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Journal of Functional Analysis

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The biradial Paley–Wiener theorem for theHelgason Fourier transform on Damek–Ricci spaces

Roberto CamporesiDipartimento di Scienze Matematiche, Politecnico di Torino,Corso Duca degli Abruzzi 24, 10129 Torino, Italy

a r t i c l e i n f o a b s t r a c t

Article history:Received 10 April 2013Accepted 25 April 2014Available online xxxxCommunicated by B. Chow

MSC:22E2543A8043A8553C30

Keywords:Damek–Ricci spacesJacobi functionsPaley–Wiener theorem

We prove the Paley–Wiener theorem for the Helgason Fouriertransform of smooth compactly supported biradial functionson a Damek–Ricci space S = NA.

© 2014 Published by Elsevier Inc.

1. Introduction

Let S = NA be a Damek–Ricci space, i.e., the semidirect product of a (connected andsimply connected) nilpotent Lie group N of Heisenberg type [18] and the one-dimensionalLie group A ∼= R

+ acting on N by anisotropic dilations. When S is equipped with a suit-able left-invariant Riemannian metric γS , S becomes a (noncompact, simply connected)homogeneous harmonic Riemannian space [8,9]. Conversely, every such space is a Damek–

E-mail address: [email protected].

http://dx.doi.org/10.1016/j.jfa.2014.04.0130022-1236/© 2014 Published by Elsevier Inc.

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Ricci space if we exclude Rn and the “degenerate” case of real hyperbolic spaces (see [13,

Corollary 1.2]). We refer to [23] for a nice introduction to the geometry and harmonicanalysis on Damek–Ricci spaces.

We use the ball model B of S, namely we identify S with the unit ball B in the Liealgebra s via the Cayley transform C [5]:

S = NAC∼= B =

{(V ′, Z ′, t′

)∈ s :

∣∣V ′∣∣2 +∣∣Z ′∣∣2 + t′ 2 < 1

}.

Here s = n⊕ a = v⊕ z⊕ a, where z is the center of n and v its orthogonal complementin n. We let p = dim v, q = dim z, Q = p/2 + q, and let Sp+q be the unit sphere in s

Sp+q = ∂B ={ω =

(V ′, Z ′, t′

)∈ s :

∣∣V ′∣∣2 +∣∣Z ′∣∣2 + t′ 2 = 1

}.

Let f ∈ C∞0 (B). The Helgason Fourier transform of f is defined by

f̃(λ, ω) =∫B

f(b)Qλ(b, ω) db(λ ∈ C, ω ∈ Sp+q

), (1.1)

where Qλ(b, ω) is the normalized Poisson kernel with parameter λ on B (see [3,4], andSection 2). The inversion formula is [3, Theorem 4.4]

f(b) = cp,q2π

∞∫0

∫Sp+q

Q−λ(b, ω)f̃(λ, ω) dω dμ(λ), (1.2)

where dω is the normalized Euclidean surface measure on Sp+q, dμ(λ) = |c(λ)|−2 dλ,where

c(λ) = 2Q−2iλΓ (2iλ)Γ (d/2)Γ (iλ + Q/2)Γ (iλ + p/4 + 1/2) , (1.3)

and

cp,q = 2q−1Γ (d/2)π−d/2, d = p + q + 1.

A C∞ function ψ(λ, ω) on C×∂B, holomorphic in λ, is called a holomorphic functionof uniform exponential type if there exists a constant R > 0 such that for each integerj � 0

sup(λ,ω)∈C×∂B

e−R|Im λ|(1 + |λ|)j∣∣ψ(λ, ω)

∣∣ < ∞. (1.4)

Theorem 1.1. The Fourier transform f(b) → f̃(λ, ω) is a bijection of C∞0 (B) onto the

set of holomorphic functions ψ(λ, ω) of uniform exponential type satisfying the condition

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∫
∂B

Q−λ(b, ω)ψ(λ, ω) dω =∫∂B

Qλ(b, ω)ψ(−λ, ω) dω (1.5)

for any b ∈ B and λ ∈ C. Moreover f̃ satisfies (1.4) if and only if f has support in theclosed ball BR = {b ∈ B : d(b, C(e)) � R}.

The direct part of this theorem, namely the fact that supp f ⊂ BR implies f̃ holomor-phic of uniform exponential type R, was proved in [3, Theorem 4.5] (in the open model).In this paper we prove as Theorem 3.3 the biradial case of Theorem 1.1. The conversepart, in particular the surjectivity statement, is proved for biradial functions f on B,i.e., functions that depend only on the norms |V ′|, |Z ′|, and t′. In this case we show thatthe map f → f̃ is a bijection onto the set of functions ψ(λ, ω) that are holomorphic ofuniform exponential type, biradial in ω, and satisfy (1.5).

We use recent results in [4] to generalize to biradial functions on B the well-knownexpansion into K-types of the symmetric case [14]. By working in geodesic polar co-ordinates (r, ω) ∈ (0,∞) × Sp+q around the origin in B, we expand both functionsω → f(r, ω) and ω → f̃(λ, ω) in Fourier series with respect to an orthogonal system ofbiradial eigenfunctions of the angular Laplacian LS(r) in L2(Sp+q). Here S(r) � Sp+q isthe geodesic sphere of radius r > 0 centered at the origin. The Fourier coefficients arethen functions of r and λ, respectively, related by a suitable Jacobi transform. Usingwell-known estimates for Jacobi functions, we prove the result.

The method followed in this paper should generalize to arbitrary functions in C∞0 (B).

In the symmetric case, the biradial eigenfunctions of the Laplacian on B of a givenK-type δ determine the radial dependence of the K-finite eigenfunctions of type δ [15,Lemma 4.2]. The reason is that these eigenfunctions separate in geodesic polar coor-dinates, with angular part given by suitable matrix coefficients of δ. Since the angularLaplacian has the same eigenvalue as in the biradial case, all K-finite eigenfunctionsof type δ must have the same radial part. One can then prove the full Paley–Wienertheorem by an easy adaptation of the biradial proof.

It is not clear if a similar behaviour holds in general. The main question is whetheror not the spectrum of the operator LS(r) (without counting multiplicities, i.e., the setof distinct eigenvalues) always coincides with its biradial part (i.e., the eigenvalues ofLS(r) acting on biradial functions, given by formula (2.12) below). A claim in this sensewas made in [24,25] for the case in which there are both symmetric and non-symmetricDamek–Ricci spaces with the same q and p (namely q = 3 and p = 4n � 8). The claimis that the geodesic spheres of the same radius r are isospectral within each finite familywith p fixed, i.e., the associated Laplacians have the same eigenvalues and the samemultiplicities. However, the opposite claim appeared in [2], where it was proved that thespectrum of geodesic spheres determines the local symmetry of a harmonic space. Theproblem in the non-symmetric case is that there is no group K acting transitively byisometries on the geodesic spheres. This makes it more difficult to construct non-biradialeigenfunctions of LS(r) as it requires the explicit form of the full angular Laplacian,which is not yet available.


