Title: Weyl calculus with respect to the Gaussian measure and restricted $L^p$-$L^q$ boundedness of the Ornstein-Uhlenbeck semigroup in complex time,
Abstract: 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
$(2\pi\tau)^{-d/2}\exp(-|x|^2/2\tau)$.