The Fourier transform on negatively curved harmonic manifolds
aa r X i v : . [ m a t h . DG ] F e b THE FOURIER TRANSFORM ON NEGATIVELY CURVEDHARMONIC MANIFOLDS
KINGSHOOK BISWAS
Abstract.
Let X be a complete, simply connected harmonic manifold withsectional curvatures K satisfying K ≤ −
1, and let ∂X denote the boundary atinfinity of X . Let h > X , and let ρ = h/
2. Fixing a basepoint o ∈ X , for ξ ∈ ∂X , let B ξ denote the Busemannfunction at ξ such that B ξ ( o ) = 0, then for λ ∈ C the function e ( iλ − ρ ) B ξ is aneigenfunction of the Laplace-Beltrami operator with eigenvalue − ( λ + ρ ).For a function f on X , we define the Fourier transform of f by˜ f ( λ, ξ ) := Z X f ( x ) e ( − iλ − ρ ) B ξ ( x ) dvol ( x )for all λ ∈ C , ξ ∈ ∂X for which the integral converges. We prove a Fourierinversion formula f ( x ) = C Z ∞ Z ∂X ˜ f ( λ, ξ ) e ( iλ − ρ ) B ξ ( x ) dλ o ( ξ ) | c ( λ ) | − dλ for f ∈ C ∞ c ( X ), where c is a certain function on R − { } , λ o is the visibilitymeasure on ∂X with respect to the basepoint o ∈ X and C > NA groups (or Damek-Ricci spaces). Contents
1. Introduction 12. Preliminaries 42.1. CAT(-1) spaces 42.2. Harmonic manifolds 93. Radial and horospherical parts of the Laplacian 104. Analysis of radial functions 124.1. Chebli-Trimeche hypergroups 124.2. The density function of a harmonic manifold 164.3. The spherical Fourier transform 195. Fourier inversion and Plancherel theorem 206. The convolution algebra of radial functions 28References 301.
Introduction A harmonic manifold is a Riemannian manifold X such that for any point x ∈ X , there exists a non-constant harmonic function on a punctured neighbourhoodof x which is radial around x , i.e. only depends on the geodesic distance from x . Copson and Ruse showed that this is equivalent to requiring that sufficientlysmall geodesic spheres centered at x have constant mean curvature, and moreoversuch manifolds are Einstein manifolds [CR40]. Hence they have constant curvaturein dimensions 2 and 3. The Euclidean spaces and rank one symmetric spacesare examples of harmonic manifolds. The Lichnerowicz conjecture asserts thatconversely any harmonic manifold is either flat or locally symmetric of rank one.The conjecture was proved for harmonic manifolds of dimension 4 by A. G. Walker[Wal48]. In 1990 Z. I. Szabo proved the conjecture for compact simply connectedharmonic manifolds [Sza90]. In 1995 G. Besson, G. Courtois and S. Gallot provedthe conjecture for manifolds of negative curvature admitting a compact quotient[BCG95], using rigidity results from hyperbolic dynamics including the work of Y.Benoist, P. Foulon and F. Labourie [BFL92] and that of P. Foulon and F. Labourie[FL92]. In 2005 Y. Nikolayevsky proved the conjecture for harmonic manifolds ofdimension 5, showing that these must in fact have constant curvature [Nik05].In 1992 however E. Damek and F. Ricci had already provided a family of coun-terexamples to the conjecture in the noncompact case, which have come to beknown as harmonic NA groups , or Damek-Ricci spaces [DR92]. These are solvableLie groups X = N A with a suitable left-invariant Riemannian metric, given bythe semi-direct product of a nilpotent Lie group N of Heisenberg type (see [Kap80])with A = R + acting on N by anisotropic dilations. While the noncompact rank onesymmetric spaces G/K may be identified with harmonic
N A groups (apart fromthe real hyperbolic spaces), there are examples of harmonic
N A groups which arenot symmetric. In 2006, J. Heber proved that the only complete simply connectedhomogeneous harmonic manifolds are the Euclidean spaces, rank one symmetricspaces, and harmonic
N A groups [Heb06].Though the harmonic
N A groups are not symmetric in general, there is still awell developed theory of harmonic analysis on these spaces which parallels that ofthe symmetric spaces
G/K . For a noncompact symmetric space X = G/K , animportant role in the analysis on these spaces is played by the well-known
HelgasonFourier transform [Hel94]. For harmonic
N A groups, F. Astengo, R. Camporesiand B. Di Blasio have defined a Fourier transform [ACB97], which reduces to theHelgason Fourier transform when the space is symmetric. In both cases a Fourierinversion formula and a Plancherel theorem hold.The aim of the present article is to generalize these results to general noncompactharmonic manifolds and initiate a study of harmonic analysis on these spaces. Ouranalysis will apply however only to the negatively curved harmonic manifolds. Ourresults may be described briefly as follows:Let X be a complete, simply connected harmonic manifold of negative sectionalcurvature K ≤ −
1. Then X is a CAT(-1) space and can be compactified by addinga boundary at infinity ∂X , defined in terms of equivalence classes of geodesic rays γ : [0 , ∞ ) → X in X . Fix a basepoint o ∈ X . Then for each boundary point ξ ∈ ∂X , there is a Busemann function B ξ at ξ such that B ξ ( o ) = 0, defined for x ∈ X by B ξ ( x ) = lim y → ξ ( d ( x, y ) − d ( o, y ))where y ∈ X converges to ξ along the geodesic ray [ o, ξ ) joining o to ξ . The levelsets of the Busemann functions are called horospheres in X . The manifold X , HE FOURIER TRANSFORM ON NEGATIVELY CURVED HARMONIC MANIFOLDS 3 being harmonic, is also asymptotically harmonic , i.e. the mean curvature of allhorospheres is equal to a constant h ≥
0. Since X is negatively curved, in fact h is positive, and, by a result of G. Knieper [Kni12], the manifold X has purelyexponential volume growth , i.e. there is a constant C >
R > B ( x, R ) in X is given by1 C e hR ≤ vol ( B ( x, R )) ≤ Ce hR We let ρ = 12 h Then it turns out that for any λ ∈ C and ξ ∈ ∂X , the function f = e ( iλ − ρ ) B ξ is aneigenfunction of the Laplace-Beltrami operator ∆ on X with eigenvalue − ( λ + ρ ).The Fourier transform of a function f ∈ C ∞ c ( X ) is then defined to be the functionon C × ∂X given by ˜ f ( λ, ξ ) = Z X f ( x ) e ( − iλ − ρ ) B ξ ( x ) dvol ( x )When X is a noncompact rank one symmetric space, this reduces to the HelgasonFourier transform.The unit tangent sphere T o X is identified with the boundary ∂X via the home-omorphism p : v ∈ T o X ξ = γ v ( ∞ ) ∈ ∂X , where γ v is the unique geodesic raywith γ ′ v (0) = v . Pushing forward the Lebesgue measure on T o X by the map p givesa measure on ∂X called the visibility measure , which we denote by λ o . We thenhave the following Fourier inversion formula: Theorem 1.1.
There is a constant C > and a function c on C − { } such thatfor any f ∈ C ∞ c ( X ) , we have f ( x ) = C Z ∞ Z ∂X ˜ f ( λ, ξ ) e ( iλ − ρ ) B ξ ( x ) dλ o ( ξ ) | c ( λ ) | − dλ for all x ∈ X . We also have a Plancherel formula:
Theorem 1.2.
For any f, g ∈ C ∞ c ( X ) , we have Z X f ( x ) g ( x ) dvol ( x ) = C Z ∞ Z ∂X ˜ f ( λ, ξ )˜ g ( λ, ξ ) dλ o ( ξ ) | c ( λ ) | − dλ The Fourier transform extends to an isometry of L ( X, dvol ) into L ((0 , ∞ ) × ∂X, C dλ o ( ξ ) | c ( λ ) | − dλ ) . The article is organized as follows. In section 2 we recall the facts about CAT(-1) spaces and harmonic manifolds which we require. In section 3 we computethe action of the Laplacian ∆ on spaces of functions constant on geodesic spheresand horospheres respectively. In section 4 we carry out the harmonic analysis ofradial functions, i.e. functions constant on geodesic spheres centered around agiven point. Unlike the well-known
Jacobi analysis [Koo84] which applies to radialfunctions on rank one symmetric spaces and harmonic
N A groups, our analysishere is based on hypergroups [BH95]. We define a spherical Fourier transform forradial functions, and obtain an inversion formula and Plancherel theorem for this
KINGSHOOK BISWAS transform. In section 5 we prove the inversion formula and Plancherel formula forthe Fourier transform. The main point of the proof is an identity expressing radialeigenfunctions in terms of an integral over the boundary ∂X . Finally in section 6we define an operation of convolution with radial functions, and show that the L radial functions form a commutative Banach algebra under convolution. Acknowledgements.
The author would like to thank Swagato K. Ray andRudra P. Sarkar for generously sharing their time and knowledge over the courseof numerous educative and enjoyable discussions.2.
Preliminaries
In this section we recall the facts about CAT(-1) spaces and harmonic manifoldswhich we will require.2.1.
CAT(-1) spaces.
