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

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- by Pradeep Boggarapu, Luz Roncal and Sundaram Thangavelu PDF
- Trans. Amer. Math. Soc.
**369**(2017), 7021-7047 Request permission

## 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) $H_{\kappa }:=-\Delta _{\kappa }+|x|^2$, 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.## References

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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
- MR Author ID: 769603
- 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
- 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
- © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**369**(2017), 7021-7047 - MSC (2010): Primary 42C10; Secondary 43A90, 42B08, 42B35, 33C45
- DOI: https://doi.org/10.1090/tran/6861
- MathSciNet review: 3683101