Jean-Philippe Anker

Hong-Wei Zhang

WAVE EQUATION ON GENERAL NONCOMPACT SYMMETRIC SPACES

Keywords: 2020 Mathematics Subject Classification. 22E30, 35L05, 43A85, 43A90 noncompact symmetric space of higher rank, semilinear wave equation, dispersive property, Strichartz inequality, global well-posedness

We establish sharp pointwise kernel estimates and dispersive properties for the wave equation on noncompact symmetric spaces of general rank. This is achieved by combining the stationary phase method and the Hadamard parametrix, and in particular, by introducing a subtle spectral decomposition, which allows us to overcome a well-known difficulty in higher rank analysis, namely the fact that the Plancherel density is not a differential symbol in general. As consequences, we deduce the Strichartz inequality for a large family of admissible pairs and prove global well-posedness results for the corresponding semilinear equation with low regularity data as on hyperbolic spaces.

Given the rich Euclidean results, several works have been made in other settings. We are interested in Riemannian symmetric spaces of noncompact type, where relevant questions are now well answered in rank one, see for instance [Fon97; Ion00; Tat01; MeTa11; MeTa12; APV12; AnPi14] on hyperbolic spaces, and [START_REF] Anker | The wave equation on Damek-Ricci spaces[END_REF] on Damek-Ricci spaces. A first study of the wave equation on general symmetric spaces of higher rank was carried out in [START_REF] Hassani | Wave equation on Riemannian symmetric spaces[END_REF], where some non optimal estimates were obtained under a strong smoothness assumption. Recently, time sharp kernel estimates and dispersive properties have been proven in [START_REF] Zhang | Wave equation on certain Riemannian noncompact symmetric spaces[END_REF] on noncompact symmetric spaces G/K, with G complex. In this case, the Harish-Chandra c-function and the spherical function have elementary expressions, which is not the case in general.

In this paper, we establish pointwise wave kernel estimates and dispersive properties for the wave equation on general noncompact symmetric spaces, which are sharp in time and which extend previous results obtained on real hyperbolic spaces [APV12; AnPi14] to higher rank. The main challenge is that the Plancherel density involved in the wave kernel is not a polynomial, nor even a differential symbol in general. To bypass this problem, we consider barycentric decompositions of the Weyl chambers into subcones and differentiate in each subcone along a well chosen direction.

For suitable σ ∈ C, we consider the wave operator W σ t = (-∆) -σ 2 e it √ -∆ associated to the Laplace-Beltrami operator ∆ on a d-dimensional noncompact symmetric space X = G/K. To avoid possible singularities (see Sect. 3.2), we consider actually the analytic family of operators

W σ t = e σ 2 Γ( d+1 2 -σ) (-∆) -σ 2 e it √ -∆ (1.1)
in the vertical strip 0 ≤ Re σ ≤ d+1 2 . Let us denote by ω σ t its K-bi-invariant convolution kernel. Our first main result is the following pointwise estimate, which summarizes Theorem 3.3, Theorem 3.7 and Theorem 3.10 proved in Sect. 3.

Theorem 1.1 (Pointwise kernel estimates). Let d ≥ 3 and σ ∈ C with Re σ = d+1 2 . There exist C > 0 and N ∈ N such that the following estimates hold for all t ∈ R * and x ∈ X:

| ω σ t (x)| ≤ C(1 + |x + |) N e -ρ,x + |t| -d-1 2 if 0 < |t| < 1, |t| -D 2 if |t| ≥ 1,
where x + ∈ a + denotes the radial component of x in the Cartan decomposition, and D = + 2|Σ + r | is the so-called dimension at infinity of X. Remark 1.2. These kernel estimates are sharp in time and similar results hold obviously in the easier case where Re σ > d+1 2 . The value of N will be specified in Sect. 3. However, the polynomial (1+|x + |) N is not crucial for further computations because of the exponential decay e -ρ,x + .

By interpolation arguments, we deduce our second main result.

Theorem 1.3 (Dispersive property). Assume that d ≥ 3, 2 < q, q < +∞ and σ ≥ (d + 1) max( 1 2 -1 q , 1 2 -1 q ). Then there exists a constant C > 0 such that following dispersive estimates hold:

W σ t L q (X)→L q (X) ≤ C |t| -(d-1) max( 1 2 -1 q , 1 2 -1 q ) if 0 < |t| < 1, |t| -D 2 if |t| ≥ 1.
Remark 1.4. At the endpoint q = q = 2, t → e it √ -∆ is a one-parameter group of unitary operators on L 2 (X).

Remark 1.5. Theorem 1.1 and Theorem 1.3 generalize earlier results obtained for real hyperbolic spaces [APV12; AnPi14] (which extend straightforwardly to all noncompact symmetric spaces of rank one), or for noncompact symmetric spaces G/K with G complex [START_REF] Zhang | Wave equation on certain Riemannian noncompact symmetric spaces[END_REF]. Notice that D = 3 in rank one and that D = d if G is complex.

Remark 1.6. For simplicity, we omit the 2-dimensional case where the small time bounds in Theorem 1.1 and Theorem 1.3 involve an additional logarithmic factor, see [AnPi14, Theorem 3.2 and 4.2]. Notice that d ≥ 4 in higher rank, see (2.1).

Let us sketch the proofs of our main results. We prove the dispersive properties of W σ t by using interpolation arguments based on pointwise estimates of ω σ t , which are sharp in time. By the way, let us point out that the kernel analysis carried out on hyperbolic spaces [START_REF] Anker | Wave and Klein-Gordon equations on hyperbolic spaces[END_REF] can not be extend straightforwardly in higher rank, since the Plancherel density is not a differential symbol in general. Consider the Poisson operator P τ = e -τ √ -∆ , for all τ ∈ C with Re τ ≥ 0. Along the lines of [Sch88; GiMe90; CGM02], we can write formally our wave operator (1.1) as

W σ t = e σ 2 Γ( d+1 2 -σ) 1 Γ(σ) +∞ 0 ds s σ-1 P s-it .
Our analysis is focused on kernel estimates of the Poisson operator P s-it where s ∈ R + and t ∈ R * . We adopt different methods depending whether s, |t| and |x| |t| (x ∈ X) are small or large. Specifically,

• If s is bounded from above and |x| |t| is sufficiently small with |t| large, we develop an effective stationary phase method based on barycentric decompositions of Weyl chambers described in Sect. 2.3. In each subdivision, the Plancherel density becomes a differential symbol for a well chosen directional derivative, see Sect. 3.1.

• If s is bounded from above but |x| |t| is large (with |t| small or large), we estimate the kernel along the lines of [START_REF] Cowling | L p -L q estimates for functions of the Laplace-Beltrami operator on noncompact symmetric spaces[END_REF], where Cowling, Guilini and Meda have studied the Poisson operator P τ for τ ∈ C with Re τ ≥ 0. Unfortunately, their estimates are not sharp when τ is large and nearly imaginary, which happens in our context when s is small and |t| is large. To deal with this case, we resume and improve slightly their method by writing down more explicitly the Hadamard parametrix on noncompact symmetric spaces along the lines of [START_REF] Bérard | On the wave equation on a compact Riemannian manifold without conjugate points[END_REF], see Sect. 3.2.

• If s is large, the kernel is estimated by using the standard stationary phase method, which is similar to the rank one analysis, see Sect. 3.3.

This paper is organized as follows. We recall spherical Fourier analysis on noncompact symmetric spaces and introduce the barycentric decomposition of Weyl chambers in Sect. 2. Next, we derive pointwise wave kernel estimates in Sect. 3. By using interpolation arguments, we prove in Sect. 4 the dispersive property for the wave operator. As consequences, we establish the Strichartz inequality for a large family of admissible pairs and obtain well-posedness results for the associated semilinear wave equation in Sect. 5. We give further results about the Klein-Gordon equation in Sect. 6. Finally, we collect in the appendices some useful results: in Appendix A, we study by the stationary phase method an oscillatory integral occurring in the wave kernel analysis; next we describe in Appendix B the Hadamard parametrix on noncompact symmetric spaces and consider its application to the Poisson operator in Appendix C.

Preliminaries

In this section, we first review briefly spherical Fourier analysis on noncompact symmetric spaces. Next we introduce a barycentric decomposition for Weyl chambers, which will be crucial for the forthcoming kernel estimates.

2.1. Notations. We adopt the standard notation and refer to [Hel78; Hel00] for more details. Let G be a semisimple Lie group, connected, noncompact, with finite center, and K be a maximal compact subgroup of G. The homogeneous space X = G/K is a Riemannian symmetric space of noncompact type. Let g = k ⊕ p be the Cartan decomposition of the Lie algebra of G. There is a natural identification between p and the tangent space of X at the origin. The Killing form of g induces a K-invariant inner product on p, hence a G-invariant Riemannian metric on X.

Fix a maximal abelian subspace a in p. The rank of X is the dimension of a. Let Σ ⊂ a be the root system of (g, a) and denote by W the Weyl group associated to Σ. Once a positive Weyl chamber a + ⊂ a has been selected, Σ + (resp. Σ + r or Σ + s ) denotes the corresponding set of positive roots (resp. positive reduced roots or simple roots). Let d be the dimension of X and D be the dimension at infinity of X:

d = + α∈Σ + m α and D = + 2|Σ + r |, (2.1)
where m α is the dimension of the positive root subspace g a . Notice that one cannot compare d and D without specifying the geometric structure of X. For example, when G is complex, we have d = D; but when X has normal real form, we have d = + |Σ + r | which is strictly smaller than D. Since we focus on the higher rank analysis, we may assume that d ≥ 3.

Let n be the nilpotent Lie subalgebra of g associated to Σ + and let N = exp n be the corresponding Lie subgroup of G. We have the decompositions

G = N (exp a) K (Iwasawa), G = K (exp a+) K (Cartan).
In the Cartan decomposition, the Haar measure on G writes

G f (x)dx = const. K dk 1 a + dx + δ(x + ) K dk 2 f (k 1 (exp x + )k 2 ), with δ(x + ) = α∈Σ + sinh α(x + ) mα α∈Σ + α, x + 1 + α, x + mα e 2ρ,x + ∀ x + ∈ a + .
Here ρ ∈ a + denotes the half sum of all positive roots α ∈ Σ + counted with their multiplicities m α :

ρ = 1 2 α∈Σ + m α α.
2.2. Spherical Fourier analysis on X. Let S(K\G/K) be the Schwartz space of K-bi-invariant functions on G. The spherical Fourier transform H is defined by

Hf (λ) = G dx ϕ -λ (x)f (x) ∀ λ ∈ a, ∀ f ∈ S(K\G/K),
where ϕ λ ∈ C ∞ (K\G/K) denotes the spherical function of index λ ∈ a C , which is a smooth K-bi-invariant eigenfunction for all invariant differential operators on X, in particular for the Laplace-Beltrami operator:

-∆ϕ λ (x) = |λ| 2 + |ρ| 2 ϕ λ (x).
In the noncompact case, spherical functions have the integral representation

ϕ λ (x) = K dk e iλ+ρ,A(kx) ∀ λ ∈ a C , (2.2)
where A(kx) denotes the a-component in the Iwasawa decomposition of kx. It satisfies the basic estimate

|ϕ λ (x)| ≤ ϕ 0 (x) ∀ λ ∈ a, ∀ x ∈ G,
where

ϕ 0 (exp x + ) α∈Σ + r 1 + α, x + e -ρ,x + ∀ x + ∈ a + .
Denote by S(a) W the subspace of W -invariant functions in the Schwartz space S(a). Then H is an isomorphism between S(K\G/K) and S(a) W . The inverse spherical Fourier transform is given by

f (x) = C 0 a dλ |c(λ)| -2 ϕ λ (x) Hf (λ) ∀ x ∈ G, ∀ f ∈ S(a) W ,
where C 0 > 0 is a constant depending only on the geometric structure of X, and which has been computed explicitly for instance in [AnJi99, Theorem 2.2.2]. By using the Gindikin & Karpelevič formula of the Harish-Chandra c-function (see [START_REF] Helgason | Groups and geometric analysis: integral geometry, invariant differential operators, and spherical functions[END_REF] or [START_REF] Gangolli | Harmonic analysis of spherical functions on real reductive groups[END_REF]), we can write the Plancherel density as

|c(λ)| -2 = α∈Σ + r |c α ( α, λ )| -2 , (2.3) with c α (v) = Γ( α,ρ α,α + 1 2 mα) Γ( α,ρ α,α ) Γ( 1 2 α,ρ α,α + 1 4 mα+ 1 2 m2α) Γ( 1 2 α,ρ α,α + 1 4 mα) Γ(iv) Γ(iv+ 1 2 mα) Γ( i 2 v+ 1 4 mα) Γ( i 2 v+ 1 4 mα+ 1 2 m2α) .
Since |c α | -2 is a homogeneous symbol on R of order m α + m 2α for every α ∈ Σ + r , then |c(λ)| -2 is a product of one-dimensional symbols, but not a symbol on a in general. The Plancherel density satisfies

|c(λ)| -2 α∈Σ + r α, λ 2 (1 + | α, λ |) mα+m2α-2 |λ| D- if |λ| ≤ 1, |λ| d- if |λ| ≥ 1,
together with all its derivatives.

