Bounded eigenfunctions in the real Hyperbolic space

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Bounded eigenfunctions in the real Hyperbolic space

Sandrine Grellier, Jean-Pierre Otal

To cite this version:

Sandrine Grellier, Jean-Pierre Otal. Bounded eigenfunctions in the real Hyperbolic space. interna-

tional mathematical research notices, 2005, 62, pp.3867-3897. �hal-00022000�

2005,No. x

Bounded Eigenfunctions in the Real Hyperbolic Space

Sandrine Grellier and Jean-Pierre Otal

1 Introduction

LetHⁿ be the real hyperbolic space of dimensionn, that is,the complete and simply connected Riemannian manifold of constant curvature−1. Using the Poincar ´e model,we identifyHⁿwith the unit ball ofRⁿand its boundary at infinity∂Hⁿwith the unit sphere Sⁿ⁻¹ofRⁿ.

We denote byΔthe Laplace-Beltrami operator acting on functions onHⁿ.

Definition 1.1. Forλ ∈ C,say thatf : Hⁿ → Cis a λ-eigenfunction whenf is C²and satisfiesΔf+λf=0. Denote byE_λthe space of theλ-eigenfunctions which are bounded.

The purpose of this paper is to describe the spaceEλ. The Harnack inequality(see,e.g., [6,page 199])implies that the gradient of a boundedλ-eigenfunctionfis bounded by a constant which only depends onλand on the sup norm off. This implies,using the Ascoli theorem,that the spaceE_λwith the sup norm is a Banach space.

Fors ∈ C,we setλ_s = s(n−1−s). The basic example of aλ_s-eigenfunction is provided byx→ P^s(x, ξ),whereP(x, ξ)is the Poisson kernel ofHⁿandξis any point of the boundary at infinity∂Hⁿ ofHⁿ. Anyλ_s-eigenfunctions can be described in terms of hyperfunctions on∂Hⁿ. Namely, for anyλ_s-eigenfunction, there is a hyperfunction on

∂Hⁿsuch that

f(x)=P^s(T)(x)=

T, P^s(x,·)

; (1.1)

Received 25 March 2005. Revision received 26 October 2005.

Communicated by Peter Sarnak.

this is a theorem of Helgason and Minemura(cf.[7,9]),which was also proven later by Agmon[1].

Definition 1.2. Call the functionx → T, P^s(x,·)the Poisson-Helgason transform of T. Denote it byP^s(T).

Furthermore,one knows that the hyperfunctionT is unique,except for the cases whens= −1,−2, . . . ,−k, . . .(see[9,Proposition 2.1]). Moreover,iffgrows slowly,that is, at most exponentially with the hyperbolic distance,thenT is a distribution. This result has been proven initially by ¯Oshima and Sekiguchi[10]using microlocal techniques. In [15],the authors give a new proof within the framework of asymptotic expansions(see [15,Theorem 2-2]). In particular,the hyperfunction which represents a bounded eigen- function is indeed a distribution. Thus,to describe the space of bounded eigenfunctions is equivalent to describe the distributionsT onSⁿ⁻¹such thatP^s(T)is bounded.

WhenTis equal to the normalized Lebesgue measuredσon∂Sⁿ⁻¹,then the eigen- functionP^s(dσ)is the spherical λ_s-eigenfunction: this is the unique eigenfunction h_s such thath_s(o)=1,and which is invariant under the subgroup(O(n))of the isometries ofHⁿwhich fix the origino.

First of all,it is easy to see that the only values ofssuch thatEλs ={0}are con- tained in the strip{s∈C; (s)∈[0, n−1]}. Indeed,letf∈E_λsbe a nonzero function and letx∈Hⁿbe a point wheref(x)=0. Up to precomposingfwith an isometry which sends otox,we can assume thatf(o) =0. Then,by averagingfover the subgroup(O(n))of all rotations which fixo,we obtain aλ_s-eigenfunction which is bounded,nonzero,and invariant under the orthogonal groupO(n). But the unicity of the spherical functionh_s implies that this function is a multiple ofh_s;therefore the spherical functionh_shas to be bounded. It is now easy to see that the only values ofsfor which this can occur are exactly those such thats∈[0, n−1].

We denote byHthe vertical strip{s ∈C | s∈ ]0, n−1[}. The method we use in this paper does not allow to deal with the values ofssuch thats=0orn−1. Therefore, we will focus on the values ofs∈H.

Whenn= 2,there is a characterization of the distributionsT such thatP^s(T)is bounded[11]: fors = 0,they are exactly the distributional derivatives of(s)-H ¨older functions on∂H². In higher dimension,the notion of derivatives depends on the data of a vector field,and so does not have easily an intrinsic meaning. One way to avoid this dif- ficulty is to use the spherical LaplacianΔ_σacting on functions onSⁿ⁻¹. If a distribution TsatisfiesT,1=0,there is an integerkand a continuous functionHsuch that the dis- tributionTcan be written asΔ^k_σ(H). Then the regularity ofTcan be measured in terms of the integerkand of the regularity ofH. We use the Lipschitz spaces of orderαforα > 0

in order to measure the regularity of a functionH:Sⁿ⁻¹→C. Let us recall the definition and the basic properties of Lipschitz spaces.

