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Sharp endpoint estimates for some operators associated with the Laplacian with drift in Euclidean space

Part of: Harmonic analysis in several variables Partial differential equations on manifolds; differential operators

Published online by Cambridge University Press:  16 June 2020

Hong-Quan Li  and
Peter Sjögren
Hong-Quan Li
Affiliation:
School of Mathematical Sciences, Fudan University, 220 Handan Road, Shanghai200433, People’s Republic of China e-mail: hongquan_li@fudan.edu.cnhong_quanli@yahoo.fr
Peter Sjögren*
Affiliation:
Mathematical Sciences, University of Gothenburg, and Mathematical Sciences, Chalmers, SE-412 96Göteborg, Sweden
*
e-mail: peters@chalmers.se
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Abstract

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Let $v \ne 0$ be a vector in ${\mathbb {R}}^n$ . Consider the Laplacian on ${\mathbb {R}}^n$ with drift $\Delta _{v} = \Delta + 2v\cdot \nabla $ and the measure $d\mu (x) = e^{2 \langle v, x \rangle } dx$ , with respect to which $\Delta _{v}$ is self-adjoint. This measure has exponential growth with respect to the Euclidean distance. We study weak type $(1, 1)$ and other sharp endpoint estimates for the Riesz transforms of any order, and also for the vertical and horizontal Littlewood–Paley–Stein functions associated with the heat and the Poisson semigroups.

Keywords

Riesz transformLittlewood–Paley–Stein operatorsHeat semigroupLaplacian with drift

MSC classification

Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)
Secondary: 42B25: Maximal functions, Littlewood-Paley theory 58J35: Heat and other parabolic equation methods
Type
Article
Information
Canadian Journal of Mathematics , Volume 73 , Issue 5 , October 2021 , pp. 1278 - 1304
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© Canadian Mathematical Society 2020

Footnotes

H.-Q. Li is partially supported by NSF of China (Grants No. 11625102 and No. 11571077) and The Program of Shanghai Academic Research Leader (18XD1400700). Both authors profited from a grant from the Gothenburg Centre for Advanced Studies in Science and Technology.

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