On theorems of Chernoff and Ingham on the Heisenberg group

Abstract

We prove an analogue of Chernoff's theorem for the sublaplacian on the Heisenberg group and use it prove a version of Ingham's theorem for the Fourier transform on the same group.

Full text

arXiv:2009.14230v2 [math.FA] 8 Feb 2021
ON THEOREMS OF CHERNOFF AND INGHAM
ON THE HEISENBERG GROUP
SAYAN BAGCHI, PRITAM GANGULY, JAYANTA SARKAR
AND SUNDARAM THANGAVELU
Abstract. We prove an analogue of Chernoff’s theorem for the sublaplacian on the Heisen-
berg group and use it to prove a version of Ingham’s theorem for the Fourier transform on
the same group.
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 manifesta-
tions of this principle are known: Heisenberg-Pauli-Weyl inequality, Paley-Wiener theorem,
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 that is
allowed of a function of compact support? An interesting answer to this question is provided
by the following theorem of Ingham [10].
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)having Fourier transform b
fsatisfying the estimate |b
f(y)| ≤
Ce−|y|Θ(y)if and only if Θsatisfies R∞
1Θ(t)t−1dt < ∞.
This theorem of Ingham and its close relatives Paley -Wiener ([20, 21]) and Levinson ([14])
theorems have received considerable attention in recent years. In [2] Bhowmik et al proved
analogues of the above theorem for Rn,the n-dimensional torus Tnand step two nilpotent
2010 Mathematics Subject Classification. Primary: 43A80. Secondary: 22E25, 33C45, 26E10, 46E35.
Key words and phrases. Heisenberg group, sublaplacian, quasi-analyticity, sobolev spaces, Chernoff’s
theorem, Ingham’s theorem.
1

2 BAGCHI, GANGULY, SARKAR AND THANGAVELU
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 fon Hn
let b
f(λ) be the operator valued Fourier transform of findexed by non-zero real 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-vairable and can be
easily deduced from Hardy’s theorem on R, see Theorem 2.9.1 in [31]. Compare this with
the following version, Theorem 2.9.2 in [31]. 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
characterisation 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 [12], Pfannschmidt’s theorem [33]
and the extension problem for the sublaplacian [23] on the Heisenberg group. In connection
with the study of Poisson integrals, it has been noted in [32] 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
tube domain in the complexification of Hnas a holomorphic function and hence the vanishing

CHERNOFF AND INGHAM ON THE HEISENBERG GROUP 3
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 analogue of Ingham’s theorem for the Fourier transform on Hn.
Theorem 1.3. Let Θ(λ)be a nonnegative even function on Rsuch that Θ(λ)decreases to
zero when λ→ ∞ and satisfies the condition R∞
1Θ(t)t−1dt < ∞.Then there exists a nonzero
compactly supported continuous function fon Hnwhose Fourier transform b
fsatisfies the
estimate
b
f(λ)∗b
f(λ)≤Ce−2Θ(√H(λ))√H(λ).(1.3)
Conversely, if there exists a nontrivial fsatisfying (1.3) and the extra assumption f(z, t) =
f(|(z, t)||z|−1z, 0) where |(z, t)|= (|z|4+t2)1/4is the Koranyi norm on Hnwhich vanishes
on a neighbourhood of 0then it is necessary that R∞
1Θ(t)t−1dt < ∞.
We believe that the above result is true without the extra assumption on f. As the proof
requires a general version of Chernoff’s theorem for the sublaplacian which is yet to be
proved (see Theorem 1.5 below), we only have the above result at present. However, the
class of functions to which the above theorem applies is relatively large. To see this, consider
the Heisenberg coordinates (ρ, ω, θ) of a point (z, t)∈Hndefined by z=rω, r > 0, ω ∈
S2n−1, t +ir2=ρ2eiθ ,0≤θ≤πso that f(z, t) = f(ρω√sin θ, ρ2cos θ).The functions f
satisfying the extra assumption in Theorem 1.3 are precisely those which are independent of
θin the Heisenberg variables. Any function gon Cngives rise to such a function on Hnby
the prescription f(z, t) = g(ρω).
Theorem 1.1 was proved in [10] 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 1.4. [8, Chernoff] Let fbe a smooth function on Rn.Assume that ∆mf∈L2(Rn)
for all m∈Nand P∞
m=1 k∆m
Rnfk−1
2m
2=∞.If fand all its partial derivatives vanish at 0,
then fis identically zero.
This theorem shows how partial differential operators generate the class of quasi-analytic
functions. Recently, Bhowmik-Pusti-Ray [3] have established an analogue of Chernoff’s the-
orem for the Laplace-Beltrami operator on non-compact Riemannian symmetric spaces and
use the same in proving a version of Ingham’s theorem for the Helgason Fourier transform. It

4 BAGCHI, GANGULY, SARKAR AND THANGAVELU
is therefore natural to look for an analogue of this result for sublaplacian on the Heisenberg
group. In this paper, we prove the following result.
Theorem 1.5. Let Lbe the sublaplacian on the Heisenberg group and let fbe a smooth
function on Hnsuch that f(ρ ω√sin θ, ρ2cos θ) = f(ρ ω, 0) in the Heisenberg coordinates.
Assume that Lmf∈L2(Hn)for all m∈Nand P∞
m=1 kLmfk−1
2m
2=∞. If fand all its
partial derivatives vanish at 0, then fis identically zero.
An immediate corollary of this theorem is the following, which can be seen as an L2
version of the classical Denjoy-Carleman theorem on the Heisenberg group using iterates of
sublaplacian.
Corollary 1.6. Let {Mk}kbe a log convex sequence. Define C({Mk}k,L,Hn)to be the class
of all smooth functions fon Hnsatisfying the condition f(ρ ω√sin θ, ρ2cos θ) = f(ρ ω, 0)
such that Lmf∈L2(Hn)for all k∈Nand kLkfk2≤Mkλkfor some constant λ(may
depend on f). Suppose that P∞
k=1 M−1
2k
k=∞.Then every member of that class is quasi-
analytic.
We conclude the introduction by briefly describing the organisation 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 sublaplacian (Theorem 1.5) in
Section 3. In section 4, we use this version of Chernoff’s theorem to prove Ingham’s theorem
on the Heisenberg group i.e., Theorem 1.3.
2. Preliminaries
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 [28] for the preliminaries of harmonic analysis on
the Heisenberg group. However, we will be closely following the notations of the books of
Thangavelu [30] and [31].
2.1. Heisenberg group and Fourier transform. Let Hn:= Cn×Rdenote the (2n+ 1)-
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.

