Revisiting Hardy's Theorem for the Heisenberg Group

Math. Z. 242,761–779 (2002) Digital Object Identiﬁer (DOI) 10.1007/s002090100379

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: [email protected])

Received: 9 January 2001 / in ﬁnal form: 17 April 2001 Published 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 Fourier transform on the Heisenberg group. Let fˆ(λ) be the Fourier transform of a n ˆ ∗ ˆ function f on H and assume f(λ) f(λ) ≤ c pˆ2b(λ) where ps is the heat kernel associated to the sublaplacian. We show that if |f(z,t)|≤cpa(z,t) then f =0whenever abthere are inﬁnitely many linearly independent functions f satisfying both conditions on f λ and fˆ(λ).

1 Introduction and the main results

The aim of this paper is to establish an analogue of Hardy’s theorem for the Fourier transform on the Heisenberg group. This is a continuation of the author’s earlier paper [22] in which a version of Hardy’s theorem was proved. That theorem in turn has its roots in the ‘Hardy’s theorem for the Weyl transform’ which appeared in the author’s monograph [21]. If this is so one wonders,as the author often does himself,why he is visiting Hardy’s theorem again and again,each time reformulating and reﬁning his earlier results. We have no explanation to offer except possibly the compulsion of the demanding critic within himself who never leaves the author at peace until his theorems,if not their proofs,are straight from the ‘Book’. So,like a painter,with an extra stroke or two we try to change the shade a bit here and there in an attempt to draw closer to the truth. 762 S. Thangavelu

We can think of Hardy’s theorem as an uncertainty principle which roughly says that a function and its Fourier transform both cannot be de- caying fast at infinity unless,of course,the function is identically zero. Our first result,Theorem 1.1 below gives the optimal version of this uncertainty principle on the Heisenberg group. As the Fourier transform on the Heisen- berg group is operator valued we have to measure the decay of the Fourier transform in terms of the Hermite semigroup. We can also think of Hardy’s theorem as one characterising the heat kernel associated to the sublaplacian. Unlike the Euclidean case the heat kernel on the Heisenberg group is not explicitly known. What is known is its partial Fourier transform in the cen- tral variable and our result Theorem 1.2 characterises exactly this partial Fourier transform of the heat kernel. Consider the Fourier transform fˆ(λ),λ ∈ R,λ =0of a function f n n on the Heisenberg group H . Let pt(z,s), (z,s) ∈ H ,t > 0 be the heat n kernel associated to the sublaplacian L on H . The Fourier transform of pt −tH(λ) 2 2 is given by pˆt(λ)=e where H(λ)=−∆ + λ |x| is the (scaled) Hermite operator on Rn. As an analogue of Hardy’s theorem for the group Fourier transform on Hn we offer: Theorem 1.1. Let f be a measurable function on Hn which satisfies the 2 m estimate |f(z,s)|≤c (1 + |z| ) pa(z,s) for some a>0,m≥ 0. Further ˆ ∗ ˆ m assume that f(λ) f(λ) ≤ cH(λ) pˆ2b(λ), for some b>0 and for all λ =0. Then f =0whenever a

2 2 |f(x)|≤c(1 + |x|2)me−a|x| , |fˆ(ξ)|≤c(1 + |ξ|2)me−b|ξ| (1.1) where fˆ(ξ) is the Fourier transform of f given by − n −ix·ξ fˆ(ξ)=(2π) 2 f(x)e dx

