An analogue of Ingham's theorem on the Heisenberg group

Abstract

We prove an exact analogue of Ingham's uncertainty principle for the group Fourier transform on the Heisenberg group. This is accomplished by explicitly constructing compactly supported functions on the Heisenberg group whose operator-valued Fourier transforms have suitable Ingham type decay and proving an analogue of Chernoff's theorem for the family of special Hermite operators.

Full text

arXiv:2204.10017v1 [math.CA] 21 Apr 2022
AN ANALOGUE OF INGHAM’S THEOREM
ON THE HEISENBERG GROUP
SAYAN BAGCHI, PRITAM GANGULY, JAYANTA SARKAR
AND SUNDARAM THANGAVELU
Abstract. We prove an exact analogue of Ingham’s uncertainty principle for the group
Fourier transform on the Heisenberg group. This is accomplished by explicitly constructing
compactly supported functions on the Heisenberg group whose operator valued Fourier
transforms have suitable Ingham type decay and proving an analogue of Chernoff’s theorem
for the family of special Hermite operators.
1. Introduction
Roughly speaking, the uncertainty principle for the Fourier transform on Rnsays that a
function fand its Fourier transform b
fcannot both have rapid decay. Several manifestations
of this principle are known: Heisenberg-Pauli-Weyl inequality, Paley-Wiener theorem and
Hardy’s uncertainty principle are some of the most well known. But there are lesser known
results such as theorems of Ingham and Levinson. The best decay a non trivial function can
have is vanishing identically outside a compact set and for such functions it is well known
that their Fourier transforms extend to Cnas entire functions and hence cannot vanish on
any open set. For any such function of compact support, its Fourier transform cannot have
any exponential decay for a similar reason: if |b
f(ξ)| ≤ Ce−a|ξ|for some a > 0, then it follows
that fextends to a tube domain in Cnas a holomorphic function and hence it cannot have
compact support. So it is natural to ask the question: what is the best possible decay, on
the Fourier transform side, that is allowed of a function of compact support? An interesting
answer to this question is provided by the following theorem of Ingham [12].
Theorem 1.1 (Ingham).Let Θ(y)be a nonnegative even function on Rsuch that Θ(y)
decreases to zero when y→ ∞.There exists a nonzero continuous function fon R,equal
to zero outside an interval (−a, a)whose Fourier transform b
fsatisfies the estimate |b
f(y)| ≤
Ce−|y|Θ(y)if and only if R∞
1Θ(t)t−1dt < ∞.
This theorem of Ingham and its close relatives Paley -Wiener ([22, 23]) and Levinson ([16])
theorems have received considerable attention in recent years. In [2], Bhowmik et al proved
2010 Mathematics Subject Classification. Primary: 43A80. Secondary: 22E25, 33C45, 26E10, 46E35.
Key words and phrases. Heisenberg group, special Hermite operators, quasi-analyticity, bi-graded spher-
ical harmonics, Chernoff’s theorem, Ingham’s theorem.
1

2 BAGCHI, GANGULY, SARKAR AND THANGAVELU
analogues of the above theorem for Rn,the n-dimensional torus Tnand step two nilpotent
Lie groups. See also the recent work of Bowmik-Pusti-Ray [3] for a version of Ingham’s
theorem for the Fourier transform on Riemannian symmetric spaces of non-compact type.
As we are interested in Ingham’s theorem on the Heisenberg group, let us recall the result
proved in [2]. Let Hn=Cn×Rbe the Heisenberg group. For an integrable function f
on Hn, let b
f(λ) be the operator valued Fourier transform of findexed by non-zero reals λ.
Measuring the decay of the Fourier transform in terms of the Hilbert-Schmidt operator norm
kb
f(λ)kHS Bhowmik et al. have proved the following result.
Theorem 1.2 (Bhowmik-Ray-Sen).Let Θ(λ)be a nonnegative even function on Rsuch that
Θ(λ)decreases to zero when λ→ ∞.There exists a nonzero, compactly supported continuous
function fon Hn,whose Fourier transform satisfies the estimate kb
f(λ)kHS ≤C|λ|n/2e−|λ|Θ(λ)
if the integral R∞
1Θ(t)t−1dt < ∞.On the other hand, if the above estimate is valid for a
function fand the integral R∞
1Θ(t)t−1dt diverges, then the vanishing of fon any set of the
form {z∈Cn:|z|< δ} × Rforces fto be identically zero.
As the Fourier transform on the Heisenberg group is operator valued, it is natural to
measure the decay of b
f(λ) by comparing it with the Hermite semigroup e−aH(λ)generated by
H(λ) = −∆Rn+λ2|x|2.In this connection, let us recall the following two versions of Hardy’s
uncertainty principle. Let pa(z, t) stand for the heat kernel associated to the sublaplacian
Lon the Heisenberg group whose Fourier transform turns out to be the Hermite semigroup
e−aH(λ).The version in which one measures the decay of b
f(λ) in terms of its Hilbert-Schmidt
operator norm reads as follows. If
|f(z, t)| ≤ Ce−a(|z|2+t2),kb
f(λ)kHS ≤Ce−bλ2(1.1)
then f= 0 whenever ab > 1/4.This is essentially a theorem in the t-variable and can be
easily deduced from Hardy’s theorem on R, see Theorem 2.9.1 in [33]. Compare this with
the following version [33, Theorem 2.9.2]. If
|f(z, t)| ≤ Cpa(z, t),b
f(λ)∗b
f(λ)≤Ce−2bH(λ)(1.2)
then f= 0 whenever a < b. This latter version is the exact analogue of Hardy’s theorem for
the Heisenberg group, which we can view not merely as an uncertainty principle but also as a
characterization of the heat kernel. Hardy’s theorem in the context of semi-simple Lie groups
and non-compact Riemannian symmetric spaces are also to be viewed in this perspective.
We remark that the Hermite semigroup has been used to measure the decay of the Fourier
transform in connection with the heat kernel transform [14], Pfannschmidt’s theorem [35]
and the extension problem for the sublaplacian [25] on the Heisenberg group. In connection
with the study of Poisson integrals, it has been noted in [34] that when the Fourier transform
of fsatisfies an estimate of the form b
f(λ)∗b
f(λ)≤Ce−a√H(λ),then the function extends to a

INGHAM’S THEOREM ON THE HEISENBERG GROUP 3
tube domain in the complexification of Hnas a holomorphic function and hence the vanishing
of fon an open set forces it to vanish identically. It is therefore natural to ask if the same
conclusion can be arrived at by replacing the constant ain the above estimate by an operator
Θ(pH(λ)) for a function Θ decreasing to zero at infinity. Our investigations have led us to
the following exact analogue of Ingham’s theorem for the Fourier transform on Hn.
Theorem 1.3. Let Θ(λ)be a nonnegative function on [0,∞)which decreases to zero as
λ→ ∞.Then there exists a nonzero compactly supported continuous function fon Hn
whose Fourier transform b
fsatisfies the estimate
b
f(λ)∗b
f(λ)≤Ce−2Θ(√H(λ))√H(λ), λ 6= 0,(1.3)
if and only if Θsatisfies the condition R∞
1Θ(t)t−1dt < ∞.
Under the assumption that R∞
1Θ(t)t−1dt =∞, the above theorem demonstrates that any
compactly supported function whose Fourier transform satisfies (1.3) vanishes identically.
This can be viewed as an uncertainty principle in the sense mentioned in the first paragraph.
Recently this aspect of Ingham’s theorem has been proved in the context of higher dimen-
sional Euclidean spaces and Riemannian symmetric spaces with a much weaker hypothesis
on the function. As observed in [11], for the Heisenberg group case, the hypothesis can
be weakened considerably if we slightly strengthen the condition (1.3). More precisely, the
second and the last author proved the following theorem in this context.
Theorem 1.4. [11] Let Θ(λ)be a nonnegative function on [0,∞)such that it decreases
to zero as λ→ ∞, and satisfies the conditions R∞
1Θ(t)t−1dt =∞.Let fbe an integrable
function on Hnwhose Fourier transform satisfies the estimate
ˆ
f(λ)∗ˆ
f(λ)≤C e−2|λ|Θ(|λ|)e−2√H(λ) Θ(√H(λ)).(1.4)
Then fcannot vanish on any nonempty open set unless it is identically zero.
Comparing the decay condition (1.3) and (1.4), it is not difficult to see that the Theorem
1.3 is a significant improvement of the Theorem 1.4 in terms of the Ingham type decay
condition. However, we believe that the necessary part of the Theorem 1.3 is true under the
weaker hypothesis on the function as in the Theorem 1.4. In what follows, we shed more
light on the difficulties in this regard.
The sufficiency part of Theorem 1.3 is proved in Section 4.1 by explicitly constructing
compactly supported functions whose Fourier transforms satisfy the stated decay condition.
Though at present we are not able to prove the necessary part of the theorem under the
assumption that fvanishes on an open set, a slightly different version can be proved. Recall

