Wave and Klein-Gordon equations on certain locally symmetric spaces

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Wave and Klein-Gordon equations on certain locally symmetric spaces

Hong-Wei Zhang

To cite this version:

Hong-Wei Zhang. Wave and Klein-Gordon equations on certain locally symmetric spaces. The Journal of Geometric Analysis, Springer, In press, �10.1007/s12220-019-00246-8�. �hal-01872928v2�

WAVE AND KLEIN-GORDON EQUATIONS ON CERTAIN LOCALLY SYMMETRIC SPACES

HONG-WEI ZHANG

Abstract. This paper is devoted to study the dispersive properties of the linear Klein-Gordon and wave equations on a class of locally symmetric spaces.

As a consequence, we obtain the Strichartz estimate and prove global well- posedness results for the corresponding semilinear equation with low regularity data as on real hyperbolic spaces.

1. Introduction

LetM be a Riemannian manifold and denote by∆the Laplace-Beltrami operator onM. The theory is well established for the following wave equation onM =Rⁿ,

(∂²_tu(t, x)−∆u(t, x) =F(t, x), u(0, x) =f(x), ∂t|t=0u(t, x) =g(x), (1.1)

where the solutionsusatisify the Strichartz estimates: ¹

k∇_R×Rnuk_Lp(I;H^−σ,q(Rⁿ)).kfk_H1(Rⁿ)+kgk_L2(Rⁿ)+kFk_Lp˜0

(I;H^σ,˜˜q0(Rⁿ)), on any intervalI⊆Runder the assumptions

σ=n+ 1 2

1 2 −1

q

, eσ= n+ 1 2

1 2 −1

˜ q

,

and the couples(p, q),(˜p,q)˜ ∈(2,+∞]×[2,2ⁿ⁻¹_n−3)fulfill the admissibility conditions:

1

p =n−1 2

1 2 −1

q

, 1

˜

p= n−1 2

1 2 −1

˜ q

.

These estimates serve as a tool for finding minimal regularity conditions on the initial data ensuring well-posedness for corresponding semilinear wave equations, which is addressed in [23], and almost fully answered in [11,17,24,27].

Analogous results have been found for the Klein-Gordon equation (∂_t²u(t, x)−∆u(t, x) +cu(t, x) =F(t, x),

u(0, x) =f(x), ∂t|t=0u(t, x) =g(x).

(1.2)

2000Mathematics Subject Classification. 35Q55, 43A85, 22E30, 35P25, 47J35, 58D25.

Key words and phrases. Locally symmetric space, wave operator, dispersive estimate, semilin- ear wave equation, semilinear Klein-Gordon equation.

This work is part of the author’s Ph.D. thesis, supervised by J.-Ph. Anker and N. Burq.

The author thanks Marc Peigné for helpful hints about convex cocompact subgroups and Michel Marias for stimulating discussions.

1The symbol., let us recall, means precisely that there exists a constant0< C <+∞such that k∇_R×Rnuk_Lp(I;H−σ,q(Rⁿ)) ≤C

kfk_H1(Rⁿ)+kgk_L2(Rⁿ)+kFk

L^p˜0(I;H^˜σ,˜q0(Rⁿ))

, where p˜⁰ is the dual exponent ofp, defined by the formula˜ ¹_p˜+_p˜¹0 = 1, so isq˜⁰.

1

withc= 1, see [6,18,30,31].

Given the rich Euclidean theory, it is natural to look at the corresponding equa- tions on more general manifolds. We consider in the present paper a class of non- compact locally symmetric spacesM, on which we study the Klein-Gordon equation (1.2) withc ≥ −ρ², where ρis a positive constant depending on the structure of M and defined in the next section. Due to large-scale dispersive effects in nega- tive curvature, we expect stronger results than in the Euclidean setting, as on real hyperbolic space, see [3,4].

In the critical case c =−ρ², (1.2) is called the shifted wave equation. To our knowledge, it was first considered in [14, 15] in low dimensions n = 2 and n = 3. In [4, 5], a detailed analysis of the shifted wave equation was carried out on real hyperbolic spaces and on Damek-Ricci spaces, which contains all rank one symmetric spaces of noncompact type. In the non-shifted case c > −ρ², similar results on real hyperbolic spaces were obtained in [3].

