Nonlinear Schrödinger Equation on Real Hyperbolic Spaces

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Nonlinear Schrödinger equation on real hyperbolic spaces

Jean-Philippe Anker, Vittoria Pierfelice

Université d’Orléans & CNRS, Laboratoire MAPMO (UMR 6628), Fédération Denis Poisson (FR 2964), Bâtiment de Mathématiques, B.P. 6759, 45067 Orléans cedex 2, France

Received 22 June 2008; accepted 31 January 2009 Available online 23 February 2009

Abstract

We consider the Schrödinger equation with no radial assumption on real hyperbolic spacesHⁿ. We obtain in all dimensions n2 sharp dispersive and Strichartz estimates for a large family of admissible pairs. As a first consequence, we obtain strong well-posedness results for NLS. Specifically, for small initial data, we proveL²andH¹global well-posedness for any subcritical power (in contrast with the Euclidean case) and with no gauge invariance assumption on the nonlinearityF. On the other hand, ifF is gauge invariant,L²charge is conserved and hence, as in the Euclidean case, it is possible to extend localL²solutions to global ones. The corresponding argument inH¹requires conservation of energy, which holds under the stronger condition thatF is defocusing. Recall that global well-posedness in the gauge invariant case was already proved by Banica, Carles and Staffilani, for small radialL²data or for large radialH¹data. The second application of our global Strichartz estimates isscatteringfor NLS both inL²and inH¹, with no radial or gauge invariance assumption. Notice that, on Euclidean spacesRⁿ, this is only possible for the critical powerγ =1+⁴_n and can be false for subcritical powers while, on hyperbolic spacesHⁿ, global existence and scattering of smallL²solutions hold for all powers 1< γ 1+⁴_n. If we restrict to defocusing nonlinearitiesF, we can extend theH¹scattering results of Banica, Carles and Staffilani to the nonradial case. Also there is no distinction anymore between short range and long range nonlinearities: the geometry of hyperbolic spaces makes every power-like nonlinearity short range.

©2009 Elsevier Masson SAS. All rights reserved.

Résumé

Nous étudions l’équation de Schrödinger sur les espaces hyperboliques réelsHⁿ, sans aucune hypothèse de radialité. Nous com- mençons par établir une inégalité dispersive optimale en toute dimensionn2, ainsi qu’une inégalité de Strichartz pour une grande famille de paires admissibles. Nous en déduisons que l’équation semi-linéaire est fortement bien posée dansL²ou dansH¹, pour des données initiales petites et pour des non-linéarités relativement générales, en particulier pour toutes les puissance sous-critiques (contrairement au cas euclidien) et sans hypothèse d’invariance par changement de jauge. Dans ce dernier cas, on a conservation de la charge et les solutionsL²locales se prolongent en solutionsL²globales ; le phénomène analogue dansH¹repose sur la conservation de l’énergie, qui est vérifiée pour des non-linéarités défocalisantes. Rappelons que Banica, Carles et Staffilani avaient précédemment montré que l’équation semi-linéaire était globalement bien posée pour des non-linéarités invariantes par change- ment de jauge et pour des données radiales petites dansL²ou arbitraires dansH¹. Comme seconde application, nous montrons qu’il y a diffusion (scattering) dansL²et dansH¹, à nouveau sans hypothèse de radialité ou d’invariance par changement de jauge. Rappelons que dansRⁿceci n’est possible que pour l’exposant critiqueγ =1+⁴_n et peut être faux pour des exposants sous-critiques, tandis que sur l’espace hyperboliquesHⁿ, on a existence globale et diffusion pour tout exposant 1< γ1+_n⁴

* Corresponding author.

E-mail addresses:[email protected] (J.-P. Anker), [email protected] (V. Pierfelice).

0294-1449/$ – see front matter ©2009 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2009.01.009

(et pour des conditions initiales petites dansL²). Dans le cas défocalisant, nous pouvons étendre au cas non radial les résultats de diffusionH¹de Banica, Carles, Staffilani. Observons également que, sur l’espace hyperbolique, toutes les non-linéarités de type puissance n’ont qu’un effet à courte portée, contrairement au cas euclidien.

