The Weyl calculus with respect to the Gaussian measure and restricted Lp-Lq boundedness of the Ornstein-Uhlenbeck semigroup in complex time

In this paper, we introduce a Weyl functional calculus a↦a(Q,P) for the position and momentum operators Q and P associated with the Ornstein-Uhlenbeck operator L=−Δ+x⋅∇, and give a simple criterion for restricted Lp-Lq 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 Lp(Rd,γα) to Lq(Rd,γβ) for suitable values of z∈C with Rez>0, p,q∈[1,∞), and α,β>0. Here, γτ denotes the centered Gaussian measure on Rd with density (2πτ)−d/2exp(−|x|2/2τ)