CHERNOFF AND INGHAM ON THE HEISENBERG GROUP 5
For each non zero real number λwe have an infinite dimensional representation πλrealised
on the Hilbert space L2(Rn).These are explicitly given by
πλ(z, t)ϕ(ξ) = eiλt ei(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]) upto unitary
equivalence these account for all the infinite dimensional irreducible unitary representations
of Hnwhich act as eiλtIon the center. Also there is another class of finite dimensional
irreducible representations. 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).It is known 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 denote 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
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 function fon Hnwe have the following inversion formula
f(z, t) = Z∞
−∞
tr(πλ(z, t)∗b
f(λ))dµ(λ).
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∞
−∞
eiλ.tf(z, t)dt.
This motivates the following operator. Given a function gon Cn, we consider the following
operator valued function defined by
Wλ(g) := ZCn
g(z)πλ(z, 0)dz.

6 BAGCHI, GANGULY, SARKAR AND THANGAVELU
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).Moreover, the
Fourier transform bahaves well with the convolution of two functions defined by
f∗g(x) := ZHn
f(xy−1)g(y)dy.
Infact, for any f, g ∈L1(Hn), directly from the definition it follows that [
f∗g(λ) = ˆ
f(λ)ˆg(λ).
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 [29]). 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 fis invariant under the action of U(n) i.e., Rσf=ffor all σ∈U(n).The Fourier
transforms of such radial integrable funtions are functions of the Hermite operator H(λ) . In
fact, if H(λ) = P∞
k=0(2k+n)|λ|Pk(λ) stands for 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).
More explicitly, Pk(λ) stands for the orthogonal projection of L2(Rn) onto the kth eigenspace
spanned by scaled Hermite functions Φλ
αfor |α|=k. The coefficients Rk(λ, f) are explicitly
given by
Rk(λ, f ) = k!(n−1)!
(k+n−1)! ZCn
fλ(z)ϕn−1
k,λ (z)dz. (2.1)
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

CHERNOFF AND INGHAM ON THE HEISENBERG GROUP 7
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 [29, Lemma 1.5.3]. The
estimates in [29, Lemma 1.5.3] are sharp, see [15, Section 2] and [16, 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 Sobolev spaces on Hn.We let hnstand for the Heisenberg
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 .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

8 BAGCHI, GANGULY, SARKAR AND THANGAVELU
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
dilations given by δr(z, t) = (rz, r2t).The sublaplacian is also invariant undrer rotation i.e.,
Rσ◦L =L◦Rσ, σ ∈U(n).It is convenient for our purpose to represent the sublaplacian in
terms of another set of vector fields defined as follows:
Zj:= 1
2(Xj−iYj) = ∂
∂zj
+i
4¯zj
∂
∂t ,¯
Zj:= 1
2(Xj+iYj) = ∂
∂¯zj
+i
4zj
∂
∂t
where ∂
∂zj=1
2∂
∂xj−i∂
∂yjand ∂
∂¯zj=1
2∂
∂xj+i∂
∂yj.Now an easy calculation yields
L=−1
2
n
X
j=1 Zj¯
Zj+¯
ZjZj.
The action of Fourier transform on Zjf,¯
Zjfand T f are well-known and are given by
d
Zjf(λ) = ib
f(λ)Aj(λ),d
¯
Zjf(λ) = ib
f(λ)Aj(λ)∗and c
T f (λ) = −iλ b
f(λ) (2.2)
where Aj(λ) and A∗
j(λ) are the annihilation and creation operators given by
Aj(λ) = −∂
∂ξj
+iλξj, A∗
j(λ) = ∂
∂ξj
+iλξj.
These along with the above representation of the sublaplacian yield the relation c
Lf(λ) =
b
f(λ)H(λ).
We can define the spaces Ws,2(Hn) for any s∈Ras the completion of C∞
c(Hn) under
the norm kfk(s)=k(I+L)s/2fk2where the fractional powers (I+L)s/2are defined using
spectral theorem. To study these spaces, it is better to work with the following expression
of the norm kfk(s)for f∈C∞
c(Hn).In view of Plancherel theorem for the Fourier transform
kfk2
(s)= (2π)−n−1Z∞
−∞ kb
f(λ)(1 + H(λ))s/2k2
HS |λ|ndλ
which is valid for any s∈R.Here we have made use of the fact that c
Lf(λ) = b
f(λ)H(λ).
Computing the Hilbert-Schmidt norm in terms of the Hermite basis, we have the more
explicit expression:
kfk2
(s)= (2π)−n−1Z∞
−∞ X
α∈NnX
β∈Nn
(1 + (2|α|+n)|λ|)s|hb
f(λ)Φλ
α,Φλ
βi|2|λ|ndλ.
Consider b
Hn=R∗×Nn×Nnequipped with the measure µ×νwhere νis the counting measure
on Nn×Nn.The above shows that, for f∈C∞
c(Hn) the function m(λ, α, β) = hb
f(λ)Φλ
α,Φλ
βi.

