THE SPHERICAL TRANSFORM OF A SCHWARTZ FUNCTION ON THE HEISENBERG GROUP

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1 THE SPHERICAL TRANSFORM OF A SCHWARTZ FUNCTION ON THE HEISENBERG GROUP CHAL BENSON, JOE JENKINS, AND GAIL RATCLIFF Abstract. Suppose that K U(n is a compact Lie group acting on the (2n+1- dimensional Heisenberg group H n. We say that (K, H n is a Gelfand pair if the convolution algebra L 1 K (H n of integrable K-invariant functions on H n is commutative. In this case, the Gelfand space (K, H n is equipped with the Godement- Plancherel measure, and the spherical transform : L 2 K (H n L 2 ( (K, H n is an isometry. The main result in this paper provides a complete characterization of the set S K (H n { f f S K (H n } of spherical transforms of K-invariant Schwartz functions on H n. We show that a function F on (K, H n belongs to S K (H n if and only if the functions obtained from F via application of certain derivatives and difference operators satisfy decay conditions. We also consider spherical series expansions for K-invariant Schwartz functions on H n modulo its center. Contents 1. Introduction 1 2. Notation and Preliminaries 6 3. K-spherical series expansions on Generalized binomial coefficients K-Invariant Schwartz functions on K-Invariant Schwartz functions on H n Analysis of boundary terms 32 References Introduction Given a complex vector space of dimension n with Hermitian inner product,, one forms the Heisenberg group H n R with group law ( (z, t(z, t z + z, t + t 1 2 Im z, z Mathematics Subject Classification. Primary 22E30, 43A55. Any opinions, findings, conclusions, or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the National Science Foundation. 1

2 2 C. BENSON, J. JENKINS, AND G. RATCLIFF In order to present explicit formulae, one can use an orthonormal basis to identify with C n so that z, z z z for z, z C n. The group U(n of unitary transformation of (,, acts by automorphisms on H n via k (z, t : (kz, t and on the space of polynomials C[ ] via for k U(n and (z, t H n k p(z : p(k 1 z for k U(n, p C[ ] and z. If K is a compact Lie subgroup of U(n then we say that (K, H n is a Gelfand pair when the algebra L 1 K (H n of K-invariant L 1 -functions on H n is commutative under convolution. One has the following result: Theorem 1.1 (cf. [Car87], [BJR90]. (K, H n is a Gelfand pair if and only if the representation of K on C[ ] is multiplicity free. Using Theorem 1.1, one sees that the group U(n and many of its proper subgroups K yield Gelfand pairs. Working from Theorem 1.1, one obtains a complete classification of all such subgroups [Kac80, BR96]. This classification shows that the theory of Gelfand pairs associated with Heisenberg groups is quite rich. There are, for example, twenty distinct families of Gelfand pairs (K, H n where K is connected and acts irreducibly on. The current paper concerns analysis in this setting. There is a well developed theory of spherical functions associated with Gelfand pairs of the form (K, H n. We denote the space of bounded K-spherical functions, equipped with the compact-open topology, by (K, H n. There are two types of bounded K-spherical functions: Type 1: 1 (K, H n {φ,λ Λ, λ R }. Here Λ is a countable index set that parameterizes the decomposition C[ ] Λ P of C[ ] into K-irreducible subspaces. 1 (K, H n is dense and of full Godement- Plancherel measure in (K, H n. Type 2: 2 (K, H n {η Kw w }, where Kw is a K-orbit in. The reader will find the relevant definitions below in Section 2 along with a summary of some results from earlier work. The K-spherical transform f : (K, H n C for a function f L 1 K (H n is defined by f(ψ : f(z, tψ(z, t dzdt, H n were dzdt denotes Haar measure for the group H n. f belongs to the space C ( (K, H n of continuous functions on (K, H n that vanish at infinity. The spherical functions for the Gelfand pairs (U(n, H n and (T n, H n have been obtained from a variety of viewpoints in works including [BJR92, Far87, HR80, Kor80,

3 SCHWARTZ FUNCTIONS ON THE HEISENBERG GROUP 3 Ste88, Str91]. The systematic study of other Gelfand pairs (K, H n and their spherical functions is less standard but no less natural. In any case, as explained below in Section 2.3, the K-spherical functions are joint eigenfunctions for / t and the Heisenberg sub-laplacian. As shown in [Str89, Str91], the joint spectral theory for these operators is central to harmonic analysis on the Heisenberg group and the spherical transform is its main tool. In [HR80] the spherical functions and spherical transform for (T n, H n are used to establish a tangential convergence theorem for bounded harmonic functions on the hyperbolic space SU(1, N + 1/U(n + 1. The reader will find further applications of radial functions and the spherical transform in J. Faraut s book [Far87]. What can one say about the spherical transform of a K-invariant Schwartz function on H n? More precisely, letting S K (H n denote the space of K-invariant Schwartz functions on H n, we seek to characterize the subspace { } S K (H n f f SK (H n of C ( (K, H n. The main result in this paper is Theorem 6.1 below, which provides a complete solution to this problem. Before describing the contents of this theorem we wish to provide some background and motivation for the study of S K (H n via the spherical transform. Schwartz functions have played an important role in harmonic analysis with nilpotent groups since the work of Kirilov [Kir62]. Let N be a connected and simply connected nilpotent Lie group with Lie algebra n. The exponential map exp : n N is a polynomial diffeomorphism and one defines the (Fréchet space S(N of Schwartz functions on N via identification with the usual space S(n of Schwartz functions on the vector space n: S(N : {f : N C f exp S(n}. S(N is dense in L p (N for each p and carries an algebra structure given by the convolution product. Moreover, it is known that the primitive ideal space for S(N is isomorphic to that of both L 1 (N and C (N [Lud88]. The Heisenberg groups H n are the simplest groups for which S(N is non-abelian.. Recall that the group Fourier transform for a function f L 1 (N associates to π N, an irreducible unitary representation of N, the bounded operator π(f f(xπ(x dx N in the representation space of π. This generalizes the usual Euclidean Fourier transform for the case N R n. The importance of Schwartz functions in Euclidean harmonic analysis arises from the fact that S(R n is preserved by the Fourier transform. It is thus very natural to seek a characterization of S(N via the group Fourier transform; a problem solved by R. Howe in [How77]. A related result for the case N H n can be found in D. Geller s paper [Gel77].