1 −a|x|2 then (i) f =0whenever ab > 4 ; (ii) f(x)=cp(x)e where p(x) is a 1 polynomial of degree ≤ 2m when ab = 4 and (iii) there are infinitely many 1 linearly independent functions satisfying the conditions (1.1) when ab < 4 . To see the analogy of this result with Theorem 1.1 we only have to rewrite conditions (1.1) in terms of the heat kernel − n − 1 |x|2 pt(x)=(4πt) 2 e 4t associated to the standard Laplacian ∆ on Rn. The above result for the Euclidean Fourier transform was first established by Hardy [6] for n =1and m =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 Theorem b 1.1 when m =0and a< 2 . The optimal result aorder to state our results for these cases we need to establish some more notation. Weare going to replace the condition on f by a condition on the Euclidean Fourier transform f λ(z) of f(z,t) in the t-variable. For each pair of non- negative integers (p, q) let Spq be the space of bigraded spherical harmonics 2 2n−1 of degree (p, q). Then L (S ) is the orthogonal direct sum of Spq,p,q ≥ j 0. Fix an orthonormal basis {Ypq : p, q ≥ 0,j =1, 2,...d(p, q)} for 2 2n−1 j L (S ) with Ypq ∈Spq. (We say more about these spherical harmonics in Sect. 2). Theorem 1.2. Let f be an integrable function on Hn whose Fourier trans- ˆ ∗ ˆ form satisfies the condition f(λ) f(λ) ≤ c pˆ2a(λ) for some a>0 and for all λ =0. Further assume that f λ(z) satisfies λ j j λ f (|z|z )Ypq(z )dz ≤ cpq pa(z) (1.2) S2n−1 j for all Ypq and λ =0. Then f = ϕ ∗3 pa where ϕ is a tempered distribution ∞ on R with ϕˆ ∈ L (R) and ∗3 stands for the convolution in the t-variable. It would be ideal to say something in Theorem 1.1 when a = b but unfortunately we are not able to draw any conclusion. Even in Theorem 1.2 we are not able to conclude that f(z,t)=cpa(z,t); this is not surprising since the hypotheses are satisfied by f ∗3 ϕ as well and hence we can only conclude that f = ϕ ∗3 pa. The case a>bis still excluded and our next theorem treats this case. We are going to weaken the condition ˆ ∗ ˆ f(λ) f(λ) ≤ c pˆ2b(λ) for which some more definitions are needed. λ n n Let Φα,α ∈ N be the (scaled) Hermite functions on R which are eigenfunctions of H(λ). Let Hpq be the space of all polynomials of the p+q form P (z)=|z| Y (z ) with Y ∈Spq and define Wλ(P ) to be the λ Weyl correspondence of P . Let Ek be the finite dimensional subspace of 2 n λ L (R ) spanned by {Φα : |α| = k}.IfT and S are bounded operators 2 n λ on L (R ),then for their restrictions to Ek there is a natural inner product (T,S)k defined by Geller [5]. Under this inner product the operators j j p+q j Wλ(Ppq),Ppq(z)=|z| Ypq(z ) form an orthogonal system. They can be considered as the operator analogues of Spherical harmonics. (Details of this will be given in Sect. 2). With these notations we prove Theorem 1.3. Let f be an integrable function on Hn for which holds the λ λ estimate |f (z)|≤cpa(z). Further assume that ˆ p+q+n−1 −(2k+n)|λ|b |(f(λ),Wλ(P ))k|≤c|λ| e (1.3) 764 S. Thangavelu

p+q j for every k ∈ N and P (z)=|z| Ypq(z ). Then (i) when a = b, f = ∞ ϕ ∗3 pa with ϕˆ ∈ L (R); (ii) when a>bthere are infinitely many linearly independent 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 to the Helgason’s treatment of the Paley-Wiener theorem [7]. The above two theorems are analogues of these refinements. Since the perfect symmetry between f and fˆ is lost when we move away from the Euclidean Fourier transform we have two versions as above instead of one in the Euclidean case. The problem of establishing an analogue of Hardy’s theorem for Fourier transforms on Lie groups started with the work of Sitaram and Sundari [18]. For other versions of Hardy’s theorem for semi-simple Lie groups see Cowling et al [3] and Sengupta [16]. Analogues of Hardy’s theorem for the Heisenberg 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 and Ray [2] and Astengo et al [1]. General nilpotent Lie groups were considered in Kaniuth and Kumar [9]. Symmetric spaces were treated in Narayanan and Ray [12],solvable extensions of H-type groups by [1]. See also the recent works [14] and [15] of Sarkar for semi-simple Lie groups. Analogues of Theorems 1.2 and 1.3 have been established for noncompact rank one symmetric spaces in [23]. There we have also proved a version of Hardy’s theorem for the spectral projections associated to the Laplace- Beltrami operator. We will state and prove a similar theorem for the spectral projections associated to the sublaplacian in Sect. 4. The plan of the paper is as 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 refer to 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 manuscript and valuable suggestions.

