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$L^{p}$–$L^{q}$ OFF-DIAGONAL ESTIMATES FOR THE ORNSTEIN–UHLENBECK SEMIGROUP: SOME POSITIVE AND NEGATIVE RESULTS

Part of: Abstract harmonic analysis Groups and semigroups of linear operators, their generalizations and applications

Published online by Cambridge University Press:  06 February 2017

ALEX AMENTA  and
JONAS TEUWEN
ALEX AMENTA*
Affiliation:
Delft Institute of Applied Mathematics, Delft University of Technology, PO Box 5031, 2628 CD Delft, The Netherlands email amenta@fastmail.fm
JONAS TEUWEN
Affiliation:
Division of Radiation Oncology, Netherlands Cancer Institute/Antoni van Leeuwenhoek, Plesmanlaan 121, 1066 CX Amsterdam, The Netherlands Department of Imaging Physics, Optics Research Group, Delft University of Technology, PO Box 5031, 2628 CD Delft, The Netherlands email jonasteuwen@gmail.com
*
✉amenta@fastmail.fm
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Abstract

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We investigate $L^{p}(\unicode[STIX]{x1D6FE})$–$L^{q}(\unicode[STIX]{x1D6FE})$ off-diagonal estimates for the Ornstein–Uhlenbeck semigroup $(e^{tL})_{t>0}$. For sufficiently large $t$ (quantified in terms of $p$ and $q$), these estimates hold in an unrestricted sense, while, for sufficiently small $t$, they fail when restricted to maximal admissible balls and sufficiently small annuli. Our counterexample uses Mehler kernel estimates.

Keywords

Ornstein–Uhlenbeck semigroupoff-diagonal estimatesMehler kernel

MSC classification

Primary: 47D06: One-parameter semigroups and linear evolution equations
Secondary: 43A99: None of the above, but in this section
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 96 , Issue 1 , August 2017 , pp. 154 - 161
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

Footnotes

The first author acknowledges financial support from the Australian Research Council Discovery Grant DP120103692 and the ANR project ‘Harmonic analysis at its boundaries’ ANR-12-BS01-0013. The second author acknowledges partial financial support from the Netherlands Organisation for Scientific Research (NWO) by the NWO-VICI grant 639.033.604.

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