On a theorem of Chernoff on rank one Riemannian symmetric spaces

Abstract

In 1975, P.R. Chernoff used iterates of the Laplacian on Rn\mathbb{R}^n to prove an L2L^2 version of the Denjoy-Carleman theorem which provides a sufficient condition for a smooth function on Rn\mathbb{R}^n to be quasi-analytic. In this paper, we prove an exact analogue of Chernoff's theorem for all rank one Riemannian symmetric spaces (of noncompact and compact types) using iterates of the associated Laplace-Beltrami operators.

Full text

arXiv:2103.13888v1 [math.FA] 25 Mar 2021
ON A THEOREM OF CHERNOFF ON RANK ONE
RIEMANNIAN SYMMETRIC SPACES
PRITAM GANGULY, RAMESH MANNA AND SUNDARAM THANGAVELU
Abstract. In 1975, P.R. Chernoff used iterates of the Laplacian on Rnto prove an L2
version of the Denjoy-Carleman theorem which provides a sufficient condition for a smooth
function on Rnto be quasi-analytic. In this paper we prove an exact analogue of Chernoff’s
theorem for all rank one Riemannian symmetric spaces (of noncompact and compact types)
using iterates of the associated Laplace-Beltrami operators.
1. Introduction and the main results
The paramount property of an analytic function is that it is completely determined by
its value and the values of all its derivatives at a single point. Borel first perceived that
there is a more larger class of smooth functions than that of analytic functions which has
this magnificent property. He coined the term quasi-analytic for such class of functions. In
exact terms a subset of smooth functions on an interval (a, b) is called a quasi-analytic class
if for any function ffrom that set and x0∈(a, b), dn
dxnf(x0) = 0 for all n∈Nimplies f= 0.
Now recall that a smooth function on an interval Iis analytic provided its Taylor series
converges to the function on Iwhich naturally restricts the growth of derivatives of that
function. In fact, if for every n,kdn
dxnfkL∞(I)≤Cn!Anfor some constant Adepending on
fthen the Taylor series of fconverges to funiformly and the converse is also true. This
drives an analytic mind to investigate whether relaxing growth condition on the derivatives
generates quasi-analytic class. In 1912 Hadamard proposed the problem of finding sequence
{Mn}nof positive numbers such that the class C{Mn}of smooth functions on Isatisfying
kdn
dxnfkL∞(I)≤An
fMnfor all f∈C{Mn}is a quasi-analytic class. A solution to this problem
is provided by a theorem of Denjoy and Carleman where they showed that C{Mn}is quasi-
analytic if and only if P∞
n=1 M−1/n
n=∞. As a matter of fact Denjoy [11] first proved a
sufficient condition and later Carleman [7] completed the theorem giving a necessary and
sufficient condition. A short proof of this theorem based on complex analytic ideas can
be found in Rudin [26]. A several variable analogue of this theorem has been obtained by
Bochner and Taylor [4] in 1939.
Later in 1950, instead of using all partial derivatives, Bochner used iterates of the Laplacian
∆ and proved an analogue of Denjoy-Carleman theorem which reads as follows: if f∈
C∞(Rn) satisfies P∞
m=1 k∆mfk−1/m
∞=∞,then the condition ∆mf(x) = 0 for all m≥
0 and for all xin a set Uof analytic determination implies f= 0.Building upon the
works of Masson- McClary [24] and Nussbaum [25], in 1972 Chernoff [8] used operator
theoretic arguments to study quasi-analytic vectors. As an application he improved the
above mentioned result of Bochner by proving the following very interesting result in 1975.
2010 Mathematics Subject Classification. Primary: 43A85, 43A25 , Secondary:22E30, 33C45.
Key words and phrases. Chernoff’s theorem, Riemannian symmetric spaces, Helgason Fourier transform,
Jacobi analysis.
1

2 GANGULY, MANNA AND THANGAVELU
Theorem 1.1. [9, Chernoff] Let fbe a smooth function on Rn.Assume that ∆mf∈L2(Rn)
for all m∈Nand P∞
m=1 k∆mfk−1
2m
2=∞.If fand all its partial derivatives vanish at a
point a∈Rn, then fis identically zero.
In this paper we prove an analogue of Chernoff’s theorem for the Laplace-Beltrami operator
on rank one symmetric spaces of both compact and noncompact types. In order to state
our results we first need to introduce some notations. Let Gbe a connected, noncompact
semisimple Lie group with finite centre and Ka maximal compact subgroup of G. Let
X=G/K be the associated symmetric space which is assumed to have rank one. The origin
oin the symmetric space is given by the identity coset eK where eis the identity element
in G. We know that Xis a Riemannian manifold equipped with a Ginvariant metric on it.
We denote by ∆Xthe Laplace-Beltrami operator associated to X.
The Iwasawa decomposition of Greads as G=KAN where Ais abelian and Nis a
nilpotent Lie group. Let gand astand for the Lie algebras corresponding to Gand A
respectively. Here ais one dimensional since Xis of rank one. It is well known that every
element of ggives rise to a left invariant vector field on G. Let Hbe the left invariant
vector field corresponding to a fixed basis element of a.We will describe all these notations
in detail in the next section. As an exact analogue of Chernoff’s theorem for Xwe prove
the following:
Theorem 1.2. Let X=G/K be a rank one symmetric space of noncompact type. Suppose
f∈C∞(X)satisfies ∆m
Xf∈L2(X)for all m≥0and P∞
m=1 k∆m
Xfk−1
2m
2=∞.If Hlf(eK) =
0for all l≥0then fis identically zero.
As an immediate consequence of the above result we obtain an analogue of the L2version
of the classical Denjoy-Carleman theorem using iterates of the Laplace-Beltrami operator on
X=G/K.
Corollary 1.3. Let X=G/K be a rank one symmetric space of noncompact type. Let
{Mk}kbe a log convex sequence. Define C({Mk}k,∆X, X)to be the class of all smooth
functions fon Xsatisfying ∆m
Xf∈L2(X)for all m∈Nand k∆k
Xfk2≤Mkλ(f)kfor some
constant λ(f)depending on f. Suppose that P∞
k=1 M−1
2k
k=∞.Then every member of that
class is quasi-analytic.
As Chernoff’s theorem is a useful tool in establishing uncertainty principles of Ingham’s
type, proving analogues of Theorem 1.1 in contexts other than Euclidean spaces have received
considerable attention in recent years. Recently, an analogue of Chernoff’s theorem for the
sublaplacian on the Heisenberg group has been proved in [1]. For noncompact Riemannian
symmetric spaces X=G/K, without any restriction on the rank, the following weaker
version of Theorem 1.2 has been proved in Bhowmik-Pusti-Ray [2].
Theorem 1.4 (Bhowmik-Pusti-Ray).Let X=G/K be a noncompact Riemannian sym-
metric space and let ∆Xbe the associated Laplace-Beltrami operator. Suppose f∈C∞(X)
satisfies ∆m
Xf∈L2(X)for all m≥0and P∞
m=1 k∆m
Xfk−1
2m
2=∞.If fvanishes on a non
empty open set, then fis identically zero.
In proving the above theorem, the authors have made use of a result of de Jeu [15]. In the
case of rank one symmetric spaces, a different proof was given by the first and third authors
of this article by making use of spherical means and an analogue of Chernoff’s theorem

