Exponential integrability in Gauss space

Abstract

Talagrand showed that finiteness of ‫ޅ‬ e |∇ f (X )| 2 /2 implies finiteness of ‫ޅ‬ e f (X ‫ޅ-)‬ f (X ) , where X is the standard Gaussian vector in ‫ޒ‬ n and f is a smooth function.However, in this paper we show that finiteness of ‫ޅ‬ X ) , and we also obtain quantitative boundsMoreover, the extra factor (1is the best possible in the sense that there is a smooth f withAs an application we show corresponding dual inequalities for the discrete time dyadic martingales and their quadratic variations.is the best possible inequality one may seek for someAccording to a discussion on page 8 in [Bobkov and Götze 1999], Talagrand showed that even though (1-1) fails at the endpoint exponent α = 1 2 , surprisingly, the finiteness of ‫ޅ‬e |∇ f (X )| 2 /2 still implies finiteness of ‫ޅ‬e f for X ∼ N (0, I n×n ).We are not aware of Talagrand's proof as it was never published; we do not know if he solved the problem only for n = 1 or for all n ≥ 1.

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