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Contents lists available atScienceDirect

Journal of Functional Analysis

www.elsevier.com/locate/jfa

On a theorem of Chernoff on rank one Riemannian symmetric spaces

Pritam Gangulya,∗, Ramesh Mannab, Sundaram Thangavelua

aDepartmentofMathematics,IndianInstituteofScience,560012Bangalore, India

bSchoolofMathematicalSciences,NationalInstituteofScienceEducationand ResearchBhubaneswar,HBNI,Jatni752050,India

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received17June2021 Accepted4December2021 Availableonline14December2021 CommunicatedbyCamilMuscalu

MSC:

primary43A85,43A25 secondary22E30,33C45

Keywords:

Chernoff’stheorem

Riemanniansymmetricspaces HelgasonFouriertransform Jacobianalysis

In 1975, P.R. Chernoff used iterates of the Laplacian on Rn toproveanL2 versionoftheDenjoy-Carlemantheorem which provides a sufficient conditionfor a smooth function onRn tobequasi-analytic.Inthispaperweproveanexact analogueof Chernoff’stheoremfor allrankoneRiemannian symmetricspaces of noncompact type using iterates of the associated Laplace-Beltrami operators. Moreover, we also proveananalogueofChernoff’stheoremforthespherewhich isarankonecompactsymmetricspace.

©2021ElsevierInc.Allrightsreserved.

1. Introductionandthemain results

Theparamount property of ananalytic functionis thatit is completely determined byits value and thevalues ofall its derivativesat asingle point. Borel firstperceived thatthereisalargerclassofsmoothfunctionsthanthatofanalyticfunctionswhichhas

* Correspondingauthor.

E-mailaddresses:pritamg@iisc.ac.in(P. Ganguly),rameshmanna@niser.ac.in(R. Manna), veluma@iisc.ac.in(S. Thangavelu).

https://doi.org/10.1016/j.jfa.2021.109351

0022-1236/©2021ElsevierInc. Allrightsreserved.

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this magnificentproperty. Hecoinedthetermquasi-analyticforsuchclass offunctions.

Inexacttermsasubsetofsmoothfunctionsonaninterval(a,b) iscalledaquasi-analytic class if for any function f from that set and x0 (a,b), dxdnnf(x0) = 0 for all n N implies f = 0. Nowrecall thatasmooth functiononan intervalI is analytic provided its Taylor series converges to the function on I which naturally restricts the growth of derivatives of thatfunction. In fact, iffor every n, dxdnnfL(I) Cn!An for some constant A depending on f then the Taylor series of f converges to f uniformly and the converse is also true.This drives ananalytic mind to investigatewhether relaxing growth condition on the derivatives generates quasi-analytic class. In 1912 Hadamard proposedtheproblemoffindingsequence{Mn}nofpositivenumberssuchthattheclass C{Mn}ofsmoothfunctionsonIsatisfyingdxdnnfL(I)≤AnfMn forallf ∈C{Mn}is aquasi-analyticclass.AsolutiontothisproblemisprovidedbyatheoremofDenjoyand CarlemanwheretheyshowedthatC{Mn}isquasi-analyticifandonlyif

n=1Mn1/n=

.AsamatteroffactDenjoy[10] firstprovedasufficientconditionandlaterCarleman [6] completed thetheorem givinganecessary andsufficient condition.A short proof of this theorem based on complex analytic ideas can be found in Rudin [26]. A several variable analogueofthistheoremhasbeenobtainedbyBochnerand Taylor[4] in1939.

Later in 1950, instead of using all partial derivatives, Bochner used iterates of the LaplacianΔ andprovedananalogueofDenjoy-Carlemantheoremwhichreadsasfollows:

iff ∈C(Rn) satisfies

m=1Δmf−1/m =,thentheconditionΔmf(x)= 0 forall m≥0 andforallxinasetU ofanalyticdeterminationimpliesf = 0.Buildinguponthe works ofMasson- McClary [24] andNussbaum [25], in1972Chernoff [7] used operator theoreticargumentsto studyquasi-analyticvectors.As anapplicationheimprovedthe above mentioned result of Bochner by proving the following very interesting result in 1975.

Theorem 1.1.[8, Chernoff] Let f be a smooth function on Rn. Assume that Δmf L2(Rn) forall m∈N and

m=1Δmf2 2m1 =∞. If f and allits partial derivatives vanish atapointa∈Rn,thenf isidenticallyzero.

As Chernoff’s theorem is a useful tool in establishing uncertaintyprinciples of Ing- ham’s type([14]), proving analogues of Theorem 1.1 incontexts 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]. See also [13] for an analogue of Chernoff’s theorem for the full Laplacian on the Heisenberg group. In this paper we prove an analogue of Chernoff’s theorem for the Laplace-Beltramioperatoronrankonesymmetricspacesofbothcompactandnoncom- pact types.

