Digital Object Identifier (DOI) 10.1007/s002090100379Math. Z. 242, 761–779 (2002)

Revisiting Hardy’s theorem for the Heisenberg group

S. Thangavelu

Stat-Math Division, Indian Statistical Institute, 8th Mile Mysore Road,Bangalore, 560 059, India (e-mail: veluma@isibang.ac.in)

Received: 9 January 2001 / in final form: 17 April 2001Published online: 1 February 2002 –c© Springer-Verlag 2002

Dedicated to Eli Stein on his 70th birthday

Abstract. We establish several versions of Hardy’s theorem for the Fouriertransform on the Heisenberg group. Letf(λ) be the Fourier transform of afunctionf onHn and assumef(λ)∗f(λ) ≤ c p2b(λ) whereps is the heatkernel associated to the sublaplacian. We show that if|f(z, t)| ≤ c pa(z, t)thenf = 0 whenevera < b. Whena ≥ b we replace the condition onf by|fλ(z)| ≤ c pλa(z)wherefλ(z) is the Fourier transform off in thet-variable.Under suitable assumptions on the ‘spherical harmonic coefficients’ off(λ)we prove: (i)fλ(z) = c(λ)pλa(z) whena = b; (ii) when a > b there areinfinitely many linearly independent functionsf satisfying both conditionsonfλ andf(λ).

1 Introduction and the main results

The aim of this paper is to establish an analogue of Hardy’s theorem forthe Fourier transform on the Heisenberg group. This is a continuation ofthe author’s earlier paper [22] in which a version of Hardy’s theorem wasproved. That theorem in turn has its roots in the ‘Hardy’s theorem for theWeyl transform’ which appeared in the author’s monograph [21]. If this isso one wonders, as the author often does himself, why he is visiting Hardy’stheorem again and again, each time reformulating and refining his earlierresults. We have no explanation to offer except possibly the compulsion ofthe demanding critic within himself who never leaves the author at peaceuntil his theorems, if not their proofs, are straight from the ‘Book’. So, likea painter, with an extra stroke or two we try to change the shade a bit hereand there in an attempt to draw closer to the truth.

762 S. Thangavelu

We can think of Hardy’s theorem as an uncertainty principle whichroughly says that a function and its Fourier transform both cannot be de-caying fast at infinity unless, of course, the function is identically zero. Ourfirst result, Theorem 1.1 below gives the optimal version of this uncertaintyprinciple on the Heisenberg group. As the Fourier transform on the Heisen-berg group is operator valued we have to measure the decay of the Fouriertransform in terms of the Hermite semigroup. We can also think of Hardy’stheorem as one characterising the heat kernel associated to the sublaplacian.Unlike the Euclidean case the heat kernel on the Heisenberg group is notexplicitly known. What is known is its partial Fourier transform in the cen-tral variable and our result Theorem 1.2 characterises exactly this partialFourier transform of the heat kernel.

Consider the Fourier transformf(λ), λ ∈ R, λ �= 0 of a functionfon the Heisenberg groupHn. Let pt(z, s), (z, s) ∈ Hn, t > 0 be the heatkernel associated to the sublaplacianL onHn. The Fourier transform ofptis given bypt(λ) = e−tH(λ) whereH(λ) = −∆ + λ2|x|2 is the (scaled)Hermite operator onRn. As an analogue of Hardy’s theorem for the groupFourier transform onHn we offer:

Theorem 1.1. Let f be a measurable function onHn which satisfies theestimate|f(z, s)| ≤ c (1 + |z|2)mpa(z, s) for somea > 0,m ≥ 0. Furtherassume thatf(λ)∗f(λ) ≤ c H(λ)mp2b(λ), for someb > 0 and for allλ �= 0. Thenf = 0 whenevera < b.

This is the analogue of classical Hardy’s theorem for the Fourier trans-form onR

n which says that if

|f(x)| ≤ c(1 + |x|2)me−a|x|2 , |f(ξ)| ≤ c(1 + |ξ|2)me−b|ξ|2 (1.1)

wheref(ξ) is the Fourier transform off given by

f(ξ) = (2π)− n2

∫Rn

f(x)e−ix·ξdx

then (i)f = 0 wheneverab > 14 ; (ii) f(x) = c p(x)e−a|x|2 wherep(x) is a

polynomial of degree≤ 2mwhenab = 14 and (iii) there are infinitely many

linearly independent functions satisfying the conditions (1.1) whenab < 14 .

To see the analogy of this result with Theorem 1.1 we only have to rewriteconditions (1.1) in terms of the heat kernel

pt(x) = (4πt)− n2 e−

14t

|x|2

associated to the standard Laplacian∆ onRn.

The above result for the Euclidean Fourier transform was first establishedby Hardy [6] forn = 1 andm = 0. Now several versions of this interesting

Revisiting Hardy’s theorem for the Heisenberg group 763

result are known, see [13] and [23]. In [22] we have established Theorem1.1 whenm = 0 anda < b

2 . The optimal resulta < b will be establishedin this paper. As in the Euclidean case it is natural to ask what happens ifa ≥ b in Theorem 1.1. In order to state our results for these cases we needto establish some more notation.

We are going to replace the condition onf by a condition on the EuclideanFourier transformfλ(z) of f(z, t) in the t-variable. For each pair of non-negative integers(p, q) letSpq be the space of bigraded spherical harmonicsof degree(p, q). ThenL2(S2n−1) is the orthogonal direct sum ofSpq, p, q ≥0. Fix an orthonormal basis{Y j

pq : p, q ≥ 0, j = 1, 2, . . . d(p, q)} forL2(S2n−1) with Y j

pq ∈ Spq. (We say more about these spherical harmonicsin Sect. 2).