In the recent paper [16], the Schrödinger equation was considered on certain locally symmetric spaces. In the present paper, we study the wave and Klein- Gordon equations in the same spirit.

1.1. Notations

We adopt the standard notation (see for instance [20], [8]). LetGbe a semisimple Lie group, connected, noncompact, with finite center, andKbe a maximal compact subgroup ofG. The homogenous spaceX =G/Kis a Riemannian symmetric space of noncompact type, whose dimension is denoted byn. Letg=k⊕pbe the Cartan decomposition of its Lie algebra. The Killing form ofginduces aK-invariant inner product onp, and hence aG-invariant Riemannian metric onG/K.

Fix a maximal abelian subspacea inp. The symmetric spaceX is said to have rank one if dima = 1. Denote by a^∗ the real dual of a, let Σ ⊂ a^∗ be the root system of(g,a)and denote byW the Weyl group associated toΣ. Choose a setΣ⁺ of positive roots, leta⁺⊂abe the corresponding positive Weyl chamber anda⁺its closure. Denote byρthe half sum of positive roots counted with their multiplicities:

ρ= 1 2

X

α∈Σ⁺

m_αα,

wheremα is the dimension of root spacega={Y ∈g|[H, Y] =α(H)Y,∀H∈a }.

Let Γ be a discrete torsion-free subgroup of G. The locally symmetric space M = Γ\X, equipped with the Riemannian structure inherited fromX becomes a Riemannian manifold. We say thatM has rank one ifX has rank one. Moreover Γis called convex cocompact if the quotient groupΓ\Conv(Λ_Γ)is compact, where Conv(Λ_Γ)is the convex hull of the limit setΛ_ΓofΓ. We denote by∆the Laplace- Beltrami operator, by d(·,·) the Riemannian distance, and by dx the associated measure, both onX andM. Consider the Poincaré series

P(s;x, y) =X

γ∈Γ

e^−sd(x,γy), s >0, x, y∈X,

and denote byδ(Γ)its critical exponent:

δ(Γ) = inf{s >0 |P(s;x, y)<+∞}.

1.2. Assumptions

In this paper, M = Γ\X is a rank one locally symmetric space such that Γ is convex cocompact andδ(Γ)< ρ.

Let us comment a few words on these assumptions. Wave type equations on noncompact rank one symmetric spaces are well understood. Sharp pointwise esti- mates of wave kernels on X(see Section 2.2), which were obtained in [3, 4], allow us to deal with wave kernels on a locally symmetric space M. Notice that such information is lacking in higher rank.

The rank one symmetric spaces of the noncompact type are the hyperbolic spaces Hⁿ(F)with F =R,C,H or H²(O). In particular, we have a^∗ =a anda⁺ ∼=R^∗+, henceρis just a positive constant depending on the structure ofX. Specifically, as a direct consequence of the assumptionδ(Γ)< ρ, the series (3.1) defining the wave kernel on M is absolutely convergent, see Proposition 3.1. In addition, according to [9], the bottom λ0 of the L²-spectrum of −∆ on M is equal to ρ², as on X. Consequently, we obtain an analogousL²Kunze-Stein phenomenon onM without further assumptions, see Proposition3.2. Notice thatλ0=ρ²>0impliesVol(M) = +∞, whileλ0= 0 ifM is a lattice.

At last, the convex cocompactness assumption implies a uniform upper bound of the Poincaré series, see Lemma3.3, which is crucial for theL¹→L^∞boundedness of wave propagators onM.

Remark 1.1. The Schrödinger equation is studied in [16] under slightly different assumptions, our well-posedness results hold also in that setting.

1.3. Statement of the results Consider the operatorD =p

−∆−ρ²+κ² withκ >0, then the Klein-Gordon equations (1.2) becomes

(∂²_tu(t, x) +D²_xu(t, x) =F(t, x), u(0, x) =f(x), ∂t|t=0u(t, x) =g(x).

