Schrödinger equation on noncompact symmetric spaces

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Schrödinger equation on noncompact symmetric spaces

Jean-Philippe Anker, Stefano Meda, Vittoria Pierfelice, Maria Vallarino, Hong-Wei Zhang

To cite this version:

Jean-Philippe Anker, Stefano Meda, Vittoria Pierfelice, Maria Vallarino, Hong-Wei Zhang.

Schrödinger equation on noncompact symmetric spaces. 2021. �hal-03187413v2�

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JEAN-PHILIPPE ANKER, STEFANO MEDA, VITTORIA PIERFELICE, MARIA VALLARINO AND HONG-WEI ZHANG

Abstract. We consider the Schrödinger equation on Riemannian symmetric spaces of noncompact type. Previous studies in rank one included sharp-in-time pointwise estimates for the Schrödinger kernel, dispersive properties, Strichartz inequalities for a large family of admissible pairs, and global well-posedness and scattering, both for small initial data. In this paper we establish analogous results in the higher rank case. The kernel estimates, which is our main result, are obtained by combining a subordination formula, an improved Hadamard parametrix for the wave equation, and a barycentric decomposition initially developed for the wave equation, which allows us to overcome a well-known problem, namely the fact that the Plancherel density is not always a differential symbol.

Contents

1. Introduction. . . 1

2. Preliminaries. . . 3

2.1. Noncompact symmetric spaces. . . 3

2.2. Spherical Fourier analysis on symmetric spaces. . . 4

2.3. Barycentric decomposition of the Weyl chamber. . . 5

3. Pointwise estimates of the Schrödinger kernel. . . 6

3.1. Large time kernel estimate. . . 6

3.2. Small time kernel estimate. . . 7

4. Dispersive estimates and Strichartz inequalities on symmetric spaces. . . 14

5. Well-posedness and scattering for the semilinear Schrödinger equation. . . 15

References. . . 16

1. Introduction

In this paper, we establish sharp-in-time kernel estimates and dispersive properties for the Schrödinger equation on general Riemannian symmetric spaces of noncompact type. Such estimates allow us to prove a global-in-time Strichartz inequality for a large family of admissible couples. This inequality has two important applications in the study of nonlinear Schrödinger equations. On the one hand, it serves as a tool for finding minimal regularity conditions on the initial data ensuring well-posedness. On the other hand, it is one of the key ingredient to prove scattering results.

The Schrödinger equation was studied extensively in the Euclidean setting. We refer to [Caz03;Tao06] and the references therein for more details. A basic theme in analysis of PDEs is the influence of geometry on the behavior of solutions. On a compact or nonnegatively curved manifold, one gets in general a Strichartz inequality with some loss of derivatives, which implies weaker well-posedness and scattering results than in the Euclidean case, see for instance [Bou93;

BGT04;GePi10;Zha20].

2020Mathematics Subject Classification. 22E30, 35J10, 35P25, 43A85, 43A90.

Key words and phrases. Noncompact symmetric space, semilinear Schrödinger equation, pointwise kernel estimates, dispersive property, Strichartz inequality, global well-posedness.

1

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In this paper we consider general Riemannian symmetric spaces of noncompact type, which form an important class of non-positively curved Riemannian manifolds with exponential volume growth. For such geometries, one expects stronger dispersive phenomena than in the Euclidean setting and hence better well-posedness and scattering results. This was indeed brought to light in the study of the Schrödinger equation on real hyperbolic spaces, which are the simplest symmetric spaces of noncompact type and rank one [Pie06; Ban07; BCS08; AnPi09; IoSt09], and subsequently on Damek-Ricci spaces [APV11], a class of harmonic manifolds which includes all noncompact symmetric spaces of rank one.

Extending these rank one results to higher rank was a challenging problem, which required new ideas. Surprisingly, large time estimates are obtained rather easily, while the main difficulty lies in the kernel estimates at infinity for small time. We have finally overcome it by a clever combination of the following tools, the last two being borrowed from [AnZh20] :

• the subordination principle for the Schrödinger propagator in terms of the wave propagator;

• an improved Hadamard parametrix for the wave equation;

• a barycentric decomposition of the inverse spherical Fourier transform, which allows us to integrate by parts as if the Plancherel density were a differentiable symbol.

Let us state the main two results obtained in this paper. Consider the Schrödinger propagator e^it∆ associated to the Laplace-Beltrami operator ∆ defined on a d-dimensional noncompact symmetric space X=G/K and let us denote byst itsK-bi-invariant convolution kernel.

Theorem 1.1(Pointwise kernel estimates). There exist C >0and N >0such that the following estimates hold, for all t∈R^∗ and x∈X:

|s_t(x)| ≤ C(1 +|x|)^Ne^−hρ,x⁺ⁱ







|t|⁻^d² if 0<|t|<1,

|t|⁻^D² if |t| ≥1,

where x⁺ ∈ a⁺ denotes the radial component of x in the Cartan decomposition, ρ ∈ a⁺ the half sum of all positive roots and D=`+ 2|Σ⁺_r| the so-called dimension at infinity or pseudo- dimension of X.

