Imaginary Powers of the Dunkl Harmonic Oscillator

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.

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Datum:	2009
Hauptverfasser:	Nowak, A., Stempak, K.
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Sprache:	English
Veröffentlicht:	Інститут математики НАН України 2009
Schriftenreihe:	Symmetry, Integrability and Geometry: Methods and Applications
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spelling	irk-123456789-1492442019-02-20T01:26:49Z Imaginary Powers of the Dunkl Harmonic Oscillator Nowak, A. Stempak, K. 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. 2009 Article Imaginary Powers of the Dunkl Harmonic Oscillator / A. Nowak, K. Stempak // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 15 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 42C10; 42C20 http://dspace.nbuv.gov.ua/handle/123456789/149244 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	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.
format	Article
author	Nowak, A. Stempak, K.
spellingShingle	Nowak, A. Stempak, K. Imaginary Powers of the Dunkl Harmonic Oscillator Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Nowak, A. Stempak, K.
author_sort	Nowak, A.
title	Imaginary Powers of the Dunkl Harmonic Oscillator
title_short	Imaginary Powers of the Dunkl Harmonic Oscillator
title_full	Imaginary Powers of the Dunkl Harmonic Oscillator
title_fullStr	Imaginary Powers of the Dunkl Harmonic Oscillator
title_full_unstemmed	Imaginary Powers of the Dunkl Harmonic Oscillator
title_sort	imaginary powers of the dunkl harmonic oscillator
publisher	Інститут математики НАН України
publishDate	2009
url	http://dspace.nbuv.gov.ua/handle/123456789/149244
citation_txt	Imaginary Powers of the Dunkl Harmonic Oscillator / A. Nowak, K. Stempak // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 15 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT nowaka imaginarypowersofthedunklharmonicoscillator AT stempakk imaginarypowersofthedunklharmonicoscillator
first_indexed	2025-07-12T21:40:18Z
last_indexed	2025-07-12T21:40:18Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 5 (2009), 016, 12 pages Imaginary Powers of the Dunkl Harmonic Oscillator? Adam NOWAK and Krzysztof STEMPAK Instytut Matematyki i Informatyki, Politechnika Wroc lawska, Wyb. Wyspiańskiego 27, 50–370 Wroc law, Poland E-mail: Adam.Nowak@pwr.wroc.pl, Krzysztof.Stempak@pwr.wroc.pl URL: http://www.im.pwr.wroc.pl/∼anowak/, http://www.im.pwr.wroc.pl/∼stempak/ Received October 14, 2008, in final form February 08, 2009; Published online February 11, 2009 doi:10.3842/SIGMA.2009.016 Abstract. In this paper we continue the study of spectral properties of the Dunkl harmo- nic oscillator in the context of a finite reflection group on Rd isomorphic to Zd 2. 