The Clifford Deformation of the Hermite Semigroup

This paper is a continuation of the paper [De Bie H., Ørsted B., Somberg P., Souček V., Trans. Amer. Math. Soc. 364 (2012), 3875–3902], investigating a natural radial deformation of the Fourier transform in the setting of Clifford analysis. At the same time, it gives extensions of many results obtai...

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Datum:	2013
Hauptverfasser:	De Bie, H., Ørsted, B., Somberg, P., Souček, V.
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spelling	irk-123456789-1491982019-02-20T01:28:25Z The Clifford Deformation of the Hermite Semigroup De Bie, H. Ørsted, B. Somberg, P. Souček, V. This paper is a continuation of the paper [De Bie H., Ørsted B., Somberg P., Souček V., Trans. Amer. Math. Soc. 364 (2012), 3875–3902], investigating a natural radial deformation of the Fourier transform in the setting of Clifford analysis. At the same time, it gives extensions of many results obtained in [Ben Saïd S., Kobayashi T., Ørsted B., Compos. Math. 148 (2012), 1265–1336]. We establish the analogues of Bochner's formula and the Heisenberg uncertainty relation in the framework of the (holomorphic) Hermite semigroup, and also give a detailed analytic treatment of the series expansion of the associated integral transform. 2013 Article The Clifford Deformation of the Hermite Semigroup / H. De Bie, B. Ørsted, P. Somberg, V. Souček // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 29 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33C52; 30G35; 43A32 DOI: http://dx.doi.org/10.3842/SIGMA.2013.010 http://dspace.nbuv.gov.ua/handle/123456789/149198 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	This paper is a continuation of the paper [De Bie H., Ørsted B., Somberg P., Souček V., Trans. Amer. Math. Soc. 364 (2012), 3875–3902], investigating a natural radial deformation of the Fourier transform in the setting of Clifford analysis. At the same time, it gives extensions of many results obtained in [Ben Saïd S., Kobayashi T., Ørsted B., Compos. Math. 148 (2012), 1265–1336]. We establish the analogues of Bochner's formula and the Heisenberg uncertainty relation in the framework of the (holomorphic) Hermite semigroup, and also give a detailed analytic treatment of the series expansion of the associated integral transform.
format	Article
author	De Bie, H. Ørsted, B. Somberg, P. Souček, V.
spellingShingle	De Bie, H. Ørsted, B. Somberg, P. Souček, V. The Clifford Deformation of the Hermite Semigroup Symmetry, Integrability and Geometry: Methods and Applications
author_facet	De Bie, H. Ørsted, B. Somberg, P. Souček, V.
author_sort	De Bie, H.
title	The Clifford Deformation of the Hermite Semigroup
title_short	The Clifford Deformation of the Hermite Semigroup
title_full	The Clifford Deformation of the Hermite Semigroup
title_fullStr	The Clifford Deformation of the Hermite Semigroup
title_full_unstemmed	The Clifford Deformation of the Hermite Semigroup
title_sort	clifford deformation of the hermite semigroup
publisher	Інститут математики НАН України
publishDate	2013
url	http://dspace.nbuv.gov.ua/handle/123456789/149198
citation_txt	The Clifford Deformation of the Hermite Semigroup / H. De Bie, B. Ørsted, P. Somberg, V. Souček // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 29 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 9 (2013), 010, 22 pages The Clifford Deformation of the Hermite Semigroup Hendrik DE BIE †, Bent ØRSTED ‡, Petr SOMBERG § and Vladimir SOUČEK § † Department of Mathematical Analysis, Ghent University, Galglaan 2, 9000 Gent, Belgium E-mail: Hendrik.DeBie@UGent.be ‡ Department of Mathematical Sciences, University of Aarhus, Building 530, Ny Munkegade, DK 8000, Aarhus C, Denmark E-mail: orsted@imf.au.dk § Mathematical Institute of Charles University, Sokolovská 83, 186 75 Praha, Czech Republic E-mail: somberg@karlin.mff.cuni.cz, soucek@karlin.mff.cuni.cz Received September 21, 2012, in final form January 29, 2013; Published online February 05, 2013 http://dx.doi.org/10.3842/SIGMA.2013.010 Abstract. This paper is a continuation of the paper [De Bie H., Ørsted B., Somberg P., Souček V., Trans. Amer. Math. Soc. 364 (2012), 3875–3902], investigating a natural radial deformation of the Fourier transform in the setting of Clifford analysis. At the same time, it gives extensions of many results obtained in [Ben Säıd S., Kobayashi T., Ørsted B., Compos. Math. 148 (2012), 1265–1336]. We establish the analogues of Bochner’s formula and the Heisenberg uncertainty relation in the framework of the (holomorphic) Hermite semigroup, and also give a detailed analytic treatment of the series expansion of the associated integral transform. Key words: Dunkl operators; Clifford analysis; generalized Fourier transform; Laguerre polynomials; Kelvin transform; holomorphic semigroup 2010 Mathematics Subject Classification: 33C52; 30G35; 43A32 1 Introduction It is well-known that the classical Dirac operator and its Fourier symbol generate via Clifford multiplication a natural Lie superalgebra osp(1\|2) contained in the Clifford–Weyl algebra. More surprisingly, this carries over to a natural family of deformations of the Dirac operator, see [7]. Moreover, it is possible to define a Fourier transform naturally associated to the deformed family. The novelty of the present article is that we let group theory be the guiding principle in defining operators and transformations, in the next step followed by a study of explicit (analytic) properties for naturally arising eigenfunctions and kernel functions. Thus the main aim is to find the kernel function for the Fourier transform connected with our deformation, and also to study its associated holomorphic semigroup regarded as a particular