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
c Springer-Verlag 2002
Published online: 1 February 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
function f on H n and assume fˆ(λ)∗ fˆ(λ) ≤ c p̂2b (λ) where ps is the heat
kernel associated to the sublaplacian. We show that if |f (z, t)| ≤ c pa (z, t)
then f = 0 whenever a < b. When a ≥ b we replace the condition on f by
|f λ (z)| ≤ c pλa (z) where f λ (z) is the Fourier transform of f in the t-variable.
Under suitable assumptions on the ‘spherical harmonic coefficients’ of fˆ(λ)
we prove: (i) f λ (z) = c(λ)pλa (z) when a = b; (ii) when a > b there 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 decaying 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 Heisenberg 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 central 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 = 0 of a function f
on the Heisenberg group H n . Let pt (z, s), (z, s) ∈ H n , t > 0 be the heat
kernel associated to the sublaplacian L on H n . The Fourier transform of pt
is given by p̂t (λ) = e−tH(λ) where H(λ) = −∆ + λ2 |x|2 is the (scaled)
Hermite operator on Rn . As an analogue of Hardy’s theorem for the group
Fourier transform on H n we offer:
Theorem 1.1. Let f be a measurable function on H n which satisfies the
estimate |f (z, s)| ≤ c (1 + |z|2 )m pa (z, s) for some a > 0, m ≥ 0. Further
assume that fˆ(λ)∗ fˆ(λ) ≤ c H(λ)m p̂2b (λ), for some b > 0 and for all
λ = 0. Then f = 0 whenever a < b.
This is the analogue of classical Hardy’s theorem for the Fourier transform on Rn which says that if
2
2
|f (x)| ≤ c(1 + |x|2 )m e−a|x| , |fˆ(ξ)| ≤ c(1 + |ξ|2 )m e−b|ξ|
(1.1)
where fˆ(ξ) is the Fourier transform of f given by
−n
ˆ
2
f (ξ) = (2π)
f (x)e−ix·ξ dx
Rn
2
then (i) f = 0 whenever ab > 14 ; (ii) f (x) = c p(x)e−a|x| where p(x) is a
polynomial of degree ≤ 2m when ab = 14 and (iii) there are infinitely many
linearly independent functions satisfying the conditions (1.1) when ab < 14 .
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
pt (x) = (4πt)− 2 e− 4t |x|
2
associated to the standard Laplacian ∆ on Rn .
The above result for the Euclidean Fourier transform was first established
by Hardy [6] for n = 1 and 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
1.1 when m = 0 and a < 2b . The optimal result a < b will 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.
We are 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 nonnegative integers (p, q) let Spq be the space of bigraded spherical harmonics
of degree (p, q). Then L2 (S 2n−1 ) is the orthogonal direct sum of Spq , p, q ≥
j
: p, q ≥ 0, j = 1, 2, . . . d(p, q)} for
0. Fix an orthonormal basis {Ypq
j
∈ Spq . (We say more about these spherical harmonics
L2 (S 2n−1 ) with Ypq
in Sect. 2).
Theorem 1.2. Let f be an integrable function on H n whose Fourier transform satisfies the condition fˆ(λ)∗ fˆ(λ) ≤ c p̂2a (λ) for some a > 0 and for
all λ = 0. Further assume that f λ (z) satisfies
λ
j
f (|z|z )Ypq (z )dz ≤ cjpq pλa (z)
(1.2)
2n−1
S
j
Ypq
and λ = 0. Then f = ϕ ∗3 pa where ϕ is a tempered distribution
for all
on R with ϕ̂ ∈ 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) = c pa (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 > b is still excluded and
our next theorem treats this case. We are going to weaken the condition
fˆ(λ)∗ fˆ(λ) ≤ c p̂2b (λ) for which some more definitions are needed.