4 BAGCHI, GANGULY, SARKAR AND THANGAVELU
that the Fourier transform b
fis defined by integrating fagainst the Schr¨odinger representa-
tions πλ:
b
f(λ) = ZHn
f(z, t)πλ(z, t)dz dt.
Since πλ(z, t) = eiλt πλ(z, 0), it follows that b
f(λ) = πλ(fλ),where fλ(z) is the inverse Fourier
transform of f(z, t) in the central variable and
Wλ(fλ) = ZCn
fλ(z)πλ(z, 0)dz
is the Weyl transform of fλ. With these notations we prove the following improvement on
the necessary part of Theorem 1.3.
Theorem 1.5. Let Θ(λ)be a nonnegative function on [0,∞)such that it decreases to zero
when λ→ ∞, and satisfies the condition R∞
1Θ(t)t−1dt =∞.Let fbe an integrable function
on Hnwhose Fourier transform b
fsatisfies the estimate
b
f(λ)∗b
f(λ)≤Ce−2Θ(√H(λ))√H(λ), λ 6= 0.(1.5)
If for every λ6= 0,there exists an open set Uλ⊂Cnon which fλvanishes, then f= 0.
Remark 1.1.Note that when fis compactly supported the function fλis also compactly
supported and hence vanishes on an open set. The same is true if we assume that fis
supported on a cylindrical set {z∈Cn:|z|< a} × R.As b
f(λ) = πλ(fλ), the above can be
considered as a result for the Weyl transform of functions on Cn.
Theorem 1.1 was proved in [12] by Ingham by making use of Denjoy-Carleman theorem
on quasi-analytic functions. In [2], the authors have used Radon transform and a several
variable extension of Denjoy-Carleman theorem due to Bochner and Taylor [5] in order to
prove the n-dimensional version of Theorem 1.1. An L2variant of the result of Bochner-
Taylor which was proved by Chernoff in [8] has turned out to be very useful in establishing
Ingham type theorems.
Theorem 1.6. [8, Chernoff] Let fbe a smooth function on Rn.Assume that ∆mf∈L2(Rn)
for all m∈Nand that P∞
m=1 k∆m
Rnfk−1
2m
2=∞.If fand all its partial derivatives vanish at
0, then fis identically zero.
As the Laplacian is translation invariant, 0 can be replaced by any other point in the
above theorem. As a matter of fact, this theorem shows how partial differential operators
generate the class of quasi-analytic functions. Recently, Bhowmik-Pusti-Ray [3] have estab-
lished an analogue of Chernoff’s theorem for the Laplace-Beltrami operators on non-compact
Riemannian symmetric spaces and use the same in proving a version of Ingham’s theorem
for the Helgason Fourier transform.

INGHAM’S THEOREM ON THE HEISENBERG GROUP 5
In the context of the Heisenberg group, we prove Theorem 1.5, and hence Theorem 1.3,
by using the following analogue of Chernoff’s theorem for the family of special Hermite
operators Lλ.These operators on Cnare defined via the relation L(f(z)eiλt ) = eiλtLλf(z)
where Lis the sublaplacian on Hn.Observe that when λ= 0,the special Hermite operator
Lλreduces to ∆ on Cn.
Theorem 1.7. For any fixed λ∈R,let f∈C∞(Cn)be such that Lm
λf∈L2(Cn)for all
m≥0and that P∞
m=1 kLm
λfk−1
2m
2=∞.If fand all its partial derivatives vanish at some
w∈Cn, then fis identically zero.
When λ= 0, the above is just Chernoff’s theorem for the Laplacian on Cn. For λ= 1, a
weaker version of the theorem, namely under the assumption that fvanishes on an open set,
has been proved in [10, Theorem 4.1]. The weaker version is in fact good enough to prove
Theorems 1.5 and 1.3. However, in this paper, we prove the above improvement which is
the exact analogue of Theorem 1.6 for the special Hermite operators and the second main
result of this article.
We conclude the introduction by briefly describing the organization of the paper. After
recalling the required preliminaries regarding harmonic analysis on Heisenberg group in
Section 2, we prove an analogue of Chernoff’s theorem for the special Hermite operators
(Theorem 1.7) in Section 3. In section 4, we prove the Ingham’s theorems on the Heisenberg
group, namely Theorems 1.3, and 1.5.
2. Preliminaries on Heisenberg groups
In this section, we collect the results which are necessary for the study of uncertainty
principles for the Fourier transform on the Heisenberg group. We refer the reader to the
two classical books Folland [9] and Taylor [30] for the preliminaries of harmonic analysis on
the Heisenberg group. However, we will be closely following the notations of the books of
Thangavelu [32] and [33].
2.1. Heisenberg group and Fourier transform. Let Hn:= Cn×Rdenote the (2n+ 1)-
dimensional Heisenberg group equipped with the group law
(z, t).(w, s) := z+w, t +s+1
2ℑ(z. ¯w),∀(z, t),(w, s)∈Hn.
This is a step two nilpotent Lie group where the Lebesgue measure dzdt on Cn×Rserves as
the Haar measure. The representation theory of Hnis well-studied in the literature. In order
to define Fourier transform, we use the Schr¨odinger representations as described below.

6 BAGCHI, GANGULY, SARKAR AND THANGAVELU
For each non-zero real number λ, we have an infinite dimensional representation πλrealised
on the Hilbert space L2(Rn).These are explicitly given by
πλ(z, t)ϕ(ξ) = eiλt eiλ(x·ξ+1
2x·y)ϕ(ξ+y),
where z=x+iy and ϕ∈L2(Rn).These representations are known to be unitary and
irreducible. Moreover, by a theorem of Stone and Von-Neumann (see e.g., [9]), these account,
upto unitary equivalence, for all the infinite dimensional irreducible unitary representations
of Hnwhich act as eiλtI, λ 6= 0, on the center. Also, there is another class of one dimensional
irreducible representations that corresponds to the case λ= 0.As they do not contribute to
the Plancherel measure we will not describe them here.
The Fourier transform of a function f∈L1(Hn) is the operator valued function obtained
by integrating fagainst πλ:
ˆ
f(λ) = ZHn
f(z, t)πλ(z, t)dzdt.
Note that ˆ
f(λ) is a bounded linear operator on L2(Rn).Now, by definition of πλand ˆ
f(λ),
it is easy to see that
b
f(λ) = ZCn
fλ(z)πλ(z, 0)dz,
where fλstands for the inverse Fourier transform of fin the central variable:
fλ(z) := Z∞
−∞
eiλtf(z, t)dt.
This motivates the following definition. Given a function gon Cn, we consider the following
operator defined by
Wλ(g) := ZCn
g(z)πλ(z, 0)dz.
With these notations, we note that ˆ
f(λ) = Wλ(fλ).These transforms are called the Weyl
transforms and for λ= 1, they are simply denoted by W(g) instead of W1(g).We have the
following Plancherel formula for the Weyl transforms (See [33, 2.2.9, Page no-49])
kWλ(g)k2
HS |λ|n= (2π)nkgk2
2, g ∈L2(Cn).(2.1)
This, in view of the relation between the group Fourier transform and the Weyl transform,
proves that when f∈L1∩L2(Hn), its Fourier transform is actually a Hilbert-Schmidt
operator and one has
ZHn|f(z, t)|2dzdt = (2π)−(n+1) Z∞
−∞ kb
f(λ)k2
HS |λ|ndλ,
where k.kHS denotes the Hilbert-Schmidt norm. The above allows us to extend the Fourier
transform as a unitary operator between L2(Hn) and the Hilbert space of Hilbert-Schmidt