In order to state ourresults we first need to introducesome notations. Let Gbe a connected, noncompactsemisimpleLie groupwithfinite centre andK amaximal com- pactsubgroupofG.LetX =G/K betheassociatedsymmetricspacewhichisassumed to have rank one. The origin o in the symmetric space is given by the identity coset

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eK where e is the identity element inG. We know thatX is a Riemannian manifold equippedwithaGinvariantmetriconit.WedenotebyΔX theLaplace-Beltramiopera- torassociatedtoX.FornoncompactRiemanniansymmetricspacesX =G/K,without anyrestrictionontherank,thefollowingweakerversionofTheorem1.1hasbeenproved inBhowmik-Pusti-Ray[2].

Theorem 1.2(Bhowmik-Pusti-Ray). LetX =G/K be anoncompact Riemannian sym- metric space and let ΔX be the associated Laplace-Beltrami operator. Suppose f C(X)satisfiesΔmXf ∈L2(X)forallm≥0and

m=1ΔmXf2 2m1 =∞.Iff vanishes onanonempty openset,thenf isidenticallyzero.

Inprovingtheabovetheorem,theauthorshavemadeuseofaresultofdeJeu[15].In thecaseofrankonesymmetricspaces,adifferentproofwasgivenbythefirstandthird authors ofthis articlebymaking useofsphericalmeans andan analogueofChernoff’s theorem for the Jacobi transform proved in [12]. 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 1.7 proved in this paper is built upon the ideas used in [12].

Inaveryrecentpreprint [3],Bhowmik-Pust-Rayhaveprovedthefollowing improve- mentoftheirTheorem1.2.InwhatfollowsletD(G/K) denotethealgebraofdifferential operatorsonG/K whichareinvariantunderthe(left)actionofG.

Theorem 1.3(Bhowmik-Pusti-Ray). LetX =G/K be anoncompact Riemannian sym- metric space and let ΔX be the associated Laplace-Beltrami operator. Suppose f C(G/K) be a left K-invariant function on X which satisfies ΔmXf L2(X) for all m 0and

m=1ΔmXf22m1 = ∞.If there is an x0 ∈X such that Df(x0) vanishes forallD∈D(G/K)thenf is identicallyzero.

Remark 1.4.Observe that in the above theorem the function f is assumed to be K- biinvariant.It is well-known that, forrank onesymmetric spaces of noncompact type, D(G/K) consistsofonly polynomials intheLaplace-Beltramioperatorwhich issome- what analogous to the Euclidean case. In fact, D(G/K) is the counterpart of the set ofalltranslationandrotationinvariantdifferentialoperatorsonRn whicharegivenby polynomials inLaplacian onRn. Inboth thecases, it hasbeen shown thatChernoff’s theorem doesnothold with vanishing conditionassumed onlyon thepowersof Lapla- cian.Indeed,forexplicit examplessee[8,Page-645] for theEuclidean spaceandsee [2, Example 3.7] for the rank onesymmetric spaces. As a matter of fact,it is natural to expect that, for the right analogue of Chernoff’s theorem,one needs to consider more derivatives.

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Inordertomotivatewhatwedointhecaseofrankonesymmetricspaces,letusrevisit theEuclideancase.ForfunctionsonRnconsiderthecondition(drd)kf(rω)|r=0= 0 where x=rω,r >0,ω∈Sn−1is thepolarcoordinatesofx∈Rn.Ascanbe easilychecked

d dr

k

f(rω) =

|α|=k

αf(rω)ωα

andhence(drd)kf(rω)|r=0= 0 forallkifandonlyifαf(0)= 0 forallα.Thisobservation playsanimportantroleinformulatingtherightanalogueChernoff’stheoremforcompact Riemanniansymmetricspaces.Inviewoftheaboveobservation,Chernoff’stheorem for theLaplacianonRn canbestatedinthefollowingform.

Theorem 1.5. Letf be a smooth function on Rn. Assume that Δmf L2(Rn) for all m∈N and

m=1Δmf22m1 =∞.If(drd)kf(rω)|r=0= 0forallkandω∈Sn−1,then f isidenticallyzero.

We can give a proof of the above theorem by reducing it to a theorem for Bessel operators.RecallthatwritteninpolarcoordinatestheLaplaciantakestheform

Δ = 2

∂r2+n−1 r

∂r+ 1

r2ΔSn−1 (1.1)

where ΔSn−1 is the spherical Laplacian on the unit sphere Sn−1. By expanding the function F(r,ω) =f(rω) in terms of sphericalharmonics on Sn−1 and making use of Hecke-Bochner formula, we can easily reduce Theorem 1.5 to a sequence of theorems for theBesseloperatorr2+ (n+ 2m+ 1)r−1r for variousvaluesof m∈N.This idea hasbeen alreadyusedinthepaper[12]. Asimilarexpansioninthecaseofnoncompact Riemannian symmetric spaces leadsto Jacobi operators as done in [12] which will be used inprovingTheorem1.7.Astheproofoftheabovetheorem issimilartoandeasier than thatofTheorem1.7,wewillnotpresentithere.

Remark 1.6. We remark in passing thatthe above theorem canalso be proved in the context of Dunkl Laplacian on Rn associated to root systems. We would also like to mention thatanalogues of Chernoff’stheorem canbe provedfor the Hermiteoperator H onRn and the specialHermiteoperator Lon Cn. Againtheidea isto make useof Hecke-Bochner formulafor theHermite and specialHermite projections(associated to theirspectral decompositions).