4 4 C. BENSON, J. JENKINS, AND G. RATCLIFF One can sometimes obtain abelian subalgebras of S(N by considering radial functions. This is of interest even when N R n. Indeed, the algebra S O(n (R n of radial Schwartz functions on R n can be identified with S(R + and the Fourier transform becomes a Hankel transform on S(R +. This is the spherical transform for the Gelfand pair obtained from the action of the orthogonal group O(n on R n [Hel84]. In the case N H n, we are led to consider the abelian subalgebras S K (H n arising from Gelfand pairs (K, H n. The group Fourier transform on S K (H n then reduces to the K-spherical transform as follows. The unitary dual of H n can be written as Ĥ n {π λ λ R } {χ w w } where each π λ is an infinite dimensional representation and each χ w is one dimensional. One can realize π λ in a Fock space that contains C[ ] as a dense subspace. For f S K (H n, π λ (f is diagonalized by the decomposition C[ ] Λ P and one has that π λ (f P f(φ λ, and χ w (f f(η Kw. Thus, characterizing S K (H n enables one to construct smooth functions on H n which decay rapidly at infinity whose group Fourier transforms are prescribed in advance subject to some conditions. This provides an alternative to the application of the results from [How77] and to Dixmier s functional calculus [Dix60] in the Heisenberg group setting. J. Ludwig has recently applied the group Fourier transform and Dixmier s functional calculus to the study of hull minimal ideals in the algebra S(H n [Lud]. The authors believe that Theorem 6.1 (and Corollary 6.3 will provide a new technique to approach problems of this nature. We hope to pursue this idea elsewhere. Theorem 6.1 provides conditions that are both necessary and sufficient for a function F on (K, H n to belong to the space S K (H n : 1. F is continuous on (K, H n. 2. w F (η Kw is a Schwartz function on. 3. λ F (φ,λ is smooth on R and the functions m λ F (φ,λ satisfy certain decay conditions. In particular, m λ F (φ,λ is a rapidly decreasing sequence in for each fixed λ R. 4. Certain derivatives of F also satisfy the three conditions above. These are defined on 1 (K, H n as specific combinations of λ and difference operators which play the role of differentiation in the discrete parameter Λ. The precise formulation of these conditions can be found in Section 6. The derivatives of functions in S K (H n referred to above are operators corresponding to multiplication of functions in S K (H n by certain polynomials. The difference operators in the discrete parameter Λ are linear operators whose coefficients are generalized binomial coefficients. These coefficients were introduced by Z. Yan in [Yan] and appear in many formulas concerning the type 1 spherical functions. A summary of their properties is given below in Section 4. A more complete discussion of generalized

5 SCHWARTZ FUNCTIONS ON THE HEISENBERG GROUP 5 binomial coefficients will appear in [BR]. We remark that for the case K T n, difference operators similar to the ones used here appear in the papers [Gel77], [dmm79] and [MS94]. One consequence of the estimates involved in our characterization of S K (H n is that f S K (H n can be recovered from F f via inversion of the spherical transform. One has ( n+1 1 (1.1 f(z, t dim(p F (φ,λ φ,λ (z, t λ n dλ. 2π R Λ The first two conditions in our characterization of S K (H n play a rather subtle role in the sufficiency proof. Indeed, the bounded K-spherical functions η Kw of type 2 do not appear in Formula 1.1. In order to control the growth of f(z, t in the central direction, we introduce a λ-derivative into the right hand side of Formula 1.1. An application of integration by parts produces two boundary terms at λ 0 and we need to show that these cancel. This is done in Section 7 where we prove that ( n 1 lim dim(p λ n F (φ,λ φ,λ (z, 0 F (η Kw η Kw (zdw. λ 0 ± 2π Λ (See Proposition 7.1. It is here that the continuity and behavior of F on 2 (K, H n comes into play. We also consider the related problem of characterizing the the space S K ( of K-invariant Schwartz functions on via the spherical transform. The functions φ on defined by φ (z φ (z, 0 form a complete orthogonal system in the space L 2 K ( of square integrable K-invariant functions on (see Proposition 3.1. The φ s are eigenfunctions for differential operators that arise from K-invariant and leftinvariant differential operators on the Heisenberg group and (modulo dilations for the Fourier transform. The coefficients in the L 2 -expansion of f L 2 K ( as a series in {φ Λ} are (up to normalization f( : f, φ 2. Theorem 5.1 asserts that F belongs to S K ( { f f S K ( } if and only if {F ( Λ} is a rapidly decreasing sequence. The remainder of this paper is structured as follows. Section 2 contains preliminary material and summarizes results from previous work concerning spherical functions. Section 3 concerns the spherical series expansions that arise by suppressing the central direction in the Heisenberg group. The material in this section appeared in [Yan] but we have provided new and complete proofs here. Section 4 concerns combinatorial properties of the generalized binomial coefficients. Our characterizations of S K ( and S K (H n appear in Section 5 and Section 6 respectively. As a corollary to Theorem 6.1, we show that for K U(n or T n, one can construct a function f S K (H n whose spherical transform f has a pre-determined compact support in

6 6 C. BENSON, J. JENKINS, AND G. RATCLIFF (K, H n. The sufficiency proof for Theorem 6.1 is completed in Section 7, which concerns the analysis of boundary terms. The first and third authors wish to thank the Université de Metz for their support during the completion of this research. 2. Notation and Preliminaries We need to establish notation and recall some results concerning spherical functions associated with Gelfand pairs (K, H n. We refer the reader to [BJR90], [BJR92] and [BJRW] for complete details on this material. Throughout this paper, K will always denote a closed Lie subgroup of U(n for which (K, H n is a Gelfand pair Decomposition of C[ ]. We decompose C[ ] into K-irreducible subspaces P, (2.1 C[ ] Λ P where Λ is some countably infinite index set. Theorem 1.1 ensures that this decomposition is canonical. Since the representation of K on C[ ] preserves the space P m ( of homogeneous polynomials of degree m, each P is a subspace of some P m (. We write for the degree of homogeneity of the polynomials in P, so that P P (. We will write d for the dimension of P and denote by 0 Λ the index for the scalar polynomials P 0 P 0 ( C Invariant polynomials. Since the representation of K on C[ ] is multiplicity free, there can be no non-constant K-invariant holomorphic polynomials. One does, however, have invariant polynomials on the underlying real vector space R for. We denote the set of these by C[ R ] K. One obtains a canonical basis for C[ R ] K as follows. Given Λ and any orthonormal basis {v 1,..., v d } for P, we let (2.2 p (z : 1 d d j1 v j (zv j (z. The polynomial p on R is R + -valued, K-invariant, and homogeneous of degree 2. The definition of p does not depend on the choice of basis for P and {p Λ} is a vector space basis for C[ R ] K. One computes that (2.3 d p (z 1 m! γ(zm where γ(z is defined by m γ(z : z 2 /2.

7 SCHWARTZ FUNCTIONS ON THE HEISENBERG GROUP 7 A result in [HU91] ensures that the algebra C[ R ] K is freely generated by a canonical finite subset {γ 1, γ 2,..., γ d } {p Λ}, (2.4 C[ R ] K C[γ 1, γ 2,..., γ d ]. We call the generators {γ 1,..., γ d } the fundamental invariants for the action of K on. When K acts irreducibly on, the polynomial γ is one of the fundamental invariants and we can suppose that γ 1 γ Invariant differential operators. One basis for the Lie algebra of left invariant vector fields on H n is written as {Z 1, Z 2,..., Z n, Z 1, Z 2,..., Z n, T } where (2.5 and Z j 2 + i z j z j 2 t, T : t. Z j 2 i z j z j 2 t, With these conventions one has [Z j, Z j ] 2iT. The first order operators Z 1,..., Z n, Z 1,..., Z n, T generate the algebra D(H n of left-invariant differential operators on H n. We denote the subalgebra of K-invariant differential operators by D K (H n : {D D(H n D(f k D(f k for k K, f C (H n }. Since (K, H n is a Gelfand pair, D K (H n is an abelian algebra. A result due to Thomas [Tho82] shows that the converse is also true, at least when K is connected. One differential operator will play a key role in this paper. This is the Heisenberg sub-laplacian defined by L : 1 n ( (2.6 Zj Z j + Z j Z j. 2 j1 L is U(n-invariant and hence belongs to D K (H n for all Gelfand pairs (K, H n. Since the formal adjoints of Z j and Z j as operators on L 2 (H n are Zj Z j and Z j Z j, we see that L is essentially self-adjoint on L 2 (H n. The algebra D K (H n is generated by {L γ1,..., L γd, T }, where L γj is the operator obtained from γ j (Z, Z by symmetrization Spherical functions. A smooth function ψ : H n C is called K-spherical if 1. ψ is K-invariant, 2. ψ is an eigenfunction for every D D K (H n, and 3. ψ(0, 0 1. We write D(ψ for the eigenvalue of D D K (H n on a K-spherical function ψ, that is D(ψ D(ψψ. We denote the set of positive definite K-spherical functions on H n by (K, H n. In [BJR90], it is shown that every bounded K-spherical function is positive definite, so (K, H n is also the set of bounded K-spherical functions. We remark that this