The outline of this paper is as follows. In Section 2 we review some results in [4] aboutnon radial eigenfunctions of the Laplace–Beltrami operator on B. We give a differentproof of the formula for the biradial part of the angular Laplacian. We also prove thePoisson integral representation of biradial eigenfunctions using the Radon transform andthe method of “descent” to complex hyperbolic spaces, already employed in [10,21] inthe radial case. In Section 3 we write down the Fourier transform and prove the biradialcase of the theorem.

2. Biradial eigenfunctions on B

2.1. The Cayley transform and the biradial Laplacian

We denote by 〈·,·〉 and | · | the fixed inner product and associated norm on s, andby (V,Z, t) ∈ s the element exp(V + Z) exp(tH) of S, where V ∈ v, Z ∈ z, t ∈ R, andH ∈ a is a unit vector. For each Z ∈ z we have the linear map JZ : v → v defined by〈JZV, V ′〉 = 〈Z, [V, V ′]〉, ∀V, V ′ ∈ v. The Cayley transform is defined by

C : NA → B, (V,Z, t) →(V ′, Z ′, t′

),

where ⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

V ′ = (1+et+14 |V |2)V−JZV

(1+et+14 |V |2)2+|Z|2

Z ′ = 2Z(1+et+1

4 |V |2)2+|Z|2

t′ = −1+(et+14 |V |2)2+|Z|2

(1+et+14 |V |2)2+|Z|2

,

(2.1)

with inverse ⎧⎪⎪⎨⎪⎪⎩V = 2 (1−t′)V ′+JZ′V ′

(1−t′)2+|Z′|2

Z = 2Z′

(1−t′)2+|Z′|2

et = 1−R2

(1−t′)2+|Z′|2 ,

(2.2)

where R =√|V ′|2 + |Z ′|2 + t′2 (see [23, (18), (19), sect. 4.4]).

We also have a generalized stereographic projection

C0 : N → Sp+q \ {o}(o = (0, 0, 1)

)obtained by letting at = et = 0, i.e., t = −∞, in (2.1):

C0(n) = lim C(nat) ∈ ∂B
t→−∞


(see [23, section 4.6]). Its inverse C−10 is given by the first two lines in (2.2).

In the ball model B of S, equipped with the transported metric γB = C−1∗(γS), wehave C(Expe rω) = th r

2ω for r � 0 and ω ∈ Sp+q. Thus the geodesics through the originare the diameters, and the Riemannian sphere S(r) of radius r (centered at the origin)is just the Euclidean sphere S(R) of radius R = th( r2 ) [23, Thm. 10].

In geodesic polar coordinates (r, ω) around the origin x0 = C(e) = (0, 0, 0) in B, theRiemannian measure is given by

db = 2p+q

(sh r

2

)p+q(ch r

2

)q

vol(Sp+q

)dr dω = J(r) dr dω, (2.3)

where vol(Sn) = 2π(n+1)/2/Γ ((n + 1)/2).Let M be the group of orthogonal automorphisms of NA, namely the automorphisms

of S that preserve the inner product on the Lie algebra s. By conjugating with theCayley map C, we obtain a group of transformations of B that we still denote by M . Itis easy to check that this action of M on B is just the action of M on s = v⊕ z⊕ a, i.e.,m · (V ′, Z ′, t′) = (m · V ′,m ·Z ′, t′), where M is trivial on a and leaves v and z invariant.It is known that M acts transitively on the unit sphere Sq−1 in z (see [5, Remark 6.3]).However, M may or may not be transitive on the unit sphere Sp−1 in v, depending onthe Heisenberg-type group N [7,22].

Let f be a biradial function on B, i.e., f depends only on |V ′|, |Z ′| and t′. Then f isM -invariant, but the converse is not necessarily true. We denote by C∞

0 (B)� the subspaceof biradial functions in C∞

0 (B). In geodesic polar coordinates we write f(b) = f(r, ω).For each r > 0, the function ω = (V ′, Z ′, t′) → f(r, ω) is biradial on Sp+q, i.e., it dependsonly on |Z ′| and t′. We introduce polar coordinates (ρ, φ) by{

t′ = ρ cosφ|Z ′| = ρ sinφ,

and let D+ be the upper-half unit disk defined by 0 � ρ � 1 and 0 � φ � π. Anyω ∈ Sp+q can then be written in the form

ω =(V ′, Z ′, t′

)=

(√1 − ρ2ω1, ρ sinφω2, ρ cosφ

),

where ω1 ∈ Sp−1, ω2 ∈ Sq−1, and ρ2 = 1−|V ′|2 = |Z ′|2 + t′ 2. We write ω = (ρ, φ, ω1, ω2)and refer to this as a system of bispherical coordinates on Sp+q. The choices of ω1, ω2and φ are unique except when V ′ = 0, or Z ′ = 0, or (Z ′, t′) = (0, 0).

For f biradial, we write f = f(r, ρ, φ). The Laplace–Beltrami operator on B ingeodesic polar coordinates reads

L = Lrad + LS(r), (2.4)

where Lrad is the radial part, given by


Lrad = ∂2r +

(p2 cth r

2 + q cth r)∂r (∂r = ∂/∂r), (2.5)

and LS(r) is the angular part, i.e., the Laplacian on the Riemannian sphere S(r) withrespect to the induced metric. We identify S(r) with Sp+q by the map C(Expe rω) → ω,i.e., th r

2ω → ω, for any fixed r > 0.

Theorem 2.1. Let χ be a function on Sp+q � S(r) depending only on ρ and φ. Then theangular Laplacian LS(r) acting on χ is given by

LS(r)χ = 14 sh2 r

2D1χ− 1

4 ch2 r2D2χ, (2.6)

where D1 and D2 are the differential operators{D1 = (1 − ρ2)∂2

ρ + ( qρ − (p + q)ρ)∂ρ + 1ρ2 (∂2

φ + (q − 1) cotφ∂φ)D2 = ∂2

φ + (q − 1) cotφ∂φ.(2.7)

Proof. This was proved in [4, Theorem 4.1], by a calculation in bispherical coordinateson Sp+q using the explicit form of the induced metric γB |S(r) obtained there. We outlinehere a different proof, which is somewhat simpler.

Let {E1, . . . , Ep, Y1, . . . , Yq, H} be an orthonormal basis of s = v⊕ z⊕ a with respectto the Euclidean inner product 〈 , 〉 on s. We identify H = ∂t and regard Ei, Yj asleft-invariant vector fields on the group N , given at (V,Z) by

Ei = ∂vi + 12

q∑j=1

〈JYjV,Ei〉∂zj , Yj = ∂zj ,

if we write V =∑

i viEi and Z =∑

j zjYj . Then the Laplace–Beltrami operator on S isgiven in the usual NA-chart by [6]

L = etp∑

i=1E2

i + e2tq∑

j=1Y 2j + ∂2

t −Q∂t.