INGHAM’S THEOREM ON THE HEISENBERG GROUP 7
operator valued functions on Rwhich are square integrable with respect to the Plancherel
measure dµ(λ) = (2π)−n−1|λ|ndλ. We polarize the above identity to obtain
ZHn
f(z, t)g(z, t)dzdt =Z∞
−∞
tr(b
f(λ)bg(λ)∗)dµ(λ).
Also for suitable functions fon Hnwe have the inversion formula
f(z, t) = Z∞
−∞
tr(πλ(z, t)∗b
f(λ))dµ(λ).
Moreover, the Fourier transform behaves well with the convolution of two functions defined
by
f∗g(x) := ZHn
f(xy−1)g(y)dy.
In fact, for any f, g ∈L1(Hn), it follows from the definition that
[
f∗g(λ) = ˆ
f(λ)ˆg(λ).
In the following subsection, we describe the role of special functions in the harmonic analysis
on Hnand show that the group Fourier transform of a suitable class of functions take a nice
form.
2.2. Special functions and Fourier transform. For each λ6= 0, we consider the following
family of scaled Hermite functions indexed by α∈Nn:
Φλ
α(x) := |λ|n
4Φα(p|λ|x), x ∈Rn,
where Φαdenote the n−dimensional Hermite functions (see [31]). It is well-known that these
scaled functions Φλ
αare eigenfunctions of the scaled Hermite operator H(λ) := −∆Rn+λ2|x|2
with eigenvalue (2|α|+n)|λ|and {Φλ
α:α∈Nn}forms an orthonormal basis for L2(Rn). As
a consequence,
kb
f(λ)k2
HS =X
α∈Nnkb
f(λ)Φλ
αk2
2.
In view of this, the Plancheral formula takes the following very useful form
ZHn|f(z, t)|2dzdt =Z∞
−∞ X
α∈Nnkb
f(λ)Φλ
αk2
2dµ(λ).
Given σ∈U(n), we define Rσf(z, t) = f(σ.z, t). We say that a function fon Hnis
radial if it is invariant under the action of U(n) i.e., Rσf=ffor all σ∈U(n).The Fourier
transforms of such radial integrable functions are functions of the Hermite operator H(λ).
In fact, if H(λ) = P∞
k=0(2k+n)|λ|Pk(λ) is the spectral decomposition of this operator, then
for a radial intrgrable function fwe have
b
f(λ) = ∞
X
k=0
Rk(λ, f )Pk(λ).

8 BAGCHI, GANGULY, SARKAR AND THANGAVELU
Here, Pk(λ) stands for the orthogonal projection of L2(Rn) onto the kth eigenspace spanned
by scaled Hermite functions Φλ
αwith |α|=k. The coefficients Rk(λ, f ) are given by
Rk(λ, f ) = k!(n−1)!
(k+n−1)! ZCn
fλ(z)ϕn−1
k,λ (z)dz. (2.2)
In the above formula, ϕn−1
k,λ are the Laguerre functions of type (n−1):
ϕn−1
k,λ (z) = Ln−1
k(1
2|λ||z|2)e−1
4|λ||z|2,
where Ln−1
kdenotes the Laguerre polynomial of type (n−1). For the purpose of estimating
the Fourier transform we need good estimates for the Laguerre functions ϕn−1
k,λ .In order to
get such estimates, we use the available sharp estiamtes of standard Laguerre functions as
described below in more general context.
For any δ > −1, let Lδ
k(r) denote the Laguerre polynomials of type δ. The standard
Laguerre functions are defined by
Lδ
k(r) = Γ(k+ 1)Γ(δ+ 1)
Γ(k+δ+ 1) 1
2Lδ
k(r)e−1
2rrδ/2
which form an orthonormal system in L2((0,∞), dr). In terms of Lδ
k(r),we have
ϕδ
k(r) = 2δΓ(k+ 1)Γ(δ+ 1)
Γ(k+δ+ 1) −1
2r−δLδ
k1
2r2.
Asymptotic properties of Lδ
k(r) are well-known in the literature, see [31, Lemma 1.5.3]. The
estimates in [31, Lemma 1.5.3] are sharp, see [17, Section 2] and [18, Section 7]. For our
convenience, we restate the result in terms of ϕn−1
k,λ (r).
Lemma 2.1. Let ν(k) = 2(2k+n)and Ck,n =k!(n−1)!
(k+n−1)! 1
2.For λ6= 0,we have the estimates
Ck,n |ϕn−1
k,λ (r)| ≤ C(rp|λ|)−(n−1)



















(1
2ν(k)r2|λ|)(n−1)/2,0≤r≤√2
√ν(k)|λ|
(1
2ν(k)r2|λ|)−1
4,√2
√ν(k)|λ|≤r≤√ν(k)
√|λ|
ν(k)−1
4(ν(k)1
3+|ν(k)−1
2|λ|r2|)−1
4,√ν(k)
√|λ|≤r≤√3ν(k)
√|λ|
e−1
2γr2|λ|, r ≥√3ν(k)
√|λ|,
where γ > 0is a fixed constant and Cis independent of kand λ.
2.3. The sublaplacian and special Hermite operators. We let hnstand for the Heisen-
berg Lie algebra consisting of left invariant vector fields on Hn.A basis for hnis provided by
the 2n+ 1 vector fields
Xj=∂
∂xj
+1
2yj
∂
∂t , Yj=∂
∂yj−1
2xj
∂
∂t , j = 1,2, ..., n, and T=∂
∂t .

INGHAM’S THEOREM ON THE HEISENBERG GROUP 9
These correspond to certain one parameter subgroups of Hn.The sublaplacian on Hnis
defined by
L:= −∞
X
j=1
(X2
j+Y2
j)
which can be explicitly calculated as
L=−∆Cn−1
4|z|2∂2
∂t2+N∂
∂t ,
where ∆Cnstands for the Laplacian on Cnand Nis the rotation operator defined by
N=
n
X
j=1 xj
∂
∂yj−yj
∂
∂xj.
This is a sub-elliptic operator and homogeneous of degree 2 with respect to the non-isotropic
dilation given by δr(z, t) = (rz, r2t).The sublaplacian is also invariant under rotation i.e.,
Rσ◦ L =L ◦ Rσ, σ ∈U(n).For each λ6= 0, special Hermite operator Lλis defined via the
relation
L(eiλtf(z)) = eiλt Lλf(z).
Furthermore, it is not hard to see that (Lf)λ(z) = Lλfλ(z).It turns out that Lλis explicitly
given by
Lλ=−∆Cn+1
4λ2|z|2+iλN.
This family of special Hermite operators has a useful translation invariance property coming
from the sublaplacian.
Recall that the sublaplacian Lis invariant under the left translations defined by τyf(x) :=
f(y−1x), x, y ∈Hn.In other words, τy(Lf) = L(τyf).Now, with x= (w, 0) ∈Hn,taking
inverse Fourier transform in the central variable gives us
(τx(Lf))λ(z) = Lλ(τxf)λ(z)
which, after simplification leads to
eiλ
2ℑ(w.¯z)Lλfλ(z−w) = Lλ(eiλ
2ℑ(w.¯z)fλ(z−w)).
This observation in turn implies that the special Hermite operator Lλis invariant under the
λ-twisted translation Tλ
w, w ∈Cn, defined by
Tλ
wg(z) := eiλ
2ℑ(w.¯z)g(z−w).(2.3)
In other words,
Tλ
w(Lλg) = Lλ(Tλ
wg), w ∈Cn.(2.4)
It is also known that these Lλ’s are elliptic operators on Cnwith an explicit spectral decom-
position. The spectrum consists of the real numbers of the form (2k+n)|λ|,k≥0, and the
eigenspaces associated to each of these eigenvalues are infinite dimensional.

