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Exponential integrability in Gauss space

Paata Ivanisvili and Ryan Russell

Vol. 16 (2023), No. 5, 1271–1288
Abstract

Talagrand showed that finiteness of 𝔼 e|f(X)|22 implies finiteness of 𝔼 ef(X)𝔼f(X), where X is the standard Gaussian vector in n and f is a smooth function. However, in this paper we show that finiteness of 𝔼 e|f|22 (1 + |f|)1 implies finiteness of 𝔼 ef(X)𝔼f(X), and we also obtain quantitative bounds

log 𝔼 ef𝔼f 10 𝔼 e|f|22 (1 + |f|)1.

Moreover, the extra factor (1 + |f|)1 is the best possible in the sense that there is a smooth f with 𝔼 ef𝔼f = but 𝔼 e|f|22 (1 + |f|)c < 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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