CHERNOFF AND INGHAM ON THE HEISENBERG GROUP 9
belongs to the weighted space
Ws,2(b
Hn) = L2(b
Hn, wsd(µ×ν))
where ws(λ, α) = (1 + (2|α|+n)|λ|)s.As these weighted L2spaces are complete, we can
identify Ws,2(Hn) with Ws,2(b
Hn).It is then clear that for any s > 0 we have
Ws,2(b
Hn)⊂W0,2(b
Hn)⊂W−s,2(b
Hn)
and the same inclusion holds for Ws,2(Hn).It is clear that any m∈Ws,2(b
Hn) can be written
as m(λ, α, β) = (1 + (2|α|+n)|λ|)−s/2m0(λ, α, β) where m0∈W0,2(b
Hn) = L2(b
Hn) for
any s∈R.Consequently, any f∈Ws,2(Hn) can be written as f= (I+L)−s/2f0, where
f0∈L2(Hn) is the function which corresponds to m0which is given explicitly by
f0(z, t) = Zb
Hn
m0(λ, α, β)e−λ
α,β(z, t)dν(α, β)dµ(λ).
Thus we see that f∈Ws,2(Hn) if and only if there is an f0∈L2(Hn) such that f=
(I+L)−s/2f0.The inner product on Ws,2(Hn) is given by
hf, gis=h(I+L)s/2f, (I+L)s/2gi=hf0, g0i
where hf, giis the inner product in L2(Hn).This has the following interesting consequence.
Given f∈Ws,2(Hn) and g∈W−s,2(Hn),let f0, g0∈L2(Hn) be such that f= (I+L)−s/2f0
and g= (I+L)s/2g0.The duality bracket (f, g) defined by
(f, g) = h(I+L)−s/2f0,(I+L)s/2g0i=hf0, g0i
allows us to identify the dual of Ws,2(Hn) with W−s,2(Hn).This is also clear from the
identification of Ws,2(Hn) with Ws,2(b
Hn).Thus for every g∈W−s,2(Hn) there is a linear
functional Λg:Ws,2(Hn)→Cgiven by Λg(f) = hf0, g0i.
The following observation is also very useful in applications. For s > 0 every member
f∈Ws,2(Hn) defines a distribution on Hn.The same is true for every g∈W−s,2(Hn) as
well. To see this, consider the map taking f∈C∞
c(Hn) into the duality bracket (f, g) which
satisfies
|(f, g)| ≤ kfk(s)kgk(−s)≤ kgk(−s)k(I+L)mfk2
where m > s/2 is an integer. From the above it is clear that Λg(f) = (f, g) is a distribution.
If g∈W−s,2(Hn) is such a distribution, it is possible to define its Fourier transform as
an unbounded operator valued function on R∗.Indeed, let g0∈L2(Hn) be such that g=
(I+L)s/2g0then we define bg(λ) = bg0(λ)(1 + H(λ))s/2which is a densely defined operator
whose action on Φλ
αis given by
bg(λ)Φλ
α= (1 + (2|α|+n)|λ|)s/2bg0(λ)Φλ
α.

10 BAGCHI, GANGULY, SARKAR AND THANGAVELU
Thus we see that when g∈W−s,2(Hn) we have
Z∞
−∞ X
α∈NnX
β∈Nn
(1 + (2|α|+n)|λ|)−s|hbg(λ)Φλ
α,Φλ
βi|2dµ(λ) = ZHn|g0(z, t)|2dzdt < ∞.(2.3)
Remark 2.1.When g∈W−s,2(Hn) is a compactly supported distribution, then we already
have a definition of bg(λ) given by hbg(λ)Φλ
α,Φλ
βi= (g, eλ
α,β),the action of gon the smooth
function eλ
α,β (z, t).The two definitions agree as eλ
α,β are eigenfunctions of Lwith eigenvalues
(2|α|+n)|λ|.
3. Chernoff’s theorem on the Heisenberg group
In this section we prove Theorem 1.5 for the sublaplacian on the Heisenberg group. For the
proof we need to recall some properties of the so called Stieltjes vectors for the sublaplacian.
Let Xbe a Banach space and A, a linear operator on Xwith domain D(A)⊂X. A vector
x∈Xis called a C∞- vector or smooth vector for Aif x∈ ∩∞
n=1D(An).AC∞- vector xis
said to be a Stieltjes vector for Aif P∞
n=1 kAnxk−1
2n=∞.These vectors were first introduced
by Nussbaum [19] and independently by Masson and Mc Clary [17]. We denote the set of
all Stieltjes vector for Aby DSt(A).The following theorem summarises the interconnection
between the theory of Stieltjes vectors and the essential self adjointness of certain class of
operators.
Theorem 3.1. Let Abe a semibounded symmetric operator on a Hilbert space H. Assume
that the set DSt (A)has a dense span. Then Ais essentially self adjoint.
A very nice simplified proof this theorem can be found in Simon [26]. In 1975, P.R.Chernoff
used this result to prove an L2-version of the classical Denjoy-Carleman theorem regarding
quasi-analytic functions on Rn.
The above theorem talks about essential self adjointness of operators. Let us quickly
recall some relevant definitions from operator theory. By an operator Aon a Hilbert space
Hwe mean a linear mapping whose domain D(A) is a subspace of Hand whose range
Ran(A)⊂ H. We say that an operator Sis an extension of Aif D(A)⊂D(S) and
Sx =Ax for all x∈D(A). An operator Ais called closed if the graph of Adefined by
G={(x, Ax) : x∈D(A)}is a closed subset of H×H.We say that an operator Ais closable
if it has a closed extension. Every closable operator has a smallest closed extension, called
its closure, which we denote by ¯
A. An operator Ais said to be densely defined if D(A) is
dense in Hand it is called symmetric if hAx, yi=hx, Ayifor all x, y ∈D(A). A densely
defined symmetric operator Ais called essentially self adjoint if its closure ¯
Ais self adjoint.
It is easy to see that an operator Ais essentially self adjoint if and only if Ahas unique