10 BAGCHI, GANGULY, SARKAR AND THANGAVELU
In the following, we describe the spectral decomposition for the case when λ= 1. For the
sake of simplicity, we write Linstead of L1and f×ginstead of f∗1g. Thus,
f×g(z) = ZCn
f(z−w)g(w)ei
2ℑ(z·¯w)dw.
It is known that ([33, page no. 58]) the special Hermite expansion of a function f∈L2(Cn)
and Parseval’s identity reads as
f(z) = (2π)−n∞
X
k=0
f×ϕn−1
k(z),kfk2
2= (2π)−n∞
X
k=0 kf×ϕn−1
kk2
2(2.5)
and each f×ϕn−1
kis an eigenfunction of the operator Lwith eigenvalue (2k+n).Now if
g(z) = g0(|z|) is a radial function on Cn, then Lλgtakes the form Lλg=Lλ,n−1g0where
Lλ,n−1is the scaled Laguerre operator of type (n−1) given by
Lλ,n−1:= −d2
dr2−2n−1
r
d
dr +1
4λ2r2.
In what follows, when λ= 1, we simply denote the radial part of the special Hermite
operator L1,n−1by Ln−1.Also, in order to prove Chernoff’s theorem for the special Hermite
operator, we need to use Laguerre operators of more general type and eigenfunction expansion
associated with them. In the following subsection, we develop notations and record required
results related to Laguerre expansion in this connection.
2.4. Laguerre expansion. To start with, we first recall the definition of Laguerre polyno-
mials. For any δ≥ −1
2,the Laguerre polynomials of type δare defined by
e−ttδLδ
k(t) = 1
k!
dk
dtk(e−ttk+δ)
for t > 0, and k≥0.The explicit form of Lδ
k(t) which is a polynomial of degree k, is given
by
Lδ
k(t) =
k
X
j=0
Γ(k+δ+ 1)
Γ(j+δ+ 1)Γ(k−j+ 1)
(−t)j
j!.
We now introduce the normalised Laguerre functions Lδ
kdefined as follows.
Lδ
k(t) = Γ(k+ 1)
Γ(k+ 1 + δ)1
2
e−t
2tδ
2Lδ
k(t), t > 0.
Then it is well-known that for any fixed δ≥ −1
2,Lδ
k∞
k=0 is an orthonormal basis for
L2(R+, dt).Now, fix δ≥ −1
2, and consider the following Laguerre functions of type δdefined
by
ψδ
k(r) := Γ(k+ 1)Γ(δ)
Γ(k+δ+ 1) Lδ
k(1
2r2)e−1
4r2, r > 0.

INGHAM’S THEOREM ON THE HEISENBERG GROUP 11
It turns out that ψδ
k(0) = 1, and these are eigenfunctions of the following Laguerre operator
of type δgiven by
Lδ:= −d2
dr2−2δ+ 1
r
d
dr +1
4r2
with eigenvalue (2k+δ+ 1) i.e., Lδψδ
k= (2k+δ+ 1)ψδ
k.This can be checked using the
relations [31, 1.1.48, 1.1.49] satisfied by the Laguerre polynomials. We will see later that
for δ=n−1, Lδcorresponds to the radial part of the special Hermite operator. Now,
using the orthogonality property of the functions Lδ
k(mentioned above), it is not difficult
to see that {ψδ
k:k≥0}forms an orthogonal basis for L2(R+, r2δ+1 dr).In view of this, for
f∈L2(R+, r2δ+1dr) we have
f(r) = ∞
X
k=0
cδ
kRδ
k(f)ψδ
k(r),kfk2
2=∞
X
k=0
cδ
k|Rδ
k(f)|2,(2.6)
where (cδ
k)−1:= R∞
0|ψδ
k(r)|2r2α+1dr, and Rδ
k(f) denotes the Laguerre coefficients of fgiven
by
Rδ
k(f) = Z∞
0
f(r)ψδ
k(r)r2δ+1dr, k ≥0.
We have the following Chernoff type theorem for Lδ:
Theorem 2.2. Let δ≥ −1
2and f∈L2(R+, r2δ+1dr)be such that Lm
δf∈L2(R+, r2δ+1dr)for
all m≥0, and satisfies the Carleman condition P∞
m=1 kLm
δfk−1/(2m)
2=∞.If Lm
δf(0) = 0
for all m≥0, then fis identically zero.
For a proof of this result, we refer the reader to Theorem 2.4 and the Remark 2.5 after
that in [10].
3. An analogue of Chernoff’s theorem for the special Hermite operator
Our next aim is to prove Theorem 1.7. For the sake of simplicity, we assume that λ= 1
and prove the Theorem 1.7 for L. In proving the weaker version of Chernoff’s theorem for L,
in [10], the authors used twisted spherical means and a Chernoff type theorem for its radial
part which is a Laguerre operator of type (n−1).However, in this case, we have to consider
Laguerre operators of a more general type, as well as the eigenfunction expansion that goes
with them, which has already been described at end of the previous section. Furthermore,
we will use Hecke-Bochner type identity for special Hermite projections, which requires some
preparations. To begin with, closely following the notations of [33, Section 5, Chapter 2] we
describe bi-graded spherical harmonics on Cn.

12 BAGCHI, GANGULY, SARKAR AND THANGAVELU
Bi-graded spherical harmonics: Let pand qbe two non-negative integers. Suppose
Pp,q denotes the set of all polynomials in zand ¯zof the form
P(z) = X
|α|≤p, |β|≤q
cα,β zα¯zβ
which clearly has the following homogeneity property: P(λz) = λp¯
λqP(z), λ ∈C.Now,
in terms of the vector fields ∂
∂zj,∂
∂¯zj, j = 1,2, .., n, the Laplacian on Cnhas the form
∆Cn= 4 Pn
j=1
∂2
∂zj∂¯zj.In view of this, it can be checked that ∆Cn:Pp,q → Pp−1,q−1. We
denote the kernel of ∆Cnby Hp,q.More precisely,
Hp,q := {P∈ Pp,q : ∆CnP= 0},
which is called the set of all bi-graded solid harmonics of degree (p, q).We define
Sp,q := {P|S2n−1:P∈ Hp,q}.
The elements of Sp,q are called the bi-graded spherical harmonics of degree (p, q). This turns
out be a Hilbert space under the usual inner-product of L2(S2n−1).Let d(p, q) denote the
dimension of this Hilbert space. Now, it is well-known that we can choose an orthonormal
basis Bp,q := {Sj
p,q : 1 ≤j≤d(p, q)}for Sp,q, for each pair of non-negative integers (p, q) such
that B:= ∪p,q≥0Bp,q forms an orthonormal basis for L2(S2n−1).For our purpose, we require
the following Hecke-Bochner type identity in the context of special Hermite projections.
Theorem 3.1. Suppose f∈L1(Cn)has the form f=P g where gis radial and P∈ Hp,q
for some p, q ≥0.Then f×ϕn−1
k= 0 unless k≥p, in which case
f×ϕn−1
k(z) = (2π)−ng×ϕn+p+q−1
k−p(z)P(z),
where the twisted convolution on the right hand side is on Cn+p+q.
For a proof of this result, we refer the reader to [33, Theorem 2.6.1]. We are now in a
position to prove the Theorem 1.7.
Proof of Theorem 1.7:Let fbe as in the statement. The main idea is to reduce the
matters to radial case by expanding fin terms of bi-graded spherical harmonics and then
use Chernoff’s theorem for Laguerre operator of suitable type. The proof will be completed
in the following steps.
Step 1:(Reduction of vanishing condition) Suppose fand all its partial derivatives vanish
at a point 0 6=w∈Cn.Consider the function gdefined by g=T1
−wf, which is nothing but
the twisted translation of fby −w(See (2.3)). In the following, we will be using standard