descendant of the Gelfand–Gindikin program analyzing representations of reductive Lie groups, see, e.g., [22] and the discussion in [2]. Let us now recall the basic setup and results from [7] and also discuss further aspects of our construction. The deformation family of Dunkl–Dirac operators D = r1− a 2Dκ + br− a 2 −1x+ cr− a 2 −1xE, a, b, c ∈ R, together with the radial deformation of the coordinate function xa = r a 2 −1x, r = √√√√ m∑ i=1 x2i , mailto:Hendrik.DeBie@UGent.be mailto:orsted@imf.au.dk mailto:somberg@karlin.mff.cuni.cz mailto:soucek@karlin.mff.cuni.cz http://dx.doi.org/10.3842/SIGMA.2013.010 2 H. De Bie, B. Ørsted, P. Somberg and V. Souček forms a realization of osp(1\|2) in the Clifford–Weyl algebra. HereDκ = m∑ i=1 eiTi with Ti the Dunkl operators, x = m∑ i=1 eixi and E = m∑ i=1 xi∂xi . The ei are generators of the Clifford algebra Clm. See also the next section for more details. We will show in Proposition 3.2 that this realization builds a Howe dual pair with G̃. Here the group G̃ is the double cover (contained in the Pin group) of the finite reflection group G used in the construction of the Dunkl operators. The Fourier transform is then defined by FD = e iπ 2 ( 1 2 + µ−1 a(1+c) ) e −iπ 2a(1+c)2 (D2−(1+c)2x2a), where L = D2 − (1 + c)2x2a is the generalized Hamiltonian and µ the Dunkl dimension. The main aim of the present paper is to find an integral expression for this Fourier transform, FD(f)(y) = ∫ Rm K(x, y)f(x)h(rx)dx with h(rx)dx the measure associated to D and K(x, y) the integral kernel to be determined. Note that this ties in with recent work on generalized Fourier transforms in different contexts, e.g., analysis on minimal representations of reductive groups (see [19, 20, 21]) or integral transforms in Clifford analysis (see [6, 8]). The deformation of the classical Hamiltonian for the harmonic oscillator is visualized in the following figure: ∆κ − \|x\|2 ∆− \|x\|2 Dunkl deformation OO Clifford deformation {{ a-deformation "" D2 + (1 + c)2\|x\|a \|x\|2−a∆− \|x\|a The Dunkl deformation is by now quite standard and described for example in [11]. The a- deformation is the subject of the paper [2] and is a scalar radial deformation of the harmonic oscillator. Our Clifford deformation is also a radial deformation but richer in the sense that Clifford algebra- (or spinor)-valued functions are involved. In this paper we will thus find a series representation of the kernel function for our new Fourier transform FD, and also study the holomorphic semigroup with generator L. The main results are Theorem 6.1 on the operator properties of the semigroup, Theorem 7.2 on the Fourier transform intertwining the Dirac operator and the Clifford multiplication, Proposition 7.2 on the Bochner identities, and Proposition 7.3 on the Heisenberg uncertainty relation. Finally in Theorem 7.3 we give the analogue of what is sometimes called the “Master formula” in the context of Dunkl operators (see, e.g., [26, Lemma 4.5(1)] or [4]). The paper is organized as follows. In Section 2 we repeat basic notions on Clifford algebras and Dunkl operators needed in the rest of the paper. In Section 3 we construct intertwining operators to reduce our radially deformed Dirac operator to its simplest form. Subsequently, The Clifford Deformation of the Hermite Semigroup 3 in Section 4 we discuss the representation theoretic content of our deformation and solve the spectral problem of the associated Hamiltonian. In Section 5, we obtain the reproducing kernels for spaces of spherical monogenics, which allows us to construct the kernel of the holomorphic semigroup in Section 6. Section 7 contains the results on the (deformed) Fourier transform. Further properties are collected in Section 8. Finally, we summarize some results on special functions used in the paper in Appendix A and give a list of notations in Appendix B. 2 Preliminaries In this section we collect some basic results on Clifford algebras and Dunkl operators. 2.1 Clifford algebras Let V be a vector space of dimension m with a given negative definite quadratic form and let Clm be the corresponding Clifford algebra. If {ei} is an orthonormal basis of V, then Clm is generated by ei, i = 1, . . . ,m, with the relations eiej + eiej = 0, i 6= j, e2i = −1. The algebra Clm has dimension 2m as a vector space over R. It can be decomposed as Clm = ⊕mk=0Clkm with Clkm the space of k-vectors defined by Clkm := span{ei1 · · · eik , i1 < · · · < ik}. The projection on the space of k-vectors is denoted by [·]k. The operator .̄ is the main anti-involution on the Clifford algebra Clm defined by ab = ba, ei = −ei, i = 1, . . . ,m. Similarly we have the automorphism ε given by ε(ab) = ε(a)ε(b), ε(ei) = −ei, i = 1, . . . ,m. In the sequel, we will always consider functions f taking values in Clm, unless explicitly mentioned. Such functions can be decomposed as f(x) = f0(x) + m∑ i=1 eifi(x) + ∑ i<j eiejfij(x) + · · ·+ e1 · · · emf1...m(x) with f0, fi, fij , . . . , f1...m all real-valued functions. Several important groups can be embedded in the Clifford algebra. Note that the space of 1-vectors in Clm is canonically isomorphic to V ∼= Rm. Hence we can define Pin(m) = { s1s2 · · · sn \|n ∈ N, si ∈ Cl1m such that s2i = −1 } , i.e., the Pin group is the group of products of unit vectors in Clm. This group is a double cover of the orthogonal group O(m) with covering map p : Pin(m) → O(m), which we will describe explicitly in the next section. Similarly we define Spin(m) = { s1s2 · · · s2n \|n ∈ N, si ∈ Cl1m such that s2i = −1 } , i.e., the Spin group is the group of even products of unit vectors in Clm. This group is a double cover of SO(m). For more information about Clifford algebras and analysis, we refer the reader to [9, 16]. 