Theorem 1.2. Letf be an integrable function onHn whose Fourier trans-form satisfies the conditionf(λ)∗f(λ) ≤ c p2a(λ) for somea > 0 and forall λ �= 0. Further assume thatfλ(z) satisfies∣∣∣∣∣∣

∫S2n−1

fλ(|z|z′)Y jpq(z

′)dz′

∣∣∣∣∣∣ ≤ cjpq pλa(z) (1.2)

for all Y jpq andλ �= 0. Thenf = ϕ ∗3 pa whereϕ is a tempered distribution

onR with ϕ ∈ L∞(R) and∗3 stands for the convolution in thet-variable.

It would be ideal to say something in Theorem 1.1 whena = b butunfortunately we are not able to draw any conclusion. Even in Theorem 1.2we are not able to conclude thatf(z, t) = c pa(z, t); this is not surprisingsince the hypotheses are satisfied byf ∗3 ϕ as well and hence we canonly conclude thatf = ϕ ∗3 pa. The casea > b is still excluded andour next theorem treats this case. We are going to weaken the conditionf(λ)∗f(λ) ≤ c p2b(λ) for which some more definitions are needed.

Let Φλα, α ∈ N

n be the (scaled) Hermite functions onRn which areeigenfunctions ofH(λ). Let Hpq be the space of all polynomials of theform P (z) = |z|p+qY (z′) with Y ∈ Spq and defineWλ(P ) to be theWeyl correspondence ofP . Let Eλ

k be the finite dimensional subspace ofL2(Rn) spanned by{Φλ

α : |α| = k}. If T andS are bounded operatorson L2(Rn), then for their restrictions toEλ

k there is a natural inner prod-uct (T, S)k defined by Geller [5]. Under this inner product the operatorsWλ(P j

pq), Pjpq(z) = |z|p+qY j

pq(z′) form an orthogonal system. They can beconsidered as the operator analogues of Spherical harmonics. (Details ofthis will be given in Sect. 2). With these notations we prove

Theorem 1.3. Let f be an integrable function onHn for which holds theestimate|fλ(z)| ≤ c pλa(z). Further assume that

|(f(λ),Wλ(P ))k| ≤ c|λ|p+q+n−1e−(2k+n)|λ|b (1.3)

764 S. Thangavelu

for everyk ∈ N andP (z) = |z|p+qY jpq(z′). Then (i) whena = b, f =

ϕ ∗3 pa with ϕ ∈ L∞(R); (ii) whena > b there are infinitely many linearlyindependent functions satisfying the conditions of the theorem.

Refinements of the classical Hardy theorem have been established in[23] using spherical harmonics. These results in [23] are similar in spirit tothe Helgason’s treatment of the Paley-Wiener theorem [7]. The above twotheorems are analogues of these refinements. Since the perfect symmetrybetweenf and f is lost when we move away from the Euclidean Fouriertransform we have two versions as above instead of one in the Euclideancase.

The problem of establishing an analogue of Hardy’s theorem for Fouriertransforms on Lie groups started with the work of Sitaram and Sundari[18]. For other versions of Hardy’s theorem for semi-simple Lie groups seeCowling et al [3] and Sengupta [16]. Analogues of Hardy’s theorem for theHeisenberg group have been studied in Sitaram et al [17] and Thangavelu[21] and [22]. For step two nilpotent Lie groups see the works of Bagchi andRay [2] and Astengo et al [1]. General nilpotent Lie groups were consideredin Kaniuth and Kumar [9]. Symmetric spaces were treated in Narayanan andRay [12], solvable extensions ofH-type groups by [1]. See also the recentworks [14] and [15] of Sarkar for semi-simple Lie groups.

Analogues of Theorems 1.2 and 1.3 have been established for noncom-pact rank one symmetric spaces in [23]. There we have also proved a versionof Hardy’s theorem for the spectral projections associated to the Laplace-Beltrami operator. We will state and prove a similar theorem for the spectralprojections associated to the sublaplacian in Sect. 4. The plan of the paper isas follows. In Sect. 2 we collect relevant material on the Heisenberg group.Most results we need are contained in the paper [5] of Geller. We also referto the monograph of Folland [4] and that of the author [21]. Theorems 1.1,1.2 and 1.3 will be proved in Sect. 3.

We are extremely thankful to Ms. Ashalata for typing the manuscript.We also wish to thank the referee for his careful reading of the manuscriptand valuable suggestions.

2 Preliminaries on the Heisenberg group

The Heisenberg groupHn is justCn × R equipped with the group law

(z, t)(w, s) =(z + w, t+ s+

12Im(z · w)

).

For eachλ ∈ R, λ �= 0 there is an irreducible unitary representationπλ(z, t)ofHn realised onL2(Rn). These representations are explicitly given by

πλ(z, t)ϕ(ξ) = eiλteiλ(x·ξ+ 12x·y)ϕ(ξ + y),

Revisiting Hardy’s theorem for the Heisenberg group 765

whereϕ ∈ L2(Rn) andz = x + iy. Each infinite dimensional irreducibleunitary representation is equivalent to one of these. The group Fourier trans-form of a functionf ∈ L1(Hn) is defined to be the operator valued function

f(λ) =∫

Hn

f(z, t)πλ(z, t)dz dt.