In the case of rank one symmetric spaces we look for the counterpart of (drd)kf(rω)|r=0 = 0. The Iwasawa decomposition of G reads as G = KAN where A is abelianandN isanilpotent Liegroup.LetgandastandfortheLiealgebras corre- sponding toG andA respectively.Here aisonedimensionalsinceX is ofrankone.It iswell knownthateveryelementofggivesrisetoaleftinvariantvectorfieldonG. Let

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H betheleft invariantvectorfieldcorresponding toafixed basiselementof a.Wewill describeallthesenotationsindetailinthenextsection.

We consider the condition Hl(eK) = 0 as the analogue of (drd)kf(rω)|r=0 = 0. To justifytheanalogy,weremarkthatthisconditionisequivalenttotheconditionVl(eK)= 0 forallV pwhere pis thetangentspace ofX =G/K at theidentity coset.This is clearfromthefactsthatp=k∈KAdkaand

Hlf(k) = dl

dtlf(kexp(tH)) = dl

dtlf(kexp(tH)k1) = dl

dtlf(exp(AdkH))

for any k K. Note that none of these vector fields belong to D(G/K). It would be interestingtoseewhetheranalogousvectorfieldscanbeusedinthevanishing condition inthehigherrankcaseto obtainacorrectanalogueofChernoff’stheorem withoutany extra assumptions on the function side. As an analogue of the version of Chernoff’s theoremstatedinTheorem 1.5forrankonesymmetricspacesX weofferthefollowing:

Theorem1.7. LetX =G/K bearankonesymmetricspaceofnoncompacttype.Suppose f C(X) satisfies ΔmXf L2(X) for all m 0 and

m=1ΔmXf22m1 = ∞. If Hlf(eK)= 0foralll≥0thenf isidenticallyzero.

NowinviewofthetranslationinvarianceofLaplace-BeltramioperatorΔX andH,it isnotdifficulttoseethattheabovetheoremholdstrueifwetakeanyx0∈X inplaceof eK.As animmediateconsequenceof theaboveresult weobtainananalogueoftheL2 versionoftheclassicalDenjoy-CarlemantheoremusingiteratesoftheLaplace-Beltrami operatoronX =G/K.

Corollary 1.8. Let X = G/K be a rank one symmetric space of noncompact type. Let {Mk}k be alogconvex sequence. Define C({Mk}k,ΔX,X)tobe the classof allsmooth functions f on X satisfying ΔmXf L2(X) for all m N and ΔkXf2 Mkλ(f)k for some constant λ(f) depending on f. Suppose that

k=1M

1 2k

k = ∞. Then every memberof thatclassisquasi-analytic.

SofarwehaveonlyconsiderednoncompactRiemanniansymmetric spaces,butnow we turn our attention to proving an analogue of Theorem 1.5 for compact, rank one symmetric spaces. As we will see, our investigation is complete only for the case of spheres.

Let(U,K) be acompactsymmetricpairandS =U/K be theassociated symmetric space. Here U is a compact semisimple Lie group and K is a connected subgroup of U. Weassume thatS hasrank one.Being a compactRiemannian manifold, S admits aLaplace-Beltramioperator ΔS. In[28], H.C. Wang hascompletely classified allrank onecompactsymmetricspaces.Tobemoreprecise,S isoneofthefollowings:Theunit

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sphere Sq =SO(q+ 1)/SO(q), thereal projective spacePq(R)=SO(q+ 1)/O(q), the complex projectivespace Pl(C),the quaternionprojective space Pl(H) and theCayley projective spaceP2(Cay)=F4/Spin(9). Ineachcase,S comes upwithan appropriate polarform(0,π)×Sqwhereq=qS dependsonthesymmetricspaceS.Asaconsequence, functionsonS canbeidentifiedwith functionsontheproductspaceY = (0,π)×Sq.

In order to motivate the formulation of our result for spheres, and in general for compactrankonesymmetricspaces, letusreturnto theEuclidean case.Inthecontext ofTheorem1.5,byidentifyingRn with(0,)×Sn1,everyfunctionf onRngivesrise to afunctionF(r,ω) on(0,)×Sn1 andinviewof (1.1),theactionofΔ on f takes theform,

Δf(r, ω) = 2

∂r2F(r, ω) +n−1 r

∂rF(r, ω) + 1

r2ΔSn−1F(r, ω).