4 H. De Bie, B. Ørsted, P. Somberg and V. Souček 2.2 Dunkl operators Denote by 〈·, ·〉 the standard Euclidean scalar product in Rm and by \|x\| = 〈x, x〉1/2 the associated norm. For α ∈ Rm \ {0}, the reflection rα in the hyperplane orthogonal to α is given by rα(x) = x− 2 〈α, x〉 \|α\|2 α, x ∈ Rm. A root system is a finite subset R ⊂ Rm of non-zero vectors such that, for every α ∈ R, the associated reflection rα preserves R. We will assume that R is reduced, i.e. R ∩ Rα = {±α} for all α ∈ R. Each root system can be written as a disjoint union R = R+ ∪ (−R+), where R+ and −R+ are separated by a hyperplane through the origin. R+ is called a positive subsystem of the root system R. The subgroup G ⊂ O(m) generated by the reflections {rα\|α ∈ R} is called the finite reflection group associated with R. We will also assume that R is normalized such that 〈α, α〉 = 2 for all α ∈ R. For more information on finite reflection groups we refer the reader to [18]. If we identify α with a 1-vector in Clm (and hence α/ √ 2 with an element in Pin(m)), we can rewrite the reflection rα as rα(x) = 1 2 αxα with x = m∑ i=1 eixi. Generalizing this map gives us the covering map p from Pin(m) to O(m) as p(s)(x) = ε(s)xs−1, s ∈ Pin(m). In particular, we obtain a double cover of the reflection group G as G̃ = p−1(G) (see also the discussion in [1]). A multiplicity function κ on the root system R is a G-invariant function κ : R → C, i.e. κ(α) = κ(hα) for all h ∈ G. We will denote κ(α) by κα. We will always assume that the multiplicity function is real and satisfies κ ≥ 0. This assumption is, e.g., necessary to obtain the subsequent formula (2.1), which is crucial for the sequel. Fixing a positive subsystem R+ of the root system R and a multiplicity function κ, we introduce the Dunkl operators Ti associated to R+ and κ by (see [10, 13]) Tif(x) = ∂xif(x) + ∑ α∈R+ κααi f(x)− f(rα(x)) 〈α, x〉 , f ∈ C1(Rm). An important property of the Dunkl operators is that they commute, i.e. TiTj = TjTi. The Dunkl Laplacian is given by ∆κ = m∑ i=1 T 2 i , or more explicitly by ∆κf(x) = ∆f(x) + 2 ∑ α∈R+ κα ( 〈∇f(x), α〉〈α, x〉 − f(x)− f(rα(x)) 〈α, x〉2 ) with ∆ the classical Laplacian and ∇ the gradient operator. We also define the constant µ = 1 2 ∆κ\|x\|2 = m+ 2 ∑ α∈R+ κα, called the Dunkl-dimension. The Clifford Deformation of the Hermite Semigroup 5 It is possible to construct an intertwining operator Vκ connecting the classical derivatives ∂xj with the Dunkl operators Tj such that TjVκ = Vκ∂xj (see, e.g., [12]). Note that explicit formulae for Vκ are only known in a few special cases. The weight function related to the root system R and the multiplicity function κ is given by wκ(x) = ∏ α∈R+ \|〈α, x〉\|2κα . For suitably chosen functions f and g one then has the following property of integration by parts (see [11])∫ Rm (Tif)gwκ(x)dx = − ∫ Rm f (Tig)wκ(x)dx. (2.1) For more information about Dunkl operators we refer the reader to [13, 25]. The starting point in the subsequent analysis is the Dunkl–Dirac operator, given by Dκ = m∑ i=1 eiTi. Together with the vector variable x = m∑ i=1 eixi this Dunkl–Dirac operator generates a copy of osp(1\|2), see [23] or the subsequent Theorem 3.1. In particular, we have D2 κ = −∆κ and x2 = −\|x\|2 = −r2 = − m∑ i=1 x2i . 3 Intertwining operators Let, for a, b ∈ R, P and Q be two operators defined by Pf(x) = rbf ((a 2 ) 1 a xr 2 a −1 ) , Qf(x) = r− ab 2 f (( 2 a ) 1 2 xr a 2 −1 ) . These two operators act as generalized Kelvin transformations. Indeed, one can easily compute their composition QP = PQ = ( 2 a ) b 2 . We will show that these operators allow to reduce the Dirac operator D to a simpler form. We have the following proposition, where E = m∑ i=1 xi∂xi denotes the Euler operator. Recall also from the introduction that xa = r a 2 −1x. Proposition 3.1. One has the following intertwining relations (a 2 ) b−1 2 Q ( Dκ + br−2x+ cr−2xE ) P = r1− a 2Dk + βr− a 2 −1x+ γr− a 2 −1xE,(a 2 ) b+1 2 QxP = xa with β = 2b+ bc, γ = 2 a(1 + c)− 1. 6 H. De Bie, B. Ørsted, P. Somberg and V. Souček Proof. In [7, Proposition 3], we already proved that(a 2 ) b−1 2 Q (Dκ)P = r1− a 2Dk + br− a 2 −1x+ ( 2 a − 1 ) r− a 2 −1xE = xa. Similarly we obtain(a 2 ) b−1 2 Q ( r−2x ) P = r− a 2 −1x and (a 2 ) b−1 2 Q ( r−2xE ) P = br− a 2 −1x+ ( 2 a ) r− a 2 −1xE. This completes the proof of the proposition. � So we are reduced to the study of the operator D = Dκ + br−2x+ cr−2xE, where b, c ∈ R, c 6= −1. Here, the term br−2x can also be removed. Indeed, we have r−α ( Dκ + br−2x+ cr−2xE ) rα = Dκ + cr−2xE, when α = −b/(1 + c). As a result of the previous discussion, we see that it is sufficient to study the function theory for the operator D = Dκ + cr−2xE, where we have put a = 2, b = 0. Furthermore, we will restrict ourselves to the case c > −1 for reasons that will become clear in Proposition 3.3. Similarly, we no longer need to consider xa but can restrict ourselves to x. Now we repeat the basic facts concerning this operator we need in the sequel. All the results are taken from [7], putting a = 2, b = 0. Theorem 3.1. The operators D and x generate a Lie superalgebra, isomorphic to osp(1\|2), with the following relations {x,D} = −2(1 + c) ( E + δ 2 ) , [ E + δ 2 ,D ] = −D, [ x2,D ] = 2(1 + c)x, [ E + δ 2 , x ] = x, [ D2, x ] = −2(1 + c)D, [ E + δ 2 ,D2 ] = −2D2, [ D2, x2 ] = 4(1 + c)2 ( E + δ 2 ) , [ E + δ 2 , x2 ] = 2x2, where δ = 1 + µ−1 1+c . Note that the square of D is a complicated operator, given by D2 = −∆κ − (cµ) r−1∂r − ( c2 + 2c ) ∂2r + cr−2 ∑ i xiTi − cr−2 ∑ i<j eiej(xiTj − xjTi). If κ = 0, the formula for D2 simplifies a bit as now ∑ i xiTi = r∂r = E. The Clifford Deformation of the Hermite Semigroup 7 Remark 3.1. The operator D = Dκ+cr−2xE is also considered from a very different perspective in [3] (in