INGHAM’S THEOREM ON THE HEISENBERG GROUP 13
multi-index notations. Using the product rule of partial derivatives, an easy calculation
shows that ∂αg(z) is equal to
∂α(e−i
2ℑ(w.¯z)f(z+w)) = X
β≤αα
β∂β(e−i
2ℑ(w.¯z))∂α−β(f(z+w))
=X
β≤αα
β(e−i
2ℑ(w.¯z))Pβ(w, ¯w)∂α−β(f(z+w)),
where Pβ(w, ¯w) is some polynomial in wand ¯wwhose explicit form is not required for our
purpose. Note that for any multi-index α, we have from the equation above
∂αg(0) = X
β≤αα
βPβ(w, ¯w)∂α−βf(w) = 0
by the the assumption that ∂αf(w) = 0 for all α. Furthermore, using the twisted translation
invariance of L(See (2.4)), it is not hard to see that kLmgk2=kLmfk2, whence kLmgk2
also satisfy the Carleman condition. Therefore, if fand all its partial derivatives vanish at
any point, we can simply work with a suitable twisted translate of f. So, there is no loss of
generality in assuming that fand all its partial derivatives vanish at 0.
Step 2: (Spherical harmonic coefficients of Lmf) The spherical harmonic expansion of f
reads as
f(z) = ∞
X
p,q=0
d(p,q)
X
j=1
(f(r.), Sj
p,q)L2(S2n−1)Sj
p,q(ω), z =rω.
Writing fj
p,q(r) = r−p−q(f(r.), Sj
p,q)L2(S2n−1), and Pj
p,q(z) = |z|p+qSj
p,q(ω), we observe from the
above that
Lmf(z) = ∞
X
p,q=0
d(p,q)
X
j=1
Lm(fj
p,qPj
p,q)(z) = ∞
X
p,q=0
d(p,q)
X
j=1
LmFj
p,q(z),
where we have written Fj
p,q(z) := fj
p,q(|z|)Pj
p,q(z).Let us calculate the special Hermite pro-
jections of Fj
p,q. In view of the Theorem 3.1, we see that for k≥p,
Fj
p,q ×ϕn−1
k(z) = (2π)−nPj
p,q(z)fj
p,q ×ϕn+p+q−1
k−p(z)
= (2π)−nPj
p,q(z)Rδ(p,q)
k−p(fj
p,q)ϕδ(p,q)
k−p(z)
where δ(p, q) := n+p+q−1.In the last equality, we have used the fact that fj
p,q can be
thought of as a radial function on Cn+p+q.Therefore, we obtain from the special Hermite
expansion of Fthat
LmFj
p,q(z) = (2π)−n∞
X
k=0
(2k+n)mFj
p,q ×ϕn−1
k(z)

14 BAGCHI, GANGULY, SARKAR AND THANGAVELU
= (2π)−2nPj
p,q(z)∞
X
k=p
(2k+n)mRδ(p,q)
k−p(fj
p,q)ϕδ(p,q)
k−p(z)
= (2π)−2nPj
p,q(z)∞
X
k=0
(2k+ 2p+n)mRδ(p,q)
k(fj
p,q)ϕδ(p,q)
k(z)
= (2π)−2nSj
p,q(ω)rp+q∞
X
k=0
(2k+ 2p+n)mRδ(p,q)
k(fj
p,q)ϕδ(p,q)
k(r), z =rω. (3.1)
Thus, for a fixed mthe spherical harmonic coefficients of Lmf(r·) are given by
Gj
p,q(r) := (Lmf(r.), Sj
p,q)L2(S2n−1)= (2π)−2nrp+q∞
X
k=0
(2k+ 2p+n)mRδ(p,q)
k(fj
p,q)ϕδ(p,q)
k(r)
(3.2)
for any r > 0.Now, by using the orthogonality of the Laguerre functions, we get from (3.2)
that Z∞
0
Gj
p,q(r)ϕδ(p,q)
k(r)r2n+p+q−1dr
= (2π)−2n(2k+ 2p+n)mRδ(p,q)
k(fj
p,q)kϕδ(p,q)
kk2
2
=cn(2k+ 2p+n)mk!(n+p+q−1)!
(k+n+p+q−1)!Rδ(p,q)
k(fj
p,q).(3.3)
Step 3:(Carleman condition) We consider the function fj
p,q for fixed j, p and q. With δ(p, q)
as above, in view of the Plancherel formula (2.6), we see that for any m≥1,
kLm
δ(p,q)fj
p,qk2
2=∞
X
k=0
(2k+δ(p, q) + 1)2mcδ(p,q)
k|Rδ(p,q)
k(fj
p,q)|2
=∞
X
k=0
C(k, m, p, q, n)Z∞
0
Gj
p,q(r)ϕδ(p,q)
k(r)r2n+p+q−1dr
2
,(3.4)
where we have used (3.3). Here, C(k, m, p, q, n) is given by
C(k, m, p, q, n) := 2k+δ(p, q) + 1
2k+ 2p+ 1 2m
cδ(p,q)
k(k+n+p+q−1)!
k!(n+p+q−1)! 2
.
Now, using the value of δ(p, q), we see that
2k+δ(p, q) + 1
2k+ 2p+ 1 ≤1 + q
p:= ap,q.
Using this, we have from (3.4) that
kLm
δ(p,q)fj
p,qk2
2≤a2m
p,q
∞
X
k=0
cδ(p,q)
kZ∞
0
r−p−qGj
p,q(r)ψδ(p,q)
k(r)r2n+2p+2q−1dr
2
.(3.5)

INGHAM’S THEOREM ON THE HEISENBERG GROUP 15
We observe that the expression inside the modulus sign on the right hand side is the Laguerre
coefficient Rδ(p,q)
k(r−p−qGj
p,q).Therefore, by the Plancherel formula (2.6), we obtain
kLm
δ(p,q)fj
p,qk2
2≤a2m
p,q Z∞
0|r−p−qGj
p,q(r)|2r2n+2p+2q−1dr
=a2m
p,q Z∞
0|Gj
p,q(r)|2r2n−1dr.
Recalling the definition of Gj
p,q(r), we observe that
|Gj
p,q(r)|=|(Lmf(r.), Sj
p,q)L2(S2n−1)| ≤ kLmf(r.)kL2(S2n−1).
Using this in the equation above and integrating in polar coordinates, we obtain
kLm
δ(p,q)fj
p,qk2
2≤a2m
p,q kLmfk2
2.(3.6)
Thus, the Carleman condition on Lmfimplies the Carleman condition
∞
X
m=1 kLm
δ(p,q)fj
p,qk−1/(2m)
2=∞(3.7)
for any spherical harmonic coefficient fj
p,q.
Step 4:(Vanishing condition) We have assumed that fand all its partial derivatives vanish
at the origin. However, for our purpose, it is more convenient to work with the following
equivalent vanishing condition written in terms of polar coordinates:
d
dr m
f(rω)|r=0 = 0,for all ω∈S2n−1, m ≥0.(3.8)
Indeed, it can be checked that
d
dr k
f(rω) = X
|α|=k
∂αf(rω)ωα.
Hence, ( d
dr )kf(rω)|r=0 = 0, for all kif and only if ∂αf(0) = 0, for all α. We recall that fj
p,q
is explicitly given by
fj
p,q(r) = r−p−qZS2n−1
f(rω)Sj
p,q(ω)dσ(ω).
In view of the vanishing condition (3.8), a calculation using repeated application of L’Hospital
rule, we verify that all the derivatives of fj
p,q at 0 are zero. Thus, Lm
δ(p,q)fj
p,q(0) = 0, for all
m≥0. Hence, by Chernoff’s theorem for Lδ(p,q)(See Theorem 2.2), we have fj
p,q = 0, for all
j, p, q. Therefore, we conclude that f= 0,thereby completing the proof. 

16 BAGCHI, GANGULY, SARKAR AND THANGAVELU
4. Ingham’s theorem on the Heisenberg group
In this section we prove Theorems 1.3, and 1.5 using Chernoff’s theorem for the special
Hermite operator. We first show the existence of a compactly supported function fon Hn
whose Fourier transform has a prescribed decay as stated in Theorem 1.3. This proves the
sufficiency part of the condition on the function Θ appearing in the hypothesis. We then use
this part of the theorem to prove the necessity of the condition on Θ.We begin with some
preparations.
4.1. Construction of F.The Koranyi norm of x= (z, t)∈Hn, is defined by |x|=|(z, t)|=
(|z|4+t2)1
4.In what follows, we work with the following left invariant metric defined by
d(x, y) := |x−1y|, x, y ∈Hn.Given a∈Hnand r > 0, the open ball of radius rwith centre
at ais defined by
B(a, r) := {x∈Hn:|a−1x|< r}.
With this definition, we note that if f, g :Hn→Care such that supp(f)⊂B(0, r1) and
supp(g)⊂B(0, r2), then we have
supp(f∗g)⊂B(0, r1).B(0, r2)⊂B(0, r1+r2),
where f∗g(x) = RHnf(xy−1)g(y)dy is the convolution of fwith g.
Suppose {ρj}jand {τj}jare two sequences of positive real numbers such that both the
series P∞
j=1 ρjand P∞
j=1 τjare convergent. We let BCn(0, r) stand for the ball of radius r
centered at 0 in Cnand let χSdenote the characteristic function of a set S. For each j∈N,
we define functions fjon Cnand gjon Rby
fj(z) := ρ−2n
jχBCn(0,aρj)(z), z ∈Cn;
gj(t) := τ−2
jχ[−τ2
j/2,τ2
j/2](t), t ∈R,
where the positive constant ais chosen so that kfjkL1(Cn)= 1.We now consider the functions
Fj:Hn→Cdefined by
Fj(z, t) := fj(z)gj(t),(z, t)∈Hn.
In the following lemma, we record some useful, but easily proven properties of these functions.
Lemma 4.1. Let Fjbe as above and define GN=F1∗F2∗..... ∗FN.Then we have
(1) kFjkL∞(Hn)≤ρ−2n
jτ−2
j,kFjkL1(Hn)= 1,
(2) supp(Fj)⊂BCn(0, aρj)×[−τ2
j/2, τ2
j/2] ⊂B(0, aρj+cτj), where 4c4= 1.
(3) For any N∈N, supp(GN)⊂B(0, a PN
j=1 ρj+cPN
j=1 τj),kGNk1= 1.
(4) Given x∈Hn, and N∈N,F2∗F3..... ∗FN(x)≤ρ−2n
2τ−2
2.