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 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 with Lk by considering the imaginary powers L−iγ k , γ ∈ R. Our objective is to study Lp mapping properties of the operators L−iγ k , and the principal tool is the general Calderón– Zygmund operator theory. The main result we get (Theorem 1) partially extends the result obtained recently 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 multiplicity function k : R → [0,∞) on a root system R ⊂ Rd associated with the reflections of G, the Dunkl differential-difference operators T k j , j = 1, . . . , d, are defined by T k j f(x) = ∂jf(x) + ∑ β∈R+ k(β)βj f(x)− f(σβx) 〈β, x〉 , f ∈ C1(Rd); here ∂j is the jth partial derivative, 〈·, ·〉 denotes the standard inner product in Rd, R+ is a fixed 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, when k ≡ 0. Moreover, T k j are homogeneous of degree −1 on P, the space of all polynomials in Rd. This means that T k j Pm ⊂ Pm−1, where m ∈ N = {0, 1, . . .} and Pm denotes the subspace of P consisting of polynomials of total degree m (by convention, P−1 consists only of the null function). ?This paper is a contribution to the Special Issue on Dunkl Operators and Related Topics. The full collection is available at http://www.emis.de/journals/SIGMA/Dunkl operators.html mailto:Adam.Nowak@pwr.wroc.pl mailto:Krzysztof.Stempak@pwr.wroc.pl http://www.im.pwr.wroc.pl/~anowak/ http://www.im.pwr.wroc.pl/~stempak/ http://dx.doi.org/10.3842/SIGMA.2009.016 http://www.emis.de/journals/SIGMA/Dunkl_operators.html 2 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 L2(Rd, 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]. Due to the form of Lk, it is reasonable 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 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 = xn1 1 · · · · · xnd d (and similarly xα for x ∈ Rd + and α ∈ Rd); ‖x‖ denotes the Euclidean norm of x ∈ Rd, and ej is the jth coordinate vector in Rd. Given x ∈ Rd and r > 0, B(x, r) is the Euclidean ball in Rd centered at x and of radius r. For a nonnegative weight function w on Rd, by Lp(Rd, w), 1 ≤ p <∞, we denote the usual Lebesgue spaces related to the measure dw(x) = w(x)dx (in the sequel we will often abuse slightly the notation and use the same symbol w to denote the measure induced by a density w). Writing X . Y indicates that X ≤ CY with a positive constant C independent of significant quantities. We shall write X ' Y when X . Y and Y . X. 2 Preliminaries In the setting of general Dunkl’s theory Rösler [11] constructed systems of naturally associated multivariable generalized Hermite polynomials and Hermite functions. The system of generalized Hermite polynomials {Hk n : n ∈ Nd} is orthogonal and complete in L2(Rd, e−‖·‖ 2 wk), while the system {hk n : n ∈ Nd} of generalized Hermite functions hk n(x) = ( 2\|n\|ck )−1/2 exp(−‖x‖2/2)Hk n(x), x ∈ Rd, n ∈ Nd, is an orthonormal basis in L2(Rd, wk), cf. [11, Corollary 3.5 (ii)]; here the normalizing constant ck equals to ∫ Rd exp(−‖x‖2)wk(x) dx. Moreover, hk n are eigenfunctions of Lk, Lkh k n = (2\|n\|+ 2τ + d)hk n, where τ = ∑ β∈R+ k(β). For k ≡ 0, h0 n are the usual multi-dimensional Hermite functions, see for instance [14] or [15]. Imaginary Powers of the Dunkl Harmonic Oscillator 3 Let 〈·, ·〉k be the canonical inner product in L2(Rd, wk). The operator Lkf = ∑ n∈Nd (2\|n\|+ 2τ + d)〈f, hk n〉k hk n, defined on the domain Dom(Lk) = { f ∈ L2(Rd, wk) : ∑ n∈Nd ∣∣(2\|n\|+ 2τ + d)〈f, hk n〉k ∣∣2 <∞ } , is a self-adjoint extension of Lk considered on C∞ c (Rd) as the natural domain (the inclusion C∞ c (Rd) ⊂ Dom(Lk) may be easily verified). The spectrum of Lk is the discrete set {2m+2τ+d : m ∈ N}, and the spectral decomposition of Lk is Lkf = ∞∑ m=0 (2m+ 2τ + d)Pk mf, f ∈ Dom(Lk), where the spectral projections are Pk mf = ∑ \|n\|=m 〈f, hk n〉k hk n. By Parseval’s identity, for each γ ∈ R the operator L−iγ k f = ∞∑ m=0 (2m+ 2τ + d)−iγ Pk mf is an isometry on L2(Rd, wk). Consider the finite reflection group generated by σj , j = 1, . . . , d, σj(x1, . . . , xj , . . . , xd) = (x1, . . . ,−xj , . . . , xd), and isomorphic to Zd 2 = {0, 1}d. The reflection σj is in the hyperplane orthogonal to ej . Thus R = {± √ 2ej : j = 1, . . . , d}, R+ = { √ 2ej : j = 1, . . . , d}, and for a nonnegative multiplicity function k : R → [0,∞) which is Zd 2-invariant only values of k on R+ are essential. Hence we may think k = (α1 + 1/2, . . . , αd + 1/2), αj ≥ −1/2. We write αj + 1/2 in place of seemingly more appropriate αj since, for the sake of clarity, it is convenient for us to stick to the notation used in [8] and [9]. In what follows the symbols Tα j , ∆α, wα, Lα, Lα, hα n, and so on, denote the objects introduced earlier and related to the present Zd 2 group setting. Thus the Dunkl differential-difference operators are now given by Tα j f(x) = ∂jf(x) + (αj + 1/2) f(x)− f(σjx) xj , f ∈ C1(Rd), and the explicit formula for the Dunkl Laplacian is ∆αf(x) = d∑ j=1 ( ∂2f ∂x2 j (x) + 2αj + 1 xj ∂f ∂xj (x)− (αj + 1/2) f(x)− f(σjx) x2 j ) . The corresponding weight wα has the form wα(x) = d∏ j=1 \|xj \|2αj+1 ' ∏ β∈R+ \|〈β, x〉α\|2k(β), x ∈ Rd. 4 A. Nowak and K. Stempak Given α ∈ [−1/2,∞)d, the associated generalized Hermite functions are tensor products hα n(x) = hα1 n1 (x1) · · · · · hαd nd (xd), x = (x1, . . . , xd) ∈ Rd, n = (n1, . . . , nd) ∈ Nd, where hαi ni are the one-dimensional functions (see Rosenblum [10]) hαi 2ni (xi) = d2ni,αie −x2 i /2Lαi ni ( x2 i ) , hαi 2ni+1(xi) = d2ni+1,αie −x2 i /2xiL αi+1 ni ( x2 i ) ; here Lαi ni denotes the Laguerre polynomial of degree ni and order αi, cf. [5, p. 76], and d2ni,αi = (−1)ni ( Γ(ni + 1) Γ(ni + αi + 1) )1/2 , d2ni+1,αi = (−1)ni ( Γ(ni + 1) Γ(ni + αi + 2) )1/2 . For α = (−1/2, . . . ,−1/2) we obtain the usual Hermite functions. The system {hα n : n ∈ Nd} is an orthonormal basis in L2(Rd, wα) and Lαh α n = (2\|n\|+ 2\|α\|+ 2d)hα n, where by \|α\| we denote \|α\| = α1 + · · ·+ αd (thus \|α\| may be negative). The semigroup Tα t = exp(−tLα), t ≥ 0, generated by Lα is a strongly continuous semigroup of contractions on L2(Rd, wα). By the spectral theorem, Tα t f = ∞∑ m=0 e−t(2m+2\|α\|+2d)Pα mf, f ∈ L2(Rd, wα). The integral representation of Tα t on L2(Rd, wα) is Tα t f(x) = ∫ Rd Gα t (x, y)f(y) dwα(y), x ∈ Rd, t > 0, where the heat kernel {Gα t }t>0 is given by Gα t (x, y) = ∞∑ m=0 e−t(2m+2\|α\|+2d) ∑ \|n\|=m hα n(x)hα n(y). (1) In dimension one, for α ≥ −1/2 it is known (see for instance [11, Theorem 3.12] and [11, p. 523]) that Gα t (x, y) = 1 2 sinh 2t exp ( −1 2 coth(2t) ( x2 + y2 ))[Iα ( xy sinh 2t ) (xy)α + xy Iα+1 ( xy sinh 2t ) (xy)α+1 ] , with Iν being the modified Bessel function of the first kind and order ν, Iν(z) = ∞∑ k=0 (z/2)ν+2k Γ(k + 1)Γ(k + ν + 1) . Here we consider the function z 7→ zν , and thus also the Bessel function Iν(z), as an analytic function defined on C\{ix : x ≤ 0} (usually Iν is considered as a function on C cut along the half-line (−∞, 0]). Note that Iν , as a function on R+, is real, positive and smooth for any ν > −1, see [5, Chapter 5]. Imaginary Powers of the Dunkl Harmonic Oscillator 5 Therefore, in d dimensions, Gα t (x, y) = ∑ ε∈Zd 2 Gα,ε t (x, y), where the component kernels are Gα,ε t (x, y) = 1 (2 sinh 2t)d exp ( −1 2 coth(2t) ( ‖x‖2 + ‖y‖2 )) d∏ i=1 (xiyi)εi Iαi+εi ( xiyi sinh 2t ) (xiyi)αi+εi . Note that Gα,ε t (x, y) is given by the series (1), with the summation in n restricted to the set of multi-indices Nε = { n ∈ Nd : ni is even if εi = 0 or ni is odd if εi = 1, i = 1, . . . , d } . To verify this fact it is enough to restrict to the one-dimensional case and then use the Hille– Hardy formula, cf. [5, (4.17.6)]. In the sequel we will make use of the following technical result concerning Gα,ε t (x, y). The corresponding proof is given at the end of Section 4. Lemma 1. Let α ∈ [−1/2,∞)d and let ε ∈ Zd 2. Then, with x, y ∈ Rd + fixed, x 6= y, the kernel Gα,ε t (x, y) decays rapidly when either t → 0+ or t → ∞. Further, given any disjoint compact sets E,F ⊂ Rd +, we have∫ ∞ 0 ∣∣∂tG α,ε t (x, y) ∣∣ dt . 1, (2) uniformly in x ∈ E and y ∈ F . We end this section with pointing out that there is a general background for the facts conside- red here for an arbitrary reflection group, see [13] for a comprehensive account. In particular, the heat (or Mehler) kernel (1) has always a closed form involving the so-called Dunkl kernel, and is always strictly positive. This implies that the corresponding semigroup is contractive on L∞(Rd, wk), and as its generator is self-adjoint and positive in L2(Rd, wk), the semigroup is also contractive on the latter space. Hence, by duality and interpolation, it is in fact contractive on all Lp(Rd, wk), 1 ≤ p ≤ ∞. 3 Main result From now on we assume γ ∈ R, γ 6= 0, to be fixed. Recall that the operator L−iγ α is given on L2(Rd, wα) by the spectral series, L−iγ α f = ∑ n∈Nd (2\|n\|+ 2\|α\|+ 2d)−iγ〈f, hα n〉α hα n. Our main result concerns mapping properties of L−iγ α f on Lp spaces. Theorem 1. Assume α ∈ [−1/2,∞)d. Then L−iγ α , defined initially on L2(Rd, wα), extends uniquely to a bounded operator on Lp(Rd, wα), 1 < p < ∞, and to a bounded operator from L1(Rd, wα) to L1,∞(Rd, wα). 