CHERNOFF AND INGHAM ON THE HEISENBERG GROUP 11
self adjoint extension. The following is a very important characterization of essentially self
adjoint operators.
Theorem 3.2. ([22]) Let Abe a positive, densely defined symmetric operator. The followings
are equivalent: (i) Ais essentially self adjoint (ii) Ker(A∗+I) = {0}and (iii) Ran(A+I)
is dense in H.
We apply the above theorem to study the essential self adjointness of Lconsidered on
a domain inside the Sobolev space Ws,2(Hn), s > 0.Let Astand for the sublaplacian L
restricted to the domain D(A) consisting of all smooth functions fsuch that for all α, β ∈
Nn, j ∈Nthe derivatives XαYβTjfare in L2(Hn) and vanish at the origin. Since Xj, Yj
agree with ∂
∂xj,∂
∂yjat the origin, we can also define D(A) in terms of ordinary derivatives
∂α
x∂β
y∂j
t.
Proposition 3.3. Let Aand D(A)be defined as above where (n−1) < s ≤(n+ 1).Then
Ais not essentially self adjoint.
Proof. In view of Theorem 3.2 it is enough to show that for sas in the statement of the
proposition, D(A) is dense in Ws,2(Hn) but (I+A)D(A) is not. These are proved in the
following lemmas. 
Lemma 3.4. D(A)is dense in Ws,2(Hn)for any 0≤s≤(n+ 1).
Proof. If we let Ω = Hn\ {0}so that C∞
c(Ω) ⊂D(A),it is enough to show that the smaller
set is dense in Ws,2(Hn).This will follow if we can show that the only linear functional that
annihilates C∞
c(Ω) is the zero functional (see chapter 3 of [24]). Let Λ ∈(Ws,2(Hn))′, the
dual of Ws,2(Hn), be such that Λ(C∞
c(Ω)) = 0.Then there exists g∈W−s,2(Hn) such that
Λ = Λgand hence Λg(φ) = 0 for any φ∈C∞
c(Ω) Notice that for φ∈C∞
c(Hn) the linear map
φ֌Λg(φ) defines a distribution. Indeed, the estimate
|Λg(φ)| ≤ kgk(−s)kφk(s)≤ kgk(−s)k(I+L)mφk2
for any integer m > s/2 shows that it is indeed a distribution. As it vanishes on Ω it is
supported at the origin. The structure theory of such distributions allow us to conclude that
Λgis a finite linear combination of derivatives of Dirac δat the origin, Λg=P|a|≤Nca∂aδ,
see e.g Chapter 6 of [24].
Since Xaδ=∂a
xδ, and Ybδ=∂b
yδin the above representation we can also use XaYbTj.
It is even more convenient to write them in terms of the complex vector fields defined by
Zj=1
2(Xj−iYj),Zj=1
2(Xj+iYj).Thus we have g=P|a|+|b|+2j≤Nca,b,j ZaZbTjδ. If
g0∈L2(Hn) is such that (I+L)−s/2g=g0then by (2.3) we have
Z∞
−∞ X
α∈NnX
β∈Nn
(1 + (2|α|+n)|λ|)−s|hbg(λ)Φλ
α,Φλ
βi|2dµ(λ)<∞.

12 BAGCHI, GANGULY, SARKAR AND THANGAVELU
Since gis compactly supported we can calculate the Fourier transform of gas in Remark
2.1. In view of the relations 2.2 we have
hbg(λ)Φλ
α,Φλ
βi=X
|a|+|b|+2j≤N
ca,b,jλjhA(λ)a(A(λ)∗)bΦλ
α,Φλ
βi
By defining m(λ, α, β) to be the expression on the right hand side of the above equation we
see that Z∞
−∞ X
α∈NnX
β∈Nn
(1 + (2|α|+n)|λ|)−s|m(λ, α, β)|2|λ|ndλ < ∞.(3.1)
The action of A(λ)aand (A(λ)∗)bon Φλ
αare explicitly known, see ([31]). It is therefore easy
to see that
m((2|α|+n)−1λ, α, β) = X
|a|+|b|+2j≤N
Ca,b,j(α, β)λj+(|a|+|b|)/2
where the coefficients Ca,b,j(α, β) are uniformly bounded in both variables. We also remark
that for a given αthe function Ca,b,j (α, β ) is non zero only for a single value of β. By making
a change of variables in (3.1) we see that
X
α∈NnX
β∈Nn
(2|α|+n)−n−1Z∞
−∞ X
|a|+|b|+2j≤N
Ca,b,j(α, β)λj+(|a|+|b|)/22|λ|n
(1 + |λ|)sdλ < ∞.
As we are assuming that 0 ≤s≤(n+ 1) the above integral cannot be finite unless all the
coefficients ca,b,j = 0.Hence g= 0 proving the density of D(A).
Lemma 3.5. For any s > (n−1),(I+A)D(A)is not dense in Ws,2(Hn).
Proof. For any f∈D(A) the inversion formula for the Fourier transform on Hnshows that
Z∞
−∞
tr(b
f(λ))dµ(λ) = f(0) = 0.
Let gbe the functions defined by bg(λ) = (1 + H(λ))−s−1we can rewrite the above as
h(I+L)f, gis=Z∞
−∞
tr(b
f(λ))dµ(λ) = 0.
So all we need to do is to check g∈Ws,2(Hn),or equivalently
Z∞
−∞ ∞
X
k=0
(1 + (2k+n)|λ|)−s−2kPk(λ)k2
HS |λ|ndλ < ∞.
It is known that kPk(λ)k2
HS =(k+n−1)!
k!(n−1)! ≤C(2k+n)n−1and so by making a change of variables
the above integral is bounded by
∞
X
k=0
(2k+n)−2Z∞
−∞
(1 + |λ|)−s−2|λ|ndλ.
As we assume that s > (n−1) the integral is finite which proves that g∈Ws,2(Hn).Hence
the lemma. 