References for the material in this section include [BH99],[Bou95], [Bou96].Given a real number κ , a CAT( κ ) space X is a metric space with a syntheticnotion of having curvature bounded above by κ . In the following we will assumehowever that κ ≤
0. It is required firstly that X be a geodesic metric space , i.e. anytwo points x, y in X can be joined by a geodesic , which is an isometric embeddingof an interval of length l = d ( x, y ) in R into X . Secondly, one requires that geodesictriangles in X should be ”thinner” than the corresponding triangles in the modelspace M κ of constant curvature κ (for κ = 0, M κ = R , while for κ < M κ is realhyperbolic space H with the distance function scaled by a factor of 1 / √− κ ) in thefollowing sense:Given a geodesic triangle ∆ in X formed by three geodesic segments [ x, y ], [ y, z ],[ z, x ], a comparison triangle is a geodesic triangle ∆ in M κ formed by geodesicsegments [ x, y ], [ y, z ], [ z, x ] of the same lengths as those of ∆ (such a triangle existsand is unique up to isometry). A point p ∈ [ x, y ] is called a comparison point for p ∈ [ x, y ] if d ( x, p ) = d ( x, p ). We say X is a CAT( κ ) space if, in addition to beinggeodesic, X satisfies the ”CAT( κ ) inequality”: for all geodesic triangles ∆ in Xd ( p, q ) ≤ d ( p, q ) for all comparison points p, q ∈ ∆ ⊂ M κ . Alexandrov proved thata complete, simply connected Riemannian manifold is a CAT( κ ) space if and onlyif it has sectional curvature bounded above by κ .In a CAT( κ ) space X , there is a unique geodesic segment [ x, y ] joining any twopoints x, y , and moreover X is contractible. Given x ∈ X and two points y, z ∈ X distinct from x , the comparison angle θ x ( y, z ) is defined to be the angle at thevertex x in a comparison triangle in M κ corresponding to the geodesic triangle in X with vertices x, y, z .The boundary at infinity ∂X of a CAT( κ ) space X is defined to be the set ofequivalence classes of geodesic rays in X , where a geodesic ray is an isometricembedding γ : [0 , ∞ ) → X and two rays γ , γ are defined to be equivalent if { d ( γ ( t ) , γ ( t )) : t ≥ } is bounded (when X is a rank one symmetric space G/K or a harmonic
N A group, then the boundary is identified with
K/M and N ∪ {∞} respectively). For a geodesic ray γ , we denote by γ ( ∞ ) ∈ ∂X the equivalence class[ γ ] of γ . Then for any x ∈ X and ξ ∈ ∂X there is a unique geodesic ray γ joining HE FOURIER TRANSFORM ON NEGATIVELY CURVED HARMONIC MANIFOLDS 5 x to ξ , i.e. such that γ (0) = x and γ ( ∞ ) = ξ . We denote this ray by [ x, ξ ). Theset X := X ∪ ∂X can be given a topology called the cone topology , such that X isan open dense subset of X , and for any geodesic ray γ , γ ( t ) converges to γ ( ∞ ) as t → ∞ . Moreover, X is compact if and only if X is proper, i.e. closed balls in X are compact. We will assume for the rest of this section that X is proper, so that ∂X is compact.We now restrict ourselves to the case κ <
0, in which case after rescaling themetric by a constant we may assume κ = − X is a CAT(-1) space. Inthis case any two distinct points ξ, η ∈ ∂X can be joined by a unique bi-infinitegeodesic γ : R → X , i.e. γ ( −∞ ) = ξ, γ ( ∞ ) = η . We denote this bi-infinite geodesicby ( ξ, η ). We remark that this is a characteristic feature of negative curvature (orCAT(-1) spaces) as opposed to just nonpositive curvature (or CAT(0) spaces), forexample in higher rank symmetric spaces (which are CAT(0) but not CAT(-1)) notwo points in the boundary of a maximal flat can be joined by a geodesic. ForCAT(-1) spaces, we also have a natural family { ρ x : x ∈ X } of metrics on ∂X called visual metrics which are compatible with the cone topology on ∂X . Theseare defined as follows:Let x ∈ X be a basepoint. Given ξ, η ∈ ∂X , as points y, z in X convergeto ξ, η along the geodesic rays [ x, ξ ) , [ x, η ) the comparison angles θ x ( y, z ) increasemonotonically, and hence their limit exists; we define the comparison angle θ x ( ξ, η )to be this limit. The visual metric ρ x on ∂X based at x is then defined by ρ x ( ξ, η ) = sin (cid:18) θ x ( ξ, η ) (cid:19) The above formula does indeed define a metric on ∂X , which is of diameter one.For example, if we take X to be the unit ball model B n of n -dimensional realhyperbolic space and the point x is the center of the ball, then ∂X is naturallyidentified with S n − and the visual metric ρ x is simply half the chordal metric on S n − . In general, two points ξ, η ∈ ∂X are at visual distance one from each otherif and only if they are antipodal with respect to the basepoint x , in the sense that x lies on the geodesic ( ξ, η ).The Gromov inner product of two points y, z ∈ X with respect to a basepoint x ∈ X is defined by ( y | z ) x = 12 ( d ( x, y ) + d ( x, z ) − d ( y, z ))When X is a metric tree, the Gromov inner product equals the length of the segmentcommon to the geodesics [ x, y ] and [ x, z ]. For points ξ, η ∈ X ∪ ∂X , the Gromovinner product ( ξ | η ) x is defined by( ξ | η ) x = lim y → ξ,z → η ( y | z ) x (the limit exists taking values in [0 , + ∞ ], with ( ξ | η ) x = + ∞ if and only if ξ = η ∈ ∂X ). The Gromov inner product ( ξ | η ) x measures, up to a bounded error, thedistance of the geodesic ( ξ, η ) from the point x : there exists a constant C > | d ( x, ( ξ, η )) − ( ξ | η ) x | ≤ C KINGSHOOK BISWAS (for ξ = η ). The visual metric ρ x can be expressed in terms of the Gromov innerproduct as ρ x ( ξ, η ) = e − ( ξ | η ) x for ξ, η ∈ ∂X . It follows that for any ǫ >
0, there is a constant M = M ( ǫ ) > ρ x ( ξ, η ) ≥ ǫ then the geodesic ( ξ, η ) intersects the ball of radius M around x . The Busemann cocycle B : ∂X × X × X → R is defined by B ( x, y, ξ ) := lim z → ξ ( d ( x, z ) − d ( y, z ))It satisfies the cocycle identity B ( x, z, ξ ) = B ( x, y, ξ ) + B ( y, z, ξ )By the triangle inequality, | B ( x, y, ξ ) | ≤ d ( x, y ), and in fact equality occurs ifand only if x, y lie on a geodesic ray with endpoint ξ . For any ξ ∈ ∂X and x ∈ X , the Busemann function at ξ based at x is the function on X definedby B ξ,x ( y ) := B ( y, x, ξ ). Note that B ξ,x ( x ) = 0. The level sets of a Busemannfunction B ξ,x are called horospheres centered at ξ . A key property of CAT(-1)spaces is the exponential convergence of geodesics : given two points p, q in thesame horosphere centered at ξ ∈ ∂X , if x, y are points on the rays [ p, ξ ) , [ q, ξ ) with d ( p, x ) = d ( q, y ) = t >
0, then d ( x, y ) ≤ Ce − t for some constant C only depending on d ( p, q ). Thus the distance between thegeodesic rays [ p, ξ ) , [ q, ξ ), which is a priori only bounded, must in fact tend to zeroexponentially fast.The visual metrics satisfy the following Geometric Mean-Value Theorem : for x, y ∈ X and ξ, η ∈ ∂X , we have ρ y ( ξ, η ) = e B ( x,y,ξ ) e B ( x,y,η ) ρ x ( ξ, η ) , in particular they are all bi-Lipschitz equivalent to each other.We will need the following formula for Busemann functions: Lemma 2.1.
For o, x ∈ X, ξ ∈ ∂X , the limit of the comparison angles θ o ( x, z ) exists as z converges to ξ along the geodesic ray [ o, ξ ) . Denoting this limit by θ o ( x, ξ ) , it satisfies e B ξ,o ( x ) = cosh( d ( o, x )) − sinh( d ( o, x )) cos( θ o ( x, ξ )) Proof:
Consider a comparison triangle in H with side lengths a = d ( o, x ) , b = d ( o, z ) , c = d ( x, z ) and angle θ = θ ( x, z ) at the vertex corresponding to o . By thehyperbolic law of cosine we havecosh c = cosh a cosh b − sinh a sinh b cos θ As z → ξ , we have b, c → ∞ , and c − b → B ξ,o ( x ), thuscos θ = cosh a cosh b sinh a sinh b − cosh c sinh a sinh b → cosh a sinh a − e B ξ,o ( x ) sinh a HE FOURIER TRANSFORM ON NEGATIVELY CURVED HARMONIC MANIFOLDS 7 hence the angle θ converges to a limit. Denoting this limit by θ o ( x, ξ ), by the aboveit satisfies e B ξ,o ( x ) = cosh( d ( o, x )) − sinh( d ( o, x )) cos( θ o ( x, ξ )) ⋄ We will need the following lemma relating the asymptotics of Busemann func-tions to the Gromov inner product:
Lemma 2.2.
Let o ∈ X . Given ǫ > , there exists R = R ( ǫ ) > such that for any ξ, η ∈ ∂X with ρ o ( ξ, η ) ≥ ǫ and for any x ∈ [ o, η ) with r := d ( o, x ) ≥ R , | ( B ξ,o ( x ) − r ) + 2( ξ | η ) o | ≤ ǫ In particular, for distinct points ξ, η ∈ ∂X , B ξ,o ( x ) − r → − ξ | η ) o as x → η along the geodesic ray [ o, η ) . Proof:
Given ǫ >
0, let ξ, η ∈ ∂X be such that ρ o ( ξ, η ) ≥ ǫ , let x ∈ [ o, η ) and r = d ( o, x ). It is straightforward from the definitions that B ξ,o ( x ) − r = − ξ | x ) o Let x ′ ∈ X be the unique point of intersection of the geodesic ( ξ, η ) with thehorosphere centered at η passing through x , so that B ( x, x ′ , η ) = 0. Let y, z bepoints on the rays [ o, ξ ) , [ o, η ) converging to ξ and η respectively, and let y ′ , z ′ be thepoints of intersection of the geodesic ( ξ, η ) with the horospheres based at ξ and η passing through the points y, z respectively, then d ( y, y ′ ) , d ( z, z ′ ) → y → ξ, z → η by exponential convergence of geodesics. Using d ( y ′ , z ′ ) = d ( y ′ , x ′ ) + d ( x ′ , z ′ ), itfollows that d ( y, z ) = d ( y, x ′ ) + d ( x ′ , z ) + o (1), thus we have | ξ | η ) o − ξ | x ) o | = lim y → ξ,z → η | ( d ( o, y ) + d ( o, z ) − d ( y, z )) − ( d ( o, y ) + d ( o, x ) − d ( x, y )) | = lim y → ξ,z → η | ( d ( o, z ) − d ( o, x )) − d ( y, z ) + d ( x, y ) | = lim y → ξ,z → η | d ( x, z ) − ( d ( y, x ′ ) + d ( x ′ , z ) + o (1)) + d ( x, y ) | = lim y → ξ,z → η | ( d ( x, z ) − d ( x ′ , z )) + ( d ( x, y ) − d ( x ′ , y )) + o (1) | = | B ( x, x ′ , η ) − B ( x, x ′ , ξ ) | = | B ( x, x ′ , ξ ) |≤ d ( x, x ′ )Thus it remains to estimate d ( x, x ′ ) for r large.Since ρ o ( ξ, η ) ≥ ǫ , there exists M = M ( ǫ ) > p ∈ ( ξ, η ) suchthat d ( o, p ) ≤ M . Let p ′ be the point of intersection of the geodesic ( ξ, η ) withthe horosphere centered at η passing through o , so that d ( p, p ′ ) = | B ( p, o, η ) | and B ( p ′ , o, η ) = 0. Then d ( p ′ , x ′ ) = d ( o, x ) = r , and d ( o, p ′ ) ≤ d ( o, p ) + d ( p, p ′ ) ≤ M ,so by exponential convergence of geodesics there is C = C ( M ) > M such that d ( x, x ′ ) ≤ Ce − r KINGSHOOK BISWAS hence we can choose R = R ( ǫ ) > ǫ such that d ( x, x ′ ) ≤ ǫ for r ≥ R . ⋄ For x ∈ X and ξ ∈ ∂X , let B x ( ξ, ǫ ) ⊂ ∂X denote the ball of radius ǫ > ξ with respect to the visual metric ρ x . The following two lemmas will be neededlater in section 5 when estimating visual measures of the visual balls B x ( ξ, ǫ ). Lemma 2.3.