Now let f be a biradial function on NA, i.e., f depends only on |V |, |Z|, and t. Let uswrite f = f(|V |, |Z|, t). The biradial part of L, i.e., the part involving only derivativeswith respect to |V |, |Z|, t, is easily computed to be [9]

Lf = et(∂2|V | + p−1

|V | ∂|V | + 14 |V |2

(∂2|Z| + q−1

|Z| ∂|Z|))f

+ e2t(∂2|Z| + q−1

|Z| ∂|Z|)f +

(∂2t −Q∂t

)f.

By taking the norms in (2.1) and (2.2), we obtain the relationship between |V |, |Z|, tand |V ′|, |Z ′|, t′. Since 〈V, JZV 〉 = 0 and |JZV | = |Z||V |, we find

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩
|V ′|2 = |V |2

(1+et+14 |V |2)2+|Z|2

|Z ′| = 2|Z|(1+et+1

4 |V |2)2+|Z|2

t′ = −1+(et+14 |V |2)2+|Z|2

(1+et+14 |V |2)2+|Z|2

,

with inverse ⎧⎪⎪⎨⎪⎪⎩|V |2 = 4|V ′|2

(1−t′)2+|Z′|2

|Z| = 2|Z′|(1−t′)2+|Z′|2

et = 1−R2

(1−t′)2+|Z′|2 ,

where R =√

|V ′|2 + |Z ′|2 + t′2. It is clear from these formulas that the function f isbiradial on S if and only if the function f ◦ C−1 is biradial on B.

We can then rewrite ∂|V |, ∂|Z| and ∂t in terms of ∂|V ′|, ∂|Z′| and ∂t′ , by disregardingthe “angular” derivatives with respect to ω1 ∈ Sp−1 and ω2 ∈ Sq−1 in the change ofvariables (V,Z, t) → (V ′, Z ′, t′) → (|V ′|, |Z ′|, t′, ω1, ω2). Finally, we can change variablesfrom (|V ′|, |Z ′|, t′) to (R, ρ, φ), where⎧⎪⎨⎪⎩

|V ′| = R√

1 − ρ2

|Z ′| = Rρ sinφ

t′ = Rρ cosφ,

and compute the biradial part L̃ of the Laplace–Beltrami operator on B in terms of ∂R,∂ρ and ∂φ. The result of a very long calculation is

L̃ = 14(1 −R2)2∂2

R + 12(p+q2R +

(q2 − 1

)R)(

1 −R2)∂R+ 1−R2

4R2 D1 − 1−R2

4 D2,

where D1 and D2 are given by (2.7). The first line is precisely Lrad in (2.5) since R =th( r2 ), so the result follows. �Remark 1. In the symmetric case, i.e., when S is a rank-1 symmetric space G/K, we havethe “polar coordinate” decomposition of the Laplace–Beltrami operator on G/K givenin [15, Theorem 3.1] (for arbitrary rank). Formulas (2.4) through (2.7) can be obtainedfrom this result by specializing to rank one, using the identity

a

sh2 r2

+ b

sh2 r=

a + b4

sh2 r2−

b4

ch2 r2,

and observing that the round Laplacian LSp+q on Sp+q � K/M is induced by (a multipleof) the Casimir element ωα + 1ω2α, whereas (a multiple of) 1ω2α induces the “vertical”
4 4


Laplacian along the fibres of the Hopf fibration of Sp+q. In fact the operators D1 andD2 in (2.7) are precisely the biradial parts of these two Laplacians on Sp+q, i.e., thosecontaining only derivatives with respect to ρ and φ (see [4, Theorem 4.1 and Remark 5]).

2.2. Separation of variables

For α > β > −1/2 and k, l ∈ Z, k � l � 0, define (cf. [20, (8.3)])

χ(α,β)k,l (ρ, φ) = R

(α−β−1,β+k−l)l

(2ρ2 − 1

)ρk−lR

(β−1/2,β−1/2)k−l (cosφ), (2.8)

where R(α,β)n (x) is a Jacobi polynomial P (α,β)

n (x) normalized so that R(α,β)n (1) = 1. Then

the set {χ(α,β)k,l , k ∈ Z

+, l = 0, 1, . . . , k} is an orthogonal system on D+ with respect tothe measure

dmα,β(ρ, φ) = 2Γ (α + 1)Γ (1

2 )Γ (α− β)Γ (β + 12)

(1 − ρ2)α−β−1(ρ sinφ)2βρ dρ dφ.

The L2-norm squared

∥∥χ(α,β)k,l

∥∥2 =1∫

0

π∫0

(χ

(α,β)k,l (ρ, φ)

)2dmα,β(ρ, φ)

is computed to be (π(α,β)k,l )−1, where (cf. [20, (8.4)], with a factor of 2 corrected)

π(α,β)k,l = (2k − 2l + 2β)(k + l + α)(α− β)l(2β + 1)k−l(α + 1)k

(k − l + 2β)(k + α)l!(k − l)!(β + 1)k, (2.9)

(a)k being defined by (a)0 = 1, (a)n = Γ (a + n)/Γ (a) = a(a + 1) · · · (a + n− 1).Let from now on

α = p + q − 12 , β = q − 1

2 , (2.10)

and

πk,l = π(α,β)k,l , dm(ρ, φ) = dmα,β(ρ, φ), χk,l = χ

(α,β)k,l .

For a function χ on Sp+q depending only on ρ, φ we have (writing χ(ω) = χ(ρ, φ))

∫Sp+q

χ(ω) dω = vol(Sp−1) vol(Sq−1)vol(Sp+q)

1∫0

π∫0

χ(ρ, φ)ρq(1 − ρ2) p

2−1(sinφ)q−1 dρ dφ

=1∫ π∫

χ(ρ, φ) dm(ρ, φ)
0 0


(see [4, (4.10)]). By defining χk,l(ω) = χk,l(ρ, φ) for ω = (ρ, φ, ω1, ω2) ∈ Sp+q, we canrewrite the orthogonality relations for the system {χk,l} as∫

Sp+q

χk,l(ω)χk′,l′(ω) dω = δkk′δll′π−1k,l . (2.11)

It follows from (2.6)–(2.7) that the functions χk,l are eigenfunctions of LS(r),

LS(r)χk,l =(− (k + l)(k + l + p + q − 1)

4 sh2 r2

+ (k − l)(k − l + q − 1)4 ch2 r

2

)χk,l, (2.12)

and {χk,l} is a complete system for biradial functions in L2(S(r)) � L2(Sp+q) [4].