CHERNOFF’S THEOREM FOR SYMMETRIC SPACES 3
for the Jacobi transform proved in [13]. In fact, we only need to use the one dimensional
version of de Jeu’s theorem which is equivalent to the Denjoy-Carleman theorem. Our proof
of Theorem 1.2 is built upon the ideas used in [13]. In a very recent preprint, Bhowmik-
Pust-Ray have proved the following improvement of their Theorem 1.4. In what follows let
D(G/K) denote the algebra of differential operators on G/K which are invariant under the
(left) action of G.
Theorem 1.5 (Bhowmik-Pusti-Ray).Let X=G/K be a noncompact Riemannian symmet-
ric space and let ∆Xbe the associated Laplace-Beltrami operator. Suppose f∈C∞(G/K)
be a left K-invariant function on Xwhich satisfies ∆m
Xf∈L2(X)for all m≥0and
P∞
m=1 k∆m
Xfk−1
2m
2=∞.If there is an x0∈Xsuch that Df (x0)vanishes for all D∈D(G/K)
then fis identically zero.
Remark 1.6.Observe that in the above theorem the function fis assumed to be K-biinvariant.
The problem of proving the same for all functions on Xis still open. However, in the case
of rank one symmetric spaces we have proved Theorem 1.2 for all functions f. Moreover, we
only require that Hlf(eK) = 0 for all l≥0.Here we can also take any x0∈Xin place of
eK using translation invariance of Laplacian and H.
We remark that the condition Hlf(eK ) = 0 is the counterpart of ( d
dr )kf(rω)|r=0 = 0
where x=rω, r > 0, ω ∈Sn−1is the polar decomposition of x∈Rn.Indeed, as can be easily
checked d
dr k
f(rω) = X
|α|=k
∂αf(rω)ωα
and hence ( d
dr )kf(rω)|r=0 = 0 for all kif and only if ∂αf(0) = 0 for all α. This observation
plays an important role in formulating the right analogue Chernoff’s theorem for compact
Riemannian symmetric spaces. In view of the above observation, Chernoff’s theorem for the
Laplacian on Rncan be stated in the following form.
Theorem 1.7. Let fbe a smooth function on Rn.Assume that ∆mf∈L2(Rn)for all
m∈Nand P∞
m=1 k∆mfk−1
2m
2=∞.If (d
dr )kf(rω)|r=0 = 0 for all kand ω∈Sn−1,then fis
identically zero.
We can give a proof of the above theorem by reducing it to a theorem for Bessel operators.
Recall that written in polar coordinates the Laplacian takes the form
∆ = ∂2
∂r2+n−1
r
∂
∂r +1
r2∆Sn−1(1.1)
where ∆Sn−1is the spherical Laplacian on the unit sphere Sn−1.By expanding the function
F(r, ω) = f(rω) in terms of spherical harmonics on Sn−1and making use of Hecke-Bochner
formula, we can easily reduce Theorem 1.7 to a sequence of theorems for the Bessel operator
∂2
r+ (n+ 2m+ 1)r−1∂rfor various values of m∈N.This idea has been already used in the
paper [13]. A similar expansion in the case of noncompact Riemannian symmetric spaces
leads to Jacobi operators as done in [13] which will be used in proving Theorem 1.2. As the
proof of the above theorem is similar to and easier than that of Theorem 1.2, we will not
present it here.
Remark 1.8.We remark in passing that the above theorem can also be proved in the context
of Dunkl Laplacian on Rnassociated to root systems. We would also like to mention that

4 GANGULY, MANNA AND THANGAVELU
analogues of Chernoff’s theorem can be proved for the Hermite operator Hon Rnand the
special Hermite operator Lon Cn.Again the idea is to make use of Hecke-Bochner formula for
the Hermite and special Hermite projections (associated to their spectral decompositions).
So far we have only considered non compact Riemannian symmteric spaces, but now we
turn our attention to proving an analogue of Theorem 1.2 for compact, rank one symmetric
spaces. We make use of the well known classification of such spaces in formulating and
proving a Chernoff theorem for the Laplace-Beltrami operator. It turns out that we only
need to prove such a result for the spherical Laplacian on spheres in Euclidean spaces.
Let (U, K) be a compact symmetric pair and S=U/K be the associated symmetric space.
Here Uis a compact semisimple Lie group and Kis a connected subgroup of U. We assume
that Shas rank one. Being a compact Riemannian manifold, Sadmits a Laplace-Beltrami
operator ˜
∆S.It is customary to add a suitable constant ρSand work with ∆S=−˜
∆S+ρ2
S.
This way we can arrange that ∆S≥ρ2
S>0.In [28] H.C.Wang has completely classified all
rank one compact symmetric spaces. To be more precise, Sis one of the followings: The
unit sphere Sq=SO(q+ 1)/SO(q), the real projective space Pq(R) = SO(q+ 1)/O(q),
the complex projective space Pl(C), the quaternion projective space Pl(H) and the Cayley
projective space P2(Cay) = F4/Spin(9).In each case, Scomes up with an appropriate polar
form (0, π)×SkSwhere kSdepends on the symmetric space S. As a consequence, functions
on Scan be identified with functions on the product space Y= (0, π)×SkS,see Section 4
for more details. We prove the following analogue of Chernoff’s theorem:
Theorem 1.9. Let Sbe a rank one Riemannian symmetric space of compact type. Suppose
f∈C∞(S)satisfies ∆m
Sf∈L2(S)for all m≥0and P∞
m=1 k∆m
Sfk−1
2m
2=∞.If the function
Fon (0, π)×SkSassociated to fon Ssatisfies ∂m
∂θmθ=0F(θ, ξ) = 0 for all m≥0, then fis
identically zero.
In the context of Theorem 1.7, by identifying Rnwith (0,∞)×Sn−1every function fon
Rngives rise to a function F(r, ω) on (0,∞)×Sn−1and in view of 1.1, the action of ∆ on f
takes the form,
∆f(r, ω) = ∂2
∂r2F(r, ω) + n−1
r
∂
∂r F(r, ω) + 1
r2∆Sn−1F(r, ω).
There is a similar decomposition of ∆Sas a sum of a Jacobi operator on (0, π) and the
spherical Laplacian ∆SkSand this justifies our formulation of Theorem 1.9.
We complete this introduction with a brief description of the plan of the paper. In Section
2 we recall the requisite preliminaries on noncompact Riemannian symmetric spaces and in
Section 3 we prove our version of Chernoff’s theorem for the Laplace-Beltrami operator. In
Section 4, after recalling necessary results from the theory of compact symmetric spaces and
setting up the notations, we prove Theorem 1.9. We refer the reader to the papers [12] and
[13] for related ideas.
2. Preliminaries on Riemannian symmetric spaces of non-compact type
In this section we describe the relevant theory regrading the harmonic analysis on rank
one Riemannian symmetric spaces of noncompact type. General references for this section
are the monographs of Helgason [18] and [19].
Let Gbe a connected, noncompact semisimple Lie group with finite centre. Suppose
gdenotes its Lie algebra. With respect to a fixed Cartan involution θon gwe have the

