Sharp Gaussian divergence and Bernstein-Markov inequalities

Abstract

We prove two dimension-free estimates in Gaussian space. The first is an optimal Meyer-type inequality for the Gaussian divergence: for $p\ge 2$, \[ \|δV\|_{L^p(γ_d)} \leq \sqrt p\,|\mathbb E V| +Cp\,\|DV\|_{L^p(γ_d;\mathrm{HS}_d)}\, . \] Here ${\mathrm{HS}}_d$ is the space of $d\times d$ matrices with its Hilbert-Schmidt norm. Its main ingredient is a dimension-free weak-type $(1,1)$ bound for the second-order transform $D^2{\mathcal{N}}^{-1}$, where $\mathcal{N}:=-\Delta +x\cdot D$ is the Ornstein-Uhlenbeck operator. We also include a direct change-of-variables proof of the weaker bounded derivative divergence inequality in the appendix. The second result is a Gaussian Bernstein-Markov inequality. If $P$ is a polynomial, $p\ge 2$, and $m_p:=(\mathbb{E}|g{|}^p{)}^{1/p}$ for a standard Gaussian $g$, then \[ \|DP\|_{L^p(γ_d;\mathbb R^d)} \leq\frac{2\sqrt{2e\, \mathrm{deg}(P)}}{m_p}\,\|P\|_{L^p(γ_d)}\, . \] The factor $2$ is unnecessary when $p$ is even, or $P$ is even or odd. Thus, the constant is of order $\sqrt{\mathrm{deg}(P)/p}$ throughout the range $p\ge 2$.