There isasimilardecompositionofΔSq asasumofaJacobioperatoron(0,π) andthe spherical Laplacian ΔSq−1 and this justifies our formulation of Chernoff’s theorem for spheres.Inordertodescribethisweworkwiththegeodesicpolarcoordinatesystemon Sq.Notethatgivenξ∈Sq, wecanwriteξ= (cosθ)e1+ξ1(sinθ)e2+...+ξq(sinθ)eq+1 for some θ (0,π) and ξ = (ξ1,...,ξq) Sq1 where {e1,e2,...,eq+1}is the standard basis for Rq+1. This observationdrives us to consider the map ϕ: (0,π)×Sq1 Sq definedby

ϕ(θ, ξ) = (cosθ, ξ1sinθ, . . . , ξqsinθ)

which induces the geodesic polar coordinate system on Sq. This also provides a polar decompositionofthenormalisedmeasureqonSqasfollows:Givenasuitablefunction f onSq wehave

Sq

f(ξ)dσq(ξ) = π 0

Sq−1

F(θ, ξ) (sinθ)q−1q1)dθ

where F =f ◦ϕ. Alsointhis coordinate system,we havethe followingrepresentation of theLaplace-Beltramioperator

ΔSq = 2

∂θ2 + (q1)cosθ sinθ

∂θ+ 1

sin2θΔSq−1.

Withthese notationswehavethefollowinganalogueofChernoff’stheoremforspheres.

Theorem 1.9. Letf ∈C(Sq)be such that ΔmSdf ∈L2(Sq) forall m≥0 andsatisfies the Carlemancondition

m=1ΔmSqf22m1 =∞.If ∂θmm

θ=0F(θ,ξ)= 0for allm≥0 and forallξSq1,thenf is identicallyzero.

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ObservethatthevanishingconditionimposedonFsimplymeansthatwhenexpressed inthegeodesicpolarcoordinatesystem,allthederivativesoff withrespecttoθvanish on all great circles about the origin. Thus, we see thatTheorem 1.9 serves as a good analogueofChernofftheoremwhenthesymmetricspaceS isthesphereSq.However,in thegeneralcaseofprojectivespaceswearenotluckyenoughtoproveanexactanalogue ofChernoff’stheorem.

The geometry of the projective spaces is much more complicated than that of the sphere. It is almost impossible to find the exact vanishing condition as in the sphere case, for theprojective spaces from the availableinformation about those spaces from thepointofviewof Chernoff’stheorem.Asamatteroffact,theproblemofprovingan exactanalogueofChernoff’stheorem forthose spacesis stillopen andwe plan totake upthis problem insomefuture work.However,drawing analogy from the spherecase, fortheotherprojective spaces,weusethecorresponding product spaceY toformulate atheorem which is somewhat related to Chernoff’s theorem.Although thetheorem is notanexactcounterpartof Chernoff’stheorem forprojectivespaces, itilluminatesthe difficultiesinobtaininganexactanalogueforthetheorem inconsideration.Thiswillbe doneintheappendix.

We complete this introduction with a brief descriptionof theplan 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 spherical harmonics and Jacobi polynomials and setting up the notations, we prove Theorem1.9. Wereferthereadertothepapers[11] and [12] forrelatedideas.

2. PreliminariesonRiemanniansymmetricspacesof non-compacttype

In this section we describe the relevant theory regarding the harmonic analysis on rankoneRiemanniansymmetricspacesofnoncompacttype.Generalreferencesforthis sectionarethemonographsofHelgason[18] and [19].

LetGbeaconnected,noncompact semisimpleLiegroupwith finitecentre.Suppose gdenotesitsLiealgebra.Withrespect toafixedCartaninvolution θongwehavethe decompositiong=kp. Herekandparethe+1 and 1 eigenspacesofθ respectively.

Let a be the maximal abelian subspace of p. Also assume that the dimension of a is one. Now we know thatthe involution θ induces an automorphism Θ on G and K = {g∈G: Θ(g)=g}isamaximalcompactsubgroupofG.Weconsiderthehomogeneous space X = G/K which ais asmooth manifold endowed with a G-Riemannianmetric inducedbytherestrictionoftheKillingformBofgonp.ThisturnsX intoarankone Riemanniansymmetricspaceofnoncompact typeandeverysuchspacecanbe realised thisway.

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Leta denotethedualofa.Givenα∈a wedefine

gα:={X g: [Y, X] =α(Y)X, Y a}.

Now Σ:={α∈a:gα={0}}isthesetof allrestrictedrootsofthepair(g,a).LetΣ+ denotethesetofallpositiverootswithrespecttoafixedWeylchamber.Itisknownthat n := α∈Σ+gα is anilpotent subalgebra of gand we have theIwasawa decomposition g=kan.Now writingN = expnand A= expaweobtainG=KAN where Ais abelian andN isanilpotent subgroupofG.Moreover,Anormalizes N. Inviewofthis decomposition everyg G can be uniquely writtenas g = k(g) expH(g)n(g) where H(g) belongstoa.AlsowehaveG=N AK and withrespectto thisdecompositionwe write g ∈NexpA(g)K where thefunctionsA andH are relatedviaA(g)=−H(g1).

Now in the rank one case when dimension of a is one, Σ is given by either {±γ} or {±γ,±}where γ belongsto Σ+. Letρ:= (mγ +m)/2 where mγ and m denote themultiplicitiesoftherootsγ and2γrespectively.TheHaarmeasuredgonGisgiven by

G

f(g)dg=

K

A

N

f(katn)e2ρtdkdtdn.

Themeasure dxonX isinducedfromtheHaarmeasuredg viatherelation

G

f(gK)dg=

X

f(x)dx.