(1.3)

with c = κ² −ρ² > −ρ². Notice that (1.3) is the wave equation when κ = ρ and becomes the shifted wave equation in the limit case κ= 0. Consider another operatorDe=p

−∆−ρ²+eκ²witheκ > ρ. We denote byω_t^σthe radial convolution kernel of the wave operatorW_t^σ:=De^−σe^itD on the symmetric spaceX:

W_t^σf(x) =f∗ω^σ_t(x) = Z

G

ω^σ_t(y⁻¹x)f(y)dy, (1.4)

where f is any reasonable function on X, see Section 2.2 for more details. By K-bi-invariance of the kernelω^σ_t, we deduce thatW_t^σf is leftΓ-invariant and right K-invariant if f is defined on the locally symmetric space M. Thus the wave operator onM, denoted byWd_t^σ, is also defined by (1.4). Consider the wave kernel ωc_t^σ onM, which is given by

ωc_t^σ(x, y) =X

γ∈Γ

ω^σ_t(y⁻¹γx), ∀x, y∈X.

Then the wave operatorWd_t^σ onM is an integral operator:

Wd_t^σf(x) = Z

M

ωc^σ_t(x, y)f(y)dy,

see Proposition 3.1. The aim of this paper is to prove the following dispersive properties:

Theorem 1.2. Forn≥3,2< q <+∞andσ≥(n+ 1)

1 2−¹_q

,

Wd_t^σ

_Lq0

(M)→L^q(M).

(|t|⁻⁽ⁿ⁻¹⁾(¹2−¹_q) if 0<|t|<1,

|t|⁻³² if |t| ≥1.

(1.5)

Remark 1.3. In dimension n= 2, there is an additional logarithmic factor in the small time bound, which becomes|t|⁻(¹₂⁻¹_q)(1−log|t|)^1−2q, see Theorem2.1in the next section.

Remark 1.4. At the endpointq= 2,t7→e^itDis a one-parameter group of unitary operators onL²(M).

By applying the classical T T^∗ method and by using the previous dispersive properties, we obtain the Strichartz estimate

k∇_R×Muk_Lp(I;H^−σ,q(M)).kfk_H1(M)+kgk_L2(M)+kFk_Lp˜0

(I;H^˜σ,˜q0(M))

for the solutionsuof (1.3), see Section 4for more information about the Sobolev spacesH^−σ,q(M). HereI⊂Ris any time interval, possibly unbounded,

σ≥n+ 1 2

1 2 −1

q

, eσ≥ n+ 1 2

1 2 −1

˜ q

,

and the couples (p, q)and (˜p,q)˜ are admissible, which means that

1 p,¹_q

,

1

˜ p,¹_q˜ belong, in dimensionn≥4 (see Section4for the lower dimensions) to the triangle 1

p,1 q

∈

0,1 2

×

0,1 2

1

p ≥n−1 2

1 2 −1

q

[ 0,1

2

, 1

2,1 2 − 1

n−1

.

1 p 1

q 1 2 1 2 −_n−1¹

0 ₁

1 2

p = ⁿ⁻¹₂

1 2 −¹_q

Figure 1. Admissibility in dimensionn≥4.

Notice that the admissible set for M is larger than the admissible set for Rⁿ which corresponds only to the lower edge of the triangle. In comparison withX, we loose the right edge of the triangle, which corresponds to the critical case ¹_p =¹₂

and ¹_q > ¹₂ − _n−1¹ , this will be explained in Section 4. Notice that we obtain nevertheless the same well-posedness results as onX.

This paper is organized as follows. In Section2, we review spherical analysis on noncompact symmetric spaces, and recall pointwise estimates of wave kernels on rank one symmetric space obtained in [3]. In Section3, after proving the necessary lemmas, we prove the dispersive estimate by an interpolation argument. As a consequence, we deduce the Strichartz estimate and obtain well-posedness results for the semilinear Klein-Gordon equation in Section4.