Let Φλα , α ∈ Nn be the (scaled) Hermite functions on Rn which are
eigenfunctions of H(λ). Let Hpq be the space of all polynomials of the
form P (z) = |z|p+q Y (z ) with Y ∈ Spq and define Wλ (P ) to be the
Weyl correspondence of P . Let Ekλ be the finite dimensional subspace of
L2 (Rn ) spanned by {Φλα : |α| = k}. If T and S are bounded operators
on L2 (Rn ), then for their restrictions to Ekλ there is a natural inner product (T, S)k defined by Geller [5]. Under this inner product the operators
j
j
j
Wλ (Ppq
), Ppq
(z) = |z|p+q 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 H n for which holds the
estimate |f λ (z)| ≤ c pλa (z). Further assume that
(1.3)
|(fˆ(λ), Wλ (P ))k | ≤ c|λ|p+q+n−1 e−(2k+n)|λ|b
764
S. Thangavelu
j
for every k ∈ N and P (z) = |z|p+q Ypq
(z ). Then (i) when a = b, f =
∞
ϕ ∗3 pa with ϕ̂ ∈ L (R); (ii) when a > b there 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 LaplaceBeltrami 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 H n is just C n × R equipped with the group law
1
(z, t)(w, s) = z + w, t + s + Im(z · w) .
2
For each λ ∈ R, λ = 0 there is an irreducible unitary representation πλ (z, t)
of H n realised on L2 (Rn ). These representations are explicitly given by
1
πλ (z, t)ϕ(ξ) = eiλt eiλ(x·ξ+ 2 x·y) 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 (H n ) is defined to be the operator valued function
f (z, t)πλ (z, t)dz dt.
fˆ(λ) =
Hn
The representation πλ satisfies πλ (z, t) = eiλt πλ (z, 0) and therefore
(2.1)
fˆ(λ) = f λ (z)πλ (z)dz,
Cn
where we have written πλ (z) = πλ (z, 0) and
∞
f (z) =
f (z, t)eiλt dt
λ
−∞
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 C n . These are defined by
(2.2)
Gλ (g) = g(z)πλ (z)dz.
Cn
If f ∗ g is the convolution of two functions on H n defined by
f ((z, t)(w, s)−1 )g(w, s)dw ds
f ∗ g(z, t) =
Hn
then it is easily checked that
(f ∗ g)λ (z) = f λ ∗λ g λ (z)
where the λ-twisted convolution of f λ and g λ is given by
λ
λ
λ
f ∗λ g (z) = f λ (z − w)g λ (w)ei 2 Im(z·w) dw.
Cn
It then follows that Gλ (f λ ∗λ g λ ) = Gλ (f λ )Gλ (g λ ). Let Fλ f be the λsymplectic Fourier transform of a function f on C n given by
λ
(2.3)
Fλ f (z) = f (z − w)ei 2 Im(z·w) dw.
Cn
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S. Thangavelu
We define the Weyl correspondence of a function f on C n by Wλ (f ) =
Gλ (Fλ−1 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
P (z) =
cαβ z α z β
(2.4)
|α|=p |β|=q
where z ∈ C n , α, β ∈ Nn . Elements of Hpq are called bigraded solid
harmonics of degree (p, q). Let Spq be the space of all restrictions of elements
of Hpq to the unit sphere S 2n−1 . The elements of Spq are called bigraded
spherical harmonics. Then L2 (S 2n−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 Hermite function which is an eigenfunction of the Hermite operator H with
eigenvalue (2|α| + n) where |α| = α1 + . . . + αn . Define Φλα (x) =
n
1
|λ| 4 Φα (|λ| 2 x) for λ = 0 so that H(λ)Φλα = (2|α| + n)|λ|Φλα where
H(λ) = −∆ + λ2 |x|2 . We say that an operator T acting on L2 (Rn ) is
radial if it is diagonalised by Φλα and T Φλα = cλ|α| Φλα . Let Lδk , δ > −1 be the
Laguerre polynomials of type δ. We refer to Szego [20] for various properties
of Lδk . Define the Laguerre functions by
(z)
ϕn−1
k
=
Ln−1
k
1 2 − 1 |z|2
|z| e 4
2
(2.5)
for z ∈ C n . With these notations we are now in a position to state Geller’s
result.
Theorem 2.1. Suppose gP ∈ L1 (C n ) or L2 (C n ) where g is a radial function and P ∈ Hpq . Then for λ > 0, Gλ (gP ) = (−1)q Wλ (P )S where S is a
radial operator whose action on Φλα is given as follows : SΦλα = cλ|α| (g)Φλα
where cλk (g) = 0 for k < p and for k ≥ p it is given by
cλk (g) =
√
(k − p)!(n − 1)!
g(z)ϕn−1+p+q
( λz)dz.
k−p
(k + q + n − 1)!