©2009 Elsevier Masson SAS. All rights reserved.

MSC:primary 35Q55, 43A85; secondary 22E30, 35P25, 47J35, 58D25

Keywords:Nonlinear Schrödinger equation; Hyperbolic space; Dispersive inequality; Strichartz estimate; Well-posedness; Scattering

1. Introduction

The nonlinear Schrödinger equation (NLS) in Euclidean spaceRⁿ i∂_tu(t, x)+u(t, x)=F

u(t, x) ,

u(0, x)=f (x) (1)

has motivated a number of mathematical results in the last 30 years. Indeed, this equation (especially in thecubiccase F (u)= ±u|u|²) seems ubiquitous in physics and appears in many different contexts, including nonlinear optics, the theory of Bose–Einstein condensates and of water waves. In particular a detailed scattering theory for NLS has been developed.

An essential tool in the study of (1) is thedispersive estimate e^it f

L^∞(Rⁿ)C|t|⁻ⁿ²fL¹(Rⁿ)

for the linear homogeneous Cauchy problem i∂tu+u=0, t∈R, x∈Rⁿ,

u(0)=f. (2)

This estimate is classical and follows directly from the representation formula for the fundamental solution. A well- known procedure (introduced by Kato [15], Ginibre and Velo [9], and perfected by Keel and Tao [16]) then leads to theStrichartz estimates

uL^p(I;L^q(Rⁿ))CfL²(Rⁿ)+CF_Lp˜(I;L^q˜(Rⁿ)) (3) for the linear inhomogeneous Cauchy problem

i∂_tu(t, x)+u(t, x)=F (t, x), u(0, x)=f (x).

The estimates (3) hold for any bounded or unbounded time intervalI⊆Rand for all pairs(p, q), (p,˜ q)˜ ∈ [2,∞] × [2,∞)satisfying theadmissibility condition

2 p+n

q =n

2. (4)

Notice that both endpoints(p, q)=(∞,2)and(p, q)=(2,_n²ⁿ₋₂)are included in dimensionn3 while only the first one is included in dimensionn=2.

The issue of well-posedness for the nonlinear Cauchy problem (1) has been widely studied, at least for a power nonlinearityF (u)= ±|u|^γ orF (u)= ±|u|^γ−1uand for suitable ranges of the exponentγ >1. Here is a brief account of the classical theory. In the model caseF (u)= |u|^γ, we have

• local well-posedness inL²in the subcritical caseγ <1+_n⁴;

• global well-posedness inL²in the critical caseγ=1+⁴_n for small data;

• local well-posedness inH¹in the subcritical caseγ <1+_n−⁴₂;

• global well-posedness inH¹in the critical caseγ=1+_n−⁴₂ for small data.

Notice that the value of the critical exponent depends on the dimensionn. On the other hand, in the model case F (u)= |u|^γ−1u, Eq. (1) isgauge invariantanddefocusing, which impliesL²andH¹ conservation laws. Thus, in addition to the previous results, we have

• global well-posedness inL²in the subcritical caseγ <1+⁴_n;

• global well-posedness inH¹in the subcritical caseγ <1+_n−⁴₂.

Global existence for arbitrary data in the critical case remains an open problem, although several results are available (Bourgain [5], Tao, Visan and Zhang [19], . . . ).

The results above are proved essentially by a fixed point argument in a suitable mixed spaceL^p(R;L^q(Rⁿ)), using Strichartz estimates in combination with conservation laws when available.

As a byproduct, this method shows that solutions u(t, x) to (1) are small in a suitableL^qx sense as t → ±∞. Hence, asymptotically, the contribution of the nonlinearity is dominated by the linear part and thenonlinearequation (1) becomes close to thelinearequation (2). This basic observation is at the origin of scattering theory for NLS. By L² scatteringwe mean that, for every global solutionu∈C(R, L²(Rⁿ)), there existscattering data u_±∈L²(Rⁿ) such that

u(t )−e^it u_±

L²_x →0 ast→ ±∞. The definition ofH¹scatteringis analogous.