Let x ∈ X and ξ ∈ ∂X . For ǫ > , let y be the point on the geodesicray [ x, ξ ) at a distance r = log(1 /ǫ ) from x . Then:(1) For all η ∈ B x ( ξ, ǫ ) we have | B ( x, y, η ) − r | ≤ C (2) There is a universal constant δ > such that B y ( ξ, δ ) ⊂ B x ( ξ, ǫ ) Proof: (1): Let η ∈ B x ( ξ, ǫ ). Let z be the point on the geodesic ray [ x, η ) atdistance r = log(1 /ǫ ) from x , so that d ( x, y ) = d ( x, z ) = r . Let θ = θ x ( y, z )be the comparison angle at x between y, z , then θ ≤ θ x ( ξ, η ) (by monotonicityof comparison angles along geodesics), so sin( θ/ ≤ ρ x ( ξ, η ) ≤ ǫ . Applying thehyperbolic law of cosine to a comparison triangle ∆ for the triangle ∆ with vertices x, y, z gives cosh( d ( y, z )) = 1 + 2 sinh r sin (cid:18) θ (cid:19) ≤ r ) ǫ ≤ ǫ = e − r and (sinh r ) e − r ≤ / d ( y, z ) ≤
2, thus, using B ( x, z, η ) = r and the cocycle identity for the Busemann cocycle we have | B ( x, y, η ) − r | = | B ( x, y, η ) − B ( x, z, η ) | = | B ( y, z, η ) |≤ d ( y, z ) ≤ δ = sup { δ > | B y ( ξ, δ ) ⊂ B x ( ξ, ǫ ) } . Then B y ( ξ, δ ) ⊂ B x ( ξ, ǫ ), andthere exists η ∈ B y ( ξ, δ ) such that ρ x ( ξ, η ) = ǫ . Using the Geometric Mean-ValueTheorem and part (1) above this gives δ ≥ ρ y ( ξ, η ) ≥ e B ( x,y,ξ )+ B ( x,y,η ) ρ x ( ξ, η ) ≥ e r +( r − e − r = e − so δ ≥ e − . Thus if we put δ = (1 / e − then B y ( ξ, δ ) ⊂ B x ( ξ, ǫ ). ⋄ Lemma 2.4.
Let X be a complete, simply connected manifold of pinched negativecurvature, − b ≤ K ≤ − . For x ∈ X and ξ, η ∈ ∂X , let θ Rx ( ξ, η ) ∈ [0 , π ] denotethe Riemannian angle between the geodesic rays [ x, ξ ) , [ x, η ) at the point x . Then ρ x ( ξ, η ) b ≤ sin (cid:18) θ Rx ( ξ, η )2 (cid:19) ≤ ρ x ( ξ, η ) Proof:
Since K ≤ −
1, Topogonov’s theorem implies that the Riemannian angle θ Rx ( ξ, η ) is bounded above by the comparison angle θ x ( ξ, η ), hencesin (cid:18) θ Rx ( ξ, η )2 (cid:19) ≤ sin (cid:18) θ x ( ξ, η )2 (cid:19) = ρ x ( ξ, η )For the lower bound, let y, z be points on the geodesic rays [ x, ξ ) , [ x, η ) respectively,and let θ bx ( y, z ) denote the comparison angle between y, z at x in the model space H − b of constant curvature − b , i.e. the angle in a comparison triangle in H − b at the vertex corresponding to x . Then the limit of the angles θ bx ( y, z ) exists as y → ξ, z → η , and, denoting the limit by θ bx ( ξ, η ), it satisfiessin (cid:18) θ bx ( ξ, η )2 (cid:19) = ρ x ( ξ, η ) b ([Bis17], Lemma 3.6). The lower curvature bound − b ≤ K gives, by Topogonov’stheorem, that the comparison angles θ bx ( y, z ) are bounded above by the Riemannianangle θ Rx ( ξ, η ), hence ρ x ( ξ, η ) b = sin (cid:18) θ bx ( ξ, η )2 (cid:19) ≤ sin (cid:18) θ Rx ( ξ, η )2 (cid:19) ⋄ Harmonic manifolds.
References for this section include [RWW61], [Sza90],[Wil93] and [KP13].Throughout the rest of this paper, (
X, g ) will denote a complete, simply con-nected Riemannian manifold of negative sectional curvature K satisfying K ≤ − x ∈ X the exponential mapexp x : T x X → X is a diffeomorphism. Then X is a CAT(-1) space with boundary ∂X . It is known [HH77] that in this case Busemann functions B ξ,o are C on X ,and |∇ B ξ,o | = 1, so their level sets, the horospheres, are C submanifolds of X .Let T X denote the unit tangent bundle of X with fibres T x X , x ∈ X . For any r > T x X to the geodesicsphere S ( x, r ) of radius r around x , w exp x ( rw ). For any v ∈ T x X and r > A ( v, r ) denote the Jacobian of this map at the point v . We say that X is a harmonic manifold if A ( v, r ) does not depend on v , i.e. there is a function A on(0 , ∞ ) such that A ( v, r ) = A ( r ) for all v ∈ T X . The function A is called the density function of the harmonic manifold. The density function A is increasing in r , and the quantity A ′ ( r ) /A ( r ) ≥ S ( x, r ) of radius r , which decreases monotonically as r → ∞ to a constant h ≥ X . In particular, a harmonicmanifold is asymptotically harmonic , i.e. the mean curvature of horospheres is constant. Let ∆ denote the Laplace-Beltrami operator or Laplacian on X , then themean curvature of horospheres centered at ξ ∈ ∂X is given by ∆ B ξ,o , so∆ B ξ,o ≡ h for all ξ ∈ ∂X, o ∈ X . Since Busemann functions are strictly convex in negativecurvature (i.e. their Hessian is positive definite), it follows that in fact the constant h is positive.A function f on X is said to be radial around a point x of X if f is constanton geodesic spheres centered at x . For each x ∈ X , we can define a radializationoperator M x , defined for a continuous function f on X by( M x f )( z ) = Z S ( x,r ) f ( y ) dσ ( y )where S ( x, r ) denotes the geodesic sphere around x of radius r = d ( x, z ), and σ denotes surface area measure on this sphere (induced from the metric on X ),normalized to have mass one. The operator M x maps continuous functions tofunctions radial around x , and is formally self-adjoint, meaning Z X ( M x u )( z ) v ( z ) dvol ( z ) = Z X u ( z )( M x v )( z ) dvol ( z )for all continuous functions u, v with compact support.For x ∈ X , let d x denote the distance function from the point x , i.e. d x ( y ) = d ( x, y ). Then X is a harmonic manifold if and only if any of the following equivalentconditions hold:(1) For any x ∈ X , ∆ d x is radial around x .(2) The Laplacian commutes with all the radialization operators M x , i.e. M x ∆ u =∆ M x u for all smooth functions u on X and all x ∈ X .(3) For any smooth function u and any x ∈ X , if u is radial around x then ∆ u isradial around x .When X is harmonic, note that the mean curvature ( A ′ /A )( r ) of the geodesicsphere S ( x, r ) at a point z ∈ S ( x, r ) equals ∆ d x ( z ), hence we have∆ d x = A ′ A ◦ d x Radial and horospherical parts of the Laplacian
Let X be a complete, simply connected, negatively curved harmonic manifold(as before we assume sectional curvatures K ≤ − h > X , let ρ = h , and let A : (0 , ∞ ) → R denote thedensity function of X . Lemma 3.1.
For f a C function on X and u a C ∞ function on R , we have ∆( u ◦ f ) = ( u ′′ ◦ f ) |∇ f | + ( u ′ ◦ f )∆ f HE FOURIER TRANSFORM ON NEGATIVELY CURVED HARMONIC MANIFOLDS 11
Proof:
Let γ be a geodesic, then ( u ◦ f ◦ γ ) ′ ( t ) = ( u ′ ◦ f )( γ ( t )) < ∇ f, γ ′ ( t ) > , so( u ◦ f ◦ γ ) ′′ ( t ) = ( u ′′ ◦ f )( γ ( t )) < ∇ f, γ ′ ( t ) > +( u ′ ◦ f )( γ ( t )) < ∇ γ ′ ∇ f, γ ′ ( t ) > Now let { e i } be an orthonormal basis of T x X , and let γ i be geodesics with γ ′ i (0) = e i .Then∆( u ◦ f )( x ) = n X i =1 < ∇ e i ∇ ( u ◦ f ) , e i > = n X i =1 ( u ◦ f ◦ γ i ) ′′ (0)= ( u ′′ ◦ f )( x ) n X i =1 < ∇ f, e i > +( u ′ ◦ f )( x ) n X i =1 < ∇ e i ∇ f, e i > = ( u ′′ ◦ f )( x ) |∇ f ( x ) | + ( u ′ ◦ f )( x )∆ f ( x ) ⋄ Any C ∞ function on X radial around x ∈ X is of the form f = u ◦ d x for someeven C ∞ function u on R , where d x denotes the distance function from the point x ,while any C ∞ function which is constant on horospheres at ξ ∈ ∂X is of the form f = u ◦ B ξ,x for some C ∞ function u on R . The following proposition says that theLaplacian ∆ leaves invariant these spaces of functions, and describes the action ofthe Laplacian on these spaces: Proposition 3.2.
Let x ∈ X, ξ ∈ ∂X .(1) For u a C ∞ function on (0 , ∞ ) , ∆( u ◦ d x ) = ( L R u ) ◦ d x where L R is the differential operator on (0 , ∞ ) defined by L R = d dr + A ′ ( r ) A ( r ) ddr (2) For u a C ∞ function on R , ∆( u ◦ B ξ,x ) = ( L H u ) ◦ B ξ,x where L H is the differential operator on R defined by L H = d dt + 2 ρ ddt Proof:
Noting that |∇ d x | = 1 , |∇ B ξ,x | = 1, and ∆ d x = ( A ′ /A ) ◦ d x , ∆ B ξ,x = 2 ρ ,the Proposition follows immediately from the previous Lemma. ⋄ Accordingly, we call the differential operators L R and L H the radial and horo-spherical parts of the Laplacian respectively. It follows from the above propositionthat a function f = u ◦ d x radial around x is an eigenfunction of ∆ with eigenvalue σ if and only if u is an eigenfunction of L R with eigenvalue σ . Similarly, a function f = u ◦ B ξ,x constant on horospheres at ξ is an eigenfunction of ∆ with eigenvalue σ if and only if u is an eigenfunction of L H with eigenvalue σ . In particular, wehave the following: Proposition 3.3.
Let ξ ∈ ∂X, x ∈ X . Then for any λ ∈ C , the function f = e ( iλ − ρ ) B ξ,x is an eigenfunction of the Laplacian with eigenvalue − ( λ + ρ ) satisfying f ( x ) = 1 . Proof:
This follows from the fact that the function u ( t ) = e ( iλ − ρ ) t on R is aneigenfunction of L H with eigenvalue − ( λ + ρ ), and B ξ,x ( x ) = 0 gives f ( x ) = 1. ⋄ Analysis of radial functions
As we saw in the previous section, finding radial eigenfunctions of the Laplacianamounts to finding eigenfunctions of its radial part L R . When X is a rank onesymmetric space G/K , or more generally a harmonic
N A group, then the volumedensity function is of the form A ( r ) = C (cid:0) sinh (cid:0) r (cid:1)(cid:1) p (cid:0) cosh (cid:0) r (cid:1)(cid:1) q , for a constant C > p, q ≥
0, and so the radial part L R = d dr + ( A ′ /A ) ddr falls intothe general class of Jacobi operators L α,β = d dr + ((2 α + 1) coth r + (2 β + 1) tanh r ) ddr for which there is a detailed and well known harmonic analysis in terms of eigen-functions (called Jacobi functions ) [Koo84]. For a general harmonic manifold X ,the explicit form of the density function A is not known, so it is unclear whether theradial part L R is a Jacobi operator. However, there is a harmonic analysis, basedon hypergroups ([Che74], [Che79], [Tri81], [Tri97b], [Tri97a], [BX95], [Xu94]), formore general second-order differential operators on (0 , ∞ ) of the form(1) L = d dr + A ′ ( r ) A ( r ) ddr where A is a function on [0 , ∞ ) satisfying certain hypotheses which allow one toendow [0 , ∞ ) with a hypergroup structure, called a Chebli-Trimeche hypergroup .We first recall some basic facts about Chebli-Trimeche hypergroups, and then showthat the density function of a harmonic manifold satisfies the hypotheses requiredin order to apply this theory.4.1.