C n+p+q
(2.6)
When λ < 0 the roles of p and q are reversed in the above definition of
cλk (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
Hpq spaces when restricted to each sphere rS 2n−1 are orthogonal and
L2 (rS 2n−1 ) is the orthogonal direct sum of these spaces. Let Ekλ be the
span of Φλα , |α| = k and let B(Ekλ ) be the space of bounded linear operators
from Ekλ into L2 (Rn ). On B(Ekλ ) we can define an inner product by setting
n−1 1
(T, S)k =
(T Φλα , SΦλα ).
(2.7)
|λ|
2
|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 ∈ Pp q and that p ≤ p or q ≤ q.
Then for λ > 0
(Wλ (Q), Wλ (P ))k
p+q+n−1
(k + q + n − 1)!
n −1 1
|λ|
(Q, P )
= (2π )
2
(k − p)!
(2.8)
where (Q, P ) is the inner product in L2 (S 2n−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
p+q
√
1
n−1+p+q
λ
(πλ (z), Wλ (P ))k = (−1)q
P (z)ϕk−p
( λz). (2.9)
2
We will make use of this formula in the proof of Theorem 1.2.
Given a continuous function f onC n we can expand fr (z ) = f (rz ), r >
0, z ∈ S 2n−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
S 2n−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 S 2n−1 defines a unitary
representation of U (n) on the Hilbert space L2 (S 2n−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∈K̂
K
where K = U (n) and K̂ 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 unitarily 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)
leaves invariant the subspaces Ekλ and commute with the projections Pk (λ)
associated to Ekλ . 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
L=−
n
j=1
(Xj2 + Yj2 ).
Revisiting Hardy’s theorem for the Heisenberg group
769
Here
∂
1 ∂
+ yj
∂xj
2 ∂t
∂
1 ∂
Yj =
− xj
∂yj
2 ∂t
j = 1, 2, . . . n are the left invariant vector fields on H n which alongwith
∂
T = ∂t
form an orthonormal basis for the Heisenberg Lie algebra. This
second order differential operator plays the role of Laplacian for H n , 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
Xj =
1
2
pλt (z) = cn λn (sinh(tλ))−n e− 4 λ(coth(tλ))|z| .
(2.14)
See Hulanicki [8] for a derivation of this formula. The group Fourier transform of pt is given by p̂t (λ) = e−tH(λ) . The kernel satisfies the pointwise
estimate
A
2
|pt (z, s)| ≤ c t−n−1 e− t |(z,s)| ,
(2.15)
1
4
where |(z, s)| = (|z|4 + s2 ) 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
pλa (z) gives us the estimate
1
2
|f λ (z)| ≤ cλ e− 4 (coth(aλ))|z| .
(3.1)
Recalling the definition of the inner product (T, S)k on B(Ekλ ) we have for
j
P = Ppq
n−1 1
(fˆ(λ), Wλ (P ))k =
(fˆ(λ)Φλα , Wλ (P )Φλα )
|λ|
2
|α|=k
which is given by the integral


n−1 1
|λ|
f λ (z) 
(πλ (z)Φλα , Wλ (P )Φλα ) dz.
2
|α|=k
Cn
(3.2)
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S. Thangavelu
Without loss of generality assume λ > 0 and use formula (2.9) to get
m−1 √
q 1
ˆ
f λ (z)P (z)ϕm−1
( λz)dz, (3.3)
(f (λ), Wλ (P ))k = (−1)
λ
k−p
2
Cn
where we have written m = n + p + q.
j
j
Note that P (z) = Ppq
(z) = |z|p+q Ypq
(z ). Defining
λ
j
fpqj
(|z|) =
f λ (z)Ypq
(z )dz (3.4)
S 2n−1
the expression (3.3) reads
√
1
λ
(fˆ(λ), Wλ (P ))k = (−1)q ( λ)m−1
fpqj
(|z|)|z|−p−q ϕm−1
( λz)dz
k−p
2
Cm
λ (|z|) as radial function onC m . Thus the hypothesis
where we are treating fpqj
on (fˆ(λ), Wλ (P ))k gives us the estimate
√
1 m−1 m−1
λ
( λ)
gpqj (z)ϕk ( λz)dz ≤ cλ e−(2k+m)λb
(3.5)
2
C m
λ (z) = |z|−p−q f λ (|z|).
for the Laguerre coefficients of the function gpqj
pqj
λ . Let us
We will use this to estimate the symplectic Fourier transform of gpqj
λ for the sake of simplicity of notation.
write g in place of gpqj
Consider the Laguerre expansion of g on C m given by
m ∞
√
k!