the case κ = 0), where the eigenfunctions of this operator are studied. Let us now discuss the symmetry of the generators of osp(1\|2). First we define the action of the Pin group on C∞(Rm)⊗ Clm for s ∈ Pin(m) as ρ(s) : C∞(Rm)⊗ Clm → C∞(Rm)⊗ Clm, f ⊗ b→ f(p ( s−1 ) x)⊗ sb. We then have Proposition 3.2. Let s ∈ G̃ and define sgn(s) := sgn(p(s)). Then one has ρ(s)x = sgn(s)xρ(s), ρ(s)D = sgn(s)Dρ(s). Proof. This follows immediately from the definition of ρ and the G-equivariance of the Dunkl operators. � So up to sign, the Dirac operator D is G̃-equivariant. At this point it is interesting to remark that an algebraic analog of the Dunkl–Dirac operator D for graded affine Hecke algebras is intro- duced in [1] with the motivation to prove a version of Vogan’s Conjecture for Dirac cohomology. The formulation is based on a uniform geometric parametrization of spin representations of Weyl groups. This Dirac operator is an algebraic variant of our family deformation of the differential Dirac operator for special values of the deformation parameters. Moreover, it satisfies the same symmetry as in Proposition 3.2, see [1, Lemma 3.4]. There is a measure naturally associated with D. Indeed, one has Proposition 3.3. If c > −1, then for suitable differentiable functions f and g one has∫ Rm (Df)gh(r)wκ(x)dx = ∫ Rm f(Dg)h(r)wκ(x)dx with h(r) = r1− 1+µc 1+c , provided the integrals exist. In this proposition, ·̄ is the main anti-involution on the Clifford algebra Clm. 4 Representation space for the deformation family of the Dunkl–Dirac operator The function space we will work with is L2κ,c(Rm) = L2(Rm, h(r)wκ(x)dx) ⊗ Clm. This space has the following decomposition L2κ,c(Rm) = L2 ( R+, r µ−1 1+c dr ) ⊗ L2(Sm−1, wκ(ξ)dσ(ξ))⊗ Clm, where on the right-hand side the topological completion of the tensor product is understood and with dσ(ξ) the Lebesgue measure on the sphere Sm−1. The space L2(Sm−1, wκ(ξ)dσ(ξ)) ⊗ Clm can be further decomposed into Dunkl harmonics and subsequently into Dunkl monogenics. This leads to L2 ( Sm−1, wκ(ξ)dσ(ξ) ) ⊗ Clm = ∞⊕ `=0 (M` ⊕ xM`) ∣∣ Sm−1 , whereM` = kerDκ∩(P` ⊗ Clm) is the space of Dunkl monogenics of degree `, with P` the space of homogeneous polynomials of degree ` (see also [5] for more details on Dunkl monogenics). 8 H. De Bie, B. Ørsted, P. Somberg and V. Souček Using this decomposition, we have obtained in [7] a basis for L2κ,c(Rm). This basis is given by the set {φt,`,m} (t, ` ∈ N and m = 1, . . . ,dimM`), defined as φ2t,`,m = 22t(1 + c)2tt!L γ` 2 −1 t (r2)rβ`M (m) ` e−r 2/2, φ2t+1,`,m = −22t+1(1 + c)2t+1t!L γ` 2 t (r2)xrβ`M (m) ` e−r 2/2 with Lβα the Laguerre polynomials and β` = − c 1 + c `, γ` = 2 1 + c ( `+ µ− 2 2 ) + c+ 2 1 + c , and where M (m) ` (m = 1, . . . ,dimM`) forms an orthonormal basis of M`, i.e.[∫ Sm−1 M (m1) ` (ξ)M (m2) ` (ξ)wκ(ξ)dσ(ξ) ] 0 = δm1m2 with [·]0 the projection on the scalar part of the Clifford algebra. The dimension ofM` is given by dimRM` = dimR Clm dimR P` ( Rm−1 ) = 2m (`+m− 2)! `!(m− 2)! with P` ( Rm−1 ) the space of homogeneous polynomials of degree ` in m− 1 variables (see [9]). Using formula (4.10) in [7] and the proof of Theorem 3 in [7], one obtains the following formulae for the action of D and x on the generalized Laguerre functions 2Dφt,`,m= φt+1,`,m+ C(t, `)φt−1,`,m, −2(1 + c)xφt,`,m= φt+1,`,m− C(t, `)φt−1,`,m (4.1) with C(2t, `) = 4(1 + c)2t, C(2t+ 1, `) = 2(1 + c)2(γ` + 2t). These formulae determine the action of osp(1\|2) on L2κ,c(Rm). Recall also that the action of G̃ on L2κ,c(Rm) is given by ρ (see Section 3). Subsequently, we can define a creation and annihilation operator in this setting by A+ = D− (1 + c)x, A− = D + (1 + c)x (4.2) satisfying A+φt,`,m = φt+1,`,m, A−φt,`,m = C(t, `)φt−1,`,m. Now we introduce the following inner product 〈f, g〉 = [∫ Rm f cgh(r)wκ(x)dx ] 0 , where h(r) is the measure associated to D (see Proposition 3.3) and f c is the complex conjugate of f . It is easy to check that this inner product satisfies 〈Df, g〉 = 〈f,Dg〉, 〈xf, g〉 = −〈f, xg〉. (4.3) The related norm is defined by \|\|f \|\|2 = 〈f, f〉. The Clifford Deformation of the Hermite Semigroup 9 Theorem 4.1. We have 〈φt1,`1,m1 , φt2,`2,m2〉 = c(t1, `1)δt1t2δ`1`2δm1m2 , where c(t, `) is a constant depending on t and `. The functions φt,`,m are eigenfunctions of the Hamiltonian of a generalized harmonic oscilla- tor. Theorem 4.2. The functions φt,`,m satisfy the following second-order PDE( D2 − (1 + c)2x2 ) φt,`,m = (1 + c)2(γ` + 2t)φt,`,m. Proof. This follows immediately from the formula (4.1). � Theorem 4.2 combined with the definition of A+, A− in (4.2) allows us to decompose the space L2κ,c(Rm) under the action of osp(1\|2). Clearly the odd elements A+ and A− generate osp(1\|2) as they are linear combinations of D and x. Moreover, they act between two basis vectors {φt,`,m} of L2κ,c(Rm), so it is sufficient to consider vectors in an irreducible representation of osp(1\|2) inside the functional space. This is achieved as follows – for fixed ` and m each vector φ0,`,m generates the irreducible representation φ0,`,m A+ // L UU φ1,`,m L UU A+ // A− oo φ2,`,m L UU A+ // A− oo φ3,`,m L UU A+ // A− oo φ4,`,m L UU A+ // A− oo . . . A− oo where L = 1 2 {A+, A−} = D2 − (1 + c)2x2 with the action given in Theorem 4.2. In fact this highest weight representation is labeled by ` only and we will denote it π(`). In conclusion, we obtain the decomposition of our func- tional space L2κ,c(Rm) into a discrete direct sum of highest weight (infinite-dimensional) Harish- Chandra modules for osp(1\|2): L2κ,c(Rm) = ∞⊕ `=0 π(`)⊗M`. These results should be compared with Theorem 3.19 and Section 3.6 in [2] (where one uses sl2 instead of osp(1\|2)). Also notice that the claim should be understood as an assertion on the deformation of the Howe dual pair for osp(1\|2) inside the Clifford–Weyl algebra on Rm acting on a fixed vector space L2κ,c(Rm). In particular, we have the following result. Recall that an operator T is essentially selfadjoint on a Hilbert space H if T is a symmetric operator with a dense domain D(T ) ⊂ H such that for a complete orthogonal set {fn}n in H with fn ∈ D(H), there exist {µn}n solving Tfn = µnf for all n ∈ N. Proposition 4.1. Let c > −1 and κ > 0. The operator L acting on L2κ,c(Rm) is essentially self- adjoint (i.e. symmetric and its closure is a selfadjoint operator). Moreover, L has no continuous spectrum and its discrete spectrum is given by Spec(L) = {2(1 + c)`+ 2(1 + c)2t+ (1 + c)(µ+ c) \| `, t ∈ N}. 