8 8 C. BENSON, J. JENKINS, AND G. RATCLIFF result contrasts with the situation for symmetric spaces. (See eg. [G88]. As shown in [BJR92], the bounded K-spherical functions can be derived from the representation theory for H n together with the action of K on. The infinite dimensional irreducible unitary representations of H n can be realized in Fock space. This is the space F consisting of entire functions f : C which are square integrable with respect to e z 2 /2 dz with Hilbert space structure ( n 1 f, g f(zg(ze z 2 /2 dz. F 2π Here dz denotes Lebesgue measure on R R 2n. The holomorphic polynomials C[ ] form a dense subspace in F. One has an irreducible unitary representation π of H n on F defined as For Λ let (2.7 (π(z, tf(w e it 1 2 <w,z> 1 4 z 2 f(w + z. φ (z, t : 1 d d j1 π(z, tv j, v j F, where {v 1,..., v d } is an orthonormal basis for P. This description of φ does not depend on our choice of basis {v j }. Define φ,λ for λ R and Λ by ( (2.8 φ,λ (z, t : φ λ z, λt, so that φ φ,1. The φ,λ s are distinct bounded K-spherical functions. We refer to these elements of (K, H n as the spherical functions of type 1. From Equation 2.7 one can show that φ has the general form φ (z, t e it q (ze 1 4 z 2 e it q (ze γ(z/2 where q is a K-invariant polynomial on R with homogeneous component of highest degree given by ( 1 p. As for the p s, the set {q Λ} is a basis for the vector space C[ R ] K. The eigenvalues of the Heisenberg sub-laplacian L on the type 1 K-spherical functions are given by (2.9 L (φ,λ λ (2 + n. This follows from Proposition 3.20 in [BJR92] together with Lemma 3.4 in [BJRW]. The key point here is that the quantum harmonic oscillator π(l on Fock space F acts via the scalar (2m + n on P m (. In addition to the K-spherical functions of type 1, there are K-spherical functions which arise from the one dimensional representations of H n. For w, let (2.10 η w (z, t : e ire w,kz dk e ire z,kw dk K K

9 SCHWARTZ FUNCTIONS ON THE HEISENBERG GROUP 9 where dk denotes normalized Haar measure on K. These are the bounded K- spherical functions of type 2. Note that η 0 is the constant function 1 and η w η w if and only if Kw Kw. Thus we have one K-spherical function for each K-orbit in and sometimes write η Kw in place of η w. It is shown in [BJR92] that every bounded K-spherical function is of type 1 or type 2. Thus we have: Theorem 2.1. The bounded K-spherical functions on H n are parameterized by the set ( Λ R (/K via (K, H n 1 (K, H n 2 (K, H n where 1 (K, H n { φ,λ Λ, λ R } and 2 (K, H n { η Kw w } Topology on (K, H n. We give (K, H n the (compact-open topology of uniform convergence on compact sets. A detailed discussion of this topology can be found in [BJRW]. A key fact is that 1 (K, H n is dense in (K, H n. Moreover, the space (K, H n can be embedded in (R + d R. Indeed, let L γ1,..., L γd D K (H n be differential operators obtained from the fundamental invariants via symmetrization and let T t. Theorem 2.2 (cf. [BJRW]. The map E : (K, H n (R + d R defined by E(ψ ( Lγ1 (ψ,..., Lγd (ψ, i T (ψ is a homeomorphism onto its image. The eigenvalues L γj (ψ for the L γj s on (K, H n are real numbers with constant sign and one can describe the map E more explicitly as follows. Let e j ( Lγj (φ and 2m j be the degree of γ j. Then E(φ,λ ( e 1 ( λ m 1,..., e d ( λ m d, λ and E(η w ( γ 1 (w,..., γ d (w, 0. We can use the map E and a metric on (R + d R to produce a metric on (K, H n which induces the compact-open topology. For example, d ( d(ψ 1, ψ 2 Lγ1 (ψ j L γ1 (ψ j ( T (ψ1 T (ψ 2 j1 is one such metric on (K, H n The K-spherical transform. The K-spherical transform for f L 1 K (H n is the function f on (K, H n defined by f(ψ : f(z, tψ(z, t dzdt. H n Here dzdt denotes Haar measure for the group H n, which is simply Euclidean measure on R R. f is a bounded function with f f 1

10 10 C. BENSON, J. JENKINS, AND G. RATCLIFF for f L 1 K (H n. This follows immediately from the fact that for ψ (K, H n one has ψ(z, t ψ(0, 0 1, since ψ is positive definite. The compact-open topology on (K, H n is the smallest topology which makes all of the maps { f f L 1 K (H n} continuous. Since L 1 K (H n is a Banach -algebra (with involution defined via f (z, t : f( z, t, f belongs to the space C ( (K, H n of continuous functions on (K, H n that vanish at infinity Godement-Plancherel measure. Godement s Plancherel Theory for Gelfand pairs (G, K (cf. [God62], or section 1.6 in [G88] ensures that there exists a unique positive Borel measure dµ on the space (K, H n for which (2.12 f(z, t 2 dzdt f(ψ 2 dµ(ψ H n (K,H n for all continuous functions f L 1 K (H n L 2 K (H n. If f L 1 K (H n L 2 K (H n is continuous and f is integrable with respect to dµ then one has the Inversion Formula: (2.13 f(z, t f(ψψ(z, t dµ(ψ. (K,H n The spherical transform f f extends uniquely to an isomorphism between L 2 K (H n and L 2 ( (K, H n, dµ. The following result makes the Godement-Plancherel measure on (K, H n explicit. Given F : (K, H n C, we write F (, λ in place of F (φ,λ. The reader can find proofs of Theorem 2.3 in [BJRW] and [Yan]. The result for K U(n is also discussed in [Far87] and [Str91]. Theorem 2.3. The Godement-Plancherel measure dµ on (K, H n is given by ( n+1 1 F (ψ dµ(ψ d F (, λ λ n dλ. 2π (K,H n R Λ Note that 2 (K, H n is a set of measure zero in (K, H n Fourier transforms. We define the Fourier transform F (f : C of f L 1 ( by F (f(w : f(ze ire z,w dz where dz denotes Euclidean measure on R. With this normalization one has F (f 2 (2π n f 2 and the inversion formula reads F (f(we ire z,w dw (2π 2n f(z