1
λ
ck (g)ϕm−1
(
λz)
g(z) =
k
2
(k + m − 1)!
k=0
where we have written ck (g) to stand for the integral on the left hand side of
(3.5). The Laguerre functions ϕm−1
satisfy the generating function identity
k
(see Szego [20])
∞
k=0
1 1+r
2
rk ϕm−1
(z) = (1 − r)−m e− 4 1−r |z| .
k
√
( λz) are eigenfunctions of the λ¿From this it is easily seen that ϕm−1
k
symplectic Fourier transform with eigenvalues (−1)k . Therefore, the Laguerre expansion of Fλ g is given by
Fλ g(z) = cλ
∞
k=0
√
k!
ck (g)(−1)k ϕm−1
( λz).
k
(k + m − 1)!
Revisiting Hardy’s theorem for the Heisenberg group
771
Applying Cauchy-Schwarz inequality and using the estimate (3.5) we get
|Fλ g(z)|2
(3.6)
∞
√
2
k!
(2k + m)2 e−2(2k+m)λb ϕm−1
( λz) .
≤ cλ
k
(k + m − 1)!
k=0
Defining
F (t) =
∞
k=0
√ 2
k!
(
λz)
e−(2k+m)t ϕm−1
k
(k + m − 1)!
we note that |Fλ g(z)|2 ≤ cλ |F (2λb)|.
The Laguerre functions satisfy another generating function identity,
namely
2
k!
2 − 12 s2
Lm−1
(s
)e
rk
k
(k + m − 1)!
k=0
2√ 1+r
2is r
−1
4 −( m−1
) − 12 1−r s2
2
,
= (1 − r) (−s r)
e
Jm−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
1
d −α
Jα (t)) = −t−α Jα+1 (t) and the estimate |Jα (it)| ≤ c t− 2 et , t → ∞
dt (t
satisfied by all Bessel functions Jα (t), we can easily get the estimate
1
|Fλ g(z)| ≤ cλ Q(z, z)e− 4 λ(tanh(bλ))|z|
2
(3.8)
where Q(z, z) is a polynomial in z and z.
Let g̃(z) stand for the Euclidean Fourier transform on C n which can
be expressed in terms of the λ-symplectic Fourier transform Fλ g(z). This
leads to the estimate
|g̃(z)| ≤ cλ Q1 (z, z)e−
tanh(bλ)
|z|2
λ
,
(3.9)
where Q1 is another polynomial. Since f λ (z) satisfies the estimate (3.1) it
λ (z)|z|−p−q also satisfies the estimate
follows that g(z) = fpqj
1
|g(z)| ≤ cλ e− 4 λ(coth(aλ))|z|
2
(3.10)
as |z| → ∞. Now we can appeal to Hardy’s theorem on C m . When a = b
it follows from (3.9) and (3.10) that
1
2
λ
(z) = cpqj (λ)e− 4 λ(coth(aλ))|z| .
gpqj
772
S. Thangavelu
λ (z) = f λ (z)|z|−p−q , the above is not compatible with the estiSince gpqj
pqj
mate
1
λ
|fpqj
(z)| ≤ cλ e− 4 λ(coth(aλ))|z|
2
unless cpqj (λ) = 0 for all (p, q) = (0, 0). This simply means that
f λ (z) = c0 (λ)pλa (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 H n+p+q . This is a radial function on z and therefore we can define a
function f on H n by
j
(z )|z|p+q p
f (z, s) = Ypq
(p,q)
a
1+
(z, s).
