Geodesic
Imaginary Powers of the Dunkl Harmonic Oscillator
Symmetry, integrability and geometry: methods and applications, Tome 5 (2009) Cet article a éte moissonné depuis la source Math-Net.Ru

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In this paper we continue the study of spectral properties of the Dunkl harmonic oscillator in the context of a finite reflection group on $\mathbb R^d$ isomorphic to $\mathbb Z^d_2$. We prove that imaginary powers of this operator are bounded on $L^p$, $1$, and from $L^1$ into weak $L^1$.
Keywords: Dunkl operators; Dunkl harmonic oscillator; imaginary powers; Calderón–Zygmund operators. 
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Adam Nowak; Krzysztof Stempak. Imaginary Powers of the Dunkl Harmonic Oscillator. Symmetry, integrability and geometry: methods and applications, Tome 5 (2009). http://geodesic.mathdoc.fr/item/SIGMA_2009_5_a15/

[1] Askey R., Wainger S., “Mean convergence of expansions in Laguerre and Hermite series”, Amer. J. Math., 87 (1965), 695–708 | DOI | MR | Zbl

[2] Dunkl C. F., “Reflection groups and orthogonal polynomials on the sphere”, Math. Z., 197 (1988), 33–60 | DOI | MR | Zbl

[3] Dunkl C. F., “Differential-difference operators associated to reflection groups”, Trans. Amer. Math. Soc., 311 (1989), 167–183 | DOI | MR | Zbl

[4] Duoandikoetxea J., Fourier analysis, Graduate Studies in Mathematics, 29, American Mathematical Society, Providence, RI, 2001 | MR | Zbl

[5] Lebedev N. N., Special functions and their applications, Dover Publications, Inc., New York, 1972 | MR | Zbl

[6] Muckenhoupt B., “On certain singular integrals”, Pacific J. Math., 10 (1960), 239–261 | MR | Zbl

[7] Muckenhoupt B., “Mean convergence of Hermite and Laguerre series. II”, Trans. Amer. Math. Soc., 147 (1970), 433–460 | DOI | MR

[8] Nowak A., Stempak K., “Riesz transforms for multi-dimensional Laguerre function expansions”, Adv. Math., 215 (2007), 642–678 | DOI | MR | Zbl

[9] Nowak A., Stempak K., “Riesz transforms for the Dunkl harmonic oscillator”, Math. Z., 262:3 (2009), 539–556 ; arXiv:0802.0474 | DOI | MR | Zbl

[10] Rosenblum M., “Generalized Hermite polynomials and the Bose-like oscillator calculus”, Nonselfadjoint Operators and Related Topics (Beer Sheva, 1992), Oper. Theory Adv. Appl., 73, Birkhäuser, Basel, 1994, 369–396 | MR | Zbl

[11] Rösler M., “Generalized Hermite polynomials and the heat equation for Dunkl operators”, Comm. Math. Phys., 192 (1998), 519–542 ; q-alg/9703006 | DOI | MR | Zbl

[12] Rösler M., “One-parameter semigroups related to abstract quantum models of Calogero types”, Infinite Dimensional Harmonic Analysis (Kioto, 1999), Gräbner, Altendorf, 2000, 290–305 | MR | Zbl

[13] Rösler M., “Dunkl operators: theory and applications”, Orthogonal Polynomials and Special Functions (Leuven, 2002), Lecture Notes in Math., 1817, Springer, Berlin, 2003, 93–135 ; math.CA/0210366 | MR | Zbl

[14] Stempak K., Torrea J. L., “Poisson integrals and Riesz transforms for Hermite function expansions with weights”, J. Funct. Anal., 202 (2003), 443–472 | DOI | MR | Zbl

[15] Stempak K., Torrea J. L., “Higher Riesz transforms and imaginary powers associated to the harmonic oscillator”, Acta Math. Hungar., 111 (2006), 43–64 | DOI | MR | Zbl