Mixed norm estimates for the Riesz transforms associated to Dunkl harmonic oscillators
Annales Mathématiques Blaise Pascal, Tome 22 (2015) no. 1, pp. 89-120.

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.

DOI : https://doi.org/10.5802/ambp.347
Classification : 42C10,  47G40,  26A33,  43A90,  42B20,  42B35,  33C44
Mots clés : Reflection groups, Dunkl operators, Hermite and generalised Hermite functions, Riesz transforms, singular integrals, weighted inequalities.
@article{AMBP_2015__22_1_89_0,
     author = {Boggarapu Pradeep and Sundaram  Thangavelu},
     title = {Mixed norm estimates for the {Riesz} transforms associated to {Dunkl} harmonic oscillators},
     journal = {Annales Math\'ematiques Blaise Pascal},
     pages = {89--120},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {22},
     number = {1},
     year = {2015},
     doi = {10.5802/ambp.347},
     zbl = {1327.42031},
     language = {en},
     url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.347/}
}
Boggarapu Pradeep; Sundaram  Thangavelu. Mixed norm estimates for the Riesz transforms associated to Dunkl harmonic oscillators. Annales Mathématiques Blaise Pascal, Tome 22 (2015) no. 1, pp. 89-120. doi : 10.5802/ambp.347. https://ambp.centre-mersenne.org/articles/10.5802/ambp.347/

[1] Béchir Amri Riesz transforms for Dunkl Hermite expansions, J. Math. Anal. Appl., Volume 423 (2015) no. 1, pp. 646-659 | Article | MR 3273200

[2] Béchir Amri; Mohamed Sifi Singular integral operators in Dunkl setting, J. Lie Theory, Volume 22 (2012) no. 3, pp. 723-739 | MR 3012151 | Zbl 1248.42018

[3] George E. Andrews; Richard Askey; Ranjan Roy Special functions, Encyclopedia of Mathematics and its Applications, Volume 71, Cambridge University Press, Cambridge, 1999, xvi+664 pages | Article | MR 1688958 | Zbl 0920.33001

[4] Bahattin Cengiz On the duals of Lebesgue-Bochner L p spaces, Proc. Amer. Math. Soc., Volume 114 (1992) no. 4, pp. 923-926 | Article | MR 1027088 | Zbl 0778.46018

[5] Óscar Ciaurri; Luz Roncal The Riesz transform for the harmonic oscillator in spherical coordinates, Constr. Approx., Volume 40 (2014) no. 3, pp. 447-472 | Article | MR 3274383 | Zbl 1305.42028

[6] Charles F. Dunkl Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc., Volume 311 (1989) no. 1, pp. 167-183 | Article | MR 951883 | Zbl 0652.33004

[7] Charles F. Dunkl; Yuan Xu Orthogonal polynomials of several variables, Encyclopedia of Mathematics and its Applications, Volume 81, Cambridge University Press, Cambridge, 2001, xvi+390 pages | Article | MR 1827871 | Zbl 0964.33001

[8] Javier Duoandikoetxea Fourier analysis, Graduate Studies in Mathematics, Volume 29, American Mathematical Society, Providence, RI, 2001, xviii+222 pages (Translated and revised from the 1995 Spanish original by David Cruz-Uribe) | MR 1800316 | Zbl 0969.42001

[9] Javier Duoandikoetxea; Adela Moyua; Osane Oruetxebarria; Edurne Seijo Radial A p weights with applications to the disc multiplier and the Bochner-Riesz operators, Indiana Univ. Math. J., Volume 57 (2008) no. 3, pp. 1261-1281 | 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., Volume 36 (1984) no. 4, pp. 615-684 | Article | MR 756538 | Zbl 0596.46034

[11] Carl Herz; Nestor Rivière Estimates for translation-invariant operators on spaces with mixed norms, Studia Math., Volume 44 (1972), pp. 511-515 (Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, V) | 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., Volume 158 (2003) no. 3, pp. 239-268 | Article | MR 2014301 | Zbl 1065.42021

[13] Adam Nowak; Krzysztof Stempak Riesz transforms for multi-dimensional Laguerre function expansions, Adv. Math., Volume 215 (2007) no. 2, pp. 642-678 | Article | MR 2355604 | Zbl 1166.42016

[14] Adam Nowak; Krzysztof Stempak Riesz transforms for the Dunkl harmonic oscillator, Math. Z., Volume 262 (2009) no. 3, pp. 539-556 | Article | MR 2506306 | Zbl 1168.44002

[15] Teresa E. Pérez; Miguel A. Piñar; Yuan Xu Weighted Sobolev orthogonal polynomials on the unit ball, J. Approx. Theory, Volume 171 (2013), pp. 84-104 | Article | MR 3053718 | Zbl 1281.42026

[16] Margit Rösler Generalized Hermite polynomials and the heat equation for Dunkl operators, Comm. Math. Phys., Volume 192 (1998) no. 3, pp. 519-542 | Article | MR 1620515 | Zbl 0908.33005

[17] José Luis Rubio de Francia Transference principles for radial multipliers, Duke Math. J., Volume 58 (1989) no. 1, pp. 1-19 | Article | MR 1016410 | Zbl 0676.42009

[18] P. K. Sanjay; Sundaram Thangavelu Revisiting Riesz transforms on Heisenberg groups, Rev. Mat. Iberoam., Volume 28 (2012) no. 4, pp. 1091-1108 | Article | MR 2990135 | Zbl 1258.22004

[19] Krzysztof Stempak; José Luis Torrea Poisson integrals and Riesz transforms for Hermite function expansions with weights, J. Funct. Anal., Volume 202 (2003) no. 2, pp. 443-472 | Article | MR 1990533 | Zbl 1040.42015

[20] Sundaram Thangavelu Lectures on Hermite and Laguerre expansions, Mathematical Notes, Volume 42, Princeton University Press, Princeton, NJ, 1993, xviii+195 pages (With a preface by Robert S. Strichartz) | MR 1215939 | Zbl 0791.41030

[21] Sundaram Thangavelu; Yuan Xu Convolution operator and maximal function for the Dunkl transform, J. Anal. Math., Volume 97 (2005), pp. 25-55 | Article | MR 2274972 | Zbl 1131.43006