Remark 2. In the symmetric case, the functions χk,l are essentially the spherical func-tions ψδk,l

= 〈δk,l(·)v, v〉 on K/M = Sp+q, where δk,l is a K-type with an M -fixedvector v. They were first determined in [16,17] by a case-by-case computation, using thedecomposition of L2(K/M) = L2(Sp+q) into K-irreducible subspaces and the biradialpart of the Euclidean Laplacian on R

p+q+1. In [12,20] the functions χk,l were recognizedto be in the general class χ

(α,β)k,l for α and β given by (2.10) (with q = 1, 3, 7 and p the

multiplicity of the short restricted root). The relationship between the integers k, l inthe present paper and the integers p, q in the parametrization of Johnson and Wallach in[17, Theorem 3.1] (for F = C,H), is as follows. For F = C we have p = k and q = l, butwe take k � l (instead of k, l ∈ Z

+) since in this case χk,l = Reψδk,l= Reψδ̌k,l

, whereδ̌k,l � δl,k is the contragredient representation of δk,l (see [20, p. 65]). For F = H we havep = k+ l, q = k− l, so p−q is even, and χk,l = ψδk,l

. Finally, the case F = O was treatedin [16, Theorem 3.1]. Changing notations from (m, l) to (p, q) in that theorem, we getthe same relationship as for F = H. Note that for F = H,O, πk,l = dim δk,l, whereas forF = C, πk,k = dim δk,k, and πk,l = 2 dim δk,l for k > l.

Using (2.12) one finds that the biradial eigenfunctions of the Laplacian on B, solutionsof

Lf = −(λ2 + Q2

4)f

(λ ∈ C, Q = p

2 + q),

that separate in geodesic polar coordinates, are given by [4, Theorem 5.1]

fλ,k,l(b) = fλ,k,l(r, ω) = φλ,k,l(r)χk,l(ω), (2.13)

where the functions φλ,k,l are the associated Jacobi functions

φλ,k,l(r) = φ(α,β)2λ,k,l

(r2)

= qk,l(λ)(2 sh r

2)k+l(2 ch r

2)k−l

φ(α+k+l,β+k−l)2λ

(r2), (2.14)

qk,l(λ) being a normalization constant and φ(a,b)λ (t) a Jacobi function


φ(a,b)λ (t) = F

(a+b+1−iλ

2 , a+b+1+iλ2 , a + 1,− sh2 t

)(2.15)

(F (a, b, c, z) is the hypergeometric function). The functions fλ,k,l are radial if and only ifk = l = 0, in which case they reduce to the spherical functions φλ(r) = φ

(α,β)2λ ( r2 ) (being

χ00(ω) = 1 and q00(λ) = 1, see (2.30)). In particular, if

b = C(at) = C(exp tH) = C(Expe tH) = th t2H, with H = (0, 0, 1) ∈ a,

then the geodesic polar coordinates of b are (r, ω) = (|t|, sgn(t)H). Now H correspondsto (ρ, φ) = (1, 0) and χk,l(1, 0) = 1, whereas −H corresponds to (ρ, φ) = (1, π) andχk,l(1, π) = (−1)k−l. By (2.14) the functions t → φλ,k,l(t) extend to negative values byφλ,k,l(−t) = (−1)k+lφλ,k,l(t). Thus we find, for all t ∈ R, fλ,k,l(C(at)) = φλ,k,l(t).

Remark 3. In the symmetric case, the functions t → fλ,k,l(C(at)) (with qk,l(λ) givenby (2.31)) coincide with the restrictions φλ,δk,l

|A, where φλ,δ(g) = d−1/2δ 〈πλ(g)e1, eδ〉

(g ∈ G) are the spherical functions of type δ on G/K ([20, (3.22), (4.15)], [12, section 4]).Here δ = δk,l ∈ K̂(M) (a K-type with an M -fixed vector), dδ = dim δ, (πλ,Hλ) is aspherical principal series representation of G, e1 ∈ Hλ is a K-fixed unit vector, andeδ ∈ Vδ ⊂ Hλ is an M -fixed unit vector. The factorization (2.13) is then equivalent to theformula φλ,δ(kat) = φλ,δ(at)ψδ(k), for k ∈ K and at ∈ A, where ψδ(k) = 〈δ(k)eδ, eδ〉 isthe spherical function on K/M corresponding to δ ([20, (3.18)]; [12, (4.3)]). The functionst → φλ,δk,l

(at) were first computed in terms of hypergeometric functions by Helgason in[15, Theorem 4.5]. See also [14, Theorem 11.2, p. 344, and Ex. B2, p. 372].

2.3. Poisson integral representation

Let P(x, n) be the Poisson kernel on NA, given by [6]

P(at, n) = cp,q

(et

(et + 14 |V |2)2 + |Z|2

)Q

(2.16)

for x = at = exp(tH) ∈ A, and by

P(nat, n

′) = P(at, n

−1n′) (n, n′ ∈ N

)for general x = nat ∈ S. Define the normalized Poisson kernel with parameter λ ∈ C onNA as the following function on NA×N [3]:

Qλ(x, n) = Pλ(x, n)Pλ(e, n) (x ∈ NA, n ∈ N), (2.17)

where

Pλ(x, n) =(P(x, n)

) 12− iλ

Q . (2.18)


We define a kernel Q̃λ on B by

Q̃λ

(C(x), C0(n)

)= Qλ(x, n) (x ∈ NA, n ∈ N),

i.e.,

Q̃λ(b, ω) = Qλ

(C−1(b), C−1

0 (ω)) (

b ∈ B, ω ∈ ∂B \ {o}).

From now on we write Qλ(b, ω) in place of Q̃λ(b, ω). The kernel Qλ(b, ω) extends to asmooth kernel on B × ∂B. For example for b = C(at) = th t

2H and ω = (ρ, φ, ω1, ω2),we have

Qλ

(C(at), ω

)=

∣∣ch t2 − ρeiφ sh t

2∣∣2iλ−Q

(cf. [4, (5.20)]). In particular, for ω = H = (0, 0, 1),

Qλ

(C(at), H

)= et(Q/2−iλ).

For a suitable choice of the constant qk,l(λ) in (2.14), one has the following expansionof ω → Qλ(C(at), ω) in terms of χk,l [4, Theorem 5.2]:

Qλ

(C(at), ω

)=

∞∑k=0

k∑l=0

πk,lφλ,k,l(t)χk,l(ω), (2.19)

where πk,l is given by (2.9), and the double series converges absolutely and uniformly incompact subsets of its domain. Conversely, we have (using (2.11))

φλ,k,l(t) = fλ,k,l(C(at)

)=

∫Sp+q

Qλ

(C(at), ω

)χk,l(ω) dω. (2.20)

For generic b ∈ B, the kernel Qλ(b, ω) is not given by the right-hand side of (2.19)with fλ,k,l(b) in place of φλ,k,l(t) = fλ,k,l(C(at)). This is clear a priori since the functionsfλ,k,l(b) are biradial, while the kernel Qλ(b, ω) is not biradial in b or ω separately. Interms of M -invariance we have, as in the symmetric case,

Qλ(m · b, ω) = Qλ

(b,m−1 · ω

)(m ∈ M), (2.21)

or

Qλ(m · b,m · ω) = Qλ(b, ω).

Nevertheless, the following Poisson integral representation holds for any b in B.


Theorem 2.2. For all b ∈ B we have

fλ,k,l(b) =∫

Sp+q

Qλ(b, ω)χk,l(ω) dω. (2.22)

Equivalently, if we define fλ,k,l(b) by (2.22), then (2.13) holds.