The classical scattering theory for NLS, in the defocusing caseF (u)= |u|^γ−1u, can be summarized as follows:

• scattering inL²holds in the critical caseγ=1+⁴_n for small data;

• scattering inH¹holds for 1+⁴_n< γ <1+_n−⁴₂;

• scattering inH¹fails for 1< γ 1+²_n.

This paper is a contribution to the study of Strichartz estimates and NLS on a manifoldM. Several results have been obtained for this problem and quite general classes of manifolds. The geometry ofMplays obviously an essential role: on a compact or positively curved manifold, one expects weaker decay properties and hence weaker results for NLS; on the other hand, on a noncompact negatively curved manifold, one expects better dispersion properties than in the Euclidean case and hence stronger well-posedness and scattering results for NLS.

The compact case has been studied extensively by Burq, Gérard and Tzvetkov [6] after earlier results by Bourgain [5] on the torus. In general one obtains Strichartz estimates

e^it f

L^p(I;L^q(M))C(I )fH^1/p(M)

which are local in time and with a loss of smoothness in space. As a consequence, the results for NLS are weaker than onRⁿ. Let us mention in particular the local well-posedness theory inH^s(Tⁿ)developed by Bourgain in the early nineties, extended ten years later to general compact manifolds by Burq, Gérard and Tzvetkov, and improved in some special cases such as spheresSⁿ[6] or 4-dimensional compact manifolds [8].

In this paper we shall restrict our attention to real hyperbolic spacesM=Hⁿof dimensionn2. Actually our re- sults extend straightforwardly to all hyperbolic spaces i.e. Riemannian symmetric spaces of noncompact type and rank one (they extend furthermore to Damek–Ricci spaces and this will be the subject of a forthcoming work). Consider the following linear Cauchy problem onHⁿ:

i∂_tu(t, x)+_Hⁿu(t, x)=F (t, x), t∈R, x∈Hⁿ,

u(0, x)=f (x). (5)

On one hand, Banica [3] (see also [17]) obtained the following weighted dispersive estimate, for radial solutions to the homogeneous equation (5) in dimensionn3:

w(x)u(t, x)C

|t|⁻ⁿ² + |t|⁻³²

Hⁿ

f (y)w(y)⁻¹dy.

Here w(x)= ^sinhr_r , where r denotes the geodesic distance fromx to the origin. On the other hand, Pierfelice [18]

obtained the following sharp weighted Strichartz estimate, for radial solutions to the inhomogeneous equation (5) in dimensionn3:

w^12−q1u

L^p_tL^q_xCfL²_x +Cw

1˜ q−¹2F

L^p_t^˜L^q_x^˜.

Herew(r)=(^sinh_r ^r)ⁿ⁻¹is the jacobian of the exponential map and(_p¹,¹_q),(_p¹_˜,¹_q˜)belong to the intervalIn= {(¹_p,_q¹)∈ [0,¹₂] ×(0,¹₂] |_p²+ⁿ_q=ⁿ₂}. Actually this result was established in the more general setting of Damek–Ricci spaces and it implies unweighted estimates for a wider range of indices, as pointed out by Banica, Carles and Staffilani [4].

Our first main result is the following dispersive estimate (Theorem 3.4), which holds for general functions (no radial assumption) in dimensionn2.

Dispersive estimate.Letq,q˜∈(2,∞]. Then, for 0<|t|<1, we have u(t )

L^q(Hⁿ)C|t|^{−max{12−1q,12−1q˜}n}f_Lq˜(Hⁿ)

while, for|t|1, we have u(t )

L^q(Hⁿ)C|t|⁻³²f_Lq˜(Hⁿ).

Ifq= ˜q=2, we have of courseL²conservationu(t )L²(Hⁿ)= fL²(Hⁿ)for allt∈R. Our second main result is the following Strichartz estimate (Theorem 3.6), which is deduced from the previous estimate and holds under the same general assumptions.

Strichartz estimate.Assume that(_p¹,_q¹)and(¹_p˜,¹_q˜)belong to the triangleT_n= {(_p¹,_q¹)∈(0,¹₂] ×(0,¹₂)|_p²+ⁿ_q

n

2} ∪ {(0,¹₂)}. Then

u_L^p_tL^q_xCfL²_x+CF_Lp˜ t L^q_x^˜.