2 Preliminaries on the Heisenberg group The Heisenberg group Hn is just Cn × R equipped with the group law 1 (z,t)(w, s)= z + w, t + s + Im(z · w) . 2

For each λ ∈ R,λ=0there is an irreducible unitary representation πλ(z,t) of Hn realised on L2(Rn). These representations are explicitly given by iλt iλ(x·ξ+ 1 x·y) πλ(z,t)ϕ(ξ)=e e 2 ϕ(ξ + y), Revisiting Hardy’s theorem for the Heisenberg group 765 where ϕ ∈ L2(Rn) and z = x + iy. Each inﬁnite dimensional irreducible unitary representation is equivalent to one of these. The group Fourier transform of a function f ∈ L1(Hn) is deﬁned to be the operator valued function ˆ f(λ)= f(z,t)πλ(z,t)dz dt. Hn

iλt The representation πλ satisﬁes πλ(z,t)=e πλ(z,0) and therefore ˆ λ f(λ)= f (z)πλ(z)dz, (2.1) C n where we have written πλ(z)=πλ(z,0) and ∞ f λ(z)= f(z,t)eiλtdt

−∞ is the inverse Fourier transform of f in the t-variable. The formula (2.1) suggests that we consider Weyl tranforms of functions g on Cn. These are deﬁned by

Gλ(g)= g(z)πλ(z)dz. (2.2) C n If f ∗ g is the convolution of two functions on Hn deﬁned by f ∗ g(z,t)= f((z,t)(w, s)−1)g(w, s)dw ds

Hn then it is easily checked that λ λ λ (f ∗ g) (z)=f ∗λ g (z) where the λ-twisted convolution of f λ and gλ is given by λ λ λ λ i λ Im(z·w) f ∗λ g (z)= f (z − w)g (w)e 2 dw. C n

λ λ λ λ It then follows that Gλ(f ∗λ g )=Gλ(f )Gλ(g ). Let Fλf be the λ- symplectic Fourier transform of a function f on Cn given by i λ Im(z·w) Fλf(z)= f(z − w)e 2 dw. (2.3) C n 766 S. Thangavelu

n We deﬁne the Weyl correspondence of a function f on C by Wλ(f)= −1 Gλ(Fλ f). The most important result we need is the Hecke-Bochner type identity for the Weyl transform Gλ. In order to state this important formula we need to recall some deﬁnitions. For each pair of non-negative integers (p, q) let Hpq be the space of all harmonic polynomials of the form α β P (z)= cαβz z (2.4) |α|=p |β|=q

n n where z ∈ C ,α,β ∈ N . Elements of Hpq are called bigraded solid harmonics of degree (p, q). Let Spq be the space of all restrictions of elements 2n−1 of Hpq to the unit sphere S . The elements of Spq are called bigraded spherical harmonics. Then L2(S2n−1) is the orthogonal direct sum of the j spaces Spq,p,q ≥ 0. Let {Ypq :1≤ j ≤ d(p, q)} be an orthonormal basis j for Spq. The corresponding elements of Hpq are denoted by Ppq. n n For each multiindex α ∈ N ,x∈ R ,let Φα(x) be the normalised Her- mite function which is an eigenfunction of the Hermite operator H with λ eigenvalue (2|α| + n) where |α| = α1 + ... + αn. Deﬁne Φα(x)= n 1 λ λ |λ| 4 Φα(|λ| 2 x) for λ =0so that H(λ)Φα =(2|α| + n)|λ|Φα where H(λ)=−∆ + λ2|x|2. We say that an operator T acting on L2(Rn) is λ λ λ λ δ radial if it is diagonalised by Φα and TΦα = c|α|Φα. Let Lk,δ >−1 be the Laguerre polynomials of type δ. We refer to Szego [20] for various properties δ of Lk. Deﬁne the Laguerre functions by n−1 n−1 1 2 − 1 |z|2 ϕ (z)=L |z| e 4 k k 2 (2.5) for z ∈ Cn. With these notations we are now in a position to state Geller’s result.

Theorem 2.1. Suppose gP ∈ L1(Cn) or L2(Cn) where g is a radial func- q tion and P ∈Hpq. Then for λ>0,Gλ(gP)=(−1) Wλ(P )S where S is a λ λ λ λ radial operator whose action on Φα is given as follows : SΦα = c|α|(g)Φα λ where ck(g)=0for k

When λ<0 the roles of p and q are reversed in the above deﬁnition of λ ck(g). Revisiting Hardy’s theorem for the Heisenberg group 767

In [5] Geller has studied operator analogues of the spaces Hpq which are given by the operators Wλ(P ) as P ranges over Hpq. Note that the 2n−1 Hpq spaces when restricted to each sphere rS are orthogonal and 2 2n−1 λ L (rS ) is the orthogonal direct sum of these spaces. Let Ek be the λ λ span of Φα, |α| = k and let B(Ek ) be the space of bounded linear operators λ 2 n λ from Ek into L (R ).OnB(Ek ) we can deﬁne an inner product by setting 1 n−1 (T,S) = |λ| (TΦλ,SΦλ). k 2 α α (2.7) |α|=k