Proof. For b = C(at) this is just (2.20). In the symmetric case, the geodesic polarcoordinates of b = kat · x0 are (r, ω) = (|t|, sgn(t) Ad(k)H), and one can easily prove(2.22) using an elementary substitution in the integral over Sp+q = K/M , rewritten asan integral over K. (See, e.g., [15, Lemma 4.2], with j = 1.)

In the general case there is no analogue of the group K acting transitively on thegeodesic spheres of S. In fact, in the non-symmetric case, the isometries of S that preservethe origin are the elements of M , and the isometry group of S is just the semidirectproduct of S and M [7]. Since S(r) � Sp+q is no longer homogeneous, formula (2.22)for generic b in B cannot be obtained by a simple substitution as in the symmetric case.A possible proof involves using the Radon transform and the method of “descent” tocomplex hyperbolic spaces, as explained in [10,21] in the radial case (k = l = 0).

First note that the right-hand side of (2.22) can be rewritten as a matrix coefficientof a suitable representation πλ of S in L2(N). For λ ∈ C set

(πλ(na)f

)(n′) = aiλ−

Q2 f

(a−1n−1n′a

) (a ∈ A, n, n′ ∈ N

).

This defines a representation of NA in L2(N) which is unitary for real λ, with respectto the usual inner product on L2(N):

〈f, g〉 =∫N

f(n)g(n) dn.

Then (2.22) is equivalent to [4, Proposition 5.3]

fλ,k,l(b) =⟨πλ

(C−1(b)

)Pλ(e, ·),Pλ(e, ·)(χk,l ◦ C0)

⟩. (2.23)

In particular for k = l = 0, one has the following formula for the spherical functionson B:

φλ(b) =∫

Sp+q

Qλ(b, ω) dω =⟨πλ

(C−1(b)

)Pλ(e, ·),Pλ(e, ·)

⟩. (2.24)

This was proved in [10, p. 639] using the Radon transform (see also [21, Theorem 3.3]).Another proof of (2.24) can be found in [1, pp. 654–655].

We now give an outline of the proof for general (k, l) using the Radon transform.


Let us go back to the open model S = NA. Given f ∈ C∞0 (S) and a unit vector ω ∈ z,

we define the Radon transform of f by

Rωf(V, η, t) = e1−q2 t

∫ω⊥

f(V, ηω + Z̃, t) dZ̃, (2.25)

where η ∈ R and ω⊥ is the orthogonal complement of ω in z, with Lebesgue measure dZ̃.Let zo = Rω, and consider the subspaces no = v ⊕ zo and so = no ⊕ a of s, with the

scalar product induced from that on s. Then no is a Heisenberg-type Lie algebra if onedefines the commutator [V, V ′]o = po([V, V ′]), where po is the orthogonal projection of zonto zo. The associated Lie group No is the classical Heisenberg group of dimension p+1.The Damek–Ricci space So = NoA can be identified with the complex hyperbolic spaceHp/2+1(C) � Go/Ko, where Go = SU(p2 +1, 1) � NoAKo, and Ko = S(U(p2 +1)×U(1)).The index o will denote quantities relative to So. For example we let Qo = p/2 + 1,Bo = {(V, η, t) ∈ so : R2

o = |V |2 + η2 + t2 < 1} (the unit ball in so), Co : So → Bo (theCayley transform), Sp+1 = ∂Bo = {(V, η, t) ∈ so : |V |2+η2+t2 = 1} � Ko/Mo (the unitsphere in so), C(o)

0 : No → ∂Bo \ {(0, 0, 1)} (the generalized stereographic projection),and denote by L, Lo the Laplace–Beltrami operators on S, So, respectively.

The function Rωf is in C∞0 (So), and Rω : C∞

0 (S) → C∞0 (So) is injective. By [23,

Proposition 16, p. 81], we have

Rω

(L + Q2

4)f =

(Lo + Q2

o

4)Rωf. (2.26)

By [9, Proposition 4.12], f is radial on S if and only if Rωf does not depend on ω andis radial on So. More generally, f is biradial on S if and only if Rωf does not dependon ω but only on |V |, |η| and t, i.e., it is biradial on So. We shall write Rf in place ofRωf for f biradial.

In [21, Theorem 3.3], it is proved that the Radon transform of the spherical functionφλ on S is well defined for −Qo/2 < Im λ < 0 (i.e., the integral in (2.25) is convergent),and is related to the spherical function φ

(o)λ on So by

Rφλ = α(λ)φ(o)λ ,

for some constant α(λ). From the asymptotic behaviour of the functions φλ,k,l in (2.14),one proves in a similar way that R(fλ,k,l ◦ C) is well defined for −Qo/2 < Im λ < 0.By (2.26), the function R(fλ,k,l ◦ C) is a biradial eigenfunction of Lo with eigenvalue−λ2 − Q2

o

4 . Then it can be expanded as

R(fλ,k,l ◦ C) =∑′ ′

αk′,l′

k,l (λ)f (o)λ,k′,l′ ◦ Co, (2.27)

k �l


for suitable constants αk′,l′

k,l (λ), where the functions f (o)λ,k′,l′ on Bo (k′, l′ ∈ Z, k′ � l′ � 0)

are given by (2.13), (2.14) and (2.8) with q = 1. More precisely, in geodesic polar coor-dinates (ro, ωo) ∈ (0,∞) × ∂Bo around the origin in Bo, we have (dropping primes)

f(o)λ,k,l(ro, ωo) = φ

(o)λ,k,l(ro)χ

(o)k,l (ωo),

φ(o)λ,k,l(ro) = φ

(αo,βo)2λ,k,l

(ro2)

= q(o)k,l (λ)

(2 sh ro

2)k+l(2 ch ro

2)k−l

φ(αo+k+l,k−l)2λ

(ro2),

χ(o)k,l (ωo) = χ

(o)k,l (ρo, φo) = R

(αo−1,k−l)l

(2ρ2

o − 1)ρk−lo cos(k − l)φo,

where t = Roρo cosφo and |η| = Roρo sinφo for (V, η, t) ∈ Bo, with Ro = th ro2 , and

αo = p/2, βo = 0. Moreover q(o)k,l (λ) is given by (2.30) with Q → Qo and d → do = p+ 2.