11 SCHWARTZ FUNCTIONS ON THE HEISENBERG GROUP 11 for suitable f : C. Similarly, the Euclidean Fourier transform F H (f of f L 1 (H n is defined by (2.14 F H (f(w, s f(z, te ire z,w e its dzdt. H n If f L 1 K (H n then the Fourier and spherical transforms are related via (2.15 f(η w F H (f(w, 0. Indeed, for w, one has f(η w f(z, te ire kz,w dkdzdt H n K f(kz, te ire z,w dkdzdt H n K f(z, te ire z,w dzdt H n F H (f(w, K-spherical series expansions on The dependence of a K-spherical function ψ(z, t on the central variable t is rather trivial. Thus, it is natural to suppress this direction and consider functions on. The Godement-Plancherel Theory in section 2.6 leads to orthogonal eigenfunction expansions for K-invariant functions on. The ideas here are quite standard and well known. For f : H n C, we define f : C by f (z : f(z, 0. Given g : C we define g 1 : H n C by g 1 (z, t : g(ze it. If f is a K-invariant function on H n then f is K-invariant on. If g is K-invariant on then g 1 is K-invariant on H n and (g 1 g. For D D(H n define a differential operator on D on R by D (g : (D (g 1 for g C ( R. It is easy to verify that if D D K (H n then D is a K-invariant differential operator on R. Moreover D (φ (D ((φ 1 (D (φ D (φ φ. Thus φ (z q (ze γ(z/2 is an eigenfunction for D whenever D D K (H n. As an example, consider the Heisenberg sub-laplacian L given by Equation 2.6. Using Equations 2.5 one computes that (3.1 L 4 γ/2 + E

12 12 C. BENSON, J. JENKINS, AND G. RATCLIFF where : n j1 z j z j (so that 4 is the usual Laplace operator on R and E : n j1 (z j z j z j z j. A homogeneity argument shows that E(φ 0 for Λ. Equation 2.9 thus yields (3.2 Proposition 3.1 (cf. [Yan]. {φ L 2 K ( with φ 2 2 (2π n /d. (4 γ/2 (φ L (φ (2 + nφ. Λ} is a complete orthogonal system in Proof. The functional form for φ,λ shows that φ q (ze 1 4 z 2 for some q C[ R ] K. Thus φ is a K-invariant Schwartz function and hence belongs to L 2 K ( R. For v F, define M v (z : π(z, 0v, v. Since π is a square-integrable representation, M v is square-integrable. Let {v 1,..., v d }, {u 1,..., u d } be orthonormal bases F for P, P. One has φ 1 d i M v i, φ 1 d j M u j so that φ, φ 2 1 M vi, M uj d d 2. Standard facts concerning square-integrable representations ensure that 2 M vi, M uj 2 c v i, u j F for some constant c. Thus we have φ, φ 2 cδ, d, and hence the φ s are pair-wise orthogonal. To compute the L 2 -norm of φ, one must determine c. As 1 is a unit vector in F, one has i,j c M 1, M 1 2 e 1 4 z e 1 2 z 2 dz 2πn (n 1! (2π n. 0 e r2 /2 r 2n 1 dr It remains to show that {φ Λ} is complete in L 2 K (. Suppose that f L 2 K ( and that f, φ 2 0 for all Λ. Since φ (z q (ze z 2 /4 where {q Λ} is a basis for C[ R ] K, we see that f(zp(ze z 2 /4 dz 0 for all polynomials p C[ R ]. It follows that f 0 almost everywhere and hence that {φ Λ} is complete.

13 SCHWARTZ FUNCTIONS ON THE HEISENBERG GROUP 13 Note that φ (z is bounded by φ (0 1. For f L p K ( and Λ let f( be defined by f( f, φ 2 f(zφ (zdz. Corollary 3.2. If f L 2 K ( then f ( φ f( n 1 d φ Λ 2 f(φ 2 2π Λ f 2 2 f( ( 2 n 1 d φ Λ 2 2 2π f( 2. Λ in L 2 ( and It is shown in [Yan] that the φ s are essentially eigenfunctions for F. precisely, Proposition 3.3 (cf. [Yan]. The function φ (z : φ ( 2z satisfies F ( φ (2π n ( 1 φ. More The factor of 2 appears above but not in [Yan] because different coordinates on H n are used there. The key fact used in Yan s proof of Proposition 3.3 is that φ can be expressed in terms of the Hermite-Weber transform of p. One can also prove Proposition 3.3 by using the fact that a suitable multiple of the Hermite operator 4 + 2γ is an infinitesimal generator for the Fourier transform. (See for example, pg 122 in [HT92]. With the notational conventions here, this fact can be written as F In view of Equation 3.2, we have and hence (2πi n exp(iπ( γ/2. ( γ/2 φ 2 + n φ 2 F ( φ (2πi n e iπ(2 +n/2 φ (2π n ( 1 φ as stated. For r > 0, we let δ r : be the dilation δ r (z : rz. Since one has F (f δ r ( 1 r 2n F (f δ 1/r, we obtain the formulas (3.3 (3.4 F (φ (4π n ( 1 φ δ 2 and more generally ( n 4π F (φ δ r ( 1 φ δ 2/r. r 2

14 14 C. BENSON, J. JENKINS, AND G. RATCLIFF In summary, the set {φ Λ} forms a complete orthogonal system in L 2 K (. Each φ is a simultaneous eigenfunction for the differential operators {D D D K (H n } and (modulo dilation for the Fourier transform F. 4. Generalized binomial coefficients For Λ, the functions {p } form a basis for the space of K-invariant polynomials on R of degree at most 2. Since q belongs to this space, we can write: q [ ] (4.1 ( 1 p [ ] for some well defined numbers. We call these values generalized binomial coefficients for the action of K on. They will play a crucial role in the formulation and proof of our characterization of K-invariant Schwartz functions via the spherical transform. The results that we will need concerning generalized binomial coefficients are contained in this section. Since the functions q and p are all real valued, the generalized binomial [ ] coefficients are real numbers. It is shown in [BR], moreover, that the s are non-negative. [ Since ] ( 1 p is the [ homogeneous ] component of highest degree in q, we see that 1 and that 0 for with. We extend the [ ] definition of generalized binomial coefficients by setting 0 when >. [ ] Since q (0 1 p 0 and p (0 0 for > 0, we see that 1. Here recall 0 that 0 Λ denotes the index with P 0 C. The generalized binomial coefficients were introduced by Z. Yan in [Yan], where one finds the important identity (4.2 γ k k! p +k d d [ ] p. For the case where K U(n, decomposition 2.1 reads C[ ] m0 P m( and the U(n-invariant polynomial p m associated with P m ( is p m (n 1! (m+n 1! γm (see Proposition 6.2 in [BJR92]. Equation 4.2 shows that in this case we have [ ] ( m + k m + k (4.3. m k

15 SCHWARTZ FUNCTIONS ON THE HEISENBERG GROUP 15 The generalized binomial coefficients are studied further in [BR], where the reader will find proofs of the following identities: γq [ ] d q d + (2 + nq [ ] (4.4 q, ( [ ][ ] δ l 1 ( δ! ( l!(l δ![ δ Letting δ 0 in Equation 4.5, we see that [ ] ( (4.6, l l [ ] [ ] [ ] since 1. It follows that for fixed, has polynomial growth 0 0 in. Equation 4.6 gives, in particular, [ ] ( Evaluating Equation 4.4 at z 0 and using Equation 4.7 yields also [ ] d (4.8 + n. d +1 Our characterization of S K (H n, presented below in Section 6, involves the application of difference operators D +, D defined as follows. Definition 4.1. Given a function g on Λ, D + g and D g are the functions on Λ defined by D + g( [ ] d g( ( + ng( [ ] d (g( g( d d +1 D g( g( 1 for > 0 and D g(0 0. [ ] g( 1 +1 [ ]. ] (g( g( The two formulae presented for D ± g agree in view of Equations 4.8 and 4.7. Using this notation, Equation 4.4 can be rewritten as (4.9 γq (D + D q. Since φ (z q (ze γ(z/2 and φ,λ (z, t φ ( λ ze iλt, we obtain also (4.10 γφ (D + D φ, γφ,λ 1 λ (D+ D φ,λ.