2.3. Barycentric decomposition of the Weyl chamber. Let Σ + s = {α 1 , . . . , α } be the set of positive simple roots, and let {Λ 1 , . . . , Λ } be the dual basis of a, which is defined by where Remark 2.1. S j is the convex hull of the points

α j , Λ k = δ jk ∀ 1 ≤ j, k ≤ . (2.4) Notice that a + = R + Λ 1 + • • • + R + Λ and recall that    α j , α k ≤ 0 ∀ 1 ≤ j = k ≤ Λ j , Λ k ≥ 0 ∀ 1 ≤ j, k ≤ (2.
S j = {λ ∈ S ∩ a + | α j , λ = max 1≤j≤ α k , λ }. O a + α 2 α 1 Λ 1 Λ 2 S1 S2 B O Λ 2 Λ 1 Λ 3 B S1 S2 S3 a +
Λ k1 + • • • + Λ kr
r where {Λ k1 , . . . , Λ kr } runs through all subsets of {Λ 1 , . . . , Λ } containing Λ j .

Lemma 2.2. Let w ∈ W and 1 ≤ j ≤ l. Then (i) a root α ∈ Σ is orthogonal to some vectors in the tile w.S j if and only if α is orthogonal to its vertex w.λ j .

(ii) w.Λ j , λ ≥ 1 |Λ j | 2 for every λ ∈ w.S j .

Proof. (i) Let us show that α, w.Λ j = 0 if there exists λ ∈ w.S j such that α, λ = 0. By symmetry, we may assume that w = id and that α is a positive root. On the one hand, since α is spanned by the positive simple roots α 1 , . . . , α , we have

α = 1≤k≤ α, Λ k α k with α, Λ k ∈ N.
On the other hand, since α 1 , λ , . . . , α , λ are the barycentric coordinates of λ ∈ S ∩ a + , we have

λ = 1≤k≤ α k , λ Λ k (2.7)
which is a convex combination. In particular, α j , λ > 0 for all λ ∈ S j . Hence the inner product

α, λ = 1≤k≤ α, Λ k ≥0 α k , λ ≥0 α k , Λ k =1
cannot vanish unless α, Λ j = 0.

(ii) By symmetry, we may assume again that w = id. By taking the inner product of Λ j with (2.7), we obtain

Λ j , λ = 1≤k≤ Λ j , Λ k α k , λ = |Λ j | 2 α j , λ ≥ 1 + k =j Λ j , Λ k ≥0 α k , λ ≥0 ≥ 1 |Λ j | 2 ,
according to the property (2.5), and the fact that α j , λ is the largest barycentric coordinates for λ ∈ S j . Now, consider the tiling of the unit sphere obtained by projecting (2.6):

S(a) = w∈W 1≤j≤
w.S j where S j are the projections of the barycentric subdivisions S j on the unit sphere.

We establish next a smooth version of the partition of unity w∈W 1≤j≤ Let χ : R → [0, 1] be a smooth cut-off function such that χ(r) = 1 when r ≥ 0 and χ(r) = 0 when r ≤ -c 1 , where c 1 > 0 will be specified in Remark 2.5. For every w ∈ W and 1 ≤ j ≤ , we define Proposition 2.3. Let w ∈ W and 1 ≤ j ≤ . For all λ ∈ a {0}, we have (i) χ w.Sj (w.λ) = χ Sj (λ) and χ is W -invariant.

1 w.Sj λ |λ| = 1 a.e.. O Λ 2 Λ 1 Λ 3 S1 S2 S3 S1 S2 S 3
χ w.Sj (λ) = 1≤k≤ ,k =j χ w.α k , λ |λ| χ w.α j , λ -w.α k , λ |λ| ∀ λ ∈ a {0},
(ii) χ w.Sj = 1 on w.S j and χ ≥ 1 on a {0}.

(iii) χ w.Sj and χ are homogeneous symbols of order 0.

Proof. (i) follows from immediately from the definitions. In order to prove (ii), we may assume that w = id by symmetry. For all λ ∈ S j , we have

α k , λ ≥ 0 and α j , λ ≥ α k , λ
for every 1 ≤ k ≤ with k = j, hence χ Sj (λ) = 1. We deduce straightforwardly that χ ≥ 1 on a {0}. (iii) is obvious, since χ w.α k ,λ |λ| and χ w.αj ,λ -w.α k ,λ |λ| are homogeneous symbols of order 0 for all λ ∈ a {0} and 1 ≤ k ≤ .

For every w ∈ W and 1 ≤ j ≤ , we set χ w.Sj = χ w.S j χ on a {0}. It follows from Proposition 2.3 that χ w.Sj (w.λ) = χ Sj (λ) and that χ w.Sj is a homogeneous symbol of order 0. In particular, we have w∈W 1≤j≤ χ w.Sj = 1 on a {0}.

(2.8)

In addition, the vectors in the support of χ w.Sj satisfy further properties, which require some preliminaries.

Lemma 2.4. There exits c 2 > 0 such that, if λ ∈ a satisfies

-c 2 |λ| ≤ α k , λ ≤ α j , λ + c 2 |λ| ∀ k ∈ {1, . . . , } {j},
for some 1 ≤ j ≤ , then α j , λ ≥ c 2 |λ|.

Proof. By homogeneity, we may reduce to |λ| = 1. Since all norms are equivalent on a, there exists

c 3 > 0 such that 1≤j≤ | α k , λ | ≥ c 3 ∀ λ ∈ S(a).
(2.9)

Set c 2 = c3 2 . On the other hand, if -c 2 ≤ α k , λ ≤ 2c 2 ∀ k ∈ {1, . . . , } {j}, then α j , λ ≥ 2c 2 . Otherwise, 1≤j≤ | α k , λ | = | α j , λ | <2c2 + k =j | α k , λ | ≤2c2 < 2 c 2 = c 3 ,
which contradicts (2.9). On the other hand, if

2c 2 ≤ α k , λ ≤ α j , λ + c 2
for some k ∈ {1, . . . , } {j}, then α j , λ ≥ c 2 is obvious.

Remark 2.5. We clarify in this remark all constants appearing in this subsection. Denote by L 1 the highest root length and by L 2 the sum of lengths of the dual basis

L 1 = max α∈Σ + 1≤k≤ α, Λ k and L 2 = 1≤k≤ |Λ k |.
In addition, we denote by M 1 and M 2 the shortest and the longest generators

M 1 = min 1≤k≤ |Λ k | and M 2 = max 1≤k≤ |Λ k |. Then we choose c 1 > 0 such that c 1 < c 2 min{ 1 L1 , M 2 1 M2L2 }, where c 2 = c3 2 with c 3 defined in (2.9). Let c 4 = c 2 -L 1 c 1 and c 5 = M 2 1 c 2 -M 2 L 2 c 1 . Notice that L 1 ∈ N * , c 1 < c 2 ,
c 4 > 0 and c 5 > 0. All these constants depend only on the geometric structure of the roots system corresponding to X.

The following result is an analog of Lemma 2.2 for the wider regions supp χ ω.Sj .

Proposition 2.6. Let w ∈ W and 1 ≤ j ≤ . Then (i) a root α ∈ Σ satisfies either α, w.λ j = 0 or

| α, λ | ≥ c 4 |λ| ∀ λ ∈ supp χ w.Sj , (2.10) (ii) | w.Λ j , λ | ≥ c 5 |λ| for every λ ∈ supp χ w.Sj .
Proof. (i) By symmetry, we may assume that w = id and that α is a positive root. Notice that α, Λ j is a nonnegative integer, we suppose that α, Λ j > 0 and let us prove (2.10). As

-c 1 |λ| ≤ α k , λ ≤ α j , λ + c 1 |λ| ∀ λ ∈ supp χ Sj , ∀ k ∈ {1, . . . , } {j}, we have indeed α, λ = 1≤k≤ α, Λ k α k , λ = α, Λ j ≥1 α j , λ ≥c2|λ| + k =j α, Λ k α k , λ ≥-c1|λ| ≥ (c 2 -L 1 c 1 )|λ| = c 4 |λ|, according to Lemma 2.4 since c 1 < c 2 .
(ii) By symmetry, we assume again w = id. By taking the inner product of Λ j with (2.7), we obtain, for every λ ∈ supp χ Sj ,

Λ j , λ = 1≤k≤ Λ j , Λ k α k , λ = |Λ j | 2 ≥M 2 1 α j , λ ≥c2|λ| + k =j Λ j , Λ k ≤|Λj ||Λ k | α k , λ ≥-c1|λ| ≥ (M 2 1 c 2 -M 2 L 2 c 1 )|λ| = c 5 |λ|.
Remark 2.7. The partition of unity (2.8) plays an important role in the kernel analysis carried out in Sect. 3. It allows us to overcome a well-known problem in spherical Fourier analysis in higher rank, namely the fact that the Plancherel density is not a symbol in general. This new tool should certainly help solving other problems.

Pointwise estimates of the wave kernel

In this section, we derive pointwise estimates for the K-bi-invariant convolution kernel ω σ t of the operator W σ t = (-∆) -σ 2 e it √ -∆ on the symmetric space X:

W σ t f (x) = f * ω σ t (x) = G dy ω σ t (y -1 x)f (y)
for suitable exponents σ ∈ C. By using the inverse formula of the spherical Fourier transform, we have

ω σ t (x) = C 0 a dλ |c(λ)| -2 ϕ λ (x)(|λ| 2 + |ρ| 2 ) -σ 2 e it √ |λ| 2 +|ρ| 2
Let us point out that the analysis of this oscillatory integral carried out on hyperbolic spaces or on symmetric spaces G/K with G complex (see [AnPi14; Zha20]) does not hold in general since the Plancherel density |c(λ)| -2 is no more a differential symbol. We write

ω σ t (x) = 1 Γ(σ) +∞ 0 ds s σ-1 C 0 a dλ |c(λ)| -2 ϕ λ (x)e -(s-it) √ |λ| 2 +|ρ| 2 ps-it(x)
. according to the formula

r -σ = 1 Γ(σ) +∞ 0 ds s s σ e -sr ∀ r > 0.
Here p s-it is the K-bi-invariant convolution kernel of the Poisson operator P s-it .

