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

TL;DR

This paper establishes a sharp inequality for derivatives of polynomials with high-frequency spectrum under Gaussian measures, demonstrating rapid growth of the Ornstein-Uhlenbeck operator on Gaussian tail spaces and extending results to multidimensional analytic polynomials.

Contribution

It proves a universal lower bound for the derivative of high-frequency polynomials under Gaussian measures, confirming a Gaussian analogue of a question by Mendel and Naor.

Findings

Proved a universal inequality for derivatives of polynomials with spectrum at least d.

Confirmed the rapid growth of the Ornstein-Uhlenbeck operator on Gaussian tail spaces.

Extended the bound to gradients of analytic polynomials in multiple dimensions.

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 $$(\mathbb{E} |h'(X)|^{p})^{1/p} \geq C_{p} \sqrt{d} (\mathbb{E} |h(X)|^{p})^{1/p}$$ 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.