REVISITING THE FOURIER TRANSFORM ON THE HEISENBERG GROUP

A recent theorem of S. Alesker, S. Artstein-Avidan and V. Milman characterises the Fourier transform on Rn as essentially the only transform on the space of tempered distributions which interchanges convolutions and pointwise products. In this note we study the image of the Schwartz space on the Heisenberg group under the Fourier transform and obtain a similar characterisation for the Fourier transform on the Heisenberg group. 2010 Mathematics Subject Classification: 43A30, 43A70, 35K08.


Introduction
Let H n = C n × R denote the Heisenberg group equipped with the group law (z, t)(w, s) = z + w, t + s + 1 2 (z • w) .
The group Fourier transform on H n is defined in terms of the Schrödinger representations π λ (z, t).Recall that π λ (z, t) are irreducible unitary representations of H n realised on the same Hilbert space, namely L 2 (R n ); their action is explicitly given by π λ (z, t)ϕ(ξ) = e iλt e iλ(x•ξ+ 1 2 x•y) ϕ(ξ + y), for ϕ ∈ L 2 (R n ).In the above λ ∈ R * = R \ {0} and according to a theorem of Stone and von Neumann any irreducible unitary representation ρ of H n such that ρ(0, t) = e iλt I is unitarily equivalent to π λ .The group Fourier transform of a function f on H n , say f ∈ L 1 (H n ), is defined to be the operator valued function Note that π λ (f ) is a bounded operator on L 2 (R n ) and π λ (f ) op ≤ f 1 .It is well known that when f ∈ L 1 ∩ L 2 (H n ), π λ (f ) is a Hilbert-Schmidt operator and we have the Plancherel theorem where dµ(λ) = c n |λ| n dλ is the Plancherel measure.Let S 2 stand for the space of Hilbert-Schmidt operators on L 2 (R n ) and define L 2 (R * , S 2 , dµ) to be the space of all functions on R * taking values in S 2 , and square integrable with respect to dµ(λ).Then the Plancherel theorem can be restated as: the Fourier transform is a unitary operator from L 2 (H n ) onto L 2 (R * , S 2 , dµ).Properties of the group Fourier transform have been studied by several authors starting from the classic work of D. Geller [4].
Apart from the Schrödinger representations π λ , which are all irreducible, we also have certain one dimensional representations on H n .For each ζ ∈ C n , these are given by the character χ ζ (z, t) = e i z•ζ .As we have noted above, the representations χ ζ do not occur in the Plancherel theorem as they form a set of zero Plancherel measure.However, we can consider both π λ and χ ζ together and consider the representations parametrised by (λ, ζ) ∈ R * ×C n .Consequently, we can define the group Fourier transform of a function f on H n by This operator valued function ζ → f (λ, ζ) has been introduced in [9] in connection with a Paley-Wiener theorem for the Heisenberg group.
In this paper we will use the above modified Fourier transform, which we call the Fourier-Weyl transform to characterise the image of the Schwartz space under the Fourier transform.Analogous to this, for S ∈ L 2 (R * , S 2 , dµ), we let Here F stands for the Euclidean Fourier transform.Moreover, F takes the Schwartz space S(R n ) onto itself; also it takes S (R n ), the space of tempered distributions onto itself.It has been shown by S. Alesker, S. Artstein-Avidan and V. Milman that these properties completely characterise the Euclidean Fourier transform.
Theorem 1.1 (S.Alesker, S. Artstein-Avidan and V. Milman [1]).Assume we are given a transform T : Then, T is essentially the Fourier transform F: that is, for some In an earlier paper [7], we obtained some characterisations of the Weyl transform and the Heisenberg group Fourier transform assuming the map to be a continuous linear operator satisfying certain conditions.In this paper, we obtain another characterisation, analogous to Theorem 1.1, for the Heisenberg group Fourier transform where we do not assume the map to be linear or continuous, but we only assume it to be a bijection.
Let S(H n ) stand for the Schwartz space on H n .Denote by S(H n ) the image of S(H n ) under the group Fourier transform f → π λ (f ).It is well known that the Fourier transform is an isometry from In Section 2, we give a characterisation of S(H n ).
For functions f, g ∈ L 1 (H n ) their convolution defined by is also integrable on H n and satisfies Given f and g from L 1 (H n ) we denote by f * 3 g the convolution in the central variable; thus We will show below that the Fourier-Weyl transform satisfies where F −1 is the inverse Euclidean Fourier transform on C n × R. We consider the above as the analogue of for the Euclidean Fourier transform.If we denote by R (z,t) , the right translation on H n , which gives rise to a translation on functions on H n given by (R (z,t) f )(w, s) = f ((w, s)(z, t)), then the Fourier transform satisfies We show that the properties (1.1), (1.2) and (1.3) characterise the Fourier transform on H n .The Heisenberg group is equipped with the non-isotropic dilation δ r (z, t) = (rz, r 2 t) with respect to which we have where d r ϕ(ξ) = ϕ(rξ) is the standard dilation.In the next theorem we let T g(λ, w) stand for π Then there exists a map ζ : Remark 1.1.The above theorem is the analogue of Theorem 1.1 for the group Fourier transform on H n .A similar result for the Weyl transform was proved in [6].Note that we do not assume anything on the action of the transform on the space of tempered distributions.The proof of the above theorem indicates that similar assumptions on the space of tempered distributions are not necessary in the characterisation of the Weyl transform obtained in [6].
Corollary 1.3.In addition to the hypotheses of Theorem 1.2, if the map T satisfies Corollary 1.4.If T is as in Theorem 1.2 and T satisfies As mentioned above we obtain a characterisation of S(H n ).As the results are technical we do not state them here.In the last section we consider the subspace of L 2 (H n ) consisting of functions of the form f * 3 q s where q s is the heat kernel on H n associated to the sublaplacian.
By slightly changing the notation we write the Fourier-Weyl transform of a function g as With this notation we address the question of characterising functions of the form g = f * 3 q s for a fixed s > 0 under the Fourier-Weyl transform.We prove the following result.For λ ∈ R * and s > 0 we let We prove this theorem in Section 4.