(3.11)
Then it is clear that |f λ (z)| ≤ cλ pλa (z). In view of Theorem 2.1 we know
j
that fˆ(λ) = Gλ (f λ ) = (−1)q Wλ (P )S where P (z) = Ypq
(z )|z|p+q and
S is radial. Since
∞
1
2
(p,q)
eiλs pt (z, s)ds = cn+p+q λn+p+q (sinh λt)−(n+p+q) e− 4 λ coth(tλ)|z|
−∞
which equals a constant times
|λ|n+p+q
∞
k=0
e−(2k+n+p+q)|λ|t ϕkn−1+p+q ( |λ|z)
we know that
fˆ(λ)Pk (λ) = cn
a
(k − p)!
e−(2k+n+p+q)|λ|( 1+ ) Wλ (P ).
(k + q + n − 1)!
(3.12)
Using the result of Theorem 2.2 we see that (fˆ(λ), Wλ (Q))k = 0 if (Q, P ) =
0 and when P = Q
a
(fˆ(λ), Wλ (P ))k = cn |λ|n+p+q−1 e−(2k+n)|λ|( 1+ )
which gives the estimate
|(fˆ(λ), Wλ (P ))k | ≤ c |λ|n+p+q−1 e−(2k+n)|λ|b
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
f (σz, s)χpq (σ)dσ
fpq (z, s) =
U (n)
we calculate its Fourier transform to be



f (σz, s)πλ (z, s)dz ds χpq (σ)dσ.
fˆpq (λ) =
U (n)
Hn
Since πλ (σ ∗ z, s) = µλ (σ)πλ (z, s)µλ (σ)∗ we have
ˆ
fpq (λ) =
µλ (σ)fˆ(λ)µλ (σ)∗ χpq (σ)dσ.
U (n)
The condition fˆ(λ)∗ fˆ(λ) ≤ c p̂2a (λ) means that fˆ(λ)eaH(λ) is bounded
with norm independent of λ. Here eaH(λ) is an unbounded operator defined
on finite linear combinations of Φλα . As p̂a (λ) = e−aH(λ) commutes with
µλ (σ) we infer that fˆpq (λ)eaH(λ) is bounded which leads to the estimate
fˆpq (λ)Φλα 2 ≤ c e−(2|α|+n)|λ|a .
(3.13)
We want to use these estimates to get upper bounds for the Laguerre coefλ.
ficients of fpq
λ is given by
The spherical harmonic expansion of fpq
λ
(z)
fpq
d(p,q)
=
λ
j
fpqj
(|z|)Ypq
(z )
j=1
λ (|z|) are defined in (3.4). The hypothesis on f λ (z) leads us to the
where fpqj
estimate
λ
(|z|)| ≤ cpqj pλa (z).
|fpqj
(3.14)
j
j
(z) = |z|p+q Ypq
(z ) and using Theorem 2.1 we have the forDefining Ppq
mula
fˆpq (λ) = (−1)q
d(p,q)
j
j
Wλ (Ppq
)Spq
(λ)
j=1
j
(λ) are given by
where the radial operators Spq
j
Spq
(λ)
= cn
∞
k=p
(k − p)!
λ
ck−p (gpqj
)Pk (λ).
(k + q + n − 1)!
774
S. Thangavelu
λ ) are the Laguerre coefficients of g λ (z) = f λ (|z|)|z|−p−q .
Here ck (gpqj
pqj
pqj
More precisely,
λ
λ
ck (gpqj ) =
gpqj
(z)ϕn−1+p+q
(
|λ|z)dz.
(3.15)
k
C n+p+q
j
By Theorem 2.2 the restrictions of Wλ (Ppq
) to Ekλ are mutually orthogonal and hence we get
j
λ
fˆpq (λ), Wλ (Ppq
) = cn |λ|p+q+n−1 ck−p (gpqj
)
k
which leads us to the expression
λ
j
fˆpq (λ)Φλα , Wλ (Ppq
) = cn |λ|−p−q
)Φλα .
ck (gpqj
(3.16)
|λ|=k+p
Applying Cauchy-Schwarz inequality and using the estimate (3.13) we get
λ
j
)| ≤ c|λ|−p−q e−(2k+n)|λ|a
Wλ (Ppq
)Φα )2 . (3.17)
|ck (gpqj
|λ|=k+p
Another application of Cauchy-Schwarz gives
2

j

Wλ (Ppq
)Φλα 2 
|α|=k+p
≤c
(k + p + n − 1)! −n+1 j
j
|λ|
Wλ (Ppq
), Wλ (Ppq
) k+p . (3.18)
(k + p)!