Chebli-Trimeche hypergroups.
A hypergroup ( K, ∗ ) is a locally compactHausdorff space K such that the space M b ( K ) of finite Borel measures on K isendowed with a product ( µ, ν ) µ ∗ ν turning it into an algebra with unit, and K is endowed with an involutive homeomorphism x ∈ K ˜ x ∈ K , such that theproduct and the involution satisfy certain natural properties (see [BH95] Chapter1 for the precise definition). A motivating example relevant to the following isthe algebra of finite radial measures on a noncompact rank one symmetric space G/K under convolution; as radial measures can be viewed as measures on [0 , ∞ ),this endows [0 , ∞ ) with a hypergroup structure (with the involution being theidentity). It turns out that this hypergroup structure on [0 , ∞ ) is a special case of ageneral class of hypergroup structures on [0 , ∞ ) called Sturm-Liouville hypergroups (see [BH95], section 3.5). These hypergroups arise from Sturm-Liouville boundaryproblems on (0 , ∞ ). We will be interested in a particular class of Sturm-Liouvillehypergroups called Chebli-Trimeche hypergroups . These arise as follows (we referto [BH95] for proofs of statements below):
HE FOURIER TRANSFORM ON NEGATIVELY CURVED HARMONIC MANIFOLDS 13 A Chebli-Trimeche function is a continuous function A on [0 , ∞ ) which is C ∞ and positive on (0 , ∞ ) and satisfies the following conditions:(H1) A is increasing, and A ( r ) → + ∞ as r → + ∞ .(H2) A ′ /A is decreasing, and ρ = lim r →∞ A ′ ( r ) /A ( r ) > r > A ( r ) = r α +1 B ( r ) for some α > − / C ∞ function B on R such that B (0) > L be the differential operator on C (0 , ∞ ) defined by equation (1), where A satisfies conditions (H1)-(H3) above. Define the differential operator l on C ((0 , ∞ ) )by l [ u ]( x, y ) = ( L ) x u ( x, y ) − ( L ) y u ( x, y )= (cid:18) u xx ( x, y ) + A ′ ( x ) A ( x ) u x ( x, y ) (cid:19) − (cid:18) u yy ( x, y ) + A ′ ( y ) A ( y ) u y ( x, y ) (cid:19) For f ∈ C ([0 , ∞ ) ) denote by u f the solution of the hyperbolic Cauchy problem l [ u f ] = 0 ,u f ( x,
0) = u f (0 , x ) = f ( x ) , ( u f ) y ( x,
0) = 0 , ( u f ) x (0 , y ) = 0 for x, y ∈ [0 , ∞ )For x ∈ [0 , ∞ ), let ǫ x denote the Dirac measure of mass one at x . Then for all x, y ∈ [0 , ∞ ), there exists a probability measure on [0 , ∞ ) denoted by ǫ x ∗ ǫ y suchthat Z ∞ f d ( ǫ x ∗ ǫ y ) = u f ( x, y )for all even, C ∞ functions f on R . We have ǫ x ∗ ǫ y = ǫ y ∗ ǫ x for all x, y , and theproduct ( ǫ x , ǫ y ) ǫ x ∗ ǫ y extends to a product on all finite measures on [0 , ∞ ) whichturns [0 , ∞ ) into a commutative hypergroup ([0 , ∞ ) , ∗ ) (with the involution beingthe identity), called the Chebli-Trimeche hypergroup associated to the function A .Any hypergroup has a Haar measure, which in this case is given by the measure A ( r ) dr on [0 , ∞ ).For a commutative hypergroup K with a Haar measure dk , a Fourier analysis canbe carried out analogous to the Fourier analysis on locally compact abelian groups.There is a dual space ˆ K of characters, which are bounded multiplicative functionson the hypergroup χ : K → C satisfying χ (˜ x ) = χ ( x ), where multiplicative meansthat Z K χd ( ǫ x ∗ ǫ y ) = χ ( x ) χ ( y )for all x, y ∈ K . For f ∈ L ( K ), the Fourier transform of f is the function ˆ f on ˆ K defined by ˆ f ( χ ) = Z K f χdk The Levitan-Plancherel Theorem states that there is a measure dχ on ˆ K called thePlancherel measure, such that the mapping f ˆ f extends from L ( K ) ∩ L ( K ) to an isometry from L ( K ) onto L ( ˆ K ). The inverse Fourier transform of a function σ ∈ L ( ˆ K ) is the function ˇ σ on K defined byˇ σ ( k ) = Z ˆ K σ ( χ ) χ ( k ) dχ The Fourier inversion theorem then states that if f ∈ L ( K ) ∩ C ( K ) is such thatˆ f ∈ L ( ˆ K ), then f = ( ˆ f )ˇ, i.e. f ( x ) = Z ˆ K ˆ f ( χ ) χ ( x ) dχ for all x ∈ K .For the Chebli-Trimeche hypergroup, it turns out that the multiplicative func-tions on the hypergroup are given precisely by eigenfunctions of the operator L .For any λ ∈ C , the equation(2) Lu = − ( λ + ρ ) u has a unique solution φ λ on (0 , ∞ ) which extends continuously to 0 and satisfies φ λ (0) = 1 (note that the coefficient A ′ /A of the operator L is singular at r = 0 soexistence of a solution continuous at 0 is not immediate). The function φ λ extendsto a C ∞ even function on R . Since equation (2) reads the same for λ and − λ , byuniqueness we have φ λ = φ − λ .The multiplicative functions on [0 , ∞ ) are then exactly the functions φ λ , λ ∈ C .The functions φ λ are bounded if and only if | Im λ | ≤ ρ . Furthermore, the involutionon the hypergroup being the identity, the characters of the hypergroup are real-valued, which occurs for φ λ if and only if λ ∈ R ∪ i R . Thus the dual space of thehypergroup is given by ˆ K = { φ λ | λ ∈ [0 , ∞ ) ∪ [0 , iρ ] } which we identify with the set Σ = [0 , ∞ ) ∪ [0 , iρ ] ⊂ C .The hypergroup Fourier transform of a function f ∈ L ([0 , ∞ ) , A ( r ) dr ) is givenby ˆ f ( λ ) = Z ∞ f ( r ) φ λ ( r ) A ( r ) dr for λ ∈ Σ (when the hypergroup arises from convolution of radial measures on a rankone symmetric space
G/K , then this is the well-known Jacobi transform [Koo84]).The Levitan-Plancherel and Fourier inversion theorems for the hypergroup givethe existence of a Plancherel measure σ on Σ such that the Fourier transformdefines an isometry from L ([0 , ∞ ) , A ( r ) dr ) onto L (Σ , σ ), and, for any function f ∈ L ([0 , ∞ ) , A ( r ) dr ) ∩ C ([0 , ∞ )) such that ˆ f ∈ L (Σ , σ ), we have f ( r ) = Z Σ ˆ f ( λ ) φ λ ( r ) dσ ( λ )for all r ∈ [0 , ∞ ).In [BX95], it is shown that under certain extra conditions on the function A , thesupport of the Plancherel measure is [0 , ∞ ) and the Plancherel measure is absolutelycontinuous with respect to Lebesgue measure dλ on [0 , ∞ ), given by dσ ( λ ) = C | c ( λ ) | − dλ HE FOURIER TRANSFORM ON NEGATIVELY CURVED HARMONIC MANIFOLDS 15 where C > c is a certain complex function on C − { } . Therequired conditions on A are as follows:Making the change of dependent variable v = A / u , equation (2) becomes(3) v ′′ ( r ) = ( G ( r ) − λ ) v ( r )where the function G is defined by(4) G ( r ) = 14 (cid:18) A ′ ( r ) A ( r ) (cid:19) + 12 (cid:18) A ′ A (cid:19) ′ ( r ) − ρ If the function G tends to 0 fast enough near infinity, then it is reasonable toexpect that equation (3) above has two linearly independent solutions asymptoticto exponentials e ± iλr near infinity. Bloom and Xu show that this is indeed the case[BX95] under the following hypothesis on the function G :(H4) For some r >
0, we have Z ∞ r r | G ( r ) | dr < + ∞ and G is bounded on [ r , ∞ ).Under hypothesis (H4), for any λ ∈ C − { } , there are unique solutions Φ λ , Φ − λ of equation (2) on (0 , ∞ ) which are asymptotic to exponentials near infinity [BX95],Φ ± λ ( r ) = e ( ± iλ − ρ ) r (1 + o (1)) as r → + ∞ The solutions Φ λ , Φ − λ are linearly independent, so, since φ λ = φ − λ , there exists afunction c on C − { } such that φ λ = c ( λ )Φ λ + c ( − λ )Φ − λ for all λ ∈ C − { } . We will call this function the c -function of the hypergroup.We remark that if the hypergroup ([0 , ∞ ) , ∗ ) is the one arising from convolution ofradial measures on a noncompact rank one symmetric space G/K , then this functionagrees with Harish-Chandra’s c -function only on the half-plane { Im λ ≤ } and noton all of C .If we furthermore assume the hypothesis | α | 6 = 1 /
2, then Bloom-Xu show thatthe function c is non-zero for Im λ ≤ , λ = 0, and prove the following estimates:There exist constants C, K > C | λ | ≤ | c ( λ ) | − ≤ C | λ | , | λ | ≤ K C | λ | α + ≤ | c ( λ ) | − ≤ C | λ | α + , | λ | ≥ K Moreover they prove the following inversion formula: for any even function f ∈ C ∞ c ( R ), f ( r ) = C Z ∞ ˆ f ( λ ) φ λ ( r ) | c ( λ ) | − dλ where C > It follows that the Plancherel measure σ of the hypergroup is supported on[0 , ∞ ), and absolutely continuous with respect to Lebesgue measure, with densitygiven by C | c ( λ ) | − . Bloom-Xu also show that the c -function is holomorphic onthe half-plane { Im λ < } .4.2. The density function of a harmonic manifold.