Letθ∈[−π, π]andR_θbe a rotation of angleθacting onSⁿ⁻¹. LetHbe a function onSⁿ⁻¹. We denote byD_θ(H)=D¹_θ(H)the functionH◦R_θ−H,and forn≥1,we define inductivelyDⁿ_θ(H)=D_θ◦Dⁿ_θ⁻¹(H).

Letα > 0and let[α]be its integer part. Denote byΛ_α(Sⁿ⁻¹)the space of(equiv- alence classes of)bounded measurable functions onSⁿ⁻¹for which there exists a con- stantC > 0,such that for any rotationR_θof angleθ,one has

D^[α]_θ ⁺¹(H)

∞ ≤C|θ|^α. (1.2)

One can define a norm on the spaceΛ_α(Sⁿ⁻¹),by settingH_α equal to the infimum of theL^∞-norm ofHand of the constantsCwith this property. With this norm,the space Λ_α(Sⁿ⁻¹)becomes a Banach space: it is called the Lipschitz space of orderα.

Ifα ∈]0, 1[,the spaceΛ_α(Sⁿ⁻¹)is the classical H ¨older space of orderα,and the norm is equivalent to the classical H ¨older norm. In fact,a smoothing argument shows that any function inΛ_α(Sⁿ⁻¹)is almost everywhere equal to a H ¨older-continuous func- tion of orderα. Any function inΛ₁(Sⁿ⁻¹)admits a continuous representative as well(see [14,page 331]).

Ifα > 1,the following holds. Let Xbe any vector field onSⁿ⁻¹. Then,forH ∈ Λ_α(Sⁿ⁻¹),the Lie derivativeLXHis contained inΛ_α−1(Sⁿ⁻¹),with a norm bounded in terms ofHαand the norm of the vector fieldX. By induction,one can characterize,for α > 1,α /∈N,the spaceΛ_α(Sⁿ⁻¹)as the space of functions which are[α]-times differen- tiable and which have all mixed derivatives of order[α]−1inΛ_α−[α]+1(Sⁿ⁻¹).

We will use the following important property of the Lipschitz spaces. For any integerm≥ α,one can replace in the definition ofΛ_α(Sⁿ⁻¹)the quantityD^[α]_θ ⁺¹(H)_∞ byD^m_θ⁺¹(H)_∞. The space defined in that way is equal toΛ_α(Sⁿ⁻¹)and the two norms are equivalent(see,e.g., [14,page 331]). In particular,forα∈]0, n−1[,one can define the spaceΛ_α(Sⁿ⁻¹)as the space of functionsH∈L^∞(Sⁿ⁻¹)such thatDⁿ_θ(H)∞ ≤C|θ|^α,for some constantCindependent ofθ.

In the following,we denote by Λ⁰_α(Sⁿ⁻¹) the space of functionsH ∈ Λ_α(Sⁿ⁻¹) such that

Sⁿ⁻¹H dσ=0.

Given an integerj≥0,we denote byΔ^j_σΛ⁰_α(Sⁿ⁻¹)the space of those distributions onSⁿ⁻¹which can be written asΔ^j_σ(H)for a functionH∈Λ⁰_α(Sⁿ⁻¹).

The paper is devoted to prove the following theorem,which describes the spaces Eλsfors∈H.

Theorem 1.3. Lets=δ+it∈H.

If n is odd, then P_s induces an isomorphism between the Banach spaces Δ⁽ⁿ_σ^−1)/2Λ⁰_δ(Sⁿ⁻¹)⊕CdσandEλs;

If n is even, then Ps induces an isomorphism between the Banach spaces

Δ^n/2_σ Λ⁰_δ+1(Sⁿ⁻¹)⊕CdσandEλs.

In this theorem,theλ_s-eigenfunctions which vanish at the origin are exactly the image ofΔ⁽ⁿ_σ^−1)/2Λ⁰_δ(Sⁿ⁻¹)by the Poisson-Helgason transform. The image of the complex lineCdσis equal to the multiples of the spherical functionh_swhich equals1at the origin o.

Note thatTheorem 1.3does not include the bounded harmonic functions onHⁿ; they correspond to the cases = n−1. The boundary values of those are characterized by the Fatou theorem: they are exactly the functions inL^∞(Sⁿ⁻¹). However,derivatives of order n−1 of functions in Λⁿ⁻¹(Sⁿ⁻¹)are not in L^∞(Sⁿ⁻¹), in general. Therefore, Theorem 1.3cannot hold in the cases=n−1.

The proof ofTheorem 1.3contains two parts. The first one amounts to show that P^srestricted toΔ⁽ⁿ_σ^−1)/2Λ⁰_δ(Sⁿ⁻¹)⊕Rdσor toΔ^n/2_σ Λ⁰_δ+1(Sⁿ⁻¹)⊕Rdσaccording to the par- ity ofntakes range inEλsand that it is a continuous operator. This is done inSection 2.

If the real part ofsis not of the formn−1−2jwithj∈N,the proof is a direct computation using easy estimates on the Poisson kernel. The other cases are obtained by interpolation of analytic families of operators.

