Title: Tangential Markov inequalities on real algebraic varieties

Authors: L. P. Bos, N. Levenberg, P. D. Milman and B. A. Taylor

Issue: Volume 47 (1998), Issue 4, 1257-1272

Abstract: We say that a smooth compact submanifold $M$ of $\mathbb{R}^n$ admits a tangential Markov inequality of exponent $\ell$ if there is a constant $C > 0$ such that for all polynomials $P \in \mathbb{R}[x_1 , \ldots , x_n]$ and points $a \in M$, \[ |D_TP(a)| \leq C(\text{deg}(P))^{\ell} \sup_{x \in M} |P(x)| .\] In a previous paper, the authors have shown that $M$ admits a tangential Markov inequality of exponent $\ell = 1$ if and only if $M$ is algebraic. Here we show that if $M$ is a smooth, locally closed manifold, then $M$ is algebraic iff $M$ admits a local weighted version of this inequality with $\ell = 1$. We also give an example to show that in the non-smooth case, the exponent $\ell$ must depend on the singularities.