Properties of the Fourier transform
In this section we recall some properties of the group Fourier transform on H n most of which are well known.We also establish the property (1.2) for the Fourier transform.Let H n be the (2n+1)-dimensional Heisenberg group with the underlying manifold C n × R, and the group law given by The Haar measure on H n is the Lebesgue measure dz dt on C n × R. The Stone-von Neumann theorem states that all the infinite-dimensional irreducible unitary representations of H n , acting on L 2 (R n ), are parametrised by λ ∈ R * , and are given by When λ = 1, we call this the Weyl transform of g.For functions f, g ∈ L 1 (C n ), their λ-twisted convolution is defined as The operators W λ are continuous, linear and map For an integrable function f on H n , its Fourier transform is the operator-valued function defined by The Fourier transform is a bounded operator on L 2 (R n ) and it satisfies Let S 2 denote the space of Hilbert-Schmidt operators on L 2 (R n ).Then S 2 is a Hilbert space with the inner product (T, S) = tr(T S * ).Let S HS denote the Hilbert-Schmidt norm of an operator S ∈ S 2 .Let dµ(λ) = (2π) −n−1 |λ| n dλ and let L 2 (R * , S 2 , dµ) denote the space of functions on R * taking values in S 2 and square integrable with respect to the measure dµ.
The following well-known result summarises an important property of the Fourier transform, see [10].
We denote by F −1 f (z, λ) the inverse Euclidean Fourier transform of the function f on C n × R: For λ ∈ R, the partial inverse Fourier transform of f in the third variable, denoted by f λ , is defined as Let d r be the standard dilation on R n and δ r the non-isotropic dilation on H n defined by δ r (z, t) = (rz, r 2 t).We also let (δ r f )(z, t) = f (δ r (z, t)).An easy calculation shows that π λ (rz In view of the above formula for π λ (rz) we get We can now prove the following theorem for the Fourier-Weyl transform.
Then by the definition of g(λ, w) we have The theorem follows as the Euclidean Fourier transform converts products into convolutions.
In order to study the image of S(H n ) under the Fourier transform we need to introduce the following operators (see [4]).Let A be a (possibly unbounded) densely defined operator.Then for any bounded operator S on L 2 (R n ) which maps dom(A) into itself we define the derivation For j = 1, 2, . . ., n, let Q j ϕ(ξ) = ξ j ϕ(ξ) and P j ϕ(ξ) = ∂ ∂ξj ϕ(ξ) for a function ϕ on R n .We let ∂ Pj and ∂ Qj stand for the corresponding derivations.For j = 1, 2, . . ., n, we also define On the Heisenberg group we have the following left invariant vector fields: We let X j and Y j stand for the corresponding right invariant vector fields: The following proposition is implicit in the work of Geller [4].
Proposition 2.3.For f ∈ S(H n ) and λ ∈ R * , we have, for j = 1, 2, . . ., n, We would also like to express π λ (itf ) in terms of π λ (f ).Let Λ be the operator defined by We refer to Geller [4] for a proof.We remark that Propositions 2.3 and 2.4 are easily proved starting with the equation We leave the details to the reader.For multi-indices α, β, µ, ν ∈ N n we define ∂ α P , ∂ β Q , M µ Q and M ν P in the usual way, e.g.
. .∂ αn Pn , etc. Let S(λ) be an operator valued function on R * .For each N ∈ N we define a seminorm where the supremum is taken over all multi-indices µ, ν, α, β ∈ N n and j, k ∈ N such that |µ| stand for the set of all operator valued functions S on R * for which the seminorms S N are finite for all N ∈ N. We can make S(R * , S 2 ) into a topological vector space using these seminorms.
Proof: Let f ∈ S(H n ) and α, β, µ, ν ∈ N n , j, k ∈ N. Then the function Therefore, in view of Propositions 2.3 and 2.4, Thus the group Fourier transform takes S(H n ) into S(R * , S 2 ).Also note that there is a one to one correspondence between seminorms defining the respective topologies on S(H n ) and S(R * , S 2 ).
To prove the surjectivity, suppose S ∈ S(R * , S 2 ).Then we get Consequently, we infer that f ∈ S(H n ), which proves the surjectivity.
The above theorem states that S(H n ) = S(R * , S 2 ).

