Dimension-free Markov--Bernstein inequalities for product measures

Abstract

For even integer exponents $p$, we obtain dimension-free Markov--Bernstein inequalities for polynomials with respect to product probability measures $\mu=\mu_1\otimes\cdots\otimes\mu_n$. When the measures $\mu_j$ have unimodal densities, we prove \[ \|\nabla f\|_{L^p(\mu)} \le C(p) d^2 \|f\|_{L^p(\mu)} \] for polynomials $f$ of degree at most $d$. The dependence on $d$ is optimal already for the uniform distribution on the unit cube. For products of one-dimensional Freud measures with densities proportional to $e^{-|t|^{2m}}$, the factor $d^2$ can be replaced by $d^{1-\frac{1}{2m}}$. In the Gaussian case, for all $p\ge4$, we prove that \[ \|\nabla f\|_{L^p(\gamma^n)} \le C(p)d^{\frac12+\theta_p}\|f\|_{L^p(\gamma^n)} \] for every polynomial $f$ of degree at most $d$, where $\theta_p\le \frac{2}{3p}$ and $\theta_p=0$ whenever $p$ is an even integer. Thus, in the even-integer case, we establish the sharp dependence on the degree conjectured by Eskenazis--Ivanisvili. For general $p\ge4$, the estimate improves upon their dimension-free inequality.