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

Journal article (2018)

Authors

J.M.A.M. van Neerven Analysis - EEMCS
Pierre Portal Université de Lille, Australian National University
Research Group
Analysis (EEMCS) (TU Delft)
Copyright: © 2018 J.M.A.M. van Neerven, Pierre Portal
DOI
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https://doi.org/10.24033/bsmf.2771
Mehler kernel Weyl functional calculus Canonical commutation relation Schur estimate Ornstein-Uhlenbeck operator Restricted Lp-Lq-boundedness Restricted Sobolev embedding
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Published Date

2018

Language

English

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Faculty

Electrical Engineering, Mathematics and Computer Science

Department

Delft Institute of Applied Mathematics

Research Group

Analysis

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 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)