Dimension-free Markov--Bernstein inequalities for product measures

Abstract

We study dimension-free Markov--Bernstein inequalities for polynomials with respect to product probability measures. In the Gaussian case, for $p\ge4$, we prove that \[ \|\nabla f\|_{L^p(γ^n)} \le C(p)d^{\frac12+θ_p} \|f\|_{L^p(γ^n)} \] for every polynomial $f$ of degree at most $d$, where $θ_p\le \frac{2}{3p}$ and $θ_p=0$ whenever $p$ is an even integer. Thus, for even integer exponents, we establish the sharp dependence on the degree conjectured by Eskenazis--Ivanisvili. For general $p\ge4$, the estimate improves upon their dimension-free inequality. We also obtain dimension-free Markov--Bernstein inequalities with sharp dependence on the degree for even integer exponents beyond the Gaussian setting. We first prove such estimates for the uniform distribution on the unit cube and then extend them to products of absolutely continuous measures with unimodal densities. Finally, we treat products of one-dimensional Freud measures with densities proportional to $e^{-|t|^{2m}}$.