The representationπλ satisfiesπλ(z, t) = eiλtπλ(z, 0) and therefore

f(λ) =∫

Cn

fλ(z)πλ(z)dz, (2.1)

where we have writtenπλ(z) = πλ(z, 0) and

fλ(z) =

∞∫−∞f(z, t)eiλtdt

is the inverse Fourier transform off in thet-variable.The formula (2.1) suggests that we consider Weyl tranforms of functions

g onCn. These are defined by

Gλ(g) =∫

Cn

g(z)πλ(z)dz. (2.2)

If f ∗ g is the convolution of two functions onHn defined by

f ∗ g(z, t) =∫

Hn

f((z, t)(w, s)−1)g(w, s)dw ds

then it is easily checked that

(f ∗ g)λ(z) = fλ ∗λ gλ(z)

where theλ-twisted convolution offλ andgλ is given by

fλ ∗λ gλ(z) =

∫

Cn

fλ(z − w)gλ(w)eiλ2 Im(z·w)dw.

It then follows thatGλ(fλ ∗λ gλ) = Gλ(fλ)Gλ(gλ). Let Fλf be theλ-

symplectic Fourier transform of a functionf onCn given by

Fλf(z) =∫

Cn

f(z − w)eiλ2 Im(z·w)dw. (2.3)

766 S. Thangavelu

We define the Weyl correspondence of a functionf on Cn byWλ(f) =Gλ(F−1

λ f).The most important result we need is the Hecke-Bochner type identity

for the Weyl transformGλ. In order to state this important formula we needto recall some definitions. For each pair of non-negative integers(p, q) letHpq be the space of all harmonic polynomials of the form

P (z) =∑|α|=p

∑|β|=q

cαβzαzβ (2.4)

wherez ∈ Cn, α, β ∈ Nn. Elements ofHpq are called bigraded solid

harmonics of degree(p, q). LetSpq be the space of all restrictions of elementsof Hpq to the unit sphereS2n−1. The elements ofSpq are called bigradedspherical harmonics. ThenL2(S2n−1) is the orthogonal direct sum of thespacesSpq, p, q ≥ 0. Let {Y j

pq : 1 ≤ j ≤ d(p, q)} be an orthonormal basisfor Spq. The corresponding elements ofHpq are denoted byP j

pq.For each multiindexα ∈ N

n, x ∈ Rn, letΦα(x) be the normalised Her-

mite function which is an eigenfunction of the Hermite operatorH witheigenvalue(2|α| + n) where |α| = α1 + . . . + αn. DefineΦλ

α(x) =|λ|n

4Φα(|λ| 12x) for λ �= 0 so thatH(λ)Φλ

α = (2|α| + n)|λ|Φλα where

H(λ) = −∆ + λ2|x|2. We say that an operatorT acting onL2(Rn) isradial if it is diagonalised byΦλ

α andTΦλα = cλ|α|Φ

λα. LetLδ

k, δ > −1 be theLaguerre polynomials of typeδ. We refer to Szego [20] for various propertiesof Lδ

k. Define the Laguerre functions by

ϕn−1k (z) = Ln−1

k

(12|z|2

)e−

14 |z|2 (2.5)

for z ∈ Cn. With these notations we are now in a position to state Geller’sresult.

Theorem 2.1. SupposegP ∈ L1(Cn) or L2(Cn) whereg is a radial func-tion andP ∈ Hpq. Then forλ > 0, Gλ(gP ) = (−1)qWλ(P )S whereS is aradial operator whose action onΦλ

α is given as follows :SΦλα = cλ|α|(g)Φ

λα

wherecλk(g) = 0 for k < p and fork ≥ p it is given by

cλk(g) =(k − p)!(n− 1)!(k + q + n− 1)!

∫

Cn+p+q

g(z)ϕn−1+p+qk−p (

√λz)dz. (2.6)

Whenλ < 0 the roles ofp and q are reversed in the above definition ofcλk(g).

Revisiting Hardy’s theorem for the Heisenberg group 767

In [5] Geller has studied operator analogues of the spacesHpq whichare given by the operatorsWλ(P ) asP ranges overHpq. Note that theHpq spaces when restricted to each sphererS2n−1 are orthogonal andL2(rS2n−1) is the orthogonal direct sum of these spaces. LetEλ

k be thespan ofΦλ

α, |α| = k and letB(Eλk ) be the space of bounded linear operators

fromEλk intoL2(Rn). OnB(Eλ

k ) we can define an inner product by setting

(T, S)k =(

12|λ|

)n−1 ∑|α|=k

(TΦλα, SΦ

λα). (2.7)

With this notation the following result has been proved in [5]. LetPpq standfor the space of all polynomials of the form (2.4).

Theorem 2.2. SupposeP ∈ Hpq, Q ∈ Pp′q′ and thatp′ ≤ p or q′ ≤ q.Then forλ > 0

(Wλ(Q),Wλ(P ))k

= (2πn)−1(

12|λ|

)p+q+n−1 (k + q + n− 1)!(k − p)! (Q,P ) (2.8)

where(Q,P ) is the inner product inL2(S2n−1). Whenλ < 0, the roles ofp andq are interchanged.