By explicit computation, one can show that there is only a finite number of termsin the sum in (2.27), the “highest” one being for (k′, l′) = (k, l). This means that thefunction R(fλ,k,l ◦C) is Ko-finite on Go/Ko. For example one can compute R(fλ,k,l ◦C)at the points (0, η, 0) of the center of No. Here we have ρo = 1, sh2 ro

2 = 14η

2, andcosφo = th ro

2 . At the points (0, ηω + Z̃, 0) along the center of N we have, similarly,ρ = 1, sh2 r

2 = 14(η2 + |Z̃|2), and cosφ = th r

2 . Then, for −Qo/2 < Imλ < 0, we get

R(fλ,k,l ◦ C)(0, η, 0)

= vol(Sq−2)22k+β−1qk,l(λ)

∞∫ro

(sh r

2)k+l(ch r

2)k−l

× φ(α+k+l,β+k−l)2λ

(r2)R

(β−1/2,β−1/2)k−l

(th r

2)(ch r − ch ro)β−1 sh r dr, (2.28)

where β = (q − 1)/2. For q odd we can compute this integral using the differentiationformula for Jacobi functions and repeated integrations by parts. For q even we also needwell-known fractional integral representations for Jacobi functions. (Use [19, formula(2.15)], for both functions Φ

(α,β)λ (t) and Φ

(α,β)−λ (t), and then substitute in (2.5). This

yields a formula for φ(α,β)λ (t) which holds for |Im λ| < Re(α + β + 1).) In all cases we

make use of Gauss relations between contiguous hypergeometric functions to rewrite theresult in terms of the functions f

(o)λ,k′,l′ . For example for (k, l) = (1, 0), (1, 1), we obtain

from (2.28):

(R(fλ,1,0 ◦ C)

)◦ C−1

o = α(λ)f (o)λ,1,0 + β(λ)f (o)

λ,0,0,(R(fλ,1,1 ◦ C)

)◦ C−1

o = γ(λ)f (o)λ,1,1 + δ(λ)f (o)

λ,0,0,

for suitable constants α(λ), β(λ), γ(λ), δ(λ). Thus R(fλ,1,0◦C) is Ko-finite of type (δ1,0⊕δ0,1)⊕δ0,0, and R(fλ,1,1 ◦C) is Ko-finite of type δ1,1⊕δ0,0, in the notations of Remark 2.

This accounts for the Radon transform of the left-hand side of (2.23). Consider nowthe right-hand side of (2.23) in the open model, say gλ,k,l(nat). It can be rewritten as aconvolution over N , namely


gλ,k,l(nat) =(P−λ,k,l ∗ Pλ(at, ·)

)(n),

where

Pλ,k,l(n) = Pλ(e, n)χk,l

(C0(n)

).

Using the same notation R for the Radon transform applied to functions on N , we have

R(f ∗ g) = Rf ∗o Rg,

∗o denoting convolution over the group No. In [10] it was computed that for Imλ >

−Qo/2,

(RPλ(at, ·)

)(no) = β(λ)P(o)

λ (at, no),

for some constant β(λ) and no ∈ No. In a similar way, we compute that for Imλ < Qo/2,

(RP−λ,k,l)(no) =∑k′�l′

βk′,l′

k,l (λ)P (o)−λ,k′,l′(no)

=∑k′�l′

βk′,l′

k,l (λ)P(o)−λ(e, no)χ(o)

k′,l′(C

(o)0 (no)

),

for suitable constants βk′,l′

k,l (λ). Again there is only a finite number of terms in the sum.It follows that for |Im λ| < Qo/2,

Rgλ,k,l(noat) =∑k′�l′

βk′,l′

k,l (λ)β(λ)(P

(o)−λ,k′,l′ ∗o P

(o)λ (at, ·)

)(no)

=∑k′�l′

βk′,l′

k,l (λ)β(λ)⟨π

(o)λ (noat)P(o)

λ (e, ·),P(o)λ

(e, ·)(χ

(o)k′,l′ ◦ C

(o)0

)⟩L2(No).

(2.29)

In the ball model Bo of NoA, the scalar product in (2.29) is just the Poisson integral∫∂Bo

Q(o)λ (bo, ωo)χ(o)

k′,l′(ωo) dωo.

From symmetric space theory, this is proportional to f(o)λ,k′,l′(bo) (see [15, Lemma 4.2],

with j = 1). One can prove that if (2.20) holds, so that qk,l(λ) is given by (2.30) below,then the constants αk′,l′

k,l (λ) in (2.27) and βk′,l′

k,l (λ)β(λ) in (2.29) match up. The resultfλ,k,l = gλ,k,l follows for −Qo/2 < Imλ < 0, and then for all λ by analytic continuation,using the injectivity of the Radon transform. This concludes the proof of the theorem. �


For the proof of the Paley–Wiener theorem in the biradial case, we shall need theprecise value of the constant qk,l(λ) in (2.14) such that (2.19)–(2.20) hold. One finds

qk,l(λ) = (−iλ + Q/2)k(−iλ + Qo/2)l(d/2)k+l

. (2.30)

This can also be written as a ratio of c-functions, namely

qk,l(λ) = cα,β(−2λ)cα+k+l,β+k−l(−2λ) , (2.31)

where

cα,β(λ) = 2α+β+1−iλΓ (iλ)Γ (α + 1)Γ ( iλ+α+β+1

2 )Γ ( iλ+α−β+12 )

. (2.32)

(See [20, (2.18), (4.15), (8.5) and (8.7)]. For the symmetric case, see [20, (8.13), and thelast part of section 8.1]. See also [12, Theorem 7 and remark in p. 277].) Note that thec-function c(λ) in (1.3) is precisely cα,β(2λ) if α = (p + q − 1)/2 and β = (q − 1)/2, aswe continue to assume.

3. The Helgason Fourier transform in the biradial case

3.1. Fourier series expansion of f and f̃

Let f be a biradial function in C∞0 (B)�, with supp f ⊂ BR. The function ω → f(r, ω)

can be expanded in the Fourier series

f(r, ω) =∑k�l

πk,lak,l(r)χk,l(ω), (3.1)

where the Fourier coefficients

ak,l(r) =∫

Sp+q

f(r, ω)χk,l(ω) dω (3.2)

are smooth functions of the geodesic distance r supported in the interval [0, R].Let f̃(λ, ω) be the Fourier transform of f given by (1.1). By the direct part of The-

orem 1.1 (that was proved in [3, Theorem 4.5]), the function f̃(λ, ω) is holomorphic ofuniform exponential type with constant R.

Lemma 3.1. Let f ∈ C∞0 (B)�. Then, for each λ ∈ C, the function ω → f̃(λ, ω) is biradial

on Sp+q.


Proof. First suppose M is transitive on Sp−1, so that biradiality is equivalent toM -invariance. Then, by (2.21), we have, for m ∈ M ,

f̃(λ,m · ω) =∫B

f(b)Qλ(b,m · ω) db

=∫B

f(b)Qλ

(m−1 · b, ω

)db

=∫B

f(m · b)Qλ(b, ω) db = f̃(λ, ω),

by the M -invariance of the Riemannian measure.Consider now the general case. For simplicity, we write f in place of f ◦ C. Let

f̃(λ, n) =∫S

f(x)Qλ(x, n) dx

be the normalized Helgason Fourier transform of f in S = NA, and let

f̂(λ, n) =∫S

f(x)Pλ(x, n) dx

be the unnormalized Helgason Fourier transform of f , so that (cf. (2.17))

f̂(λ, n) = Pλ(e, n)f̃(λ, n).

