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Laguerre operator and its associated weighted Besov and Triebel-Lizorkin spaces


Authors: The Anh Bui and Xuan Thinh Duong
Journal: Trans. Amer. Math. Soc. 369 (2017), 2109-2150
MSC (2010): Primary 60J10, 42B20, 42B25
DOI: https://doi.org/10.1090/tran/6745
Published electronically: May 17, 2016
MathSciNet review: 3581229
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Abstract: Consider the space $ X=(0,\infty )$ equipped with the Euclidean distance and the measure $ d\mu _\alpha (x)=x^{\alpha }dx$ where $ \alpha \in (-1,\infty )$ is a fixed constant and $ dx$ is the Lebesgue measure. Consider the Laguerre operator $ \displaystyle L=-\frac {d^2}{dx^2} -\frac {\alpha }{x}\frac {d}{dx}+x^2$ on $ X$. The aim of this article is threefold. Firstly, we establish a Calderón reproducing formula using a suitable distribution of the Laguerre operator. Secondly, we study certain properties of the Laguerre operator such as a Harnack type inequality on the solutions and subsolutions of Laplace equations associated to Laguerre operators. Thirdly, we establish the theory of the weighted homogeneous Besov and Triebel-Lizorkin spaces associated to the Laguerre operator. We define the weighted homogeneous Besov and Triebel-Lizorkin spaces by the square functions of the Laguerre operator, then show that these spaces have an atomic decomposition. We then study the fractional powers $ L^{-\gamma }, \gamma >0$, and show that these operators map boundedly from one weighted Besov space (or one weighted Triebel-Lizorkin space) to another suitable weighted Besov space (or weighted Triebel-Lizorkin space). We also show that in particular cases of the indices, our new weighted Besov and Triebel-Lizorkin spaces coincide with the (expected) weighted Hardy spaces, the weighted $ L^p$ spaces or the weighted Sobolev spaces in Laguerre settings.


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Additional Information

The Anh Bui
Affiliation: Department of Mathematics, Macquarie University, New South Wales 2109, Australia – and – Department of Mathematics, University of Pedagogy, HoChiMinh City, Vietnam
Email: the.bui@mq.edu.au, bt_anh80@yahoo.com

Xuan Thinh Duong
Affiliation: Department of Mathematics, Macquarie University, New South Wales 2109, Australia
Email: xuan.duong@mq.edu.au

DOI: https://doi.org/10.1090/tran/6745
Keywords: Laguerre operator, Besov space, Triebel-Lizorkin space, atomic decomposition
Received by editor(s): September 1, 2014
Received by editor(s) in revised form: April 7, 2015
Published electronically: May 17, 2016
Article copyright: © Copyright 2016 American Mathematical Society