The second part amounts to show that the restriction ofP^sis surjective. This is proven in two steps. First,we consider the case whenδ = s ≥ (n−1)/2. Under this assumption,we assign inSection 4 to any functionf ∈ Eλs a “boundary value”: it is a distributionT onSⁿ⁻¹which is contained in the expected space,that is,it satisfiesT = Δ⁽ⁿ_σ^−1)/2Hfor some Lipschitz functionHinΛ_δ(Sⁿ⁻¹)ifnis odd orT = Δ^n/2_σ Hfor some Lipschitz functionHinΛ_δ+1(Sⁿ⁻¹)ifnis even. The proof relies on estimates of a certain differential operatorDλacting on functions on the real line;this operator is the radial part of the Laplace operator onHⁿ, written in polar coordinates. These estimates are obtained inSection 3;we use in an essential way the assumption thatδ≥(n−1)/2.

InSection 5,one proves,under the assumptionδ =s≥(n−1)/2,that the dis- tributionT reproducesf,that is, thatP^s(T) = f. This shows thatP^s is onto under the assumptions ofTheorem 1.3ands≥(n−1)/2.

InSection 6,we extend this to the caseδ <(n−1)/2by using a relation between the(n−1−s)th power of the Poisson kernel and itssth power: they are related by a fractional integration(resp.,derivation)operator which is studied in Proposition 6.1.

We study some continuity properties of this operator acting on Lipschitz spaces

(Proposition 6.1(4)). This result is similar to the one for fractional integration(resp., derivation)operator acting on functions onRⁿ,but as we did not find any reference,we propose an elementary proof of it.

Finally,sinceP^sis continuous,onto,and injective,the isomorphism statement in Theorem 1.3follows from the Banach closed graph theorem.

The computations in Section 3to Section 5 use heavily estimates on the Green function and on the spherical function of the operatorΔ+λ_sI. These estimates are well known,but in order to make the paper self-contained,we give the proofs in an appendix.

In all the sequel,we assume thatn≥3.

2 Continuity properties of the Poisson-Helgason transformP^s

In order to prove the direct part ofTheorem 1.3,we use the following estimate on the derivatives of the Poisson kernel.

Claim 2.1. Letj ∈ Nands = δ+itinC. There exists a polynomialQin one complex variable so that for anyζandξinSⁿ⁻¹,for any pointx∈Hⁿat distancerfromoon the ray[oζ[,one has

Δ^j_σP^s(x, ξ)≤C(s) e^−δr

e^−r+|ζ−ξ|_2δ+2j, (2.1)

whereC(s)=|Q(s)|.

Proof. In the ball model,the Poisson kernel can be written as

P(x, ξ)= 1−|x|²

|x−ξ|². (2.2)

In Euclidean polar coordinates,one hasx=(tanh(r/2))ζ. Therefore,

P(x, ξ)=

1−

tanh r

2 ₂

tanh

r 2

₂

+1−2tanh r

2 ζ, ξ

. (2.3)

The statement follows after differentiating this formula.

Proposition 2.2. Letn≥3. Lets∈H.

Ifnis odd,thenP^smaps continuouslyΔ⁽ⁿ_σ^−1)/2Λ_δ(Sⁿ⁻¹)toL^∞(Hⁿ). Ifnis even, thenP^smaps continuouslyΔ^n/2_σ Λ_δ+1(Sⁿ⁻¹)toL^∞(Hⁿ).

As it was said inSection 1,this result is false fors = n−1since,by the Fatou theorem,the Poisson-Helgason transformPⁿ⁻¹is an isomorphism fromL^∞(Sⁿ⁻¹)toE₀. Proof. Assume first thatnis odd. Letf∈Λ_δ(Sⁿ⁻¹). We choose the integerj≥1such that n−1−δ=2j−αwithα∈]0, 2]. Suppose also thatn−1−δis not an even integer,that is, thatα=2.

The Green formula implies that for anyx∈Hⁿ,one has

I:=P^s

Δ⁽ⁿ_σ^−1)/2f (x)=

Δ^j_σP^s(x,·), Δ⁽ⁿ_σ^−1)/2−jf

. (2.4)

But the functionh=Δ⁽ⁿ_σ^−1)/2−jfbelongs toΛ_α(Sⁿ⁻¹)with norm bounded byCf_δ. Asj≥1,for anyζ∈Sⁿ⁻¹,one has

I=

Sⁿ⁻¹Δ^j_σP^s(x,·)h(·)dσ=

Sⁿ⁻¹Δ^j_σP^s(x,·)

h(·)−h(ζ)

dσ. (2.5)

Let nowζbe the end of the rayoxon the sphere and denote byrthe hyperbolic distance fromxtoo. Forθ∈[0, π],denote byB(ζ, θ)⊂Sⁿ⁻¹the spherical ball of radiusθ aroundζand setP(x, θ)=P(x, ξ)for anyξ∈∂B(ζ, θ).

Using geodesic polar coordinates onSⁿ⁻¹around the pointζ,this last integral also equals

_π

0

Δ^j_σP^s(x, θ)

∂B(ζ,θ)

h(·)−h(ζ)

dσ_θ dθ, (2.6)

wheredσ_θis the Riemannian measure on the sphere∂B(ζ, θ)for the induced Riemannian metric. Denoting byξthe symmetric ofξ∈∂B(ζ, θ)with respect to the centerζ,one has

∂B(ζ,θ)

h(·)−h(ζ) dσ_θ

= 1 2

∂B(ζ,θ)

h(ξ)+h(ξ)−2h(ζ) dσ_θ(ξ)

. (2.7) Sinceh∈Λ_α(Sⁿ⁻¹)withα∈]0, 2[,|h(ξ)+h(ξ)−2h(ζ)|≤ hα|ξ−ζ|^α. So,the left integral above is bounded byChαθ^n−2+α.

