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)
| Original language | English |
|---|---|
| Pages (from-to) | 691-712 |
| Number of pages | 22 |
| Journal | Bulletin de la Sociéte Mathématique de France |
| Volume | 146 |
| Issue number | 4 |
| DOIs | |
| Publication status | Published - 2018 |
Bibliographical note
Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project https://www.openaccess.nl/en/you-share-we-take-careOtherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.
Keywords
- Weyl functional calculus
- canonical commutation relation
- Schur estimate
- Ornstein-Uhlenbeck operator
- Mehler kernel
- restricted Lp-Lq-boundedness
- restricted Sobolev embedding