Notice that the setT_nof admissible pairs forHⁿis much wider than the corresponding setI_nforRⁿ(which is just the lower edge of the triangle). This striking phenomenon was already observed in [4] for radial solutions. It can be regarded as an effect of hyperbolic geometry on dispersion.

Next we apply these estimates to study well-posedness and scattering for the nonlinear Cauchy problem i∂_tu+u=F (u),

u(0)=f. (6)

Throughout our paper, we shall use the following (standard) terminology about the nonlinearityF =F (u):

• F ispower-likeif there exist constantsγ >1 andC0 such that F (u)C|u|^γ,

F (u)−F (v)C

|u|^γ−1+ |v|^γ−1

|u−v|, (7)

• F isgauge invariantif Im

F (u)u =0, (8)

• F isdefocusingif there exists aC¹functionG=G(v)0 such that F (u)=G

|u|²

u. (9)

Notice that gauge invariance impliesL²conservation ofchargeormass:

Hⁿ

u(t, x)²dx=

Hⁿ

f (x)²dx ∀t,

while the defocusing assumption impliesH¹conservation ofenergy:

Hⁿ

∇u(t, x)²dx+

Hⁿ

Gu(t, x)²

dx=constant.

If we specialize to the model casesF = ±|u|^γ andF = ±|u|^γ−1u, then the gauge invariant nonlinearities are F = ±|u|^γ−1u

and the defocusing ones F = +|u|^γ−1u.

Let us first summarize our well-posedness results (Theorems 4.2 and 4.4).

Well-posedness for NLS.Consider the Cauchy problem (6) with a power-like nonlinearityF of orderγ.

• Assumeγ1+⁴_n. Then the problem is globally well-posed for smallL²data. For arbitraryL²data, it is locally well-posed ifγ <1+_n⁴.

• Assume γ1+_n−⁴₂. Then the problem is globally well-posed for smallH¹data. For arbitraryH¹data, it is locally well-posed ifγ <1+_n−⁴₂.

• IfF is gauge invariant andγ <1+⁴_n, the problem is globally well-posed for arbitraryL²data. IfF is defocusing andγ <1+_n−⁴₂, the problem is globally well-posed for arbitraryH¹data.

Similar results were obtained in [4] for radial functions and nonlinearities F = |u|^γ−1u. As expected, they are better for hyperbolic spaces than for Euclidean spaces. For instance, onHⁿ we have global well-posedness for small L²data, for any power 1< γ1+_n⁴, while onRⁿwe must assume in addition gauge invariance. Of course, under this condition, we can also handle arbitrarily large data, using conservation laws, as in the Euclidean case.

Let us next summarize our scattering results.

Scattering for NLS.Consider the Cauchy problem (6) with a power-like nonlinearityF of orderγ.

• Assumeγ1+⁴_n. Then, for all small dataf ∈L², the unique global solutionu(t, x)has the scattering property:

there existu_±∈L²such that u(t )−e^{it Hn}u_±

L²(Hⁿ)→0 ast→ ±∞.

• Assume γ 1+ _n−⁴₂. Then, for all small data f ∈H¹, the unique global solution u(t, x) has the scattering property: there existu_±∈H¹such that

u(t )−e^{it Hn}u_±

H¹(Hⁿ)→0 ast→ ±∞.

• Assumeγ <1+_n−⁴₂ and the defocusing condition. Then, for all dataf ∈H¹att= ±∞, the NLS has a unique global solutionu(t, x)with the following scattering property:

u(t )−e^{it Hn}f

H¹(Hⁿ)→0 ast→ ±∞.

Notice that onHⁿwe have small data scattering for all powers 1< γ 1+⁴_n (respectively 1< γ 1+_n−⁴₂). This is in sharp contrast withRⁿ, where scattering is known to fail for the range 1< γ1+²_n.

The results in this paper were presented by the second author at theConvegno Nazionale di Analisi Armonica (Caramanico, 22–25 May 2007) and by the first author at the Conference (in honor of Sigurdur Helgason on the

occasion of his 80th birthday)Integral Geometry, Harmonic Analysis and Representation Theory (Reykjavik, 15–

18 August 2007) and at the DFG–JSPS Joint Seminar Infinite Dimensional Harmonic Analysis IV (Tokyo, 10–14 September 2007).