2. Preliminairies

2.1. Spherical analysis on noncompact symmetric spaces

We review in this section some elementary facts about noncompact symmetric spaces. We refer to [1,2, 13, 20] for more details.

Recall thata⁺ is the closure of the positive Weyl chambera⁺. Denote byn= P

α∈Σ⁺gα the nilpotent Lie subalgebra of g associated with Σ⁺, and by N the corresponding Lie subgroup ofG. Then we have the following two decompositions ofG:

(G=N(expa)K (Iwasawa), G=K(expa+)K (Cartan).

In the Cartan decomposition, the Haar measure onGwrites Z

G

f(g)dg=const.

Z

K

dk1

Z

a+

Y

α∈Σ⁺

(sinhα(H))^mαdH Z

K

f(k1(expH)k2)dk2. (2.1)

In the rank one case, which we consider in this paper, Z

a₊

Y

α∈Σ⁺

(sinhα(H))^mαdH =const.

Z +∞

0

(sinhr)^mα(sinh 2r)^m2αdr, where

(sinhr)^mα(sinh 2r)^m2α.e^2ρr, ∀r >0.

(2.2)

Denote byS(K\G/K)the Schwartz space ofK-bi-invariant functions onG. The spherical Fourier transformHis defined by

Hf(λ) = Z

G

f(x)ϕ_λ(x)dx, ∀λ∈a^∗∼=R, ∀f ∈ S(K\G/K).

Here ϕλ ∈ C^∞(K\G/K) is a spherical function, which can be characterized as a radial eigenfunction of the negative Laplace-Beltrami operator−∆satisfying (2.3)

(−∆ϕλ(x) = λ²+ρ² ϕλ(x), ϕ_λ(e) = 1.

In the noncompact case, the spherical function is characterized by ϕ_λ(x) =

Z

K

e(iλ+ρ)A(kx)dk, λ∈a^∗_C, (2.4)

whereA(kx)is the uniquea-component in the Iwasawa decomposition ofkx.

Denote byS(a^∗)^W the subspace ofW-invariant functions in the Schwartz space S(a^∗). ThenHis an isomorphism between S(K\G/K)andS(a^∗)^W. The inverse spherical Fourier transform is defined by

f(x) =const.

Z

a^∗

Hf(λ)ϕ_−λ(x)|c(λ)|⁻²dλ, ∀x∈G, ∀f ∈ S(a^∗)^W, wherec(λ)is the Harish-Chandrac-function.

2.2. Pointwise estimates of the wave kernel on symmetric spaces We recall in this section the pointwise wave kernel estimates on rank one symmet- ric space obtained in [3] and [5]. Via the spherical Fourier transform and (2.3), the negative Laplace-Beltrami operator−∆corresponds toλ²+ρ², hence the operators D=p

−∆−ρ²+κ² andDe =p

−∆−ρ²+eκ² to pλ²+κ² and p

λ²+ ˜κ².

By the inverse spherical Fourier transform, the radial convolution kernel ω_t^σ of W_t^σ=De^−σe^itD onX is given by

ω_t^σ(r) =const.

Z +∞

−∞

(λ²+ ˜κ²)^−σ2e^it

√λ²+κ²ϕ_λ(r)|c(λ)|⁻²dλ

for suitable exponentsσ∈R. Consider smooth even cut-off functionsχ0 and χ∞

onRsuch that







χ0(λ) +χ_∞(λ) = 1, χ₀(λ) = 1, ∀|λ| ≤1, χ_∞(λ) = 1, ∀|λ| ≥2.

Let us split up

ω_t^σ(r) =ω^σ,0_t (r) +ω^σ,∞_t (r)

=const.

Z +∞

−∞

χ0(λ)(λ²+ ˜κ²)^−σ2e^it

√λ²+κ²ϕλ(r)|c(λ)|⁻²dλ

+const.

Z +∞

−∞

χ∞(λ)(λ²+ ˜κ²)^−σ2e^it

√λ²+κ²ϕλ(r)|c(λ)|⁻²dλ.