INGHAM’S THEOREM ON THE HEISENBERG GROUP 17
We also recall a result about Hausd¨orff measure which will be used in the proof of the
next theorem. Let Hn(A) denote the n-dimensional Hausdorff measure of A⊂Rn.Hausd¨orff
measure coincides with the Lebesgue measure for Lebesgue measurable sets. For sets in Rn
with sufficiently nice boundaries, the (n−1)-dimensional Hausdorff measure is same as the
surface measure. For more about this, we refer the reader to [29, Chapter 7 ]. Let A∆B
stands for the symmetric difference between any two sets Aand B. See [27] for a proof of
the following theorem.
Theorem 4.2. Let A⊂Rnbe a bounded set. Then for any ξ∈Rn,
Hn(A∆(A+ξ)) ≤ |ξ|Hn−1(∂A),
where A+ξis the translation of Aby ξand ∂A is the boundary of A.
Theorem 4.3. The sequence defined by Gk=F1∗F2∗..... ∗Fkconverges to a compactly
supported non-trivial function F∈L2(Hn)in L2(Hn).
Proof. In order show that {Gk}is Cauchy in L2(Hn), we first estimate kGk+1 −GkkL∞(Hn).
As all the functions Fjhave unit L1norm, we have for any x∈Hn
Gk+1(x)−Gk(x) = ZHn
Gk(xy−1)Fk+1(y)dy −Gk(x)ZHn
Fk+1(y)dy
=ZHnGk(xy−1)−Gk(x)Fk+1(y)dy.
Since Fj’s are even, we can change yinto y−1in the above and estimate the same as
|Gk+1(x)−Gk(x)| ≤ ZHn|Gk(xy)−Gk(x)|Fk+1(y)dy. (4.1)
By defining Hk−1=F2∗F3...... ∗Fk, we note that Gk=F1∗Hk−1. Thus,
Gk(xy)−Gk(y) = ZHnF1(xyu−1)−F1(xu−1)Hk−1(u)du.
Using the estimate (4) in Lemma 4.1, we get that
|Gk(xy)−Gk(x)| ≤ ρ−2n
2τ−2
2ZHnF1(xyu−1)−F1(xu−1)du. (4.2)
The change of variables u→ux transforms the integral in the right hand side of the inequality
above into ZHnF1(xyu−1)−F1(xu−1)du =ZHnF1(xyx−1u−1)−F1(u−1)du.
Since the group Hnis unimodular, another change of variables u→u−1yields
ZHnF1(xyx−1u−1)−F1(u−1)du =ZHnF1(xyx−1u)−F1(u)du.

18 BAGCHI, GANGULY, SARKAR AND THANGAVELU
Let x= (z, t) = (z, 0)(0, t), y = (w, s) = (w, 0)(0, s).As (0, t) and (0, s) belong to the
center of Hn, an easy calculation shows that xyx−1= (w, 0)(0, s +ℑ(z·¯w)).With u= (ζ, τ )
we have
xyx−1u= (w+ζ , 0)(0, τ +s+ℑ(z·¯w)−(1/2)ℑ(ζ·¯w)).
Since F1(z, t) = f1(z)g1(t), we see that the integrand F1(xyx−1u)−F1(u) in the above integral
takes the form
f1(w+ζ)g1(τ+s+ℑ(z·¯w)−(1/2)ℑ(ζ·¯w)) −f1(ζ)g1(τ).
By setting b=b(s, z, w, ζ) = s+ℑ(z·¯w)−(1/2)ℑ(ζ·¯w), we can rewrite the above as
f1(w+ζ)−f1(ζ)g1(τ+b) + f1(ζ)g1(τ+b)−g1(τ).(4.3)
In order to estimate the contribution of the second term in (4.3) to the integral under
consideration, we first estimate the integral in τ-variable as follows:
Z∞
−∞ |g1(τ+b)−g1(τ)|dτ =τ−2
1|(−b+Kτ)∆Kτ|,
where Kτ= [−1
2τ2
1,1
2τ2] is the support of g1.For ζin the support of f1,we have |ζ| ≤ aρ1,
and hence
|(−b+Kτ)∆Kτ| ≤ 2|b(s, z, w, ζ)| ≤ (2|s|+|z||w|+aρ1|w|).
Thus, we have proved the following estimate
ZHn
f1(ζ)|g1(τ+b)−g1(τ)|dζdτ ≤C2|s|+ (aρ1+|z|)|w|.(4.4)
As the integral of g1is one, the contribution of the first term in (4.3) is given by
ZCn|f1(w+ζ)−f1(ζ)|dζ =ρ−2n
1H2n((−w+BCn(0, aρ1))∆BCn(0, aρ1)) .
By appealing to Theorem 4.2 in estimating the above, we obtain
ZHn|f1(w+ζ)−f1(ζ)|g(τ+b)dζdτ ≤C|w|.(4.5)
Using the estimates (4.4) and (4.5) in (4.2) we obtain
|Gk(xy)−Gk(x)| ≤ Cρ−2n
2τ−2
2|s|+ (c1+c2|z|)|w|).
This estimate, when used in (4.1), in turn gives us
|Gk+1(z, t)−Gk(z, t)| ≤ CZHn|s|+ (c1+c2|z|)|w|)Fk+1(w, s)dw ds (4.6)
where the constants c1, c2and Cdepend only on n. Recalling that on the support of
Fk+1(w, s) = fk+1(w)gk+1(s), |w| ≤ ρk+1 and |s| ≤ τ2
k+1, the above yields the estimate
|Gk+1(z, t)−Gk(z, t)| ≤ Cτ2
k+1 + (c1+c2|z|)ρk+1.(4.7)

INGHAM’S THEOREM ON THE HEISENBERG GROUP 19
It is easily seen that the support of Gk+1 −Gkis contained in B(0, aρ+cτ ) where ρ=P∞
j=1 ρj
and τ=Pτj.Consequently, from the above we conclude that
kGk+1 −Gkk2≤ kGk+1 −Gkk∞|B(0, aρ +cτ )|1/2≤Cτ2
k+1 +c3ρk+1.
From the above, it is clear that {Gk}is Cauchy in L2(Hn), and hence converges to a function
F∈L2(Hn) whose support is contained in B(0, aρ +cτ ).The same argument shows that
{Gk}converges to Fin L1.As kGkk1= 1 for any k, it follows that kF||1= 1 and hence F
is nontrivial. 
4.2. Estimating the Fourier transform of F.Suppose now that Θ is an even, decreasing
function on Rfor which R∞
1Θ(t)t−1dt < ∞.We want to choose two sequences of positive
real numbers {ρj}and {τj}in terms of Θ so that the series P∞
j=1 ρjand P∞
j=1 τjboth
converge. We can then construct a function Fas in Theorem 4.3 which will be compactly
supported. Having done the construction we now want to compute the Fourier transform of
the constructed function Fand compare it with e−Θ(√H(λ))√H(λ).This can be achieved by
a judicious choice of the sequences {ρj}and {τj}.As Θ is given to be decreasing, it follows
that P∞
j=1
Θ(j)
j<∞.It is then possible to choose a decreasing sequence {ρj}such that
ρj≥c2
ne2Θ(j)
j(for a constant cnto be chosen later) and P∞
j=1 ρj<∞.Similarly, we choose
another decreasing sequence {τj}such that P∞
j=1 τj<∞.
In the proof of the following lemma we require good estimates for the Laguerre coefficients
of the function fj(z) = ρ−2n
jχBCn(0,aρj)(z) where achosen so that kfjk1= 1.These coefficients
are defined by
Rn−1
k(λ, fj) = k!(n−1)!
(k+n−1)! ZCn
fj(z)ϕn−1
k,λ (z)dz. (4.8)
Lemma 4.4. There exists a constant cn>0such that
|Rn−1
k(λ, fj)| ≤ cnρjp(2k+n)|λ|−n+1/2.
Proof. By abuse of notation we write ϕn−1
k,λ (r) in place of ϕn−1
k,λ (z) when |z|=r. As fjis
defined as the dilation of a radial function, the Laguerre coefficients are given by the integral
Rn−1
k(λ, fj) = 2πn
Γ(n)
k!(n−1)!
(k+n−1)! Za
0
ϕn−1
k,λ (ρjr)r2n−1dr. (4.9)
When a≤(ρjp(2k+n)|λ|)−1we use the bound k!(n−1)!
(k+n−1)! |ϕn−1
k,λ (r)| ≤ 1 to estimate
2πn
Γ(n)
k!(n−1)!
(k+n−1)! Za
0
ϕn−1
k,λ (ρjr)r2n−1dr ≤πnan+1/2
Γ(n+ 1)ρjp(2k+n)|λ|−n+1/2.
When a > (ρjp(2k+n)|λ|)−1we split the integral into two parts, one of which gives the
same estimate as above. To estimate the integral taken over (ρjp(2k+n)|λ|)−1< r < a,

