Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 096, 39 pages      arXiv:1602.05369

On Harmonic Analysis Operators in Laguerre-Dunkl and Laguerre-Symmetrized Settings

Adam Nowak a, Krzysztof Stempak b and Tomasz Z. Szarek a
a) Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656 Warszawa, Poland
b) Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Wyb. Wyspiańskiego 27, 50-370 Wrocław, Poland

Received May 25, 2016, in final form September 23, 2016; Published online September 29, 2016

We study several fundamental harmonic analysis operators in the multi-dimensional context of the Dunkl harmonic oscillator and the underlying group of reflections isomorphic to $\mathbb{Z}_2^d$. Noteworthy, we admit negative values of the multiplicity functions. Our investigations include maximal operators, $g$-functions, Lusin area integrals, Riesz transforms and multipliers of Laplace and Laplace-Stieltjes type. By means of the general Calderón-Zygmund theory we prove that these operators are bounded on weighted $L^p$ spaces, $1$ < $p$ < $\infty$, and from weighted $L^1$ to weighted weak $L^1$. We also obtain similar results for analogous set of operators in the closely related multi-dimensional Laguerre-symmetrized framework. The latter emerges from a symmetrization procedure proposed recently by the first two authors. As a by-product of the main developments we get some new results in the multi-dimensional Laguerre function setting of convolution type.

Key words: Dunkl harmonic oscillator; generalized Hermite functions; negative multiplicity function; Laguerre expansions of convolution type; Bessel harmonic oscillator; Laguerre-Dunkl expansions; Laguerre-symmetrized expansions; heat semigroup; Poisson semigroup; maximal operator; Riesz transform; $g$-function; spectral multiplier; area integral; Calderón-Zygmund operator.

pdf (662 kb)   tex (46 kb)


