Weyl calculus with respect to the Gaussian measure and restricted $L^p$-$L^q$ boundedness of the Ornstein-Uhlenbeck semigroup in complex time

In this paper, we introduce a Weyl functional calculus $a \mapsto a(Q,P)$ for the position and momentum operators $Q$ and $P$ associated with the Ornstein-Uhlenbeck operator $ L = -\Delta + x\cdot \nabla$, and give a simple criterion for restricted $L^p$-$L^q$ boundedness of operators in this functional calculus. The analysis of this non-commutative functional calculus is simpler than the analysis of the functional calculus of $L$. It allows us to recover, unify, and extend, old and new results concerning the boundedness of $\exp(-zL)$ as an operator from $L^p(\mathbb{R}^d,\gamma_{\alpha})$ to $L^q(\mathbb{R}^d,\gamma_{\beta})$ for suitable values of $z\in \mathbb{C}$ with $\Re z>0$, $p,q\in [1,\infty)$, and $\alpha,\beta>0$. Here, $\gamma_\tau$ denotes the centred Gaussian measure on $\mathbb{R}^d$ with density $(2\pi\tau)^{-d/2}\exp(-|x|^2/2\tau)$.