Sincen−2+α=δ+2j−1,it follows fromClaim 2.1that for anyx∈Hⁿ,

|I|≤ hα

c₁

_e^−r

0

e^−δr

e^−(2δ+2j)rθ^δ+2j−1dθ+c_{2 π}

e^−r

e^−δrθ^δ+2j−1

θ^2δ+2j dθ . (2.8) Therefore,|I|≤Chαfor a constantCwhich does not depend onh. So,P^s(Δ⁽ⁿσ^−1)/2f)∞

≤CΔ⁽ⁿσ^−1)/2−jfα≤Cfδ. The result of the proposition is proved except whenn−1−δ is an even integer.

To deal with the caseα=2,that is,when(n−1)−δis even,we use interpolation of analytic families of operators. We give first a complement to the last proof. Letδ₀ = n−1−2j. Chooseδ₂in]0, n−1[withδ₀< δ₂< δ₀+2. Lets=δ+it,withδ∈]0, δ2].

Then,for f ∈ Λ_δ2, one hasP^s(Δ⁽ⁿσ^−1)/2f)_∞ ≤ C(s)fδ2 for a constantC(s) which is independent offand is the absolute value of some polynomialRat points. In- deed,writing δ₂ = n−1−2j+α₂, the argument above gives, for f ∈ Λ_δ2(Sⁿ⁻¹)and h=Δ⁽ⁿ_σ^−1)/2−jf,

P^s

Δ⁽ⁿ_σ^−1)/2f (x)

≤C(s)hα2

c₁

_e^−r

0

e^−δr

e^−(2δ+2j)rθ^δ2+2j−1dθ+c_{2 π}

e^−r

e^−δrθ^δ2+2j−1 θ^2δ+2j dθ ,

(2.9)

forr=d(o, x). Therefore,ifδandδ₂satisfyδ₂< 2δ,one has

P^s

Δ⁽ⁿ_σ^−1)/2f

(x)≤C(s)fδ2e^{−(δ2−δ)r}, (2.10)

whereC(s)can be chosen to be the modulus of a polynomialRwhich does not vanish in the strips≥0.

Choose now δ₁ andδ₂ with δ₁ < δ₀ < δ₂ and sufficiently close so that δ₂ <

2δ₁. Consider the Lipschitz spacesΛ_δ1(Sⁿ⁻¹)andΛ_δ2(Sⁿ⁻¹);it is known that the spaces Λ_δ(Sⁿ⁻¹)for δ ∈ [δ1, δ₂]form an interpolating family[2]. On the closed stripB = {s | s∈[δ1, δ₂]},consider the analytic family of operatorsT_s =(1/R(s))P^s◦Δ⁽ⁿ_σ^−1)/2. Those operators are defined onΛ_δ2(Sⁿ⁻¹)and satisfy the following properties:

(1) each operator T_s maps continuously Λ_δ2(Sⁿ⁻¹) to the Banach (separable) spaceC^b(Hⁿ)of bounded continuous functions onHⁿ(with the uniform norm);

(2) for allf∈ Λ_δ2(Sⁿ⁻¹),for allx ∈Hⁿ,the functions→ T_s(f)(x)is continuous, analytic in the interior ofBand bounded overB;

(3) forf∈Λ_δ2(Sⁿ⁻¹),one hasTδ1+itf ≤ fδ1andTδ2+itf ≤ fδ2.

Then the hypotheses of the interpolation theorem(see,e.g., [3,Theorem 1])are satisfied.

This theorem gives that fors=δ∈B,T_sis continuous fromΛ_δ(Sⁿ⁻¹)toC^b(Hⁿ). Making δ=δ₀finishes the proof ofProposition 2.2for odd values ofn.

The proof for even values ofnis the same.

3 Estimates of the solutions ofDλf=g

In this section,we study the differential operatorDλwhich is defined as acting on func- tions on]0,∞[by

Dλ=(sinhr)² ∂²

∂r²+(n−1)cothr ∂

∂r+λId . (3.1)

Recall that the hyperbolic Laplacian has the following form in polar hyperbolic coordinates(r, ζ)aroundo. For aC²functionu=u(r, ζ):Hⁿ\ {o}→C,one has

Δu=∂²u

∂r² +(n−1)cothr∂u

∂r + 1

(sinhr)²Δ_σu(r,·). (3.2)

From this expression,the solutions ofDλu=0are precisely the radialλ-eigenfunctions.

In Propositions7.1and7.2,we will recall the behaviour near∞of the two basic solutions of the eigenfunction equations,that is,the spherical functionh_sand the Green function g_s. The next result studies similarly the behaviour of a solutionuofDλu=vin terms of the behaviour ofv.

To simplify notations,we writeλ=λ_s=s(n−1−s).

Throughout this section,we make the assumption thatδ=s≥(n−1)/2.