Let us split up

ω σ t (x) = ω σ,0 t (x) + ω σ,∞ t (x) with ω σ,0 t (x) = 1 Γ(σ) 1 0 ds s σ p s-it (x) and ω σ,∞ t (x) = 1 Γ(σ) +∞ 1 ds s σ p s-it (x).
We shall see in Sect. 3.2 that the kernel ω σ,0 t (x) has a logarithmic singularity on the sphere |x| = t when σ = d+1 2 . To bypass this problem, we consider the analytic family of operators

W σ,0 t = e σ 2 Γ( d+1 2 -σ)Γ(σ) C σ,d 1 0 ds s σ-1 P s-it (3.1)
in the vertical strip 0 ≤ Re σ ≤ d+1 2 and the corresponding kernels

ω σ,0 t (x) = C σ,d 1 0 ds s σ-1 p s-it (x) ∀ x ∈ X.
Notice that the Gamma function Γ( d+1 2 -σ) allows us to deal with the boundary point σ = d+1 2 , while the exponential function ensures boundedness at infinity in the vertical strip. More precisely, by using the inequality

|Γ(z)| ≥ Γ(Re z) cosh(π Im z) -1 2 ∀ z ∈ C with Re z ≥ 1 2
(see for instance [DLMF, Eq. 5.6.7]), we can estimate

|C σ,d | |σ| |σ -d+1 2 | e π | Im σ|-(Im σ) 2 (3.2) for all σ ∈ C with 0 ≤ Re σ ≤ d+1 2 .
We divide the argument into three parts depending whether |t| and |x| |t| are small or large. When |t| is large but |x| |t| is sufficiently small, we estimate ω σ,0 t in Theorem 3.3 by combining the method of stationary phase with our barycentric decomposition of Weyl chambers; when |x| |t| is large, we estimate ω σ,0 t in Theorem 3.7 by using the Hadamard parametrix along the lines of [START_REF] Cowling | L p -L q estimates for functions of the Laplace-Beltrami operator on noncompact symmetric spaces[END_REF]; ω σ,∞ t (x) is easily handled by a standard stationary phase argument, see Theorem 3.10.

3.1.

Estimates of ω σ,0 t (x) when |t| is large and |x| |t| is sufficiently small. According to the integral expression (2.2) of the spherical functions, we write

ω σ,0 t (x) = C σ,d C 0 K dk e ρ,A(kx) 1 0 ds s σ-1 I(s, t, x), where
I(s, t, x) = a dλ |c(λ)| -2 e -s √ |λ| 2 +|ρ| 2 e itψt(λ)
is an oscillatory integral with phase

ψ t (λ) = |λ| 2 + |ρ| 2 + A(kx) t , λ . (3.3)
Let us split up

I(s, t, x) = I -(s, t, x) + I + (s, t, x) = a dλ χ ρ 0 (λ) • • • + a dλ χ ρ ∞ (λ) • • •
by using smooth radial cut-off functions χ ρ 0 and χ ρ ∞ = 1 -χ ρ 0 , where χ ρ 0 (λ) equals 1 when |λ| ≤ |ρ| and vanishes if |λ| ≥ 2|ρ|. Then we have the following estimates for I -and I - ∞ . Proposition 3.1. There exists 0 < C Σ ≤ 1 2 such that the following estimates hold when 0 < s < 1, |t| ≥ 1 and |x| |t| ≤ C Σ :

|I -(s, t, x)| |t| -D 2 (1 + |x|) D- 2 , (3.4)
and

|I + (s, t, x)| |t| -N , (3.5)
for every N ∈ N.

Remark 3.2. C Σ is a small constant depending only on the geometric structure of the root system Σ, which will be specified later in the proof of (3.5). Notice that the upper bounds (3.4) of I -and (3.5) of I + hold uniformly in s ∈ (0, 1).

Proof of the estimate (3.4). Recall that

I -(s, t, x) = a dλ a 0 (s, λ)e itψt(λ)
is an oscillatory integral with amplitude

a 0 (s, λ) = χ ρ 0 (λ)|c(λ)| -2 e -s √ |λ| 2 +|ρ| 2
and phase ψ t (λ) which is defined by (3.3). The amplitude a 0 (s, λ) is compactly supported for |λ| ≤ 2|ρ|, and the phase ψ t has, in the support of χ ρ 0 , a single nondegenerate critical point λ 0 which is given by

(|λ 0 | 2 + |ρ| 2 ) -1 2 λ 0 = -A t (3.6)
where A = A(kx), and which satisfies

|λ 0 | = |ρ| |A| |t| (1 -|A| 2 t 2 ) -1 2 ≤ |ρ| |x| t (1 -|x| 2 t 2 ) -1 2 < |ρ| √ 3 , (3.7)
as |A| ≤ |x| and |x| t ≤ C Σ < 1 2 . We conclude by resuming straightforwardly the computations carried out in the proof of [Zha20, Theorem 3.1]. For the sake of completeness and for the reader's convenience, we include a detailed study of the oscillatory integral I -in Appendix A (see Lemma A.1).

Let us turn to the oscillatory integral

I + (s, t, x) = a dλ χ ρ ∞ (λ)|c(λ)| -2 e -s √ |λ| 2 +|ρ| 2 e itψt(λ) ,
which vanishes unless |λ| > |ρ|. According to (3.7), ψ t has no critical point in the support of χ ρ ∞ . In rank one or in higher rank with G complex, one can handle this integral by performing several integrations by parts. This approach fails in general since the Plancherel density |c(λ)| -2 is not a differential symbol. To get around this problem, we split up the Weyl chamber according to the barycentric decomposition carried out in Sect. 2.3, and perform integrations by parts based along a well chosen directional derivative in each component.

Proof of the estimate (3.5). According to the partition of unity (2.8), we split up

I + (s, t, x) = w∈W 1≤j≤ I w.Sj (iτ, x)
with τ = s -it, and we estimate

I w.Sj (iτ, x) = a dλ χ w.Sj (λ) χ ρ ∞ (λ) |c(λ)| -2 e -τ ψiτ (λ) (3.8)
by performing integrations by parts based on

e -τ ψiτ (λ) = -1 τ 1 ∂w.Λ j ψiτ (λ) ∂ w.Λj e -τ ψiτ (λ) .
(3.9)

Notice that

∂ w.Λj ψ iτ (λ) = w.Λ j , λ √ |λ| 2 +|ρ| 2 -i A(kx)
τ is a symbol of order 0, which satisfies in addition

|∂ w.Λj ψ iτ (λ)| ≥ | w.Λj ,λ | √ |λ| 2 +|ρ| 2 -w.Λ j , A(kx) τ ≥ c 5 |λ| √ |λ| 2 +|ρ| 2 -|Λ j | |A(kx)| |τ | ≥ c5 √ 2 -M 2 |x| |t| on (supp χ w.Sj ) ∩ (supp χ ρ ∞ )
according to Proposition (2.6), where the constants c 5 and M 2 are specified in Remark 2.5. By choosing

C Σ = min{ c5 2M2 , 1 2 } , we obtain |∂ w.Λj ψ iτ (λ)| ≥ √ 2-1 2 c 5 > 0.
Let us return to (3.8), which becomes

I w.Sj (iτ, x) = τ -N a dλ e -τ ψiτ (λ) × ∂ w.Λj • 1 ∂w.Λ j ψiτ (λ) N χ w.Sj (λ)χ ρ ∞ (λ)|c(λ)| -2 ,
after N integrations by parts based on (3.9). If some derivatives hit χ ρ ∞ (λ), the above integral is reduced to the spherical shell |λ| |ρ| and thus converges. Assume that no derivative is applied to χ ρ ∞ (λ) and that

• N 1 derivatives are applied to the factors

1 ∂w.Λ j ψiτ (λ) ,
• N 2 derivatives are applied to χ w.Sj (λ),

• N 3 derivatives are applied to |c(λ)| -2 , with N = N 1 + N 2 + N 3 . The contribution of the first item is O(|λ| -N1
), as ∂ w.Λj ψ iτ (λ) is a symbol or order 0, which stays away from 0. Similarly, the contribution of the second item is O(|λ| -N2 ), as χ w.Sj (λ) is a symbol of order 0 according to Proposition 2.3. As far as the third item is concerned, the derivatives (∂ w.Λj ) N3 are applied to the various factors in (2.3). According to Proposition 2.6, for every λ in the support of χ w.Sj , any root α ∈ Σ satisfies either α, w.Λ j = 0 or | α, λ | |λ|. On the one hand, if α, w.Λ j = 0, all derivatives

(∂ w.Λj ) Nα |c α ( α, λ )| -2 ∀ N α ∈ N *
vanish. On the other hand, if α, w.Λ j = 0, we use the fact that |c α | -2 is a symbol on R of order m α + m 2α , together with (2.10), in order to estimate

(∂ w.Λj ) Nα |c α ( α, λ )| -2 | α, λ | mα+m2α-Nα |λ| mα+m2α-Nα ∀ N α ∈ N * for λ ∈ (supp χ w.Sj ) ∩ (supp χ ρ ∞ ). Hence (∂ w.Λj ) N3 |c(λ)| -2 = O(|λ| d--N3 ) ∀ λ ∈ (supp χ w.Sj ) ∩ (supp χ ρ ∞ ). In conclusion, |I w.Sj (iτ, x)| |τ | -N a dλ |λ| d--N1-N2-N3 |t| -N
provided that N > d, and consequently

I + (s, t, x) = O(|t| -N ).
We deduce from (3.4) and (3.5) that, for all

0 < s < 1, |t| ≥ 1 and |x| |t| ≤ C Σ , |I(s, t, x)| |t| -D 2 (1 + |x|) D- 2 (3.10) uniformly in s. Notice that ∂ ∂s I(s, t, x) = - a dλ |c(λ)| -2 |λ| 2 + |ρ| 2 e -s √ |λ| 2 +|ρ| 2 e itψt(λ)
has the same phase as I(s, t, x). Hence the estimate (3.10) holds for ∂ ∂s I(s, t, x) by similar computations. Since

1 0 ds s σ-1 I(s, t, x) = 1 σ s σ I(s, t, x) 1 0 -1 σ 1 0 ds s σ ∂ ∂s I(s, t, x),
we deduce that

C σ,d 1 0 ds s σ-1 I(s, t, x) |t| -D 2 (1 + |x|) D- 2
according to (3.2). Then we obtain the following kernel estimate of ω σ,0 t . Theorem 3.3. There exists 0 < C Σ ≤ 1 2 such that the following estimate holds, when |t| ≥ 1 and |x| |t| ≤ C Σ , uniformly in the vertical strip 0 ≤ Re σ ≤ d+1 2 :

| ω σ,0 t (x)| |t| -D 2 (1 + |x|) D- 2 ϕ 0 (x).
(3.11) 3.2. Estimates of ω σ,0 t (x) in the remaining range. Recall that τ = s -it with t ∈ R * and s ∈ (0, 1) throughout this subsection. We are looking for pointwise estimates of

ω σ,0 t (x) = C σ,d 1 0 ds s σ-1 C 0 a dλ |c(λ)| -2 ϕ λ (x) pτ (λ) e -τ √ |λ| 2 +|ρ| 2 pτ (x) ∀ x ∈ X,
where p τ (x) is the Poisson kernel and p τ (λ) denotes its spherical Fourier transform. This subsection focuses on pointwise estimates of p τ along the lines of [CGM01, pp.1054-1063].

Remark 3.4. Notice that the Gaussian factor ensures the convergence of the integral defining p τ , but yields a large negative power s -d . Then ω σ,0 t converges under the strong smoothness assumption Re σ ≥ d. We will sharpen it to Re σ = d+1 2 . Notice that the stationary phase method carried out in the previous subsection fails since the critical point can be very large when |x| |t| is not bounded from above.