With this notation the following result has been proved in [5]. Let Ppq stand for the space of all polynomials of the form (2.4). Theorem 2.2. Suppose P ∈Hpq,Q ∈Ppq and that p ≤ p or q ≤ q. Then for λ>0

(Wλ(Q),Wλ(P ))k 1 p+q+n−1 (k + q + n − 1)! =(2πn)−1 |λ| (Q, P ) 2 (k − p)! (2.8) where (Q, P ) is the inner product in L2(S2n−1). When λ<0, the roles of p and q are interchanged.

The above result shows that the spaces Wλ(Hpq) are mutually orthogonal in the above inner product. In the course of the proof of Theorem 2.1 the following formula has been established: for P ∈Hpq,λ>0 1 p+q √ (π (z),W (P )) =(−1)q λ P (z)ϕn−1+p+q( λz). λ λ k 2 k−p (2.9)

We will make use of this formula in the proof of Theorem 1.2. n Given a continuous function f onC we can expand fr(z )=f(rz ),r > 0,z ∈ S2n−1 in terms of spherical harmonics obtaining f(rz )= fpq(rz ) p,q

with fpq(rz ) coming from Spq. The projections fpq(rz ) are given by   d(p,q)  j  j fpq(rz )= f(rw )Ypq(w )dw Ypq(z ). (2.10) j=1 S2n−1

We can express fpq in terms of certain representations of the unitary group U(n). 768 S. Thangavelu

The natural action of U(n) on the unit sphere S2n−1 deﬁnes a unitary representation of U(n) on the Hilbert space L2(S2n−1). When restricted to Spq it deﬁnes an irreducible representation of U(n) denoted by δpq. Let χpq be the character of δpq. We claim that fpq(z)=d(p, q) f(σz)χpq(σ)dσ. (2.11) U(n)

To see this we apply Peter-Weyl theorem to the function F (σ)=f(σz) to get the expansion f(z)= d(δ) f(σz)χδ(σ)dσ δ∈Kˆ K ˆ where K = U(n) and K is the unitary dual of K. Let K0 = U(n − 1) considered as a subgroup of U(n). Then we can show that (see Helgason [7]) the integral f(σz)χδ(σ)dσ is non-zero only if the group δ(K0) has a non-zero K ﬁxed vector. Each δpq is such a representation and all such representations are accounted for by δpq. Thus we get f(z)= d(p, q) f(σz)χpq(σ)dσ (2.12) p,q U(n) and by the uniqueness of spherical harmonic expansion we can identify each piece with fpq. We also need to make use of some properties of the metaplectic representations. For each σ ∈ U(n) the representation πλ(σz, t) agrees with πλ(z,t) at the centre and so by Stone-von Neumann theorem they are uni- tarily equivalent. Hence there is a unitary operator µλ(σ) such that ∗ πλ(σz, t)=µλ(σ) πλ(z,t)µλ(σ). (2.13)

This correspondence σ → µλ(σ) extends to a unitary representatin of the double cover of U(n) called the metaplectic representation. Each µλ(σ) λ leaves invariant the subspaces Ek and commute with the projections Pk(λ) λ associated to Ek . We refer to Folland [4] for more about these representations. Finally,we recall some properties of the heat kernel associated to the sublaplacian L which is deﬁned by n 2 2 L = − (Xj + Yj ). j=1 Revisiting Hardy’s theorem for the Heisenberg group 769

Here ∂ 1 ∂ Xj = + yj ∂xj 2 ∂t ∂ 1 ∂ Yj = − xj ∂yj 2 ∂t j =1, 2,...n are the left invariant vector ﬁelds on Hn which alongwith ∂ T = ∂t form an orthonormal basis for the Heisenberg Lie algebra. This second order differential operator plays the role of Laplacian for Hn,is hypoelliptic,self-adjoint and non-negative. It generates a diffusion semigroup with kernel pt(z,s). Its Fourier transform in the t-variable is explicitly given by λ n −n − 1 λ(coth(tλ))|z|2 pt (z)=cnλ (sinh(tλ)) e 4 . (2.14) See Hulanicki [8] for a derivation of this formula. The group Fourier trans- −tH(λ) form of pt is given by pˆt(λ)=e . The kernel satisﬁes the pointwise estimate −n−1 − A |(z,s)|2 |pt(z,s)|≤ct e t , (2.15)

