The 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 → 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 L ^p -L ^q boundedness of operators in this functional calculus. The analysis of this noncommutative 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(- _z L) as an operator from L ^p (ℝ ^d , γα) to L ^q (ℝ ^d , γ β) for suitable values of z ∈ C with Re z > 0, p, q ∈ [1, ∞), and α β > 0. Here, γ τdenotes the centered Gaussian measure on ℝ ^d with density (2π τ) ^-d/2 exp(-|x| ² /2τ).