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 $(2\pi \tau {)}^{-d/2}\exp (-|x{|}^2/2\tau )$.

Some typos have been corrected and the case p=1 has been added to Corollary 5.1

References in corpus (1)

• Dimension dependent hypercontractivity for Gaussian kernels