Let $X$ be a standard Gaussian random variable. For any $p \in (1, \infty)$, we prove the existence of a universal constant $C_{ p}>0$ such that the inequality $$(\mathbb{ E} |h'(X)|^{ p})^{ 1/p} \geq C_{ p} \sqrt{ d} (\mathbb{ E} |h(X)|^{ p})^{ 1/p}$$ holds for all $d\geq 1$ and all polynomials $h : \mathbb{ R} \to \mathbb{ C}$ whose spectrum is supported on frequencies at least $d$, that is, $\mathbb{ E} h(X) X^{ k}=0$ for all $k=0,1, \ldots, d-1$. As an application of this optimal estimate, we obtain an affirmative answer to the Gaussian analogue of a question of Mendel and Naor (2014) concerning the growth of the Ornstein-Uhlenbeck operator on tail spaces of the real line. We also show the same bound for the gradient of analytic polynomials in an arbitrary dimension.
Categories: math.CA math.FA math.PR