CHERNOFF AND INGHAM ON THE HEISENBERG GROUP 13
We now proceed to investigate some properties of the set DSt(A) of Stieltjes vectors for
the operator A. The following lemma about series of real numbers will be helpful in proving
some properties of Stieltjes vectors for the sublaplacian (see lemma 3.2 of [7]).
Lemma 3.6. If {Mn}nis sequence of non-negetive real numbers such that P∞
n=1 M−1
n
n=∞
and 0≤Kn≤aMn+bn, then P∞
n=1 K−1
n
n=∞.
For r > 0 the non-isotropic dilation of fis defined by by δrf(z, t) = f(rz, r2t) and for
σ∈U(n) we define the rotation Rσf(z, t) = f(σz, t) for all (z, t)∈Hn.
Lemma 3.7. Suppose f∈D(A)satisfies the condition P∞
m=0 kLmfk−1
2m
2=∞.Then f∈
DSt(A).Moreover, δrf, Rσfare also Stieltjes vectors for A.
Proof. We first recall that L ◦ δr=r2δr◦ L and L ◦ Rσ=Rσ◦ L,see e.g [30]. Therefore, it
follows that if a function satisfies P∞
m=0 kLmfk−1
2m
2=∞then the same is true of δrfand
Rσf. So we only need to prove our claim for f; i.e., when fsatisfies the above condition then
we also have P∞
m=0 kLmfk−1
2m
(s)=∞.To see this, we use
hLmf, Lmfi(s)=hL2mf, (1 + L)sfi ≤ k(I+L)sfk2kL2mfk2.
Thus we have kLmfk−1
2m
(s)≥C−1
4mkL2mfk−1
4m
2where C=k(I+L)sfk2.In view of Lemma
3.6 it is enough to prove the divergence of P∞
m=0 kL2mfk−1
4m
2.Without loss of generality we
can assume that kfk2= 1.But then kLmfk−1
2mis a decreasing function of m, see Lemma
2.1 in [7]. Consequently, the required divergence follows from the assumption on f. 
Before stating the next lemma, let us recall some properties of the matrix coefficients
eλ
α,β(z, t) = hπλ(z, t)Φλ
α,Φλ
β) of the Schr¨odinger representations. These are eigenfunctions of
the sublaplacian with eigenvalues (2|α|+n)|λ|.Moreover, they satisfy
Zjeλ
α,β =i(2αj+ 2)1
2|λ|1
2eλ
α+ej,β, Zjeλ
α,β =i(2αj)1
2|λ|1
2eλ
α−ej,β (3.2)
where ejare the coordinate vectors in Cn.We also recall that the sublaplacian is expressed
as L=−1
2Pn
j=1(ZjZj+ZjZj) in terms of Zjand Zj.
Lemma 3.8. If fsatisfies the hypothesis in Lemma 3.7 , then eλ
α,βf∈DSt (A),for any
α, β ∈Nnand λ∈R∗.
Proof. As noted in the previous lemma, it suffices to show that P∞
m=1 kLm(eλ
α,βf)k−1
2m
2=∞.
Since L=−1
2Pn
j=1(ZjZj+ZjZj) in terms of Zj, a simple calculation shows that
L(fg) = (Lf)g+f(Lg)−1
2
n
X
j=1 Zjf¯
Zjg+¯
ZjfZjg.(3.3)

14 BAGCHI, GANGULY, SARKAR AND THANGAVELU
By taking g=eλ
α,β and making use of (3.2) along with the estimate keλ
α,β k∞≤1 we infer
that kL(feλ
α,β)k2is bounded by
kLfk2+ (2|α|+n)|λ|kfk2+1
√2
n
X
j=1 q(αj+ 1)|λ|kZjfk2+qαj|λ|k¯
Zjfk2.
As the operators ZjL−1/2and ¯
ZjL−1/2are bounded on L2(Hn) with norms at most √2, we
see that the third term above can be estimated as
n
X
j=1 q(αj+ 1)|λ|+qαj|λ|kL1/2fk2≤2
n
X
j=1 (2αj+ 1)|λ|kL1/2fk2.
Finally using the fact that kL1/2(1 + L)−1fk2≤ kfk2we get the estimate
kL(feλ
α,β)k2≤(2|α|+n)|λ|(2kLfk2+ 3kfk2) + kLfk2.
By defining aλ(α) = (2|α|+n)|λ|), bλ(α) = (2|α|+n+ 1)|λ|) and cλ(α) = 3bλ(α) + 1,we
rewrite the above as
kL(feλ
α,β)k2≤cλ(α)kLfk2+kfk2.
In order to prove the lemma it is enough to show for any non-negative integer mthe
following estimate holds:
kLm(feλ
α,β)k2≤2m−1cλ(α)mkLmfk2+kfk2).(3.4)
We prove this by induction. Assuming the result for any m, we write Lm+1(fg) = LmL(f g)
and make use of (3.3) with g=eλ
α,β.The first two terms Lm(Lfg) and Lm(fLg) together
give the estimate
2m−1cλ(α)mkLm+1fk2+kLfk2+aλ(α)2m−1cλ(α)m(kLmfk2+kfk2.
The boundedness of L(1 + Lm+1)−1and Lm(1 + Lm+1)−1allows us to bound the above by
2mcλ(α)m(1 + bλ(α))kLm+1fk2+kfk2.(3.5)
We now turn our attention to the estimation of the term
1
√2
n
X
j=1 q(αj+ 1)|λ|Lm(Zjfeλ
α+ej,β) + qαj|λ|Lm(eλ
α−ej,β ¯
Zjf).
By using the induction hypothesis along with the fact that the operators LmZj(1 + Lm+1 )−1
and Lm¯
Zj(1 + Lm+1)−1are bounded with norm at most √2 the L2norm of the above is
bounded by
2m−1
n
X
j=1 cλ(α+ej)mq(αj+ 1)|λ|+cλ(α−ej)mqαj|λ|kLm+1fk2+kfk2.