Remark 1.2. Contrarily to the Euclidean setting, the Schrödinger kernel satisfies no rescaling and behaves differently for small and large times. Notice that the large polynomial factor (1 +

|x|)^N is harmless for the dispersive estimates because of the exponential factor e^−hρ,x⁺ⁱ.

Theorem 1.3 (Dispersive properties). Let 2< q,q <e +∞. Then there exists a constantC >0 such that following dispersive estimates hold for all t∈R^∗ :

ke^it∆k_L_eq0

(X)→L^q(X)≤C







|t|⁻^max(¹²⁻¹^q^,¹²⁻¹^q^e⁾^d if 0<|t|<1,

|t|⁻^D² if |t| ≥1, where qeandqe⁰ are dual indices in the sense that ¹

qe+ ¹

qe⁰ = 1.

Remark 1.4. At the endpoint q = qe = 2, t 7→ e^it∆ is a one-parameter group of unitary operators on L²(X). Theorem 1.1 andTheorem 1.3 extend earlier results obtained in rank one (whereD= 3) to higher rank. Notice that we obtain stronger dispersive properties in our setting than in Euclidean spaces, especially in large time, where we have a fast time decay independent of q or q.e

These sharp-in-time estimates are significantly different from those in the Euclidean setting.

The large scale geometry of symmetric spaces yield better dispersive properties, in the sense that the large time decay |t|^−D/2 is independent of q. Such an improvement allows us to establish global-in-time Strichartz inequalities for a large family of admissible pairs. As an application to

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the Schrödinger equation with power-like nonlinearities and small initial data, we deduce global well-posedness and scattering, both in L² and in H¹, for any subcritical power without further assumptions (in contrast with the Euclidean case).

Our paper is organized as follows. After reviewing spherical Fourier analysis on noncompact symmetric spaces and the barycentric decomposition of the Weyl chamber in Sect. 2, we prove in Sect. 3 our main result, namely the pointwise kernel estimates. In Sect. 3, we deduce the dispersive properties and the Strichartz inequalities for a large family of admissible pairs, by adapting straightforwardly the method carried out in rank one. Such stronger estimates imply better well-posedness and scattering results for nonlinear Schrödinger equations, which are listed inSect. 5.

Throughout this paper, the symbolA.Bbetween two positive expressions means that there is a constant C >0 such thatA≤CB. The symbol AB means thatA.B andB .A.

2. Preliminaries

In this section we review briefly harmonic analysis on Riemannian symmetric spaces of noncompact type. We adopt the standard notation and refer to [Hel78;Hel00;GaVa88] for more details.

2.1. Noncompact symmetric spaces. LetGbe a semisimple Lie group, connected, noncompact, with finite center, and K be a maximal compact subgroup ofG. The homogeneous space X=G/K is a Riemannian symmetric space of noncompact type. Letg=k⊕p be the Cartan decomposition of the Lie algebra ofG, the Killing form ofginduces aK-invariant inner product h. , .i 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. We identify a with its dual a^∗ by means of the inner product inherited from p. Let Σ ⊂a be the root system of (g,a) and denote byW 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 dbe the dimension ofX and letD be the so-called dimension at infinity or pseudo-dimension of X:

d = ` + P

α∈Σ⁺ m_α and D = ` + 2|Σ⁺_r| (2.1) wherem_α is the dimension of the positive root subspaceg_α. Notice that these two dimensions behave quite differently. For example, D= 3 while d≥2 is arbitrary in rank one, D=d if G is complex, andD > d (actuallyD= 2d−`) if Gis split.

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

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

In the Cartan decomposition, the Haar measure on Gwrites Z

G

dx f(x) = const.

Z

K

dk₁ Z

a⁺

dx⁺δ(x⁺) Z

K

dk₂f(k₁(expx⁺)k₂), with density

δ(x⁺) = Q

α∈Σ⁺(sinhhα, x⁺i)^m^α Q

α∈Σ⁺

hα,x⁺i 1+hα,x⁺i

mα

e^2hρ,x⁺ⁱ ∀x⁺∈a⁺.

Here ρ∈a⁺ denotes the half sum of all positive rootsα∈Σ⁺ counted with their multiplicities m_α:

ρ = ¹₂ P

α∈Σ⁺ m_αα.

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2.2. Spherical Fourier analysis on symmetric spaces. Let S(K\G/K) be the Schwartz space ofK-bi-invariant functions on G. The spherical Fourier transformHis defined by

Hf(λ) = Z

G

dx ϕ−λ(x)f(x) ∀λ∈a, ∀f ∈ S(K\G/K),

where ϕλ ∈ C^∞(K\G/K) denotes the spherical function of index λ ∈ a, which is a smooth K-bi-invariant eigenfunction of all invariant differential operators on X, in particular of the Laplace-Beltrami operator:

−∆ϕ_λ(x) = (|λ|²+|ρ|²)ϕ_λ(x).