10 H. De Bie, B. Ørsted, P. Somberg and V. Souček Using Theorem 4.2 we can now define the holomorphic semigroup for the deformed Dirac operator by FωD = e ω ( 1 2 + µ−1 2(1+c) ) e −ω 2(1+c)2 (D2−(1+c)2x2) . Here, ω takes values in the right half-plane of C. The special boundary value ω = iπ/2 corre- sponds to the Fourier transform. In that case, we will use the notation FD. The functions φt,`,m are eigenfunctions of FωD satisfying FωD(φt,`,m) = e−ωte − ω` (1+c)φt,`,m. (4.4) Note that in the special case κ = 0, c = 0 the operator FωD reduces to the classical Hermite semigroup (see, e.g., [17]). Remark 4.1. One can also consider more general deformations of the Dirac operator, by adding suitable odd powers of Γ = −xDκ − E to D as follows D = Dκ + cr−2xE + ∑̀ j=0 cjr −1 ( Γ− µ− 1 2 )2j+1 , cj ∈ R. This does not alter the osp(1\|2) relations, as Γ− µ−1 2 anti-commutes with x and has the correct homogeneity. In particular, Γ − µ−1 2 can be seen as the square root of the Casimir of osp(1\|2), see [15, Example 2 in Section 2.5]. In the sequel of the paper, we will always assume κ = 0 or in other words, we do not consider the Dunkl deformation. This is to simplify the notation of the results. Most statements can be generalized to the Dunkl case by a suitable composition with the Dunkl intertwining operator Vκ, except the results obtained in Section 8. Recall that for κ = 0, the Dunkl–Dirac operator Dκ reduces to the orthogonal Dirac operator ∂x = m∑ i=1 ei∂xi and the Dunkl dimension µ to the ordinary dimension m. 5 Reproducing kernels In this section we determine the reproducing kernels for Mk and xMk. We start with an auxiliary Lemma, which can be thought of as a Clifford analogue of the Funk–Hecke transform. We define the wedge product of two vectors as x ∧ y := ∑ j<k ejek(xjyk − xkyj). Lemma 5.1. Put x = rx′ and y = sy′ with x′, y′ ∈ Sm−1. Furthermore, put λ = (m− 2)/2 and σm = 2πm/2/Γ(m/2). Then one has, with Ml ∈M`∫ Sm−1 Cλk (〈x′, y′〉)M`(x ′)dσ(x′) = σm λ λ+ k δk,`M`(y ′),∫ Sm−1 Cλk (〈x′, y′〉)x′M`(x ′)dσ(x′) = σm λ λ+ k δk,`+1y ′M`(y ′),∫ Sm−1 (x′ ∧ y′)Cλ+1 k−1 (〈x′, y′〉)M`(x ′)dσ(x′) = −σm k 2(λ+ k) δk,`M`(y ′),∫ Sm−1 (x′ ∧ y′)Cλ+1 k−1 (〈x′, y′〉)x′M`(x ′)dσ(x′) = σm k + 2λ 2(λ+ k) δk,`+1y ′M`(y ′), where Cλk (〈x′, y′〉) is the k-th Gegenbauer polynomial in the variable 〈x′, y′〉. The Clifford Deformation of the Hermite Semigroup 11 Proof. The first integral is trivial: M` is a spherical harmonic of degree ` and Cλk (〈x′, y′〉) is the reproducing kernel for spherical harmonics of degree k (see, e.g., [13]). The second integral immediately follows, because x′M`(x ′) ∈ H`+1. The other two integrals are a bit more complicated. We show how to obtain the last one. First rewrite (x′ ∧ y′)x′ = y′ − 〈x′, y′〉x′. The first term then follows from the first integral. For the second term, we use the recursive property of Gegenbauer polynomials: wCλ+1 n−1(w) = n 2(n+ λ) Cλ+1 n (w) + n+ 2λ 2(n+ λ) Cλ+1 n−2(w). The result then follows by collecting everything. � We can use this lemma to determine the reproducing kernels. This is the subject of the following proposition. Proposition 5.1. For k ∈ N∗ put Pk(x ′, y′) = k + 2λ 2λ Cλk (〈x′, y′〉)− (x′ ∧ y′)Cλ+1 k−1 (〈x′, y′〉), Qk−1(x ′, y′) = k 2λ Cλk (〈x′, y′〉) + (x′ ∧ y′)Cλ+1 k−1 (〈x′, y′〉) with P0(x ′, y′) = Cλ0 (0) = 1. Then∫ Sm−1 Pk(x ′, y′)M`(x ′)dσ(x′) = σmδk,`M`(y ′),∫ Sm−1 Pk(x ′, y′)x′M`(x ′)dσ(x′) = 0 and ∫ Sm−1 Qk−1(x ′, y′)M`(x ′)dσ(x′) = 0,∫ Sm−1 Qk−1(x ′, y′)x′M`(x ′)dσ(x′) = σmδk,`+1y ′M`(y ′). Proof. This follows immediately from Lemma 5.1. � Remark 5.1. Note that, as expected, Pk(x ′, y′) + Qk−1(x ′, y′) = λ+k λ Cλk (〈x′, y′〉), which is the reproducing kernel for the space of spherical harmonics of degree k. Remark 5.2. When the dimension m = 2 and hence λ = 0, the reproducing kernel is still well-defined by using the well-known relation [27, (4.7.8)] lim λ→0 λ−1Cλk (w) = (2/k) cos kθ, w = cos θ, k ≥ 1. We will also need the following lemma. Lemma 5.2. The reproducing kernels satisfy the following properties, for all k, l ∈ N:∫ Sm−1 Pk(y ′, x′)P`(z ′, y′)dσ(y′) = σmδk`P`(z ′, x′),∫ Sm−1 Pk(y ′, x′)Q`(z ′, y′)dσ(y′) = 0,∫ Sm−1 Qk(y ′, x′)Q`(z ′, y′)dσ(y′) = σmδk`Q`(z ′, x′). Proof. This follows immediately using Lemma 7.6 and 7.10 from [8]. � Remark 5.3. Mind the order of the variables in the previous lemma. The kernels Pk(x ′, y′) and Qk(x ′, y′) are not symmetric. 12 H. De Bie, B. Ørsted, P. Somberg and V. Souček 6 The series representation of the holomorphic semigroup The aim of the present section is to investigate basic properties of the holomorphic semigroup defined by FωD = e ω ( 1 2 + µ−1 2(1+c) ) e −ω 2(1+c)2 (D2−(1+c)2x2) , Reω ≥ 0, acting on the space L20,c(Rm). We start with the following general statement. Theorem 6.1. Suppose c > −1. Then 1. For any t, ` ∈ N and m ∈ {1, . . . ,dimM`}, the function φt,`,m is an eigenfunction of the operator FωD: FωD(φt,`,m) = e−ωte − ω` (1+c)φt,`,m. 2. FωD is a continuous operator on L20,c(Rm) for all ω with Reω ≥ 0, in particular \|\|FωD(f)\|\| ≤ \|\|f \|\| for all f ∈ L20,c(Rm). 