4 2 1 where |(z,s)| =(|z| + s ) 4 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 stated in the introduction. We begin with a proof of Theorem 1.3. In what follows cλ will stand for constants depending on λ and other parameters which will vary from one inequality to another. The hypothesis on f λ(z) together with the explicit formula (2.14) for λ pa(z) gives us the estimate λ − 1 (coth(aλ))|z|2 |f (z)|≤cλe 4 . (3.1) λ Recalling the deﬁnition of the inner product (T,S)k on B(Ek ) we have for j P = Ppq 1 n−1 (fˆ(λ),W (P )) = |λ| (fˆ(λ)Φλ,W (P )Φλ) λ k 2 α λ α |α|=k which is given by the integral   1 n−1 |λ| f λ(z)  (π (z)Φλ,W (P )Φλ) dz. 2 λ α λ α (3.2) C n |α|=k 770 S. Thangavelu

Without loss of generality assume λ>0 and use formula (2.9) to get 1 m−1 √ (fˆ(λ),W (P )) =(−1)q λ f λ(z)P (z)ϕm−1( λz)dz, λ k 2 k−p (3.3) C n where we have written m = n + p + q. j p+q j Note that P (z)=Ppq(z)=|z| Ypq(z ). Deﬁning λ λ j fpqj(|z|)= f (z)Ypq(z )dz (3.4) S2n−1 the expression (3.3) reads 1 √ (fˆ(λ),W (P )) =(−1)q( λ)m−1 f λ (|z|)|z|−p−qϕm−1( λz)dz λ k 2 pqj k−p C m λ m where we are treating fpqj(|z|) as radial function onC . Thus the hypothesis ˆ on (f(λ),Wλ(P ))k gives us the estimate 1 √ ( λ)m−1 gλ (z)ϕm−1( λz)dz ≤ c e−(2k+m)λb 2 pqj k λ (3.5) C m λ −p−q λ for the Laguerre coefﬁcients of the function gpqj(z)=|z| fpqj(|z|). λ We will use this to estimate the symplectic Fourier transform of gpqj. Let us λ write g in place of gpqj for the sake of simplicity of notation. Consider the Laguerre expansion of g on Cm given by 1 m ∞ k! √ g(z)= λ c (g)ϕm−1( λz) 2 (k + m − 1)! k k k=0 where we have written ck(g) to stand for the integral on the left hand side of m−1 (3.5). The Laguerre functions ϕk satisfy the generating function identity (see Szego [20]) ∞ − 1 1+r |z|2 k m−1 −m 4 1−r r ϕk (z)=(1− r) e . k=0 √ m−1 ¿From this it is easily seen that ϕk ( λz) are eigenfunctions of the λ- symplectic Fourier transform with eigenvalues (−1)k. Therefore,the La- guerre expansion of Fλg is given by ∞ k! √ F g(z)=c c (g)(−1)kϕm−1( λz). λ λ (k + m − 1)! k k k=0 Revisiting Hardy’s theorem for the Heisenberg group 771

Applying Cauchy-Schwarz inequality and using the estimate (3.5) we get 2 |Fλg(z)| (3.6) ∞ k! √ 2 ≤ c (2k + m)2e−2(2k+m)λb ϕm−1( λz) . λ (k + m − 1)! k k=0 Deﬁning ∞ k! √ 2 F (t)= e−(2k+m)t ϕm−1( λz) (k + m − 1)! k k=0

2 we note that |Fλg(z)| ≤ cλ|F (2λb)|. The Laguerre functions satisfy another generating function identity, namely ∞ 2 k! m−1 2 − 1 s2 k L (s )e 2 r (k + m − 1)! k k=0 2√ −1 4 −( m−1 ) − 1 1+r s2 2is r =(1− r) (−s r) 2 e 2 1−r J , m−1 1 − r (3.7) where Jm−1 is the Bessel function of order (m − 1). This formula gives an expression for the function F (t). Taking two derivatives of F (t) and using d −α −α − 1 t dt (t Jα(t)) = −t Jα+1(t) and the estimate |Jα(it)|≤ct 2 e ,t→∞ satisﬁed by all Bessel functions Jα(t), we can easily get the estimate