Since n → Pλ(e, n) is biradial (cf. (2.16) and (2.18)), it is enough to prove thatn → f̂(λ, n) is biradial. Given a unit vector ω2 in z, i.e., ω2 ∈ Sq−1 (q > 1), consider theRadon transform Rω2 in N of the function n → f̂(λ, n). This is related to the HelgasonFourier transform in So = NoA of the function Rω2f , where Rω2f is the Radon transformof f in NA. Indeed by [26, Proposition 5.1], we have

(Rω2 f̂(λ, ·)

)(no) = cqBp,q(λ)P(o)

λ (e, no)(Rω2f)˜(λ, no)

= cqBp,q(λ)(Rω2f)̂(λ, no) (no ∈ No),

where cq is a constant depending only on q, and Bp,q(λ) is the meromorphic function

Bp,q(λ) =Γ ( q−1

2 )Γ (p4 + 12 − iλ)

Γ (Q2 − iλ).

Since f is biradial in S, the function Rω2f is biradial in So, and its Helgason Fouriertransform (normalized or not) is biradial in no ∈ No (since Mo is transitive on the unit


sphere Sp−1 in vo = v). It follows that Rω2 f̂(λ, ·) is biradial in no, and finally f̂(λ, n) isbiradial in n. �

By this lemma, we have the Fourier expansion

f̃(λ, ω) =∑k�l

πk,lbk,l(λ)χk,l(ω), (3.3)

where the coefficients

bk,l(λ) =∫

Sp+q

f̃(λ, ω)χk,l(ω) dω

are holomorphic functions of λ of exponential type R.The functions ak,l(r) and bk,l(λ) are related as follows. Let α′ = α + k + l and

β′ = β + k − l, where α and β are given by (2.10). Define cα′,β′(λ) by (2.32). LetJ (α′,β′)(g) be the Jacobi transform of g ∈ C∞

R (R)even, defined by [20]

J (α′,β′)(g)(λ) =∞∫0

g(t)φ(α′,β′)λ (t)(2 sh t)2α

′+1(2 ch t)2β′+1 dt,

with inverse (see [20, Theorem 2.3])

(J (α′,β′))−1(h)(t) = 1

2π

∞∫0

h(λ)φ(α′,β′)λ (t) |cα′,β′(2λ)|−2 dλ.

Proposition 3.2. We have

bk,l(λ) = vol(Sp+q)2q qk,l(λ)

∞∫0

ak,l(r)φ(α′,β′)2λ

(r

2

)(2 sh r

2

)p+q+k+l(2 ch r

2

)q+k−l

dr

(3.4)

= vol(Sp+q)2q−1 qk,l(λ)J (α′,β′)

(ak,l(2t)

(2 sh t)k+l(2 ch t)k−l

)(2λ), (3.5)

where qk,l(λ) is given by (2.30), and conversely

ak,l(r) = cp,q2π

(2 sh r

2

)k+l(2 ch r

2

)k−l∞∫0

qk,l(−λ)φ(α′,β′)2λ

(r

2

)bk,l(λ) dμ(λ) (3.6)

= 2q−1

vol(Sp+q)

(2 sh r

2

)k+l(2 ch r

2

)k−l(J (α′,β′))−1

(bk,l(λ

′

2 )λ′

)(r

2

)(3.7)

qk,l( 2 )


= 2q

vol(Sp+q)

(2 sh r

2

)k+l(2 ch r

2

)k−l 12π

×∞∫0

bk,l(λ)qk,l(λ)φ

(α′,β′)2λ

(r

2

)∣∣cα′,β′(2λ)∣∣−2

dλ. (3.8)

Proof. We have, by (2.22) and (2.13),

bk,l(λ) =∫

Sp+q

(∫B

Qλ(b, ω)f(b) db)χk,l(ω) dω

=∫B

( ∫Sp+q

Qλ(b, ω)χk,l(ω) dω)f(b) db

=∫B

fλ,k,l(b)f(b) db

=∞∫0

∫Sp+q

fλ,k,l(r, ω)f(r, ω)J(r) dr dω

=∞∫0

( ∫Sp+q

f(r, ω)χkl(ω) dω)φλ,k,l(r)J(r) dr

=∞∫0

ak,l(r)φλ,k,l(r)J(r) dr.

Recalling (2.3) and (2.14) we obtain (3.4). Note that, since r → ak,l(r) ∈ C∞R ([0,∞)) and

λ → φ(α+k+l,β+k−l)2λ ( r2 ) is entire, the integral in (3.4) is an entire function of λ. Since λ →

qk,l(λ) is a polynomial (cf. (2.30)), both functions λ → bk,l(λ) and λ → bk,l(λ)/qk,l(λ)are entire of exponential type R (see [14, Lemma 5.13, p. 288]), the second one beingeven.

To prove (3.5) we just rewrite (3.4) as

bk,l(λ) = vol(Sp+q)2q−1 qk,l(λ)

∞∫0

ak,l(2t)(2 sh t)k+l(2 ch t)k−l

φ(α+k+l,β+k−l)2λ (t)

× (2 sh t)2(α+k+l)+1(2 ch t)2(β+k−l)+1 dt

= vol(Sp+q)2q−1 qk,l(λ)J (α+k+l,β+k−l)

(ak,l(2t)

(2 sh t)k+l(2 ch t)k−l

)(2λ).

On the other hand, by substituting the Fourier series (3.3) in the inversion formula(1.2), and using (2.22), we get


f(b) = cp,q2π

∞∫0

∫Sp+q

Q−λ(b, ω)(∑

k�l

πk,lbk,l(λ)χk,l(ω))dμ(λ) dω

= cp,q2π

∑k�l

πk,l

∞∫0

( ∫Sp+q

Q−λ(b, ω)χk,l(ω) dω)bk,l(λ) dμ(λ)

= cp,q2π

∑k�l

πk,l

∞∫0

f−λ,k,l(b)bk,l(λ) dμ(λ),

i.e., in geodesic polar coordinates (cf. (2.13)),

f(r, ω) = cp,q2π

∑k�l

πk,lχk,l(ω)∞∫0

φ−λ,k,l(r)bk,l(λ)dμ(λ).

Comparing with (3.1) we see that

ak,l(r) = cp,q2π

∞∫0

φ−λ,k,l(r)bk,l(λ) dμ(λ), (3.9)

and by (2.14) we get (3.6). Now (3.6) is precisely the formula one gets by inverting theJacobi transform in (3.5). Indeed, recalling (2.31), we have from (3.5):

ak,l(r)(2 sh r/2)k+l(2 ch r/2)k−l

= 2q−1

vol(Sp+q)(J (α′,β′))−1

(bk,l(λ

′

2 )qk,l(λ

′

2 )

)(r

2

)

= 2q−1

vol(Sp+q)12π

∞∫0

bk,l(λ′

2 )qk,l(λ

′

2 )φ

(α′,β′)λ′

(r

2

)∣∣cα′,β′(λ′)∣∣−2

dλ′

= 2q

vol(Sp+q)12π

∞∫0

bk,l(λ)qk,l(λ)φ

(α′,β′)2λ

(r

2

)∣∣cα′,β′(2λ)∣∣−2

dλ

= 2q

vol(Sp+q)12π

∞∫0

bk,l(λ)φ(α′,β′)2λ

(r

2

)cα′,β′(−2λ)cα,β(−2λ)

1cα′,β′(2λ)cα′,β′(−2λ) dλ

= 2q

vol(Sp+q)12π

∞∫0

bk,l(λ) cα,β(2λ)cα′,β′(2λ)φ

(α′,β′)2λ

(r

2

)1

cα,β(2λ)cα,β(−2λ) dλ

= 2q

vol(Sp+q)12π

∞∫bk,l(λ)qk,l(−λ)φ(α+k+l,β+k−l)

2λ

(r

2

)∣∣cα,β(2λ)∣∣−2

dλ.