16 16 C. BENSON, J. JENKINS, AND G. RATCLIFF The right sides in these equations denote the functions on and H n obtained by applying the difference operators to the -index. Thus, for example, D φ,λ denotes the function on H n defined by D φ,λ (z, t φ,λ (z, t [ ] 1 φ,λ (z, t. ( λ z ( λ z Lemma 4.1. λ q 1 λ D q for λ 0. Proof. Equation 4.1 shows that ( q λ z [ ( ]( 1 p λ z [ ] ( λ p (z. Suppose here that λ > 0. We compute that ( λ q λ z [ ] ( 1 λ 1 p (z 1 [ ] ( 1 λ λ p (z [ ] ( λ λ p (z 1 [ ] ( ( λ λ p (z ( λ q λ z 1 [ ( ]( ( 1 λ p λ z [ ] Equation 4.5 shows that ( [ ][ ] δ δ 1. Hence δ ( λ q λ z ( λ q λ z 1 [ ][ δ ( ]( 1 λ δ p λ z δ 1 ( λ q λ z 1 [ ] [ δ ( ]( 1 λ δ p λ z δ 1 ( λ q λ z 1 [ ( ]q λ δ δ λ z δ 1 1 ( λ D q λ z. A similar analysis applies when λ < 0. Using Lemma 4.1 we obtain that for λ > 0, λ φ,λ (z, t λ [ q ( λ z e λγ(z/2 e iλt ] 1 λ D φ,λ (z, t γ(z 2 φ,λ(z, t+itφ,λ (z, t.

17 SCHWARTZ FUNCTIONS ON THE HEISENBERG GROUP 17 In view of Equation 4.10 we can also write λ φ,λ (z, t 1 λ D+ φ,λ (z, t + γ(z 2 φ,λ(z, t + itφ,λ (z, t. Thus we have { 1 λ φ,λ λ D φ,λ γ φ } 2,λ + itφ (4.11,λ 1 λ D+ φ,λ + γ φ for λ > 0, 2,λ + itφ,λ and similarly (4.12 Equivalently (4.13 { 1 λ φ,λ λ D φ,λ + γ φ } 2,λ + itφ,λ 1 λ D+ φ,λ γ φ 2,λ + itφ,λ ( γ { ( 2 + it λ φ,λ ( 1 D+ φ λ,λ for λ > 0 λ 1 D φ λ,λ for λ < 0 for λ < 0. } and (4.14 ( γ { ( 2 it λ φ,λ 1 D φ λ,λ for λ > 0 ( λ 1 D+ φ λ,λ for λ < 0 }. 5. K-Invariant Schwartz functions on We let S( denote the space of Schwartz functions on and S(H n the Schwartz functions on H n R. As usual, S K ( and S K (H n will denote the K-invariant elements in S( and S(H n respectively. Note that the φ s belong to S K (. We would like to characterize the spaces S K ( and S K (H n in terms of the spherical transform. The problem for S K ( is related to that for S K (H n as follows. Since U(n acts trivially on the center of H n, we have that S K (H n S K ( S(R. Moreover, for g S K (, h S(R and (, λ Λ R one computes easily that (5.1 (g h (, λ ( 1 λ n ( g δ 1 λ (F(h(λ where F(h(λ h(te iλt dt is the one dimensional Fourier transform in the central direction. Thus the problem of characterizing S K (H n via the spherical transform leads us to seek conditions on maps f : Λ C that are necessary and sufficient to ensure that f L 1 K ( is in fact a Schwartz function. This problem, which is of interest in its own right, is answered cleanly by Theorem 5.1 below. In Section 6 we will return to the problem of characterizing S K (H n via the spherical transform. Definition 5.1. We say that a function F : Λ C is rapidly decreasing if for every sequence ( m m1 in Λ with lim m m and every N Z + one has lim F ( m m N 0. m

18 18 C. BENSON, J. JENKINS, AND G. RATCLIFF Equivalently, for each N Z +, there is a constant C N for which F ( C N (2 + n N. Theorem 5.1. If f S K ( then f is rapidly decreasing on Λ. Conversely, if F is rapidly decreasing on Λ then F f for some f S K (. Moreover, the map : S K ( {F F is rapidly decreasing on Λ} is a bijection. Lemma 5.2. If F is rapidly decreasing on Λ then so are D + F and D F. Proof. Let c m : max m F (. The function m c m is rapidly decreasing on Z +. Using Equation 4.8 and the fact that the generalized binomial coefficients are non-negative (proved in [BR] we obtain D + F ( +1 c +1 d d [ +1 ] F ( + ( + n F ( d d [ ( + n ( c + c +1, ] + c ( + n which shows that D + F is rapidly decreasing. Using Equation 4.7 in a similar fashion yields showing that D F is rapidly decreasing. D F ( ( c 1 + c, Lemma 5.3. Let F be a rapidly decreasing function on Λ and G be a bounded function on Λ. Then d ( D + n F (G(, d F (D + G( Λ Λ ( d D + + n F (G(. Λ d F (D G( Λ Here all four series converge absolutely.

19 SCHWARTZ FUNCTIONS ON THE HEISENBERG GROUP 19 Proof. We compute formally that d F ( Λ d F (D + G( Λ Λ +1 d Λ [ ] d G( ( + ng( d ] F (G( d ( + nf (G( Λ +1 d [ 1 [ Λ d ( D n F (G(. ] F ( G( d ( + nf (G( Λ The hypotheses on F and G ensure that the terms in each series above are products of rapidly decreasing functions with functions of polynomial growth in. Thus all of these series converge absolutely and the above rearrangements are justified. Indeed, suppose that G( M for all Λ. Equations 4.7 and 4.8 show that +1 d d [ ] G( ( + ng( 2M( + n. Moreover, since P P (, we have that d dim ( P ( ( + n 1 (5.2 ( + n 1n 1, (n 1! a polynomial bound on d. Hence d ( + ng( is bounded by M( + n n and d G( by M( + n 1 n 1. Finally note that [ ] 1 F ( is bounded by dim ( P 1 ( H( ( + n 2 n 1 H( where H( max{ F ( : 1} is rapidly decreasing in. The second identity follows from the first by interchanging the roles of F and G. Proof of Theorem 5.1. First suppose that f S K ( and let L denote the Heisenberg sub-laplacian given by Equation 2.6. The associated differential operator L on, given in Equation 3.1, is self-adjoint and preserves S K (. In view of Equation 3.2, we have that for N Z +, (2 + n N f( ( (2 + n N f, φ 2 f, (L N φ 2 (L N f, φ 2 (L N f 1