  1. Álvarez López J.A., Calaza M., Embedding theorems for the Dunkl harmonic oscillator on the line, SIGMA 10 (2014), 004, 16 pages, arXiv:1301.4196.
  2. Álvarez López J.A., Calaza M., A perturbation of the Dunkl harmonic oscillator on the line, SIGMA 11 (2015), 059, 33 pages, arXiv:1412.4655.
  3. Amri B., Riesz transforms for Dunkl Hermite expansions, J. Math. Anal. Appl. 423 (2015), 646-659, arXiv:1201.1209.
  4. Amri B., Sifi M., Riesz transforms for Dunkl transform, Ann. Math. Blaise Pascal 19 (2012), 247-262, arXiv:1105.1427.
  5. Amri B., Sifi M., Singular integral operators in Dunkl setting, J. Lie Theory 22 (2012), 723-739.
  6. Amri B., Tayari H., The $L^p$-continuity of imaginary powers of the Dunkl harmonic oscillator, Indian J. Pure Appl. Math. 46 (2015), 239-249.
  7. Ben Salem N., Samaali T., Hilbert transforms associated with Dunkl-Hermite polynomials, SIGMA 5 (2009), 037, 17 pages, arXiv:0903.4369.
  8. Betancor J.J., Castro A.J., Nowak A., Calderón-Zygmund operators in the Bessel setting, Monatsh. Math. 167 (2012), 375-403, arXiv:1012.5638.
  9. Betancor J.J., Fariña J.C., Rodríguez-Mesa L., Testoni R., Torrea J.L., Fractional square functions and potential spaces, J. Math. Anal. Appl. 386 (2012), 487-504.
  10. Betancor J.J., Fariña J.C., Rodríguez-Mesa L., Testoni R., Torrea J.L., Fractional square functions and potential spaces, II, Acta Math. Sin. (Engl. Ser.) 31 (2015), 1759-1774.
  11. Betancor J.J., Molina S.M., Rodríguez-Mesa L., Area Littlewood-Paley functions associated with Hermite and Laguerre operators, Potential Anal. 34 (2011), 345-369, arXiv:1001.3814.
  12. Boggarapu P., Roncal L., Thangavelu S., Mixed norm estimates for the Cesàro means associated with Dunkl-Hermite expansions, Trans. Amer. Math. Soc., to appear, arXiv:1410.2162.
  13. Boggarapu P., Thangavelu S., Mixed norm estimates for the Riesz transforms associated to Dunkl harmonic oscillators, Ann. Math. Blaise Pascal 22 (2015), 89-120, arXiv:1407.1644.
  14. Castro A.J., Szarek T.Z., On fundamental harmonic analysis operators in certain Dunkl and Bessel settings, J. Math. Anal. Appl. 412 (2014), 943-963, arXiv:1304.2904.
  15. Christ M., Lectures on singular integral operators, CBMS Regional Conference Series in Mathematics, Vol. 77, Amer. Math. Soc., Providence, RI, 1990.
  16. Dunkl C.F., Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1989), 167-183.
  17. Duoandikoetxea J., Fourier analysis, Graduate Studies in Mathematics, Vol. 29, Amer. Math. Soc., Providence, RI, 2001.
  18. Forzani L., Sasso E., Scotto R., Maximal operators associated with generalized Hermite polynomial and function expansions, Rev. Un. Mat. Argentina 54 (2013), 83-107.
  19. Grafakos L., Modern Fourier analysis, Graduate Texts in Mathematics, Vol. 250, 3rd ed., Springer, New York, 2014.
  20. Harboure E., de Rosa L., Segovia C., Torrea J.L., $L^p$-dimension free boundedness for Riesz transforms associated to Hermite functions, Math. Ann. 328 (2004), 653-682.
  21. Johnson W.P., The curious history of Faà di Bruno's formula, Amer. Math. Monthly 109 (2002), 217-234.
  22. Langowski B., Harmonic analysis operators related to symmetrized Jacobi expansions, Acta Math. Hungar. 140 (2013), 248-292, arXiv:1210.1342.
  23. Langowski B., On potential spaces related to Jacobi expansions, J. Math. Anal. Appl. 432 (2015), 374-397, arXiv:1410.6635.
  24. Langowski B., Potential and Sobolev spaces related to symmetrized Jacobi expansions, SIGMA 11 (2015), 073, 17 pages, arXiv:1505.01653.
  25. Langowski B., Harmonic analysis operators related to symmetrized Jacobi expansions for all admissible parameters, Acta Math. Hungar. 150 (2016), 49-82, arXiv:1512.08948.
  26. Nefzi W., Higher order Riesz transforms for the Dunkl harmonic oscillator, Taiwanese J. Math. 19 (2015), 567-583.
  27. Nowak A., Stempak K., $L^2$-theory of Riesz transforms for orthogonal expansions, J. Fourier Anal. Appl. 12 (2006), 675-711.
  28. Nowak A., Stempak K., Riesz transforms for multi-dimensional Laguerre function expansions, Adv. Math. 215 (2007), 642-678.
  29. Nowak A., Stempak K., Imaginary powers of the Dunkl harmonic oscillator, SIGMA 5 (2009), 016, 12 pages, arXiv:0902.1958.
  30. Nowak A., Stempak K., Riesz transforms for the Dunkl harmonic oscillator, Math. Z. 262 (2009), 539-556, arXiv:0802.0474.
  31. Nowak A., Stempak K., Negative powers of Laguerre operators, Canad. J. Math. 64 (2012), 183-216, arXiv:0912.0038.
  32. Nowak A., Stempak K., A symmetrized conjugacy scheme for orthogonal expansions, Proc. Roy. Soc. Edinburgh Sect. A 143 (2013), 427-443, arXiv:1009.1767.
  33. Nowak A., Stempak K., Sharp estimates for potential operators associated with Laguerre and Dunkl-Laguerre expansions, Potential Anal. 44 (2016), 109-136, arXiv:1402.2522.
  34. Nowak A., Szarek T.Z., Calderón-Zygmund operators related to Laguerre function expansions of convolution type, J. Math. Anal. Appl. 388 (2012), 801-816.
  35. Rösler M., Dunkl operators: theory and applications, in Orthogonal Polynomials and Special Functions (Leuven, 2002), Lecture Notes in Math., Vol. 1817, Springer, Berlin, 2003, 93-135, math.CA/0210366.
  36. Rubio de Francia J.L., Ruiz F.J., Torrea J.L., Calderón-Zygmund theory for operator-valued kernels, Adv. Math. 62 (1986), 7-48.
  37. Ruiz F.J., Torrea J.L., Vector-valued Calderón-Zygmund theory and Carleson measures on spaces of homogeneous nature, Studia Math. 88 (1988), 221-243.
  38. Sasso E., Functional calculus for the Laguerre operator, Math. Z. 249 (2005), 683-711.
  39. Segovia C., Wheeden R.L., On certain fractional area integrals, J. Math. Mech. 19 (1969), 247-262.
  40. Stempak K., Torrea J.L., On $g$-functions for Hermite function expansions, Acta Math. Hungar. 109 (2005), 99-125.
  41. Stempak K., Torrea J.L., Higher Riesz transforms and imaginary powers associated to the harmonic oscillator, Acta Math. Hungar. 111 (2006), 43-64.
  42. Szarek T., Littlewood-Paley-Stein type square functions based on Laguerre semigroups, Acta Math. Hungar. 131 (2011), 59-109, arXiv:1001.3579.
  43. Szarek T.Z., Multipliers of Laplace transform type in certain Dunkl and Laguerre settings, Bull. Aust. Math. Soc. 85 (2012), 177-190, arXiv:1101.4139.
  44. Szarek T.Z., On Lusin's area integrals and $g$-functions in certain Dunkl and Laguerre settings, Math. Nachr. 285 (2012), 1517-1542, arXiv:1011.0898.
  45. Thangavelu S., Lectures on Hermite and Laguerre expansions, Mathematical Notes, Vol. 42, Princeton University Press, Princeton, NJ, 1993.
  46. Wróbel B., Multivariate spectral multipliers for the Dunkl transform and the Dunkl harmonic oscillator, Forum Math. 27 (2015), 2301-2322.

Previous article  Next article   Contents of Volume 12 (2016)