As in [START_REF] Cowling | L p -L q estimates for functions of the Laplace-Beltrami operator on noncompact symmetric spaces[END_REF], let us denote by p R τ (v) = τ π(τ 2 +v 2 ) the Poisson kernel on R. We may write

p τ (λ) = e -τ √ |λ| 2 +|ρ| 2 = R dv p R τ (v) cos(v |λ| 2 + |ρ| 2 ) ∀λ ∈ a.
Consider a smooth even cut-off function χ : R →

[0, 1], which is supported in [- √ 2, √ 2],
and equals

1 on [-1, 1]. Denote by χ T = χ( • 2T ) with T = √ 2 (uni- formly in t) when |t| ≤ 1 or T = √ 2|t| when |t| ≥ 1. Then χ T is supported in [-2 √ 2T, 2 √ 2T ] ⊂ (-3T, 3T
). We denote by a τ and b τ the K-bi-invariant kernels of operators

A τ = +∞ ∞ dv χ T (v) p R τ (v) cos(v √ -∆)
and

B τ = ∞ -∞ dv {1 -χ T (v)} p R τ (v) cos(v √ -∆).
Then p τ = a τ + b τ and a τ is supported in a ball of radius 3T in X by finite propagation speed. b τ is easily estimated by straightforward computations, see Proposition 3.6. In order to analyze a τ , we will expand cos(v √ -∆) by using Hadamard parametrix.

Let {R z

+ | z ∈ C} be the analytic family of Riesz distributions on R defined by

R z + (r) =    Γ(z) -1 r z-1 if r > 0, 0 if r ≤ 0,
for Re z > 0. The K-bi-invariant convolution kernel Φ v of the operator cos(v √ -∆) has the asymptotic expansion

Φ v (exp H) = J(H) -1 2 [d/2] k=0 4 -k |v| U k (H) R k-d-1 2 + (v 2 -|H| 2 ) + E Φ (v, H) (3.12)
where

J(H) = α∈Σ + sinh α, H α, H
mα denotes the Jacobian of the exponential map from p equipped with Lebesgue measure to X equipped with Riemannian measure. Moreover, the coefficients satisfy

∇ n p U k = O(1) (3.13)
for every k, n ∈ N, and the remainder is estimated as

|E Φ (v, H)| (1 + v) 3( d 2 +1) e -ρ,H . (3.14)
The Hadamard parametrix has been described and applied in various settings, see for instance [Bér77; Hor94; CGM01]. For the reader's convenience, we give in Appendix B some details about this construction in the particular case of noncompact symmetric spaces. By resuming the proof of Lemma 3.3 in [START_REF] Cowling | L p -L q estimates for functions of the Laplace-Beltrami operator on noncompact symmetric spaces[END_REF] (see Appendix C for details), we deduce the following expansion of the K-bi-invariant convolution kernel a τ of the operator A τ :

a τ (exp H) = τ π J(H) -1 2 [d/2] k=0 4 -k U k (H) Γ d+1 2 -k (|H| 2 + τ 2 ) k-d+1 2 + E(τ, H) (3.15)
where

|E(τ, H)| |T | 3( d 2 +1) (log T -log s) e -ρ,H ∀ H ∈ a + .
(3.16)

Remark 3.5. As a consequence, we may deduce that

|a τ (exp H)| s -d+1 2 e -ρ,H    |t| -d-1 2 if |t| is small, |t| 3( d 2 +1) log |t| if |t| is large,
for all H ∈ a + . However, we cannot apply straightforwardly such estimates to study the kernel ω σ t , since it kills the imaginary part of σ and yields a logarithmic singularity on the sphere |x| = t when σ ∈ C with Re σ = d+1 2 .

The following proposition concerning the estimate of b τ will be proved by straightforward computations.

Proposition 3.6. Let N > d be an even integer. Then

|b τ (x)| (1 + |t|) -N ϕ 0 (x)
(3.17)

for every x ∈ X and for every τ = s -it with s ∈ (0, 1] and t ∈ R * .

Proof. Let is study

B τ (λ) = 2τ π +∞ 0 dv {1 -χ T (v)} 1 τ 2 + v 2 cos(v |λ| 2 + |ρ| 2 ),
which vanishes unless v > 2T . By performing N integrations by parts based on

cos(v |λ| 2 + |ρ| 2 ) = - 1 |λ| 2 + |ρ| 2 ∂ 2 ∂v 2 cos(v |λ| 2 + |ρ| 2 ), we obtain B τ (λ) = 2τ π (-1) -N 2 (|λ| 2 + |ρ| 2 ) -N 2 +∞ 0 dv cos(v |λ| 2 + |ρ| 2 ) ∂ ∂v ) N ( 1-χ T (v) τ 2 +v 2 . Since v > 2T , we have |τ 2 + v 2 | v 2 uniformly in τ = s -it. Hence B τ (λ) |τ | (|λ| 2 + |ρ| 2 ) -N 2 +∞ 2T dv v -2-N |T | -N (|λ| 2 + |ρ| 2 ) -N 2 .
By the inverse formula of the spherical Fourier transform, we deduce

|b τ (x)| = a dλ |c(λ)| -2 ϕ λ (x) B τ (λ) |T | -N ϕ 0 (x) a dλ |c(λ)| -2 (|λ| 2 + |ρ| 2 ) -N 2
where the last integral converges provided that N > d.

According to the asymptotic expansion (3.15) of a τ and to the estimate (3.17) of b τ , we establish the pointwise estimates of ω σ,0 t in the case where |x| |t| is bounded from below.

Theorem 3.7. Let σ ∈ C with Re σ = d+1

2 . The following estimates hold for all t ∈ R * and x ∈ X.

(i

) If 0 < |t| < 1, then | ω σ,0 t (x)| |t| -d-1 2 (1 + |x + |) max{d,D}- 2 e -ρ,x + . (ii) If |t| ≥ 1 and |x| |t| > C Σ , then | ω σ,0 t (x)| |t| -N1 (1 + |x + |) N2 e -ρ,x + , for every N 1 ∈ N and N 2 ≥ N 1 + 2(d + 1) + max{d,D}- 2 .
Proof. Recall that we are looking for a pointwise estimate of the kernel

ω σ,0 t (x) = C σ,d 1 0 ds s σ-1 p τ (x),
where τ = s-it with s ∈ (0, 1) and t ∈ R * . According to the Cartan decomposition, for every x ∈ X, there exist k 1 , k 2 ∈ K and x + ∈ a + such that x = k 1 (exp x + )k 2 . Then

p τ (x) = a τ (exp x + ) + b τ (exp x + )
by the K-bi-invariance. According to the expansion (3.15), we split up

ω σ,0 t (x) = I 1 (t, x + ) + I 2 (t, x + ) + I 3 (t, x + ) = 1 π J(x + ) -1/2 [d/2] k=0 4 -k U k (x + ) Γ d+1 2 -k I 1,k (t, x + ) + C σ,d 1 0 ds s σ-1 E(τ, x + ) + C σ,d 1 0 ds s σ-1 b τ (exp x + )
where

I 1,k (t, x + ) = C σ,d 1 0 ds s σ-1 τ (|x + | 2 + τ 2 ) k-d+1 2 satisfies |I 1,k (t, x + )| 1 + |t| -d-1
2 according to next lemma. Hence

I 1 (t, x + ) ( |t| + |t| -d-1 2 ) J(x + ) -1/2 ∀ t ∈ R * . (3.18)
The last two terms I 2 (t, x + ) and I 3 (t, x + ) are easily handled: on the one hand, we have

|I 2 (t, x + )| 1 0 ds s Re σ-1 |E(τ, x + )| (1 + |t|) 3( d 2 +1) log (2 + |t|) e -ρ,x + , (3.19)
according to (3.16); on the other hand, (3.17) yields

|I 3 (t, x + )| 1 0 ds s Re σ-1 |b τ (exp x + )| (1 + |t|) -N ϕ 0 (exp x + ) (3.20)
for all σ ∈ C with Re σ = d+1 2 . By summing up the estimates (3.18), (3.19) and (3.20), we deduce, on the one hand,

| ω σ,0 t (x)| |t| -d-1 2 (1 + |x + |) max{d,D}- 2 e -ρ,x + if |t| < 1,
and on the other hand,

| ω σ,0 t (x)| |t| 3( d 2 +1) log (2 + |t|) (1 + |x + |) max{d,D}- 2 e -ρ,x + if |t| ≥ 1. Since |x| |t| is bounded from below, we obtain finally | ω σ,0 t (x)| |t| -N1 (1 + |x + |) N2 e -ρ,x + ∀ |t| ≥ 1 for every N 1 ∈ N and N 2 ≥ N 1 + 2(d + 1) + max{d,D}- 2 .
Remark 3.8. Notice that the above method works only in small time, or in large time under the assumption that |x| |t| is bounded from below. The large polynomial growth in |x + | appearing in the estimate is not crucial for further computations because of the exponential decay e -ρ,x + . Lemma 3.9. For every integer 0 ≤ k < d+1 2 , the integral

I 1,k (t, x + ) = C σ,d 1 0 ds s σ-1 τ (|x + | 2 + τ 2 ) k-d+1 2 satisfies |I 1,k (t, x + )| 1 + |t| k-d-1 2 ∀ t ∈ R * , ∀ x ∈ a + uniformly in σ ∈ C with Re σ = d+1 2 . Proof. Since τ = s -it, we write I 1,k (t, x + ) = P 1 + P 2 with P 1 = C σ,d 1 0 ds s σ (s 2 + |x + | 2 -t 2 -2sti) k-d+1 2 and P 2 = C σ,d (-it) 1 0 ds s σ-1 (s 2 + |x + | 2 -t 2 -2sti) k-d+1 2 .
Notice that

s 2 + |x + | 2 -t 2 -2sti = s 4 + 2s 2 (|x + | 2 + t 2 ) + (|x + | 2 -t 2 ) 2 , then s 2 + |x + | 2 -t 2 -2sti ≥        s 2 (3.21) s|t| (3.22) |x + | 2 -t 2 (3.23)
P 1 is easily estimated. By using (3.22), we obtain first

|P 1 | |t| k-d+1 2 1 0 ds s k ≤ |t| k-d+1 2 ∀ t ∈ R * .
By using in addition (3.21), we obtain next, for |t| < 1,

|P 1 | |t| k-d+1 2 |t| 0 ds s k + 1 |t| ds s 2k-d+1 2 1 + |t| 2k-d-1 2
We deduce that

|P 1 |    1 + |t| 2k-d-1 2 if |t| < 1, |t| k-d+1 2 if |t| ≥ 1.
(3.24)

Let us turn to P 2 . Consider first the easy case where 1 ≤ k < d+1 2 . By using (3.22) again, we get

|P 2 | |t| • |t| k-d+1 2 1 0 ds s Re σ-1+k-d+1 2 |t| k-d-1 2 (3.25)
for all σ ∈ C with Re σ = d+1 2 . In order to estimate P 2 in the remaining case where k = 0, we write

P 2 = C σ,d (-it) 1 0 ds s i Im σ-1 s s 2 +|x + | 2 -t 2 -2sti d+1 2 .
By performing an integration by parts, P 2 becomes the sum of P - 2 and P + 2 where

P - 2 = C σ,d Im σ (-it) s Im σ s s 2 +|x + | 2 -t 2 -2sti d+1 2
1 0 and

P + 2 = C σ,d Im σ (it) 1 0 ds s Im σ ∂ ∂s s s 2 +|x + | 2 -t 2 -2sti d+1 2
.