The above result shows that the spacesWλ(Hpq) are mutually orthogonalin the above inner product. In the course of the proof of Theorem 2.1 thefollowing formula has been established: forP ∈ Hpq, λ > 0

(πλ(z),Wλ(P ))k = (−1)q

(12λ

)p+q

P (z)ϕn−1+p+qk−p (

√λz). (2.9)

We will make use of this formula in the proof of Theorem 1.2.Givenacontinuous functionf onCn wecanexpandfr(z′) = f(rz′), r >

0, z′ ∈ S2n−1 in terms of spherical harmonics obtaining

f(rz′) =∑p,q

fpq(rz′)

with fpq(rz′) coming fromSpq. The projectionsfpq(rz′) are given by

fpq(rz′) =d(p,q)∑j=1

∫

S2n−1

f(rw′)Y jpq(w

′)dw′

Y j

pq(z′). (2.10)

We can expressfpq in terms of certain representations of the unitary groupU(n).

768 S. Thangavelu

The natural action ofU(n) on the unit sphereS2n−1 defines a unitaryrepresentation ofU(n) on the Hilbert spaceL2(S2n−1). When restricted toSpq it defines an irreducible representation ofU(n) denoted byδpq. Letχpq

be the character ofδpq. We claim that

fpq(z) = d(p, q)∫

U(n)

f(σz)χpq(σ)dσ. (2.11)

To see this we apply Peter-Weyl theorem to the functionF (σ) = f(σz) toget the expansion

f(z) =∑δ∈K

d(δ)∫K

f(σz)χδ(σ)dσ

whereK = U(n) andK is the unitary dual ofK. LetK0 = U(n− 1) con-sidered as a subgroup ofU(n). Then we can show that (see Helgason [7]) theintegral

∫K

f(σz)χδ(σ)dσ is non-zero only if the groupδ(K0) has a non-zero

fixed vector. Eachδpq is such a representation and all such representationsare accounted for byδpq. Thus we get

f(z) =∑p,q

d(p, q)∫

U(n)

f(σz)χpq(σ)dσ (2.12)

and by the uniqueness of spherical harmonic expansion we can identify eachpiece withfpq.

We also need to make use of some properties of the metaplectic rep-resentations. For eachσ ∈ U(n) the representationπλ(σz, t) agrees withπλ(z, t) at the centre and so by Stone-von Neumann theorem they are uni-tarily equivalent. Hence there is a unitary operatorµλ(σ) such that

πλ(σz, t) = µλ(σ)∗πλ(z, t)µλ(σ). (2.13)

This correspondenceσ → µλ(σ) extends to a unitary representatin of thedouble cover ofU(n) called the metaplectic representation. Eachµλ(σ)leaves invariant the subspacesEλ

k and commute with the projectionsPk(λ)associated toEλ

k . We refer to Folland [4] for more about these representa-tions.

Finally, we recall some properties of the heat kernel associated to thesublaplacianL which is defined by

L = −n∑

j=1

(X2j + Y 2

j ).

Revisiting Hardy’s theorem for the Heisenberg group 769

Here

Xj =∂

∂xj+

12yj∂

∂t

Yj =∂

∂yj− 1

2xj∂

∂t

j = 1, 2, . . . n are the left invariant vector fields onHn which alongwithT = ∂

∂t form an orthonormal basis for the Heisenberg Lie algebra. Thissecond order differential operator plays the role of Laplacian forHn, ishypoelliptic, self-adjoint and non-negative. It generates a diffusion semi-group with kernelpt(z, s). Its Fourier transform in thet-variable is explicitlygiven by

pλt (z) = cnλn(sinh(tλ))−ne−14λ(coth(tλ))|z|2 . (2.14)

See Hulanicki [8] for a derivation of this formula. The group Fourier trans-form of pt is given bypt(λ) = e−tH(λ). The kernel satisfies the pointwiseestimate

|pt(z, s)| ≤ c t−n−1e−At

|(z,s)|2 , (2.15)

where|(z, s)| = (|z|4 + s2)14 is the homogeneous norm on the Heisenberg

group.

3 Proofs of the main results

In this section we prove all the three versions of the Hardy’s theorem statedin the introduction. We begin with a proof of Theorem 1.3. In what followscλ will stand for constants depending onλ and other parameters which willvary from one inequality to another.

The hypothesis onfλ(z) together with the explicit formula (2.14) forpλa(z) gives us the estimate

|fλ(z)| ≤ cλe−14 (coth(aλ))|z|2 . (3.1)

Recalling the definition of the inner product(T, S)k onB(Eλk ) we have for

P = P jpq

(f(λ),Wλ(P ))k =(

12|λ|

)n−1 ∑|α|=k

(f(λ)Φλα,Wλ(P )Φλ

α)

which is given by the integral

(12|λ|

)n−1 ∫

Cn

fλ(z)

∑

|α|=k

(πλ(z)Φλα,Wλ(P )Φλ

α)

dz. (3.2)

770 S. Thangavelu

Without loss of generality assumeλ > 0 and use formula (2.9) to get

(f(λ),Wλ(P ))k = (−1)q

(12λ

)m−1 ∫

Cn

fλ(z)P (z)ϕm−1k−p (

√λz)dz, (3.3)

where we have writtenm = n+ p+ q.Note thatP (z) = P j

pq(z) = |z|p+qY jpq(z′). Defining

fλpqj(|z|) =

∫S2n−1

fλ(z)Y jpq(z

′)dz′ (3.4)

the expression (3.3) reads

(f(λ),Wλ(P ))k = (−1)q(12λ)m−1

∫

Cm

fλpqj(|z|)|z|−p−qϕm−1

k−p (√λz)dz

where we are treatingfλpqj(|z|) as radial function onCm. Thus the hypothesis

on (f(λ),Wλ(P ))k gives us the estimate

(12λ)m−1

∣∣∣∣∣∣∣∫

Cm

gλpqj(z)ϕm−1k (

√λz)dz

∣∣∣∣∣∣∣≤ cλe−(2k+m)λb (3.5)

for the Laguerre coefficients of the functiongλpqj(z) = |z|−p−qfλpqj(|z|).