6 A. Nowak and K. Stempak The proof we give relies on splitting L−iγ α in L2(Rd, wα) into a finite number of suitable L2-bounded operators and then treating each of the operators separately. More precisely, we decompose L−iγ α = ∑ ε∈Zd 2 L−iγ α,ε , where (with the set Nε introduced in the previous section) L−iγ α,ε f = ∑ n∈Nε (2\|n\|+ 2\|α\|+ 2d)−iγ〈f, hα n〉α hα n, f ∈ L2(Rd, wα). Clearly, each L−iγ α,ε is a contraction in L2(Rd, wα). It is now convenient to introduce the following terminology: given ε ∈ Zd 2, we say that a function f on Rd is ε-symmetric if for each i = 1, . . . , d, f is either even or odd with respect to the ith coordinate according to whether εi = 0 or εi = 1, respectively. Thus f is ε-symmetric if and only if f ◦ σi = (−1)εif , i = 1, . . . , d. Any function f on Rd can be split uniquely into a sum of ε-symmetric functions fε, f = ∑ ε∈Zd 2 fε, fε(x) = 1 2d ∑ η∈{−1,1}d ηεf(ηx). For f ∈ L2(Rd, wα) this splitting is orthogonal in L2(Rd, wα). Finally, notice that hα n is ε- symmetric if and only if n ∈ Nε. Consequently, L−iγ α,ε is invariant on the subspace of L2(Rd, wα) of ε-symmetric functions and vanishes on the orthogonal complement of that subspace. Observe that in order to prove Theorem 1 it is sufficient to show the analogous result for each L−iγ α,ε . Moreover, since L−iγ α f = ∑ ε∈Zd 2 L−iγ α,ε f = ∑ ε∈Zd 2 L−iγ α,ε fε and since for a fixed 1 ≤ p <∞ (recall that wα(ξx) = wα(x), ξ ∈ {−1, 1}d) ‖f‖Lp(Rd,wα) ' ∑ ε∈Zd 2 ‖fε‖Lp(Rd +,w+ α ), it is enough to restrict the situation to the space (Rd +, w + α ), where w+ α is the restriction of wα to Rd +. Thus we are reduced to considering the operators L−iγ α,ε,+f = ∑ n∈Nε (2\|n\|+ 2\|α\|+ 2d)−iγ〈f, hα n〉L2(Rd +,w+ α ) h α n, f ∈ L2(Rd +, w + α ), (3) which are bounded on L2(Rd +, w + α ) since the system {2d/2hα n : n ∈ Nε} is orthonormal in L2(Rd +, w + α ). Now, Theorem 1 will be justified once we prove the following. Lemma 2. Assume that α ∈ [−1/2,∞)d and ε ∈ Zd 2. Then L−iγ α,ε,+, defined initially on L2(Rd +, w + α ), extends uniquely to a bounded operator on Lp(Rd +, w + α ), 1 < p < ∞, and to a bounded operator from L1(Rd +, w + α ) to L1,∞(Rd +, w + α ). The proof of Lemma 2 will be furnished by means of the general Calderón–Zygmund theory. In fact, we shall show that each L−iγ α,ε,+ is a Calderón–Zygmund operator in the sense of the space of homogeneous type (Rd +, w + α , ‖·‖). It is well known that the classical Calderón–Zygmund theory Imaginary Powers of the Dunkl Harmonic Oscillator 7 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−tλtiγ−1 dt, λ > 0, suggests that L−iγ α,ε,+ should be associated with the kernel Kα,ε γ (x, y) = 1 Γ(iγ) ∫ ∞ 0 Gα,ε t (x, y)tiγ−1 dt, x, y ∈ Rd + (4) (note that for x 6= y the last integral is absolutely convergent due to the decay of Gα,ε t (x, y) at t→ 0+ and t→∞, see Lemma 1). The next result shows that this is indeed the case, at least in the Calderón–Zygmund theory sense. Proposition 1. Let α ∈ [−1/2,∞)d and ε ∈ Zd 2. Then for f, g ∈ C∞ c (Rd +) with disjoint supports 〈L−iγ α,ε,+f, g〉L2(Rd +,w+ α ) = ∫ Rd + ∫ Rd + Kα,ε γ (x, y)f(y)g(x) dw+ α (y) dw+ α (x). (5) Proof. We follow the lines of the proof of [15, Proposition 4.2], see also [14, Proposition 3.2]. By Parseval’s identity and (3), 〈L−iγ α,ε,+f, g〉L2(Rd +,w+ α ) = ∑ n∈Nε (2\|n\|+ 2\|α\|+ 2d)−iγ 〈f, hα n〉L2(Rd +,w+ α ) 〈h α n, g〉L2(Rd +,w+ α ). (6) To finish the proof it is now sufficient to verify that the right-hand sides of (5) and (6) coincide. This task means justifying the possibility of changing the order of integration, summation and differentiation in the relevant expressions, see the proof of Proposition 4.2 in [15]. The details are rather elementary and thus are omitted. The key estimate∫ Rd + ∫ Rd + ∫ ∞ 0 ∣∣∂tG α,ε t (x, y) ∣∣ dt \|g(x)f(y)\| dy dx <∞ is easily verified by means of Lemma 1. Another important ingredient (implicit in the proof of [15, Proposition 4.2]) is a suitable estimate for the growth of the underlying eigenfunctions. In the present setting it is sufficient to know that \|hα n(x)\| . d∏ i=1 Φαi ni (xi), x ∈ Rd +, where Φαi ni (xi) = x −αi−1/2 i { 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 Kα,ε γ (x, y) satisfies standard estimates in the sense of the homogeneous space (Rd +, w + α , ‖ ·‖). The corresponding proof is located in Section 4 below. Denote B+(x, r) = B(x, r) ∩ Rd +. 8 A. Nowak and K. Stempak Theorem 2. Given α ∈ [−1/2,∞)d and ε ∈ Zd 2, the kernel Kα,ε γ (x, y) satisfies the growth condition \|Kα,ε γ (x, y)\| . 1 w+ α (B+(x, ‖y − x‖)) , x, y ∈ Rd +, x 6= y, and the smoothness condition ‖∇x,yK α,ε γ (x, y)‖ . 1 ‖x− y‖ 1 w+ α (B+(x, ‖y − x‖)) , x, y ∈ Rd +, x 6= y. From Theorem 2 and Proposition 1 we conclude that L−iγ α,ε,+ is a Calderón–Zygmund operator. Thus Lemma 2 follows from the general theory, see [4]. Remark 1. The results of this section can be generalized in a straightforward manner by con- sidering weighted Lp spaces. By the general theory, each L−iγ α,ε,+ extends to a bounded operator on Lp(Rd +,Wdw+ α ), W ∈ Aα p , 1 < p < ∞, and to a bounded operator from L1(Rd +,Wdw+ α ) to L1,∞(Rd +,Wdw+ α ), W ∈ Aα 1 ; here Aα p stands for the Muckenhoupt class of Ap weights associated with the space (Rd +, w + α , ‖ · ‖). Consequently, analogous mapping properties hold for L−iγ α , with reflection invariant weights satisfying Aα p conditions when restricted to Rd + (or, equivalently, satisfying Ap conditions related to the whole space (Rd, wα, ‖ · ‖)). Remark 2. With the particular ε0 = (0, . . . , 0) the operator L−iγ α,ε0,+ coincides, up to a constant factor, with the same imaginary power of the Laguerre Laplacian investigated in [8]. Therefore the results of this section deliver also analogous results in the setting of [8]. 