0


This proves (3.7)–(3.8), since cp,q = 2q/ vol(Sp+q) and dμ(λ) = |cα,β(2λ)|−2 dλ, andconcludes the proof of the proposition. �Remark 4. From (3.5)–(3.7) and the Paley–Wiener theorem for the Jacobi trans-form [20, Theorem 2.1], it is clear that the map ak,l(r) → bk,l(λ) is, ∀k, l, a bi-jection from the space of smooth functions ak,l on [0,∞) compactly supported in[0, R] and such that the function r → ak,l(r)

(sh r/2)k+l(ch r/2)k−l extends to C∞R (R)even, i.e.,

ak,l ∈ (sh r2 )k+l(ch r

2 )k−lC∞R (R)even, onto the space of holomorphic functions bk,l on C

such that λ → bk,l(λ)/qk,l(λ) is in PWR(C)even, i.e., bk,l ∈ qk,l(·) PWR(C)even. HerePWR(C)even is the space of even entire functions on C of exponential type R. The proofof the converse part of Theorem 1.1 in the biradial case (see below) implies that theFourier coefficients bk,l(λ) of a function ψ(λ, ω) satisfying (1.4), (1.5) and biradial in ω

are indeed in qk,l(·) PWR(C)even.

3.2. The biradial Paley–Wiener theorem

We now prove Theorem 1.1 in the biradial case.

Theorem 3.3. The Fourier transform f(b) → f̃(λ, ω) is a bijection of C∞0 (B)� onto the

set of holomorphic functions ψ(λ, ω) of uniform exponential type, biradial in ω, andsatisfying the condition∫

∂B

Q−λ(b, ω)ψ(λ, ω) dω =∫∂B

Qλ(b, ω)ψ(−λ, ω) dω (3.10)

for any b ∈ B and λ ∈ C. Moreover f̃ satisfies (1.4) if and only if f has support in theclosed ball BR = {b ∈ B : d(b, C(e)) � R}.

Proof. In view of Lemma 3.1 and [3, Theorem 4.5], we only need to prove the conversepart, in particular the onto statement.

Let ψ(λ, ω) be a holomorphic function of uniform exponential type such that (1.4)and (3.10) hold, and such that the map ω → ψ(λ, ω) is biradial on Sp+q, ∀λ ∈ C. Definebk,l(λ) =

∫Sp+q ψ(λ, ω)χk,l(ω) dω, so that (cf. (3.3))

ψ(λ, ω) =∑k�l

πk,lbk,l(λ)χk,l(ω). (3.11)

Then λ → bk,l(λ) is holomorphic of exponential type R. Using (3.11) in the integral∫∂B

Q−λ(b, ω)ψ(λ, ω) dω we get by (2.22)∫Q−λ(b, ω)ψ(λ, ω) dω =

∑k�l

πk,lbk,l(λ)f−λ,k,l(b). (3.12)
∂B


From (3.10), (2.13)–(2.14) and (3.12), it follows that the function λ → bk,l(λ)qk,l(−λ) iseven. Thus the function

λ → bk,l(λ)qk,l(λ) = bk,l(λ)qk,l(−λ)

qk,l(λ)qk,l(−λ)

is also even. Define f by the inversion formula (1.2):

f(b) = cp,q2π

∞∫0

∫Sp+q

Q−λ(b, ω)ψ(λ, ω) dω dμ(λ).

Then f is smooth and biradial on B (by (3.12)). Define ak,l(r) by (3.1)–(3.2). Then weget again (3.9), (3.6), and (3.8).

By (2.30) we see that the function λ → bk,l(λ)/qk,l(λ) in (3.8) has no poles forIm λ � 0. Then, using the exponential type conditions for the functions bk,l(λ) and thewell-known asymptotic estimates for the functions φ(α′,β′)

2λ ( r2 ) in (3.8), we can prove thatak,l(r) = 0 for r > R.

In more detail, we use

φ(α′,β′)2λ

(r2)

= cα′,β′(2λ)Φ(α′,β′)2λ

(r2)

+ cα′,β′(−2λ)Φ(α′,β′)−2λ

(r2),

where the function λ → Φ(α′,β′)λ (t) is holomorphic in C \ {−iN} for each t > 0 (cf. [11,

Proposition 1]), to rewrite the integral in (3.8) for r > 0 as

F (r) =∞∫

−∞

bk,l(λ)qk,l(λ)

Φ(α′,β′)2λ ( r2 )

cα′,β′(−2λ) dλ.

Now (cα′,β′(−2λ))−1 has no poles for Imλ � 0, and the integrand is holomorphic forIm λ � 0. Thus we obtain by Cauchy’s theorem

F (r) =∞∫

−∞

bk,l(ξ + iη)qk,l(ξ + iη)

Φ(α′,β′)2(ξ+iη)(

r2 )

cα′,β′(−2(ξ + iη)) dξ

for any η � 0. We now use the estimates for Φ(α′,β′)λ (t) and cα′,β′(λ) given in [11,

Theorem 2] (see also [20, (6.4) and (6.5)]), namely for any c > 0 there exists K1 > 0such that for all t � c and all λ ∈ C with Imλ � 0,

∣∣Φ(α′,β′)λ (t)

∣∣ � K1e−(Im λ+α′+β′+1)t.

Moreover, there exists K2 > 0 such that for all λ ∈ C with Imλ � 0,


∣∣cα′,β′(−λ)∣∣−1 � K2

(1 + |λ|

)α′+ 12 .

Using the exponential type conditions for bk,l(λ) and the following inequality (which iseasily proved from (2.30)):∣∣∣∣ 1

qk,l(ξ + iη)

∣∣∣∣ � (d/2)k+l

(Q/2)k(p/4 + 1/2)l(∀ξ, ∀η � 0),

we find (as in [11, p. 157])

∣∣F (r)∣∣ � Ke−(2η+Q+2k) r2

∞∫−∞

∣∣bk,l(ξ + iη)∣∣(1 + 2|ξ + iη|

)p+q2 +k+l

dξ

� K ′e−(2η+Q+2k) r2 eηR � K ′eη(R−r)

for suitable constants K,K ′. Since this holds for all η � 0, we get F (r) = 0 for r > R,as claimed. It follows from (3.1) that f(b) = f(r, ω) has support in BR. The proof iscompleted by showing that the Fourier transform of f is just f̃(λ, ω) = ψ(λ, ω). In fact,the Fourier coefficients of ω → f̃(λ, ω) are just the bk,l(λ). �References

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