20 20 C. BENSON, J. JENKINS, AND G. RATCLIFF since φ (z 1 for all z. Hence we obtain a bound of the form f( C N (2 + n N for each N Z + where C N (L N f 1. It follows immediately that f is a rapidly decreasing function on Λ. Conversely, suppose that F is a rapidly decreasing function on Λ. The estimate given above in Equation 5.2 shows that d is bounded by a polynomial function of. It follows that the series Λ d (F ( p converges absolutely for all p Z +. As φ (z 1, we conclude that the series Λ d F (φ (z converges absolutely and uniformly in z. Define a function f on via ( n 1 (5.3 f(z d F (φ 2π (z. Λ Since Λ d F ( 2 converges, Corollary 3.2 shows that f L 2 K ( with f F. Moreover, as Λ d F ( converges, and each φ is a Schwartz function, and the φ s are uniformly bounded by 1, it follows easily that lim z f(z 0. To prove that f is a Schwartz function, it suffices to show that f is smooth and that lim z γ(z a ( b f(z 0 for all nonnegative integers a, b. In view of the preceding analysis, this follows by induction from Lemma 5.4 below. To complete the proof, note that if f S K ( and F ( f( 0 for all Λ then Equation 5.3 implies that f 0 in L 2 K (. As f is continuous, it follows that f(z 0 for all z. This shows that f f is injective on S K (. n Lemma 5.4. Let f ( 1 2π Λ d F (φ where F is rapidly decreasing. Then 1. γf can be written in the form ( 1 n Λ d G(φ where G is rapidly decreasing, and 2. f is twice differentiable and f can be written in the form ( 1 where H is rapidly decreasing. Proof. We use Equation 4.10 together with Lemma 5.3 to write ( n 1 γ(zf(z d F (γ(zφ 2π (z Λ ( n 1 d F ( ( D + D φ 2π (z Λ ( n 1 ( d D D + F (φ 2π (z. 2π Λ 2π n Λ d H(φ

21 SCHWARTZ FUNCTIONS ON THE HEISENBERG GROUP 21 In view of Lemma 5.2, G (D D + F is rapidly decreasing. This establishes (1 in the statement of Lemma 5.4. Equation 3.2 shows that (4 γ/2φ (2 + nφ and hence φ γ 8 φ 2 + n φ 4. Thus formal application of term-wise to the series ( 1 yields the series ( 1 n 2π Λ d H(φ where H( G( n F ( 4 2π n Λ d F (φ for f and γf ( 1 n 2π Λ d G(φ as above. As both F and G are rapidly decreasing, so is H. As both Λ d F (φ (z and Λ d H(φ (z converge uniformly in z, we conclude that f is twice differentiable with f ( 1 n 2π Λ d H(φ. This establishes (2 in the statement of Lemma K-Invariant Schwartz functions on H n In this section we return to the problem of characterizing the space S K (H n via the spherical transform. Theorem 6.1 solves this problem and is our main result. Several definitions are required in the formulation of this theorem. Definition 6.1. Let F be a function on (K, H n. We say that F is rapidly decreasing on (K, H n if F is continuous on (K, H n, the function F on defined by F (w F (η w belongs to S K (, the map λ F (, λ is smooth on R (, 0 (0, for each fixed Λ, for each m, N 0 there exists a constant C m,n for which for all (, λ Λ R. m λ F (, λ C m,n λ m+n (2 + n N We say that a continuous function on 1 (K, H n is rapidly decreasing if it extends to a rapidly decreasing function on (K, H n 1 (K, H n 2 (K, H n. Since 1 (K, H n is dense in (K, H n, such an extension is necessarily unique. Note that if F is rapidly decreasing on (K, H n then F (, λ is rapidly decreasing on Λ, in the sense of Definition 5.1, for each λ 0. We see that F is bounded by letting m N 0 and one can show, moreover, that F vanishes at infinity by letting m 0 and N 1. We remark that the functions λ m F (, λ defined on 1 (K, H n need not extend continuously across 2 (K, H n. Example 6.1 below illustrates this behavior.

22 22 C. BENSON, J. JENKINS, AND G. RATCLIFF Definition 6.2. Let F be a function on 1 (K, H n which is smooth in λ. M + F and M F are the functions on 1 (K, H n defined by { ( M + λ F (, λ 1 D+ } ( F (, λ for λ > 0 λ λ 1 D and F (, λ for λ < 0 λ M F (, λ { ( λ 1 λ D F (, λ for λ > 0 ( λ 1 λ D+ F (, λ for λ < 0 We remind the reader that the difference operators D ± are defined by D + F (, λ [ ] d F (, λ ( + nf (, λ, d +1 D F (, λ F (, λ [ ] F (, λ. 1 Definition 6.3. Ŝ(K, H n is the set of all functions F : (K, H n C for which (M + l (M m F is rapidly decreasing for all l, m 0. If F is rapidly decreasing on (K, H n then λ F (, λ is smooth on R and we have well defined functions (M + l (M m F on 1 (K, H n. F belongs to Ŝ(K, H n if and only if these functions extend continuously to rapidly decreasing functions on (K, H n. Theorem 6.1. If f S K (H n then f Ŝ(K, H n. Conversely, if F Ŝ(K, H n then F f for some f S K (H n. Moreover, the map : S K (H n Ŝ(K, H n is a bijection. If f S K (H n and f 0 then the inversion formula for the spherical transform (Equation 2.13 yields that f 0. Thus the spherical transform is injective on S K (H n. To prove Theorem 6.1, it remains to show that S K (H n Ŝ(K, H n and that Ŝ(K, H n S K (H n. This will require most of the remainder of this section. First, however, we will present an example. Example 6.1. Consider the case where K U(n. Decomposition 2.1 reads C[ ] m0 P m( and one has (U(n, H n ( R Z + R +. Using Equation 4.3, one computes that the difference operators D ± appearing in the definition of the set Ŝ(U(n, H n are given by D + g(m (m + n(g(m + 1 g(m, D g(m m(g(m g(m 1 for functions g : Z + C. Let f S K (H n be defined by f(z, t g(zh(t where g(z e z 2 and h S(R. We will compute the spherical transform f : (U(n, H n C. }.

23 SCHWARTZ FUNCTIONS ON THE HEISENBERG GROUP 23 The polynomial q m associated with P m ( is a suitably normalized generalized Laguerre polynomial which can be written explicitly as (see eg. [BJR92] m ( ( m 1 q m (z (n 1! z 2 j. j (n + j 1! 2 Thus, for λ 0 we have (g δ 1/ λ (m j0 (n 1! where K λ. One has λ 4 4 λ z 2j e K z 2 dz 2πn (n 1! and hence ( g δ 1/ λ (m ( π K e z 2 / λ q m (ze z 2 /4 dz m ( m ( 1 j j 2 j (n + j 1! j0 0 n m j0 ( π K n ( 1 1 2K z 2j e K z 2 dz e Kr2 r 2(n+j 1 dr πn (n + j 1! (n 1!K n+j ( ( m 1 j 2K m j ( 4π λ 4 + λ Thus we have (see Equation 5.1 ( n ( 1 f(m, λ g δ 1 (mf(h(λ λ λ ( n ( m 4π 4 λ F(h(λ. 4 + λ 4 + λ For w one has (see Equation 2.15 f(w F H (f(w, 0 F (g(wf(h(0 (π n h(tdt e w 2 /4. n ( m 4 λ. 4 + λ One can verify directly that f Ŝ(U(n, H n, but we will not do this here. In particular, the functions λ f(λ, m are differentiable on R for each m Z +, and the derivatives λ f(λ, m satisfy estimates as in Definition 6.1. Note, however, that these derivatives do not agree as λ 0. Specifically, lim λ 0 λ f(λ, m depends on m. Hence there is no function g L 1 U(n (H n whose U(n-spherical transform satisfies ĝ(λ, m λ f(λ, m on R Z +.

