## Imaginary Powers of the Dunkl Harmonic Oscillator ⋆ (2008)

### BibTeX

@MISC{Nowak08imaginarypowers,

author = {Adam Nowak and Krzysztof Stempak},

title = {Imaginary Powers of the Dunkl Harmonic Oscillator ⋆},

year = {2008}

}

### OpenURL

### Abstract

doi:10.3842/SIGMA.2009.016 Abstract. In this paper we continue the study of spectral properties of the Dunkl harmonic oscillator in the context of a finite reflection group on R d isomorphic to Z d 2. We prove that imaginary powers of this operator are bounded on L p, 1 < p < ∞, and from L 1 into weak L 1.

### Citations

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116 |
Differential-difference operators associated to reflection groups
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(Show Context)
Citation Context ...ixed positive subsystem of R, and σβ denotes the reflection in the hyperplane orthogonal to β. The Dunkl operators T k j , j = 1, . . . , d, form a commuting system (this is an important feature, see =-=[3]-=-) of the first order differential-difference operators, and reduce to ∂j, j = 1, . . . , d, are homogeneous of degree −1 on P, the space of all polynomials when k ≡ 0. Moreover, T k j in Rd . This mea... |

38 | Generalized Hermite polynomials and the heat equation for Dunkl operators
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- 1998
(Show Context)
Citation Context ...ymmetric in L 2 (R d , wk), where wk(x) = ∏ β∈R+ |〈β, x〉| 2k(β) , if considered initially on C∞ c (Rd). Note that wk is G-invariant. The study of the operator Lk = −∆k + ‖x‖ 2 was initiated by Rösler =-=[11, 12]-=-. It occurs that Lk (or rather its self-adjoint extension Lk) has a discrete spectrum and the corresponding eigenfunctions are the generalized Hermite functions defined and investigated by Rösler [11]... |

29 |
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(Show Context)
Citation Context ... functions are tensor products where h αi ni h α n(x) = h α1 n1 (x1) · · · · · h αd nd (xd), x = (x1, . . . , xd) ∈ R d , n = (n1, . . . , nd) ∈ N d , are the one-dimensional functions (see Rosenblum =-=[10]-=-) here L αi ni h αi 2ni (xi) = d2ni,αie−x2i /2 L αi ( ) 2 ni xi , h αi 2ni+1 (xi) = d2ni+1,αie−x2i /2 xiL αi+1( ) 2 ni xi ; denotes the Laguerre polynomial of degree ni and order αi, cf. [5, p. 76], a... |

25 |
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- 1988
(Show Context)
Citation Context ...l Issue on Dunkl Operators and Related Topics. The full collection is available at http://www.emis.de/journals/SIGMA/Dunkl operators.html2 A. Nowak and K. Stempak In Dunkl’s theory the operator, see =-=[2]-=-, ∆k = d∑ j=1 (T k j ) 2 plays the role of the Euclidean Laplacian (notice that ∆ comes into play when k ≡ 0). It is homogeneous of degree −2 on P and symmetric in L 2 (R d , wk), where wk(x) = ∏ β∈R+... |

19 | Dunkl operators: Theory and applications
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(Show Context)
Citation Context ...n what follows we will use the notation introduced there and invoke certain arguments from [8]. For basic facts concerning Dunkl’s theory we refer the reader to the excellent survey article by Rösler =-=[13]-=-. Throughout the paper we use a fairly standard notation. Given a multi-index n ∈ Nd , we write |n| = n1 + · · · + nd and, for x, y ∈ Rd , xy = (x1y1, . . . , xdyd), xn = x n1 1 · · · · · x nd d (and ... |

15 |
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(Show Context)
Citation Context ...−αi−1/2 i Φ αi ni (xi), x ∈ R d +, { 1, 0 < xi ≤ 4(ni + αi + 1); exp(−cxi), xi > 4(ni + αi + 1). This follows from Muckenhoupt’s generalization [7] of the classical estimates due to Askey and Wainger =-=[1]-=-. � The theorem below says that the kernel K α,ε γ (x, y) satisfies standard estimates in the sense of the homogeneous space (R d +, w + α , ‖·‖). The corresponding proof is located in Section 4 below... |

8 |
Mean Convergence of Hermite and Laguerre series I
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(Show Context)
Citation Context ...to know that where |h α n(x)| � d∏ i=1 Φ αi ni (xi) = x −αi−1/2 i Φ αi ni (xi), x ∈ R d +, { 1, 0 < xi ≤ 4(ni + αi + 1); exp(−cxi), xi > 4(ni + αi + 1). This follows from Muckenhoupt’s generalization =-=[7]-=- of the classical estimates due to Askey and Wainger [1]. � The theorem below says that the kernel K α,ε γ (x, y) satisfies standard estimates in the sense of the homogeneous space (R d +, w + α , ‖·‖... |