CHERNOFF AND INGHAM ON THE HEISENBERG GROUP 15
Since bλ(α+ej)≤2bλ(α),we have cλ(α+ej)≤2cλ(α),and so the above sum is bounded by
2aλ(α)2mcλ(α)mkLm+1fk2+kfk2.(3.6)
Combining (3.5) and (3.6), using aλ(α)≤bλ(α) and recalling the definition of cλwe obtain
(3.4), proving the lemma. 
Proposition 3.9. Let Abe as in Proposition 3.3 where we have assumed that (n−1) <
s≤(n+ 1).Assume that DSt (A)contains a subset Vwhich has the following properties:
(i) every element of Vsatisfies the hypothesis of Lemma 3.7 (ii) for every (z, t)∈Hnthere
exists f∈Vsuch that f(z, t)6= 0.Then the linear span of DS t(A)is dense in Ws,2(Hn).
Proof. We first observe that if f∈Vthen δrfand Rσfare also in V. We will show that the
closed linear span of DSt(A) equals Ws,2(Hn).To prove this, let us take g∈Ws,2(Hn) which
is orthogonal to DSt(A).By Lemma 3.8 we know that f eλ
α,β ∈DSt(A) for all α, β ∈Nnand
λ∈R∗.Thus,
h(I+L)sg, eλ
αβfiL2=hg, eλ
αβfi(s)= 0.
By defining p(z, t) = f(z, t)(I+L)sg(z, t), the above translates into
hbp(λ)Φλ
α,Φλ
βi=ZHn
p(z, t)(πλ(z, t)Φλ
α,Φλ
β)dzdt = 0.
By the inversion formula for the Fourier transform on Hnwe conclude that p= 0 which
means (1 + L)sgvanishes on the support of f. Under the assumption on Vit follows that
(1 + L)sg(z, t) = 0 for every (z, t)∈Hnfrom which we can conclude that g= 0 as the
operator (1 + L)sis invertible. This proves the density. 
Finally, we are in a position to prove the analogue of Chernoff’s theorem for the sublapla-
cian on the Heisenberg group.
Proof of Theorem 1.5 Consider the operator Adefined in Proposition 3.3. We have
already shown that it is not essentially self adjoint. Suppose there exists a nontrivial f
satisfying the hypothesis of Theorem 1.5. Then by Lemma 3.7 we know that falong with
δrfand Rσfbelong to DSt(A).We take V={Rσ(δrf) : σ∈U(n), r > 0}.It is clear that
Vis invariant under Rσand δr.If we can show that Valso satisfies the condition (ii) in
Proposition 3.9 we know that the linear span of DS t is dense in Ws,2(Hn).By Theorem 3.1
this allows us to conclude that Ais essentially self-adjoint. As this is not the case, fhas to
be trivial which proves the theorem.
Thus it remains to prove the claim. Let (w, s) = (ρ′√sin θ′ω′,(ρ′)2cos θ′) be such that
f(w, s)6= 0.For any (z, t)∈Hnwrite (z, t) = (ρ√sin θ ω, ρ2cos θ) and choose σ∈U(n) such
that σ·ω=ω′.Then it is clear that
Rσ(δρ′/ρf)(z, t) = f(ρ′√sin θ ω′,(ρ′)2cos θ)) = f(ρ′√sin θ′ω′,(ρ′)2cos θ′).

16 BAGCHI, GANGULY, SARKAR AND THANGAVELU
The extra assumption on fmeans that Rσ(δρ′/ρf)(z, t) = f(w, s)6= 0.Hence the claim.
4. Ingham’s theorem on the Heisenberg group
In this section we prove Theorem 1.3 using Chernoff’s theorem for the sublaplacian. We
first show the existence of a compactly supported function fon Hnwhose 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)∈Hnis 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 τjon Rby
fj(z) := ρ−2n
jχBCn(0,aρj)(z), gj(t) := τ−2
jχ[−τ2
j/2,τ2
j/2](t)
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, easy to prove, 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∈Hnand N∈N,F2∗F3..... ∗FN(x)≤ρ−2n
2τ−2
2.

CHERNOFF AND INGHAM 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
intuitive surface area. For more about this see [27, Chapter 7 ] . Let A∆Bstand for the
symmetric difference between any two sets Aand B. See [25] 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 F∈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, for any x∈Hnwe have
Gk+1(x)−Gk(x) = ZHn
Gk(xy−1)Fk+1(y)dy −Gk(x)(x)ZHn
Fk+1(y)dy
=ZHnGk(xy−1)−Gk(x)Fk+1(y)dy.
As Fjare 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...... ∗Fkso that Gk=F1∗Hk−1,we have
Gk(xy)−Gk(y) = ZHnF1(xyu−1)−F1(xu−1)Hk−1(u)du
Using the estimate in Lemma 4.1 (4) we now estimate
|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 above
equation 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 to the integral under consideration
we first estimate the τintegral 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(z, w, ζ)| ≤ (2|s|+|z||w|+aρ1|w|).
Thus we have proved the estimate
ZHn
f1(ζ)|g1(τ+b)−g1(τ)|dζdτ ≤C2|s|+ (aρ1+|z|)|w|(4.4)
As g1integrates to 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)

CHERNOFF AND INGHAM 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 Gkis 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
Gkconverges to Fin L1.As kGkk1= 1 for any k, it follows that kF||1= 1 and hence Fis
nontrivial. 
4.2. Estimating the Fourier transform of F.Suppose now that Θ is an even, decreas-
ing function on Rfor which R∞
1Θ(t)t−1dt < ∞.We want to choose two sequences ρjand
τjin terms of Θ so that the series P∞
j=1 ρjand P∞
j=1 τjboth converge. We can then con-
struct 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 ρjand 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 ρjsuch that ρj≥c2
ne2Θ(j)
j(for a constant cnto be
chosen later) and P∞
j=1 ρj<∞.Similarly, we choose another decreasing sequence τjsuch
that P∞
j=1 τj<∞.
In the proof of the following theorem 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 [31]. 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)

CHERNOFF AND INGHAM 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)
jwhere 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 ρjis 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 ρjit 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 prove Theorem 1.3. Since half of the theorem has
been already proved, we only need to prove the following.
Theorem 4.6. Let Θ : R→[0,∞)be an even, decreasing function with lim|λ|→∞ Θ(λ) = 0
and I=R∞
1Θ(λ)λ−1dλ =∞.Suppose the Fourier transform of f∈L1(Hn)satisfies
ˆ
f(λ)∗ˆ
f(λ)≤e−Θ(√H(λ))√H(λ), λ 6= 0.
Further assume that f(ρ ω√sin θ, ρ2cos θ) = f(ρ ω, 0) in the Heisenberg coordinates. If f
vanishes on a non-empty open set containing 0then f= 0 a.e.
Proof. Without loss of generality we can assume that fvanishes on B(0, δ).First we assume
that Θ(λ)≥2|λ|−1
2,|λ| ≥ 1.In view of Plancherel theorem for the group Fourier transform
on the Heisenberg group we have
kLmfk2
2= (2π)−(n+1) Z∞
−∞ kˆ
f(λ)H(λ)mk2
HS |λ|ndλ.

