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Boggarapu Pradeep; Sundaram Thangavelu
Mixed norm estimates for the Riesz transforms associated to Dunkl harmonic oscillators
Annales mathématiques Blaise Pascal, 22 no. 1 (2015), p. 89-120, doi: 10.5802/ambp.347
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Class. Math.: 42C10, 47G40, 26A33, 43A90, 42B20, 42B35, 33C44
Keywords: Reflection groups, Dunkl operators, Hermite and generalised Hermite functions, Riesz transforms, singular integrals, weighted inequalities.

Résumé - Abstract

In this paper we study weighted mixed norm estimates for Riesz transforms associated to Dunkl harmonic oscillators. The idea is to show that the required inequalities are equivalent to certain vector valued inequalities for operator defined in terms of Laguerre expansions. In certain cases the main result can be deduced from the corresponding result for Hermite Riesz transforms.

Bibliography

[1] Béchir Amri. Riesz transforms for Dunkl Hermite expansions. J. Math. Anal. Appl., 423(1):646-659, 2015. Article |  MR 3273200
[2] Béchir Amri and Mohamed Sifi. Singular integral operators in Dunkl setting. J. Lie Theory, 22(3):723-739, 2012.  MR 3012151 |  Zbl 1248.42018
[3] George E. Andrews, Richard Askey and Ranjan Roy. Special functions, volume 71 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1999 Article |  MR 1688958 |  Zbl 0920.33001
[4] Bahattin Cengiz. On the duals of Lebesgue-Bochner $L^p$ spaces. Proc. Amer. Math. Soc., 114(4):923-926, 1992. Article |  MR 1027088 |  Zbl 0778.46018
[5] Óscar Ciaurri and Luz Roncal. The Riesz transform for the harmonic oscillator in spherical coordinates. Constr. Approx., 40(3):447-472, 2014. Article |  MR 3274383 |  Zbl 1305.42028
[6] Charles F. Dunkl. Differential-difference operators associated to reflection groups. Trans. Amer. Math. Soc., 311(1):167-183, 1989. Article |  MR 951883 |  Zbl 0652.33004
[7] Charles F. Dunkl and Yuan Xu. Orthogonal polynomials of several variables, volume 81 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 2001 Article |  MR 1827871 |  Zbl 0964.33001
[8] Javier Duoandikoetxea. Fourier analysis, volume 29 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2001  MR 1800316 |  Zbl 0969.42001
[9] Javier Duoandikoetxea, Adela Moyua, Osane Oruetxebarria and Edurne Seijo. Radial $A_p$ weights with applications to the disc multiplier and the Bochner-Riesz operators. Indiana Univ. Math. J., 57(3):1261-1281, 2008. Article |  MR 2429092 |  Zbl 1268.42018
[10] Daryl Geller. Spherical harmonics, the Weyl transform and the Fourier transform on the Heisenberg group. Canad. J. Math., 36(4):615-684, 1984. Article |  MR 756538 |  Zbl 0596.46034
[11] Carl Herz and Nestor Rivière. Estimates for translation-invariant operators on spaces with mixed norms. Studia Math., 44:511-515, 1972.  MR 342959 |  Zbl 0269.43006
[12] Adam Nowak. Heat-diffusion and Poisson integrals for Laguerre and special Hermite expansions on weighted $L^p$ spaces. Studia Math., 158(3):239-268, 2003. Article |  MR 2014301 |  Zbl 1065.42021
[13] Adam Nowak and Krzysztof Stempak. Riesz transforms for multi-dimensional Laguerre function expansions. Adv. Math., 215(2):642-678, 2007. Article |  MR 2355604 |  Zbl 1166.42016
[14] Adam Nowak and Krzysztof Stempak. Riesz transforms for the Dunkl harmonic oscillator. Math. Z., 262(3):539-556, 2009. Article |  MR 2506306 |  Zbl 1168.44002
[15] Teresa E. Pérez, Miguel A. Piñar and Yuan Xu. Weighted Sobolev orthogonal polynomials on the unit ball. J. Approx. Theory, 171:84-104, 2013. Article |  MR 3053718 |  Zbl 1281.42026
[16] Margit Rösler. Generalized Hermite polynomials and the heat equation for Dunkl operators. Comm. Math. Phys., 192(3):519-542, 1998. Article |  MR 1620515 |  Zbl 0908.33005
[17] José Luis Rubio de Francia. Transference principles for radial multipliers. Duke Math. J., 58(1):1-19, 1989. Article |  MR 1016410 |  Zbl 0676.42009
[18] P. K. Sanjay and Sundaram Thangavelu. Revisiting Riesz transforms on Heisenberg groups. Rev. Mat. Iberoam., 28(4):1091-1108, 2012. Article |  MR 2990135 |  Zbl 1258.22004
[19] Krzysztof Stempak and José Luis Torrea. Poisson integrals and Riesz transforms for Hermite function expansions with weights. J. Funct. Anal., 202(2):443-472, 2003. Article |  MR 1990533 |  Zbl 1040.42015
[20] Sundaram Thangavelu. Lectures on Hermite and Laguerre expansions, volume 42 of Mathematical Notes. Princeton University Press, Princeton, NJ, 1993  MR 1215939 |  Zbl 0791.41030
[21] Sundaram Thangavelu and Yuan Xu. Convolution operator and maximal function for the Dunkl transform. J. Anal. Math., 97:25-55, 2005. Article |  MR 2274972 |  Zbl 1131.43006