− 1 λ(tanh(bλ))|z|2 |Fλg(z)|≤cλQ(z,z)e 4 (3.8) where Q(z,z) is a polynomial in z and z. Let g˜(z) stand for the Euclidean Fourier transform on Cn which can be expressed in terms of the λ-symplectic Fourier transform Fλg(z). This leads to the estimate

− tanh(bλ) |z|2 |g˜(z)|≤cλQ1(z,z)e λ , (3.9)

λ where Q1 is another polynomial. Since f (z) satisﬁes the estimate (3.1) it λ −p−q follows that g(z)=fpqj(z)|z| also satisﬁes the estimate

− 1 λ(coth(aλ))|z|2 |g(z)|≤cλe 4 (3.10) as |z|→∞. Now we can appeal to Hardy’s theorem on Cm. When a = b it follows from (3.9) and (3.10) that

λ − 1 λ(coth(aλ))|z|2 gpqj(z)=cpqj(λ)e 4 . 772 S. Thangavelu

λ λ −p−q Since gpqj(z)=fpqj(z)|z| , the above is not compatible with the estimate λ − 1 λ(coth(aλ))|z|2 |fpqj(z)|≤cλe 4

unless cpqj(λ)=0for all (p, q) =(0, 0). This simply means that

λ λ f (z)=c0(λ)pa(z).

Finally,by (3.1) the function c0(λ) is bounded. Let ϕ be the inverse Fourier transform of c0(λ) to get f(z,t)=ϕ ∗3 pa(z,t). Let us now consider the case a>b. We can choose 8>0 so that (p,q) a>(1+8)b. Let pt (z,s) be the heat kernel associated to the sublaplacian L on Hn+p+q. This is a radial function on z and therefore we can deﬁne a function f on Hn by

j p+q (p,q) f(z,s)=Ypq(z )|z| p a (z,s). (3.11) 1+

λ λ Then it is clear that |f (z)|≤cλpa(z). In view of Theorem 2.1 we know ˆ λ q j p+q that f(λ)=Gλ(f )=(−1) Wλ(P )S where P (z)=Ypq(z )|z| and S is radial. Since ∞ iλs (p,q) n+p+q −(n+p+q) − 1 λ coth(tλ)|z|2 e pt (z,s)ds = cn+p+qλ (sinh λt) e 4 −∞

which equals a constant times ∞ n+p+q −(2k+n+p+q)|λ|t n−1+p+q |λ| e ϕk ( |λ|z) k=0 we know that

(k − p)! −(2k+n+p+q)|λ|( a ) fˆ(λ)P (λ)=c e 1+ W (P ). k n (k + q + n − 1)! λ (3.12) ˆ Using the result of Theorem 2.2 we see that (f(λ),Wλ(Q))k =0if (Q, P )= 0 and when P = Q ˆ n+p+q−1 −(2k+n)|λ|( a ) (f(λ),Wλ(P ))k = cn|λ| e 1+

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

as a>(1 + 8)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. Deﬁning fpq by

fpq(z,s)= f(σz, s)χpq(σ)dσ U(n) we calculate its Fourier transform to be   ˆ   fpq(λ)= f(σz, s)πλ(z,s)dz ds χpq(σ)dσ. U(n) Hn ∗ ∗ Since πλ(σ z,s)=µλ(σ)πλ(z,s)µλ(σ) we have ˆ ˆ ∗ fpq(λ)= µλ(σ)f(λ)µλ(σ) χpq(σ)dσ. U(n)

ˆ ∗ ˆ ˆ aH(λ) The condition f(λ) f(λ) ≤ c pˆ2a(λ) means that f(λ)e is bounded with norm independent of λ. Here eaH(λ) is an unbounded operator defined λ −aH(λ) on finite linear combinations of Φα.Aspâ(λ)=e commutes with ˆ aH(λ) µλ(σ) we infer that fpq(λ)e is bounded which leads to the estimate ˆ λ −(2|α|+n)|λ|a fpq(λ)Φα2 ≤ ce . (3.13) We want to use these estimates to get upper bounds for the Laguerre coef- λ ficients of fpq. λ The spherical harmonic expansion of fpq is given by d(p,q) λ λ j fpq(z)= fpqj(|z|)Ypq(z ) j=1