Let X be a simply con-nected, n -dimensional negatively curved harmonic manifold as before, and let A be the density function of X . We check that A is a Chebli-Trimeche function,so that we obtain a commutative hypergroup ([0 , ∞ ) , ∗ ), and that the conditionsof Bloom-Xu are met so that the Plancherel measure is given by C | c ( λ ) | − dλ on[0 , ∞ ).The function A ( r ) equals, up to a constant factor, the volume of geodesic spheres S ( x, r ), which is increasing in r and tends to infinity as r tends to infinity, socondition (H1) is satisfied. As stated in section 2.2, the function A ′ ( r ) /A ( r ) equalsthe mean curvature of geodesic spheres S ( x, r ), which decreases monotonically toa limit 2 ρ which is positive (and equals the mean curvature of horospheres), socondition (H2) is satisfied.Fixing a point x ∈ X , for r >
0, the density function A ( r ) is given by theJacobian of the map φ : v exp x ( rv ) from the unit tangent sphere T x X to thegeodesic sphere S ( x, r ). Let T be the map v rv from the unit tangent sphere T x X to the tangent sphere of radius r , T rx X ⊂ T x M , then φ = exp x ◦ T , so theJacobian of φ is given by the product of the Jacobians of T and exp x , hence A ( r ) = r n − B ( r )where the function B is given by B ( r ) = det( D exp x ) rv where v is any fixed vector in T x X . Since B is independent of the choice of v , inparticular is the same for vectors v and − v , the function B is even, and C ∞ on R with B (0) = 1. Thus condition (H3) holds for the function A , with α = ( n − / A is thus a Chebli-Trimeche function, so we obtain a hy-pergroup structure on [0 , ∞ ), which we call the radial hypergroup of the harmonicmanifold X (the reason for this terminology will become clear from the the followingsections).We proceed to check that condition (H4) is satisfied. For this we will need thefollowing theorem of Nikolayevsy: Theorem 4.1. [Nik05]
The density function of a harmonic manifold is an expo-nential polynomial, i.e. a function of the form A ( r ) = k X i =1 ( p i ( r ) cos( β i r ) + q i ( r ) sin( β i r )) e α i r where p i , q i are polynomials and α i , β i ∈ R , i = 1 , . . . , k . HE FOURIER TRANSFORM ON NEGATIVELY CURVED HARMONIC MANIFOLDS 17
It will be convenient to rearrange terms and write the density function in theform(5) A ( r ) = l X i =1 m i X j =0 f ij ( r ) r j e α i r where α < α < · · · < α l , and each f ij is a trigonometric polynomial, i.e. a finitelinear combination of functions of the form cos( βr ) and sin( βr ), β ∈ R , with f im i not identically zero, for i = 1 , . . . , l . For an exponential polynomial written in thisform, we will call the largest exponent α l which appears in the exponentials the exponential degree of the exponential polynomial. Lemma 4.2.
With the density function as above, we have α l = 2 ρ, m l = 0 and f l = C for some constant C > . Thus the density function is of the form A ( r ) = Ce ρr + P ( r ) where P is an exponential polynomial of exponential degree δ < ρ . Proof:
Since X is CAT(-1), in particular X is Gromov-hyperbolic, so by a resultof Knieper [Kni12] X has purely exponential volume growth , i.e. there exists aconstant C > C ≤ A ( r ) e ρr ≤ C for all r ≥
1. If α l < ρ , then A ( r ) /e ρr → r → ∞ , contradicting (6)above, so we must have α l ≥ ρ . On the other hand, if α l > ρ , then since f lm l is atrigonometric polynomial which is not identically zero, we can choose a sequence r m tending to infinity such that f lm l ( r m ) → α = 0. Then clearly A ( r m ) /e ρr m → ∞ ,again contradicting (6). Hence α l = 2 ρ .Using (5) and α l = 2 ρ , we have A ′ ( r ) A ( r ) − ρ = f ′ lm l ( r ) + o (1) f lm l ( r ) + o (1)as r → ∞ , thus f ′ lm l ( r ) + o (1) = ( f lm l ( r ) + o (1)) (cid:18) A ′ ( r ) A ( r ) − ρ (cid:19) → r → ∞ since f lm l is bounded and A ′ ( r ) /A ( r ) − ρ → r → ∞ . Thus f ′ lm l isa trigonometric polynomial which tends to 0 as r → ∞ , so it must be identicallyzero, hence f lm l = C for some non-zero constant C .It follows that A ( r ) = Cr m l e ρr (1 + o (1))as r → ∞ . If m l ≥ A ( r ) /e ρr → ∞ as r → ∞ , so we must have m l = 0. ⋄ Lemma 4.3.
Condition (H4) holds for the density function A , i.e. Z ∞ r r | G ( r ) | dr < + ∞ and G is bounded on [ r , ∞ ) for any r > , where G ( r ) = 14 (cid:18) A ′ ( r ) A ( r ) (cid:19) + 12 (cid:18) A ′ A (cid:19) ′ ( r ) − ρ Proof:
By the previous lemma, A ( r ) = Ce ρr + P ( r ), where P is an exponentialpolynomial of exponential degree δ < ρ . We then have A ′ ( r ) A ( r ) − ρ = P ′ ( r ) − ρP ( r ) Ce ρr + P ( r )= Q ( r ) Ce ρr + P ( r )where Q is an exponential polynomial of exponential degree less than or equal to δ . Putting α = (2 ρ − δ ) /
2, it follows that A ′ ( r ) /A ( r ) − ρ = O ( e − αr ) as r → ∞ .Differentiating, we obtain (cid:18) A ′ A (cid:19) ′ ( r ) = ( Ce ρr + P ( r )) Q ′ ( r ) − Q ( r )(2 ρCe ρr + P ′ ( r ))( Ce ρr + P ( r )) = Q ( r )( Ce ρr + P ( r )) where Q is an exponential polynomial of exponential degree less than or equal to(2 ρ + δ ). Since the denominator of the above expression is of the form ke ρr + P ( r )with P an exponential polynomial of exponential degree strictly less than 4 ρ , itfollows that ( A ′ /A ) ′ ( r ) = O ( e − αr ) as r → ∞ .Now we can write the function G as G ( r ) = 14 (cid:18) A ′ ( r ) A ( r ) − ρ (cid:19) (cid:18) A ′ ( r ) A ( r ) + 2 ρ (cid:19) + 12 (cid:18) A ′ A (cid:19) ′ ( r )Since ( A ′ ( r ) /A ( r ) + 2 ρ ) is bounded, it follows from the previous paragraph that G ( r ) = O ( e − αr ) as r → ∞ . This immediately implies that condition (H4) holds. ⋄ In order to apply the result of Bloom-Xu on the Plancherel measure for thehypergroup, it remains to check that | α | 6 = 1 /
2. Since α = ( n − /
2, this means n = 3. Now the Lichnerowicz conjecture holds in dimensions n ≤ X = G/K , for which as mentioned earlier the Jacobianalysis applies, and the Plancherel measure of the hypergroup is well known to begiven by C | c ( λ ) | − dλ where c is Harish-Chandra’s c -function. Thus in our casewe may as well assume that X has dimension n ≥
6, so that | α | 6 = 1 /
2, and we maythen apply the results of Bloom-Xu stated in the previous section.
HE FOURIER TRANSFORM ON NEGATIVELY CURVED HARMONIC MANIFOLDS 19
The spherical Fourier transform.
Let φ λ denote as in section 4.1 theunique function on [0 , ∞ ) satisfying L R φ λ = − ( λ + ρ ) φ λ and φ λ (0) = 1. For x ∈ X let d x denote as before the distance function from the point x , d x ( y ) = d ( x, y ). Wedefine the following eigenfunction of ∆ radial around x : φ λ,x := φ λ ◦ d x The uniqueness of φ λ as an eigenfunction of L R with eigenvalue − ( λ + ρ ) andtaking the value 1 at r = 0 immediately implies the following lemma: Lemma 4.4.
The function φ λ,x is the unique eigenfunction f of ∆ on X witheigenvalue − ( λ + ρ ) which is radial around x and satisfies f ( x ) = 1 . Note that for λ ∈ R , the functions φ λ,x are bounded. Let dvol denote theRiemannian volume measure on X . Definition 4.5.
Let f ∈ L ( X, dvol ) be radial around the point x ∈ X . We definethe spherical Fourier transform of f by ˆ f ( λ ) := Z X f ( y ) φ λ,x ( y ) dvol ( y ) for λ ∈ R . For f a function on X radial around the point x , let f = u ◦ d x where u is afunction on [0 , ∞ ), then evaluating the integral over X in geodesic polar coordinatesgives Z X | f ( y ) | dvol ( y ) = Z ∞ | u ( r ) | A ( r ) dr thus f ∈ L ( X ) if and only if u ∈ L ([0 , ∞ ) , A ( r ) dr ). In that case, again integratingin polar coordinates givesˆ f ( λ ) = Z ∞ u ( r ) φ λ ( r ) A ( r ) dr = ˆ u ( λ )where ˆ u is the hypergroup Fourier transform of the function u . Moreover f ∈ C ∞ c ( X ) if and only if u extends to an even function on R such that u ∈ C ∞ c ( R ).Applying the Fourier inversion formula of Bloom-Xu for the radial hypergroupstated in section 4.1 to the function u then leads immediately to the followinginversion formula for radial functions: Theorem 4.6.
Let f ∈ C ∞ c ( X ) be radial around the point x ∈ X . Then f ( y ) = C Z ∞ ˆ f ( λ ) φ λ,x ( y ) | c ( λ ) | − dλ for all y ∈ X . Here c denotes the c -function of the radial hypergroup and C > isa constant. Proof:
As shown in the previous section, all the hypotheses required to apply theinversion formula of Bloom-Xu are satisfied, hence u ( r ) = C Z ∞ ˆ u ( λ ) φ λ ( r ) | c ( λ ) | − dλ Since f = u ◦ d x , this gives f ( y ) = u ( d x ( y ))= C Z ∞ ˆ u ( λ ) φ λ ( d x ( y )) | c ( λ ) | − dλ = C Z ∞ ˆ f ( λ ) φ λ,x ( y ) | c ( λ ) | − dλ ⋄ The Plancherel theorem for the radial hypergroup leads to the following:
Theorem 4.7.
Let L x ( X, dvol ) denote the closed subspace of L ( X ) consisting ofthose functions in L ( X ) which are radial around the point x . For f ∈ L ( X, dvol ) ∩ L x ( X, dvol ) , we have Z X | f ( y ) | dvol ( y ) = C Z ∞ | ˆ f ( λ ) | | c ( λ ) | − dλ The spherical Fourier transform f ˆ f extends to an isometry from L x ( X, dvol ) onto L ([0 , ∞ ) , C | c ( λ ) | − dλ ) . Proof:
The map u f = u ◦ d x defines an isometry of L ([0 , ∞ ) , A ( r ) dr ) onto L ( X, dvol ) x , which maps L ([0 , ∞ ) , A ( r ) dr ) ∩ L ([0 , ∞ ) , A ( r ) dr ) onto L ( X, dvol ) ∩ L x ( X, dvol ). The statements of the theorem then follow from the Levitan-Planchereltheorem for the radial hypergroup and from the fact that the Plancherel measureis supported on [0 , ∞ ), given by C | c ( λ ) | − dλ . ⋄ Fourier inversion and Plancherel theorem
We proceed to the analysis of non-radial functions on X . Our definition ofFourier transform will depend on the choice of a basepoint x ∈ X . Definition 5.1.