We will use this to estimate the symplectic Fourier transform ofgλpqj . Let uswrite g in place ofgλpqj for the sake of simplicity of notation.

Consider the Laguerre expansion ofg onCm given by

g(z) =(

12λ

)m ∞∑k=0

k!(k +m− 1)!

ck(g)ϕm−1k (

√λz)

where we have writtenck(g) to stand for the integral on the left hand side of(3.5). The Laguerre functionsϕm−1

k satisfy the generating function identity(see Szego [20])

∞∑k=0

rkϕm−1k (z) = (1 − r)−me−

14

1+r1−r

|z|2 .

¿From this it is easily seen thatϕm−1k (

√λz) are eigenfunctions of theλ-

symplectic Fourier transform with eigenvalues(−1)k. Therefore, the La-guerre expansion ofFλg is given by

Fλg(z) = cλ∞∑

k=0

k!(k +m− 1)!

ck(g)(−1)kϕm−1k (

√λz).

Revisiting Hardy’s theorem for the Heisenberg group 771

Applying Cauchy-Schwarz inequality and using the estimate (3.5) we get

|Fλg(z)|2 (3.6)

≤ cλ∞∑

k=0

k!(k +m− 1)!

(2k +m)2e−2(2k+m)λb(ϕm−1

k (√λz)

)2.

Defining

F (t) =∞∑

k=0

k!(k +m− 1)!

e−(2k+m)t(ϕm−1

k (√λz)

)2

we note that|Fλg(z)|2 ≤ cλ|F ′′(2λb)|.The Laguerre functions satisfy another generating function identity,

namely

∞∑k=0

k!(k +m− 1)!

(Lm−1

k (s2)e−12 s2

)2rk

= (1 − r)−1(−s4r)−(m−12 )e−

12

1+r1−r

s2Jm−1

(2is2

√r

1 − r), (3.7)

whereJm−1 is the Bessel function of order(m− 1). This formula gives anexpression for the functionF (t). Taking two derivatives ofF (t) and usingddt(t

−αJα(t)) = −t−αJα+1(t) and the estimate|Jα(it)| ≤ c t− 12 et, t→ ∞

satisfied by all Bessel functionsJα(t), we can easily get the estimate

|Fλg(z)| ≤ cλQ(z, z)e−14λ(tanh(bλ))|z|2 (3.8)

whereQ(z, z) is a polynomial inz andz.Let g(z) stand for the Euclidean Fourier transform onCn which can

be expressed in terms of theλ-symplectic Fourier transformFλg(z). Thisleads to the estimate

|g(z)| ≤ cλQ1(z, z)e−tanh(bλ)

λ|z|2 , (3.9)

whereQ1 is another polynomial. Sincefλ(z) satisfies the estimate (3.1) itfollows thatg(z) = fλ

pqj(z)|z|−p−q also satisfies the estimate

|g(z)| ≤ cλe−14λ(coth(aλ))|z|2 (3.10)

as|z| → ∞. Now we can appeal to Hardy’s theorem onCm. Whena = bit follows from (3.9) and (3.10) that

gλpqj(z) = cpqj(λ)e−14λ(coth(aλ))|z|2 .

772 S. Thangavelu

Sincegλpqj(z) = fλpqj(z)|z|−p−q, the above is not compatible with the esti-

mate

|fλpqj(z)| ≤ cλe−

14λ(coth(aλ))|z|2

unlesscpqj(λ) = 0 for all (p, q) �= (0, 0). This simply means that

fλ(z) = c0(λ)pλa(z).

Finally, by (3.1) the functionc0(λ) is bounded. Letϕ be the inverse Fouriertransform ofc0(λ) to getf(z, t) = ϕ ∗3 pa(z, t).

Let us now consider the casea > b. We can chooseε > 0 so thata > (1+ε)b. Letp(p,q)

t (z, s) be the heat kernel associated to the sublaplacianL onHn+p+q. This is a radial function onz and therefore we can define afunctionf onHn by

f(z, s) = Y jpq(z

′)|z|p+qp(p,q)

a1+ε

(z, s). (3.11)

Then it is clear that|fλ(z)| ≤ cλpλa(z). In view of Theorem 2.1 we knowthat f(λ) = Gλ(fλ) = (−1)qWλ(P )S whereP (z) = Y j

pq(z′)|z|p+q andS is radial. Since

∞∫−∞eiλsp

(p,q)t (z, s)ds = cn+p+qλ

n+p+q(sinhλt)−(n+p+q)e−14λ coth(tλ)|z|2

which equals a constant times

|λ|n+p+q∞∑

k=0

e−(2k+n+p+q)|λ|tϕn−1+p+qk (

√|λ|z)

we know that

f(λ)Pk(λ) = cn(k − p)!

(k + q + n− 1)!e−(2k+n+p+q)|λ|( a

1+ε)Wλ(P ). (3.12)

Using the result ofTheorem2.2wesee that(f(λ),Wλ(Q))k = 0 if (Q,P ) =0 and whenP = Q

(f(λ),Wλ(P ))k = cn|λ|n+p+q−1e−(2k+n)|λ|( a1+ε

)

which gives the estimate

|(f(λ),Wλ(P ))k| ≤ c |λ|n+p+q−1e−(2k+n)|λ|b

asa > (1 + ε)b. This completes the proof of Theorem 1.3.

