Dimension independent Bernstein-Markov inequalities in Gauss space

Abstract

We obtain the following dimension independent Bernstein-Markov inequality in Gauss space: for each $1\le p<\infty$ there exists a constant $C_p>0$ such that for any $k\ge 1$ and all polynomials $P$ on ${\mathbb{R}}^k$ we have $$\Vert \nabla P{\Vert }_{L^p({\mathbb{R}}^k,\mathrm{d}{\gamma }_k)}\le C_p(\mathrm{deg}P{)}^{\frac{1}{2}+\frac{1}{\pi }\arctan \left(\frac{|p-2|}{2\sqrt{p-1}}\right)}\Vert P{\Vert }_{L^p({\mathbb{R}}^k,\mathrm{d}{\gamma }_k)},$$ where $\mathrm{d}{\gamma }_k$ is the standard Gaussian measure on ${\mathbb{R}}^k$. We also show that under some mild growth assumptions on any function $B\in C^2((0,\infty ))\cap C([0,\infty ))$ with $B^{\prime },B^{\prime \prime }>0$ we have $${\int }_{{\mathbb{R}}^k}B\left(|LP(x)|\right)\mathrm{d}{\gamma }_k(x)\le {\int }_{{\mathbb{R}}^k}B\left(10(\mathrm{deg}P{)}^{{\alpha }_B}|P(x)|\right)\mathrm{d}{\gamma }_k(x)$$ where $L=\Delta -x\cdot \nabla$ is the generator of the Ornstein-Uhlenbeck semigroup and $${\alpha }_B=1+\frac{2}{\pi }\arctan \left(\frac{1}{2}\sqrt{\underset{s\in (0,\infty )}{\sup }\left\{\frac{s{B}^{\prime \prime }(s)}{B^{\prime }(s)}+\frac{B^{\prime }(s)}{s{B}^{\prime \prime }(s)}\right\}-2}\right).$$