Let x ∈ X . For f ∈ C ∞ c ( X ) , the Fourier transform of f based atthe point x is the function on C × ∂X defined by ˜ f x ( λ, ξ ) = Z X f ( y ) e ( − iλ − ρ ) B ξ,x ( y ) dvol ( y ) for λ ∈ C , ξ ∈ ∂X . Here as before B ξ,x denotes the Busemann function at ξ basedat x such that B ξ,x ( x ) = 0 . Using the formula B ξ,x = B ξ,o − B ξ,o ( x )for points o, x ∈ X , we obtain the following relation between the Fourier transformsbased at two different basepoints o, x ∈ X :(7) ˜ f x ( λ, ξ ) = e ( iλ + ρ ) B ξ,o ( x ) ˜ f o ( λ, ξ )The key to passing from the inversion formula for radial functions of section 4.3to an inversion formula for non-radial functions will be a formula expressing the HE FOURIER TRANSFORM ON NEGATIVELY CURVED HARMONIC MANIFOLDS 21 radial eigenfunctions φ λ,x as an integral with respect to ξ ∈ ∂X of the eigenfunctions e ( iλ − ρ ) B ξ,x (Theorem 5.9). This will be the analogue of the well-known formulae forrank one symmetric spaces G/K and harmonic
N A groups expressing the radialeigenfunctions φ λ,x as matrix coefficients of representations of G on L ( K/M ) and
N A on L ( N ) respectively. We first need to define the visibility measures on theboundary ∂X :Given a point x ∈ X , let λ x be normalized Lebesgue measure on the unit tan-gent sphere T x X , i.e. the unique probability measure on T x X invariant under theorthogonal group of the tangent space T x M . For v ∈ T x X , let γ v : [0 , ∞ ) → X bethe unique geodesic ray with initial velocity v . Then we have a homeomorphism p x : T x X → ∂X, v γ v ( ∞ ). The visibility measure on ∂X (with respect to thebasepoint x ) is defined to be the push-forward ( p x ) ∗ λ x of λ x under the map p x ;for notational convenience, we will however denote the visibility measure on ∂X bythe same symbol λ x .In [KP13], it is shown that the visibility measures λ x , x ∈ X are mutually abso-lutely continuous and their Radon-Nikodym derivatives are given by dλ y dλ x ( ξ ) = e − ρB ξ,x ( y ) The above formula for Radon-Nikodym derivatives leads in a standard way to thefollowing estimate for visual measures of visual balls (in the literature the estimateis usually proved for ”shadows” of balls in X and called Sullivan’s Shadow Lemma): Lemma 5.2.
There is a constant
C > such that C ǫ ρ ≤ λ x ( B x ( ξ, ǫ )) ≤ Cǫ ρ for all x ∈ X, ξ ∈ ∂X, ǫ > . Proof:
Given x ∈ X, ξ ∈ ∂X and ǫ >
0, choose y on the geodesic ray [ x, ξ ) atdistance r = log(1 /ǫ ) from x . Using the formula for the Radon-Nikodym derivativeof λ x with respect to λ y , we have λ x ( B x ( ξ, ǫ )) = Z B x ( ξ,ǫ ) e − ρB ( x,y,η ) dλ y ( η )On the other hand by Lemma 2.3 we know that | B ( x, y, η ) − r | ≤ η ∈ B x ( ξ, ǫ ). Using e − ρr = ǫ ρ , it follows that1 C λ y ( B x ( ξ, ǫ )) ǫ ρ ≤ λ x ( B x ( ξ, ǫ )) ≤ Cλ y ( B x ( ξ, ǫ )) ǫ ρ for some constant C . Since λ y is a probability measure, this gives the requiredupper bound λ x ( B x ( ξ, ǫ )) ≤ Cǫ ρ for some constant C > λ y ( B x ( ξ, ǫ )) by apositive constant. By Lemma 2.3, there is a universal constant δ > B y ( ξ, δ ) ⊂ B x ( ξ, ǫ ), so λ y ( B x ( ξ, ǫ )) ≥ λ y ( B y ( ξ, δ )) . Now the sectional curvature of X is bounded below, − b ≤ K ([Kni02], Cor. 2.12in section 1.2). Identifying ∂X with T y X via the map p y : T y X → ∂X and thenidentifying T y X with the standard sphere S n − via an isometry T : R n → T y X , themeasure λ y corresponds to Lebesgue measure on S n − , and it follows from Lemma2.4 that the image of the ball B y ( ξ, δ ) in S n − contains a ball in S n − of radius δ b with respect to the chordal metric, which has Lebesgue measure α > α only depends on δ , hence λ y ( B y ( ξ, δ )) ≥ α and the Lemma follows. ⋄ Lemma 5.3.
Let x ∈ X and ξ ∈ ∂X . Then the integral c ( λ, x, ξ ) := Z ∂X ρ x ( ξ, η ) iλ − ρ ) dλ x ( η ) converges for Im λ < and is holomorphic in λ on the half-plane { Im λ < } . Proof:
Let λ = σ − iτ , where σ ∈ R and τ >
0. Then Z ∂X | ρ x ( ξ, η ) iλ − ρ ) | dλ x ( η ) = Z ∂X ρ x ( ξ, η ) τ − ρ ) dλ x ( η )= Z ∞ λ x ( { η | ρ x ( ξ, η ) τ − ρ ) > t } ) dt If τ ≥ ρ then the set { η | ρ x ( ξ, η ) τ − ρ ) > t } is empty for t > , λ x is a probability measure.For 0 < τ < ρ using Lemma 5.2 and the fact that λ x is a probability measurewe have Z ∞ λ x ( { η | ρ x ( ξ, η ) τ − ρ ) > t } ) dt ≤ Z ∞ λ x ( B x ( ξ, (1 /t ) / ρ − τ ) )) dt ≤ C Z ∞ (cid:18) t (cid:19) ρ ρ − τ ) dt< + ∞ Thus the integral defining c ( λ, x, ξ ) converges for Im λ <
0. That c ( λ, x, ξ ) isholomorphic in λ follows from Morera’s theorem. ⋄ Lemma 5.4.
Let x ∈ X and ξ ∈ ∂X . Then for all λ ∈ C , φ λ,x = M x ( e ( iλ − ρ ) B ξ,x ) (where M x is the radialisation operator around the point x ). In particular, φ λ,x ( y ) is entire in λ for fixed y ∈ X , and is real and positive for λ such that ( iλ − ρ ) isreal and positive. Proof:
Since the function e ( iλ − ρ ) B ξ,x is an eigenfunction of the Laplacian ∆ witheigenvalue − ( λ + ρ ) and the operator M x commutes with ∆, the function f = HE FOURIER TRANSFORM ON NEGATIVELY CURVED HARMONIC MANIFOLDS 23 M x ( e ( iλ − ρ ) B ξ,x ) is also an eigenfunction of ∆ for the eigenvalue − ( λ + ρ ). Since f is radial around x and f ( x ) = 1, it follows from Lemma 4.4 that f = φ λ,x . ⋄ Proposition 5.5.
Let f ∈ C ∞ c ( X ) be radial around the point x ∈ X . Then theFourier transform of f based at x coincides with the spherical Fourier transform, ˜ f x ( λ, ξ ) = ˆ f ( λ ) for all λ ∈ C , ξ ∈ ∂X . Proof:
Let f = u ◦ d x where u ∈ C ∞ c ( R ). By Lemma 5.4 above, φ λ ( r ) = φ − λ ( r ) = Z S ( x,r ) e ( − iλ − ρ ) B ξ,x ( y ) dσ r ( y )where σ r is normalized surface area measure on the geodesic sphere S ( x, r ). Eval-uating the integral defining ˜ f x in geodesic polar coordinates centered at x we have˜ f x ( λ, ξ ) = Z ∞ Z S ( x,r ) f ( y ) e ( − iλ − ρ ) B ξ,x ( y ) dσ r ( y ) A ( r ) dr = Z ∞ u ( r ) φ λ ( r ) A ( r ) dr = ˆ f ( λ ) ⋄ Proposition 5.6.
Let c be the c -function of the radial hypergroup of X and let Im λ < . Then:(1) We have lim r →∞ φ λ ( r ) e ( iλ − ρ ) r = c ( λ ) (2) We have c ( λ ) = Z ∂X ρ x ( ξ, η ) iλ − ρ ) dλ x ( η ) for any x ∈ X, ξ ∈ ∂X . In particular the integral c ( λ, x, ξ ) above is independent ofthe choice of x, ξ . Proof: (1): For Im λ <
0, using φ λ = c ( λ )Φ λ + c ( − λ )Φ − λ and Φ ± λ ( r ) = e ( ± iλ − ρ ) r (1+ o (1)) as r → ∞ , we have φ λ ( r ) e ( iλ − ρ ) r = c ( λ )(1 + o (1)) + c ( − λ ) e − iλr (1 + o (1)) → c ( λ )as r → ∞ .(2): Let λ = it where t ≤ − ρ , so that µ := iλ − ρ ≥
0. Fix x ∈ X and ξ ∈ ∂X .For η ∈ ∂X , let y ( η, r ) ∈ X denote the point on the geodesic [ x, η ) at a distance r from x . Then the normalized surface area measure on the geodesic sphere S ( x, r )is given by the push-forward of λ x under the map η y ( η, r ), so by Lemma 5.4 φ λ ( r ) e ( iλ − ρ ) r = Z ∂X e ( iλ − ρ )( B ξ,x ( y ( η,r )) − r ) dλ x ( ξ )We will apply the dominated convergence theorem to evaluate the limit of theabove integral as r → ∞ . First note that by Lemma 2.2, for any η not equal to ξ , B ξ,x ( y ( η, r )) − r → − ξ | η ) x as r → ∞ , so the integrand converges a.e. as r → ∞ , e ( iλ − ρ )( B ξ,x ( y ( η,r )) − r ) → ρ x ( ξ, η ) iλ − ρ ) (using ρ x ( ξ, η ) = e − ( ξ | η ) x ).Now by Lemma 2.1, using µ ≥ e µ ( B ξ,x ( y ( η,r )) − r ) = (cosh r − sinh r cos( θ x ( ξ, y ( x, r )))) µ e − µr = ( e − r + 2 sinh r sin ( θ x ( ξ, y ( x, r )) / µ e − µr ≤ ( e − r + ( e r − e − r ) · µ e − µr = 1So dominated convergence applies and we conclude that φ λ ( r ) e ( iλ − ρ ) r → Z ∂X ρ x ( ξ, η ) iλ − ρ ) dλ x ( η )as r → ∞ for λ = it, t ≤ − ρ .It follows from part (1) of the proposition that c ( λ ) equals the integral above,which is c ( λ, x, ξ ), for λ = it, t ≤ − ρ . Since c ( λ ) and c ( λ, x, ξ ) are holomorphic forIm λ <
0, they must then be equal for all λ with Im λ < ⋄ Remark.
Part (2) of the above proposition is the analogue of the well-knownintegral formula for Harish-Chandra’s c -function (formula (18) in [Hel94], pg. 108).For λ ∈ C and x ∈ X , define the function ˜ φ λ,x on X by˜ φ λ,x ( y ) = Z ∂X e ( iλ − ρ ) B ξ,x ( y ) dλ x ( y )It follows from the above equation that ˜ φ λ,x ( y ) is entire in λ for fixed y ∈ X ,and is real and positive for λ such that ( iλ − ρ ) is real and positive. Moreover,by Proposition 3.3, the function ˜ φ λ,x is an eigenfunction of the Laplacian ∆ witheigenvalue − ( λ + ρ ), and ˜ φ λ,x ( x ) = 1. Lemma 5.7.