To avoid possible singularities of the kernelω_t^σ,∞, see [32, Chap 9], we consider the analytic family of operators

fW_t^σ,∞:= e^σ2

Γ ⁿ⁺¹₂ −σχ_∞(D) ˜D^−σe^itD, (2.5)

in the vertical strip0≤Reσ≤ ⁿ⁺¹₂ , and their kernels

ωe_t^σ,∞(r) =const. e^σ2 Γ ⁿ⁺¹₂ −σ

Z +∞

−∞

χ_∞(λ)(λ²+ ˜κ²)^−σ2e^it

√λ²+κ²ϕ_λ(r)|c(λ)|⁻²dλ.

The following pointwise estimates of the kernels ω_t^σ,0 and eω^σ,∞_t , which were obtained in [3] for real hyperbolic spaces, extend straightforwardly to all rank one Riemannian symmetric spaces of the noncompact type.

Theorem 2.1. For allσ∈R, the kernelω_t^σ,0 satisfies

|ω_t^σ,0(r)|.

(ϕ0(r), ∀t∈R, ∀r≥0,

|t|⁻³²(1 +r)ϕ0(r), ∀|t| ≥1, ∀0≤r≤ ^|t|₂.

For all σ∈C with Reσ= ⁿ⁺¹₂ , and for every r ≥0, the following estimates hold for the kernelωe_t^σ,∞:

|ωe_t^σ,∞(r)|.

(|t|⁻ⁿ⁻¹² e^−ρr, ∀0<|t|<1, if n≥3,

|t|^−N(1 +r)^Nϕ₀(r), ∀|t| ≥1, ∀N ∈N. In the2-dimensional case, the small time estimate ofeω_t^σ,∞ reads

|ωe^σ,∞_t (r)|.|t|⁻¹²(1−log|t|)e^−r2, ∀0<|t|<1.

3. Dispersive estimates for the wave operator on locally symmetric spaces

In this section, we prove our main result, namely Theorem 1.2. Let us first describe the wave operator Wd_t^σ on locally symmetric space M. Recall that the wave kernel onM is given by

ωc_t^σ(x, y) =X

γ∈Γ

ω^σ_t(y⁻¹γx), ∀x, y∈X.

(3.1)

Proposition 3.1. The series (3.1)is convergent for every x, y∈X, and the wave operator onM is given by

Wd_t^σf(x) = Z

M

ωc^σ_t(x, y)f(y)dy, for any reasonable functionf onM.

Proof. According to the Cartan decomposition ofG, we can writey⁻¹γx=kγ(expHγ)k⁰_γ with H_γ ∈ R+ and k_γ, k_γ⁰ ∈K. Notice that H_γ =d(x, γy). Then, by the K-bi- invariance ofω^σ_t, we have

|cω_t^σ(x, y)|=

X

γ∈Γ

ω^σ_t(expHγ)

.X

γ∈Γ

|ω_t^σ,0(expHγ)|+X

γ∈Γ

|˜ω^σ,∞_t (expHγ)|.

For the first part, Theorem2.1implies that for allHγ ≥0, X

γ∈Γ

|ω^σ,0_t (expHγ)|.X

γ∈Γ

ϕ0(expHγ).X

γ∈Γ

(1 +Hγ)e^−ρHγ.

By choosing0< ε < ρ−δ(Γ), we obtain X

γ∈Γ

|ω_t^σ,0(expHγ)|.X

γ∈Γ

e−(δ(Γ)+ε)d(x,γy)=Pδ(Γ)+ε(x, y)<+∞.

The second part is handled similarly and thus omitted. Hence the serie (3.1) is convergent. According to (1.4), we know that

Wd_t^σf(x) = Z

G

ω_t^σ(y⁻¹x)f(y)dy= Z

X

ω_t^σ(y⁻¹x)f(y)dy,

sinceω_t^σ isK-bi-invariant andf is rightK-invariant. By using Weyl’s formula and the fact thatf is left Γ-invariant, we deduce that

Wd_t^σf(x) = Z

Γ\X



 X

γ∈Γ

f(γy)ω_t^σ(y⁻¹γx)



dy= Z

M

ωc^σ_t(x, y)f(y)dy.

