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Exponential integrability for log-concave measures

Authors :
Ivanisvili, Paata
Russell, Ryan
Source :
Analysis & PDE 16 (2023) 1271-1288
Publication Year :
2020

Abstract

Talagrand observed that finiteness of $\mathbb{E}\, e^{\frac{1}{2}|\nabla f(X)|^{2}}$ implies finiteness of $\mathbb{E}\, e^{\, f(X)}$ where $X$ is the standard Gaussian vector in $\mathbb{R}^{n}$ and $f$ is a smooth function with zero average. However, in this paper we show that finiteness of $ \mathbb{E}\, e^{\frac{1}{2}|\nabla f|^{2}} (1+|\nabla f|)^{-1}$ implies finiteness of $\mathbb{E}\, e^{\, f(X)}$, and we also obtain quantitative bounds \begin{align*} \log\, \mathbb{E}\, e^{\, f} \leq 10\, \mathbb{E}\, e^{\frac{1}{2}|\nabla f|^{2}} (1+|\nabla f|)^{-1}. \end{align*} Moreover, the extra factor $(1+|\nabla f|)^{-1}$ is the best possible in the sense that there is smooth $f$ with $\mathbb{E}\, e^{\,f} =\infty$ but $\mathbb{E}\, e^{\frac{1}{2}|\nabla f|^{2}} (1+|\nabla f|)^{-c}<\infty$ for all $c>1$. As an application we show corresponding dual inequalities for the discrete time dyadic martingales and its quadratic variations.<br />Comment: We included an application showing what is the corresponding dual inequality for the dyadic square function

Details

Database :
arXiv
Journal :
Analysis & PDE 16 (2023) 1271-1288
Publication Type :
Report
Accession number :
edsarx.2004.09704
Document Type :
Working Paper
Full Text :
https://doi.org/10.2140/apde.2023.16.1271