Let $\gamma_{ d}$ be the $d$-dimensional standard Gaussian measure and $\{ Q_{ t}\}_{ t\ge 0}$ the Ornstein-Uhlenbeck semigroup acting on $L^{ 1}(\gamma_{ d})$. We show that the hypercontractivity of $\{ Q_{ t}\}_{ t\ge 0}$ is equivalent to the property that \begin{ align*} \left\{ \int_{ \mathbb{ R}^{ d}}\exp \left(e^{ 2t}Q_{ t}f\right) d\gamma_{ d} \right\} ^{ 1/e^{ 2t}} \le \int_{ \mathbb{ R}^{ d}}e^{ f}\,d\gamma_{ d}, \end{ align*} which holds for any $f\in L^{ 1}(\gamma_{ d})$ with $e^{ f}\in L^{ 1}(\gamma_{ d})$ and for any $t\ge 0$. We then derive a family of inequalities that unifies this exponential variant and the original hypercontractivity, a generalization of the Gaussian logarithmic Sobolev inequality is obtained as a corollary. A unification of the reverse hypercontractivity and the exponential variant is also provided.
Categories: math.PR math.FA