Proposition 3.1. Letr₀ > 0and letl be an integer such that l ≤ n−3−δ. Letv be a smooth function on[r0,∞[such that|v(r)| ≤ Ce^−lr forr ≥ r₀. Letu : [r0,∞[→ Cbe a smooth function such thatDλu=v. Then,ifl < n−3−δ,one has

u(r)≤κ C+u

r₀+u

r₀e^−(l+2)r. (3.3)

Ifl=n−3−δ,one has u(r)≤κ

C+u

r₀+u

r₀re^{−(n−1−δ)r}. (3.4)

In both cases,κis a constant which depends only onr₀and ons.

Proof. The solutionuofDλu= vis uniquely determined by its behaviour up to first or- der atr₀. LetaandbinCbe such thatag_s(r₀)+bh_s(r₀) = 0andag_s(r₀)+bh_s(r₀)= 1.

This solution(a, b)exists and is unique,since the determinant of this system equals the Wronskian ofg_s andh_s atr₀which does not vanish. Then,by substracting fromuthe combinationag_s+bh_s,one reduces to the case whenu(r0)=u(r0)=0. This linear com- bination grows less thanκ(|u(r0)|+|u(r0)|)e^{−(n−1−δ)r},whereκis a constant depending only onr₀. Therefore,to proveProposition 3.1,we can assume thatu(r₀)=u(r₀)=0.

LetU(r)be the vector with coordinatesu(r)andu(r)and letV(r)be the vector with coordinatesv(r)/(sinhr)²and0. Then

U(r)=A(r)U(r)+V(r) on r₀,∞

, (3.5)

where

A(r)=

−(n−1)cothr −λ

1 0

. (3.6)

LetM(r)be the solution ofM(r)= −M(r)A(r)which vanishes atr₀;then

U(r)= _r

r0

M⁻¹(r)M(t)V(t)dt. (3.7)

For any solutionY(r)of the equation Y(r) = A(r)Y(r),the vectorM(r)Y(r)is constant.

This holds in particular for the spherical vectorH_s(r)with coordinates(h_s(r), hs(r))and for the Green vectorG_s(r)=(g_s(r), g_s(r)). Now,since the WronskianW(t)does not van- ish,we may writeV(t)as a linear combination of these two vectors:V(t) = α(t)H_s(t)+ β(t)Gs(t). It comes out that

U(r)= _r

r0

α(t)dt H_s(r)+ _r

r0

β(t)dt G_s(r), (3.8)

whereα(t)andβ(t)are given,respectively,by

α(t)=c_n−1v(t)gs(t)(sinht)ⁿ⁻³, β(t)= −cn−1v(t)hs(t)(sinht)ⁿ⁻³. (3.9) Using the classical estimates onh_s andg_s recalled in the appendix,one obtains easily

the required estimates ofProposition 3.1.

Corollary 3.2. Letr₀> 0. Letj∈N,l∈Zbe such thatl+2j≤n−1−δ. Letvbe a smooth function on[r₀,∞[such that|v(r)|≤Ce^−lr. Assume thatuis a smooth function on[r₀,∞[

which solves the differential equationD^j_λu=v. Then ifl+2j < n−1−δ,one has u(r)≤κ

C+

2j−1

p=0

u^(p)

r₀ e^−(l+2j)r. (3.10)

Ifl+2j=n−1−δ,one has u(r)≤κ

C+

2j−1

p=0

u^(p)

r₀ re^{−(n−1−δ)r}, (3.11)

where the constantκdepends only onr₀and ons.

Proof. By decreasing induction onm,1 ≤ m≤ j−1,one proves,usingProposition 3.1, that

D^mu(r)≤κ_m

C+

2(j−m)−1 0

u^(p) r₀

e^{−(l+2(j−m))r} (3.12)

forr≥r₀,where the constantκ_mdepends only onr₀. Form=0,it gives the result. Notice that one needs to use the degenerate case ofProposition 3.1only at the last step of the

induction and whenl+2j=n−1−δ.

4 Construction of a distribution onSⁿ⁻¹associated to a bounded eigenfunction Now,we begin the proof of the fact thatP^sis onto. Letf∈Eλs. We are going to construct an explicit distributionT such thatf=P^s(T). Notice first that we may assume thatf(o)= 0,since up to replacingfbyf−f(o)hs,we get a bounded eigenfunction inEλswhich has L^∞-norm less than2f∞. In this section,we associate tofa distributionT_sonSⁿ⁻¹and study its Lipschitz regularity. In the next section,we will prove that it satisfiesP^s(T_s)=f.

As in the preceding section,we assume thats = δ ∈ [(n−1)/2, n−1[and we writeλ=λ_s=s(n−1−s)to simplify notations. Consider forr > 0the function

T(r, ξ)=(sinhr)ⁿ⁻¹

f∂g_s

∂r −∂f

∂rg_s (r, ξ), (4.1)

whereg_sis the Green function: notice that the Green function depends onsand not only onλ_s(cf.Section 6).

Proposition 4.1. Lets = δ+itwithδ ∈ [(n−1)/2, n−1[. Then the distributionsT(r,·) converge to a distributionT =T_sonSⁿ⁻¹. Furthermore,ifδ=n−1−2j+α,withα∈]0, 2], there is a functionH∈Λ⁰_α(Sⁿ⁻¹)such thatT = Δ^j_σ(H). This functionHsatisfiesHα ≤

Cf∞ for a constantCwhich depends only ons.