Suppose M denotes the centralizer of Ain K. Thepolar decompositionof G reads as G =KAK inviewof which we canwrite each g ∈G as g =k1ark2 with k1,k2 ∈K.

Actually themap (k1,ar,k2)→k1ark2 ofK×A×K into Ginducesadiffeomorphism of K/M ×A+×K ontoan open densesubset of Gwhere A+ = expa+ and a+ is the fixed positiveWeylchamberwhichbasicallycanbe identifiedwith(0,) in ourcase.

It is also well-knownthateach Xggives riseto aleft invariantvector fieldon G bytheprescription

Xf(g) = d dt

t=0

f(g.exp(tX)), g∈G.

Since aisonedimensional,we fixabasis{H}ofa. Byanabuseofnotation,wedenote the leftinvariantvector fieldcorrespondingto this basiselementby H. Infact,wecan write A={ar= exp(rH):r∈R}.

2.1. HelgasonFouriertransform

Define the function A : X ×K/M a by A(gK,kM) = A(k1g). Note that A is right K-invariant in g and right M-invariant in K. In what follows we denote the

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elements of X and K/M byx and b respectively. Let a denote thedual of aand aC be its complexification. Here in our case a and aC can be identified with R and C respectively. For each λ∈ aC and b K/M, the functionx e(iλ+ρ)A(x,b) is a joint eigenfunctionofall invariantdifferentialoperators onX. Forf ∈Cc(X),itsHelgason FouriertransformisafunctionfonaC×K/M definedby

f˜(λ, b) =

X

f(x)e(iλ+ρ)A(x,b)dx, λ∈aC, b∈K/M.

Moreover,weknowthatiff ∈L1(X) thenf(.,b) isacontinuousfunctiononawhichex- tendsholomorphicallytoadomaincontaininga.Theinversionformulaforf ∈Cc(X) saysthat

f(x) =cX

−∞

K/M

f(λ, b)e(iλ+ρ)A(x,b)|c(λ)|−2dbdλ

wherestandsforusualLebesguemeasureonR(i.e.,a),dbisthenormalisedmeasure onK/M and c(λ) istheHarish-Chandrac-function.TheconstantcX appearinginthe aboveformula is explicit and dependson the symmetric space X (Seee.g., [19]). Also forf ∈L1(X) withf∈L1(a×K/M,|c(λ)|2dbdλ),theaboveinversionformula holds fora.e.x∈X.Furthermore,themappingf →fextendsasanisometry ofL2(X) onto L2(a+×K/M,|c(λ)|−2dλdb) whichisknownasthePlanchereltheoremfortheHelgason Fouriertransform.

We also need to use certain irreducible representations of K with M-fixed vectors.

SupposeK0denotesthesetofallirreducibleunitaryrepresentationsofK withMfixed vectors.Letδ∈K0andVδ bethefinitedimensionalvectorsspaceonwhichδisrealised.

Ascanbeseenin[23,Theorem6],thesubspaceofVδ consistingofM-fixedvectors,has dimensionone.Hence Vδ contains auniquenormalised M-fixed vectorv1. Consideran orthonormalbasis{v1,v2,...,vdδ}forVδ.Forδ∈K0 and1≤j ≤dδ,wedefine

Yδ,j(kM) = (vj, δ(k)v1), kM ∈K/M.

ItcanbeeasilycheckedthatYδ,1(eK)= 1 andmoreover,Yδ,1isM-invariant.

Proposition 2.1.The set {Yδ,j : 1 j dδ K0} forms an orthonormal basis for L2(K/M).

Wereferthereaderto [20,Theorem 3.5, Section3,Chapter 5] foraproofof theafore- mentionedproposition.Seealsothediscussionatthebeginningof[19,Section2,Chapter 3].WecangetanexplicitrealisationofK0 byidentifyingK/M withtheunitspherein p.BylettingHmtostandforthespaceofhomogeneousharmonicpolynomialsofdegree mrestrictedtotheunitsphere,wehavethefollowingsphericalharmonicdecomposition

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L2(K/M) =m=0Hm.

We know that each Vδ is contained in some Hm and hence the functions Yδ,j can be identified withsphericalharmonics.

Given δ ∈K0 andλ∈aC(i.e.,C in our case) we consider thesphericalfunctionsof typeδdefinedby

Φλ,δ(x) :=

K

e(iλ+ρ)A(x,kM)Yδ,1(kM)dk.

TheseareeigenfunctionsoftheLaplace-BeltramioperatorΔXwitheigenvalue2+ρ2).

When δ is thetrivialrepresentation for whichYδ,1 = 1,thefunction Φλ,δ iscalled the elementarysphericalfunction,denotedbyΦλ.Moreprecisely,

Φλ(x) =

K

e(iλ+ρ)A(x,kM)dk.

