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

SIGMA 5 (2009), 016, 12 pages      arXiv:0902.1958
Contribution to the Special Issue on Dunkl Operators and Related Topics

Imaginary Powers of the Dunkl Harmonic Oscillator

Adam Nowak and Krzysztof Stempak
Instytut Matematyki i Informatyki, Politechnika Wroclawska, Wyb. Wyspianskiego 27, 50-370 Wroclaw, Poland

Received October 14, 2008, in final form February 08, 2009; Published online February 11, 2009

In this paper we continue the study of spectral properties of the Dunkl harmonic oscillator in the context of a finite reflection group on Rd isomorphic to Z2d. We prove that imaginary powers of this operator are bounded on Lp, 1 < p < ∞, and from L1 into weak L1.

Key words: Dunkl operators; Dunkl harmonic oscillator; imaginary powers; Calderón-Zygmund operators.

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  1. Askey R., Wainger S., Mean convergence of expansions in Laguerre and Hermite series, Amer. J. Math. 87 (1965), 695-708.
  2. Dunkl C.F., Reflection groups and orthogonal polynomials on the sphere, Math. Z. 197 (1988), 33-60.
  3. Dunkl C.F., Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1989), 167-183.
  4. Duoandikoetxea J., Fourier analysis, Graduate Studies in Mathematics, Vol. 29, American Mathematical Society, Providence, RI, 2001.
  5. Lebedev N.N., Special functions and their applications, Dover Publications, Inc., New York, 1972.
  6. Muckenhoupt B., On certain singular integrals, Pacific J. Math. 10 (1960), 239-261.
  7. Muckenhoupt B., Mean convergence of Hermite and Laguerre series. II, Trans. Amer. Math. Soc. 147 (1970), 433-460.
  8. Nowak A., Stempak K., Riesz transforms for multi-dimensional Laguerre function expansions, Adv. Math. 215 (2007), 642-678.
  9. Nowak A., Stempak K., Riesz transforms for the Dunkl harmonic oscillator, Math. Z., to appear, arXiv:0802.0474.
  10. Rosenblum M., Generalized Hermite polynomials and the Bose-like oscillator calculus, in Nonselfadjoint Operators and Related Topics (Beer Sheva, 1992), Oper. Theory Adv. Appl., Vol. 73, Birkhäuser, Basel, 1994, 369-396.
  11. Rösler M., Generalized Hermite polynomials and the heat equation for Dunkl operators, Comm. Math. Phys. 192 (1998), 519-542, q-alg/9703006.
  12. Rösler M., One-parameter semigroups related to abstract quantum models of Calogero types, in Infinite Dimensional Harmonic Analysis (Kioto, 1999), Gräbner, Altendorf, 2000, 290-305.
  13. 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.
  14. Stempak K., Torrea J.L., Poisson integrals and Riesz transforms for Hermite function expansions with weights, J. Funct. Anal. 202 (2003), 443-472.
  15. Stempak K., Torrea J.L., Higher Riesz transforms and imaginary powers associated to the harmonic oscillator, Acta Math. Hungar. 111 (2006), 43-64.

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