By using (3.2) together with (3.21) in small time and (3.22) in large time, we obtain

|P - 2 |    1 if |t| < 1, |t| -d-1 2 if |t| ≥ 1. (3.26) Since ∂ ∂s s s 2 +|x + | 2 -t 2 -2sti d+1 2 = d+1 2 s s 2 +|x + | 2 -t 2 -2sti d-1 2 O(|t| -d-1 2 ) |x + | 2 -t 2 -s 2 (s 2 +|x + | 2 -t 2 -2sti) 2 ,
we have

|P + 2 | |t| -d-1 2 |t| 1 0 ds |x + | 2 -t 2 -s 2 s 2 +|x + | 2 -t 2 -2sti 2 Q . (3.27)
It remains for us to estimate Q, which is bounded by the sum of

Q 1 = |t| 1 0 ds s 2 s 2 +|x + | 2 -t 2 -2sti 2 and Q 2 = |t| 1 0 ds |x + | 2 -t 2 s 2 +|x + | 2 -t 2 -2sti 2 .
Q 1 is estimated as P 1 . According to (3.22) and (3.21), we have

Q 1    |t| 1 0 ds |t| -2 = |t| -1 ≤ 1 if |t| ≥ 1, |t| |t| 0 ds |t| -2 + |t| 1 |t| ds s -2 ≤ 2 if |t| < 1.
Let us finally estimate Q 2 . On the one hand, if |x + | 2 -t 2 ≥ |t|, by using (3.23), we get

|Q 2 | |t| 1 0 ds |x + | 2 -t 2 -1 1. On the other hand, if |x + | 2 -t 2 ≤ |t|, we have Q 2 = O(1) since |t| 0≤s≤ ||x + | 2 -t 2 | |t| ds |x + | 2 -t 2 s 2 +|x + | 2 -t 2 -2sti 2 |t| 0≤s≤ ||x + | 2 -t 2 | |t| ds |x + | 2 -t 2 -1 ≤ 1
according to (3.23), and

|t| ||x + | 2 -t 2 | |t| ≤s≤1 ds |x + | 2 -t 2 s 2 +|x + | 2 -t 2 -2sti 2 ||x + | 2 -t 2 | |t| ||x + | 2 -t 2 | |t|
|P 2 | |t| k-d-1 2 ∀ t ∈ R * .
Together with (3.24), this concludes the proof.

3.3.

Estimates of ω σ,∞ t . We establish in this last subsection the pointwise esti-

mates of ω σ,∞ t . Recall that ω σ,∞ t (x) = 1 Γ(σ) +∞ 1 ds s σ-1 p s-it (x) ∀ x ∈ X, ∀ t ∈ R * .
According to the integral expression (2.2) of the spherical function, we may write

ω σ,∞ t (x) = C 0 Γ(σ) K dk e ρ,A(kx) +∞ 1 ds s σ-1 I(s, t, x),
where, let us recall,

I(s, t, x) = a dλ |c(λ)| -2 e -s √ |λ| 2 +|ρ| 2 e itψt(λ) .
We have considered this oscillatory integral in the case where s ∈ (0, 1).For s ≥ 1, the factor e -s √ |λ| 2 +|ρ| 2 plays an important role. On the one hand, for λ close to the critical point of ψ t (λ), this Gaussian produces an exponential decay in s, which ensures the convergence of the integral over s ∈ (1, +∞). For λ away from the critical point, it produces an exponential decay in |λ|, which ensures the convergence of the integral over λ ∈ a. Let us elaborate.

Theorem 3.10. The following estimate holds, uniformly in the strip 0 ≤ Re σ ≤ d+1 2 , for all t ∈ R * and x ∈ X:

|ω σ,∞ t (x)| ϕ 0 (x).
(3.28) . We study the oscillatory integral I along the lines of Sect. 3.1. Let split up again

Moreover, if |t| ≥ 1, |ω σ,∞ t (x)| |t| -D 2 (1 + |x|) D 2 ϕ 0 (x). ( 3
I(s, t, x) = I -(s, t, x) + I + (s, t, x) = a dλ χ ρ 0 (λ) • • • + a dλ χ ρ ∞ (λ) • • •
according to cut-off functions χ ρ 0 and χ ρ ∞ = 1 -χ ρ 0 , which have been defined in Sect. 3.1. Recall that χ ρ 0 (λ) = 1 when |λ| ≤ |ρ| and vanishes if |λ| ≥ 2|ρ|.

On the one hand, I -is estimated by studying the oscillatory integral

I -(s, t, x) = a dλ χ ρ 0 (λ) |c(λ)| -2 e -s √ |λ| 2 +|ρ| 2 a0(s,λ) e itψt(λ)
where the amplitude a 0 is compactly supported for |λ| ≤ 2|ρ|, and in this range, the phase ψ t , defined by (3.3), has a single critical point, which is nondegenerate and small if |x| |t| ≤ 1 2 . According to Lemma A.1, we obtain

|I -(s, t, x)| |t| -D 2 (1 + |x|) D- 2 e -|ρ| 2 s . (3.31)
On the other hand,

I + (s, t, x) = a dλ χ ρ ∞ (λ) |c(λ)| -2 e -s √ |λ| 2 +|ρ| 2 e itψt(λ)
is easily estimated with no barycentric decomposition. Let

ψ ∞ (λ) = |λ| 2 √ |λ| 2 +|ρ| 2 + A(kx) t , λ ∀ λ ∈ supp χ ρ ∞ .
Then ψ ∞ is a symbol of order 1, and satisfies

| ψ ∞ (λ)| = |λ| 2 |λ| 2 + |ρ| 2 ≥ |λ| √ 2 - λ 0 , λ |λ 0 | 2 + |ρ| 2 ≤ |λ| √ 3 ≥ ( 1 √ 2 -1 √ 3 )|ρ| > 0
on supp χ ρ ∞ according to (3.6) and (3.7). By performing N integrations by parts based on ) , we write

e itψt(λ) = 1 it ψ ∞ (λ) -1 j=1 λ j ∂ ∂λj e itψt(λ
I + ∞ (s, t, x) = (it) -N a dλ e itψt(λ) -j=1 ∂ ∂λj • λj ψ∞(λ) N χ ρ ∞ (λ) |c(λ)| -2 e -s √ |λ| 2 +|ρ| 2 .
If some derivatives hit χ ρ ∞ (λ), the range of the above integral is reduced to a spherical shell where |λ| |ρ|, and

I + (s, t, x) |t| -N e -s|ρ| .
Assume next that no derivative is applied to χ ρ ∞ and • N 1 derivatives are applied to the factors λ j / ψ ∞ (λ), which are inhomogeneous symbols of order 0, producing contributions which are O(|λ| -N1 ),

• N 2 derivatives are applied to the factor |c(λ)| -2 which is not a symbol in general, producing a contribution which is O(|λ| d-),

• N 3 derivatives are applied to the factor e -s

√ |λ| 2 +|ρ| 2 , producing a contribution which is O(s N3 e -s √ |λ| 2 +|ρ| 2 ),
with N 1 + N 2 + N 3 = N . Then we get the upper bound

|t| -N s N3 |λ|>|ρ| dλ |λ| d--N1 e -s √ |λ| 2 +|ρ| 2 ,
which yields

|I + (s, t, x)| |t| -N s N e -s 2 |ρ| |λ|>|ρ| dλ |λ| d-e -s 2 |λ| <+∞ .
Together with (3.31), we obtain

|I(s, t, x)| |t| -D 2 (1 + |x|) D- 2 s N e -s 2 |ρ| .
for all s ≥ 1 and for N ≥ D 2 . We deduce

|ω σ,∞ t (x)| |t| -D 2 (1 + |x|) D- 2 ϕ 0 (x) +∞ 1 ds s Re σ-1+N e -|ρ| 2 s
<+∞ for all x ∈ X and |t| ≥ 1.

Dispersive estimates

In this section, we prove our second main result about the L q → L q estimates for the operator W σ t = (-∆) -σ 2 e it √ -∆ . We introduce the following criterion based on the Kunze-Stein phenomenon, which is crucial for the proof of dispersive estimates.

Lemma 4.1. Let κ be a reasonable K-bi-invariant function on G. Then

• * κ L q (X)→L q (X) ≤ G dx ϕ 0 (x) |κ(x)| q 2 2 q
for every q ∈ [2, +∞). In the limit case q = ∞,

• * κ L 1 (X)→L ∞ (X) = sup x∈G |κ(x)|.
Remark 4.2. This lemma has been proved in several contexts. For q = 2, it is the so-called Herz's criterion, see for instance [START_REF] Cowling | Herz's "principe de majoration" and the Kunze-Stein phenomenon[END_REF]. For q > 2, the proof carried out on Damek-Ricci spaces [APV11, Theorem 4.2] is adapted straightforwardly in the higher rank case.

(-∆) -σ 2 e it √ -∆ L q (X)→L q (X) |t| -(d-1)( 1 2 -1 q )
for all 2 < q < +∞ and σ ≥ (d + 1)( 1 2 -1 q ). Proof. We divide the proof into two parts, corresponding to the kernel decomposition ω σ t = ω σ,0 t + ω σ,∞ t . According to Lemma 4.1 and to the pointwise estimate Theorem 3.10, we obtain on one hand

• * ω σ,∞ t L q (X)→L q (X) ≤ G dx ϕ 0 (x) |ω σ,∞ t (x)| q 2 2 q a + dx + |ϕ 0 (x + )| q 2 +1 δ(x + ) 2 q a + dx + (1 + |x + |) D- 2 ( q 2 +1) e -( q 2 -1) ρ,x + 2 q < +∞
for all q > 2. On the other hand, we use an analytic interpolation between L 2 → L 2 and L 1 → L ∞ estimates for the family of operators W σ,0 t defined by (3.1) in the vertical strip 0 ≤ Re σ ≤ d+1 2 . When Re σ = 0, the spectral theorem yields

W σ,0 t L 2 (X)→L 2 (X) = e it √ -∆ L 2 (X)→L 2 (X) = 1
for all t ∈ R * . According to Theorem 3.7, when Re σ = d+1 2 ,

W σ,0 t L 1 (X)→L ∞ (X) ω σ,0 t L ∞ (X) |t| -d-1 2 .
By Stein's interpolation theorem applied to the analytic family of operators W σ,0 t , we conclude for σ

= (d + 1)( 1 2 -1 q ) that W σ t L q (X)→L q (X) |t| -(d-1)( 1 2 -1 q ) ,
for all 0 < |t| < 1 and 2 < q < +∞.

Theorem 4.4 (Large time dispersive estimate). Assume that |t| ≥ 1, 2 < q < +∞ and σ ≥ (d + 1)( 1 2 -1 q ). Then

(-∆) -σ 2 e it √ -∆ L q (X)→L q (X) |t| -D 2 .
Proof. We divide the proof into three parts, corresponding to the kernel decomposition

ω σ t = 1 B(0,CΣ|t|) ω σ,0 t + 1 X\B(0,CΣ|t|) ω σ,0 t + ω σ,∞ t
where the constant C Σ has been specified in the proof of Theorem 3.3. The first and the last terms are estimated by straightforward computations. By combining Lemma 4.1 with the pointwise kernel estimates in Theorem 3.3 and Theorem 3.10, we obtain

• * {1 B(0,CΣ|t|) ω σ,0 t } L q (X)→L q (X) G dx ϕ 0 (x) |1 B(0,CΣ|t|) (x) ω σ,0 t (x)| q 2 2 q |t| -D 2 |x + |<CΣ|t| dx + (1 + |x + |) D- 2 (q+1) e -( q 2 -1) ρ,x + 2 q <+∞ and • * ω σ,∞ t L q (X)→L q (X) G dx ϕ 0 (x) |ω σ,∞ t (x)| q 2 2 q |t| -D 2 a + dx + (1 + |x + |) D- 2 +(D-2 ) q 2 e -( q 2 -1) ρ,x + 2 q <+∞ .
Here q ≤ 2 < ∞ and the above estimates are uniform in the strip 0 ≤ Re σ ≤ d+1 2 . As far as the middle term is concerned, we use again the analytic interpolation for the family of operators associated with the convolution kernel 1 X\B(0,CΣ|t|) ω σ,0 t . On the one hand, if Re σ = 0, then

• * 1 X\B(0,CΣ|t|) ω σ,0 t L 2 (X)→L 2 (X) ≤ • * ω σ,0 t L 2 (X)→L 2 (X) + • * 1 B(0,CΣ|t|) ω σ,0 t L 2 (X)→L 2 (X)
1.