CHERNOFF’S THEOREM FOR SYMMETRIC SPACES 5
decomposition g=k⊕p.Here kand pare the +1 and −1 eigenspaces of θrespectively. Let a
be the maximal abelian subspace of p. Also assume that the dimension of ais one. Now we
know that the involution θinduces an automorphism Θ on Gand K={g∈G: Θ(g) = g}
is a maximal compact subgroup of G. We consider the homogeneous space X=G/K which
a is a smooth manifold endowed with a G-Riemannian metric induced by the restriction of
the Killing form Bof gon p. This turns Xinto a rank one Riemannian symmetric space of
noncompact type and every such space can be realised this way.
Let a∗denote the dual of a. Given α∈a∗we define
gα:= {X∈g: [Y, X ] = α(Y)X, ∀Y∈a}.
Now Σ := {α∈a∗:gα6={0}} is the set of all resticted roots of the pair (g,a). Let
Σ+denote the set of all positive roots with respect to a fixed Weyl chamber. It is known
that n:= ⊕α∈Σ+gαis a nilpotent subalgebra of gand we have the Iwasawa decomposition
g=k⊕a⊕n.Now writing N= exp nand A= exp awe obtain G=KAN where Ais
abelian and Nis a nilpotent subgroup of G. Moreover, Anormalizes N. In view of this
decomposition every g∈Gcan be uniquely written as g=k(g) exp H(g)n(g) where H(g)
belongs to a. Also we have G=NAK and with respect to this decomposition we write
g∈Nexp A(g)Kwhere the functions Aand Hare related via A(g) = −H(g−1).Now in the
rank one case when dimension of ais one, Σ is given by either {±γ}or {±γ, ±2γ}where γ
belongs to Σ+. Let ρ:= (mγ+m2γ)/2 where mγand m2γdenote the multiplicities of the
roots γand 2γrespectively. The Haar measure dg on Gis given by
ZG
f(g)dg =ZKZAZN
f(katn)e2ρtdkdtdn.
The measure dx on Xis induced from the Haar measure dg via the relation
ZG
f(gK)dg =ZX
f(x)dx.
Suppose Mdenotes the centralizer of Ain K. The polar decomposition of Greads as
G=KAK in view of which we can write each g∈Gas g=k1ark2with k1, k2∈K.
Actually the map (k1, ar, k2)→k1ark2of K×A×Kinto Ginduces a diffeomorphism of
K/M ×A+×Konto an open dense subset of Gwhere A+= exp a+and a+is the fixed
positive Weyl chamber which basically can be identified with (0,∞) in our case.
It is also well-known that each X∈ggives rise to a left invariant vector field on Gby the
prescription
Xf(g) = d
dtt=0
f(g. exp(tX)), g ∈G.
Since ais one dimensional, we fix a basis {H}of a. By an abuse of notation, we denote
the left invariant vector field corresponding to this basis element by H. Infact, we can write
A={ar= exp(rH) : r∈R}.
2.1. Helgason Fourier transform. Define the function A:X×K/M →aby A(gK, kM ) =
A(k−1g).Note that Ais right K-invariant in gand right M-invariant in K. In what follows
we denote the elements of Xand K/M by xand brespectively. Let a∗denote the dual of a
and a∗
Cbe its complexification. Here in our case a∗and a∗
Ccan be identified with Rand C
respectively. For each λ∈a∗
Cand b∈K/M, the function x→e(iλ+ρ)A(x,b)is a joint eigen-
function of all invariant differential operators on X. For f∈C∞
c(X), its Helgason Fourier

6 GANGULY, MANNA AND THANGAVELU
transform is a function e
fon a∗
C×K/M defined by
˜
f(λ, b) = ZX
f(x)e(−iλ+ρ)A(x,b)dx, λ ∈a∗
C, b ∈K/M.
Moreover, we know that if f∈L1(X) then e
f(., b) is a continuous function on a∗which
extends holomorphically to a domain containing a∗.The inversion formula for f∈C∞
c(X)
says that
f(x) = cXZ∞
−∞ ZK/M e
f(λ, b)e(iλ+ρ)A(x,b)|c(λ)|−2dbdλ
where dλ stands for usual Lebesgue measure on R(i.e., a∗) , db is the normalised measure
on K/M and c(λ) is the Harish-Chandra c-function. The constant cXappearing in the
above formula is explicit and depends on the symmetric space X(See e.g., [19]). Also
for f∈L1(X) with e
f∈L1(a∗×K/M, |c(λ)|−2dbdλ), the above inversion formula holds
for a.e. x∈X. Furthermore, the mapping f→e
fextends as an isometry of L2(X) onto
L2(a∗
+×K/M, |c(λ)|−2dλdb) which is known as the Plancherel theorem for the Helgason
Fourier transform.
We also need to use certain irreducible representations of Kwith M-fixed vectors. Suppose
c
K0denotes the set of all irreducible unitary representations of Kwith Mfixed vectors. Let
δ∈c
K0and Vδbe the finite dimensional vectors space on which δis realised. We know
that Vδcontains a unique normalised M-fixed vector v1(See Kostant [23]). Consider an
orthonormal basis {v1, v2, ..., vdδ}for Vδ. For δ∈c
K0and 1 ≤j≤dδ, we define
Yδ,j(kM ) = (vj, δ(k)v1), kM ∈K/M.
It can be easily checked that Yδ,1(eK) = 1 and moreover, Yδ,1is M-invariant.
Proposition 2.1 ([19]).The set {Yδ,j : 1 ≤j≤dδ, δ ∈c
K0}forms an orthonormal basis for
L2(K/M).
We can get an explicit realisation of c
K0by identifying K/M with the unit sphere in p.
By letting Hmto stand for the space of homogeneous harmonic polynomials of degree m
restricted to the unit sphere, we have the following spherical harmonic decomposition
L2(K/M) = ⊕∞
m=0Hm.
Thus the functions Yδ,j can be identified with the spherical harmonics.
Given δ∈c
K0and λ∈a∗
C(i.e., Cin our case) we consider the spherical functions of type
δdefined by
Φλ,δ(x) := ZK
e(iλ+ρ)A(x,kM )Yδ,1(kM )dk.
These are eigenfunctions of the Laplace-Beltrami operator ∆Xwith eigenvalue −(λ2+ρ2).
When δis the trivial representation for which Yδ,1= 1,the function Φλ,δ is called the
elementary spherical function, denoted by Φλ. More precisely,
Φλ(x) = ZK
e(iλ+ρ)A(x,kM )dk.
Note that these functions are K-biinvariant. The spherical functions can be expressed in
terms of Jacobi functions. In fact, if x=gK and g=kark′(polar decomposition), Φλ,δ(x) =

CHERNOFF’S THEOREM FOR SYMMETRIC SPACES 7
Φλ,δ(ar). Suppose
α=1
2(mγ+m2γ−1), β =1
2(m2γ−1).(2.1)
For each δ∈c
K0there exists a pair of integers (p, q) such that
Φλ,δ(x) = Qδ(iλ +ρ)(α+ 1)−1
p(sinh r)p(cosh r)qϕ(α+p,β+q)
λ(r) (2.2)
where ϕ(α+p,β+q)
λare the Jacobi functions of type (α+p, β +q) and Qδare the Kostant
polynomials given by
Qδ(iλ +ρ) = 1
2(α+β+ 1 + iλ)(p+q)/21
2(α−β+ 1 + iλ)(p−q)/2
.(2.3)
In the above we have used the notation (z)m=z(z+ 1)(z+ 2)...(z+m−1).The following
result proved in Helgason [19] will be very useful for our purpose:
Proposition 2.2. Let δ∈c
K0and 1≤j≤dδ. Then we have
ZK
e(iλ+ρ)A(x,k′M)Yδ,j(k′M)dk′=Yδ,j (kM)Φλ,δ (ar), x =kar∈X. (2.4)
We refer the reader to the papers [16] and [17] for all the results recalled in this subsection.
2.2. Spherical Fourier transform. We say that a function fon Gis K-biinvariant if
f(k1gk2) = f(g) for all k1, k2∈K. It can be checked that if fis a K-biinvariant integrable
function then its Helgason Fourier transform e
f(λ, b) is independent of b∈K/M and by a
little abuse of notation we write this as
˜
f(λ) = ZX
f(x)Φ−λ(x)dx.
This is called the spherical Fourier transform. Now since fis Kbiinvarinat, using the polar
decomposition g=k1ark2, we can view fas a function on Aalone: f(g) = f(ar). So the
above integral takes the following polar form:
˜
f(λ) = Z∞
0
f(ar)ϕλ(r)wα,β(r)dr
where wα,β(r) = (2 sinh r)2α+1(2 cosh r)2β+1 and Φ−λ(ar) = ϕλ(r) are given by Jacobi func-
tion ϕα,β
λ(r) of type (α, β).Here αand βare associated to the symmetric space as mentioned
above. So it is clear that the spherical Fourier transform is basically Jacobi transform of
type (α, β). In the rest of the section we describe certain results from the theory of Jacobi
analysis.
Let α, β, λ ∈Cand −α /∈N.The Jacobi functions ϕ(α,β)
λ(r) of type (α, β) are solutions of
the initial value problem
(Lα,β +λ2+̺2)ϕ(α,β)
λ(r) = 0, ϕ(α,β)
λ(0) = 1
where Lα,β is the Jacobi operator defined by
Lα,β := d2
dr2+ ((2α+ 1) coth r+ (2β+ 1) tanh r)d
dr