Revisiting Hardy’s theorem for the Heisenberg group 773

We next turn our attention to a proof of Theorem 1.2. Definingfpq by

fpq(z, s) =∫

U(n)

f(σz, s)χpq(σ)dσ

we calculate its Fourier transform to be

fpq(λ) =∫

U(n)

∫

Hn

f(σz, s)πλ(z, s)dz ds

χpq(σ)dσ.

Sinceπλ(σ∗z, s) = µλ(σ)πλ(z, s)µλ(σ)∗ we have

fpq(λ) =∫

U(n)

µλ(σ)f(λ)µλ(σ)∗χpq(σ)dσ.

The conditionf(λ)∗f(λ) ≤ c p2a(λ) means thatf(λ)eaH(λ) is boundedwith norm independent ofλ. HereeaH(λ) is an unbounded operator definedon finite linear combinations ofΦλ

α. As pa(λ) = e−aH(λ) commutes withµλ(σ) we infer thatfpq(λ)eaH(λ) is bounded which leads to the estimate

‖fpq(λ)Φλα‖2 ≤ c e−(2|α|+n)|λ|a. (3.13)

We want to use these estimates to get upper bounds for the Laguerre coef-ficients offλ

pq.The spherical harmonic expansion offλ

pq is given by

fλpq(z) =

d(p,q)∑j=1

fλpqj(|z|)Y j

pq(z′)

wherefλpqj(|z|) are defined in (3.4). The hypothesis onfλ(z) leads us to the

estimate

|fλpqj(|z|)| ≤ cpqjp

λa(z). (3.14)

DefiningP jpq(z) = |z|p+qY j

pq(z′) and using Theorem 2.1 we have the for-mula

fpq(λ) = (−1)q

d(p,q)∑j=1

Wλ(P jpq)S

jpq(λ)

where the radial operatorsSjpq(λ) are given by

Sjpq(λ) = cn

∞∑k=p

(k − p)!(k + q + n− 1)!

ck−p(gλpqj)Pk(λ).

774 S. Thangavelu

Hereck(gλpqj) are the Laguerre coefficients ofgλpqj(z) = fλpqj(|z|)|z|−p−q.

More precisely,

ck(gλpqj) =∫

Cn+p+q

gλpqj(z)ϕn−1+p+qk (

√|λ|z)dz. (3.15)

By Theorem 2.2 the restrictions ofWλ(P jpq) toEλ

k are mutually orthog-onal and hence we get(

fpq(λ),Wλ(P jpq)

)k

= cn|λ|p+q+n−1ck−p(gλpqj)

which leads us to the expression

ck(gλpqj) = cn|λ|−p−q∑

|λ|=k+p

(fpq(λ)Φλ

α,Wλ(P jpq)Φ

λα

). (3.16)

Applying Cauchy-Schwarz inequality and using the estimate (3.13) we get

|ck(gλpqj)| ≤ c|λ|−p−qe−(2k+n)|λ|a ∑|λ|=k+p

‖Wλ(P jpq)Φα)‖2. (3.17)

Another application of Cauchy-Schwarz gives ∑

|α|=k+p

‖Wλ(P jpq)Φ

λα‖2

2

≤ c(k + p+ n− 1)!(k + p)!

|λ|−n+1 (Wλ(P j

pq),Wλ(P jpq)

)k+p. (3.18)

Using (2.8) we get the estimate

∑|α|=k+p

‖Wλ(P jpq)Φ

λα‖2 ≤ c|λ| 1

2 (p+q) (k + p+ q + n− 1)!k!

. (3.19)

¿From (3.19) and (3.17) we get the estimate

|ck(gλpqj)| ≤ cλ(2k +m)m−1e−(2k+m)|λ|a, (3.20)

where we have setm = n+ p+ q.Proceeding as in the proof of Theorem 1.3, using the estimates (3.14)

and (3.20) we can conclude that

fλpqj(|z|) = cpqj(λ)|z|p+qpλa(z). (3.21)

As before this is not compatible with (3.14) unlesscpqj(λ) = 0 for all(p, q) �= (0, 0) and we getfλ(z) = c0(λ)pλa(z) completing the proof ofTheorem 1.2.

Revisiting Hardy’s theorem for the Heisenberg group 775

Finally we consider Theorem 1.1. The proof is similar to the one givenin [22] for a weaker version of Theorem 1.1. The estimate given on thefunction gives

|fλ(z)| ≤ c(1 + |z|2)me−14a

|z|2 (3.22)

and the estimate (2.15) shows that forz fixedfλ(z) extends to a holomorphicfunction ofλ ∈C in a strip|Im(λ)| < A/a. Givena < b we chooseδ > 0so thata(ebδ + e−bδ) < 2b. The theorem will be proved by showing thatfλ(z) = 0 for 0 < λ < δ which will force fλ(z) = 0 for all λ and hencef(z, t) = 0.

Proceeding as in the previous proofs, the conditionf(λ)∗f(λ) ≤c H(λ)mp2b(λ) leads us to the estimate

|g(z)| ≤ cλ(1 + |z|2)m′e−

tanh(bλ)λ

|z|2 , (3.23)

whereg(z) = |z|−p−qfλpqj(z) andg is the Euclidean Fourier transform of

g. Our choice ofδ shows that for0 < λ < δ

0 < aλ <2bλ

ebδ + e−bδ<ebλ − e−bλ

ebλ + e−bλ= tanh(bλ).

In view of estimates (3.22) and (3.23) the classical Hardy’s theorem givesg = 0 sincetanh(bλ) > aλ. As this is true for everyp, q andj we getfλ(z) = 0 for 0 < λ < δ, z ∈ Cn. This completes the proof of Theorem1.1.