The spherical functions have the following integral representation ϕλ(x) =

Z

K

dk ehiλ+ρ,A(kx)i ∀λ∈a, (2.2) where A(kx) denotes the a-component in the Iwasawa decomposition of kx. They satisfy the basic estimate

|ϕ_λ(x)| ≤ ϕ₀(x) ∀λ∈a, ∀x∈ G, where

ϕ₀(x) Q

α∈Σ⁺_r(1 +hα, x⁺i) e^−hρ,x⁺ⁱ ∀x∈G.

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

f(x) = C0

Z

a

dλ|c(λ)|⁻²ϕ_λ(x)Hf(λ) ∀x∈G, ∀f ∈ S(a)^W, (2.3) where C0 > 0 is a constant depending only on the geometry of X. By using the Gindikin- Karpelevič formula of the Harish-Chandra c-function (see [Hel00] or [GaVa88]), we can write the Plancherel density as

|c(λ)|⁻² = Q

α∈Σ⁺r |c_α(hα, λi)|⁻², (2.4) with

cα(v) =

Cα

z }| {

Γ(_hα,αi^hα,ρi+¹₂mα) Γ(_hα,αi^hα,ρi)

Γ(¹₂_hα,αi^hα,ρi+¹₄mα+¹₂m2α) Γ(¹₂_hα,αi^hα,ρi+¹₄mα)

Γ(iv) Γ(iv+¹₂mα)

Γ(₂ⁱv+¹₄mα) Γ(ⁱ₂v+¹₄mα+¹₂m2α).

Notice that |c_α|⁻² is a homogeneous differential symbol on R of order mα +m2α, for every α ∈ Σ⁺_r. Hence |c(λ)|⁻² is a product of one-dimensional symbols, but not a symbol on a in general. It has the following behavior

|c(λ)|⁻² Q

α∈Σ⁺rhα, λi²(1 +|hα, λi|)^m^α^+m^2α⁻² and satisfies

|c(λ)|⁻² .

(|λ|^D−` if |λ| ≤1,

|λ|^d−` if |λ| ≥1,

(2.5) together with all its derivatives.

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O

Λ₂ Λ₁

Λ₃

S1

S2

S3

S1

S2 S₃

Figure 1. Example of barycentric subdivisions inA₃

2.3. Barycentric decomposition of the Weyl chamber. This tool plays an important role in spherical Fourier analysis in higher rank. It allows us to overcome a well-known difficulty, namely the fact that the Plancherel density is not a symbol in general. We list here only some useful properties and refer to [AnZh20, Subsection 2.2] for more details about this decomposition.

Let Σ⁺_s ={α₁, . . . , α_`} be the set of positive simple roots, and let {Λ₁, . . . ,Λ_`} be the dual basis of a, which is defined by

hα_j,Λki = δjk ∀1≤j, k≤`.

Denote by B the convex hull ofW.Λ₁t · · · tW.Λ_`, and bySits polyhedral boundary. Notice thatB∩a⁺ is the`-simplex with vertices0,Λ₁, . . . ,Λ_`, andS∩a⁺ is the(`−1)-simplex with vertices Λ₁, . . . ,Λ_`. The following tiling is obtained by regrouping the barycentric subdivisions of the simplices S∩w.a⁺:

S = [

w∈W

[

1≤j≤`

w.S_j

where

S_j = {λ∈S∩a⁺| hα_j, λi = max

1≤j≤`hα_k, λi}.

We project these subdivisions S_j onto the unit sphere and denote these projections by S_j. We establish next a smooth version of the partition of unity

X

w∈W

X

1≤j≤`

1w.Sj

λ

|λ|

= 1 a.e..

Let χ_C₁ : R → [0,1] be a smooth cut-off function such that χ_C₁(r) = 1 when r ≥ 0 and χC1(r) = 0whenr ≤ −C1, whereC1 >0 is a suitable constant (see [AnZh20, p.8]). For every w∈W and1≤j≤`, we define

χe_w.S_j(λ) = Y

1≤k≤`,k6=j

χ_C₁ ^hw.α_|λ|^k^,λi

χ_C₁ ^hw.α^j^,λi−hw.α_|λ| ^k^,λi

∀λ∈ar{0},

and

χ_w.S_j = χew.Sj

P

w∈W

P

1≤j≤`χe_w.S_j. Then,χ_w.S_j is a homogeneous symbol of order0 and

X

w∈W

X

1≤j≤`

χ_w.S_j = 1 on ar{0}. (2.6)

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In addition, the following properties hold for the support ofχ_w.S_j

Lemma 2.1. ([AnZh20, Proposition 2.6]) Let w ∈ W and 1 ≤ j ≤ `. Then a root α ∈ Σ satisfies either hα, w.Λ_ji= 0 or

|hα, λi| |λ| ∀λ∈suppχw.Sj. (2.7)

Moreover,

|hw.Λ_j, λi| |λ| ∀λ∈suppχ_w.S_j. (2.8)

3. Pointwise estimates of the Schrödinger kernel

For simplicity, we consider in this section the shifted Schrödinger propagator e^−itD² with D = p

−∆− |ρ|², and denote still by s_t its K-bi-invariant convolution kernel. By using the inverse formula of the spherical Fourier transform, we have

s_t(x) = C₀ Z

a

dλ|c(λ)|⁻²ϕ_λ(x)e^−it|λ|² ∀t∈R^∗,∀x∈X. (3.1) As usual, such an oscillatory integral makes sense by applying standard procedures (as a limit of convergent integrals and/or after performing several integrals by parts). We will study (3.1) differently, depending whether |t|is large or small. Let us begin with the easier case where |t|

is large.