8 GANGULY, MANNA AND THANGAVELU
and ̺=α+β+ 1.Thus Jacobi functions ϕ(α,β)
λare eigenfunctions of Lα,β with eigenvalues
−(λ2+̺2).These are even functions on Rand are expressible in terms of hypergeomet-
ric functions. For certain values of the parameters (α, β) these functions arise naturally
as spherical functions on Riemannian symmetric spaces of noncompact type. The Jacobi
transform of a suitable function fon R+is defined by
Jα,βf(λ) = Z∞
0
f(r)ϕ(α,β)
λ(r)wα,β(r)dr.
This is also called the Fourier-Jacobi transform of type (α, β).It can be checked that the
operator Lα,β is selfadjoint on L2(R+, wα,β (r)dr) and that
^
Lα,βf(λ) = −(λ2+̺2)˜
f(λ).
Under certain assumptions on αand βthe inversion and Plancherel formula for this transform
take a nice form as described below.
Theorem 2.3 ([22]).Let α, β ∈R,α > −1and |β| ≤ α+ 1.Suppose cα,β(λ)denotes the
Harish-Chandra c- function defined by
cα,β(λ) = 2̺−iλΓ(α+ 1)Γ(iλ)
Γ1
2(iλ +̺)Γ1
2(iλ +α−β+ 1)
(1) (Inversion) For f∈C∞
0(R)which is even we have
f(r) = 1
2πZ∞
0
Jα,βf(λ)ϕ(α,β)
λ(r)|cα,β(λ)|−2dλ
(2) (Plancherel) For f, g ∈C∞
0(R)which are even, the following holds
Z∞
0
f(r)g(r)wα,β(r)dr =Z∞
0
Jα,βf(λ)Jα,β g(λ)|cα,β(λ)|−2dλ.
The mapping f֌˜
fextends as an isometry from L2(R+, wα,β(r)dr)onto L2(R+,|cα,β (λ)|−2dλ).
We will make use of this theorem in proving an analogue of Chernoff’s theorem for the
Laplace-Beltrami operator ∆Xin the next section.
3. Chernoff’s theorem on noncompact symmetric spaces of rank one
In this section we prove our main theorem i.e., an analogue of Chernoff’s theorem for ∆X.
The main idea of the proof is to reduce the result for ∆Xto a result for Jacobi operator.
So, first we indicate a proof of Chernoff’s theorem for Jacobi operator. It has already been
discussed in the work of Ganguly-Thangavelu [13].
Theorem 3.1. Let α, β ∈R,α > −1and |β| ≤ α+ 1.Suppose f∈L2(R+, wα,β(r)dr)
is such that Lm
α,βf∈L2(R+, wα,β(r)dr)for all m∈Nand satisfies the Carleman condition
P∞
m=1 kLm
α,βfk−1/(2m)
2=∞.If Lm
α,βf(0) = 0 for all m≥0then fis identically zero.
In [13] the above result was proved under the assumption that fvanishes near 0 but a
close examination of the proof reveals that the assumption is superfluous and the same is
true as stated above. In order to prove our main result, the following estimate for the ratio
of Harish-Chandra c-functions is also needed.

CHERNOFF’S THEOREM FOR SYMMETRIC SPACES 9
Lemma 3.2. Let α, β be as in 2.1 and (p, q)be the pair of integers associated to δ∈c
K0.
Then for any λ≥0we have
|cα,β(λ)|2
|cα+p,β+q(λ)|2|Qδ(iλ +ρ)|−2≤C
where Cis a constant independent of λdepending only on the parameters (α, β)and (p, q).
Proof. First note that from the definition 2.3 of Kostant polynomials we have
|Qδ(iλ +ρ)|=
p+q
2
Y
j=0 (B1+j)2+1
4λ21
2
p−q
2
Y
j=0 (B2+j)2+1
4λ21
2
where B1=1
2(α+β+ 1) and B2=1
2(α−β+ 1). From the above expression, it can be easily
checked that |Qδ(iλ +ρ)|/(2−1λ)p→1 as λ→ ∞ so that
|Qδ(iλ +ρ)| ∼ 2−pλp, λ → ∞.(3.1)
Moreover, we also have
|Qδ(iλ +ρ)| ≥
p+q
2
Y
j=0
|B1+j|
p−q
2
Y
j=0
|B2+j|= constant.
Now using [6, Lemma 2.4] we have
|cα,β(λ)|2
|cα+p,β+q(λ)|2∼λ2p, λ → ∞ (3.2)
which together with 3.1 implies that
|cα,β(λ)|2
|cα+p,β+q(λ)|2|Qδ(iλ +ρ)|−2∼1, λ → ∞.
Also the ratio in 3.2 being a continuous function of λis bounded near the origin. Hence the
result follows. 
Proof of Theorem 1.2: Let fbe as in the statement of the theorem 1.2. We complete the
proof in the following steps.
Step 1: Using Proposition 2.1 we write
e
f(λ, k) = X
δ∈
c
K0
dδ
X
j=1
Fδ,j(λ)Yδ,j(k) (3.3)
where Fδ,j(λ) are the spherical harmonic coefficients of ˜
f(λ, ·) defined by
Fδ,j(λ) = ZK/M e
f(λ, k)Yδ,j (k)dk.
Fix δ∈c
K0and 1 ≤j≤dδ.From the definition of the Helgason Fourier transform we have
Fδ,j(λ) = ZK/M ZG/K
f(x)e(−iλ+ρ)A(x,kM)Yδ,j (kM )dxdk.

10 GANGULY, MANNA AND THANGAVELU
Now using Fubini’s theorem, in view of the Proposition 2.2 the integral on the right hand
side of above is equal to ZG/K
f(x)Yδ,j(kM)Φλ,δ (ar)dx. (3.4)
The function gδ,j(x) defined by
gδ,j(x) = ZK
f(k′x)Yδ,j(k′M)dk′, x ∈X
is clearly K-biinvariant, and hence by abuse of notation we write
gδ,j(r) = ZK
f(k′ar)Yδ,j(k′M)dk′.
Now performing the integral in 3.4 using polar coordinates we obtain
Fδ,j(λ) = Z∞
0
gδ,j(r)Φλ,δ(ar)wα,β (r)dr (3.5)
Now recall that for each δ∈c
K0there exist a pair of integers (p, q) such that
Φλ,δ(x) = Qδ(iλ +ρ)(α+ 1)−1
p(sinh r)p(cosh r)qϕ(α+p,β+q)
λ(r).
By defining
fδ,j(r) = 4−(p+q)
(α+ 1)p
gδ,j(r)(sinh r)−p(cosh r)−q(3.6)
and recalling the definition of Jacobi transforms we obtain
Fδ,j(λ) = Qδ(iλ +ρ)Jα+p,β+q(fδ,j )(λ) (3.7)
Step 2: In this step we estimate the L2norm of powers of Jacobi operator applied to fδ,j in
terms of the L2norm of corresponding powers of ∆Xapplied to f. Let m∈N.Note that
the Plancherel formula 2.3 for the Jacobi transform yields
kLm
α+p,β+q(fδ,j )kL2(R+,wα+p,β +q(r)dr)
=Z∞
0
(λ2+ρ2
δ)2m|Jα+p,β+q(fδ,j )(λ)|2|cα+p,β +q(λ)|−2dλ1
2
where where ρδ=α+β+p+q+ 1.In view of 3.7 the above integral reduces to
Z∞
0
(λ2+ρ2
δ)2m|Fδ,j(λ)|2|Qδ(iλ +ρ)|−2cα+p,β+q(λ)|−2dλ1
2
which after recalling the definition of Fδ,j(λ) reads as
Z∞
0
(λ2+ρ2
δ)2m|Qδ(iλ +ρ)|−2|ZKe
f(λ, k)Yδ,j (k)dk
2
|cα+p,β+q(λ)|−2dλ!1
2
.
By an application of Minkowski’s integral inequality, the above integral is dominated by
ZKZ∞
0
(λ2+ρ2
δ)2m|Qδ(iλ +ρ)|−2|e
f(λ, k)|2|cα+p,β+q(λ)|−2dλ1
2
|Yδ,j(k)|dk.