Remark 4.2. As a corollary,we obtain that the limit distributionTcan always be written asT = Δ⁽ⁿ_σ^−1)/2hwithh ∈ Λ⁰_δ(Sⁿ⁻¹)ifnis odd or thatT = Δ^n/2_σ hwithh ∈ Λ⁰_δ+1(Sⁿ⁻¹) ifnis even. Indeed,sinceH has mean-value zero overSⁿ⁻¹,one may write it as H = Δ⁽ⁿ_σ^−1)/2−jhifnis odd andH = Δ^n/2_σ ^−jhifnis even. As powers of Laplacians give iso- morphisms between Lipschitz spaces of different parameters(see[12]),we get thath ∈ Λ⁰_δ(Sⁿ⁻¹)ifnis odd and thathinΛ⁰_δ+1(Sⁿ⁻¹)ifnis even sinceH∈Λ⁰_α(Sⁿ⁻¹).

Proof. The following properties ofT(r,·)are easy to establish.

(1)

Sⁿ⁻¹T(r,·)dσ=0: this comes from the Green formula sincef(o)=0and since in the distributional sense,Δg_s+λsg_sequals the Dirac mass at the origin δ_o.

(2) (∂T/∂r)(r, ξ)= −(sinhr)ⁿ⁻³g_s(r)Δσf(r, ξ): this comes by differentiating the expression ofT(r, ξ)and using the expression for the Laplacian in polar coordinates.

In order to illustrate the method,we consider first the easiest case δ = s ∈ ]n−3, n−2[. By(1),there is a unique smooth functionξ →H(r, ξ)which has zero mean and such thatT(r,·)=Δ_σH(r,·). By(2),we have

∂

∂rH(r,·)= −(sinhr)ⁿ⁻³g_s(r)f(r,·). (4.2)

Fix anr₀> 0. Using the estimates ong_sand the boundedness off,one obtains that there existsC > 0which depends only onsand such that for anyr≥r₀,

(sinhr)ⁿ⁻³g_s(r)f(r,·)≤Cf_∞e^{(n−3−δ)r}. (4.3)

Therefore, H(·)=H

r₀,·

− _∞

r0

(sinht)ⁿ⁻³g_s(t)f(t,·)dt (4.4)

defines a continuous function onSⁿ⁻¹. In the distributional sense,we have Δ_σH(·)= lim

r→∞T(r,·)=:T(·). (4.5)

We are going to prove thatH ∈ Λ⁰_δ−(n−3)(Sⁿ⁻¹). By assumptionδ−(n−3) ∈ ]0, 1[so it suffices to show thatD_θH_∞ ≤ Cf_∞|θ|^δ−(n−3),for a constantCwhich only depends ons. Recall thatD_θis the first-order difference operatorD_Rθ associated to the rotation R_θ∈O(n),that is,the operator which assigns to a functionFthe functionD_RθF=F◦Rθ−F.

We have

D_θH(·)=D_θH r₀,·

− _∞

r0

(sinht)ⁿ⁻³g_s(t)Dθf(t,·)dt=I+II,

|II|≤ _∞

r0

(sinht)ⁿ⁻³g_s(t)D_θf(t,·)dt

≤

_−log|θ|

r0

(sinht)ⁿ⁻³g_s(t)Dθf(t,·)dt +

_∞

−log|θ|(sinht)ⁿ⁻³g_s(t)Dθf(t,·)dt

=(1)+(2).

(4.6)

By the Harnack inequalities,sincefis bounded,the gradient offis bounded in terms of f_∞ ands;it follows that

D_θf(r, ξ)≤Cf∞|θ|e^r (4.7)

for some constantCwhich only depends ons. This gives

(1)≤ f_∞

₋log|θ|

r0

e^{t(n−2−δ)}|θ|dt≤Cf_∞|θ|^δ−(n−3). (4.8)

Clearly,one has also,for anyr,Dθf(r,·)∞ ≤2f∞,so that one gets

(2)= _∞

−log|θ|(sinht)ⁿ⁻³g_s(t)Dθf(t,·)dt

≤Cf_{∞ ∞}

−log|θ|e^{t(n−3−δ)}dt≤Cf_∞|θ|^δ−(n−3).

(4.9)

So |II| ≤ Cf∞|θ|^δ−(n−3). It remains to estimate the term I = D_θH(r₀,·). We have Δ_σD_θH(r₀,·) = D_θf(r₀,·). By the Harnack inequality,and due to the compacity ofSⁿ⁻¹, Δ⁻_σ¹is a continuous operator for theL^∞-norm,so one has

|I|≤D_θH r₀,·

∞ ≤Cf∞|θ|≤Cf∞|θ|^δ−(n−3), (4.10) for a constantCwhich only depends ons. It follows thatH∈Λ⁰_δ−(n−3)(Sⁿ⁻¹),with a norm smaller thanCf_∞.

We now show how this method can be extended to handle the general case. We are going to use the results ofSection 3on the operatorDλ;this is why we need to assume thats=δ ∈[(n−1)/2, n−1[. Let us denote byjthe integer such thatδ=n−1−2j+α withα∈]0, 2].