3.1. Large time kernel estimate. Assume that|t| ≥1. In this case, we establish the following pointwise kernel estimate, by using the standard stationary phase method and the elementary estimate (2.5) about the Plancherel density and its derivatives.

Theorem 3.1. There exist an integerN >max{d, D} and a constantC >0 such that

|s_t(x)| ≤ C|t|⁻^D² 1 +|x|N

ϕ₀(x) ∀ |t| ≥1,∀x∈X. Proof. By using the integral expression (2.2) of the spherical function, we write

s_t(x) = C₀ Z

K

dk e^hρ,A(kx)i Z

a

dλ|c(λ)|⁻²e^−it|λ|²e^ihλ,A(kx)i

| {z }

I(t,A(kx))

(3.2)

where A(kx) denotes the a-component in the Iwasawa decomposition of kx, which satisfies

|A(kx)| ≤ |x|and which we abbreviate Ain the sequel. Theorem 3.1 will follow from

|I(t, A)| . |t|⁻^D² 1 +|A|N

. (3.3)

Let us split up

I(t, A) = I0(t, A) + I∞(t, A) = Z

a

dλ χ⁰_t(λ). . . + Z

a

dλ χ^∞_t (λ). . . whereχ⁰_t(λ) =χ(p

|t||λ|) is a radial cut-off function such that suppχ⁰_t ⊂ B(0,2|t|⁻¹²),χ⁰_t = 1 on B(0,|t|⁻¹²)and χ^∞_t = 1−χ⁰_t. On the one hand, by using (2.5), we easily estimate

|I₀(t, A)| . Z

|λ|.|t|⁻¹2

dλ|c(λ)|⁻² . |t|⁻^D². (3.4) On the other hand, after performing N integrations by parts based on

e^−it|λ|² = −_2it¹ P` j=1

λj

|λ|²

∂

∂λje^−it|λ|², (3.5)

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we obtain

I∞(t, A) = (2it)^−N Z

a

dλ e^−it|λ|² nP`

j=1 ∂

∂λj ◦ _|λ|^λ^j₂oNn

χ^∞_t (λ)|c(λ)|⁻²e^ihλ,Ai o

. (3.6) Assume that

• N₀ derivatives are applied to the cut-off function χ⁰_t(λ), which produces O(|t|^N²⁰),

• N₁ derivatives are applied to the factors _|λ|^λ^j₂, which produces O(|λ|^−N−N¹),

• N2 derivatives are applied to the Plancherel density |c(λ)|⁻², which is not a symbol in general and which produces

(O(|λ|^D−`) if |λ| ≤1, O(|λ|^d−`) if |λ| ≥1,

• N₃ derivatives are applied to the exponential factor e^ihλ,Ai, which produces O(|A|^N³), withN₀+N₁+N₂+N₃ =N. If some derivatives hit the cut-off functionχ⁰_t(λ), i.e., if N₀ ≥1, then the integral reduces to a spherical shell where |λ| |t|⁻¹², and the contribution to (3.6) is estimated by

|t|⁻^N² |t|^N²⁰ |t|^N²¹ |t|⁻^D² |A|^N³ . |t|⁻^D² 1 +|A|N

, (3.7)

since |t| ≥1. IfN0 = 0, then

|I_∞(t, A)| . |t|^−N Z

|λ|&|t|⁻¹²

dλ

∇^N_λ²|c(λ)|⁻²

|λ|^−N^−N¹|A|^N³

. |t|^−N|A|^N³nZ

|t|⁻¹2.|λ|≤1

dλ|λ|^{D−`−N−N}¹ + Z

|λ|≥1

dλ|λ|^{d−`−N−N}¹o

. |t|⁻^D² (1 +|A|)^N +|t|^−N(1 +|A|)^N . |t|⁻^D² (1 +|A|)^N (3.8) provided thatN > d and N ≥ ^D₂. In conclusion, (3.3) follows from (3.4), (3.7) and (3.8).

Remark 3.2. The analysis carried out in the proof of Theorem 3.1yields at best the following small time estimate

|s_t(x)| . |t|^−d(1 +|x|)^dϕ₀(x) ∀0<|t|<1,∀x∈X. (3.9)

3.2. Small time kernel estimate. Assume that0<|t|<1. Our aim is to reduce the negative power|t|^−d in (3.9) to|t|⁻^d². We shall use different tools, depending on the size of √^|x|

|t|. If √^|x|

|t|

is small, we decompose the Weyl chamber into several subcones according to the barycentric decompositions described inSect. 2.3, and perform in each subcone several integrations by parts along a well chosen direction. If √^|x|

|t| is large, we express in addition the Schrödinger propagator in terms of the wave propagator and use the Hadamard parametrix.