CHERNOFF’S THEOREM FOR SYMMETRIC SPACES 11
Now using Cauchy-Schwarz inequality along with the fact that kYδ,jkL2(K/M)= 1,we see
that the above integral is bounded by
ZK/M Z∞
0
(λ2+ρ2
δ)2m|Qδ(iλ +ρ)|−2|e
f(λ, k)|2|cα+p,β+q(λ)|−2dλ dk1
2
Since λ2+ρ2
δ
λ2+ρ2= 1 + ρ2
δ−ρ2
λ2+ρ2is a decreasing function of λit follows that λ2+d2
λ2+ρ2≤C(α, β) with
C(α, β) = (α+β+p+q+1)2
(α+β+1)2.This together with the Lemma 3.2 yields the following estimate for
the integral under consideration: for some constant C1=C1(α, β)
Cm
1ZK/M Z∞
0
(λ2+ρ2)2m|e
f(λ, k)|2|cα,β (λ)|−2dλ dk1
2
.
Finally, from the series of inequalities above, we obtain
kLm
α+p,β+q(fδ,j )kL2(R+,wα+p,β +q(r)dr)≤Cm
1k∆m
Xfk2.(3.8)
Hence from the hypothesis of the theorem it follows that
∞
X
m=1
kLm
α+p,β+q(fδ,j )k−1
2m
L2(R+,wα+p,β+q(r)dr)=∞.
Step 3: Finally in this step we prove that Lm
α+p,β+q(fδ,j )(0) = 0 for all m≥0.First recall
that
fδ,j(r) = 4−(p+q)
(α+ 1)p
(sinh r)−p(cosh r)−qZK
f(kar)Yδ,j (kM)dk.
As sinh rhas a zero at the origin and cosh 0 = 1, if we can show that as a function of r, the
integral RKf(kar)Yδ,j(kM )dk has a zero of infinite order at the 0,then we are done. Now
note that for any m∈N
dm
drmZK
f(kar)Yδ,j (kM )dk =ZK
dm
drmf(kar)Yδ,j(kM )dk.
But by definition of the vector fields on G, writing ar= exp(rH) we have
dm
drmf(kar)|r=0 =dm
drmf(k. exp(rH))|r=0 =Hmf(k).
Hence by the hypothesis on fwe obtain dm
drmf(kar)|r=0 = 0 for all m. Finally, proving
Lm
α+p,β+q(fδ,j )(0) = 0 is a routine matter: repeated application of L’Hospital rule gives the
desired result.
Therefore, fδ,j satisfies all the hypothesis of the Proposition 3.1 which allows us to conclude
that fδ,j = 0 i.e., Fδ,j = 0.As this is true for every δ∈c
K0and 1 ≤j≤dδwe get f= 0
completing the proof of Theorem 1.2.
4. Compact symmetric spaces
Our aim in this section is to prove an analogue of Chernoff’s theorem on compact sym-
metric spaces of rank one. To begin with, we first recall briefly some necessary background
material on rank one compact symmetric spaces. Let Sbe a compact Riemannian manifold
equipped with a Riemannian metric dS. We say that Sis a two point homogeneous space
if for any xj, yj∈S, j = 1,2 with dS(x1, x2) = dS(y1, y2), there exists g∈I(S), the group
of isometries of Ssuch that g.x1=y1, and g.x2=y2where g.x denotes the usual action of

12 GANGULY, MANNA AND THANGAVELU
I(S) on S. It is well known that compact rank one symmetric spaces are compact two point
homogeneous spaces (see Helgason[20]). Also these two point homogeneous spaces are com-
pletely classified by H-.C. Wang [28]. So, following Wang any compact rank one symmetric
space Sis one of the following:
(1) the sphere Sq⊂Rq+1, q ≥1;
(2) the real projective space Pq(R), q ≥2;
(3) the complex projective space Pl(C), l ≥2;
(4) the quaternionic projective space Pl(H), l ≥2;
(5) the Cauchy projective plane P2(Cay).
We describe the necessary preliminaries and prove Theorem 1.9 in each of the above five
cases separately. We start with a brief description of Jacobi polynomial expansions in the
following subsection.
4.1. Jacobi polynomial expansion: Let α, β > −1.The Jacobi polynomials Pα,β
nof degree
n≥0 and type (α, β) are defined by
(1 −x)α(1 + x)βPα,β
n(x) = (−1)n
2nn!
dn
dxn{(1 −x)n+α(1 + x)n+β}, x ∈(−1,1).(4.1)
By making a change of variable x= cos θ, it is convenient to work with the Jacobi trigono-
metric polynomials
P(α,β)
n(θ) = C(α, β, n)P(α,β)
n(cos θ),(4.2)
where C(α, β, n) is the normalising constant, explicitly given by
C(α, β, n)2=(2n+α+β+ 1)Γ(n+ 1)Γ(n+α+β+ 1)
Γ(n+α+ 1)Γ(n+β+ 1) .(4.3)
Also it is worth pointing out that these polynomials are closely related to Gegenbauer’s
polynomials by the following formula
Cλ
k(t) = Γ(λ+1
2)Γ(k+ 2λ)
Γ(2λ)Γ(k+λ+1
2)P(λ−1
2,λ−1
2)
k(t), λ > −1
2, t ∈(−1,1).(4.4)
These Jacobi trigonometric polynomials are the eigenfunctions of the Jacobi differential
operator given by
Lα,β =−d2
dθ2−α−β+ (α+β+ 1) cos θ
sin θ+α+β+ 1
22
with eigenvalues (n+α+β+1
2)2i.e.,
Lα,βP(α,β)
n=n+α+β+ 1
22
P(α,β)
n,
and {P(α,β)
n:n≥0}forms an orthonormal basis for the weighted L2space L2( ˜wα,β) :=
L2((0, π),˜wα,β (θ)dθ) where the weight is given by
˜wα,β(θ) = sin θ
22α+1 cos θ
22β+1
.