Sincefis aλ-eigenfunction,it satisfies

Dλf(r, ξ)= −Δ_σf(r, ξ). (4.11)

From the study of the Green functiong_s in the appendix,there is anr₀depending only onssuch thatg_s(r)does not vanish forr ≥ r₀. Therefore, by iterating the differential equation satisfied byT(r,·),we see that forr≥r₀,one has

D^j_λ⁻¹

(sinhr)³⁻ⁿ g_s(r)

∂T

∂r(r,·) =(−1)^jΔ^j_σf(r,·). (4.12)

SinceT(r,·)has zero mean,there exists a unique smooth functionH(r,·)onSⁿ⁻¹which has zero mean and such thatT(r,·) = Δ^j_σH(r,·). This uniqueness property ofH(r,·)im- plies that

D^j_λ⁻¹

(sinhr)³⁻ⁿ g_s(r)

∂H

∂r(r,·) =(−1)^jf(r,·). (4.13)

Sincefis bounded and since2(j−1)≤n−1−δ,Corollary 3.2can be applied with l=0. This gives

(sinhr)³⁻ⁿ g_s(r)

∂H

∂r(r,·) ≤C

f∞ +

2j−3

0

∂^pH

∂r^p r₀,·

e^−2(j−1)r ifα < 2, (sinhr)³⁻ⁿ

g_s(r)

∂H

∂r(r,·) ≤C

f_∞ +

2j−3

0

∂^pH

∂r^p r₀,·

re^−2(j−1)r ifα=2,

(4.14)

for a constantC,which only depends onsand onr₀. One also has

Δ^j_σ ∂^pH

∂r^p

r₀,· = ∂^pT

∂r^p r₀,·

. (4.15)

Going back to the definition ofT(r,·)and using the Harnack inequality,one finds ∂^pT

∂r^p r₀,·

∞ ≤Cf∞, (4.16)

forp≤2j−3,the constantConly depending onr₀andj. Therefore,we have ∂^pH

∂r^p

r₀,·

∞

≤Cf_∞, ∂H

∂r(r,·)

∞ ≤C(sinhr)ⁿ⁻³g_s(r)e^−2(j−1)rf∞ ≤Cf∞e^−αr ifα < 2,

≤C(sinhr)ⁿ⁻³g_s(r)re^−2(j−1)rf_∞ ≤Cf_∞re^−αr ifα=2,

(4.17)

for a constantCwhich only depends onr₀and ons. In any cases,since the last term is integrable over[r0,∞[,uniformly onSⁿ⁻¹,we may define a continuous functionHonSⁿ⁻¹ by setting

H(·)= lim

r→∞H(r,·)=H r₀,·

+ _∞

r0

∂H

∂t(t,·)dt. (4.18) In the distributional sense,this function satisfies

Δ^j_σH(·)= lim

r→∞Δ^j_σH(r,·)=:T(·). (4.19)

We are now going to prove thatH∈Λ⁰_α(Sⁿ⁻¹). LetR_θ∈O(n)be a rotation of angle θ∈]−π, π[. Sinceα∈]0, 2],we just need to prove thatD³_θH_∞ ≤Cf_∞|θ|^αfor a constant Cwhich does not depend onθnor onR_θ(see[14,page 331]). We have

D³_θH(·)=D³_θH r₀,·

+ _∞

r0

∂D³_θH

∂t (t,·)dt. (4.20) For the termD³_θH(r0,·), we note that Δ^j_σ⁻¹D³_θH(r0,·) = D³_θT(r0,·). By the Harnack in- equality,D³_θT(r0,·)_∞ is bounded byCf_∞|θ|²,for a constantCwhich depends only on r₀. Therefore,sinceΔ⁻_σ¹is continuous for theL^∞-norm,a similar bound holds also for D³_θH(r0,·): itsL^∞-norm is smaller thanCf∞|θ|²which is smaller thanCf∞|θ|^α.

It remains to deal with the term corresponding to the integral. Arguing as in the caseδ ∈]n−3, n−2[,we split this integral into the sum ofI=_−log|θ|

r0 andII=_∞

−log|θ|. We have

D^j_λ⁻¹

(sinhr)³⁻ⁿ g_s(r)

∂D³_θH

∂r (r,·) =(−1)^jD³_θf(r,·). (4.21) SinceD³_θf(r,·)_∞ ≤Cf_∞|θ|³e^3r,for a constantCindependent ofrandθ,we may apply Corollary 3.2withl= −3: it comes out that forα∈]0, 2],

∂D³_θH

∂r (r,·)

∞ ≤Cf∞|θ|³e^(3−α)r, (4.22) so that

|I|≤Cf_∞|θ|³

₋log|θ|

r0

e^(3−α)tdt≤Cf_∞|θ|^α. (4.23) Now,ifα < 2,we already proved during the construction ofHthat(∂H/∂r)(r,·)_∞

≤Cf∞e^−αr;we therefore have also(∂D³_θH/∂r)(r,·)∞ ≤Cf∞e^−αr. This gives

|II|≤Cf∞|θ|^α. (4.24)

Adding the estimates above,we deduce thatD³_θH∞ ≤Cf∞|θ|^α. This finishes the proof in the caseα < 2.