Let λ = it, t < − ρ , and let x ∈ X . Given ǫ > , there exists R = R ( ǫ ) such that for any y ∈ X with r = d ( x, y ) ≥ R , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ φ λ,x ( y ) e ( iλ − ρ ) r − c ( λ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ǫ HE FOURIER TRANSFORM ON NEGATIVELY CURVED HARMONIC MANIFOLDS 25
Proof:
Let µ = iλ − ρ >
0. We may assume ǫ > | e µδ − | ≤ µǫ for 0 ≤ δ ≤ ǫ .Let y ∈ X with r = d ( x, y ) >
0, then the geodesic segment [ x, y ] extends uniquelyto a geodesic ray [ x, y + ) for some y + ∈ ∂X . By Lemma 2.2, there exists R = R ( ǫ )such that if r ≥ R and ξ ∈ ∂X is such that ρ x ( ξ, y + ) ≥ ǫ then | ( B ξ,x ( y ) − r ) + 2( ξ | y + ) x | < ǫ and hence we can write exp µB ξ,x ( y ) e µr = ρ x ( ξ, y + ) µ e µδ ( ξ ) where δ = δ ( ξ ) satisfies | δ ( ξ ) | ≤ ǫ for ρ x ( ξ, y + ) ≥ ǫ . Now˜ φ λ,x ( y ) e µr = Z ∂X e µB ξ,x ( y ) e µr dλ x ( ξ )= Z ∂X − B x ( y + ,ǫ ) e µB ξ,x ( y ) e µr dλ x ( ξ ) + Z B x ( y + ,ǫ ) e µB ξ,x ( y ) e µr dλ x ( ξ )= I + I say. We estimate the two integrals I , I appearing above separately.For the second integral I , since µ ≥
0, the same estimate as in the proof ofProposition 5.6 gives e µB ξ,x ( y ) /e µr ≤
1, hence by Lemma 5.2 we obtain | I | ≤ Cǫ ρ for some constant C > I , by Lemma 5.6 since c ( λ ) = c ( λ, x, y + ) we have | I − c ( λ ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∂X − B x ( y + ,ǫ ) e µB ξ,x ( y ) e µr dλ x ( ξ ) − Z ∂X ρ x ( ξ, y + ) µ dλ x ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z ∂X − B x ( y + ,ǫ ) ρ x ( ξ, y + ) µ | e µδ ( ξ ) − | dλ x ( ξ ) + Z B x ( y + ,ǫ ) ρ x ( ξ, y + ) µ dλ x ( ξ ) ≤ µǫ + Cǫ ρ (where we have used ρ x ( ξ, y + ) ≤ µ >
0, and Lemma 5.2).Putting together the estimates for I and I gives (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ φ λ,x ( y ) e ( iλ − ρ ) r − c ( λ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ µǫ + 2 Cǫ ρ for all y ∈ X such that r = d ( x, y ) ≥ R ( ǫ ). Since ǫ > ⋄ Lemma 5.8.
Let λ = it, t < − ρ and let x ∈ X . Given ǫ > , there exists R = R ( ǫ ) such that for any y ∈ X with r = d ( x, y ) ≥ R , we have − ǫ ≤ ˜ φ λ,x ( y ) φ λ,x ( y ) ≤ ǫ (note that for the given value of λ both functions appearing above are real andpositive). Proof:
Note c ( λ ) > R = R ( ǫ ) such that for y ∈ X with r = d ( x, y ) ≥ R we have(1 − ǫ ) c ( λ ) ≤ ˜ φ λ,x ( y ) e ( iλ − ρ ) r ≤ (1 + ǫ ) c ( λ )and (1 − ǫ ) c ( λ ) ≤ φ λ,x ( y ) e ( iλ − ρ ) r ≤ (1 + ǫ ) c ( λ )The above two inequalities imply1 − ǫ ǫ ≤ ˜ φ λ,x ( y ) φ λ,x ( y ) ≤ ǫ − ǫ Since ǫ > ⋄ We can finally prove the required formula expressing φ λ,x as an integral over ξ ∈ ∂X of the functions e ( iλ − ρ ) B ξ,x : Theorem 5.9.
Let λ ∈ C and x ∈ X . Then φ λ,x ( y ) = Z ∂X e ( iλ − ρ ) B ξ,x ( y ) dλ x ( ξ ) for all y ∈ X (i.e. φ λ,x = ˜ φ λ,x ). Proof:
We first assume λ is of the form λ = it, t < − ρ .For notational convenience, write φ = φ λ,x and ˜ φ = ˜ φ λ,x , then φ, ˜ φ are positivefunctions. Let u be the function u = ˜ φ/φ .Using the fact that φ, ˜ φ are both eigenfunctions of ∆ for the eigenvalue − ( λ + ρ )and the classical formula∆( φu ) = (∆ φ ) u + 2 < ∇ φ, ∇ u > + φ ∆ u we have − ( λ + ρ ) ˜ φ = ∆( φu )= − ( λ + ρ ) φu + 2 < ∇ φ, ∇ u > + φ ∆ u = − ( λ + ρ ) ˜ φ + 2 < ∇ φ, ∇ u > + φ ∆ u Dividing the above equation by φ , we obtain∆ ′ u = 0where ∆ ′ is the second-order differential operator defined by∆ ′ f = ∆ f + 2 < ∇ φφ , ∇ f > for f ∈ C ∞ ( X ).Now given ǫ >
0, let R = R ( ǫ ) > r ≥ R , by Lemma 5.8 we have(1 − ǫ ) ≤ u ( y ) ≤ (1 + ǫ ) HE FOURIER TRANSFORM ON NEGATIVELY CURVED HARMONIC MANIFOLDS 27 for all y on the geodesic sphere S ( x, r ). Since the operator ∆ ′ is an elliptic secondorder differential operator without any zeroth order term, and ∆ ′ u = 0, the max-imum principle applies to u and − u ([Eva98], Theorem 1 of section 6.4.1), so weobtain (1 − ǫ ) ≤ u ( y ) ≤ (1 + ǫ )for all y in the geodesic ball B ( x, r ). Since this is true for all r ≥ R , the aboveinequality holds in fact for all y ∈ X . Since ǫ > u = 1on X . Thus ˜ φ λ,x ( y ) = φ λ,x ( y )for all y ∈ X , and λ of the form λ = it, t < − ρ . Since both sides of the aboveequation are entire in λ for fixed y , it follows that ˜ φ λ,x = φ λ,x on X for all λ asrequired. ⋄ We can now prove the Fourier inversion formula:
Theorem 5.10.
Fix a basepoint o ∈ X . Then for f ∈ C ∞ c ( X ) we have f ( x ) = C Z ∞ Z ∂X ˜ f o ( λ, ξ ) e ( iλ − ρ ) B ξ,o ( x ) dλ o ( ξ ) | c ( λ ) | − dλ for all x ∈ X (where C > is a constant). Proof:
Given f ∈ C ∞ c ( X ) and x ∈ X , the function M x f is in C ∞ c ( X ), is radialaround the point x and satisfies ( M x f )( x ) = f ( x ). By Theorem 4.6 applied to thefunction M x f we have f ( x ) = ( M x f )( x ) = C Z ∞ [ M x f ( λ ) φ λ,x ( x ) | c ( λ ) | − dλ = C Z ∞ [ M x f ( λ ) | c ( λ ) | − dλ (since φ λ,x ( x ) = 1). Now using the formal self-adjointness of the operator M x ,Theorem 5.9, the fact that φ λ,x is radial around x and φ λ,x = φ − λ,x we obtain [ M x f ( λ ) = Z X ( M x f )( y ) φ − λ,x ( y ) dvol ( y )= Z X f ( y )( M x φ − λ,x )( y ) dvol ( y )= Z X f ( y ) φ − λ,x ( y ) dvol ( y )= Z X f ( y ) (cid:18)Z ∂X e ( − iλ − ρ ) B ξ,x ( y ) dλ x ( ξ ) (cid:19) dvol ( y )= Z ∂X (cid:18)Z X f ( y ) e ( − iλ − ρ ) B ξ,x ( y ) dvol ( y ) (cid:19) dλ x ( ξ )= Z ∂X ˜ f x ( λ, ξ ) dλ x ( ξ )Using the relations ˜ f x ( λ, ξ ) = e ( iλ + ρ ) B ξ,o ( x ) ˜ f o ( λ, ξ ) and dλ x dλ o ( ξ ) = e − ρB ξ,o ( x ) we get [ M x f ( λ ) = Z ∂X e ( iλ + ρ ) B ξ,o ( x ) ˜ f o ( λ, ξ ) e − ρB ξ,o ( x ) dλ o ( ξ )= Z ∂X ˜ f o ( λ, ξ ) e ( iλ − ρ ) B ξ,o ( x ) dλ o ( ξ )Substituting this last expression for [ M x f ( λ ) in the equation f ( x ) = C R ∞ [ M x f ( λ ) | c ( λ ) | − dλ gives f ( x ) = C Z ∞ Z ∂X ˜ f o ( λ, ξ ) e ( iλ − ρ ) B ξ,o ( x ) dλ o ( ξ ) | c ( λ ) | − dλ as required. ⋄ The Fourier inversion formula leads immediately to a Plancherel theorem:
Theorem 5.11.
Fix a basepoint o ∈ X . For f, g ∈ C ∞ c ( X ) , we have Z X f ( x ) g ( x ) dvol ( x ) = C Z ∞ Z ∂X ˜ f o ( λ, ξ )˜ g o ( λ, ξ ) dλ o ( ξ ) | c ( λ ) | − dλ where C is the constant appearing in the Fourier inversion formula. The Fouriertransform f ˜ f o extends to an isometry of L ( X, dvol ) into L ([0 , ∞ ) × ∂X, C | c ( λ ) | − dλdλ o ( ξ )) . Proof:
Applying the Fourier inversion formula to the function g gives Z X f ( x ) g ( x ) dvol ( x ) = C Z X f ( x ) (cid:18)Z ∞ Z ∂X ˜ g o ( λ, ξ ) e ( − iλ − ρ ) B ξ,o ( x ) dλ o ( ξ ) | c ( λ ) | − dλ (cid:19) dvol ( x )= C Z ∞ Z ∂X (cid:18)Z X f ( x ) e ( − iλ − ρ ) B ξ,o ( x ) dvol ( x ) (cid:19) ˜ g o ( λ, ξ ) dλ o ( ξ ) | c ( λ ) | − dλ = C Z ∞ Z ∂X ˜ f o ( λ, ξ )˜ g o ( λ, ξ ) dλ o ( ξ ) | c ( λ ) | − dλ Taking f = g gives that the Fourier transform preserves L norms, || f || = || ˜ f o || for all f ∈ C ∞ c ( X ), from which it follows from a standard argument that the Fouriertransform extends to an isometry of L ( X, dvol ) into L ([0 , ∞ ) × ∂X, C | c ( λ ) | − dλdλ o ( ξ )). ⋄ The convolution algebra of radial functions
Fix a basepoint o ∈ X . We define a notion of convolution with radial functionsas follows:For a function f radial around the point o , let f = u ◦ d o , where u is a functionon R . For x ∈ X , the x-translate of f is defined to be the function τ x f = u ◦ d x HE FOURIER TRANSFORM ON NEGATIVELY CURVED HARMONIC MANIFOLDS 29
Note that if f ∈ L ( X, dvol ), then evaluating integrals in geodesic polar coordinatescentered at o and x gives || f || = Z ∞ | u ( r ) | A ( r ) dr = || τ x f || Definition 6.1.
