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

Abstract

Let X be a standard Gaussian random variable. For any p ∈ (1, ∞), we prove the existence of a universal constant Cp > 0 such that the inequality

$${\mathbb{E}\left({\left| {{h^\prime }{{\left(X \right)}^p}} \right|} \right)^{1/p}} \ge {C_p}\sqrt d {\left({\mathbb{E}{{\left| {h\left(X \right)} \right|}^p}} \right)^{1/p}}$$

holds for all d ≥ 1 and all polynomials h: ℝ → ℂ whose spectrum is supported on frequencies at least d, that is, $\mathbb{E}h\left(X \right){X^k} = 0$ for all k = 0, 1, …, 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.