20 BAGCHI, GANGULY, SARKAR AND THANGAVELU
we use the bound stated in Lemma 2.1 which leads to the estimate
2πn
Γ(n)
k!(n−1)!
(k+n−1)! Za
(ρj√(2k+n)|λ|)−1
ϕn−1
k,λ (ρjr)r2n−1dr
≤Cnρjp(2k+n)|λ|−n+1/2Za
0
rn−1/2dr =C′
nan+1/2ρjp(2k+n)|λ|−n+1/2.
Combining the two estimates we get the lemma. 
Theorem 4.5. Let Θ : R→[0,∞)be an even, decreasing function with limλ→∞ Θ(λ) = 0
for which R∞
1
Θ(λ)
λdλ < ∞.Let ρjand τjbe chosen as above. Then the Fourier transform of
the function Fconstructed in Theorem 4.3 satisfies the estimate
b
F(λ)∗b
F(λ)≤e−2Θ(√H(λ))√H(λ), λ 6= 0.
Proof. Observe that Fis radial since each Fjis radial and hence the Fourier transform b
F(λ)
is a function of the Hermite opertaor H(λ).More precisely,
b
F(λ) = ∞
X
k=0
Rn−1
k(λ, F )Pk(λ) (4.10)
where the Laguerre coefficients are explicitly given by (see (2.4.7) in [33]. There is a typo-
the factor |λ|n/2should not be there)
Rn−1
k(λ, F ) = k!(n−1)!
(k+n−1)! ZCn
Fλ(z)ϕn−1
k,λ (z)dz.
In the above, Fλ(z) stands for the inverse Fourier transform of F(z, t) in the tvariable.
Expanding any ϕ∈L2(Rn) in terms of Φλ
αit is easy to see that the conclusion b
F(λ)∗b
F(λ)≤
e−2Θ(√H(λ))√H(λ)follows once we show that
(Rn−1
k(λ, F ))2≤Ce−2Θ(√(2k+n)|λ)√(2k+n)|λ|
for all k∈Nand λ∈R∗.Now note that, by definition of gjand the choice of a, we have
|bgj(λ)|=
sin(1
2τ2
jλ)
1
2τ2
jλ≤1,|Rn−1
k(λ, fj)| ≤ 1.
The bound on Rn−1
k(λ, fj) follows from the fact that |ϕλ
k(z)| ≤ (k+n−1)!
k!(n−1)! .Since Fis constructed
as the L2limit of the N-fold convolution GN=F1∗F2...... ∗FNwe observe that for any N
(Rn−1
k(λ, F ))2≤(Rn−1
k(λ, GN))2= (ΠN
j=1Rn−1
k(λ, Fj))2
and hence it is enough to show that for a given kand λone can choose N=N(k, λ) in such
a way that
(ΠN
j=1Rn−1
k(λ, Fj))2≤Ce−2Θ(√(2k+n)|λ|)√(2k+n)|λ|.(4.11)

INGHAM’S THEOREM ON THE HEISENBERG GROUP 21
where Cis independent of N. From the definition of GNit follows that
d
GN(λ) = ΠN
j=1c
Fj(λ) = ΠN
j=1∞
X
k=0
Rn−1
k(λ, Fj)Pk(λ)
and hence Rn−1
k(λ, GN) = ΠN
j=1Rn−1
k(λ, Fj).As Fj(z, t) = fj(z)gj(t), we have
Rn−1
k(λ, GN) = ΠN
j=1 bgj(λ)ΠN
j=1Rn−1
k(λ, fj).
As the first factor is bounded by one, it is enough to consider the product ΠN
j=1Rn−1
k(λ, fj).
We now choose ρjsatisfying ρj≥c2
ne2Θ(j)
j, where cnis the same constant appearing in
Lemma 4.4. We then take N=⌊Θ(((2k+n)|λ|)1
2)((2k+n)|λ|)1
2⌋, and consider
ΠN
j=1Rn−1
k(λ, fj)≤ΠN
j=1cn(ρjp(2k+n)|λ|)−n+1/2
where we have used the estimates proved in Lemma 4.4. As {ρj}is decreasing
ΠN
j=1cn(ρjp(2k+n)|λ|)−n+1/2≤cN
nρNp(2k+n)|λ|−(n−1/2)N.(4.12)
By the choice of ρj, it follows that
ρ2
N(2k+n)|λ| ≥ c4
ne4Θ(N)2
N2(2k+n)|λ|.
As Θ is decreasing and N≤p(2k+n)|λ|), we have Θ(N)≥Θ(p(2k+n)|λ|) and so
Θ(N)2(2k+n)|λ| ≥ Θp(2k+n)|λ|2(2k+n)|λ| ≥ N2
which proves that ρ2
N(2k+n)|λ| ≥ c4
ne4.Using this in (4.12) we obtain
ΠN
j=1cnρjp(2k+n)|λ|−n+1/2≤(c2
ne2)−(n−1)Ne−N.
Finally, as N+ 1 ≥Θ(((2k+n)|λ|)1
2)((2k+n)|λ|)1
2, we obtain the estimate (4.11). 
4.3. Ingham’s theorem. We can now complete the proofs of Theorems 1.3, and 1.5. Since
half of the theorem has been already proved, as already mentioned in Section 1, we only
need to prove the Theorem 1.5.
Proof of Theorem 1.5:Fix λ6= 0.By the hypothesis, fλvanishes on an open set Uλ
in Cn.First we assume that Θ(λ)≥c|λ|−1
2,|λ| ≥ 1.In view of Plancherel formula (2.1) for
the Weyl transform, we have
(2π)nkLm
λfλk2
2=|λ|nkWλ(Lm
λfλ)k2
HS =|λ|nkb
f(λ)H(λ)mk2
HS .
Using the formula for Hilbert-Schmidt norm of an operator we have
(2π)nkLm
λfλk2
2=|λ|nX
α
((2|α|+n)|λ|)2mkˆ
f(λ)Φλ
αk2
2.