Note thatthese functionsare K-biinvariant. Thespherical functions canbe expressed in termsof Jacobi functions. Infact, ifx=gK and g =kark (polar decomposition), Φλ,δ(x)= Φλ,δ(ar).Suppose

α=1

2(mγ+m1), β= 1

2(m1). (2.1)

Foreachδ∈K0 there existsapairofintegers(p,q) suchthat

Φλ,δ(x) =Qδ(iλ+ρ)(α+ 1)−1p (sinhr)p(coshr)qϕ(α+p,β+q)λ (r) (2.2) where ϕ(α+p,β+q)λ aretheJacobifunctionsoftype(α+p,β+q) andQδ aretheKostant polynomials givenby

Qδ(iλ+ρ) = 1

2(α+β+ 1 +iλ)

(p+q)/2

1

2(α−β+ 1 +iλ)

(p−q)/2

. (2.3) Intheabovewehaveusedthenotation(z)m=z(z+1)(z+2)...(z+m1).Thefollowing resultprovedinHelgason[19] willbeveryusefulforourpurpose:

Proposition 2.2. Letδ∈K0 and1≤j ≤dδ.Thenwehave

K

e(iλ+ρ)A(x,kM)Yδ,j(kM)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.

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2.2. Spherical Fouriertransform

WesaythatafunctionfonGisK-biinvariantiff(k1gk2)=f(g) forallk1,k2∈K.It canbecheckedthatiff isaK-biinvariantintegrablefunctionthenitsHelgasonFourier transformf(λ,b) isindependentofb∈K/M andbyalittleabuseofnotationwewrite thisas

f˜(λ) =

X

f(x)Φλ(x)dx.

This is called thespherical Fouriertransform. Now since f is K biinvariant,using the polardecompositiong=k1ark2,wecanviewf asafunctiononAalone:f(g)=f(ar).

Sotheaboveintegraltakesthefollowingpolarform:

f˜(λ) = 0

f(arλ(r)wα,β(r)dr

where wα,β(r)= (2sinhr)2α+1(2coshr)2β+1 and Φ−λ(ar)=ϕλ(r) are givenby Jacobi functionϕα,βλ (r) of type(α,β).Here αandβ areassociated to thesymmetricspaceas mentionedabove. Soit isclear thatthesphericalFourier transformis basically Jacobi transformof type(α,β). Intherest of thesection wedescribe certainresultsfrom the theoryofJacobianalysis.

Letα,β,λ∈Cand−α /∈N.TheJacobifunctionsϕ(α,β)λ (r) oftype(α,β) aresolutions oftheinitialvalueproblem

(Lα,β+λ2+2(α,β)λ (r) = 0, ϕ(α,β)λ (0) = 1 whereLα,β istheJacobioperatordefinedby

Lα,β := d2

dr2 + ((2α+ 1) cothr+ (2β+ 1) tanhr) d dr

and=α+β+1.ThusJacobifunctionsϕ(α,β)λ areeigenfunctionsofLα,βwitheigenvalues

2+2).TheseareevenfunctionsonRandareexpressibleintermsofhypergeometric functions.Forcertain valuesofthe parameters(α,β) these functionsarisenaturallyas spherical functionson Riemannian symmetric spaces of noncompact type. The Jacobi transformofasuitablefunctionf onR+ isdefinedby

Jα,βf(λ) = 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 theoperatorLα,β isselfadjointonL2(R+,wα,β(r)dr) andthat

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Lα,βf(λ) =2+2) ˜f(λ).

Under certain assumptions on α and β the inversion and Plancherel formula for this transformtakeaniceform asdescribed below.

Theorem 2.3([22]).Letα,β∈R,α >−1and|β|≤α+ 1.Supposecα,β(λ)denotesthe Harish-Chandra c- functiondefined by

cα,β(λ) = 2−iλΓ(α+ 1)Γ(iλ) Γ1

2(iλ+) Γ1

2(iλ+α−β+ 1) (1) (Inversion) Forf ∈C0(R)whichiseven wehave

f(r) = 1 2π

0

Jα,βf(λ)ϕ(α,β)λ (r)|cα,β(λ)|2

(2) (Plancherel) Forf,g∈C0(R)which areeven,thefollowingholds

0

f(r)g(r)wα,β(r)dr= 0

Jα,βf(λ)Jα,βg(λ)|cα,β(λ)|−2dλ.

The mapping f f˜ extends 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 ananalogue of Chernoff’stheorem for theLaplace-BeltramioperatorΔX inthenextsection.

3. Chernoff’stheoremonrankonesymmetricspacesofnoncompacttype

Inthissectionweproveourmaintheoremi.e.,ananalogueofChernoff’stheorem for ΔX. The main idea of the proof is to reducethe result for ΔX to aresult for Jacobi operator.So,firstweindicateaproof ofChernoff’stheoremforJacobioperator.

Theorem 3.1.Letα,β∈R,α >−1and|β|≤α+ 1.Supposef ∈L2(R+,wα,β(r)dr) is suchthatLmα,βf ∈L2(R+,wα,β(r)dr)forallm∈NandsatisfiestheCarlemancondition

m=1Lmα,βf−1/(2m)2 =∞.If Lmα,βf(0)= 0 forallm≥0thenf is identicallyzero.