4 Kernel estimates This section is mainly devoted to the proof of the standard estimates stated in Theorem 2. The proof follows the pattern of the proof of Proposition 3.1 in [8], see also [9]. We use the formula Gα,ε t (x, y) = 1 2d (1− ζ2 2ζ )d+\|α\|+\|ε\| (xy)ε ∫ [−1,1]d exp ( − 1 4ζ q+(x, y, s)− ζ 4 q−(x, y, s) ) Πα+ε(ds), where q±(x, y, s) = ‖x‖2 + ‖y‖2 ± 2 d∑ i=1 xiyisi (for the sake of brevity we shall often write shortly q+ or q− omitting the arguments) and t ∈ (0,∞) and ζ ∈ (0, 1) are related by ζ = tanh t, so that t = t(ζ) = 1 2 log 1 + ζ 1− ζ ; (7) eventually, Πα denotes the product measure d⊗ i=1 Παi , where Παi is determined by the density Παi(ds) = (1− s2)αi−1/2ds√ π2αiΓ(αi + 1/2) , s ∈ (−1, 1), when αi > −1/2, and in the limiting case of αi = −1/2, Π−1/2 = 1√ 2π ( η−1 + η1 ) (η−1 and η1 denote point masses at −1 and 1, respectively). Imaginary Powers of the Dunkl Harmonic Oscillator 9 By the change of variable (7) the kernels (4) can be expressed as Kα,ε γ (x, y) = ∫ [−1,1]d Πα+ε(ds) ∫ 1 0 βd,α+ε(ζ)ψε ζ(x, y, s) dζ, (8) where ψε ζ(x, y, s) = (xy)ε exp ( − 1 4ζ q+(x, y, s)− ζ 4 q−(x, y, s) ) and βd,α(ζ) = 21−d−iγ Γ(iγ) ( 1− ζ2 2ζ )d+\|α\| 1 1− ζ2 ( log 1 + ζ 1− ζ )iγ−1 . Notice that \|βd,α(ζ)\| coincides, up to a constant factor, with β0 d,α(ζ) defined in [8, (5.4)]. The application of Fubini’s theorem that was necessary to get (8) is also justified since, in fact, the proof (to be given below) of the first estimate in Theorem 2 contains the proof of∫ [−1,1]d Πα+ε(ds) ∫ 1 0 ∣∣βd,α+ε(ζ)ψε ζ(x, y, s) ∣∣ dζ <∞, x 6= y. For proving Theorem 2 we need a specified version of [8, Corollary 5.2] and a slight extension of [8, Lemma 5.5 (b)] (the proof of the latter result in [8] is given under assumption k ≥ 1, but in fact it is also valid for any real k provided that the constant factor in the definition of βk d,α(ζ) is neglected; in particular, k = 0 can be admitted). Lemma 3. Assume that α ∈ [−1/2,∞)d. Let b ≥ 0 and c > 0 be fixed. Then, we have (a) ( \|x1 ± y1s1\|+ \|y1 ± x1s1\| ) exp ( −c1 ζ q±(x, y, s) ) . ζ±1/2, (b) ∫ 1 0 ∣∣βd,α(ζ) ∣∣ζ−b exp ( −c1 ζ q+(x, y, s) ) dζ . ( q+(x, y, s) )−d−\|α\|−b , uniformly in x, y ∈ Rd +, s ∈ [−1, 1]d, and also in ζ ∈ (0, 1) if (a) is considered. We also need the following generalization of [8, Proposition 5.9], cf. [9, Lemma 5.3]. Lemma 4. Assume that α ∈ [−1/2,∞)d and let δ, κ ∈ [0,∞)d be fixed. Then for x, y ∈ Rd +, x 6= y, (x+ y)2δ ∫ [−1,1]d Πα+δ+κ(ds) ( q+(x, y, s) )−d−\|α\|−\|δ\| . 1 w+ α (B+(x, ‖y − x‖)) and (x+ y)2δ ∫ [−1,1]d Πα+δ+κ(ds) ( q+(x, y, s) )−d−\|α\|−\|δ\|−1/2 . 1 ‖x− y‖ 1 w+ α (B+(x, ‖y − x‖)) . Proof of Theorem 2. The growth estimate is rather straightforward. Using Lemma 3 (b) with b = 0 and observing that (xy)ε ≤ (x+ y)2ε gives \|Kα,ε γ (x, y)\| . (x+ y)2ε ∫ [−1,1]d Πα+ε(ds)(q+)−d−\|α\|−\|ε\|. Now Lemma 4, taken with δ = ε and κ = (0, . . . , 0), provides the desired bound. 10 A. Nowak and K. Stempak It remains to prove the smoothness estimate. Notice that by symmetry reasons it is enough to show that \|∂x1K α,ε γ (x, y)\|+ \|∂y1K α,ε γ (x, y)\| . 