Using (2.8) we get the estimate
1
(k + p + q + n − 1)!
j
. (3.19)
Wλ (Ppq
)Φλα 2 ≤ c|λ| 2 (p+q)
k!
|α|=k+p
¿From (3.19) and (3.17) we get the estimate
λ
)| ≤ cλ (2k + m)m−1 e−(2k+m)|λ|a ,
|ck (gpqj
(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
fpqj
(|z|) = cpqj (λ)|z|p+q pλa (z).
(3.21)
As before this is not compatible with (3.14) unless cpqj (λ) = 0 for all
(p, q) = (0, 0) and we get f λ (z) = c0 (λ)pλa (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
1
|f λ (z)| ≤ c(1 + |z|2 )m e− 4a |z|
2
(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. Given a < b we choose δ > 0
so that a(ebδ + e−bδ ) < 2b. The theorem will be proved by showing that
f λ (z) = 0 for 0 < λ < δ which will force f λ (z) = 0 for all λ and hence
f (z, t) = 0.
Proceeding as in the previous proofs, the condition fˆ(λ)∗ fˆ(λ) ≤
c H(λ)m p̂2b (λ) leads us to the estimate
|g̃(z)| ≤ cλ (1 + |z|2 )m e−
tanh(bλ)
|z|2
λ
,
(3.23)
λ (z) and g̃ is the Euclidean Fourier transform of
where g(z) = |z|−p−q fpqj
g. Our choice of δ shows that for 0 < λ < δ
2bλ
ebλ − e−bλ
<
= tanh(bλ).
ebδ + e−bδ
ebλ + e−bλ
In view of estimates (3.22) and (3.23) the classical Hardy’s theorem gives
g = 0 since tanh(bλ) > aλ. As this is true for every p, q and j we get
f λ (z) = 0 for 0 < λ < δ, z ∈ C n . This completes the proof of Theorem
1.1.
0 < aλ <
4 A Hardy’s theorem for spectral projections
In this section we prove a version of Hardy’s theorem for the spectral
projecn−1
n
λ
tions associated to the sublaplacian on H . Let ϕk (z) stand for ϕk ( |λ|z)
and define
eλk (z, t) = eiλt ϕλk (z).
(4.1)
These eλk 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
∞ ∞
f (z, t) =
f ∗ eλk (z, t)dµ(λ)
(4.2)
k=0−∞
for f ∈ L2 (H n ). Here dµ(λ) = (2π)−n−1 |λ|n dλ is the Plancherel measure
for the Heisenberg group. Since Leλk = (2k + n)|λ|eλk , f ∗ eλk 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 (H n ) satisfy the condition
j
λ
f ∗ ek (z, t)Y pq (z )dz ≤ c |z|p+q e−(2k+n+p+q)|λ|b
2n−1
(4.3)
S
j
for all Ypq
with c independent of (z, t) and λ.
(i) If |f (z, t)| ≤ c pa (z, t) and a < b then f = 0
(ii) If |f λ (z)| ≤ c pλa (z) and a ≥ b then conclusions of Theorem 1.3 hold.
We remark that we can consider Theorem 4.1 as a Hardy’s theorem for
the Fourier transform on the Heisenberg motion group Gn . As shown in [21]
the spectral projections f ∗ eλk (z, t) are nothing but ρλk (f )eλk (z, t) for certain
irreducible unitary representations ρλk of Gn . However, we will not pursue
this line of thought in this paper.
The functions eλk are nothing but spherical functions for the Gelfand pair
(U (n), H n ); (see [21] for more about Gelfand pairs and associated spherical
functions). In order to prove Theorem 4.1 we need the following result which
is an addition formula for the spherical functions eλk . (Compare this result
with formula (3.25) in Koornwinder [10] which is an addition formula for
Jacobi functions). As eλk (z, t) = eiλt ϕλk (z) we state the formula in terms of
ϕλk .