λ λ where fpqj(|z|) are deﬁned in (3.4). The hypothesis on f (z) leads us to the estimate λ λ |fpqj(|z|)|≤cpqjpa(z). (3.14)

j p+q j Deﬁning Ppq(z)=|z| Ypq(z ) and using Theorem 2.1 we have the formula d(p,q) ˆ q j j fpq(λ)=(−1) Wλ(Ppq)Spq(λ) j=1 j where the radial operators Spq(λ) are given by ∞ (k − p)! Sj (λ)=c c (gλ )P (λ). pq n (k + q + n − 1)! k−p pqj k k=p 774 S. Thangavelu

λ λ λ −p−q Here ck(gpqj) are the Laguerre coefﬁcients of gpqj(z)=fpqj(|z|)|z| . More precisely, λ λ n−1+p+q ck(gpqj)= gpqj(z)ϕk ( |λ|z)dz. (3.15) C n+p+q

j λ By Theorem 2.2 the restrictions of Wλ(Ppq) to Ek are mutually orthogonal and hence we get

ˆ j p+q+n−1 λ fpq(λ),Wλ(Ppq) = cn|λ| ck−p(gpqj) k which leads us to the expression λ −p−q ˆ λ j λ ck(gpqj)=cn|λ| fpq(λ)Φα,Wλ(Ppq)Φα . (3.16) |λ|=k+p Applying Cauchy-Schwarz inequality and using the estimate (3.13) we get λ −p−q −(2k+n)|λ|a j |ck(gpqj)|≤c|λ| e Wλ(Ppq)Φα)2. (3.17) |λ|=k+p Another application of Cauchy-Schwarz gives  2  j λ  Wλ(Ppq)Φα2 |α|=k+p (k + p + n − 1)! ≤ c |λ|−n+1 W (P j ),W (P j ) . (k + p)! λ pq λ pq k+p (3.18) Using (2.8) we get the estimate j λ 1 (p+q) (k + p + q + n − 1)! W (P )Φ ≤ c|λ| 2 . λ pq α 2 k! (3.19) |α|=k+p ¿From (3.19) and (3.17) we get the estimate λ m−1 −(2k+m)|λ|a |ck(gpqj)|≤cλ(2k + m) e , (3.20) where we have set m = n + p + q. Proceeding as in the proof of Theorem 1.3,using the estimates (3.14) and (3.20) we can conclude that λ p+q λ fpqj(|z|)=cpqj(λ)|z| pa(z). (3.21)

As before this is not compatible with (3.14) unless cpqj(λ)=0for all λ λ (p, q) =(0, 0) and we get f (z)=c0(λ)pa(z) completing the proof of Theorem 1.2. Revisiting Hardy’s theorem for the Heisenberg group 775

Finally we consider Theorem 1.1. The proof is similar to the one given in [22] for a weaker version of Theorem 1.1. The estimate given on the function gives λ 2 m − 1 |z|2 |f (z)|≤c(1 + |z| ) e 4a (3.22) and the estimate (2.15) shows that for z ﬁxed f λ(z) extends to a holomorphic function of λ ∈ C in a strip |Im(λ)| < A/a.Givena0 so that a(ebδ + e−bδ) < 2b. The theorem will be proved by showing that f λ(z)=0for 0 <λ<δwhich will force f λ(z)=0for all λ and hence f(z,t)=0. Proceeding as in the previous proofs,the condition fˆ(λ)∗fˆ(λ) ≤ m cH(λ) pˆ2b(λ) leads us to the estimate

2 m − tanh(bλ) |z|2 |g˜(z)|≤cλ(1 + |z| ) e λ , (3.23) −p−q λ where g(z)=|z| fpqj(z) and g˜ is the Euclidean Fourier transform of g. Our choice of δ shows that for 0 <λ<δ 2bλ ebλ − e−bλ 0 aλ. As this is true for every p, q and j we get f λ(z)=0for 0 <λ<δ,z∈ Cn. This completes the proof of Theorem 1.1.