CHERNOFF’S THEOREM FOR SYMMETRIC SPACES 13
As a consequence we have the following Plancherel formula valid for f∈L2( ˜wα,β )
Zπ
0
|f(θ)|2˜wα,β(θ)dθ =
∞
X
n=0
|Jα,βf(n)|2(4.5)
where Jα,βf(n) denotes the Fourier-Jacobi coefficients defined by
Jα,βf(n) = Zπ
0
f(θ)P(α,β)
n(θ) ˜wα,β(θ)dθ, n ≥0.
We have the following version of Chernoff’s theorem using the iterates of the Jacobi operator
proved in Ganguly-Thangavelu [12].
Theorem 4.1. Let α, β > −1. Suppose f∈L2( ˜wα,β)is such that Lm
α,β f∈L2( ˜wα,β)for all
m∈Nand satisfies the Carleman condition P∞
m=1 kLm
α,β fk−1/(2m)
2=∞.If Lm
α,β f(0) = 0 for
all m≥0then fis identically zero.
This is the analogue of Theorem 3.1 for Jacobi polynomial expansions which plays an
important role in proving Theorem 1.9 for compact Riemannian symmetric spaces.
4.2. The unit sphere Sq.Let q≥2. The unit sphere in Rq+1 is given by
Sq:= {ξ∈Rq+1 :ξ2
1+···+ξ2
q+1 = 1}.
The spherical harmonic decomposition reads as
L2(Sq) =
∞
M
n=0
Hn(Sq)
where Hn(Sq) denotes the set of spherical harmonics of degree n. Now, for our purposes it is
more convenient to work with the geodesic polar coordinate system on Sq. Note that given
ξ∈Sq, we can write ξ= (cos θ)e1+ξ′
1(sin θ)e2+... +ξ′
q(sin θ)eq+1 for some θ∈(0, π) and
ξ′= (ξ′
1, ..., ξ′
q)∈Sq−1where {e1, e2, ..., eq+1}is the standard basis for Rq+1 .This observation
drives us to consider the map ϕ: (0, π)×Sq−1→Sqdefined by
ϕ(θ, ξ′) = (cos θ, ξ′
1sin θ,...,ξ′
qsin θ)
which induces the geodesic polar coordinate system on Sq. This also provides a polar de-
composition of the normalised measure dσqon Sqas follows: Given a suitable function fon
Sqwe have ZSq
f(ξ)dσq(ξ) = Zπ
0ZSq−1
F(θ, ξ′) (sin θ)q−1dσq−1(ξ′)dθ
where F=f◦ϕ. Also in this coordinate system, we have the following representation of the
Laplace-Beltrami operator
∆Sq=−∂2
∂θ2−(q−1) cot θ∂
∂θ +1
4(q−1)2−sin−2θ˜
∆Sq−1
The following theorem gives a representation of the spherical harmonics in this polar coor-
dinate system.
Theorem 4.2. [21, Theorem 2.4] For n≥0we have the following orthogonal decomposition
Hn(Sq) =
n
M
l=0
Hn,l(Sq)

14 GANGULY, MANNA AND THANGAVELU
where the subspaces Hn,l(Sq)are irreducible and invariant under SO(q). Moreover, functions
in Hn,l(Sq)can be represented as
S(ξ) = (sin θ)lCq/2−1/2+l
n−l(cos θ)S′
l(ξ′)
where ξ=ϕ(θ, ξ′)and S′
l∈ Hl(Sq−1).
In view of the above theorem we have the orthogonal decomposition
L2(Sq) =
∞
M
n=0
n
M
l=0
Hn,l(Sq).
Now we set
Sn,l,k(ξ) = an,l (sin θ)lCl+q−1
2
n−l(cos θ)S′
k,l(ξ′)
where {S′
l,k : 1 ≤k≤N(l)}is an orthonormal basis for Hl(Sq−1).Here an,l is the normalising
constant so that kSn,l,kkL2(Sq)= 1 and it is explicitly given by
an,l =2−(l+q−1
2)Γ(2l+q−1)Γ(n+q
2)
Γ(l+q
2)Γ(n+l+q−1) Cl+q−2
2, l +q−2
2, n −l.(4.6)
Theorem 4.3. Let f∈C∞(Sq)be such that ∆m
Sdf∈L2(Sq)for all m≥0and satisfies
∞
X
m=1
k∆m
Sqfk−1
2m
2=∞.
If ∂m
∂θmθ=0F(θ, ξ′) = 0 for all m≥0and for all ξ′∈Sq−1,then fis identically zero.
Proof. Let fbe as in the statement of the theorem. For n≥0, let Pnfdenote the projection
of fonto the space Hn(Sq).Then from the above observations we have
Pnf=
n
X
l=0
N(l)
X
k=1
(f, Sn,l,k)L2Sn,l,k.(4.7)
Also since f∈L2(Sq) we have
f=
∞
X
n=0
Pnf=
∞
X
n=0
n
X
l=0
N(l)
X
k=1
(f, Sn,l,k)L2(Sq)Sn,l.k
By interchanging the summations, we observe that
f=
∞
X
l=0
∞
X
n=l
N(l)
X
k=1
(f, Sn,l,k)L2(Sq)Sn,l.k
=
∞
X
l=0
∞
X
n=0
N(l)
X
k=1
(f, Sn+l,l,k)L2(Sq)Sn+l,l.k.
In view of this, to prove the theorem it is enough to prove that (f, Sn+l,l,k)L2(Sq)= 0 for all
n, l, k. To start with, let us first fix n, l and k. From the expansion 4.7 we observe that
(Pnf, Sn+l,l,k)L2(Sq)= (f, Sn+l,l,k)L2(Sq).(4.8)

CHERNOFF’S THEOREM FOR SYMMETRIC SPACES 15
Next we use the expression for Sn+l,l,k to show that these coefficients are nothing but Jacobi
coefficients of a suitable function. In order to do so, we write the integral on Sqin polar
coordinates to obtain
(f, Sn+l,l,k)L2(Sq)=Zπ
0ZSq−1
F(θ, ξ′)an+l,l (sin θ)l+q−1Cl+q−1
2
n(cos θ)S′
k,l(ξ′)dσq−1(ξ′)dθ
where F:= f◦ϕ. Now using 4.3, 4.4 and 4.6, a simple calculation yields
an+l,lCl+q−1
2
n(cos θ) = 2−(l+q−1
2)C(l+q
2−1, l +q
2−1, n)P(l+q
2−1,l+q
2−1)
n(cos θ) (4.9)
which transforms the above equation into
(f, Sn+l,l,k)L2(Sq)= 2−(l+q−1
2)Zπ
0
Fk,l(θ) (sin θ)l+q−1P(l+d
2−1,l+d
2−1)
n(θ)dθ (4.10)
where we have defined
Fk,l(θ) := ZSq−1
F(θ, ξ′)S′
k,l(ξ′)dσq−1(ξ′).
Now letting α=l+d
2−1 and writing sin θ= 2 sin θ
2cos θ
2we see that
(sin θ)l+q−1= 2l+q−1(sin θ)−lwα,α(θ)
which together with 4.10 yields
(f, Sn+l,l,k)L2(Sq)=Jα,α (gk,l )(n) (4.11)
where gk,l(θ) := 2 q−1
2(sin θ)−lFk,l(θ).
In view of the Plancherel formula 4.5 and the relation 4.8 we have
kLm
α,αgl,k k2
2=
∞
X
n=0 n+2α+ 1
24m
|Jα,α(gl,k )(n)|2
=
∞
X
n=0 n+2l+q−1
24mZSd
Pnf(ξ)Sn+l,l,k(x)dσq(ξ)
2
,(4.12)
By Cauchy-Schwarz inequality we note that
ZSd
Pnf(ξ)Sn+l,l,k(ξ)dσq(ξ)
2
≤ kPnfk2
L2(Sq).
Finally, using the fact that n+1
2(2α+ 1) = n+1
2(2l+q−1) ≤n+q−1
21 + 2l
q−1,from
4.12 we get the estimate
kLm
α,αgl,k k2
2≤1 + 2l
q−14m∞
X
n=0 n+q−1
24m
kPnfk2
L2(Sq).
Therefore, we have proved
kLm
α,αgl,k k2≤1 + 2l
q−12m
k∆m
SqfkL2(Sq)
which by the hypothesis on the function f, implies that
∞
X
m=1
kLm
α,αgl,k k−1
2m
2=∞.(4.13)