Proof. Given f as inthestatement of thetheorem,we consider the Borelmeasure μf definedonRgiven by

μf(E) =

E

|Jα,βf(λ)||cα,β(λ)|−2

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whereE is aBorelsubset ofR. SincetheFourier-JacobitransformJα,βf(λ) isan even functiononR,itisnothardtoseethatμf isanevenmeasure.LetM(m) stand forthe mthordermomentofthemeasure.Thenweseethat

M(2m) =

−∞

t2mf(t) =

−∞

λ2m|Jα,βf(λ)||cα,β(λ)|2

−∞

2+2)m|Jα,βf(λ)||cα,β(λ)|−2dλ.

ButbyCauchy-Schwarzinequalityfollowed byanapplicationof plancherelformula we have

M(2m)≤Cj

0

2+2)2(m+j)|Jα,βf(λ)|2|cα,β(λ)|2

1 2

=CjL(m+j)α,β f2

where Cj2 =

02 +2)2j|cα,β(λ)|2 which is finite for large enough j since

|cα,β(λ)|2haspolynomialgrowth inλ.Nowasinproofof[12,Theorem2.1] oneeasily checksthat

m=1M(2m)1/2m=togetherwith[15,Theorem2.3] provesthatpoly- nomials are dense inL1(R,f) and henceeven polynomials are dense inL1e(R,f).

Since f L2(R+,wα,β(r)dr), by Plancherel Jα,βf L2e(R,|cα,β(λ)|−2dλ) which can be used to show that Jα,βf L1e(R,f). Hencefor any ε > 0, there exists an even polynomialq suchthat

0

|Jα,βf(λ)−q(λ)||Jα,βf(λ)|w(λ)dλ < ε.˜ (3.1)

But notice that |Jα,βf(λ)|2 = (Jα,βf(λ)−q(λ))Jα,βf(λ)+q(λ)Jα,βf(λ). So from the Plancherelformula,wesee that

f22 0

|Jα,βf(λ)−q(λ)||Jα,βf(λ)||cα,β(λ)|2+ 0

q(λ)Jα,βf(λ)|cα,β(λ)|2dλ.

ButthevanishingconditionLmα,βf(0)= 0 (interpretedas limr0+Lmα,βf(r)= 0)forall m≥0 yields

0

2+2)mJα,βf(λ)ϕλ(r)|cα,β(λ)|2= 0. (3.2)

Nowsinceqisanevenpolynomial,thereexistsapolynomialpsuchthatp(λ2+2)=q(λ).

Hencetheaboveequation(3.2) alongwiththefactthatϕλ(0)= 1 gives

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0

q(λ)Jα,βf(λ)|cα,β(λ)|−2= lim

r0

0

p(λ2+2)Jα,βf(λ)ϕλ(r)|cα,β(λ)|−2= 0

which together with (3.1) proves thatf22 < ε which is truefor everyε >0,proving thatf = 0.

In[12] theaboveresultwasprovedundertheassumptionthatf vanishesnear0 which was goodenoughforusto proveIngham’stheorem whichwasourmain concernthere.

But acloseexamination ofthe proofreveals thatthe assumptionis superfluousas can be seenintheaboveproof. Inorder toproveourmain result,thefollowing estimateon theratioofHarish-Chandrac-functionsisalsoneeded.

Lemma3.2. Letα,β beasin(2.1) and(p,q)bethepairofintegersassociatedtoδ∈K0. Then foranyλ≥0wehave

|cα,β(λ)|2

|cα+p,β+q(λ)|2|Qδ(iλ+ρ)|−2≤C

where C is a constant independent of λ depending only on the parameters (α,β) and (p,q).

Proof. Firstnote thatfrom thedefinition(2.3) ofKostantpolynomialswehave

|Qδ(iλ+ρ)|=

p+q

2

j=0

(B1+j)2+1 4λ2

12 p2q

j=0

(B2+j)2+1 4λ2

12

where B1= 12(α+β+ 1) andB2=12−β+ 1).From theaboveexpression, itcanbe easily checkedthat|Qδ(iλ+ρ)|/(2−1λ)p1 asλ→ ∞ sothat

|Qδ(iλ+ρ)| ∼2pλp, λ→ ∞. (3.3) Moreover, wealsohave

|Qδ(iλ+ρ)| ≥

p+q

2

j=0

|B1+j|

pq

2

j=0

|B2+j|= constant.

Now using[5,Lemma2.4] we have

|cα,β(λ)|2

|cα+p,β+q(λ)|2 ∼λ2p, λ→ ∞ (3.4)

whichtogether with(3.3) impliesthat

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|cα,β(λ)|2

|cα+p,β+q(λ)|2|Qδ(iλ+ρ)|−21, λ→ ∞.

Alsotheratioin(3.4) beingacontinuousfunctionofλisboundedneartheorigin.Hence theresultfollows.

Proof of Theorem1.7. Letf be as inthe statementof the Theorem 1.7. Wecomplete theproofinthefollowing steps.

