Restricted hypercontractivity on the Poisson space

Nourdin, Ivan; Peccati, Giovanni; Yang, Xiaochuan

doi:10.1090/proc/14964

Restricted hypercontractivity on the Poisson space
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by Ivan Nourdin, Giovanni Peccati and Xiaochuan Yang PDF

Proc. Amer. Math. Soc. 148 (2020), 3617-3632 Request permission

Abstract:

We show that the Ornstein-Uhlenbeck semigroup associated with a general Poisson random measure is hypercontractive, whenever it is restricted to non-increasing mappings on configuration spaces. We deduce from this result some versions of Talagrand’s $L^1$-$L^2$ inequality for increasing and concave mappings, and we build examples showing that such an estimate represents a strict improvement of the classical Poincaré inequality. We complement our finding with several results of independent interest, such as gradient estimates and an inequality with isoperimetric content.

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