Exponential integrability in Gauss space

Ivanisvili, Paata; Russell, Ryan

doi:10.2140/apde.2023.16.1271

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Exponential integrability in Gauss space

Paata Ivanisvili and Ryan Russell

Vol. 16 (2023), No. 5, 1271–1288

DOI: 10.2140/apde.2023.16.1271

Abstract

Talagrand showed that finiteness of $𝔼 e^{| \nabla f (X) |^{2} ∕ 2}$ implies finiteness of $𝔼 e^{f (X) - 𝔼 f (X)}$ , where $X$ is the standard Gaussian vector in $ℝ^{n}$ and $f$ is a smooth function. However, in this paper we show that finiteness of $𝔼 e^{| \nabla f |^{2} ∕ 2} {(1 + | \nabla f |)}^{- 1}$ implies finiteness of $𝔼 e^{f (X) - 𝔼 f (X)}$ , and we also obtain quantitative bounds

\log 𝔼 e^{f - 𝔼 f} \leq 10 𝔼 e^{| \nabla f |^{2} ∕ 2} {(1 + | \nabla f |)}^{- 1} .

Moreover, the extra factor ${(1 + | \nabla f |)}^{- 1}$ is the best possible in the sense that there is a smooth $f$ with $𝔼 e^{f - 𝔼 f} = \infty$ but $𝔼 e^{| \nabla f |^{2} ∕ 2} {(1 + | \nabla f |)}^{- c} < \infty$ for all $c > 1$ . As an application we show corresponding dual inequalities for the discrete time dyadic martingales and their quadratic variations.

Keywords

exponential integrability, heat flow, Gauss space, measure concentration

Mathematical Subject Classification

Primary: 26D10, 35E10, 42B35

Milestones

Received: 19 May 2021

Revised: 25 November 2021

Accepted: 20 December 2021

Published: 12 August 2023

Authors

Paata Ivanisvili
Department of Mathematics University of California, Irvine Irvine, CA United States

Ryan Russell
Irvine, CA United States	Department of Mathematics and Statistics California State University Long Beach, CA United States

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Vol. 16, No. 5, 2023