On the other hand, if Re σ = d+1 2 , we deduce from Theorem 3.7 that

• * 1 X\B(0,CΣ|t|) ω σ,0 t L 1 (X)→L ∞ (X) = sup x∈X |1 X\B(0,CΣ|t|) (x) ω σ,0 t (x)| |t| -N
for any N ∈ N. By using Stein's interpolation theorem between above L 2 → L 2 and L 1 → L ∞ estimates, we obtain

• * 1 X\B(0,CΣ|t|) ω σ,0 t L q (X)→L q (X) |t| -N ,
for all |t| ≥ 1, 2 < q < +∞ and for any N ∈ N. This concludes the proof.

Remark 4.5. The standard T T * method used to prove the Strichartz inequality breaks down in the critical case. In order to take care of these endpoints, we need the dyadic decomposition method carried out in [START_REF] Keel | Endpoint Strichartz estimates[END_REF] and the following stronger dispersive property, which is obtained by interpolation arguments.

Corollary 4.6. Let d ≥ 3, 2 < q, q < +∞ and σ ≥ (d+1) max( 1 2 -1 q , 1 2 -1 q ). Then there exists a constant C > 0 such that the following dispersive estimates hold:

(-∆) -σ 2 e it √ -∆ L q (X)→L q (X) ≤ C    |t| -(d-1) max( 1 2 -1 q , 1 2 -1 q ) if 0 < |t| < 1, |t| -D 2 if |t| ≥ 1.

Strichartz inequality and applications

In this section, we use the dispersive properties proved in the previous section to establish the Strichartz inequality. This inequality serves as a tool for finding minimal regularity conditions on the initial data ensuring well-posedness of related semilinear wave equations. Such results were previously known to hold for real hyperbolic spaces [START_REF] Anker | Wave and Klein-Gordon equations on hyperbolic spaces[END_REF] (actually for all noncompact symmetric spaces of rank one) and for noncompact symmetric spaces G/K with G complex [START_REF] Zhang | Wave equation on certain Riemannian noncompact symmetric spaces[END_REF]. For simplicity, we may assume that ≥ 2 throughout this section, thus d ≥ 4.

Let σ ∈ R and 1 < q < ∞. Recall that the Sobolev space H σ,q (X) is the image of L q (X) under the operator (-∆) -σ 2 , equipped with the norm f H σ,q (X) = (-∆)

σ 2 f L q (X) .
If σ = N is a nonnegative integer, then H σ,q (X) coincides with the classical Sobolev space

W N,q (X) = {f ∈ L q (X) | ∇ j f ∈ L q (X) ∀ 1 ≤ j ≤ N },
defined by means of covariant derivatives. We refer to [START_REF] Triebel | Theory of function spaces II[END_REF] for more details about function spaces on Riemannian manifolds. Let us state the Strichartz inequality and some applications. The proofs are adapted straightforwardly from [APV12; AnPi14] and are therefore omitted.

Strichartz inequality. We study the linear inhomogeneous wave equation on

X    ∂ 2 t u(t, x) -∆ u(t, x) = F (t, x), u(0, x) = f (x), ∂ t | t=0 u(t, x) = g(x)
(5.1) whose solution is given by Duhamel's formula:

u(t, x) = (cos t √ -∆)f (x) + sin t √ -∆ √ -∆ g(x) + t 0 ds sin(t-s) √ -∆ √ -∆ F (s, x).
Recall that a couple (p, q) is called admissible if ( 1 p , 1 q ) belongs to the triangle

1 p , 1 q ∈ 0, 1 2 × 0, 1 2 1 p ≥ d-1 2 1 2 -1 q 0, 1 2 . 1 p 1 q 1 2 1 2 -1 d-1 0 1 2 1 p = d-1 2 1 2 -1 q Figure 3. Admissibility in dimension d ≥ 4.
Theorem 5.1. Let (p, q) and (p, q) be two admissible couples, and let

σ ≥ d+1 2 1 2 -1 q and σ ≥ d+1 2 1 2 -1 q .
Then all solutions u to the Cauchy problem (5.1) satisfy the following Strichartz inequality:

∇ R×X u L p (I;H -σ,q (X)) f H 1 (X) + g L 2 (X) + F L p (I;H σ, q (X)) .
(5.2) Remark 5.2. As have already been observed on hyperbolic spaces, the admissible set for X is much larger than the admissible set for R d , which corresponds only to the lower edge of the triangle. This is due to large scale dispersive effects in negative curvature.

The admissible range in (5.2) can be widened by using the Sobolev embedding theorem.

Corollary 5.3. Assume that (p, q) and (p, q) are two couples corresponding to the square

0, 1 2 × 0, 1 2 0, 1 2 , Let σ, σ ∈ R such that σ ≥ σ(p, q), where σ(p, q) = d+1 2 1 2 -1 q + max 0, d-1 2 1 2 -1 q -1 p ,
and similarly σ ≥ σ(p, q). Then the Strichartz inequality (5.2) holds for all solutions to the Cauchy problem (5.1).

1 p 1 q 1 2 0 1 2 1 2 -1 d-1

5.2.

Global well-posedness in L p (R, L q (X)). By combining the classical fixed point scheme with the previous Strichartz inequalities, one obtains the global wellposedness for the semilinear equation

   ∂ 2 t u(t, x) -∆u(t, x) = F (u(t, x)), u(0, x) = f (x), ∂ t | t=0 u(t, x) = g(x).
(5.3) on X with small initial data f and g and power-like nonlinearities F satisfying

|F (u)| |u| γ and |F (u) -F (v)| (|u| γ-1 + |v| γ-1 )|u -v|
where γ > 1. Let γ c = 1 + 4 d-1 be the conformal power. The global existence of solutions to the semilinear wave equation (5.3) on R d is related to the Strauss conjecture: the critical power γ 0 , i.e., the infimum of all γ ∈ (1, γ c ] such that (5.3) has global solutions for small initial data, is the positive root of the quadratic equation

(d -1)γ 2 0 -(d + 1)γ 0 -2 = 0 (d ≥ 2).
In other words,

γ 0 = 1 2 + 1 d-1 + ( 1 2 + 1 d-1 ) 2 + 2 d-1 > 1.
We refer to [Joh79; Kat80; Stra81; GLS97; Tat01] and the references therein for more details about the Strauss conjecture in the Euclidean setting. In negative curvature, the global existence for small initial data has been proved, for any γ ∈ (1, γ c ], on real hyperbolic spaces of dimension d = 3 [MeTa11; MeTa12] and then of any dimension d ≥ 2 [START_REF] Anker | Wave and Klein-Gordon equations on hyperbolic spaces[END_REF]. In other words, there is no phenomenon analogous to Strauss conjecture on such spaces. Similar results were extended later to Damek-Ricci spaces, which contain all noncompact symmetric spaces of rank one [START_REF] Anker | The wave equation on Damek-Ricci spaces[END_REF], and have been established recently on simply connected complete Riemannian manifolds with strictly negative sectional curvature [START_REF] Sire | The Strauss conjecture on negatively curved backgrounds[END_REF], and on non-trapping asymptotically hyperbolic manifolds [START_REF] Sire | Strichartz estimates and Strauss conjecture on non-trapping asymptotically hyperbolic manifolds[END_REF]. Next theorem shows that the same phenomenon holds on general noncompact symmetric spaces. More precisely, we prove that the semilinear wave equation (5.3) on X is globally wellposed.

To state the following theorem, we need to introduce some notation. Consider the following powers

γ 1 =1 + 3 d γ 2 = 1 + 2 d-1 2 + 2 d-1 γ 3 =    1 + 4 d-2 if d ≤ 5, d-1 2 + 3 d+1 -( d-3 2 + 3 d+1 ) 2 -4 d-1 d+1 if d ≥ 6,
and the following curves

σ 1 (γ) = d + 1 4 - (d + 1)(d + 5) 8d 1 γ -d+1 2d , σ 2 (γ) = d + 1 4 - 1 γ -1 , σ 3 (γ) = d 2 - 2 γ -1 .
Theorem 5.4. The semilinear Cauchy problem (5.3) is globally well-posed for small initial data in H σ,2 (X)

× H σ-1,2 (X) provided that                σ > 0 if 1 < γ ≤ γ 1 , σ ≥ σ 1 (γ) if γ 1 < γ ≤ γ 2 , σ ≥ σ 2 (γ) if γ 2 ≤ γ ≤ γ c , σ ≥ σ 3 (γ) if γ c ≤ γ ≤ γ 3 .
More precisely, in each case, there exists 2 ≤ p, q < ∞ such that for any small initial data (f, g) in H σ,2 (X) × H σ-1,2 (X), the Cauchy problem (5.3) has a unique solution in the Banach space

C(R; H σ,2 (X)) ∩ C 1 (R; H σ-1,2 (X)) ∩ L p (R; L q (X)).

Further results for Klein-Gordon equations

The kernel estimates and dispersive estimates proved above for the wave equation still hold if we replace the operator (-∆) -σ 2 e it √ -∆ by D -σ e itD , where

D = -∆ -|ρ| 2 + κ 2 with κ > 0.
Then, for every admissible couple (p, q), the operator

Tf (t, x) = D -σ 2 x e itDx f (x)
is again bounded from L 2 (X) to L p (R; L q (X)), and its adjoint

T * F (x) = +∞ -∞ ds D -σ 2 x e -isDx F (s, x)
from L p (R; L q (X)) to L 2 (X). While L 2 Sobolev spaces may be defined in terms of D, we need the operator

D = -∆ -|ρ| 2 + κ 2 with κ ≥ |ρ|,
in order to define L q Sobolev spaces when q gets large. As D -σ 2 •D σ 2 is a topological automorphism of L 2 (X), the operator

Tf (t, x) = D -σ 2 x e itDx f (x)
is also bounded from L 2 (X) to L p (R; L q (X)), and its adjoint

T * F (x) = +∞ -∞ ds D -σ 2 x e -isDx F (s, x) from L p (R; L q (X)) to L 2 (X), hence T T * F (t, x) = +∞ -∞ ds D -σ
x e i(t-s)Dx F (s, x) from L p (R; L q (X)) to L p (R; L q (X)) for all admissible couples (p, q) and ( p, q). We deduce that Theorem 5.1 and Corollary 5.3 still hold for solutions to the inhomogeneous Klein-Gordon equation

   ∂ 2 t u(t, x) + D 2 u(t, x) = F (t, x), u(0, x) = f (x), ∂ t | t=0 u(t, x) = g(x),
and that the corresponding semilinear equation is globally well-posed with low regularity data, see Theorem 5.4.

Appendix A. Oscillatory integral on a

In this appendix, we prove the following lemma which is used in the proofs of Theorem 3.3 and Theorem 3.10. Recall that A = A(kx) is the a-component of kx ∈ X in the Iwasawa decomposition, and that C Σ ∈ (0, 1 2 ] is a fixed constant. Lemma A.1. Let s ∈ R + and |t| ≥ 1. Consider the oscillatory integral

I -(s, t, x) = a dλ a 0 (s, λ) e itψt(λ)
where the phase is given by

ψ t (λ) = |λ| 2 + |ρ| 2 + A t , λ and the amplitude a 0 (s, λ) = χ ρ 0 (λ) |c(λ)| -2 e -s √ |λ| 2 +|ρ| 2 vanishes unless |λ| ≤ 2|ρ|. Then, for all x ∈ X such that |x| |t| ≤ C Σ , |I -(s, t, x)| |t| -D 2 (1 + |x|) D- 2 e -|ρ| 2 s .
Remark A.2. The proof of this lemma is similar to the proof of [Zha20, Theorem 3.1.(ii)], except that our amplitude involves the general Plancherel density and in addition a Gaussian factor depending on s.