3. If Reω > 0, then FωD is a Hilbert–Schmidt operator on L20,c(Rm). 4. If Reω = 0, then FωD is a unitary operator on L20,c(Rm). Proof. (1) is an immediate consequence of Theorem 4.2. For (2), let f be an element in L20,c(Rm) and expand it with respect to the (normalized) basis {φt,`,m} as f = ∑ t,`,m at,`,mφt,`,m. Then one has, using orthogonality, \|\|FωD(f)\|\|2 = ∑ t,`,m \|at,`,m\|2e−2(Reω)te − 2(Reω)` (1+c) ≤ ∑ t,`,m \|at,`,m\|2 = \|\|f \|\|2 because Reω ≥ 0. As for (3), we have to show that the Hilbert–Schmidt norm is finite. We compute \|\|FωD\|\|2HS = ∑ t,k,` \|\|FωD(φt,k,`)\|\|2 = ∑ t,k,` e−2(Reω)te − 2(Reω)k (1+c) = ∑ t,k e−2(Reω)te − 2(Reω)k (1+c) dimRMk = ∑ t,k e−2(Reω)te − 2(Reω)k (1+c) (k +m− 2)! k!(m− 2)! 2m = ∑ t e−2(Reω)t ∑ k e − 2(Reω)k (1+c) (k +m− 2)! k!(m− 2)! 2m. Using the ratio test, we see that these series are convergent for Reω > 0. (4) follows immediately, because when Reω = 0 the eigenvalues all have unit norm. � We have already observed that FωD is a Hilbert–Schmidt operator for Reω > 0 and a unitary operator for Reω = 0. The Schwartz kernel theorem implies that FωD can be expressed by a distribution kernel K(x, y;ω), so( FωDf ) (y) = ∫ Rm K(x, y;ω)f(x)h(rx)dx, and K(x, y;ω)h(rx) is a tempered distribution on Rm × Rm. The Clifford Deformation of the Hermite Semigroup 13 6.1 The case Reω > 0 Using the reproducing kernels of Section 5, we can now make a reasonable ansatz for the kernel of the full holomorphic semigroup. We want to write this semigroup as Fω0,c(f)(y) = σ−1m ∫ Rm K(x, y;ω)f(x)h(rx)dx with K(x, y;ω) = K0(x, y;ω) +K1(x, y;ω) and K0 = e− cothω 2 (r2+s2) +∞∑ k=0 αkz k 1+c J̃ γk 2 −1 ( iz sinhω ) Pk(x ′, y′), K1 = e− cothω 2 (r2+s2) +∞∑ k=0 βkz 1+ k 1+c J̃ γk 2 ( iz sinhω ) Qk(x ′, y′). (6.1) Here r = \|x\|, s = \|y\| and z = \|x\|\|y\|. We also used the notation J̃ν(t) = (t/2)−νJν(t). Now we determine the complex constants {αk} and {βk} such that this integral transform coincides with FωD = e ω ( 1 2 + µ−1 2(1+c) ) e −ω 2(1+c)2 (D2−(1+c)2x2) on the basis {φt,`,m}. We calculate σ−1m ∫ Rm K0(x, y;ω)φ2t,`,m(x)dx = α`M (m) ` (y′)e− cothω 2 s2s ` 1+c ( is 2 sinhω )−γ`/2+1 × ∫ +∞ 0 rγ`/2e− (cothω+1) 2 r2Jγ`/2−1 ( irs sinhω ) L γ`/2−1 t (r2)dr = α` e−2ωt2γ`/2−1 (cothω + 1)γ`/2 φ2t,`,m(y), where we used the identity (see [2, Corollary 4.6]) 2 ∫ +∞ 0 rα+1Jα(rβ)Lαj (r2)e−δr 2 dr = (δ − 1)jβα 2αδα+j+1 Lαj ( β2 4δ(1− δ) ) e− β2 4δ . Similarly, we find σ−1m ∫ Rm K0(x, y;ω)φ2t+1,`,m(x)dx = 0, σ−1m ∫ Rm K1(x, y;ω)φ2t,`,m(x)dx = 0, σ−1m ∫ Rm K1(x, y;ω)φ2t+1,`,m(x)dx = β` e−2ωt2γ`/2 (cothω + 1)γ`/2+1 φ2t+1,`,m(y). Hence we obtain by comparison with (4.4) α` = e − ω` (1+c) (cothω + 1)γ`/2 2γ`/2−1 = 2e ωδ 2 (2 sinhω)−γ`/2, β` = α` 2 sinhω . We summarize our results in the following theorem. Theorem 6.2. Let Reω > 0 and c > −1. Put K(x, y;ω) = e− cothω 2 (r2+s2) ( A(z, w) + x ∧ yB(z, w) ) with A(z, w) = +∞∑ k=0 ( αk k + 2λ 2λ z k 1+c J̃ γk 2 −1 ( iz sinhω ) + αk−1 4 sinhω k λ z k+c 1+c J̃ γk−1 2 ( iz sinhω )) Cλk (w), 14 H. De Bie, B. Ørsted, P. Somberg and V. Souček B(z, w) = +∞∑ k=1 ( −αkz k 1+c −1J̃ γk 2 −1 ( iz sinhω ) + αk−1 2 sinhω z k+c 1+c −1J̃ γk−1 2 ( iz sinhω )) Cλ+1 k−1 (w) for z = \|x\|\|y\|, w = 〈x, y〉/z, α−1 = 0 and αk = 2e ωδ 2 (2 sinhω)−γk/2. Then these series are convergent and the integral transform defined on L20,c(Rm) by Fω0,c(f)(y) = σ−1m ∫ Rm K(x, y;ω)f(x)h(rx)dx coincides with the operator FωD = e ω ( 1 2 + µ−1 2(1+c) ) e −ω 2(1+c)2 (D2−(1+c)2x2) on the basis {φt,`,m}. Proof. We have already shown that the integral transform coincides with the operator FωD = e ω ( 1 2 + µ−1 2(1+c) ) e −ω 2(1+c)2 (D2−(1+c)2x2) on the basis {φt,`,m}. So we only have to show that the series are convergent. We do this for the term +∞∑ k=0 (2 sinhω)−γk/2 k + 2λ 2λ z k 1+c J̃ γk 2 −1 ( iz sinhω ) Cλk (w) = (2 sinhω)−δ/2 +∞∑ k=0 k + 2λ 2λ ( z 2 sinhω ) k 1+c J̃ γk 2 −1 ( iz sinhω ) Cλk (w), the other ones are treated in a similar fashion. We obtain∣∣∣∣∣ +∞∑ k=0 k + 2λ 2λ ( z 2 sinhω ) k 1+c J̃ γk 2 −1 ( iz sinhω ) Cλk (w) ∣∣∣∣∣ ≤ B(λ) 2 e\|Im iz sinhω \| +∞∑ k=0 (k + 2λ) ∣∣∣ z 2 sinhω ∣∣∣ k 1+c 1 Γ(γk/2) k2λ−1 using formula (A.1) and (A.2). As the term Γ(γk/2) is dominant, the series clearly converges. � 6.2 The case Reω = 0 In this case, we have the following theorem. Theorem 6.3. Let c > −1. Then for ω = iη with η 6∈ πZ, we put K(x, y; iη) = ei cot η 2 (r2+s2) ( A(z, w) + x ∧ yB(z, w) ) with A(z, w) = +∞∑ k=0 ( αk k + 2λ 2λ z k 1+c J̃ γk 2 −1 ( z sin η ) + αk−1 4i sin η k λ z k+c 1+c J̃ γk−1 2 ( z sin η )) Cλk (w), B(z, w) = +∞∑ k=1 ( −αkz k 1+c −1J̃ γk 2 −1 ( z sin η ) + αk−1 2i sin η z k+c 1+c −1J̃ γk−1 2 ( z sin η )) Cλ+1 k−1 (w) for z = \|x\|\|y\|, w = 〈x, y〉/z, α−1 = 0 and αk = 2e iηδ 2 (2i sin η)−γk/2. These series are convergent and the unitary integral transform defined in distributional sense on L20,c(Rm) by F iη0,c(f)(y) = σ−1m ∫ Rm K(x, y; iη)f(x)h(rx)dx coincides with the operator F iηD = e iη ( 1 2 + µ−1 2(1+c) ) e −iη 2(1+c)2 (D2−(1+c)2x2) on the basis {φt,`,m}. Proof. This follows by taking the limit ω → iη in Theorem 6.2. � The Clifford Deformation of the Hermite Semigroup 15 7 The series representation of the Fourier transform The Fourier transform is the very special case of the holomorphic semigroup, evaluated at ω = iπ/2. In this case, the kernel K(x, y) = K(x, y; iπ/2) is given by the following theorem. Theorem 7.1. Put K(x, y) = A(z, w) + x ∧ yB(z, w) with A(z, w) = +∞∑ k=0 z− δ−2 2 ( αk k + 2λ 2λ J γk 2 −1(z)− iαk−1 k 2λ J γk−1 2 (z) ) Cλk (w), B(z, w) = +∞∑ k=1 z− δ 2 ( −αkJ γk 2 −1(z)− iαk−1J γk−1 2 (z) ) Cλ+1 k−1 (w) and z = \|x\|\|y\|, w = 〈x, y〉/z, α−1 = 0 and αk = e − iπk 2(1+c) . These series are convergent and the integral transform defined in distributional sense on L20,c(Rm) by F0,c(f)(y) = σ−1m ∫ Rm K(x, y)f(x)h(rx)dx