5 |
Fourier Analysis, Graduate
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- 2000
(Show Context)
Citation Context ... works, with appropriate adjustments, when the underlying space is of homogeneous type. Thus we shall use properly adjusted facts from the classic Calderón–Zygmund theory (presented, for instance, in =-=[4]-=-) in the setting of the space (Rd +, w + α , ‖ · ‖) without further comments. A formal computation based on the formula λ −iγ = 1 Γ(iγ) ∫ ∞ 0 e −tλ t iγ−1 dt, λ > 0, suggests that L −iγ α,ε,+ should b... |

3 | One-parameter semigroups related to abstract quantum models of Calogero type
- Rösler
- 1999
(Show Context)
Citation Context ...ymmetric in L 2 (R d , wk), where wk(x) = ∏ β∈R+ |〈β, x〉| 2k(β) , if considered initially on C∞ c (Rd). Note that wk is G-invariant. The study of the operator Lk = −∆k + ‖x‖ 2 was initiated by Rösler =-=[11, 12]-=-. It occurs that Lk (or rather its self-adjoint extension Lk) has a discrete spectrum and the corresponding eigenfunctions are the generalized Hermite functions defined and investigated by Rösler [11]... |

2 |
On certain singular integrals
- Muckenhoupt
(Show Context)
Citation Context ...ecently by Stempak and Torrea [15, Theorem 4.3] and corresponding to the trivial multiplicity function k ≡ 0. Imaginary powers of the Euclidean Laplacian were investigated much earlier by Muckenhoupt =-=[6]-=-. Let us briefly describe the framework of the Dunkl theory of differential-difference operators on Rd related to finite reflection groups. Given such a group G ⊂ O(Rd ) and a G-invariant nonnegative ... |

2 |
Stempak K., Riesz transforms for multi-dimensional Laguerre function expansions
- Nowak
(Show Context)
Citation Context ...able to call it the Dunkl harmonic oscillator. In fact Lk becomes the classic harmonic oscillator −∆ + ‖x‖2 when k ≡ 0. The results of the present paper are naturally related to the authors’ articles =-=[8, 9]-=-. In what follows we will use the notation introduced there and invoke certain arguments from [8]. For basic facts concerning Dunkl’s theory we refer the reader to the excellent survey article by Rösl... |

2 | Stempak K., Riesz transforms for the Dunkl harmonic oscillator
- Nowak
(Show Context)
Citation Context ...d from L 1 into weak L 1 . Key words: Dunkl operators; Dunkl harmonic oscillator; imaginary powers; Calderón– Zygmund operators 2000 Mathematics Subject Classification: 42C10; 42C20 1 Introduction In =-=[9]-=- the authors defined and investigated a system of Riesz transforms related to the Dunkl harmonic oscillator Lk. The present article continues the study of spectral properties of operators associated w... |

2 |
Torrea J.L., Poisson integrals and Riesz transforms for Hermite function expansions with weights
- Stempak
(Show Context)
Citation Context ...cf. [11, Corollary 3.5 (ii)]; here the normalizing constant ck equals to ∫ R d exp(−‖x‖ 2 )wk(x) dx. Moreover, h k n are eigenfunctions of Lk, Lkh k n = (2|n| + 2τ + d)h k n, where τ = ∑ for instance =-=[14]-=- or [15]. β∈R+ k(β). For k ≡ 0, h0 n are the usual multi-dimensional Hermite functions, seeImaginary Powers of the Dunkl Harmonic Oscillator 3 Let 〈·, ·〉k be the canonical inner product in L 2 (R d ,... |

1 |
Torrea J.L., Higher Riesz transforms and imaginary powers associated to the harmonic oscillator
- Stempak
(Show Context)
Citation Context ... Corollary 3.5 (ii)]; here the normalizing constant ck equals to ∫ R d exp(−‖x‖ 2 )wk(x) dx. Moreover, h k n are eigenfunctions of Lk, Lkh k n = (2|n| + 2τ + d)h k n, where τ = ∑ for instance [14] or =-=[15]-=-. β∈R+ k(β). For k ≡ 0, h0 n are the usual multi-dimensional Hermite functions, seeImaginary Powers of the Dunkl Harmonic Oscillator 3 Let 〈·, ·〉k be the canonical inner product in L 2 (R d , wk). Th... |