Sharp growth of the Ornstein-Uhlenbeck operator on Gaussian tail spaces

Abstract

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 $$(\mathbb{E}|h^{\prime }(X){|}^p{)}^{1/p}\ge C_p\sqrt{d}(\mathbb{E}|h(X){|}^p{)}^{1/p}$$ holds for all $d\ge 1$ 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 $k=0,1,\dots ,d-1$. 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.