Next, we introduce the following version of the L² Kunze-Stein phenomenon on locally symmetric space M, which plays an essential role in the proof of the dispersive estimate.

Proposition 3.2. Let ψ be a reasonable bi-K-invariant functions on G, e.g., in the Schwartz class. Then

k.∗ψk_L2(M)→L²(M)≤ Z

G

|ψ(x)|ϕ0(x)dx.

(3.2)

The Kunze-Stein phenomenon is a remarkable convolution property on semisim- ple Lie groups and symmetric spaces (see e.g., [26], [21], [10] and [22]), which was extended to some classes of locally symmetric spaces in [28] and [29]. Let us prove Proposition 3.2 along the lines of [29] in our setting where rankX = 1, δ(Γ)< ρ and with no additional assumption.

Proof. SinceGis a connected semisimple Lie group, it is of type I, [19,12]. Denote byGb the unitary dual ofGand by GbK the spherical subdual. We write L²(Γ\G) as the direct integral

L²(Γ\G)∼= Z ⊕

Gb

Hπdν(π) and

L²(M)∼= Z ⊕

Gb_K

(Hπ)^Kdν(π) (3.3)

accordingly, where ν is a positive measure on G, see for instance [8]. Recall thatb (Hπ)^K =Ceπ is one-dimensional for every π∈GbK. Recall moreover that, in rank one,Gb_K is parametrized by a subset ofC/±1. Specifically,Gb_K consists of

• the unitary spherical principal seriesπ_±λ (λ∈R/±1),

• the trivial representationπ_±iρ= 1,

• the complementary seriesπ_±iλ (λ∈I), where I=

((0, ρ) if X =Hⁿ(R)or Hⁿ(C), 0,^m₂^α + 1

if X=Hⁿ(H)or H²(O).

This result goes back to [25]. Under the assumptionδ(Γ)≤ρ, we know thatλ₀=ρ² is the bottom of the spectrum of −∆ onL²(M). As −∆ acts on(H_π)^K by multi- plication byλ²+ρ², we deduce that (3.3) involves only tempered representations, i.e., representations πλ with λ ∈ R/±1. Moreover, as the right convolution by ψ∈ S(K\G/K)acts on(Hπ_λ)^K by multiplication by

Hf(λ) = Z

G

f(x)ϕλ(x)dx,

whereϕλ(x) =hπλ(x)eπ_λ, eπ_λiis the spherical function (2.3), we deduce from (2.4) that

k.∗ψk_L2(M)→L²(M)≤sup

λ∈R

Z

G

ψ(x)ϕλ(x)dx

≤ Z

G

|ψ(x)|ϕ0(x)dx.

The following two lemmas are used in the proof of dispersive estimates.

Lemma 3.3. If Γ is convex cocompact, then there exists a constant C > 0 such that for all x, y∈X,

P(s;x, y)≤CP(s;0,0), where0=eK denotes the origin ofX.

Proof. Let Conv(ΛΓ) be the convex hull of the limit set ΛΓ of Γ. Recall that Γ is said to be convex cocompact if Γ\Conv(ΛΓ) is compact. Let F be a compact fundamental domain containing0for the action ofΓonConv(ΛΓ). Then, for each z∈Conv(ΛΓ), there existsγ∈Γandz⁰∈F such thatz=γz⁰.

The orthogonal projectionπ_⊥ :X →Conv(ΛΓ)is defined as follows. For every xinX,π_⊥(x)is the unique point inConv(ΛΓ)such that

d(x, π_⊥(x)) =d(x,Conv(ΛΓ)) := inf

y∈Conv(ΛΓ)

d(x, y).

Then, for allx, y∈X, we have (see [7], Chap II, Proposition 2.4.) d(π_⊥(x), π_⊥(y))≤d(x, y).

