the inequalities introduced are equivalent.