4 A Hardy’s theorem for spectral projections

In this section we prove a version of Hardy’s theorem for the spectral projec-tionsassociated to thesublaplacianonHn. Letϕλ

k(z)stand forϕn−1k (

√|λ|z)and define

eλk(z, t) = eiλtϕλk(z). (4.1)

Theseeλk are joint eigenfunctions of the sublaplacian andT = ∂t. Thejoint spectral theory of these two operators has been studied in details byStrichartz [19] where he has established the expansion

f(z, t) =∞∑

k=0

∞∫−∞f ∗ eλk(z, t)dµ(λ) (4.2)

for f ∈ L2(Hn). Heredµ(λ) = (2π)−n−1|λ|ndλ is the Plancherel measurefor the Heisenberg group. SinceLeλk = (2k + n)|λ|eλk , f ∗ eλk representsthe projection off onto the generalised eigenspace with the eigenvalue(2k + n)|λ|. For these spectral projections we prove the following result.

776 S. Thangavelu

Theorem 4.1. Letf ∈ L1(Hn) satisfy the condition∣∣∣∣∣∣∫

S2n−1

f ∗ eλk(z, t)Y jpq(z

′)dz′

∣∣∣∣∣∣ ≤ c |z|p+qe−(2k+n+p+q)|λ|b (4.3)

for all Y jpq with c independent of(z, t) andλ.

(i) If |f(z, t)| ≤ c pa(z, t) anda < b thenf = 0(ii) If |fλ(z)| ≤ c pλa(z) anda ≥ b then conclusions of Theorem 1.3 hold.

We remark that we can consider Theorem 4.1 as a Hardy’s theorem forthe Fourier transform on the Heisenberg motion groupGn. As shown in [21]the spectral projectionsf ∗eλk(z, t) are nothing butρλk(f)eλk(z, t) for certainirreducible unitary representationsρλk of Gn. However, we will not pursuethis line of thought in this paper.

The functionseλk are nothing but spherical functions for the Gelfand pair(U(n), Hn); (see [21] for more about Gelfand pairs and associated sphericalfunctions). In order to prove Theorem 4.1 we need the following result whichis an addition formula for the spherical functionseλk . (Compare this resultwith formula (3.25) in Koornwinder [10] which is an addition formula forJacobi functions). Aseλk(z, t) = eiλtϕλ

k(z) we state the formula in terms ofϕλ

k .

Proposition 4.1. For eachλ > 0 andk ∈ N

ϕλk(z − w)ei

λ2 Im(z·w)=

∞∑q=0

k∑p=0

d(p,q)∑

j=1

Y jpq(z

′)Y jpq(w

′)

(k − p)!(n− 1)!

(k + q + n− 1)!

|z|p+qϕn−1+p+qk−p (

√λz)|w|p+qϕn−1+p+q

k−p (√λw).

There is a similar formula forλ < 0 as well.

Proof: We prove this formula by appealing to Theorem 2.1. First observethat whenf = gP whereg is radial andP ∈ Hpq, f ∗λ ϕ

λk(z) has a simple

form. Indeed, a simple calculation shows that (see [11])

(gP ) ∗λ ϕλk(z) = cλk(g)P (z)ϕn−1+p+q

k−p

(√λz

)(4.4)

wherecλk(g) is as in Theorem 2.1. LetFk(z, w) be the expression on theright hand side of the formula to be proved. As finite linear combinations offunctions of the formf = gP, g radial andP ∈ Hpq are dense inL2(Cn)it is enough to show that

f ∗λ ϕλk(z) =

∫

Cn

Fk(z, w)f(w)dw (4.5)

Revisiting Hardy’s theorem for the Heisenberg group 777

wheneverf(z) = g(z)|z|p+qY jpq(z′) with g radial. But for such functions

both sides of (4.5) are same in view of (4.4). This proves the proposition.We now proceed with a proof of Theorem 4.1. A simple calculation

shows thatf ∗ eλk(z, t) = eiλtfλ ∗λ ϕλk(z) for anyf ∈ L1(Hn). Consider

fλ ∗λ ϕλk(z) =

∫

Cn

fλ(w)ϕλk(z − w)e−i λ

2 Im(z·w)dw.

In view of the addition formula stated in the proposition the above equals

∑p,q,j

(k − p)!(n− 1)!(k + q + n− 1)!

∫

Cn

fλpqj(|w|)|w|p+qϕn−1+p+q

k−p (√λw)dw

Y jpq(z

′)|z|p+qϕn−1+p+qk−p

(√λz

)(4.6)

wherefλpqj(|w|) are the functions defined in 3.4. Therefore, it follows that

∫S2n−1

fλ ∗λ ϕλk(z)Y j

pq(z′)dz′ (4.7)

=(k − p)!(n− 1)!(k + q + n− 1)!

∫

Cm

gλpqj(w)ϕm−1k−p (

√λw)dw

|z|p+qϕm−1

k−p (√λz)

wheregλpqj(w) = |w|−p−qfλpqj(w) andm = n+ p+ q.

Thus the hypothesis of the theorem gives us the estimate∣∣∣∣∣∣∣∫

Cm

gλpqj(w)ϕm−1k−p (

√λw)dw

∣∣∣∣∣∣∣(k − p)!(m− 1)!(k − p+m− 1)!

∣∣∣ϕm−1k−p (

√λz)

∣∣∣

≤ c e−(2k+m)|λ|b.