4 A Hardy’s theorem for spectral projections

In this section we prove a version of Hardy’s theorem for the spectral projec- n λ n−1 tions associated to the sublaplacian on H . Let ϕk(z) stand for ϕk ( |λ|z) and deﬁne λ iλt λ ek(z,t)=e ϕk(z). (4.1) λ These ek are joint eigenfunctions of the sublaplacian and T = ∂t. The joint spectral theory of these two operators has been studied in details by Strichartz [19] where he has established the expansion ∞ ∞ λ f(z,t)= f ∗ ek(z,t)dµ(λ) (4.2) k=0−∞ for f ∈ L2(Hn). Here dµ(λ)=(2π)−n−1|λ|ndλ is the Plancherel measure λ λ λ for the Heisenberg group. Since Lek =(2k + n)|λ|ek,f ∗ ek represents the projection of f 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. Let f ∈ L1(Hn) satisfy the condition λ j p+q −(2k+n+p+q)|λ|b f ∗ ek(z,t)Y pq(z )dz ≤ c |z| e (4.3) S2n−1 j for all Ypq with c independent of (z,t) and λ. (i) If |f(z,t)|≤cpa(z,t) and a0 and k ∈ N   ∞ k d(p,q) λ i λ Im(z·w) j j (k − p)!(n − 1)! ϕ (z − w)e 2 =  Y (z )Y (w ) k pq pq (k + q + n − 1)! q=0 p=0 j=1 √ √ p+q n−1+p+q p+q n−1+p+q |z| ϕk−p ( λz)|w| ϕk−p ( λw). There is a similar formula for λ<0 as well. Proof: We prove this formula by appealing to Theorem 2.1. First observe λ that when f = gP where g is radial and P ∈Hpq,f ∗λ ϕk(z) has a simple form. Indeed,a simple calculation shows that (see [11]) √ λ λ n−1+p+q (gP) ∗λ ϕk(z)=ck(g)P (z)ϕk−p λz (4.4)

λ where ck(g) is as in Theorem 2.1. Let Fk(z,w) be the expression on the right hand side of the formula to be proved. As ﬁnite linear combinations of 2 n functions of the form f = gP, g radial and P ∈Hpq are dense in L (C ) it is enough to show that λ f ∗λ ϕk(z)= Fk(z,w)f(w)dw (4.5) C n Revisiting Hardy’s theorem for the Heisenberg group 777

p+q j whenever f(z)=g(z)|z| Ypq(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 λ iλt λ λ 1 n shows that f ∗ ek(z,t)=e f ∗λ ϕk(z) for any f ∈ L (H ). Consider λ λ λ λ −i λ Im(z·w) f ∗λ ϕk(z)= f (w)ϕk(z − w)e 2 dw. C n In view of the addition formula stated in the proposition the above equals   (k − p)!(n − 1)!  √   f λ (|w|)|w|p+qϕn−1+p+q( λw)dw (k + q + n − 1)! pqj k−p p,q,j C n √ j p+q n−1+p+q Ypq(z )|z| ϕk−p λz (4.6)

λ where fpqj(|w|) are the functions deﬁned in 3.4. Therefore,it follows that λ λ j f ∗λ ϕk(z)Ypq(z )dz (4.7) S2n−1   (k − p)!(n − 1)!  √  √ =  gλ (w)ϕm−1( λw)dw |z|p+qϕm−1( λz) (k + q + n − 1)! pqj k−p k−p C m

λ −p−q λ where gpqj(w)=|w| fpqj(w) and m = n + p + q. Thus the hypothesis of the theorem gives us the estimate √ (k − p)!(m − 1)! √ gλ (w)ϕm−1( λw)dw ϕm−1( λz) pqj k−p (k − p + m − 1)! k−p C m ≤ ce−(2k+m)|λ|b.

Since this estimate is true for all z,we can take limit as z → 0. Noting that (k − p + m − 1)! ϕm−1(0) = k−p (k − p)!(m − 1)! we get the estimate √ gλ (w)ϕm−1( λw)dw ≤ ce−(2k+m)|λ|b. pqj k C m 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 λ j p+q −(2k+n+p+q)|λ|b f ∗ ek(z ,t)Ypq(z )dt ≤ c |λ| e (4.8) S2n−1 then proceeding as above we will get √ λ m−1 k!(m − 1)! m−1 1 − 1 |λ| g (w)ϕ ( λw)dw L |λ| e 4 pqj k (k + m − 1)! k 2 C m −(2k+m)|λ|b ≤ cλe .

For λ lying in a compact set of the form 0 <λ1 ≤|λ|≤λ2 < ∞ we know that (see Szego [20]) m−1 1 − 1 |λ| m−1 − 1 L |λ| e 4 = O k 2 4 k 2 and therefore,we get an estimate of the form √ λ m−1 m−1 −(2k+m)|λ|b g (w)ϕ ( λw)dw ≤ c (2k + m) 2 e . pqj k λ (4.9) C n Thus conclusion (i) of Theorem 4.1 is true under the weaker hypothesis λ λ (4.8). For part (ii) we can only get f (z)=c0(λ)pa(z) with c0(λ) locally bounded on IR\{0}.

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