Abstract
In this article, we prove a dimension-free upper bound for the Lp-norms of the vector of Riesz transforms in the rational Dunkl setting. Our main technique is the Bellman function method adapted to the Dunkl setting.
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B. Amri and M. Sifi, Riesz transforms for Dunkl transform, Ann. Math. Blaise Pascal 19 (2012), 247–262.
J.-P. Anker, J. Dziubański and A. Hejna, Harmonic functions, conjugate harmonic functions and the Hardy space H1in the rational Dunkl setting, J. Fourier Anal. Appl. 25 (2019), 2356–2418.
N. Arcozzi, Riesz transforms on compact Lie groups, spheres and Gauss space, Ark. Mat. 36 (1998), 201–231.
R. Bañuelos, Martingale transforms and related singular integrals, Trans. Amer. Math. Soc. 293 (1986), 547–563.
R. Bañuelos and G. Wang, Sharp inequalities for martingales with applications to the Beurling—Ahlfors and Riesz transforms, Duke Math. J. 80 (1995), 575–600.
J. J. Betancor, E. Dalmasso, J. C. Fariña and R. Scotto, Bellman functions and dimension free Lp-estimates for the Riesz transforms in Bessel settings, Nonlinear Anal. 197 (2020), Article no. 111850.
D. L. Burkholder, Boundary value problems and sharp inequalities for martingale transforms, Ann. Probab. 12 (1984), 647–702.
D. L. Burkholder, A proof of Pełczynśki’s conjecture for the Haar system, Studia Math. 91 (1988), 79–83.
D. L. Burkholder, Explorations in martingale theory and its applications, in École d’Été de Probabilités de Saint-Flour XIX—1989, Springer, Berlin, 1991, pp. 1–66.
A. Carbonaro and O. Dragičević, Bellman function and linear dimension-free estimates in a theorem of Bakry, J. Funct. Anal. 265 (2013), 1085–1104.
A. Carbonaro and O. Dragičević, Functional calculus for generators of symmetric contraction semigroups, Duke Math. J. 166 (2017), 937–974.
A. Carbonaro and O. Dragičević, Bounded holomorphic functional calculus for nonsymmetric Ornstein—Uhlenbeck operators, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 19 (2019), 1497–1533.
A. Carbonaro and O. Dragičević, Bilinear embedding for divergence-form operators with complex coefficients on irregular domains, Calc. Var. Partial Differential Equations 59 (2020), Article no. 104.
T. Coulhon, D. Müller and J. Zienkiewicz, About Riesz transforms on the Heisenberg groups, Math. Ann. 305 (1996), 369–379.
K. Dahmani, Sharp dimension free bound for the Bakry—Riesz vector, arXiv:1611.07696 [math.CA]
M. F. E. de Jeu, The Dunkl transform, Invent. Math. 113 (1993), 147–162.
K. Domelevo, S. Petermichl and J. Wittwer, A linear dimensionless bound for the weighted Riesz vector, Bull. Sci. Math. 141 (2017), 385–407.
O. Dragičević and A. Volberg, Bellman functions and dimensionless estimates of Littlewood—Paley type, J. Operator Theory 56 (2006), 167–198.
O. Dragičević and A. Volberg, Bilinear embedding for real elliptic differential operators in divergence form with potentials, J. Funct. Anal. 261 (2011), 2816–2828.
O. Dragičević and A. Volberg, Linear dimension-free estimates in the embedding theorem for Schrödinger operators, J. Lond. Math. Soc. (2) 85 (2012), 191–222.
C. F. Dunkl, Reflection groups and orthogonal polynomials on the sphere, Math. Z. 197 (1988), 33–60.
C. F. Dunkl, Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1989), 167–183.
C. F. Dunkl, Hankel transforms associated to finite reflection groups, in Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications (Tampa, FL, 1991), American Mathematical Society, Providence, RI, 1992, pp. 123–138.
J. Duoandikoetxea and J. L. Rubio de Francia, Estimations indépendantes de la dimension pour les transformées de Riesz, C. R. Acad. Sci. Paris Sér. I Math. 300 (1985), 193–196.
L. Forzani, E. Sasso and R. Scotto, Lpboundedness of Riesz transforms for orthogonal polynomials in a general context, Studia Math. 231 (2015), 45–71.
P. Graczyk, J.-J. Loeb, I. A. López, A. Nowak and W. O. Urbina-Romero, Higher order Riesz transforms, fractional derivatives, and Sobolev spaces for Laguerre expansions, J. Math. Pures Appl. (9) 84 (2005), 375–405.
P. Graczyk, T. Luks and M. Rösler, On the Green function and Poisson integrals of the Dunkl Laplacian, Potential Anal. 48 (2018), 337–360.
R. F. Gundy, Sur les transformations de Riesz pour le semi-groupe d’Ornstein—Uhlenbeck, C. R. Acad. Sci. Paris Sér. I Math. 303 (1986), 967–970.
C. E. Gutiérrez, On the Riesz transforms for Gaussian measures, J. Funct. Anal. 120 (1994), 107–134.
C. E. Gutiérrez, On the Riesz transforms for Gaussian measures, J. Funct. Anal. 120 (1994), 107–134.
C. E. Gutiérrez, A. and J. L. Torrea, Riesz transforms, g-functions, and multipliers for the Laguerre semigroup, Houston J. Math. 27 (2001), 579–592.