22 BAGCHI, GANGULY, SARKAR AND THANGAVELU
Using the formula for Hilbert-Schmidt norm of an operator we have
kLmfk2
2= (2π)−(n+1) Z∞
−∞ X
α
((2|α|+n)|λ|)2mkˆ
f(λ)Φλ
αk2
2|λ|ndλ
Now the given condition on the Fourier transform leads to the estimate
kLmfk2
2≤CZ∞
−∞ X
α
((2|α|+n)|λ|)2me−Θ(((2|α|+n)|λ|)1
2)((2|α|+n)|λ|)1
2|λ|ndλ
≤C∞
X
k=0
(2k+n)n−1Z∞
−∞
((2k+n)|λ|)2me−Θ(((2k+n)|λ|)1
2)((2k+n)|λ|)1
2|λ|ndλ
Now changing the variable from λto (2k+n)−1λwe get
kLmfk2
2≤C∞
X
k=0
(2k+n)−2Z∞
0
λ2m+ne−Θ(√λ)√λdλ.
The integral Iappearing above can be estimated as follows. Under the extra assumption
Θ(λ)≥2|λ|−1
2, on Θ we have
I=Zm8
0
λ2m+ne−Θ(√λ)√λdλ +Z∞
m8
λ2m+ne−Θ(√λ)√λdλ
≤2m8(n+1) Zm4
0
λ4m−1e−Θ(m4)λdλ + 4 Z∞
m2
λ8m+4(n+1)−1e−2λdλ.
The above is dominated by a sum of two gamma integrals which can be evaluated to get
I≤2m8(n+1)Γ(4m)Θ(m4)−4m+ 4e−m2Γ(8m+ 4(n+ 1)).
Using Stirling’s formula (see Ahlfors [1]) Γ(x) = √2πxx−1/2e−xeθ(x)/12x,0< θ(x)<1 valid
for x > 0,we observe the the second term in the estimate for Igoes to zero as mtends to
infinity and the first term (and hence Iitself ) is bounded by C(4m)4mΘ(m4)−4m.
Thus we have proved the estimate kLmfk2
2≤C(4m)4mΘ(m4)−4m.The hypothesis on Θ
namely, R∞
1
Θ(t)
tdt =∞,by a change of variable implies that R∞
1
Θ(y4)
ydy =∞.Hence by
integral test we get P∞
m=1
Θ(m4)
m=∞.Therefore, it follows that P∞
m=1 kLmfk−1
2m
2=∞.
Since it vanishes on B(0, δ), fand all its partial derivatives vanish at the origin. Therefore,
by Chernoff’s theorem for the sublaplacian we conclude that 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 radial function F∈L2(Hn) supported in B(0, δ/2) such that
ˆ
F(λ)∗ˆ
F(λ)≤e−Ψ(√H(λ))√H(λ), λ 6= 0.

CHERNOFF AND INGHAM ON THE HEISENBERG GROUP 23
As fis assumed to vanish on B(0, δ),the function h=f∗Fvanishes on the smaller ball
B(0, δ/2).This can be easily verified by looking at
f∗F(x) = ZHn
f(xy−1)F(y)dy =ZB(0,δ
2)
f(xy−1)F(y)dy.
When both x, y ∈B(0, δ/2), d(0, xy−1) = |xy−1| ≤ |x|+|y|< δ and hence f(xy−1) = 0
proving that f∗F(x) = 0.The same is true for all the derivatives of h. We now claim that
b
h(λ)∗b
h(λ)≤e−2Φ(√H(λ))√H(λ)
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.We also note that Φ(y)≥ |y|−1/2.However,
we cannot appeal to Theorem 1.5 since h=f∗Fneed not satisfy the extra assumption
h(ρ ω√sin θ, ρ2cos θ) = h(ρ ω, 0).Nevertheless, we can modify the proof of Proposition 3.9
to complete the proof of Ingham’s theorem.
Given Fas above, let us consider δrF(z, t) = F(rz, r2t).It has been shown elsewhere (see
e.g. [13]) 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)δ−1
εF(x) then it follows that Fεis an ap-
proximate identity. Moreover, Fεis supported in B(0, εδ/2) and satisfies the same hypothesis
as Fwith Ψ(y) replaced by εΨ(εy) which has the same integrability and decay conditions.

24 BAGCHI, GANGULY, SARKAR AND THANGAVELU
Returning to the proof of Proposition 3.9 we only need to produce a subset V⊂DSt(A)
satisfying the conditions (i) and (ii).Then the linear span of DSt(A) will be dense contra-
dicting the already proved result that Ais not essentially self-adjoint. Suppose the function
fgiven in Theorem 4.6 is non trivial. We consider the set
V={Rσ◦δr(f∗Fε) : r, ε > 0, σ ∈U(n)}.
As Fεsatisfies the same hypothesis as Fwith Ψ replaced by εΨ(εy) it follows that f∗Fε
satisfies the hypothesis of the theorem with Θ(y) replaced by Θ(y)+εΨ(εy). By the previous
part of the theorem it follows that P∞
m=1 kLm(f∗Fε)k−1
2m
2=∞.Hence for any 0 < ε <
1, f ∗Fε∈DS t(A) and consequently V⊂DSt(A).Thus Vsatisfies the condition (i).
To verify the condition (ii) we observe that Rσ(f∗g) = Rσf∗gfor any radial gand
δr(f∗g) = r2n+2 (δrf∗δrg). Consequently, as Fis radial it follows that Rσ◦δr(f∗Fε) =
(Rσ◦δrf)∗Fε/r.Since f(ρ ω√sin θ, ρ2cos θ) = f(ρ ω, 0),we can proceed as in the proof of
Theorem 1.5 to show that for any (z, t) we can find rand σsuch that Rσ◦δrf(z, t)6= 0.As f
is smooth and Fεis an approximate identity, (Rσ◦δrf)∗Fε/r(z, t) converges to (Rσ◦δrf)(z, t)
as ε→0.Hence for all small enough εwe have (Rσ◦δrf)∗Fε/r(z, t)6= 0.This proves our
claim that Vsatisfies condition (ii) and completes the proof. 
Acknowledgments
The authors would like to thank Prof. Swagato K. Ray for his suggestions and discussions.
The first author is supported by Inspire faculty award from D. S. T, Govt. of India. The
second author is supported by Int. Ph.D. scholarship from Indian Institute of Science. The
third author is supported by a research fellowship from Indian Statistical Institute. The last
author is supported by J. C. Bose Fellowship from D.S.T., Govt. of India as well as a grant
from U.G.C.
References
[1] Lars V. Ahlfors, Complex Analysis: An introduction to the theory of analytic functions of one complex
variable, 3rd Edition, McGraw-Hill, Inc.
[2] M. Bhowmik, S. K. Ray, and S.Sen, Around theorems of Ingham-type regarding decay of Fourier trans-
form on Rn,Tnand two step nilpotent Lie Groups, Bull. Sci. Math,155 (2019) 33-73.
[3] M. Bhowmik, S. Pusti, and S. K. Ray, Theorems of Ingham and Chernoff on Riemannian symmetric
spaces of noncompact type, Journal of Functional Analysis, Volume 279, Issue 11 (2020).
[4] M. Bhowmik and S. Sen, Uncertainty principles of Ingham and Paley-wiener on semisimple Lie groups,
Israel Journal of Mathematics 225 (2018),193-221.
[5] S. Bochner and A. E. Taylor, Some theorems on quasi-analyticity for functions of several variables,
Amer. J. Math.,61 (1939), no-2,303-329.