Step1: UsingProposition2.1wewrite f(λ, k) =

δK0

dδ

j=1

Fδ,j(λ)Yδ,j(k) (3.5)

whereFδ,j(λ) arethesphericalharmoniccoefficientsoff˜(λ,·) definedby Fδ,j(λ) =

K/M

f(λ, k)Y δ,j(k)dk.

Fixδ ∈K0 and 1≤j ≤dδ.From the definitionof theHelgason Fouriertransform we have

Fδ,j(λ) =

K/M

G/K

f(x)e(−iλ+ρ)A(x,kM)Yδ,j(kM)dxdk.

NowusingFubini’stheorem,inviewoftheProposition2.2theintegralontherighthand sideofaboveisequalto

G/K

f(x)Yδ,j(kM)Φλ,δ(ar)dx. (3.6)

Thefunctiongδ,j(x) definedby gδ,j(x) =

K

f(kx)Yδ,j(kM)dk, x∈X

isclearlyK-biinvariant,and hencebyabuseofnotationwewrite gδ,j(r) =

K

f(kar)Yδ,j(kM)dk.

Nowperformingtheintegralin(3.6) usingpolarcoordinatesweobtain

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Fδ,j(λ) = 0

gδ,j(r)Φλ,δ(ar)wα,β(r)dr (3.7)

Now recallthatforeachδ∈K0 thereexist apairofintegers(p,q) suchthat Φλ,δ(x) =Qδ(iλ+ρ)(α+ 1)p1(sinhr)p(coshr)qϕ(α+p,β+q)λ (r).

Bydefining

fδ,j(r) = 4−(p+q) (α+ 1)p

gδ,j(r)(sinhr)p(coshr)q (3.8) and recallingthedefinitionofJacobitransformsweobtain

Fδ,j(λ) =Qδ(iλ+ρ)Jα+p,β+q(fδ,j)(λ) (3.9) Step2:InthisstepweestimatetheL2normofpowersofJacobioperatorappliedtofδ,j

intermsoftheL2 normofcorresponding powersof ΔX appliedtof.Letm∈N.Note thatthePlancherelformula2.3fortheJacobitransformyields

Lmα+p,β+q(fδ,j)L2(R+,wα+p,β+q(r)dr)

=

0

2+ρ2δ)2m|Jα+p,β+q(fδ,j)(λ)|2|cα+p,β+q(λ)|2

1 2

where ρδ=α+β+p+q+ 1.Inviewof(3.9) theaboveintegralreduces to

0

2+ρ2δ)2m|Fδ,j(λ)|2|Qδ(iλ+ρ)|−2cα+p,β+q(λ)|−2

1 2

whichafter recallingthedefinitionof Fδ,j(λ) readsas

⎜⎝ 0

2+ρ2δ)2m|Qδ(iλ+ρ)|−2|

K

f(λ, k)Yδ,j(k)dk

2

|cα+p,β+q(λ)|−2

⎟⎠

1 2

.

ByanapplicationofMinkowski’sintegralinequality,theaboveintegralisdominatedby

K

0

2+ρ2δ)2m|Qδ(iλ+ρ)|−2|f(λ, k)|2|cα+p,β+q(λ)|−2

1 2

|Yδ,j(k)|dk.

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Now using Cauchy-Schwarz inequalityalong with the fact thatYδ,jL2(K/M) = 1, we seethattheaboveintegralisboundedby

⎜⎝

K/M

0

2+ρ2δ)2m|Qδ(iλ+ρ)|−2|f(λ, k)|2|cα+p,β+q(λ)|−2dλ dk

⎟⎠

1 2

Since λλ222δ2 = 1+ρλ2δ2ρ22 is adecreasingfunction of λit follows that λλ22+d22 ≤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 estimatefortheintegralunderconsideration:forsomeconstantC1=C1(α,β)

C1m

⎜⎝

K/M

0

2+ρ2)2m|f(λ, k)|2|cα,β(λ)|2dλ dk

⎟⎠

1 2

.

Finally,fromtheseriesofinequalitiesabove,weobtain

Lmα+p,β+q(fδ,j)L2(R+,wα+p,β+q(r)dr)≤C1mΔmXf2. (3.10) Hencefrom thehypothesisofthetheoremitfollowsthat

m=1

Lmα+p,β+q(fδ,j)L22m(R1 +,wα+p,β+q(r)dr)=∞.

Step3:FinallyinthisstepweprovethatLmα+p,β+q(fδ,j)(0)= 0 forallm≥0.Firstrecall that

fδ,j(r) = 4−(p+q)

(α+ 1)p(sinhr)p(coshr)q

K

f(kar)Yδ,j(kM)dk.

Assinhrhasazeroattheoriginandcosh 0= 1,ifwecanshowthatas afunctionofr, theintegral

Kf(kar)Yδ,j(kM)dkhasazeroofinfiniteorderatthe0,thenwearedone.

Nownotethatforanym∈N dm

drm

K

f(kar)Yδ,j(kM)dk=

K

dm

drmf(kar)Yδ,j(kM)dk.

Butbydefinitionofthevectorfields onG,writingar= exp(rH) wehave dm

drmf(kar)|r=0= dm

drmf(k.exp(rH))|r=0=Hmf(k).

References

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