Theorem 3.3. The following estimate holds, for 0<|t|<1 and |x| ≤p

|t|:

|s_t(x)| . |t|⁻^d² ϕ₀(x).

Proof. By resuming the notation in the proof of Theorem 3.1, we have s_t(x) = C₀

Z

K

dk e^hρ,A(kx)i I₀(t, A) +I∞(t, A) .

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Clearly,

|I₀(t, A)| = Z

a

dλ χ⁰_t(λ)|c(λ)|⁻²e^−it|λ|²e^ihλ,Ai .

Z

|λ|.|t|⁻¹²

dλ|c(λ)|⁻² . |t|⁻^d². (3.10) In order to estimate

I∞(t, A) = Z

a

dλ χ^∞_t (λ)|c(λ)|⁻²e^−it|λ|²e^ihλ,Ai, we split up

I∞(t, A) = X

w∈W

X

1≤j≤`

Z

a

dλ χ_w.S_j(λ)χ^∞_t (λ)|c(λ)|⁻²e^−it|λ|²e^ihλ,Ai

| {z }

I_w.Sj(t,x)

.

according to the barycentric decomposition (2.6). Next, we study I_w.S_j(t, x) by performing N integrations by parts based on

e^−it|λ|² = −_2it¹ _hw.Λ¹

j,λi∂_w.Λ_je^−it|λ|², which yields

Iw.Sj(t, x) = (2it)^−N Z

a

dλ e^−it|λ|² n

∂w.Λj◦_hw.Λ¹

j,λi

oNn

χw.Sj(λ)χ^∞_t (λ)|c(λ)|⁻²e^ihλ,Ai o

. As in the the proof ofTheorem 3.1, we assume that

• N0 derivatives are applied to the cut-off function χ^∞_t (λ):

∂^N_w.Λ⁰

jχ^∞_t (λ) = O(|t|^N²⁰),

• N₁ derivatives are applied to the factors _hw.Λ¹

j,λi, which produces O(|λ|^−N−N¹),

• N₂ derivatives are applied to the factor χ_w.S_j(λ), which is a homogeneous symbol of order0:

∂_w.Λ^N²

jχ_w.S_j(λ) = O(|λ|^−N²),

• N₃ derivatives are applied to the factor e^ihλ,Ai:

∂_w.Λ^N³

je^ihλ,Ai = O(|A|^N³),

• N₄ derivatives are applied to the Plancherel density |c(λ)|⁻²,

with N0+N1+N2+N3+N4 =N. According toLemma 2.1, any rootα∈Σ satisfies either hα, w.Λ_ji= 0or |hα, λi| |λ|, for every λ∈suppχ_w.S_j. On the one hand, ifhα, w.Λ_ji= 0, all derivatives

∂_w.Λ^k _j|c_α(hα, λi)|⁻² ∀k∈N^∗ vanish. On the other hand, if hα, w.Λ_ji 6= 0, we have

∂_w.Λ^k _j|c_α(hα, λi)|⁻²

. |hα, λi|^m^α^+m^2α^−k |λ|^m^α^+m^2α^−k

for anyk∈Nand for everyλ∈(suppχ_w.S_j)∩(suppχ^∞_t ). Here we have used (2.7) and the fact that|c_α|⁻² is an inhomogeneous symbol of order m_α+m_2α on R. Hence

∂_w.Λ^N⁴

j|c(λ)|⁻² = O(|λ|^d−`−N⁴) ∀λ∈(suppχw.Sj)∩(suppχ^∞_t ).

Therefore, if no derivative hits the cut-off function χ^∞_t (λ), i.e., if N₀= 0, then

|I_w.S_j(t, x)| . |t|^−N Z

|λ|&|t|⁻¹²

dλ|λ|^−N^−N¹^−N²^+d−`−N⁴|A|^N³

. |t|^−N|t|⁻^d²⁺^N²⁺^N²¹⁺^N²²⁺^N²⁴ |t|^N²³ ≤ |t|⁻^d²

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provided that N > d. If N₀ ≥ 1, then the integral is reduced to a spherical shell where

|λ| |t|⁻¹², and hence

|I_w.S_j(t, x)| . |t|^−N|t|^N²⁰ |t|⁻^d²⁺^N²⁺^N²¹⁺^N²²⁺^N²⁴ |A|^N³ ≤ |t|⁻^d², since |A| ≤ |x| ≤p

|t|. Together with (3.10), we conclude that |I(t, A)|.|t|⁻^d² and

|s_t(x)| . |t|⁻^d² ϕ0(x), for all 0<|t|<1and x∈Xsuch that|x| ≤p

|t|.

