Exponential integrability for log-concave measures

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.

We included an application showing what is the corresponding dual inequality for the dyadic square function