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

We introduce a Weyl functional calculus for the Ornstein-Uhlenbeck operator L=−Δ+x⋅∇, and give a simple criterion for L^p-L^q boundedness of operators in this functional calculus. It allows us to recover, unify, and extend, old and new results concerning the boundedness of exp(−zL)as an operator from L^p(ℝ^d,γ_α) to L^q(ℝ^d,γ_β) for suitable values of z∈ℂ with ℜz>0 and α,β>0. Here, γ_τ denotes the centred Gaussian measure on ℝ^d with density (2πτ)^−d/2 exp(−|x|²/2τ).