The above proof shows that, for every λ∈(suppχ_w.S_j)∩(suppχ^∞_t ), the Plancherel density

|c(λ)|⁻² behaves like an inhomogeneous symbol of order d−` if we differentiate it along the direction w.Λj. When |x|>p

|t|, we combine this argument with the Hadamard parametrix, which was used in the study of other equations on symmetric spaces, see for instance [CGM01;

AnZh20].

Let us express the Schrödinger propagator e^−itD² = π⁻¹² e⁻ⁱ^π⁴^{sign (t)}

| {z }

C2

|t|⁻¹² Z +∞

0

ds e^4tⁱ^s²cos(sD) in terms of the wave propagator and correspondingly

s_t(x) = C₂|t|⁻¹² Z +∞

0

ds e^4tⁱ^s²Φ_s(x) (3.11) for their K-bi-invariant convolution kernels. On the one hand, by finite propagation speed,

Φs(x) = 0 if |x|>|s|. (3.12)

On the other hand, recall the Hadamard parametrix Φ_s(expH) = J(H)⁻¹² |s|

+∞

X

k=0

4^−kU_k(H)R^k−

d−1 2

+ (s²− |H|²) ∀s∈R^∗,∀H∈p, (3.13) whereJ denotes the Jacobian of the exponential map p→G/K, which is given by

J(H) = Q

α∈Σ⁺

sinhhα,Hi hα,Hi

mα

∀H∈a⁺,

and {R₊^z |z∈C} denotes the analytic family of Riesz distributions onR, which is defined by R^z₊(r) =

(

Γ(z)⁻¹r^z−1 if r >0

0 if r≤0 ∀ Rez >0.

This parametrix was constructed and used in various settings, see for instance [Ber77;Hor94;

CGM01]. We refer to [AnZh20, Appendix B] for details about the wave propagatorcos (t√

−∆) associated to the unshifted Laplacian ∆on noncompact symmetric spaces and notice that the same results hold for cos (tD). Specifically (3.13) is an asymptotic expansion

Φ_s(expH) = J(H)⁻¹² |s|

[d/2]

X

k=0

4^−kU_k(H)R^k−

d−1 2

+ (s²− |H|²) + E(s, H) (3.14) where the coefficientsU_kareAdK-invariant smooth functions onp, which are bounded together with their derivatives, while the remainder satisfies

|E(s, H)| . (1 +|s|)³⁽^d²⁺¹⁾e^−hρ,Hi ∀s∈R^∗,∀H ∈a⁺. (3.15)

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Let us split up Z +∞

0

ds = Z +∞

0

ds χ0(_|x|^s ) + Z +∞

0

ds χ1(_|x|^s ) + Z +∞

0

ds χ∞(_|x|^s ) in (3.11) by means of a smooth partition of unity 1 =χ₀+χ₁+χ∞ onR such that











suppχ₀ ⊂ (−1,1),

suppχ₁ ⊂ (−2C₃,−¹₂) ∪ (¹₂,2C₃), suppχ∞ ⊂ (−∞,−C₃) ∪ (C3,+∞)

where the choice of C3 > 1 will be specified later. Then the contribution of the first integral vanishes according to (3.12) and we are left with

s_t(x) = C₂|t|⁻¹² Z +∞

0

ds χ₁(_|x|^s)e^4tⁱ^s²Φ_s(x)

| {z }

s¹_t(x)

+C₂|t|⁻¹² Z +∞

0

ds χ∞(_|x|^s )e^4tⁱ^s²Φ_s(x)

| {z }

s^∞_t (x)

wheres¹_t(x)ands^∞_t (x)areK-bi-invariant. Let us first studys^∞_t (x)by using again the barycentric decomposition. In comparison with the proof ofTheorem 3.3, we have now|x|>p

|t|and there is an additional integral over s∈(1,∞) to control. Let us state the theorem.

Theorem 3.4. The following estimate holds, for all 0<|t|<1 and |x|>p

|t|:

|s^∞_t (x)| . |t|⁻^d² ϕ0(x).

Proof. We express

s^∞_t (x) = ¹₂C₀C₂|t|⁻¹² Z +∞

−∞

ds χ∞(_|x|^s )e^4tⁱ^s² Z

a

dλ|c(λ)|⁻²ϕ_λ(x)e^−is|λ|

by evenness and by expressing the wave kernel Φ_s by means of the inverse spherical Fourier transform. Let us split up s^∞_t = ¹₂C0C2(s^∞,0_t +s^∞,∞_t ), where

s^∞,0_t (x) = |t|⁻¹² Z +∞

−∞

a

dλ χ⁰_t(λ)|c(λ)|⁻²ϕ_λ(x)e^−is|λ|

| {z }

I0(s,t,x)

and

s^∞,∞_t (x) = |t|⁻¹² Z +∞

−∞

a

dλ χ^∞_t (λ)|c(λ)|⁻²ϕλ(x)e^−is|λ|

| {z }

I∞(s,t,x)

.

Recall that χ⁰_t(λ) = χ(p

|t||λ|) is a radial cut-off function such that suppχ⁰_t ⊂ B(0,2|t|⁻¹²), χ⁰_t = 1 on B(0,|t|⁻¹²) and χ^∞_t = 1−χ⁰_t.

