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Mixed norm estimates for the Cesàro means associated with Dunkl-Hermite expansions


Authors: Pradeep Boggarapu, Luz Roncal and Sundaram Thangavelu
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 42C10; Secondary 43A90, 42B08, 42B35, 33C45
DOI: https://doi.org/10.1090/tran/6861
Published electronically: March 29, 2017
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Abstract: Our main goal in this article is to study mixed norm estimates for the Cesàro means associated with Dunkl-Hermite expansions on $ \mathbb{R}^d$. These expansions arise when one considers the Dunkl-Hermite operator (or Dunkl harmonic oscillator) $ H_{\kappa }:=-\Delta _{\kappa }+\vert x\vert^2$, where $ \Delta _{\kappa }$ stands for the Dunkl-Laplacian. It is shown that the desired mixed norm estimates are equivalent to vector-valued inequalities for a sequence of Cesàro means for Laguerre expansions with shifted parameter. In order to obtain such vector-valued inequalities, we develop an argument to extend these Laguerre operators for complex values of the parameters involved and apply a version of the three lines lemma.


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Additional Information

Pradeep Boggarapu
Affiliation: Department of Mathematics, BITS Pilani - K.K. Birla Goa Campus, 403726 Goa, India
Email: pradeep@math.iisc.ernet.in

Luz Roncal
Affiliation: Departamento de Matemáticas y Computación, Universidad de La Rioja, 26004 Logroño, Spain
Address at time of publication: BCAM – Basque Center for Applied Mathematics, Alameda de Mazarredo 14, 48009 Bilbao, Spain
Email: luz.roncal@unirioja.es, lroncal@bcamath.org

Sundaram Thangavelu
Affiliation: Department of Mathematics, Indian Institute of Science, 560012 Bangalore, India
Email: veluma@math.iisc.ernet.in

DOI: https://doi.org/10.1090/tran/6861
Received by editor(s): October 8, 2014
Received by editor(s) in revised form: October 14, 2015
Published electronically: March 29, 2017
Additional Notes: All three authors were supported by the J. C. Bose Fellowship of the third author from the Department of Science and Technology, Government of India. The second author was also supported by grant MTM2012-36732-C03-02 from the Spanish Government
Article copyright: © Copyright 2017 American Mathematical Society