E. Harboure, L. de Rosa, C. Segovia and J. L. Torrea, Lp-dimension free boundedness for Riesz transforms associated to Hermite functions, Math. Ann. 328 (2004), 653–682.
T. Iwaniec and G. Martin, Riesz transforms and related singular integrals, J. Reine Angew. Math. 473 (1996), 25–57.
M. Junge, T. Mei and J. Parcet, Noncommutative Riesz transforms—dimension free bounds and Fourier multipliers, J. Eur. Math. Soc. (JEMS) 20 (2018), 529–595.
M. Kucharski, Dimension-free estimates for Riesz transforms related to the harmonic oscillator, Colloq. Math. 165 (2021), 139–161.
F. Lust-Piquard, Dimension free estimates for Riesz transforms associated to the harmonic oscillator on ℝn, Potential Anal. 24 (2006), 47–62.
G. Mauceri and M. Spinelli, Riesz transforms and spectral multipliers of the Hodge—Laguerre operator, arXiv:1407.2838 [math.FA]
G. Mauceri and M. Spinelli, Riesz transforms and spectral multipliers of the Hodge—Laguerre operator, J. Funct. Anal. 269 (2015), 3402–3457.
P.-A. Meyer, Transformations de Riesz pour les lois gaussiennes, in Seminar on Probability, XVIII, Springer, Berlin, 1984, pp. 179–193.
F. Nazarov, S. Treil and A. Volberg, The Bellman functions and two-weight inequalities for Haar multipliers, J. Amer. Math. Soc. 12 (1999), 909–928.
F. L. Nazarov and S. R. Treil, The hunt for a Bellman function: applications to estimates for singular integral operators and to other classical problems of harmonic analysis, Algebra i Analiz 8 (1996), 32–162.
A. Nowak, On Riesz transforms for Laguerre expansions, J. Funct. Anal. 215 (2004), 217–240.
A. Nowak and P. Sjögren, Riesz transforms for Jacobi expansions, J. Anal. Math. 104 (2008), 341–369.
A. Nowak and K. Stempak, Riesz transforms for multi-dimensional Laguerre function expansions, Adv. Math. 215 (2007), 642–678.
A. Nowak and T. Z. Szarek, Calderón-Zygmund operators related to Laguerre function expansions of convolution type, J. Math. Anal. Appl. 388 (2012), 801–816.
A. Osekowski, Sharp Martingale and Semimartingale Inequalities, Birkhäuser, Basel, 2012.
G. Pisier, Riesz transforms: a simpler analytic proof of P.-A. Meyer’s inequality, in Séminaire de Probabilités, XXII, Springer, Berlin, 1988, pp. 485–501.
M. Rösler, Generalized Hermite polynomials and the heat equation for Dunkl operators, Comm. Math. Phys. 192 (1998), 519–542.
M. Rösler, Dunkl operators: theory and applications, in Orthogonal Polynomials and Special Functions (Leuven, 2002), Springer, Berlin, 2003, pp. 93–135.
M. Rösler and M. Voit, Markov processes related with Dunkl operators, Adv. in Appl. Math. 21 (1998), 575–643.
P. K. Sanjay and S. Thangavelu, Dimension free boundedness of Riesz transforms for the Grushin operator, Proc. Amer. Math. Soc. 142 (2014), 3839–3851.
E. M. Stein, Some results in harmonic analysis in ℝN, for n→ ∞, Bull. Amer. Math. Soc. (N.S.) 9 (1983), 71–73.
K. Stempak, Jacobi conjugate expansions, Studia Sci. Math. Hungar. 44 (2007), 117–130.
K. Stempak and B. Wróbel, Dimension free LPestimates for Riesz transforms associated with Laguerre function expansions of Hermite type, Taiwanese J. Math. 17 (2013), 63–81.
S. Thangavelu and Y. Xu, Convolution operator and maximal function for the Dunkl transform, J. Anal. Math. 97 (2005), 25–55.
S. Thangavelu and Y. Xu, Riesz transform and Riesz potentials for Dunkl transform, J. Comput. Appl. Math. 199 (2007), 181–195.
V. Vasyunin and A. Volberg, The Bellman Function Technique in Harmonic Analysis, Cambridge University Press, Cambridge, 2020.
A. Velicu, Sobolev-type inequalities for Dunkl operators, J. Funct. Anal. 279 (2020), 108695, 37.
B. Wróbel, Dimension free Lpestimates for single Riesz transforms via an H∞joint functional calculus, J. Funct. Anal. 267 (2014), 3332–3350.
B. Wróbel, Dimension-free Lpestimates for vectors of Riesz transforms associated with orthogonal expansions, Anal. PDE 11 (2018), 745–773.
Acknowledgments
The author would like to thank Błażej Wróbel and Jacek Dziubański for their helpful comments and suggestions, and Charles Dunkl for pointing out some references. The authors thank the anonymous reviewer for a careful reading of the manuscript and for helpful comments which improved the presentation of the paper.
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Research supported by the National Science Centre, Poland (Narodowe Centrum Nauki), Grant 2018/31/B/ST1/00204.
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Hejna, A. Dimension-free Lp-estimates for vectors of Riesz transforms in the rational Dunkl setting. JAMA 150, 485–528 (2023). https://doi.org/10.1007/s11854-023-0278-z
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DOI: https://doi.org/10.1007/s11854-023-0278-z