Dimension-free Markov--Bernstein inequalities for product measures

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 $$\mathrm{\parallel }\mathrm{\nabla }f{\mathrm{\parallel }}_{L^p(\mu )}\le C(p)d^2\mathrm{\parallel }f{\mathrm{\parallel }}_{L^p(\mu )}$$ 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 $e^{-\mathrm{\mid }t{\mathrm{\mid }}^{2m}}$, the factor $d^2$ can be replaced by $d^{1-\frac{1}{2m}}$. In the Gaussian case, for all $p\ge 4$, we prove that $$\mathrm{\parallel }\mathrm{\nabla }f{\mathrm{\parallel }}_{L^p({\gamma }^n)}\le C(p)d^{\frac{1}{2}+{\theta }_p}\mathrm{\parallel }f{\mathrm{\parallel }}_{L^p({\gamma }^n)}$$ 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 $p\ge 4$, the estimate improves upon their dimension-free inequality.