A characterisation of the Fourier transform
In this section we prove Theorem 1.2 stated in the introduction.If f ∈ S(H n ), then T f ∈ S(H n ) and hence we can find g ∈ S(H n ) so that T f (λ) = π λ (g).Defining U f = g, where g is as above, we get a bijection of S(H n ) onto itself such that T f (λ) = π λ (U f ) for all f ∈ S(H n ).For f, g ∈ S(H n ), the assumption (i) of Theorem 1.2 gives
We are left with deducing Corollary 1.4.
Proof of Corollary 1.4: On the one hand by (2.1), On the other hand by (1.5), Therefore, we get r

On the image of very rapidly decreasing functions under the Fourier-Weyl transform
In this section we prove Theorem 1.5 stated in the introduction.First we recall the Euclidean result and state it for the sake of convenience.Let be the heat kernel associated to the standard Laplacian on R n .The Segal-Bargmann or the heat kernel transform is the one which takes f ∈ L 2 (R n ) into the entire function Then it is well known from the work of Segal and Bargmann [2] that the image is a weighted Bergman space.More precisely, the above transformation takes L 2 (R n ) isometrically onto the space of entire functions F on C n for which In particular We rephrase this result in the following form.Consider functions of the form g(x) = f (x)p t (x) where f ∈ L 2 (R n ), for a fixed t > 0. Then Since p t (ξ) = e −t|ξ| 2 = c n,t p 1 4t (ξ) we see that g(ξ) = c n,t f * p 1 4t (ξ) and consequently g can be extended to C n as an entire function.Moreover,

Thus we have
Theorem 4.1.Let t > 0 be fixed.Then a function g ∈ L 2 (R n ) can be factored as g = f p t , f ∈ L 2 (R n ) if and only if g is entire and satisfies Theorem 1.5 stated in the introduction is the analogue of this result for functions on the Heisenberg group.We consider functions of the form g = f * 3 q s so that g λ (z) = f λ (z)q λ s (z).Note that Theorem 1.5 is a characterisation of the image of functions g for which g λ (z) has a Gaussian decay under the Fourier-Weyl transform.
Coming to the proof we make use of the fact that (see [11] for a proof).In terms of the Euclidean heat kernel we can write where s λ = λ −1 (tanh λs).Recalling the definition of the Fourier-Weyl transform we see that In view of the above expression for q λ s in terms of p s λ , we can appeal to Theorem 4.1 to conclude that ((f * 3 q s ) (λ, x, u)ϕ, ψ) extends to C n × C n as an entire function and that where z = x + iy, w = u + iv.Letting ψ run through an orthonormal basis for L 2 (R n ) we obtain Applying Plancherel theorem in the (x, u)-variable this becomes q λ s (ξ, η) −2 |g λ (ξ + iη)| 2 dξ dη < ∞ i.e., g λ (q λ s ) −1 ∈ L 2 (C n ).Let f λ = g λ (q λ s ) −1 and define f by the equation f (x, y, t) = R f λ (x, y)e −iλt dλ.