Proof. By symmetry, we may assume that t ≥ 1. Recall that the critical point λ 0 of the phase ψ is given by

(|λ 0 | 2 + |ρ| 2 ) -1 2 λ 0 = -A t (A.1)
and satisfies

|λ 0 | = |ρ| |A| |t| (1 -|A| 2 t 2 ) -1 2 ≤ |ρ| |x| t (1 -|x| 2 t 2 ) -1 2 < |ρ| √ 3 , (A.2)
as |A| ≤ |x| and |x| t ≤ 1 2 . Denote by

B(λ 0 , η) = {λ ∈ a | |λ -λ 0 | ≤ η}
the ball in a centered at λ 0 , where the radius η will be specified later. Notice that |λ| < |ρ| + η for all λ ∈ B(λ 0 , η). Let P λ be the projection onto the vector spanned by λ |λ| . Then |λ| 2 P λ = λ ⊗ λ and the Hessian matrix of ψ t is given by

Hess ψ t (λ) = (|λ| 2 + |ρ| 2 ) -1 2 I -(|λ| 2 + |ρ| 2 ) -3 2 λ ⊗ λ = (|λ| 2 + |ρ| 2 ) -3 2 {|ρ| 2 P λ + (|λ| 2 + |ρ| 2 )(I -P λ )} = (|λ| 2 + |ρ| 2 ) -3 2      |ρ| 2 0 0 (|λ| 2 + |ρ| 2 )I -1     
which is a positive definite symmetric matrix. Hence λ 0 is a nondegenerate critical point. Since ∇ a ψ t (λ 0 ) = 0, we write

ψ t (λ) -ψ t (λ 0 ) = (λ -λ 0 ) T 1 0 ds (1 -s) Hess ψ t λ 0 + s(λ -λ 0 ) M(λ) (λ -λ 0 ),
where M(λ) belongs, for every λ ∈ B(λ 0 , η), to a compact subset of the set of positive definite symmetric matrices. We introduce a new variable µ = M(λ)

1 2 (λλ 0 ), then |µ| 2 = ψ t (λ) -ψ t (λ 0 ) and µ = 0 if and only if λ = λ 0 . There exist 0 < η 1 ≤ η 2 such that µ ∈ B(0, η 1 ) implies λ ∈ B(λ 0 , η), and λ ∈ B(λ 0 , η) implies µ ∈ B(0, η 2 ). Notice that for every k ∈ N, there exists C k > 0 such that

|∇ k a M(λ) 1 2 | ≤ C k ∀ λ ∈ B(λ 0 , η). (A.3)
Denote by j(λ) the Jacobian matrix such that dµ = det[ j(λ) ]dλ, then we can choose η > 0 small enough such that

det[ j(λ) ] > 1 2 det[ M(λ) 1 2 ] ∀ λ ∈ B(λ 0 , η). (A.4)
Now, we split up

I -(s, t, x) = I - 0 (s, t, x) + I - ∞ (s, t, x) = a dλ χ η 0 (λ) a 0 (s, λ) e itψt(λ) + a dλ χ η ∞ (λ) a 0 (s, λ) e itψt(λ)
where χ η 0 : a → [0, 1] is a smooth cut-off function which vanishes unless |λ-

λ 0 | ≤ η 2 , χ η 0 (λ) = 1 if |λ -λ 0 | ≤ η 4 , and χ η ∞ = 1 -χ η 0 .
Estimate of I - 0 . We estimate I - 0 by using the stationary phase analysis described in [START_REF] Stein | Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals[END_REF]Chap.VIII 2.3]. Notice that supp χ η 0 ⊂ B(λ 0 , η). By substituting ψ t (λ) = |µ| 2 + ψ t (λ 0 ), we get

I - 0 (s, t, x) = e itψt(λ0) a dµ a(s, λ(µ)) e it|µ| 2
where the amplitude

a(s, λ(µ)) = χ η 0 (λ(µ)) χ ρ 0 (λ(µ)) |c(λ(µ))| -2 e -s √ |λ(µ)| 2 +|ρ| 2 det[ j(λ(µ)) ] -1 (A.5)
is smooth and compactly supported in B(0, η 2 ). We deduce, from (A.3) and (A.4) that a(s, λ(µ)) is bounded, together with all its derivatives. Let χ η2 ∈ C ∞ c (a) be a bump function such that χ η2 = 1 on B(0, η 2 ). Then

I - 0 (s, t, x) = e itψt(
c k µ k + R 2M (µ)
at the origin satisfy

|c k | |c(λ 0 )| -2 (1 + s) k e -s √ |λ0| 2 +|ρ| 2 ( |x| t ) D-(1 + s) k e -|ρ|s , (A.6)
according to (A.2), and the remainder satisfies

|∇ n a R 2M (µ)| |µ| 2M +1-n (1 + s) 2M +1+n e -|ρ|s ∀ 0 ≤ n ≤ 2M + 1. (A.7)
By substituting this expansion in the above integral, I - 0 (s, t, x) is the sum of following three terms:

I 1 = |k|≤2M c k a dµ µ k e it|µ| 2 e -|µ| 2 I 2 = a dµ χ η2 (µ) R 2M (µ) e it|µ| 2 e -|µ| 2 ,
and

I 3 = |k|≤2M c k a dµ {χ η2 (µ) -1} µ k e it|µ| 2 e -|µ| 2 .
To estimate I 1 , we write

I 1 = |k|≤2M c k j=1 +∞ -∞ dµ j e itµ 2 j e -µ 2 j µ kj j
where

+∞ -∞ dµ j e -(1-it)µ 2 j µ kj j = 0 if k j is odd, while +∞ -∞ dµ j e -(1-it)µ 2 j µ kj j = 2 (1 -it) -k j +1 2 +∞ 0 dz j e -z 2 j z kj j
by a change of contour if k j is even. We deduce from (A.6)

|I 1 | t -2 ( |x| t ) D-(1 + s) 2M e -|ρ|s t -D 2 |x| -D- 2 e -|ρ|
|I 3 | t -N (1 + s) 2M e -|ρ| 2 s .
By summing up the estimates of I 1 , I 2 and I 3 , we deduce that

|I - 0 (s, t, x)| t -D 2 (1 + |x|) D- 2 (1 + s) 2d+3 e -|ρ|s t -D 2 (1 + |x|) D- 2 e -|ρ| 2 s . (A.9)
Estimate of I - ∞ . Since the phase ψ t has a unique critical point λ 0 which is defined by (A.1) and satisfies (A.2), then for all λ ∈ supp χ η ∞ , we have ∇ a ψ t (λ) = 0. In order to get large time decay, we estimate

I - ∞ (s, t, x) = a dλ χ η ∞ (λ) a 0 (s, λ) e itψt(λ)
by using several integrations by parts based on

e itψt(λ) = 1 it ψ 0 (λ) -1 j=1 λj √ |λ| 2 +|ρ| 2 + Aj t ∂ ∂λj e itψt(λ) ,
where

ψ 0 (λ) = λ √ |λ| 2 +|ρ| 2 + A t 2
is a smooth function, which is bounded from below on the compact set (supp χ η ∞ ) ∩ (supp χ ρ 0 ), uniformly in A t . After performing N such integrations by parts,

I - ∞ (s, t, x) becomes const. (it) -N a dλ e itψt(λ) - j=1 ∂ ∂λ j • ψ 0 (λ) -1 λj √ |λ| 2 +|ρ| 2 + Aj t N χ η ∞ (λ) a 0 (s, λ)
where the last integral is bounded from above by (1 + s) N e -|ρ|s e -|ρ| 2 s . Hence

|I - ∞ (s, t, x)| t -N e -|ρ| 2 s (A.10)
for every N ∈ N. By combining (A.9) and (A.10), we conclude that

|I -(s, t, x)| t -D 2 (1 + |x|) D- 2 e -|ρ| 2 s .

Appendix B. Hadamard parametrix on symmetric spaces

Let Φ v be the K-bi-invariant convolution kernel of the operator cos(v √ -∆) whose spherical Fourier transform is given by Φ

v (λ) = cos(v |λ| 2 + |ρ| 2 ). Then Φ v (x) solves the following Cauchy problem    ∂ 2 v U (v, x) -∆ x U (v, x) = 0, U (0, x) = δ 0 (x), ∂ v | v=0 U (v, x) = 0.
We are looking for the asymptotic expansion of the kernel Φ v . Recall that J denotes the Jacobian of the exponential map from p equipped with Lebesgue measure to X equipped with Riemannian measure. It satisfies

J(H) -1 2 = α∈Σ + α, H sinh α, H mα 2 α∈Σ + (1 + α, H ) mα 2 e -ρ,H ∀H ∈ a + .
Let f be a K-bi-invariant function on G, then f is also Ad K-invariant on p and W -invariant on a. Recall that ∆ p and ∆ a denote the usual Laplacian on the Euclidean spaces p and a ⊂ p. The radial part of the Laplacian ∆ on X is defined by

∆ rad f (H) = ∆ a f (H) + α∈Σ + m α coth α, H ∂ α f (H) ∀H ∈ a + ,
and that of ∆ p is given by

∆ rad p f (H) = ∆ a f (H) + α∈Σ + m α α, H -1 ∂ α f (H) ∀H ∈ a + ,
see [Hel00, Propositions 3.9 and 3.11]. The following proposition provides a relation between ∆ rad and ∆ rad p , it allows us to simplify the computations about the parametrix.

Proposition B.1. Let f ∈ C ∞ (a) be a W -invariant function. Then J(H) 1 2 • ∆ rad • J(H) -1 2 f (H) = ∆ rad p + ω(H) f (H) ∀H ∈ a,
where

ω(H) = α∈Σ + m α 2 m α 2 -1 |α| 2 1 α, H - 1 sinh 2 α, H + α∈Σ + s.t. 2α∈Σ + m α m 2α 2 |α| 2 m α 2 -1 |α| 2 1 α, H - 1 sinh 2 α, H -|ρ| 2
is a smooth W -invariant function, which is uniformly bounded together with all its derivatives.

Proof. Notice that

J(H) 1 2 • ∆ rad • J(H) -1 2 f (H) =J(H) 1 2 ∆ rad J -1 2 (H)f (H) + ∆ rad f (H) + 2J(H) 1 2 ∇ a J -1 2 (H) • ∇ a f (H) ∆ rad p f (H) , since J(H) 1 2 ∇ a J -1 2 (H) = α∈Σ + m α 2 1 α, H -coth α, H α.
We deduce from the next lemma that

J(H) 1 2 ∆ rad J -1 2 (H) = ω(H) ∀H ∈ a,
and this concludes the proof.

Lemma B.2 (Cancellations lemma). The following equations hold for all H ∈ a:

α,β∈Σ + , Rα =Rβ m α m β α, β α, H β, H = 0 α,β∈Σ + , Rα =Rβ m α m β α, β coth α, H coth β, H -1 = 0
Proof. See [HaSt03, Appendix] for a detailed proof of this "folkloric" result.