For f an L function on X and g an L function on X which isradial around the point o , the convolution of f and g is the function on X definedby ( f ∗ g )( x ) = Z X f ( y )( τ x g )( y ) dvol ( y )Note that, if g = u ◦ d o , then || f ∗ g || ≤ Z X Z X | f ( y ) || ( τ x g )( y ) | dvol ( y ) dvol ( x )= Z X | f ( y ) | (cid:18)Z X | u ( d ( x, y )) | dvol ( x ) (cid:19) dvol ( y )= Z X | f ( y ) | (cid:18)Z ∞ | u ( r ) | A ( r ) dr (cid:19) dvol ( y )= || f || || g || < + ∞ so that the integral defining ( f ∗ g )( x ) exists for a.e. x, and f ∗ g ∈ L ( X, dvol ).Note if f, g ∈ C ∞ c ( X ) with g = u ◦ d o radial around o , then f ∗ g is compactlysupported. For the Fourier transform of f ∗ g based at o , using the identity B ξ,o ( x ) = B ξ,o ( y ) + B ξ,y ( x ) we have ] f ∗ g o ( λ, ξ ) = Z X (cid:18)Z X f ( y ) u ( d ( x, y )) dvol ( y ) (cid:19) e ( − iλ − ρ ) B ξ,o ( x ) dvol ( x )= Z X f ( y ) e ( − iλ − ρ ) B ξ,o ( y ) (cid:18)Z X u ( d ( x, y )) e ( − iλ − ρ ) B ξ,y ( x ) dvol ( x ) (cid:19) dvol ( y )= Z X f ( y ) e ( − iλ − ρ ) B ξ,o ( y ) ^ u ◦ d yy ( λ, ξ ) dvol ( y )= ˜ f o ( λ, ξ )ˆ u ( λ )= ˜ f o ( λ, ξ )ˆ g ( λ )where we have used the fact that for the function u ◦ d y which is radial around y we have ^ u ◦ d yy ( λ, ξ ) = ˆ u ( λ ) = ˆ g ( λ )where ˆ u is the hypergroup Fourier transform of u and ˆ g is the spherical Fouriertransform of the function g which is radial around o . Theorem 6.2.
Let L o ( X, dvol ) denote the closed subspace of L ( X, dvol ) consist-ing of those L functions which are radial around the point o . Then for f, g ∈ L o ( X, dvol ) we have f ∗ g ∈ L o ( X, dvol ) , and L o ( X, dvol ) forms a commutativeBanach algebra under convolution. Proof:
We first consider functions f, g ∈ C ∞ c ( X ) which are radial around o . For x ∈ X , the function M x φ λ,o is an eigenfunction of ∆ with eigenvalue − ( λ + ρ )which is radial around the point x and takes the value φ λ,o ( x ) at the point x , so itfollows from Lemma 4.4 that M x φ λ,o = φ λ,o ( x ) φ λ,x The above equation together with the spherical Fourier inversion formula for thefunction f gives, for y ∈ Y ,( M x f )( y ) = C Z ∞ ( M x φ λ,o )( y ) ˆ f ( λ ) | c ( λ ) | − dλ = C Z ∞ φ λ,o ( x ) φ λ,x ( y ) ˆ f ( λ ) | c ( λ ) | − dλ Now using the formal self-adjointness of M x we have( f ∗ g )( x ) = Z X f ( y )( τ x g )( y ) dvol ( y )= Z X f ( y )( M x τ x g )( y ) dvol ( y )= Z X ( M x f )( y )( τ x g )( y ) dvol ( y )= C Z ∞ φ λ,o ( x ) ˆ f ( λ ) (cid:18)Z X φ λ,x ( y )( τ x g )( y ) dvol ( y ) (cid:19) | c ( λ ) | − dλ = C Z ∞ φ λ,o ( x ) ˆ f ( λ )ˆ g ( λ ) | c ( λ ) | − dλ It follows from the above equation that f ∗ g is radial around the point o since allthe functions φ λ,o are radial around the point o , and it also follows that f ∗ g = g ∗ f .Now the inequality || f ∗ g || ≤ || f || || g || implies, by the density of smooth,compactly supported radial functions in the space L o ( X, dvol ), that for f, g ∈ L o ( X, dvol ) we have f ∗ g = g ∗ f ∈ L o ( X, dvol ), so L o ( X, dvol ) forms a commutativeBanach algebra under convolution. ⋄ Finally, we remark that the radial hypergroup of the harmonic manifold X canbe realized as the convolution algebra of finite radial measures on the manifold:convolution with radial measures can be defined, and the convolution of two radialmeasures is again a radial measure. This can be proved by approximating finiteradial measures by L radial functions and applying the previous Theorem. Theconvolution algebra L o ( X, dvol ) is then identified with a subalgebra of the hyper-group algebra of finite radial measures under convolution.
References [ACB97] F. Astengo, R. Camporesi, and B. Di Blasio. The helgason fourier transform on a classof nonsymmetric harmonic spaces.
Bull. Austral. Math. Soc. 55 , pages 405–424, 1997.[BCG95] G. Besson, G. Courtois, and S. Gallot. Entropies et rigidit´es des espaces localementsym´etriques de courbure strictement n´egative.
Geometric and Functional Analysis Vol.5 , pages 731–799, 1995.
HE FOURIER TRANSFORM ON NEGATIVELY CURVED HARMONIC MANIFOLDS 31 [Bes78] A. L. Besse. Manifolds all of whose geodesics are closed.
Ergebnisse u.i. Grenzgeb.Math., vol. 93, Springer, Berlin , 1978.[BFL92] Y. Benoist, P. Foulon, and F. Labourie. Flots d’anosov `a distributions stable et instablediffer´entiables.
J. Amer. Math. Soc. 5 (1) , pages 33–74, 1992.[BH95] W. R. Bloom and H. Heyer. Harmonic analysis of probability measures on hypergroups. de Gruyter Studies in Mathematics 20 (Walter de Gruyter, Berlin) , 1995.[BH99] M. R. Bridson and A. Haefliger. Metric spaces of non-positive curvature.
Grundlehrender mathematischen Wissenschaften, ISSN 0072-7830; 319 , 1999.[Bis17] K. Biswas. Circumcenter extension of moebius maps to cat(-1) spaces.
Preprint,https://arxiv.org/pdf/1709.09110.pdf , 2017.[Bou95] M. Bourdon. Structure conforme au bord et flot geodesique d’un cat(-1) espace.
En-seign. Math. (2) no. 41 , pages 63–102, 1995.[Bou96] M. Bourdon. Sur le birapport au bord des cat(-1) espaces.
Inst. Hautes Etudes Sci.Publ. Math. No. 83 , pages 95–104, 1996.[BX95] W. R. Bloom and Z. Xu. The hardy-littlewood maximal function for chebli-trimechehypergroups.
Contemp. Math. 183 , pages 45–69, 1995.[Che74] H. Chebli. Operateurs de translation generalisee et semi-groupes de convolution.
The-orie de potentiel et analyse harmonique, Springer Lecture Notes in Math., 404 , pages35–59, 1974.[Che79] H. Chebli. Theoreme de paley-wiener associe a un operateur differentiel singulier sur(0 , ∞ ). J. Math. Pures Appl. (9) 58 , pages 1–19, 1979.[CR40] E. T. Copson and H. S. Ruse. Harmonic riemannian spaces.
Proc. Roy. Soc. Edinburgh60 , pages 117–133, 1940.[DR92] E. Damek and F. Ricci. A class of nonsymmetric harmonic riemannian spaces.
Bull.Amer. Math. Soc., N.S. 27 (1) , pages 139–142, 1992.[Eva98] L. C. Evans. Partial differential equations.
Graduate studies in mathematics, Vol. 19,A. M. S. , 1998.[FL92] P. Foulon and F. Labourie. Sur les vari´et´es compactes asymptotiquement harmoniques.
Invent. Math. 109 (1) , pages 97–111, 1992.[Heb06] J. Heber. On harmonic and asymptotically harmonic homogeneous spaces.
Geom.Funct. Anal. 16 (4) , pages 869–890, 2006.[Hel94] S. Helgason. Geometric analysis on symmetric spaces.
Math. Surveys and Monographs39 (American Mathematical Society, Providence RI) , 1994.[HH77] E. Heintze and H. C. Im Hof. Geometry of horospheres.
Journal of Differential Geom-etry, 12 , pages 481–491, 1977.[Kap80] A. Kaplan. Fundamental solution for a class of hypoelliptic pde generated by compo-sition of quadratic forms.
Trans. Amer. Math. Soc. 258 , pages 147–153, 1980.[Kni02] G. Knieper. Hyperbolic dynamics and riemannian geometry.
Handbook of DynamicalSystems, Vol.1A, Elsevier Science B., eds. B. Hasselblatt and A. Katok , pages 453–545,2002.[Kni12] G. Knieper. New results on noncompact harmonic manifolds.
Comment. Math. Helv.87 , pages 669–703, 2012.[Koo84] T. H. Koornwinder. Jacobi functions and analysis on noncompact semisimple lie groups.
Special functions: Group theoretical aspects and applications, R. A. Askey et al (eds.),Reidel , pages 1–85, 1984.[KP13] G. Knieper and N. Peyerimhoff. Noncompact harmonic manifolds. , 2013.[Lic44] A. Lichnerowicz. Sur les espaces riemanniens completement harmoniques.
Bull. Soc.Math. France 72 , pages 146–168, 1944.[Nik05] Y. Nikolayevsky. Two theorems on harmonic manifolds.
Comment. Math. Helv. 80 ,pages 29–50, 2005.[RWW61] H. Ruse, A. Walker, and T. Willmore. Harmonic spaces.
Edizioni Cremonese , 1961.[Sza90] Z. Szabo. The lichnerowicz conjecture on harmonic manifolds.
Journal of DifferentialGeometry, 31 , pages 1–28, 1990.[Tri81] K. Trimeche. Transformation integrale de weyl et theoreme de paley-wiener associe aun operateur differentiel singulier sur (0 , ∞ ). J. Math. Pures Appl. 60 , pages 51–98,1981. [Tri97a] K. Trimeche. Generalized wavelets and hypergroups.
Gordon and Breach, Amsterdam ,1997.[Tri97b] K. Trimeche. Inversion of the lions transmutation operators using generalized wavelets.
Appl. Comput. Harmon. Anal. 4, no. 1 , pages 97–112, 1997.[Wal48] A. C. Walker. On lichnerowicz’s conjecture for harmonic 4-spaces.
J. London Math.Soc. 24 , pages 317–329, 1948.[Wil93] T. Willmore. Riemannian geometry.
Oxford University Press , 1993.[Xu94] Z. Xu. Harmonic analysis on chebli-trimeche hypergroups.
Ph.D. thesis, Murdoch Uni-versity, Australia , 1994., 1994.