Mixed norm estimates for the Riesz transforms associated to Dunkl harmonic oscillators

Pradeep, Boggarapu; Thangavelu, Sundaram

doi:10.5802/ambp.347

Pradeep, Boggarapu ; Thangavelu, Sundaram

Annales Mathématiques Blaise Pascal, Tome 22 (2015) no. 1, pp. 89-120.

Résumé

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.

Zbl 1327.42031

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 = {Pradeep, Boggarapu and Thangavelu, Sundaram},
     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 = {www.numdam.org/item/AMBP_2015__22_1_89_0/}
}

Pradeep, Boggarapu; Thangavelu, Sundaram . 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. http://www.numdam.org/item/AMBP_2015__22_1_89_0/

Bibliographie

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

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

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

[4] Cengiz, Bahattin 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] Ciaurri, Óscar; Roncal, Luz 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] Dunkl, Charles F. 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] Dunkl, Charles F.; Xu, Yuan Orthogonal polynomials of several variables, Encyclopedia of Mathematics and its Applications, Volume 81, Cambridge University Press, Cambridge, 2001, pp. xvi+390 | Article | MR 1827871 | Zbl 0964.33001

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

[9] Duoandikoetxea, Javier; Moyua, Adela; Oruetxebarria, Osane; Seijo, Edurne 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] Geller, Daryl 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] Herz, Carl; Rivière, Nestor 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] Nowak, Adam 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] Nowak, Adam; Stempak, Krzysztof 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] Nowak, Adam; Stempak, Krzysztof Riesz transforms for the Dunkl harmonic oscillator, Math. Z., Volume 262 (2009) no. 3, pp. 539-556 | Article | MR 2506306 | Zbl 1168.44002

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

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

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

[19] Stempak, Krzysztof; Torrea, José Luis 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] Thangavelu, Sundaram Lectures on Hermite and Laguerre expansions, Mathematical Notes, Volume 42, Princeton University Press, Princeton, NJ, 1993, pp. xviii+195 (With a preface by Robert S. Strichartz) | MR 1215939 | Zbl 0791.41030

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