On the other hand, for allx∈X andγ∈Γ,

d(γx, π⊥(γx)) =d(γx,Conv(ΛΓ)) =d(x,Conv(ΛΓ)), sinceΛΓ andConv(ΛΓ)areΓ-invariant. Thus

d(γx, π_⊥(γx)) =d(x, π_⊥(x)) =d(γx, γπ_⊥(x)),

which implies that, for all x ∈ X and γ ∈ Γ, π_⊥(γx) = γπ_⊥(x). Therefore, for everyx, y∈X ands >0, the Poincaré series satisfies:

P(s;x, y) =X

γ∈Γ

e^−sd(x,γy)≤X

γ∈Γ

e^{−sd(π⊥(x),π⊥(γy))}

=X

γ∈Γ

e^{−sd(π⊥(x),γπ⊥(y))}=P(s;π_⊥(x), π_⊥(y)), (3.4)

withπ_⊥(x), π_⊥(y)∈Conv(Λ_Γ). Moreover, as there existγ₁, γ₂ ∈Γ and x⁰, y⁰ ∈F such thatπ_⊥(x) =γ₁x⁰,π_⊥(y) =γ₂y⁰, we have

P(s;π_⊥(x), π_⊥(y)) =X

γ∈Γ

e^{−sd(x0,γ1−1γγ2y0)}= X

γ⁰∈Γ

e^{−sd(x0,γ0y0)}=P(s;x⁰, y⁰).

(3.5)

Sincex⁰, y⁰ ∈F, the triangular inequality yields

d(0, γ⁰0)≤d(0, x⁰) +d(x⁰, γ⁰y⁰) +d(γ⁰y⁰, γ⁰0)≤d(x⁰, γ⁰y⁰) + 2 diam(F).

Hence

P(s;x⁰, y⁰)≤e^2sdiam(F)P(s;0,0).

(3.6)

We conclude by combining (3.4), (3.5) and (3.6).

Consider the radial weight function defined by µ(x) =e(δ(Γ)+ε)d(x,0),

with0< ε < ρ−δ(Γ). We prove the following lemma by applying previous results.

Lemma 3.4. Let f be a reasonable function on M, and g be a radial reasonable function on X. Then the bilinear operator B(f, g) :=f ∗(µ⁻¹g)satisfies the fol- lowing estimate:

kB(·, g)k_Lq0

(M)→L^q(M)≤C_q Z

G

ϕ₀(x)µ⁻¹(x)|g(x)|^q/2dx 2/q

,

for all2≤q≤ ∞.

Proof. According to Proposition3.2, kB(·, g)k_L2(M)→L²(M)≤

Z

G

ϕ0(x)µ⁻¹(x)|g(x)|dx.

(3.7)

Sincef is leftΓ-invariant, we can rewrite B(f, g)(x) =

Z

X

µ⁻¹g y⁻¹x

f(y)dy

= Z

Γ\X



 X

γ∈Γ

µ⁻¹g

y⁻¹γx



f(y)dy,

with

X

γ∈Γ

µ⁻¹g

y⁻¹γx

≤||g||_∞X

γ∈Γ

e−(δ(Γ)+ε)d(x,γy)

≤||g||_∞P_δ(Γ)+ε(x, y)≤ ||g||_∞P_δ(Γ)+ε(0,0), according to Lemma3.3. Hence

kB(·, g)k_L1(M)→L^∞(M)= sup

x,y∈G

X

γ∈Γ

µ⁻¹g

y⁻¹γx

≤C||g||_∞. (3.8)

We conclude by standard interpolations between (3.7) and (3.8).

We prove now our main result.

Proof of Theorem 1.2. We split up the proof into two parts, depending whether the timetis small or large.

Dispersive estimate for small time

Assume that 0 < |t| < 1. On the one hand, by using the Lemma 3.4 with g(x) =µ(x)ω_t^σ,0(x), we have

k · ∗ω^σ,0_t k_Lq0

(M)→L^q(M)≤C_q Z

G

ϕ₀(x)µ(x)^q2−1|ω^σ,0_t (x)|^q2dx ²_q

.

Notice that the ground spherical functionϕ₀, the weightµand the kernelω^σ,0_t are allK-bi-invariant. By using the expression (2.1) of the Haar measure in the Cartan