coincides with the operator FD = e iπ 2 ( 1 2 + µ−1 2(1+c) ) e −iπ 4(1+c)2 (D2−(1+c)2x2) on the basis {φt,`,m}. Proof. Using the well-known identity (see [27, Exercise 21, p. 371])∫ +∞ 0 rα+1Jα(rs)Lαj ( r2 ) e−r 2/2dr = (−1)jsαLαj (s2)e−s 2/2 we can prove in the same way as leading to Theorem 6.2 that the integral transform F0,c coincides with FD = e iπ 2 ( 1 2 + µ−1 2(1+c) ) e −iπ 4(1+c)2 (D2−(1+c)2x2) on the basis φt,`,m. The theorem also follows as a special case of Theorem 6.2, taking the limit ω → iπ/2. � Remark 7.1. One can also define an analogue of the Schwartz space of rapidly decreasing functions in this context. Let L = D2 − (1 + c)2x2 and denote by D(L) the domain of L in L20,c(Rm). Then the Schwartz space is defined by S0,c ( Rm ) = ∞⋂ k=0 D ( Lk ) and one can check that the Fourier transform F0,c is an isomorphism of this space. Remark 7.2. In the limit case c = 0, we can check that the kernel reduces to K(x, y) = +∞∑ k=0 k + λ λ (−i)kz−λJk+λ(z)Cλk (w). This is a well-known expansion of the classical Fourier kernel (see [29, Section 11.5]): K(x, y) = e−i〈x,y〉 Γ(m/2)2 m−2 2 . 16 H. De Bie, B. Ørsted, P. Somberg and V. Souček We can now summarize the main properties of the deformed Fourier transform in the following theorem. Theorem 7.2. The operator F0,c defines a unitary operator on L20,c(Rm) and satisfies the fol- lowing intertwining relations on a dense subset: F0,c ◦D = i(1 + c)x ◦ F0,c, F0,c ◦ x = i 1 + c D ◦ F0,c, F0,c ◦ E = − (E + δ) ◦ F0,c. Moreover, F0,c is of finite order if and only if c is rational. Proof. Every f in L20,c(Rm) can be expanded in terms of the orthogonal basis φt,`,m, satisfying 〈φt1,`1,m1 , φt2,`2,m2〉 = δt1t2δ`1`2δm1m2〈φt1,`1,m1 , φt1,`1,m1〉, see Section 3. Note that the normalization can be computed explicitly (see [7, Theorem 6]). As the eigenvalues of F0,c are given by (see (4.4)) (−i)te−i π` 2(1+c) which clearly live on the unit circle, we conclude that 〈f, g〉 = 〈F0,c(f),F0,c(g)〉 and that F0,c is a unitary operator. The intertwining relations are an immediate consequence of formula (4.1) combined with the fact that φt,`,m is an eigenbasis of F0,c. The formula for E follows from the anti-commutator (see Theorem 3.1) {D, x} = −2(1 + c) ( E + δ 2 ) . The statement on the finite order of the Fourier transform is an immediate consequence of the explicit expression for the eigenvalues of the transform. � Now we collect some properties of the kernel K(x, y). Proposition 7.1. One has, with x, y ∈ Rm K(λx, y) = K(x, λy), λ > 0, K(y, x) = K(x, y), K(0, y) = 1 2γ0/2−1Γ(γ0/2) , K(sxs, sys, ) = sK(x, y)s, s ∈ Spin(m), where ·̄ is the anti-involution on the Clifford algebra Clm. Proof. The first property is trivial. The second follows because x ∧ y = −x ∧ y = y ∧ x. The third property follows from Theorem 7.1. Finally, the 4th equation follows because z and w are spin-invariant and (sxs) ∧ (sys) = s ( x ∧ y ) s. � We can also obtain Bochner identities for the deformed Fourier transform. They are given in the following proposition. The Clifford Deformation of the Hermite Semigroup 17 Proposition 7.2. Let M` ∈ M` be a spherical monogenic of degree `. Let f(x) = f(r) be a radial function. Then the Fourier transform of f(r)M` and f(r)xM` can be computed as follows F0,c(f(r)M`) = e − iπ` 2(1+c)M`(y ′) ∫ +∞ 0 r`f(r)z− δ−2 2 J γk 2 −1(z)h(r)rm−1dr, F0,c(f(r)xM`) = −ie− iπ` 2(1+c) y′M`(y ′) ∫ +∞ 0 r`+1f(r)z− δ−2 2 J γk 2 (z)h(r)rm−1dr with y = sy′, y′ ∈ Sm−1 and z = rs. Proof. This follows immediately from Theorem 7.1 combined with Proposition 5.1. � Remark 7.3. As a special case of this proposition, we reobtain the eigenfunctions of the Fourier transform by putting f(r) = L γ` 2 −1 t (r2)rβ`e−r 2/2, resp. f(r) = L γ` 2 t (r2)rβ`e−r 2/2 (see equation (4.4)). Now we prove the following lemma. Lemma 7.1. For all f ∈ L20,c(Rm) one has \|\|xf(x)\|\|2 + \|\|x (F0,cf) (x)\|\|2 ≥ δ\|\|f(x)\|\|2. The equality holds if and only if f is a multiple of e−r 2/2. Proof. Using formula (4.3) and the unitarity of F0,c, one can compute that \|\|xf(x)\|\|2 + \|\|x (F0,cf) (x)\|\|2 = 1 (1 + c)2 〈 ( D2 − (1 + c)2x2 ) f, f〉. Now use the fact that the smallest eigenvalue of 1 (1 + c)2 ( D2 − (1 + c)2x2 ) is given by δ, see Theorem 4.2. This proves the inequality. The equality holds when f is a multiple of an eigenfunction corresponding to the smallest eigenvalue, i.e. when f is a multiple of e−r 2/2. � This lemma allows us to obtain the Heisenberg inequality for the deformed Fourier transform Proposition 7.3. For all f ∈ L20,c(Rm), the deformed Fourier transform satisfies \|\|xf(x)\|\| · \|\|x (F0,cf) (x)\|\| ≥ δ 2 \|\|f(x)\|\|2. The equality holds if and only if f is of the form f(x) = λe−r 2/α. Proof. Using Lemma 7.1, we can continue in the same way as in the proof of Theorem 5.28 in [2]. � Now we can obtain the Master formula for the kernel of the Fourier transform. We use the formula (see [14, p. 50])∫ +∞ 0 Jν(at)Jν(bt)e−γ 2t2tdt = 1 2 γ−2e −a 2+b2 4γ2 Iν ( ab 2γ2 ) , Re ν > −1, Re γ2 > 0, (7.1) where Iν(z) = e−i πν 2 Jν(iz). We then obtain 18 H. De Bie, B. Ørsted, P. Somberg and V. Souček Theorem 7.3 (Master formula). Let s > 0. Then one has∫ Rm K ( y, x; i π 2 ) K ( z, y;−iπ 2 ) e−sr 2 yh(ry)dy = σme −ωδ 2 K(z, x;ω)e− \|x\|2+\|z\|2 2 1−coshω sinhω with 2s = sinhω. Proof. First observe that K ( y, x; iπ2 ) = K(y, x) and that K(z, y;−iπ2 ) is the complex conjugate of K ( z, y; iπ2 ) . We rewrite the kernel K obtained in Theorem 7.1 in terms of the reproducing kernels Pk and Qk, i.e. as K(x, y) = K0(x, y) +K1(x, y) with K0(x, y) = +∞∑ k=0 αk ( \|x\|\|y\| )− δ−2 2 J γk 2 −1(\|x\|\|y\|)Pk(x ′, y′), K1(x, y) = +∞∑ k=0 βk ( \|x\|\|y\| )− δ−2 2 J γk 2 (\|x\|\|y\|)Qk(x′, y′), where αk = e − iπk 2(1+c) and βk = −iαk. When passing to spherical co-ordinates, the