1 ‖x− y‖ 1 w+ α (B+(x, ‖y − x‖)) , x, y ∈ Rd +, x 6= y. Moreover, we can focus on estimating the x1-derivative only. This is because in the final stroke we shall use Lemma 4, where the left-hand sides are symmetric in x and y. Thus we are reduced to estimating the quantity J = ∫ [−1,1]d Πα+ε(ds) ∫ 1 0 ∣∣βd,α+ε(ζ)∂x1ψ ε ζ(x, y, s) ∣∣ dζ (passing with ∂x1 under the integral signs is legitimate, the justification being implicitly con- tained in the estimates below, see the argument in [8, pp. 671–672]). An elementary computation produces ∂x1ψ ε ζ(x, y, s) = [ (xy)ε ( − 1 2ζ (x1 + y1s1)− ζ 2 (x1 − y1s1) ) + ε1y1(xy)ε−e1 ] × exp ( − 1 4ζ q+ − ζ 4 q− ) . Hence, by Lemma 3 (a), we have∣∣∂x1ψ ε ζ(x, y, s) ∣∣ . (xy)ε(ζ−1/2 + ζ1/2) exp ( − 1 8ζ q+ − ζ 8 q− ) + ε1y1(xy)ε−e1 exp ( − 1 4ζ q+ − ζ 4 q− ) . (x+ y)2εζ−1/2 exp ( − 1 8ζ q+ ) + ε1(x+ y)2(ε−e1/2) exp ( − 1 4ζ q+ ) (notice that the second term above vanishes when ε1 = 0). Consequently, J . (x+ y)2ε ∫ [−1,1]d Πα+ε(ds) ∫ 1 0 ∣∣βd,α+ε(ζ) ∣∣ζ−1/2 exp ( − 1 8ζ q+ ) dζ + ε1(x+ y)2(ε−e1/2) ∫ [−1,1]d Πα+ε(ds) ∫ 1 0 ∣∣βd,α+ε(ζ) ∣∣ exp ( − 1 4ζ q+ ) dζ. Now, applying Lemma 3 (b) with either b = 1/2 or b = 0 leads to J . (x+ y)2ε ∫ [−1,1]d Πα+ε(ds)(q+)−d−\|α\|−\|ε\|−1/2 + ε1(x+ y)2(ε−e1/2) ∫ [−1,1]d Πα+ε(ds)(q+)−d−\|α\|−\|ε\|. Finally, Lemma 4 with either δ = ε and κ = (0, . . . , 0) or (in case ε1 = 1) δ = ε − e1/2 and κ = e1/2 delivers the required smoothness bound for J . The proof of Theorem 2 is complete. � Proof of Lemma 1. Recall that Gα,ε t (x, y) = 1 2d ( 1− ζ2 2ζ )d+\|α\|+\|ε\| (xy)ε ∫ [−1,1]d Πα+ε(ds) exp ( − 1 4ζ q+ − ζ 4 q− ) , Imaginary Powers of the Dunkl Harmonic Oscillator 11 where t and ζ are related by ζ = tanh t. Since ζ ∈ (0, 1) and ‖x− y‖2 ≤ q± ≤ ‖x+ y‖2, we see that Gα,ε t (x, y) . ( 1− ζ ζ )d+\|α\|+\|ε\| (xy)ε exp ( − 1 4ζ ‖x− y‖2 ) . From this estimate the rapid decay easily follows. To verify (2) we need first to compute ∂tG α,ε t (x, y). We get ∂tG α,ε t (x, y) = ( d+ \|α\|+ \|ε\| )1 + ζ2 ζ Gα,ε t (x, y) + 1 2d ( 1− ζ2 2ζ )d+\|α\|+\|ε\| (1− ζ2)(xy)ε × ∫ [−1,1]d Πα+ε(ds) ( 1 4ζ2 q+ − 1 4 q− ) exp ( − 1 4ζ q+ − ζ 4 q− ) (here passing with ∂t under the integral can be easily justified). Consequently, taking into account the estimates above, ∣∣∂tG α,ε t (x, y) ∣∣ . 1 ζ ( 1− ζ ζ )d+\|α\|+\|ε\| (xy)ε exp ( − 1 4ζ ‖x− y‖2 ) + (1− ζ) ( 1− ζ ζ )d+\|α\|+\|ε\| (xy)ε ‖x+ y‖2 ζ2 exp ( − 1 4ζ ‖x− y‖2 ) . This implies∫ ∞ 0 ∣∣∂tG α,ε t (x, y) ∣∣ dt = ∫ 1 0 ∣∣∣∂tG α,ε t (x, y) ∣∣ t=tanh−1 ζ ∣∣∣ dζ 1− ζ2 . (xy)ε ∫ 1 0 (1− ζ)d+\|α\|+\|ε\|−1ζ−(d+\|α\|+\|ε\|−1) exp ( − 1 4ζ ‖x− y‖2 ) dζ + (xy)ε‖x+ y‖2 ∫ 1 0 ζ−(d+\|α\|+\|ε\|−2) exp ( − 1 4ζ ‖x− y‖2 ) dζ. Now using the fact that d+ \|α\|+ \|ε\| > 0 and supu>0 u a exp(−Au) <∞ for any fixed A > 0 and a ≥ 0, leads to the bound∫ ∞ 0 ∣∣∂tG α,ε t (x, y) ∣∣ dt . (xy)ε ‖x− y‖2(d+\|α\|+\|ε\|+1) + (xy)ε‖x+ y‖2 ‖x− y‖2(d+\|α\|+\|ε\|+2) . The conclusion follows. � References [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. 12 A. Nowak and K. Stempak [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 oscil- lator, Acta Math. Hungar. 111 (2006), 43–64. http://arxiv.org/abs/0802.0474 http://arxiv.org/abs/q-alg/9703006 http://arxiv.org/abs/math.CA/0210366 1 Introduction 2 Preliminaries 3 Main result 4 Kernel estimates References

Imaginary Powers of the Dunkl Harmonic Oscillator

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