Weyl calculus with respect to the Gaussian measure and restricted L^p-L^q boundedness of the Ornstein-Uhlenbeck semigroup in complex time

2017/02/13 by van Neerven, Jan, Portal, Pierre
#47A60 #47D06 #47G30 #60H07 #81S05 #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Probability (math.PR)

paper · doi:10.48550/arxiv.1702.03602

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 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(ℝ^d,γ_α) 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_τ denotes the centred Gaussian measure on ℝ^d with density (2πτ)^-d/2exp(-|x|²/2τ).

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