Estimate of s^∞,0_t . Notice that the obvious estimate |I₀(s, t, x)| .|t|⁻^d²|ϕ₀(x) is not enough for our purpose. We need indeed to compensate on the one hand the factor|t|⁻¹² and to get on the other hand enough decay in |s|to ensure the convergence of the external integral. To this end, we perform two integrations by parts based on

e^4tⁱ^s² = −^2it_s _∂s^∂ e^4tⁱ^s² (3.16) and obtain this way

s^∞,0_t (x) = −4|t|³² Z +∞

−∞

ds e^4tⁱ^s² _∂s^∂ ¹_s _∂s^∂ ¹_sχ∞(_|x|^s )I0(s, t, x) . (3.17)

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Notice that

(_∂s^∂)^kI₀(s, t, x)

. |t|⁻^d+k² ϕ₀(x) ∀k∈N.

If any derivative hits χ∞(_|x|^s ) in (3.17), the integral reduces to two intervals where |s| |x|, and the corresponding contribution is estimated by

|t|⁻^d−3² ϕ₀(x) Z

|s||x|

ds{s⁻²|x|⁻² + s⁻³|x|⁻¹ + s⁻²|x|⁻¹|t|⁻¹²} . |t|⁻^d²ϕ₀(x), since |t|¹² <|x|. Otherwise we end up with the estimate

|t|⁻^d−3² ϕ₀(x) Z

|s|&|x|

ds{s⁻⁴ + s⁻³|t|⁻¹² + s⁻²|t|⁻¹} . |t|⁻^d² ϕ₀(x).

In conclusion,

|s^∞,0_t (x)| . |t|⁻^d² ϕ0(x), for all 0<|t|<1and x∈Xsuch that|x|>p

|t|.

Estimate of s^∞,∞_t . Let us turn to s^∞,∞_t (x) =

Z +∞

−∞

ds χ∞(_|x|^s )e^4tⁱ^s²I∞(s, t, x).

We will prove the following estimate, for any integer N > d,

|I∞(s, t, x)|. |s|^−N|t|⁻^d²⁺^N² ϕ0(x) ∀ |s| ≥C3|x|. (3.18) Then, as|x|>p

|t|, we conclude easily that

|s^∞,∞_t (x)| . |t|⁻^d²⁺^N²⁻¹² ϕ0(x) Z

|s|&|x|

ds|s|^−N . |t|⁻^d²

√

|t|

|x|

N−1

ϕ0(x) . |t|⁻^d² ϕ0(x). (3.19) In order to establish (3.18), we express

I∞(s, t, x) = Z

a

dλ χ^∞_t (λ)|c(λ)|⁻²ϕ_λ(x)e^−is|λ|

= Z

K

dk e^−hρ,Ai X

w∈W

X

1≤j≤`

Z

a

dλ χ_w.S_j(λ)χ^∞_t (λ)|c(λ)|⁻²e−i(s|λ|−hλ,Ai)

| {z }

I_w.Sj(s,t,A)

by using again the integral formula (2.2) and the barycentric decomposition (2.6). According to (2.8), we can choose C3>0such that, if |s| ≥C3|x|, then

|∂_w.Λ_j(s|λ| − hλ, Ai)| =

s^hw.Λ_|λ|^j^,λi − hw.Λ_j, Ai

≥ |s| ^|hw.Λ_|λ|^j^,λi|

| {z }

&1

− |hw.Λ_j, Ai|

| {z }

.|x|

& |s|,

for every λ ∈ (suppχ^∞_t )∩(suppχ_w.S_j). Under these assumptions, the phase function λ 7→

s|λ| − hλ, Ai has no critical point along the direction w.Λ_j. By performing N integrations by parts based on

e−i(s|λ|−hλ,Ai)

= _∂ ⁱ

w.Λj(s|λ|−hλ,Ai) ∂w.Λje−i(s|λ|−hλ,Ai)

,

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we write

I_w.S_j(s, t, A) = (is)^−N Z

a

dλ e−i(s|λ|−hλ,Ai)

n

∂_w.Λ_j ◦_∂ ^s

w.Λj(s|λ|−hλ,Ai)

oNn

χ_w.S_j(λ)χ^∞_t (λ)|c(λ)|⁻²o . Assume that

• N₀ derivatives are applied to the cut-off function χ^∞_t (λ), which produces O(|t|^N²⁰),

• N1 derivatives are applied to the factors _∂ ^s

w.Λj(s|λ|−hλ,Ai), which produces O(|λ|^−N¹),

• N2 derivatives are applied to the cut-off functionsχw.Sj(λ), which produces O(|λ|^−N²),

• N₃derivatives are applied to the Plancherel density|c(λ)|⁻², which produces O(|λ|^d−`−N³), with N₀+N₁+N₂+N₃ =N. Again, if some derivatives hit χ^∞_t (λ), i.e., if N₀ ≥1, then the integral reduces to a spherical shell where |λ| |t|⁻¹², and its contribution is estimated by

|s|^−N|t|⁻^`² |t|^N²⁰ |t|^N²¹ |t|^N²² |t|⁻^d²⁺²^`⁺^N²³ = |s|^−N|t|⁻^d²⁺^N². IfN0= 0, then

|I_w.S_j(s, t, A)| . |s|^−N Z

|λ|&|t|⁻¹²

dλ|λ|^−N¹|λ|^−N²|λ|^d−`−N³ . |s|^−N|t|⁻^d²⁺^N²

provided thatN > d. This proves (3.18) and hence (3.19).