It remains to consider the caseα=2. There we are in the critical case for applying Corollary 3.2with l = 0 and we argue differently. By the Harnack inequality,we have Dθf(r,·)∞ ≤Cf∞|θ|e^rfor a constantCindependent offandθ. ApplyingCorollary 3.2 withl= −1,we deduce that(∂DθH/∂r)(r,·)∞ ≤Cf∞|θ|e^−r. But,we have

∂D³_θH

∂r (r,·) ∞ ≤4

∂D_θH

∂r (r,·)

∞ ≤Cf∞|θ|e^−r, (4.25)

for a constantCwhich is independent ofR_θandf. It follows that _∞

−log|θ|

∂D³_θH

∂t (t,·)dt

≤Cf_∞|θ|². (4.26) It ends the proof of the caseα=2and ofProposition 4.1also.

5 Proof ofTheorem 1.3whenδ≥(n−1)/2

Letf ∈ Eλs. By a theorem of Helgason and Minemura([7,9],see also[1]),anyλ_s-eigen- function can be written asP^s(D),whereDis a hyperfunction onSⁿ⁻¹. Also,T is unique except whens∈−N^∗[9].

To prove Theorem 1.3 when s ≥ (n−1)/2, we show that the distribution T we constructed in the previous section is equal toD,that is, that one hasf = P^s(T).

Theorem 1.3will then follow fromProposition 4.1. We suppose first thatf = P^s(φ),for aC¹-functionφonSⁿ⁻¹. ByProposition 2.2,fis bounded and we can consider the cor- responding distributionT_s. We need to show thatT_s = D. To simplify the notations,we denoteT_sbyT.

Recall that the distributionTis the limit asr→ ∞of the distributions

T(r,·)=(sinhr)ⁿ⁻¹

∂f(r,·)

∂r g_s−∂g_s

∂r f(r,·) . (5.1)

With these notations,we have the following result.

Proposition 5.1. Letφbe aC¹function onSⁿ⁻¹andf=P^s(φ). Then,one has

r→∞lim

T(r,·), P^s(z,·)

=f(z). (5.2)

Proof. We begin by studying the kernel functionK^(s)_r (ζ, ξ)defined by

K^(s)_r (ζ, ξ)=(sinhr)ⁿ⁻¹ ∂P^s

∂r (rζ, ξ)g_s(r)−P^s(rζ, ξ)∂g_s

∂r (r)

. (5.3)

Lemma 5.2. The kernelK^(s)_r (ζ, ξ)has the following two properties:

(1) for anyξ∈Sⁿ⁻¹,one has

Sⁿ⁻¹K^(s)_r (ζ, ξ)dσ(ζ)=1;

(2) for(s)=δ≥(n−1)/2,andr₀> 0,there is a constantC > 0such that for all r≥r₀,for allζ,ξinSⁿ⁻¹,

K^(s)_r (ζ, ξ)≤C e^{(n−1−2δ)r}

e^−r+|ζ−ξ|_2δ, K^(s)_r (ζ, ξ)≤Ce^{r(n−3−2δ)}

|ζ−ξ|^2δ+2. (5.4)

Proof. Property(1)follows from the Green formula applied to the functionsP^s(·, ξ)and g_sand from the fact thatP^s(o, ξ)=1for allξ∈Sⁿ⁻¹.

To prove(2),we assume first thatδ > (n−1)/2. For anyr ≥ r₀ > 0,we know from the appendix that|gs(r)|≤Ce^−δrfor a constantCwhich is uniform assremains in a compact of the set{s | (s) ∈ [(n−1)/2, n−1[}. By the Harnack inequality,it follows that|∂gs(r)/∂r|≤Ce^−δr,where the constantCis uniform assremains in a compact set.

This leads first to the inequality K^(s)_r (ζ, ξ)≤C e^{(n−1−2δ)r}

e^−r+|ζ−ξ|_2δ, (5.5)

where the constantCis independent ofr ≥ r₀and ofsas long as it stays in a compact set. In particular,for allζ=ξ,K^(s)_r (ζ, ξ)→0asr→ ∞whenδ >(n−1)/2. The derivative with respect torofK^(s)_r (ζ, ξ)can be computed as we did in the previous section for the construction of the distributionT. This gives

∂K^(s)_r (ζ, ξ)

∂r =(sinhr)ⁿ⁻³g_s(r)ΔσP^s(rζ, ξ). (5.6)

Using the same estimates as above ong_sand its derivative,and onΔ_σP^s(cf.Claim 2.1), it comes out that

∂K^(s)_r (ζ, ξ)

∂r

≤C e^{r(n−3−2δ)}

e^−r+|ζ−ξ|_2δ+2. (5.7)

Integrating with respect torand using thatK_t(ζ, ξ)→0ast→ ∞,it gives that K^(s)_r (ζ, ξ)≤Ce^{(n−3−2δ)r}

|ζ−ξ|^2δ+2, (5.8)

for a constantCwhich does not depend onsas long as it stays in a compact set ofs≥ (n−1)/2.

Since the kernelK^(s)_r is continuous with respect tos,the last estimate holds also for anyswiths=(n−1)/2(for a constant which depends onr₀and ons): it suffices to approximates=(n−1)/2+itbys_k+itfor a sequences_kwhich tends to(n−1)/2from

above.

Proof ofProposition 5.1. Asf=P^s(φ),we have

T(r, ζ)=

Sⁿ⁻¹K^(s)_r (ζ, ξ)φ(ξ)dσ(ξ). (5.9)