Since this estimate is true for allz, we can take limit asz → 0. Noting that

ϕm−1k−p (0) =

(k − p+m− 1)!(k − p)!(m− 1)!

we get the estimate∣∣∣∣∣∣∣∫

Cm

gλpqj(w)ϕm−1k (

√λw)dw

∣∣∣∣∣∣∣≤ c e−(2k+m)|λ|b.

778 S. Thangavelu

Once we have this estimate, the rest of the proof proceeds as in Sect. 3.We conclude this section with the following remark. If we replace the

condition in the theorem by∣∣∣∣∣∣∫

S2n−1

f ∗ eλk(z′, t)Y jpq(z

′)dt

∣∣∣∣∣∣ ≤ c |λ|p+qe−(2k+n+p+q)|λ|b (4.8)

then proceeding as above we will get∣∣∣∣∣∣∣∫

Cm

gλpqj(w)ϕm−1k (

√λw)dw

∣∣∣∣∣∣∣k!(m− 1)!

(k +m− 1)!

∣∣∣∣Lm−1k

(12|λ|

)∣∣∣∣ e− 14 |λ|

≤ cλe−(2k+m)|λ|b.

Forλ lying in a compact set of the form0 < λ1 ≤ |λ| ≤ λ2 <∞ we knowthat (see Szego [20])

Lm−1k

(12|λ|

)e−

14 |λ| = O

(k

m−12 − 1

4

)

and therefore, we get an estimate of the form∣∣∣∣∣∣∣∫

Cn

gλpqj(w)ϕm−1k (

√λw)dw

∣∣∣∣∣∣∣≤ cλ(2k +m)

m−12 e−(2k+m)|λ|b. (4.9)

Thus conclusion (i) of Theorem 4.1 is true under the weaker hypothesis(4.8). For part (ii) we can only getfλ(z) = c0(λ)pλa(z) with c0(λ) locallybounded onIR\{0}.

References

1. F. Astengo, M. Cowling, B. Di Blasio and M. Sundari: Hardy’s uncertainty principleon some Lie groups. J. London Math. Soc.62, 461–472 (2000)

2. S. C. Bagchi and S. Ray, Uncertainty principles like Hardy’s theorem on some Liegroups, J. Aust. Math. Soc.65, 289–302 (1998)

3. M. Cowling, A. Sitaram and M. Sundari, Hardy’s uncertainty principles on semisimpleLie groups, Pacific J. Math.192, 293–296 (2000)

4. G. B. Folland, Harmonic Analysis in Phase Space, Ann. of Math. Stud. 122, PrincetonUniv. press, Princeton, 1989

5. D. Geller, Spherical harmonics, the Weyl transform and the Fourier transform on theHeisenberg group, Canad. J. Math.36, 615–684 (1984)

6. G. H. Hardy, A theorem concerning Fourier transforms, J. London Math. Soc.8, 227–231 (1933)

Revisiting Hardy’s theorem for the Heisenberg group 779

7. S. Helgason, Geometric Analysis on Symmetric Spaces, Math. Surveys and Mono-graphs, Amer. Math. Soc.39 (1994)

8. A. Hulanicki, The distribution of energy in the Brownian motion in the Gaussian fieldand analytic hypoellipticity of certain subelliptic operators on the Heisenberg group,Stud. Math.56, 165–173 (1976)

9. E. Kaniuth and A. Kumar, Hardy’s theorem for simply connected nilpotent Lie groups,preprint (1999)

10. T. H. Koornwinder, Jacobi functions and analysis on non-compact semisimple Liegroups, in ”Special functions: Group theoretical aspects and applications” (R. Askey,T. H. Koornwinder and W. Schempp, Eds.), Reidel Dordrecht (1984)

11. E. K. Narayanan and S. Thangavelu, Injectivity sets for spherical means on the Heisen-berg group, J. Math. Anal. and Appli.263, 565–579 (2001)

12. E. K. Narayanan and S. Ray, Hardy’s theorem for symmetric spaces of noncompacttype, preprint (2000)

13. C. Pfannschmidt, A generalisation of the theorem of Hardy: A most general version ofthe uncertainty principle for Fourier integrals, Math. Nachr.182, 317–327 (1996)

14. R. Sarkar, Revisiting Hardy’s theorem on semisimple Lie groups, preprint (2000)15. R. Sarkar, Revisiting Hardy’s theorem onSL(2, R), preprint (2000)16. J. Sengupta, An analogue of Hardy’s theorem for semisimple Lie groups, Proc. Amer.

Math. Soc.128, 2493–2499 (2000)17. A. Sitaram, M. Sundari and S. Thangavelu, Uncertainty principles on certain Lie groups,

Proc. Indian Acad. Sci.105(2), 135–151 (1995)18. A. Sitaram and M. Sundari, An analogue of Hardy’s theorem for very rapidly decreasing

functions on semisimple Lie groups, Pacific J. Math.177, 187–200 (1997)19. R. Strichartz,Lp harmonic analysis and Radon transform on the Heisenberg group, J.

Funct. Anal.96, 350–406 (1991)20. G. Szego, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., Providence, R.I.

(1967)21. S. Thangavelu, Harmonic analysis on the Heisenberg group, Progress in Math. 159,

Birkhauser, Boston (1998)22. S. Thangavelu, An analogue of Hardy’s theorem for the Heisenberg group, Colloq.

Math.87, 137–145 (2001)23. S. Thangavelu, Hardy’s theorem for the Helgason Fourier transform on non-compact

rank one symmetric spaces, preprint