24 24 C. BENSON, J. JENKINS, AND G. RATCLIFF This example illustrates why the characterization of S K (H n is somewhat complicated. Although the behavior of the derivatives of the functions λ f(λ, m does come into play, as expected, the space S K (H n is not closed under λ-derivatives. Indeed, one must replace λ by the operators M ± which involve both λ and the difference operators D ±. Proof that S K (H n Ŝ(K, H n. Suppose that f S K (H n and let F : f. We begin by showing that F is rapidly decreasing. F is continuous on (K, H n, as is the spherical transform of any integrable K-invariant function. Moreover, Equation 2.15 shows that F (w F H (f(w, 0. Since f is a Schwartz function, so is F (w. Thus F satisfies the first two conditions in Definition 6.1. Next we will show that F satisfies the estimates in Definition 6.1 for m 0. The argument is similar to that in the first part of the proof for Theorem 5.1. Recall that the Heisenberg sub-laplacian L is a self-adjoint operator on L 2 (H n with L(φ,λ λ (2 + nφ,λ. Thus we have λ N (2 + n N F (, λ ( λ (2 + n N f, φ,λ 2 f, L N φ,λ 2 L N f, φ,λ 2 L N f 1 since φ,λ (z, t 1 for all (z, t H n. Letting C 0,N : L N f 1, we see that the inequalities in Definition 6.1 hold for m 0. Since φ,λ (z, t is smooth in λ R for fixed (z, t and fφ,λ is a Schwartz function, F (, λ f(, λ is smooth in λ R with (6.1 λ F (, λ f(z, t λ φ,λ (z, tdzdt. Equations 4.11 and 4.12 provide formulae for λ φ,λ λ φ, λ but we require a different approach here. We write φ (z, t φ (ze it as φ (z, ze it so that We see that λ φ,λ (z, t 1 λ φ,λ (z, t φ ( λ z, ze iλt. [( n j1 z j z j φ ] ( λ z, z itφ,λ (z, t. Substituting this expression into Equation 6.1 and integrating by parts gives λ F (, λ 1 λ (Df (, λ i(tf (, λ where Df : n j1 z j (z j f. Since tf and Df are both Schwartz functions on H n, they satisfy estimates as above. Thus, given N 0, one can find constants A and B

25 with λ F (, λ SCHWARTZ FUNCTIONS ON THE HEISENBERG GROUP 25 A λ N+1 (2 + n + B N λ N+1 (2 + n C 1,N N+1 λ N+1 (2 + n N where C 1,N A+B. By induction on m we see that λ m F (, λ satisfies an estimate as in Definition 6.1. This completes the proof that F is rapidly decreasing. Equations 4.13 and 4.13 show that (( γ (( γ M + F f 2 + it and M F f 2 it 1 (K,H n Thus (M + l (M m F is the restriction of ĝ to 1 (K, H n where g ( 1 m (γ/2 + it l (γ/2 it m f. 1 (K,H n Since g S K (H n, it now follows that (M + l (M m F is rapidly decreasing. Thus F Ŝ(K, H n as desired. The following proposition is required to complete the proof of Theorem 6.1. Proposition 6.2. Let F be a bounded measurable function on (K, H n with C F (, λ λ N (2 + n N for some N n + 2 and some constant C. Then 1. F L p ( (K, H n for all p 1, and 2. F f for some bounded continuous function f L 2 K (H n. Suppose, for example, that f is a continuous function in L 1 K (H n with F f rapidly decreasing. Proposition 6.2 shows that f is square integrable and that F is integrable. Thus, the inversion formula 2.13 applies and we can recover f from F via ( n+1 1 (6.2 f(z, t d F (, λφ,λ (z, t λ n dλ. 2π R Λ In particular, we see that this formula certainly holds for any function f S K (H n. Proof of Proposition 6.2. To establish the first assertion, it suffices to prove that F L 1 ( (K, H n. Indeed, F (, λ p satisfies an inequality as in the statement of the proposition with N replaced by pn. Let A 1 : {φ,λ λ (2 + n 1} and A 2 : {φ,λ λ (2 + n > 1}. Let M be a constant for which F (ψ M for all ψ (K, H n and let ( m + n 1 (6.3 d m : dim (P m (. m Note that d d m (m + n 1 n 1. m

26 26 C. BENSON, J. JENKINS, AND G. RATCLIFF We compute A 1 F (ψ dµ(ψ ( 1 d F (, λ λ n dλ 2π Λ 0< λ < 1 2 +n 2M (2π n+1 m0 2M (n + 1(2π n+1 d m 1 2m+n 0 m0 λ n dλ ( n+1 1 d m. 2m + n Since d m O(m n 1, we see that the last series converges. Hence F is integrable over A 1. Next we compute ( 1 F (ψ dµ(ψ d F (, λ λ n dλ A 2 2π Λ λ > 1 2 +n 2C (2π n+1 m0 d m 1 2m+n 2C (N n 1(2π n+1 m0 λ n λ N (2m + n dλ N d m (2m + n. n+1 The hypothesis that N n 2 was used above to evaluate the integral of 1/λ N n over 1 < λ <. Since d 2m+n m O(m n 1, we see that the series in the last expression converges. Hence F is integrable over A 1. As A 1 A 2 1 (K, H n is a set of full measure in (K, H n, it follows that F L 1 ( (K, H n. Next let f be the function on H n defined by f(z, t F (ψψ(z, tdµ(ψ. (K,H n Since F L 1 ( (K, H n and the spherical functions are continuous and bounded by 1, we see that f is well defined, continuous and bounded by F L 1 ( (K,H n. Moreover, since F L 2 ( (K, H n and : L 2 ( (K, H n L 2 K (H n is an isometry, we have that f L 2 K (H n with f 2 2 (K,H n F (ψ 2 dµ(ψ and f F. This establishes the second assertion in Proposition 6.2. Remark 6.1. One can show that the set A 1 used in the proof of Proposition 6.2 is compact in (K, H n. Thus any bounded measurable function is integrable over A 1. This observation motivates the decomposition used in the proof. Proof that Ŝ(K, H n S K (H n. Suppose now that F Ŝ(K, H n. Proposition 6.2 shows that F f where ( n+1 1 f(z, t d F (, λφ,λ (z, t λ n dλ 2π R Λ