16 GANGULY, MANNA AND THANGAVELU
Since gl,k(θ) is related to F(θ, ξ′) via the integral
gl,k(θ) = 2 q−1
2(sin θ)−lZSq−1
F(θ, ξ′)S′
k,l(ξ′)dσq−1(ξ′)
the hypothesis ∂m
∂θmθ=0F(θ, ξ′) = 0 for all m≥0 allows us to conclude that Lm
α,αgl,k (0) = 0
for all m≥0.Hence gl,k satisfies the hypotheses of Theorem 4.1 and hence we conclude that
gl,k = 0 and consequently (f, Sn+l,l,k)L2(Sq)= 0.As this is true for any n, l, k, we conclude
that f= 0 completing the proof of the theorem. 
4.3. The real projective spaces Pq(R).Let O(q) denote the group of q×qorthogonal
matrices. Then Pq(R) can be identified with SO(q+ 1)/O(q) which makes this a compact
symmetric space. Now it is well-known that the real projective space Pq(R) can be obtained
from Sqby identifying the antipodal points i.e., Pq(R) = Sq/{±I}and the projection map
s→ ±sfrom Sqto Pq(R) is locally an isometry. So, the functions on Pq(R) can be viewed
as even functions on the corresponding sphere Sqand if feis the even function on Sqcorre-
sponding to the function fon Pq(R) then ∆Pq(R)f= ∆Sqfe.Hence the analogue of Chernoff’s
theorem on Pq(R) follows directly from the case of sphere.
4.4. The other projective spaces Pl(C),Pl(H),and P2(Cay).As pointed out by T. O.
Sherman in [27], analysis on these three projective spaces is quite similar. Closely following
the notations of [27] (see also [10]), we first describe the appropriate polar coordinate repre-
sentation of these spaces and then as in the sphere case we prove the Chernoff’s theorem for
the associated Laplace-Beltrami operators. To begin with, let Sdenote any of these three
spaces Pl(C), Pl(H),and P2(Cay). Suppose ˜
∆Sdenotes the corresponding Laplace-Beltrami
operator. Let dµ denote the normalised Riemann measure on S. We have the following
orthogonal decomposition:
L2(S, dµ) := L2(S) =
∞
M
n=0
Hn(S),
where Hn(S) are finite dimensional and eigenspaces of ˜
∆Swith eigenvalue −n(n+k+q)
where q= 2,4,8, k =l−2,2l−3,3,for Pl(C), Pl(H) and P2(Cay),respectively. However,
it is convenient to work with ∆S:= −˜
∆S+ρ2
Swhere ρS:= 1
2(k+q).As a result Hn(S)
becomes eigenspaces of ∆Swith eigenvalue n+k+q
22.
Let Ω := {x∈Rq+1 :|x| ≤ 1}be the closed unit ball in Rq+1 .We consider a weight
function wdefined by w(r) := r−1(1 −r)kfor 0 < r ≤1.With these notations we have the
following result proved in [27, Lemma 4.15].
Proposition 4.4. There is a bounded linear map E:L1(S)→L1(Ω, w(|x|)dx)satisfying
(1) For f∈L1(S),ZS
fdµ =ZΩ
E(f)(x)w(|x|)dx
(2) The norm of Eas a map from Lp(S)to Lp(Ω, w(|x|)dx)is 1 (1 ≤p≤ ∞).
The integration formula in the above proposition is very useful. In fact, integrating the
right hand side of that formula in polar coordinates we have
ZΩ
E(f)(x)w(|x|)dx =Z1
0ZSq
E(f)(rξ)w(r)rqdσq(ξ)dr.

CHERNOFF’S THEOREM FOR SYMMETRIC SPACES 17
Now a change of variables r= sin2(θ/2) allows us to conclude that
ZS
fdµ =Zπ
0ZSq
F(θ, ξ)sin θ
22q−1cos θ
22k+1
dθ dσq(ξ),(4.14)
where F(θ, ξ) = E(f)(sin2(θ/2) ξ). In [27] Sherman has described the image of Hn(S)
under the map E. It has been proved that E(Hn(S)) = Hn(Ω, w) where Hn(Ω, w) is the
orthocomplement of Pn−1(Ω) in Pn(Ω) with respect to the inner product in L2(Ω, w(|x|)dx).
Here Pn(Ω) denotes the set of all polynomials on Ω of degree up to n. Also note that in
these trigonometric polar coordinates we can identify Ω with Ω0:= (0, π)×Sqand
dω(θ, ξ) := sin θ
22q−1cos θ
22k+1
dθ dσq(ξ)
is the corresponding measure on Ω0.Basically in view of this trigonometric polar coordinates
we have Hn(Ω, w) = Hn(Ω0, ω).These spaces are eigenspaces of the following differential
operator
ΛS=−∂2
∂θ2−(q−1−k) + (q+k) cos θ
sin θ
∂
∂θ −1
sin2(θ/2) ˜
∆Sq+k+q
22
with eigenvalues (n+q+k
2)2.The relation between this operator and the Laplace-Beltrami
operator is described in the following proposition.
Proposition 4.5. Let f∈C2(S)and Ebe as in the Proposition4.4. Then we have
E(∆Sf) = ΛSE(f).
For a proof of this fact we refer the reader to [27, Lemma 4.25]. Thus we have the following
orthogonal decomposition
L2(Ω0, dω) =
∞
M
n=0
Hn(Ω0, ω).
Moreover, Hn(Ω0, ω) admits a further decomposition as Hn(Ω0, ω) = Ln
j=0 Hn,j(Ω0, ω) where
each Hn,j(Ω0, ω) is irreducible under SO(q+ 1) and spanned by {Qn,j,l : 1 ≤l≤N(j)}(see
[27, Theorem 4.22]) where for x= sin2(θ/2) ξ, θ ∈(0, π) and ξ∈Sq
Qn,j,l(x) = bn,j sin θ
22j
P(k,q−1+2j)
n−j2 sin2(θ/2) −1Sj,l(ξ)
= (−1)n−jbn,j sin θ
22j
P(q−1+2j,k)
n−j(cos θ)Sj,l(ξ).
In the second equality we have used the symmetry relation for Jacobi polynomials i.e.,
P(α,β)
n(−x) = (−1)nP(β,α)
n(x).Here {Sj,l : 1 ≤l≤N(j)}a basis for Hj(Sq), the spherical
harmonics of degree lon Sq.The constants bn,j appearing in the above expression are chosen
so that kQn,j,lk2= 1. In fact, it can be checked that bn,j =C(q−1 + 2j, k, n −j).So, clearly
{Qn,j,l :n, l ≥0,1≤N(l)}forms an orthonormal basis for L2(Ω0, dω).Now we are ready to
state and prove an analogue of Chernoff’s theorem on S.
Theorem 4.6. Let f∈C∞(S)be such that ∆m
Sf∈L2(S)for all m≥0.Assume that
∞
X
m=1
k∆m
Sfk−1
2m
2=∞.