22 BAGCHI, GANGULY, SARKAR AND THANGAVELU
Now, the given condition on the Fourier transform leads to the estimate
(2π)nkLm
λfλk2
2≤C|λ|nX
α
((2|α|+n)|λ|)2me−2Θ(((2|α|+n)|λ|)1
2)((2|α|+n)|λ|)1
2
≤C|λ|∞
X
k=0
((2k+n)|λ|)2m+n−1e−2Θ(((2k+n)|λ|)1
2)((2k+n)|λ|)1
2.(4.13)
We write the last sum as I1+I2, where
I1:= X
k≥0,(2k+n)|λ|≤m8
((2k+n)|λ|)2m+n−1e−2Θ(((2k+n)|λ|)1
2)((2k+n)|λ|)1
2,and
I2:= X
k≥0,(2k+n)|λ|>m8
((2k+n)|λ|)2m+n−1e−2Θ(((2k+n)|λ|)1
2)((2k+n)|λ|)1
2.
Now, we estimate each sum separately. Notice that when (2k+n)|λ| ≤ m8,we have Θ(((2k+
n)|λ|)1
2)≥Θ(m4), as Θ is decreasing. This shows that
I1≤X
k≥0,(2k+n)|λ|≤m8
((2k+n)|λ|)2m+n−1e−2Θ(m4)((2k+n)|λ|)1
2
which can be dominated by
X
k≥0,(2k+n)|λ|≤m8
((2k+n)|λ|)nZ(√(2k+n)|λ|+1)2
(2k+n)|λ|
x2m−1e−2Θ(m4)(√x−1)dx
≤e2Θ(m4)m8nZ∞
0
x2m−1e−2Θ(m4)√xdx.
The change of variable y= 2Θ(m4)√xtransform the last expression into
e2Θ(m4)m8n
(2Θ(m4))4mZ∞
0
y4m−1e−ydy =e2Θ(m4)m8n
(2Θ(m4))4mΓ(4m).
This along with the fact that Θ(m4)≤Θ(1) shows that
I1≤Cm8n
(2Θ(m4))4mΓ(4m).
Using Stirling’s formula (see Ahlfors [1]) Γ(x) = √2π xx−1/2e−xeθ(x)/12x,0< θ(x)<1, which
is valid for x > 0,for large m, we observe that
I1≤C2m
Θ(m4)4m
.(4.14)
Now, to estimate I2, we make use of the initial assumption that Θ(t)≥c t−1/2for t≥1.
Following the same procedure as above, we observe that I2is dominated by
X
k≥0,(2k+n)|λ|>m8Z((2k+n)|λ|)2
(2k+n)|λ|
x2m+n−1e−2c√xdx

INGHAM’S THEOREM ON THE HEISENBERG GROUP 23
≤e−c m4X
k≥0,(2k+n)|λ|>m8Z((2k+n)|λ|)2
(2k+n)|λ|
x2m+n−1e−c√xdx
=e−c m4Z∞
0
x2m+n−1e−2c√xdx.
Again the change of variables y=c√xtransforms the above integral into
2c−(4m+2n−2) Z∞
0
y4m+2n−1e−ydy = 2c−(4m+2n−2)Γ(4m+ 2n).
Hence, we obtain
I2≤2c−(4m+2n−2)Γ(4m+ 2n)e−cm4.
Now, for large m, using the fact that Γ(4m+ 2n)≤Γ(5m), and Stirling’s formula we have
I2≤C(c−4(5m)5e−cm3)m.
But the right hand side of above goes to zero as m→ ∞.Hence, in view of (4.14), for large
m, we conclude that
I1+I2≤C2m
Θ(m4)4m
(4.15)
which from (4.13) yields for large m
(2π)nkLm
λfλk2
2≤C|λ|2m
Θ(m4)4m
.
The hypothesis on Θ,namely R∞
1
Θ(t)
tdt =∞,implies that R∞
1
Θ(y4)
ydy =∞.Hence, by
integral test we get P∞
m=1
Θ(m4)
m=∞.Therefore, it follows that
∞
X
m=1 kLm
λfλk−1
2m
2=∞.
Since fλvanishes on an open set, by the Theorem 1.7 ( analogue of Chernoff’s theorem for
Lλ) we conclude that fλ= 0 which is true for all λ6= 0.Hence f= 0.
Now, we consider the general case. The function Ψ(y) = (1+ |y|)−1/2satisfies R∞
1
Ψ(y)
ydy <
∞.By Theorem 4.3 we can construct a compactly supported radial function F∈L2(Hn)
such that
ˆ
F(λ)∗ˆ
F(λ)≤e−2Ψ(√H(λ))√H(λ), λ 6= 0.
We can further arrange that supp(F)⊂BCn(0, δ)×(−a, a) for some δ, a > 0.We now
consider the function h=f∗F. Notice that
hλ(z) = (f∗F)λ(z) = ZCn
fλ(z−w)Fλ(w)eiλ
2Im(z. ¯w)dw.
As fλis assumed to vanish on Uλ, the function hλvanishes on a smaller open set Uλ,δ ⊂Uλ.
We now claim that
b
h(λ)∗b
h(λ)≤e−2Φ(√H(λ))√H(λ)

24 BAGCHI, GANGULY, SARKAR AND THANGAVELU
where Φ(y) = Θ(y) + Ψ(y).As b
h(λ) = b
f(λ)b
F(λ), for any ϕ∈L2(Rn) we have
hb
h(λ)∗b
h(λ)ϕ, ϕi=hb
f(λ)∗b
f(λ)b
F(λ)ϕ, b
F(λ)ϕi.
The hypothesis on fgives us the estimate
hb
f(λ)∗b
f(λ)b
F(λ)ϕ, b
F(λ)ϕi ≤ Che−2Θ(√H(λ))√H(λ)b
F(λ)ϕ, b
F(λ)ϕi.
As Fis radial, b
F(λ) commutes with any function of H(λ) and hence the right hand side can
be estimated using the decay of b
F(λ):
hb
F(λ)∗b
F(λ)e−Θ(√H(λ))√H(λ)ϕ, e−Θ(√H(λ))√H(λ)ϕi ≤ Che−2(Θ+Ψ)(√H(λ))√H(λ)ϕ, ϕi.
This proves our claim on b
h(λ) with Φ = Θ + Ψ.As Φ(y)≥ |y|−1/2, by the already proved
part of the theorem we conclude that h= 0.In order to conclude that f= 0 we proceed as
follows.
Given Fas above, let us consider δrF(z, t) = F(rz, r2t).It has been shown elsewhere (see
e.g. [15]) that
d
δrF(λ) = r−(2n+2)dr◦b
F(r−2λ)◦d−1
r
where dris the standard dilation on Rngiven by drϕ(x) = ϕ(rx).The property of the
function F, namely ˆ
F(λ)∗ˆ
F(λ)≤e−2Ψ(√H(λ))√H(λ)gives us
d
δrF(λ)∗d
δrF(λ)≤Cr−2(2n+2)dr◦e−2Ψ(√H(λ/r2))√H(λ/r2)◦d−1
r.
Testing against Φλ
αwe can simplify the right hand side which gives us
d
δrF(λ)∗d
δrF(λ)≤Cr−2(2n+2)e−2Ψr(√H(λ))√H(λ),
where Ψr(y) = 1
rΨ(y/r).If we let Fε(x) = ε−(2n+2)δε−1F(x), then it follows that Fεis an
approximate identity. Moreover, Fεis compactly supported and satisfies the same hypothesis
as Fwith Ψ(y) replaced by εΨ(εy) which has the same integrability and decay conditions.
Hence, working with Fεwe can conclude that f∗Fε= 0 for any ε > 0.Letting ε→0 and
noting that f∗Fεconverges to fin L1(Hn), we conclude that f= 0.This completes the
proof. 
Remark 4.1.It would be interesting to see whether the conclusion of the Theorem 1.5 still
holds true under the assumption that the function vanishes on a non-empty open subset of
Hn.A moment’s thought staring at the above proof reveals that this can be achieved if we
use an analogue of the Theorem 1.6 for the sublaplacian instead of special Hermite operators.
But it turns out that proving an analogue of Theorem 1.6 is a very interesting and difficult
open problem. We hope to revisit this in the near future.

INGHAM’S THEOREM ON THE HEISENBERG GROUP 25
Acknowledgments
The work of the first named author is supported by INSPIRE Faculty Awards from the
Department of Science and Technology. The second author is supported by Int.Ph.D. schol-
arship from Indian Institute of Science. The third named author is supported by NBHM
Post-Doctoral fellowship from the Department of Atomic Energy (DAE), Government of
India. The work of the last named author is supported by J. C. Bose Fellowship from the
Department of Science and Technology, Government of India.
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(S. Bagchi, J. Sarkar) Department of Mathematics and Statistics, Indian Institute of Science
Education and Research Kolkata, Mohanpur-741246, Nadia, West Bengal, India.
Email address:sayansamrat@gmail.com, jayantasarkarmath@gmail.com
(P. Ganguly, S. Thangavelu) Department of Mathematics, Indian Institute of Science, Bangalore-
560 012, India.
Email address:pritam1995.pg@gmail.com, veluma@iisc.ac.in