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 a<b. When a ≥ b we replace the condition on f by λ λ λ |f (z)|≤cpa(z) where f (z) is the Fourier transform of f in the t-variable. Under suitable assumptions on the ‘spherical harmonic coefficients’ of fˆ(λ) λ λ we prove: (i) f (z)=c(λ)pa(z) when a = b; (ii) when a>bthere 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<b. This is the analogue of classical Hardy’s theorem for the Fourier trans- form on Rn which says that if 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 Rn 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 a<bwill be established in this paper. As in the Euclidean case it is natural to ask what happens if a ≥ b in Theorem 1.1. In order 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 prod- uct (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].