A unification of the hypercontractivity and its exponential variant of the Ornstein-Uhlenbeck semigroup

Abstract: 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.