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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. 369 (2017), 7021-7047
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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  • [1] Bechir Amri and Mohamed Sifi, Riesz transforms for Dunkl transform, Ann. Math. Blaise Pascal 19 (2012), no. 1, 247-262 (English, with English and French summaries). MR 2978321, https://doi.org/10.5802/ambp.312
  • [2] George E. Andrews, Richard Askey, and Ranjan Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR 1688958
  • [3] Richard Askey and Stephen Wainger, Mean convergence of expansions in Laguerre and Hermite series, Amer. J. Math. 87 (1965), 695-708. MR 0182834
  • [4] Óscar Ciaurri and Juan L. Varona, Two-weight norm inequalities for the Cesàro means of generalized Hermite expansions, J. Comput. Appl. Math. 178 (2005), no. 1-2, 99-110. MR 2127873, https://doi.org/10.1016/j.cam.2004.03.027
  • [5] Feng Dai and Heping Wang, A transference theorem for the Dunkl transform and its applications, J. Funct. Anal. 258 (2010), no. 12, 4052-4074. MR 2609538, https://doi.org/10.1016/j.jfa.2010.03.006
  • [6] Feng Dai and Yuan Xu, Approximation theory and harmonic analysis on spheres and balls, Springer Monographs in Mathematics, Springer, New York, 2013. MR 3060033
  • [7] Marcel F. E. de Jeu, The Dunkl transform, Invent. Math. 113 (1993), no. 1, 147-162. MR 1223227, https://doi.org/10.1007/BF01244305
  • [8] Jan Felipe van Diejen and Luc Vinet (eds.), Calogero-Moser-Sutherland models, CRM Series in Mathematical Physics, Springer-Verlag, New York, 2000. MR 1843558
  • [9] Charles F. Dunkl, Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1989), no. 1, 167-183. MR 951883, https://doi.org/10.2307/2001022
  • [10] Charles F. Dunkl and Yuan Xu, Orthogonal polynomials of several variables, Encyclopedia of Mathematics and its Applications, vol. 81, Cambridge University Press, Cambridge, 2001. MR 1827871
  • [11] J. J. Gergen, Summability of double Fourier series, Duke Math. J. 3 (1937), no. 2, 133-148. MR 1545976, https://doi.org/10.1215/S0012-7094-37-00310-7
  • [12] Sallam Hassani and Mohamed Sifi, Spectral multipliers for the Dunkl Laplacian, Commun. Stoch. Anal. 6 (2012), no. 4, 547-563. MR 3034004
  • [13] Jean-Louis Krivine, Théorèmes de factorisation dans les espaces réticulés, Séminaire Maurey-Schwartz 1973-1974: Espaces $ L^{p}$, applications radonifiantes et géométrie des espaces de Banach, Exp. Nos. 22 et 23, Centre de Math., École Polytech., Paris, 1974, 22 pp (French). MR 0440334
  • [14] Luc Lapointe and Luc Vinet, Exact operator solution of the Calogero-Sutherland model, Comm. Math. Phys. 178 (1996), no. 2, 425-452. MR 1389912
  • [15] N. N. Lebedev, Special functions and their applications, Dover Publications, Inc., New York, 1972. Revised edition, translated from the Russian and edited by Richard A. Silverman; Unabridged and corrected republication. MR 0350075
  • [16] Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. II: Function spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Springer-Verlag, Berlin-New York, 1979. MR 540367
  • [17] Margit Rösler, Generalized Hermite polynomials and the heat equation for Dunkl operators, Comm. Math. Phys. 192 (1998), no. 3, 519-542. MR 1620515, https://doi.org/10.1007/s002200050307
  • [18] Margit Rösler, An uncertainty principle for the Dunkl transform, Bull. Austral. Math. Soc. 59 (1999), no. 3, 353-360. MR 1698045, https://doi.org/10.1017/S0004972700033025
  • [19] José Luis Rubio de Francia, Transference principles for radial multipliers, Duke Math. J. 58 (1989), no. 1, 1-19. MR 1016410, https://doi.org/10.1215/S0012-7094-89-05801-8
  • [20] Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972
  • [21] Sundaram Thangavelu, Lectures on Hermite and Laguerre expansions, Mathematical Notes, vol. 42, Princeton University Press, Princeton, NJ, 1993. With a preface by Robert S. Strichartz. MR 1215939
  • [22] Sundaram Thangavelu, Summability of Hermite expansions. I, II, Trans. Amer. Math. Soc. 314 (1989), no. 1, 119-142, 143-170. MR 958904, https://doi.org/10.2307/2001439
  • [23] Sundaram Thangavelu and Yuan Xu, Convolution operator and maximal function for the Dunkl transform, J. Anal. Math. 97 (2005), 25-55. MR 2274972, https://doi.org/10.1007/BF02807401
  • [24] Sundaram Thangavelu and Yuan Xu, Riesz transform and Riesz potentials for Dunkl transform, J. Comput. Appl. Math. 199 (2007), no. 1, 181-195. MR 2267542, https://doi.org/10.1016/j.cam.2005.02.022

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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

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