27 SCHWARTZ FUNCTIONS ON THE HEISENBERG GROUP 27 is K-invariant, continuous, bounded and square integrable. S K (H n, we will show that f is smooth and that ( γ 2 a ( b 4 + t2 c f L 2 t K(H n for all a, b, c 0. This will follow from the facts 1. f L 2 K (H n with ( f Ŝ(K, H n, 2. f t L2 K (H n with ( f t Ŝ(K, H n, and 3. ( γ 2 ± it f L 2 K (H n with (( γ 2 ± it f Ŝ(K, H n, To show that f which we prove below. Here n j1 z j z j as before. Using Equation 2.9 for the eigenvalues of the Heisenberg sub-laplacian, one obtains λ (2 + nφ,λ Lφ,λ [ 4 γ 2 4 φ,λ λ2 2 γφ,λ. Using Equation 4.10 for γφ,λ gives ( t 4 φ,λ λ 2 (D+ D φ,λ λ (2 + nφ,λ λ [ ] d φ 2 d,λ + (2 + nφ,λ + +1 Define a function F on 1 (K, H n by 2 ] φ,λ 1 F (, λ λ 2 (D+ D F (, λ λ (2 + nf (, λ λ [ ] d F (, λ + (2 + nf (, λ + 2 d +1 [ 1 ] φ,λ [. ] F (, λ. It is not hard to show that F Ŝ(K, H n. In particular, note that Equations 4.7 and 4.8 give F (, λ λ ( + n F (, λ + (2 + n F (, λ + F (, λ One uses this to show that F satisfies estimates as in Definition 6.1. Lemma 5.3 shows that for each λ 0 d (D + D F (, λφ,λ Λ d F (, λ(d + D φ,λ Λ 1 Moreover,

28 28 C. BENSON, J. JENKINS, AND G. RATCLIFF and hence also d F (, λ λ n dλ 4 d F (, λ ( φ,λ λ n dλ. R R Λ We conclude that f L 2 K (H n with 4( f F Ŝ(K, H n This proves item (1 above. Next note that the function defined on 1 (K, H n by λf (, λ belongs to Ŝ(K, H n. Since φ,λ iλφ t,λ, we see that f t L2 K (H n with ( f t iλf Ŝ(K, H n. This establishes item (2 above. We begin the proof of item (3 by setting Λ (6.4 F (z, λ Λ d λ n F (, λφ,λ(z for each λ 0, so that f(z, t ( n+1 1 2π R F (z, λe iλt dλ. Note that we can compute λ F by taking derivatives term-wise in Equation 6.4. For λ > 0 we have λ F (z, λ d λ n λ F (, λφ,λ(z Λ d nλ n 1 F (, λφ,λ(z + Λ + Λ d λ n F (, λ λ φ,λ(z. Since F Ŝ(K, H n, the estimates in Definition 6.1 can be applied to show that the first two sums converge absolutely for each λ > 0. For the third sum, we use Equations 4.11 for λ φ,λ (z λφ,λ (z, 0 together with Lemma 5.3 to derive two identities: d λ n F (, λ λ φ,λ(z Λ { γ 2 Λ d λ n F (, λφ,λ (z + Λ d λ n F (, λ 1 λ D φ,λ (z } γ 2 Λ d λ n F (, λφ,λ (z + Λ d λ n F (, λ 1 λ D+ φ,λ { (z γ F (z, λ 2 Λ d } λ n 1 (D + + nf (, λ γ F (z, λ 2 Λ d λ n 1 (D + nf (, λ Substituting these identities in the expression for λ F (z, λ gives (6.5 λ F (z, λ { γ 2 F (z, λ + Λ d λ n ( λ 1 λ D+ F (, λφ,λ (z γ 2 F (z, λ + Λ d λ n ( λ 1 λ D F (, λφ,λ (z }

29 SCHWARTZ FUNCTIONS ON THE HEISENBERG GROUP 29 both valid for λ > 0. We have similar identities for λ < 0: (6.6 λ F (z, λ { γ 2 F (z, λ + Λ d λ n ( λ 1 λ D F (, λφ,λ (z γ 2 F (z, λ + Λ d λ n ( λ 1 λ D+ F (, λφ,λ (z }. Note that ( λ 1 D± F is the restriction of M ± F to + λ 1 (K, H n {φ,λ Λ, λ > 0} and also of M F to 1 (K, H n {φ,λ Λ, λ < 0}. Since M ± F Ŝ(K, H n, M ± F is integrable on (K, H n and Equations 6.5 and 6.6 show that λ F (z, λ is integrable on R {λ λ 0}. We have, ( (2π n+1 itf(z, t F (z, λ λ e iλt dλ R 0 0 F (z, λ λ ( e iλt dλ + lim λ 0 + λ F (z, λe iλt dλ F (z, λ + lim λ 0 0 Λ F (z, λ. F (z, λ λ ( e iλt dλ λ F (z, λe iλt dλ It can be shown that the limits lim λ 0 ± F (z, λ exist and are equal. Here one needs to use the hypotheses that F is continuous across 2 (K, H n and that F is a Schwartz function. The proof, which is rather involved, is presented below in Section 7. Using Equations 6.5 and 6.6 for λ F (z, λ we obtain ( (2π n+1 ± γ 2 + it ( f(z, t d λ n λ 1 λ D± F (, λφ,λ (z, tdλ Λ d λ n ( λ 1 λ D F (, λφ,λ (z, tdλ d M ± F (, λφ,λ (z, t λ n dλ. R Λ We conclude that ( ± γ + it f L 2 K (H n with (2π (( n+1 ± γ + it f 2 Ŝ(K, H n. This completes the proof for item (3. M ± F The characterization of the spherical transform of a Schwartz function given in Theorem 6.1 can be used to construct functions on H n whose transform has a predetermined support. Recall that Theorem 2.2 asserts the existence of a map E : (K, H n (R + d R which is a homeomorphism onto its image. In Corollary 6.3 below, K U(n or T n, so that d 1 or d n. In the first case, (U(n, H n

30 30 C. BENSON, J. JENKINS, AND G. RATCLIFF {(m, λ m Z +, λ R} {s s > 0} with ( λ E(m, λ (2m + n, λ 2 and E(s ( s 2, 0. In the second case, we have (T n, H n {(m, λ m (Z + n, λ R} {s s (R + n } with ( λ E(m, λ 2 (2m 1 + n,..., λ ( 2 (2m s1 n + nλ and E(s 2,..., s n 2, 0. Corollary 6.3. Let F be a smooth function of compact support on R 2 (or R n+1. Then F F E is the spherical transform of a U(n- (or T n - invariant Schwartz function on H n. Proof. Since F has compact support, we see that F (m, λ 0 if λ and λ (2 m + n are sufficiently large. Hence our growth conditions are automatically satisfied. The only property left to verify is that M ± F extends continuously across 2 (K, H n. For K U(n, we first consider M + F (m, λ with λ 0 +, λ (2m + n s. We have M + 1 F (m, λ λ F (m, λ λ D+ F (m, λ ( ] λ λ [F (2m + n, λ m + n [ ] F (m + 1, λ F (m, λ 2 λ 2m + n ( ( λ λ 1 F (2m + n, λ + 2 F (2m + n, λ m + n [ ( ( ] λ λ F (2m + n + 2, λ F (2m + n, λ. λ 2 2 Now substitute λ(2m + n/2 s/2, to obtain M + s ( s ( s F (m, λ 2λ 1F 2, λ + 2 F 2, λ s [ ( s ( s ] F 2λ λ, λ F 2, λ n [ ( s ( s ] F 2λ 2 + λ, λ F 2, λ ( s 2 F 2, λ s [ ( s ( s ( s ] F 2λ λ, λ F 2, λ λ 1 F 2, λ n [ ( s ( s ] F 2λ 2 + λ, λ F 2, λ, which converges to ( s 2 F 2, 0 s ( s 4 2 1F 2, 0 n ( s 2 1F 2, 0