18 GANGULY, MANNA AND THANGAVELU
If the function Fdefined by F(θ, ξ) = E(f)(sin2(θ/2)ξ)satisfies ∂m
∂θmθ=0F(θ, ξ) = 0 for all
m≥0and for all ξ∈Sq,then fis identically zero.
Proof. Given a function fwith the property as in the statement of the theorem, we write
E(f)(sin2(θ/2) ξ) = F(θ, ξ),(θ, ξ)∈Ω0.So, the analysis, described above allow us to write
the projection of Fonto Hn(Ω0, ω) as
PS
nF=
n
X
j=0
N(j)
X
l=1
(F, Qn,j,l)Qn,j,l.
Now as in the sphere case, it is not hard to check that
F=
∞
X
j=0
∞
X
n=0
N(j)
X
l=1
(F, Qn+j,j,l)L2(Ω0,dω)Qn+j,j,l.(4.15)
Clearly, for each n≥0 we have
(PS
nF, Qn+j,j,l)L2(Ω0,dω)= (F, Qn+j,j,l)L2(Ω0,dω).(4.16)
As in the case of sphere, we will show that the right hand side of the above equation can be
expressed as Jacobi coefficient of a suitable function related to F. By definition, we have
(F, Qn+j,j,l) = Zπ
0ZSq
F(θ, ξ)Qn+j,j,l((sin2θ
2)ξ)sin θ
22q−1cos θ
22k+1
dθ dσq(ξ).(4.17)
Now using the expression for Qn+j,j,l we have
(F, Qn+j,j,l) = (−1)jbn+j,j Zπ
0
Fj,l(θ)sin θ
22j
P(q−1+2j,k)
n(cos θ)sin θ
22q−1cos θ
22k+1
dθ.
(4.18)
where Fj,l are defined by
Fj,l(θ) := ZSq
F(θ, ξ)Sj,l(ξ)dσq(ξ).
Writing gj,l (θ) = (−1)jFj,l (θ)(sin θ
2)−2jand using the definition of Jacobi coefficients we have
(F, Qn+j,j,l)L2(Ω0,dω)=Jα,β (gj,l)(n)
where α=q−2j+kand β=k. Now using the Plancherel formula 4.5 along with 4.16 we
obtain
kLm
α,βgj,lk2
2=
∞
X
n=0 n+α+β+ 1
24m
|Jα,β(gj,l)(n)|2
=
∞
X
n=0 n+q−2j+ 2k+ 1
24m
|(PS
nF, Qn+j,j,l)|2.(4.19)
But |(PS
nf, Qn+j,j,l)| ≤ kPS
nfkL2(Ω0,dω)and n+q−2j+2k+1
2≤C(n+q+k
2) so that we have
kLm
α,β gj,lk2
2≤C4m
∞
X
n=0 n+q+k
24m
kPS
nfk2
2=C4mkΛm
SE(f)k2
2(4.20)

CHERNOFF’S THEOREM FOR SYMMETRIC SPACES 19
In view of the Proposition 4.5 we have E(∆m
Sf) = Λm
SE(f) and using the fact that operator
norm of Eis one (see Proposition 4.4) we have
kLm
α,βgj,lk2
2≤C2mk∆m
Sfk2
Hence the given condition P∞
m=1 k∆m
Sfk−1
2m
2=∞allows us to conclude that
∞
X
m=1
kLm
α,βgj,lk−1
2m
2=∞.(4.21)
Also using the hypothesis ∂m
∂θmθ=0F(θ, ξ) = 0 for all m≥0, ξ ∈Sq,a simple calculation
shows that Lm
α,β gj,l(0) = 0 for all m≥0.Hence by Theorem 4.1, we have gj,l = 0 whence
(F, Qn+j,j,l)L2(Ω0,dω)= 0.As this is true for all n, j, l we conclude f= 0. This completes the
proof of the theorem. 
Acknowledgments
The first author is supported by Int. Ph.D. scholarship from Indian Institute of Science.
The second author is thankful to DST-INSPIRE [DST/INSPIRE/04/2019/001914] for the
financial support. The third author is supported by J. C. Bose Fellowship from the Depart-
ment of Science and Technology, Govt. of India.
References
[1] S. Bagchi, P. Ganguly, J. Sarkar and S. Thangavelu, On theorems of Chernoff and Ingham on the
Heisenberg group, arXiv:2009.14230 (2020).
[2] M. Bhowmik, S. Pusti, and S. K. Ray, Theorems of Ingham and Chernoff on Riemannian symmetric
spaces of noncompact type, J. Func. Anal., Volume 279, Issue 11 (2020).
[3] M. Bhowmik, S. Pusti, and S. K. Ray, A theorem of Chernoff for quasi-analytic functions for Riemannian
symmetric spaces, arXiv:2103.07667 (2021)
[4] 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.
[5] W. Bray, Generalized spectral projections on symmetric spaces of non-compact type: Paley–Wiener
theorems, J. Funct. Anal. 135 (1996), 206-232. MR1367630
[6] William O. Bray, Mark A. Pinsky, Pointwise Fourier Inversion on Rank One Symmetric Spaces and
Related Topics,Journal of Functional Analysis, Volume 151, Issue 2, 1997, Pages 306-333.
[7] T. Carleman, Les fonctions quasi-analytiques, volume 7 of Collection de monographies sur la theorie
des fonctions publi´ee sous la direction de M. E. Borel. Gauthier-Villars, Paris, 1926
[8] P. R. Chernoff, Some remarks on quasi analytic functions, Trans. Amer. Math. Soc. 167 (1972), 105-
113.
[9] P. R. Chernoff, Quasi-analytic vectors and quasi-analytic functions. Bull. Amer. Math. Soc. 81 (1975),
637-646.
[10] ´
O. Ciaurri, L. Roncal and P.R. Stinga Fractional integrals on compact Riemannian symmetric spaces
of rank one Adv. Math.,235 (2013), pp. 627-647.
[11] A. Denjoy, Sur les fonctions quasi-analytiques de variable reelle, C.R. Acad. Sci. Paris,173, 1329-1331
(1921).
[12] P. Ganguly and S. Thangavelu, An uncertainty principle for some eigenfunction expansions with appli-
cations , arXiv:2011.09940(2020).
[13] P. Ganguly and S. Thangavelu, An uncertainty principle for spectral projections on rank one symmetric
spaces of noncompact type, arXiv:2011.09942 (2020).
[14] A. E. Ingham, A Note on Fourier Transforms, J. London Math. Soc. S1-9 (1934), no. 1, 29-32.
MR1574706
[15] M. de Jeu, Determinate multidimensional measures, the extended Carleman theorem and quasi-analytic
weights Ann. Probab.,31 (3) (2003), pp. 1205-1227 MR1988469 (2004f:44006)

20 GANGULY, MANNA AND THANGAVELU
[16] K. D. Johnson, Composition series and intertwining operators for the spherical principal series II, Trans.
Amer. Math. Soc.,215 (1976), 269-283. MR0385012
[17] K. D. Johnson and N. Wallach, Composition series and intertwining operators for the spherical principal
series I, ibid.,229 (1977), 137-173. MR0447483
[18] S. Helgason, Groups and Geometric Analysis, Academic Press, 1984. MR0754767
[19] S. Helgason, Geometric analysis on Symmetric spaces, Math. Surveys Monographs 39, Amer. Math.
Soc., 1994.
[20] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, American Mathematical Soc.,
2001.
[21] T. H. Koornwinder, The addition formula for Jacobi polynomials and spherical harmonics, SIAM J.
Appl. Math.,25, 1973.
[22] T. H. Koornwinder, Jacobi functions and analysis on noncompact semisimple Lie groups, in: Special
Functions: Group Theoretical Aspects and Applications, R. Askey, T. H. Koornwinder and W. Schempp
(Eds.), Reidel, Dordrecht, 1984, 1-85. MR0774055
[23] B. Kostant, On the existence and irreducibility of certain series of representations, Bull. Amer. Math.
Soc.,75 (1969), 627-642. MR0245725
[24] D. Masson and W. Mc Clary, Classes of C∞vectors and essential self-adjointness, J. Functional Analysis,
10 (1972), 19-32.
[25] A. E. Nussbaum, Quasi-analytic vectors, Ark. Mat. 6(1965), 179-191. MR 33 No.3105.
[26] W. Rudin, Real and complex analysis, 3rd edition, McGraw-Hill, Inc.
[27] T. O. Sherman, The Helgason Fourier transform for compact Riemannian symmetric spaces of rank
one,Acta.Math. 164 (1990).
[28] H.-C. Wang, Two-point homogeneous spaces Ann. of Math. (2),55 (1952), pp. 177-191.
Department of Mathematics, Indian Institute of Science, 560 012 Bangalore, India
Email address:pritamg@iisc.ac.in, rameshmanna@iisc.ac.in, veluma@iisc.ac.in