CHERNOFF AND INGHAM ON THE HEISENBERG GROUP 25
[6] S. Bochner, Quasi-analytic functions, Laplace operator, positive kernels Ann. of Math. (2),51 (1950),
68-91. MR0032708 (11,334g)
[7] P. R. Chernoff, Some remarks on quasi-analytic vectors, Trans. Amer. Math. Soc. 167 (1972), 105-113.
[8] P. R.Chernoff, Quasi-analytic vectors and quasi-analytic functions. Bull. Amer. Math. Soc. 81 (1975),
637-646.
[9] G. B. Folland, Harmonic Analysis in Phase Space, Ann. Math. Stud. 122. Princeton University Press,
Princeton, N.J., 1989.
[10] A. E. Ingham, A Note on Fourier Transforms, J. London Math. Soc. S1-9 (1934), no. 1, 29-32.
MR1574706
[11] B. Kr¨otz, G. Olafasson and R. J. Stanton, The image of heat kernel transform on Riemannian symmetric
spaces of the noncompact type, Int.Math.Res.Not. (2005), no.22, 1307-1329.
[12] B. Kr¨otz, S. Thangavelu and Y.Xu, Heat kernel transform for nilmanifolds associated to the Heisenberg
group, Rev.Mat.Iberoam. 24 (2008), no.1, 243-266.
[13] R. Lakshmi Lavanya and S. Thangavelu, Revisiting the Fourier transform on the Heisenberg group,
Publ. Mat. 58 (2014), 47-63.
[14] N. Levinson, On a Class of Non-Vanishing Functions, Proc. London Math. Soc. 41 (1)(1936) 393-407.
MR1576177
[15] C. Markett, Mean Ces`aro summability of Laguerre expansions and norm estimates with shifted param-
eter, Anal. Math. 8(1982), no. 1, 19–37.
[16] B. Muckenhoupt, Mean convergence of Hermite and Laguerre series. II Trans. Amer. Math. Soc. 147
(1970), 433–460
[17] D. Masson and W. McClary, Classes of C∞vectors and essential self-adjointness, J. Functional Analysis,
10 (1972), 19-32.
[18] A. E. Nussbaum, Quasi-analytic vectors, Ark. Mat. 6(1965), 179-191. MR 33 No.3105.
[19] A. E. Nussbaum, A note on quasi-analytic vectors, Studia Math. 33 (1969), 305-309.
[20] R. E. A. C. Paley and N. Wiener, Notes on the theory and application of Fourier transforms. I, II.
Trans. Amer. Math. Soc. 35 (1933), no. 2, 348-355.
[21] R. E. A. C. Paley and N. Wiener, Fourier transforms in the complex domain (Reprint of the 1934
original) American Mathematical Society Colloquium Publications, 19. American Mathematical Society,
Providence, RI, 1987.
[22] M. Reed and B. Simon, Methods of modern mathematical physics I: Functional Analysis, Academic
Press, INC. (London) LTD.
[23] L. Roncal and S. Thangavelu, An extension problem and trace Hardy inequality for the sublaplacian
on the H-type gropus, Int.Math.Res.Not. (2020) no.14, 4238-4294.
[24] W. Rudin, Functional Analysis, 2nd edition, McGraw-Hill, Inc.
[25] D. Schymura, An upper bound on the volume of the symmetric difference of a body and a congruent
copy, Adv.Geom. 14 (2014) no. 2, 287-298.
[26] B. Simon, The theory of semi-analytic vectors: A new proof of a theorem of Masson and McClary,
Indiana Univ. Math. J. 20 (1970/71), 1145-1151. MR 44 No.7357.
[27] E. M. Stein and R. Shakarchi, Real Analysis: Measure Thery, Integration and Hilbert spaces, Princeton
Lectures in Analysis III, Princeton University Press.
[28] M. E. Taylor, Non-commutative harmonic analysis, Amer. Math.Soc., Providence, RI, (1986).

26 BAGCHI, GANGULY, SARKAR AND THANGAVELU
[29] S. Thangavelu, Lectures on Hermite and Laguerre expansions, Mathematical Notes 42. Princeton Uni-
versity Press, Princeton, NJ, 1993.
[30] S. Thangavelu, Harmonic Analysis on the Heisenberg group, Progress in Mathematics 159. Birkh¨auser,
Boston, MA, 1998.
[31] S. Thangavelu, An introduction to the uncertainty principle. Hardy’s theorem on Lie groups. With a
foreword by Gerald B. Folland, Progress in Mathematics 217. Birkh¨auser, Boston, MA, 2004
[32] S. Thangavelu, Gutzmer’s formula and Poisson integrals on the Heisenberg group, Pacific J.Math. 231
(2007), no.1, 217-237.
[33] S. Thangavelu, An analogue of Pfannschmidt’s theorem for the Heisenberg group, The Journal of
Analysis,26 (2018) 235-244.
(S. Bagchi) Department of Mathematics and Statistics, Indian Institute of Science Educa-
tion and Research Kolkata, Mohanpur-741246, Nadia, West Bengal, India.
Email address:sayansamrat@gmail.com
(P. Ganguly, S. Thangavelu) Department of Mathematics, Indian Institute of Science, Bangalore-
560 012, India.
Email address:pritamg@iisc.ac.in, veluma@iisc.ac.in
(J. Sarkar) Stat-Math Unit, Indian Statistical Institute, 203, B.T. Road, Kolkata-700108,
India
Email address:jayantasarkarmath@gmail.com