Theorem 3.5. The following estimate holds, for all 0<|t|<1 and |x|>p

|t|:

|s¹_t(x)| . |t|⁻^d² (1 +|x|)³²^d+4e^−hρ,x⁺ⁱ. Proof. Since s¹_t isK-bi-invariant, we have

s¹_t(x) = ^C₂² |t|⁻¹² J(x⁺)⁻¹²

[d/2]

X

k=0

4^−kU_k(x⁺)

Ik(t,|x|)

z }| {

Z +∞

0

d(s²)χ₁(_|x|^s)e^4tⁱ^s²R^k−

d−1

+ 2 (s²− |x|²)

+ ^C₂²|t|⁻¹² Z +∞

0

ds χ₁(_|x|^s)e^4tⁱ^s²E(s, x⁺)

| {z }

E(t,|x|)e

according to (3.14). On the one hand, the remainder estimate

|E(t,e |x|)| . |x|(1 +|x|)³⁽^d²⁺¹⁾e^−hρ,x⁺ⁱ (3.20) follows from (3.15). On the other hand, we claim that

|I_k(t,|x|)| . |t|^k−^d−1² (3.21)

if |x|>p

|t|>0. Let us first prove (3.21) whendis odd. By a change of variables and by using the fact that R^k−

d−1

+ 2 (s−1) = _∂s^∂^d−1₂ −k

R⁰₊(s−1), we obtain

|I_k(t,|x|)| = |x|^2k−d+1 Z +∞

0

ds χ₁(√ s)e^i|x|

2 4t sR^k−

d−1

+ 2 (s−1)

= |x|^2k−d+1 Z +∞

0

ds R⁰₊(s−1) −_∂s^∂^d−1₂ −k {χ₁(√

s)e^i|x|

2 4t s .

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AsR⁰₊(s−1)is the Dirac measure at s= 1, we conclude that I_k(t,|x|) = |x|^2k−d+1 −₄ⁱ^|x|_t²^d−1₂ −k

= O(|t|^k−^d−1² ).

When dis even, we obtain similarly

|I_k(t,|x|)| = π⁻¹²|x|^2k−d+1 Z +∞

1

√ds

s−1 −_∂s^∂ ^d₂−k χ1(√

s)e^i|x|

2 4t s , which is a linear combination of expressions

t^j+k−^d² |x|^1−2j Z +∞

1

√ds

s−1θ_j(s)e^i|x|

2 4t s

| {z }

Jj(t,|x|)

where0≤j≤ ^d₂−kand θ_j ∈ C_c^∞(R) withsuppθ_j ⊂(−4C₃²,4C₃²). Notice that the elementary estimateJj(t,|x|) =O(1), together with the assumption|x|>p

|t| implies that

|I_k(t,|x|)|.|x| |t|⁻^d²^+k,

which might be enough for our purpose as long as k >0. The case k= 0 requires actually a more careful analysis. Let us show that

J_j(t,|x|) .

√

|t|

|x|

by splitting up

Z +∞

1

ds =

Z 1+ |t|

|x|² 1

ds + Z +∞

1+ |t|

|x|²

ds

in the definition of J_j(t,|x|). The contribution of the first integral is easily estimated by Z 1+ |t|

|x|² 1

ds^√^ds_s−1 = 2√ s−1

s=1+ |t|

|x|²

s=1 = 2

√

|t|

|x| . After performing an integration by parts based on

e^i|x|

2

4t s = −i_|x|^4t2 ∂

∂se^i|x|

2 4t s, the contribution of the second integral is also estimated by

|t|

|x|²

Z 4C₃² 1+ |t|

|x|²

ds

(s−1)⁻¹² + (s−1)⁻³² .

√|t|

|x|

under the assumption|x|>p

|t|. Thus (3.21) holds as well whendis even. In conclusion,

|s¹_t(x)| . |t|⁻^d² J(x⁺)⁻¹² + |t|⁻¹² |x|(1 +|x|)³⁽^d²⁺¹⁾e^−hρ,x⁺ⁱ . |t|⁻^d² (1 +|x|)³²^d+4e^−hρ,x⁺ⁱ

when 0<|t|<1and x∈Xsatisfies|x|>p

|t|.

In summary, we have divided our kernel analysis into three parts and deduced Theorem 1.1 fromTheorem 3.1,Theorem 3.3and Theorem 3.5. Notice that the method used to prove small time kernel estimates can be also used for large time.