A related preprint [14] was produced independently and simultaneously by Ionescu and Staffilani. While we are mostly interested in sharp dispersive and Strichartz estimates, with applications to general nonlinearities and scatter- ing, their main aim is scattering inH¹and the Morawetz inequality in the defocusing case. Thus our works, although overlapping, are complementary rather than concurrent.

2. Real hyperbolic spaces

In this paper, we consider the simplest class of Riemannian symmetric spaces of noncompact type, namely real hy- perbolic spacesHⁿof dimensionn2. We refer to Helgason’s books [10–12] for their structure, geometric properties, and for harmonic analysis on these spaces. Recall thatHⁿcan be realized as the upper sheet

x₀²−x₁²− · · · −x_n²=1, x01,

of hyperboloid inR¹⁺ⁿ, equipped with the Riemannian metric d²= −dx₀²+dx₁²+ · · · +dx_n²,

or as the homogeneous spaceG/K, whereG=SO(1, n)⁰ andK=SO(n). In geodesic polar coordinates, the Rie- mannian metric is given by

d²=dr²+(sinhr)²d²_Sn−1, the Riemannian volume by

dv=(sinhr)ⁿ⁻¹dr dv_Sn−1, and the Laplace–Beltrami operator by

=_Hn=∂_r²+(n−1)cothr∂r+(sinhr)⁻²_Sn−1.

Inhomogeneous Sobolev spaces onHⁿ(and on more general manifolds) are defined by H^s,q

Hⁿ

=(I−) ^−s2L^q Hⁿ

(1< q <∞, s∈R).

UsingL^qspectral analysis (see for instance [1]), they can be also defined as well by H^s,q

Hⁿ

=(−) ^−s2L^q Hⁿ

.

Moreover, fors=N∈N,H^s,q(Hⁿ)coincides with W^N,q

Hⁿ

= f ∈L^q

Hⁿ ∇^jf∈L^q Hⁿ

∀0j N ,

where∇denotes the covariant derivative. Recall eventually the Sobolev embedding theorem:

H^s,q Hⁿ

⊂H^s,˜q˜ Hⁿ

ifs− ˜sn 1

q −1

˜ q

>0.

3. Dispersive and Strichartz estimates onHHHⁿ

Consider first the homogeneous linear Schrödinger equation on the hyperbolic spaceHⁿof dimensionn2:

i∂_tu(t, x)+u(t, x)=0, u(0, x)=f (x),

whose solution is given by

u(t, x)=e^itf (x)=f∗s_t(x)=

Hⁿ

s_t d(x, y)

f (y) dy.

The convolution kernel s_t is a bi-K-invariant function on G i.e. a radial function on G/K=Hⁿ, which can be expressed as an inverse spherical Fourier transform:

s_t(r)=const. e^{−i(n−12 )2t}

+∞

−∞

e^{−it λ2}ϕ_λ(r) dλ

|c(λ)|².

For hyperbolic spacesHⁿ (and more generally for Damek–Ricci spaces), this expression can be made more explicit, using the inverse Abel transform:

st(r)=const.(it )⁻¹²e^{−i(n−21)2t}

− 1 sinhr

∂

∂r ⁿ⁻¹

2

e^4ir

2

t . (10)

Here(it )⁻¹² =e^{−iπ4sign(t )}|t|⁻¹² and, in the even dimensional case, the fractional derivative reads

− 1 sinhr

∂

∂r ⁿ⁻¹

2

e^4ir

2

t = 1

√π

+∞

|r|

− 1 sinhs

∂

∂s ⁿ

2

e^i4s

2

t sinhs ds

√coshs−coshr. (11)

Proposition 3.1. There exists a constantC >0 such that the following pointwise kernel estimate holds, for every t∈R^∗andr0:

s_t(r)C

|t|^−3/2(1+r)e^−n−21r if|t|1+r,

|t|^−n/2(1+r)ⁿ⁻²¹e^−n−21r if|t|1+r. (12) Remark 3.2.In dimensionn=3, this estimate boils down to

st(r)C|t|^−3/2(1+r)e^−r

and was well known (see for instance [3]). In other dimensions, it is sharper than the kernel estimates obtained previously [3,4]. Our estimate can be rewritten as follows:

s_t(r)C

|t|^−3/2ϕ0(r) if|t|1+r,

|t|^−n/2j (r)^−1/2 if|t|1+r,

using the ground spherical function ϕ0(r) (1 +r)e^−n−21r and the jacobian of the exponential map j (r)= (^sinh_r ^r)ⁿ⁻¹(₁^e₊^r_r)ⁿ⁻¹.