Recall that {R z + | z ∈ C} denotes the analytic family of Riesz distributions on R defined by

R z + (r) =    Γ(z) -1 r z-1 if r > 0, 0 if r ≤ 0.
Consider the asymptotic expansion

Φ v (exp H) = J(H) -1 2 +∞ k=0 4 -k |v| U k (H) R k-d-1 2 + (v 2 -|H| 2 ) (B.1) where U 0 is a constant such that U 0 J(H) -1 2 |v| R -d-1 2 + (v 2 -|H| 2 ) → δ 0 (H) as v → 0 and U k ∈ C ∞ (p) are smooth Ad K-invariant functions. By expanding 0 = J(H) 1 2 ∂ 2 v -∆ rad ]Φ v (exp H) = +∞ k=0 4 -k ∂ 2 v -∆ rad p -ω(H) |v| U k (H) R k-d-1 2 + (v 2 -|H| 2 ) ,
we deduce

(k + 1) + ∂ H U k+1 (H) = ∆ rad p + ω(H) U k (H), (B.2)
for every k ∈ N. In other words,

U k+1 (H) = 1 0 ds s k ∆ rad p + ω(sH) U k (sH). (B.3)
As ω and all its derivatives are uniformly bounded, we obtain

∇ n p U k = O(1) (B.4) for any k, n ∈ N.
Next, by resuming the proof of [Bér77, Proposition 27] with our asymptotic expansion (B.1), we deduce that the remainder of the truncated expansion

Φ v (exp H) = J(H) -1 2 N k=0 4 -k |v| U k (H) R k-d-1 2 + (v 2 -|H| 2 ) + E N (v, exp H) (B.5) is a solution to the inhomogeneous Cauchy problem    ∂ 2 v -∆ rad E N (v, exp H) = J(H) -1 2 U N (v, H), lim v→0 E N (v, exp H) = 0, lim v→0 ∂E N ∂v (v, exp H) = 0, where U N (v, H) = -4 -N |v| U N (H) R N -d-1 2 + (v 2 -|H| 2 ).
Hence, by Duhamel's formula

E N (v, exp H) = v 0 du sin(v-u) √ -∆ rad √ -∆ rad {J(H) -1 2 U N (u, H)}.
According to next lemma and by L 2 conservation, we have

|E N (v, exp H)| e -ρ,H E N (v, •) H 2σ+1 (X) e -ρ,H v 0 du U N (u, •)J -1 2 H 2σ (X)
provided that 2σ + 1 > d 2 , and

U N (u, •)J -1 2 2 H 2σ (X) = ∆ σ { U N (u, •)J -1 2 } 2 L 2 (X)
= const. for all K-bi-invariant functions f ∈ H σ (X).

Proof. See [Ank92, Lemma 2.3].

Notice that, for all N > d 2 , we have

J(H) -1 2 N k=[d/2]+1 4 -k |v| U k (H)R k-d-1 2 + (v 2 -|H| 2 ) (1 + v) 2N -d+ 2 e -ρ,H .
Then we deduce the following corollary.

Corollary B.4. The K-bi-invariant convolution kernel Φ v has the asymptotic expansion Remark C.2. The proof of Proposition C.1 is similar to the proof of Lemma 3.3 in [START_REF] Cowling | L p -L q estimates for functions of the Laplace-Beltrami operator on noncompact symmetric spaces[END_REF]. Notice that the latter statement contains a minor error in the Gamma factor and that our estimates contain an additional exponential decay, which is crucial for the dispersive estimates.

Φ v (exp H) = J(H) -1 2 [d/2] k=0 4 -k |v| U k (H) R k-d-1 2 + (v 2 -|H| 2 ) + E Φ (v,
By performing the change of variables u = w-1 w and noticing that du dw = 1+ 1 w 2 1, we get

I 1 3 2 -3 2 du (u 2 + cos 2 θ) -γ 2 3 2 0 du (u + cos θ) -γ          (cos θ) -γ-1 if γ > 1, 1 -log(cos θ) if γ = 1, 1 if γ < 1.
1 π τ w 2 + z 2 =    1 π z u 2 +z 2 if ε = 1, 1 √ π z √ u 2 +z 2 if ε = 1 2 .
(C.8)

Proof. The case ε = 1 follows immediately from the fact the distribution R 0 + is equal to the Dirac measure at the origin. In the case ε = 1 2 , the formula is proved first for z > 0 and then extended straightforwardly by analytic continuation to all z ∈ C with Re z > 0. Specifically, the left hand side of (C.8) becomes with

I k (τ, H) = +∞ 0 d(v 2 ) p R τ (v) R k-d-1 2 + (v 2 -|H| 2 )
and

E(τ, H) = J(H) -1 2 [d/2] k=0 4 -k U k (H) +∞ 0 d(v 2 ) {χ T (v) -1} p R τ (v) R k-d-1 2 + (v 2 -|H| 2 ) + 2 +∞ 0 dv χ T (v) p R τ (v) E Φ (v, H)
Let ε = 1 if d is even and ε = 1 2 if d odd. Then

I k (τ, H) = -∂ ∂(|H| 2 ) [ d 2 ]-k +∞ 0 d(v 2 ) p R τ (v) R 1-ε + (v 2 -|H| 2 )
where

+∞ 0 d(v 2 ) p R τ (v) R 1-ε + (v 2 -|H| 2 ) =    1 π τ |H| 2 +τ 2 if ε = 1, 1 √ π τ √ |H| 2 +τ 2 if ε = 1 2 ,
according to (C.8). Then we obtain

I k (τ, H) = τ π Γ( d+1 2 -k) (|H| 2 +τ 2 ) d+1 2 -k .
Next, we estimate the remainder E(τ, H) whose second part is easily handled. By using (B.7), we have .

By repeating the previous calculations for I k ,

I(τ, H) = τ π -∂ ∂(|H| 2 ) [ d 2 ]-k +∞ 0 d(v 2 ) {χ T ( v 2 + |H| 2 ) -1} 1 v 2 + |H| 2 + τ 2 R 1-ε + (v 2 ).
Let's first consider the case where ε = 1, i.e., d is odd. Then

I k (τ,H) = τ π -∂ ∂(|H| 2 ) d-1 2 -k χ T (|H|) -1 1 |H| 2 +τ 2 = τ π j+j = d-1 2 -k ( d-1 2 -k)! j!j ! -∂ ∂(|H| 2 ) j χ T (|H|) -1 -∂ ∂(|H| 2 ) j 1 |H| 2 +τ 2 .

  1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1. Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2. Spherical Fourier analysis on X. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3. Barycentric decomposition of the Weyl chamber . . . . . . . . . . . . . . . . . 6 3. Pointwise estimates of the wave kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.1. Estimates of ω σ,0 t (x) when |t| is large and |x| |t| is sufficiently small. 11 3.2. Estimates of ω σ,0 t (x) in the remaining range . . . . . . . . . . . . . . . . . . . . . 14 3.3. Estimates of ω σ,∞ t associated semilinear Cauchy problems. Relevant theories are well established on Euclidean spaces, see for instance [Kap94; LiSo95; GLS97; KeTa98; DGK01], and the references therein.

Figure 1 .

 1 Figure 1. Examples of barycentric subdivisions in A 2 and in A 3 .

Figure 2 .

 2 Figure 2. Example of the projection in A 3

≤s≤1 ds s -2 ≤ 2

 2 according to (3.22). Hence Q = O(1) and we deduce from (3.27) that |P + 2 | |t| -d-1 2 for all t ∈ R * . By combining with (3.26) and (3.25), we obtain

  Theorem 4.3 (Small time dispersive estimate). Let d ≥ 3 and 0 < |t| < 1. Then

Figure 4 .

 4 Figure 4. Extended admissibility in dimension d ≥ 4.

2 ( 1

 21 rad ) σ {J(X) -1 2 U N (u, X)} 2 = const. u 2 p dX [∆ rad p + ω(X)] σ ( U N (u, X)) dX ∇ j p (u 2 -|X| 2 ) N -d+1 2 + u) 4N -d ,since ω and U N , together with all their derivatives are uniformly bounded. Here we assume N > d+1 2 + 2σ to avoid possible singularities on the sphere |X| = u. We may set 2σ = d 2 and N > d. Finally, we obtain|E N (v, exp H)| e -ρ,H v 0 du (1 + u) 2N -d 2 (1 + v) 2N -d 2 +1 e -ρ,H . Lemma B.3 (Sobolev embedding theorem for K-bi-invariant functions on X). Let σ > d 2 be an integer. Then |f (exp H)| e -ρ,H f H σ (X) ∀ H ∈ a +

  H), (B.6)where the remainder satisfies|E Φ (v, H)| (1 + v) 3( d 2 +1) e -ρ,H ∀ H ∈ a + . (B.7) Appendix C.Asymptotic expansion of the Poisson kernelThe Hadamard parametrix described above provides an asymptotic development of the kernel of the truncated Poisson operatorA τ = +∞ -∞ dv χ T (v)p R τ (v) cos (v √ -∆). Here τ = s -it with s ∈ (0, 1] and t ∈ R * , |t| ≤ 1, √ 2|t| if |t| ≥ 1, χ : R → [0, 1] is a smooth even cut-off function such that χ = 1 on [-1, 1] and supp χ ⊂ [-2 √ 2, 2 √ 2], χ T (v) = χ( v 2T ) is supported in [-2 √ 2T, 2 √ 2T ] ⊂ (-3T, 3T ), and p R τ (v) = 1 π ττ 2 +v 2 is the Poisson kernel on R (with complex time τ ). Notice that |τ | ≤ T . By resuming and improving slightly [CGM01, Lemma 3.3], we deduce the following parametrix for the kernel of A τ .Proposition C.1. The kernel a τ of the operator A τ is a smooth K-bi-invariant function on G, which is supported in the ball of radius 3T in X. Moreovera τ (exp H) satisfies |E(τ, H)| |T | 3( d 2 +1) (log T -log s) e -ρ,H ∀ H ∈ a + . (C.2)Here the coefficients U k are the same as in Corollary B.4 and are uniformly bounded.

(C. 7 )

 7 In conclusion, (C.3) is obtained by combining (C.5), (C.6) and (C.7).Lemma C.4. Let z ∈ C with Re z > 0 and u ∈ R. Then +∞ 0 d(w 2 ) R 1-ε + (w 2 -u 2 )

  z 2 after performing the change of variables w = √ u 2 + z 2 r. Proof of Proposition C.1. According to the asymptotic expansion (B.6), we writea τ (exp H) = J(H) -1 2 [d/2] k=0 4 -k U k (H) I k (τ, H) + E(τ, H)

|E 2

 2 (τ, H)| ≤ 2 +∞ 0 dv χ T (v) |p R τ (v)| |E Φ (v, H)| e -ρ,H 3T 0 dv |τ | |v 2 + τ 2 | (1 + v 3( d 2+1) ). formulas in (C.3). We deduce|E 2 (τ, H)| T 3( d 2 +1) (log T -log s)e -ρ,H . 2 ) {χ T (v) -1}p R τ (v) R k-d-1 2 + (v 2 -|H| 2 ) I k (τ,H)

This paper is devoted to prove time sharp kernel estimates and dispersive properties for the wave equation on noncompact symmetric spaces of higher rank. As consequences, we prove the Strichartz inequality and study their applications to Let us state and reprove some technical results borrowed from [START_REF] Cowling | L p -L q estimates for functions of the Laplace-Beltrami operator on noncompact symmetric spaces[END_REF].

. By performing the change of variables r = |z|w, the left hand side of (C.3) becomes

Notice that 3T

|z| > 2 and that

Let us split up

The first and the last integrals are easily estimated. According (C.4),

we deduce

and

Let us turn to the remaining integral, where 1 2 ≤ w ≤ 2. In this case we use the following improvement of (C.4):

On the one hand, the expression

In addition, it is O(T -2j ). On the other hand,

Let's then consider the case where ε = 1 2 , i.e., d is even. Then

Again, the expression

) if j > 0, and that it is estimated by for all H ∈ a + .