integral simplifies, using Lemma 5.2, to σm +∞∑ k=0 (\|x\|\|z\|)− δ−2 2 Pk(z ′, x′) ∫ +∞ 0 re−sr 2 J γk 2 −1(r\|x\|)J γk 2 −1(r\|z\|)dr + σm +∞∑ k=0 (\|x\|\|z\|)− δ−2 2 Qk(z ′, x′) ∫ +∞ 0 re−sr 2 J γk 2 (r\|x\|)J γk 2 (r\|z\|)dr. The radial integral can be computed explicitly using (7.1). Comparing with formula (6.1) and Theorem 6.2 leads to the statement of the theorem. � Remark 7.4. For the Dunkl transform (see, e.g., [24, 28]) and for the Clifford–Fourier transform (see [8]) one can compute even a more general integral of the form∫ Rm K ( y, x; i π 2 ) K ( z, y;−iπ 2 ) f(ry)h(ry)dy with f(ry) an arbitrary radial function of suitable decay. This is done by using the addition formula for the Bessel function u−λJλ(u) = 2λΓ(λ) ∞∑ k=0 (k + λ)(r2\|x\|\|z\|)−λJk+λ(r\|x\|)Jk+λ(r\|z\|)Cλk (〈x′, z′〉) with u = r √ \|x\|2 + \|z\|2 − 2〈x, z〉 instead of formula (7.1). Here, we cannot do that, as the orders of the Bessel functions do not match the order of the Gegenbauer polynomials. Remark 7.5. Theorem 7.3 is the starting point for the study of a generalized heat equation, see, e.g., [26, Lemma 4.5(1)] in the context of Dunkl operators. 8 Further results for the kernel In this section we will always be working in the non-Dunkl case, i.e. we put the multiplicity function κ = 0. Theorem 7.1 implies that the kernel of our deformed Fourier transform is a function of the type K(x, y) = f(z, w) + x ∧ yg(z, w) The Clifford Deformation of the Hermite Semigroup 19 with f , g scalar functions of the variables z = \|x\|\|y\| and w = 〈x, y〉/z. On the other hand, this kernel needs to satisfy the system of PDEs DyK(x, y) = −i(1 + c)K(x, y)x, (K(x, y)Dx) = −i(1 + c)yK(x, y), as can be deduced from Theorem 7.2. In order to rewrite this system in terms of the vari- ables z, w, we first observe that ∂xf(z, w) = r−2xz∂zf(z, w) + ( z−1y − r−2xw ) ∂wf(z, w), Ef(z, w) = z∂zf(z, w), ∂x ( x ∧ y ) = (1−m)y. Using these identities, one obtains that the kernel is determined by the following 2 PDEs: (m− 1 + c)g + (1 + c)z∂zg + 1 z ∂wf + i(1 + c)f − i(1 + c)zwg = 0, (1 + c)z∂zf − w∂wf − czwg − (1 + c)z2w∂zg + z(w2 − 1)∂wg + i(1 + c)z2g = 0. (8.1) Remark 8.1. Note that, contrary to the case of the classical Fourier transform and the Dunkl transform, where the kernel is uniquely determined by the system of PDEs Tj,xK(x, y) = iyjK(x, y), j = 1, . . . ,m this is not the case for the kernel of the radially deformed Fourier transform. In fact, one can observe that there exist several different types of solutions of (8.1). This is discussed in detail in [6] for a similar system of PDEs in the context of the so-called Clifford–Fourier transform (see [8]). Now we show that it is sufficient to solve this system in dimension m = 2 and m = 3. Recall that the kernel K(x, y) is given in Theorem 7.1. To know this kernel, it is hence sufficient to know the series Aλ = +∞∑ k=0 αk(k + λ)J γk 2 −1(z)C λ k (w), Dλ = +∞∑ k=0 αk−1J γk−1 2 (z)Cλk (w), Bλ = +∞∑ k=0 αkJ γk 2 −1(z)C λ k (w), Eλ = +∞∑ k=1 αkJ γk 2 −1(z)C λ+1 k−1 (w), Cλ = +∞∑ k=0 αk−1(k + λ)J γk−1 2 (z)Cλk (w), Fλ = +∞∑ k=1 αk−1J γk−1 2 (z)Cλ+1 k−1 (w), because then one has K = 1 2λ z− δ−2 2 (Aλ − iCλ) + 1 2 z− δ−2 2 (Bλ + iDλ)− z− δ 2x ∧ y (Eλ + iFλ) . Using the well-known property of the Gegenbauer polynomials 2λCλ+1 k−1 (w) = ∂wC λ k (w), we observe the following recursion relations Aλ+1 = e i π 2(1+c) 1 2λ ∂wAλ, Bλ+1 = e i π 2(1+c) 1 2λ ∂wBλ, Cλ+1 = e −i π 2(1+c) 1 2λ ∂wCλ, Dλ+1 = e −i π 2(1+c) 1 2λ ∂wDλ, Eλ = 1 2λ ∂wBλ, Fλ = 1 2λ ∂wDλ. We conclude that it suffices to know Aλ, Bλ, Cλ and Dλ for λ = 0, 1/2 or m = 2, 3. At this point, the problem of finding explicit expressions for these functions for special values of the deformation parameter c is still open. 20 H. De Bie, B. Ørsted, P. Somberg and V. Souček A Properties of Laguerre and Gegenbauer polynomials The generalized Laguerre polynomials L (α) k for k ∈ N are defined as L (α) k (t) = k∑ j=0 Γ(k + α+ 1) j!(k − j)!Γ(j + α+ 1) (−t)j and satisfy the orthogonality relation (when α > −1)∫ ∞ 0 tαL (α) k (t)L (α) l (t) exp(−t)dt = δkl Γ(k + α+ 1) k! . The Gegenbauer polynomials C (α) k (t) are a special case of the Jacobi polynomials. For k ∈ N and α > −1/2 they are defined as C (α) k (t) = bk/2c∑ j=0 (−1)j Γ(k − j + α) Γ(α)j!(k − 2j)! (2t)k−2j and satisfy the orthogonality relation∫ 1 −1 C (α) k (t)C (α) l (t) ( 1− t2 )α− 1 2dt = δkl π21−2αΓ(k + 2α) k!(k + α)(Γ(α))2 . One can prove that there exists a constant B(α) such that sup −1≤t≤1 ∣∣∣∣ 1αC(α) k (t) ∣∣∣∣ ≤ B(α)k2α−1, ∀ k ∈ N, (A.1) see [2, Lemma 4.9]. The Bessel function Jν(z) is defined using the following Taylor series Jν(z) = ∞∑ k=0 (−1)k k!Γ(k + ν + 1) (z 2 )2k+ν . For z ∈ C and ν ≥ −1/2 one has the inequality (see, e.g., [27])∣∣∣∣(z2)−ν Jν(z) ∣∣∣∣ ≤ 1 Γ(ν + 1) e\| Im z\|. (A.2) B List of notations List of notations used in this paper: m dimension of Rm, κ multiplicity function on root system, µ Dunkl-dimension, c deformation parameter of D, ω semigroup parameter with Reω ≥ 0, ∂x ordinary Dirac operator, Dk Dunkl Dirac operator, The Clifford Deformation of the Hermite Semigroup 21 D radially deformed Dirac operator, FωD exponential form of the holomorphic semigroup, Fω0,c integral form of the holomorphic semigroup, FD exponential form of the Fourier transform, F0,c integral form of the Fourier transform. We also have the following definitions: µ = m+ 2 ∑ α∈R+ κα, λ = m− 2 2 , σm = 2πm/2/Γ(m/2), δ = 1 + µ− 1 1 + c , β` = − c 1 + c `, ` ∈ N, γ` = 2 1 + c ( `+ µ− 2 2 ) + c+ 2 1 + c , ` ∈ N. Notations for variables. Let x and y be vector variables in Rm. Then we denote z = \|x\|\|y\|, w = 〈x, y〉/z. 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The Clifford Deformation of the Hermite Semigroup

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