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

2020/11/02 by Eskenazis, Alexandros, Ivanisvili, Paata
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Probability (math.PR)

paper · doi:10.48550/arxiv.2011.01359

Abstract

Let X be a standard Gaussian random variable. For any p ∈ (1, ∞), we prove the existence of a universal constant C_p>0 such that the inequality (𝔼 |h'(X)|^p)^1/p ≥ C_p √(d) (𝔼 |h(X)|^p)^1/p holds for all d≥ 1 and all polynomials h : ℝ → ℂ whose spectrum is supported on frequencies at least d, that is, 𝔼 h(X) 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.

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