Proposition 4.1. For each λ > 0 and k ∈ N


d(p,q)
∞ k
λ
(k − p)!(n − 1)!
j
j

ϕλk (z − w)ei 2 Im(z·w)=
Ypq
(z )Ypq
(w )
(k + q + n − 1)!
q=0 p=0
j=1
√
√
p+q n−1+p+q
|z| ϕk−p
( λz)|w|p+q ϕn−1+p+q
( λw).
k−p
There is a similar formula for λ < 0 as well.
Proof: We prove this formula by appealing to Theorem 2.1. First observe
that when f = gP where g is radial and P ∈ Hpq , f ∗λ ϕλk (z) has a simple
form. Indeed, a simple calculation shows that (see [11])
√ (gP ) ∗λ ϕλk (z) = cλk (g)P (z)ϕn−1+p+q
λz
(4.4)
k−p
where cλk (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
functions of the form f = gP, g radial and P ∈ Hpq are dense in L2 (C n )
it is enough to show that
λ
f ∗λ ϕk (z) = Fk (z, w)f (w)dw
(4.5)
Cn
Revisiting Hardy’s theorem for the Heisenberg group
777
j
whenever f (z) = g(z)|z|p+q 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
shows that f ∗ eλk (z, t) = eiλt f λ ∗λ ϕλk (z) for any f ∈ L1 (H n ). Consider
λ
λ
λ
f ∗λ ϕk (z) = f λ (w)ϕλk (z − w)e−i 2 Im(z·w) dw.
Cn
In view of the addition formula stated in the proposition the above equals


(k − p)!(n − 1)!  √

n−1+p+q
λ
fpqj
(|w|)|w|p+q ϕk−p
( λw)dw

(k + q + n − 1)!
p,q,j
Cn
√ j
Ypq
(z )|z|p+q ϕn−1+p+q
λz
(4.6)
k−p
λ (|w|) are the functions defined in 3.4. Therefore, it follows that
where fpqj
j
f λ ∗λ ϕλk (z)Ypq
(4.7)
(z )dz S 2n−1
=

(k − p)!(n − 1)! 

(k + q + n − 1)!

√
 p+q m−1 √
λ
gpqj
(w)ϕm−1
ϕk−p ( λz)
k−p ( λw)dw  |z|
Cm
λ (w) = |w|−p−q f λ (w) and m = n + p + q.
where gpqj
pqj
Thus the hypothesis of the theorem gives us the estimate
(k − p)!(m − 1)! √
m−1 √ λ
gpqj
(w)ϕm−1
(
λw)dw
(
λz)
ϕ
k−p
(k − p + m − 1)! k−p
Cm
≤ c e−(2k+m)|λ|b .
Since this estimate is true for all z, we can take limit as z → 0. Noting that
ϕm−1
k−p (0) =
(k − p + m − 1)!
(k − p)!(m − 1)!
we get the estimate
√
m−1
λ
gpqj (w)ϕk ( λw)dw ≤ c e−(2k+m)|λ|b .
Cm
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
f ∗ ek (z , t)Ypq (z )dt ≤ c |λ|p+q e−(2k+n+p+q)|λ|b
(4.8)
2n−1
S
then proceeding as above we will get
k!(m − 1)! √
− 1 |λ|
m−1
m−1 1
λ
e 4
L
gpqj (w)ϕk ( λw)dw
|λ|
(k + m − 1)! k
2
Cm
≤ cλ e−(2k+m)|λ|b .
For λ lying in a compact set of the form 0 < λ1 ≤ |λ| ≤ λ2 < ∞ we know
that (see Szego [20])
m−1 1 1
m−1 1
|λ| e− 4 |λ| = O k 2 − 4
Lk
2
and therefore, we get an estimate of the form
√
m−1
λ
gpqj
(w)ϕm−1
(
λw)dw
≤ cλ (2k + m) 2 e−(2k+m)|λ|b . (4.9)
k
Cn
Thus conclusion (i) of Theorem 4.1 is true under the weaker hypothesis
(4.8). For part (ii) we can only get f λ (z) = c0 (λ)pλa (z) with c0 (λ) locally
bounded on IR\{0}.
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