Proof of Proposition 3.1. We shall assumet >0 for simplicity and we shall resume in part the analysis carried out in [2] for the heat kernel. Consider first the odd dimensional case. Setm=ⁿ⁻₂¹ and let us expand

− 1 sinhr

∂

∂r m

e^4ir

2 t =e^4ir

2 t

m j=1

t^−jf_j(r). (13)

The functionsf_j(r)involved are linear combinations of productsϕ₁(r)· · ·ϕ_j(r), where ϕ(r)=

1 sinhr

∂

∂r

r²

and1, . . . , j∈N^∗are such that1+ · · · +j=m. Using the elementary global estimate ϕ(r)=O

(1+r)e^−r , we are lead to the conclusion:

s_t(r)t⁻¹² m j=1

1+r t

j

e^−mrt⁻¹² 1+r

t +

1+r t

m e^−mr.

Because of the fractional derivative (11), the even dimensional casen=2mis more delicate to handle. According to the above estimate of (13), we have

s_t(r)t⁻¹²

+∞

r

1+s

t +

1+s t

m

e^−ms sinhs ds

√coshs−coshr. (14)

Here and throughout the proof, we make repeated use of the following elementary estimates:

sinhs s

1+se^s, (15)

and

coshs−coshr=2 sinhs−r

2 sinhs+r 2 s−r

1+s−re^s−2r s+r 1+s+re^s+2r s−r

1+s−r s

1+se^s or

s²−r²

1+r e^r ifrsr+1,

e^s ifsr+1. (16)

Thus (14) becomes st(r)t⁻¹²

+∞

r

1+s

t +

1+s t

m

e^−(m−12)s

√1+s−r

√s−r

√s

√1+sds. (17)

After performing the change of variabless=r+uand using the trivial inequalities

√r+u

√1+r+u1, 1+r+u(1+r)(1+u), we obtain eventually

s_t(r)t⁻¹² 1+r

t +

1+r t

m

e^−(m−12)r. (18)

This allows us to conclude that st(r)Ct⁻¹²1+r

t e^−(m−12)r (19)

whent1+r. Ift1+r, the polynomial powermin (18) must be brought down tom−¹₂. For this purpose, let us rewrite more carefully the expansion

− 1 sinhs

∂

∂s m

e^4is

2

t =

0<j <m

t^−jf_j(s)e^4is

2

t +t^−(m−1)

−i 2

s sinhs

m−1

− 1 sinhs

∂

∂s

e^i4s

2 t .

The contribution of the sum (which does not occur in dimensionn=2) can be handled as above and is O

t⁻¹²

1+r

t +

1+r t

m−1

e^−(m−12)r

.

Thus it remains for us to show that the integral I (t, r)=

+∞

r

s sinhs

m−1

∂

∂se^4is

2 t

ds

√coshs−coshr

is O(t⁻¹²(1+r)^m−12e^−(m−12)r)whent1+r. Let us split I (t, r)=I₁(t, r)+I₂(t, r)+I₃(t, r)

according to

+∞

r

=

√r²+t

r

+

r+1

√r²+t

+ +∞

r+1

.

The first integral is easy to handle. We simply differentiate _∂s^∂ e^4is

2

t =₂^is_te^4is

2

t and use the elementary estimates (15), (16) together with the fact thats∈ [r, r+1]. As a result,

I1(t, r)t⁻¹(1+r)^m−12e^−(m−12)r

√r²+t

r

√s ds

s²−r²=t⁻¹²(1+r)^m−12e^−(m−12)r. Let us turn to the second and third integrals, that we integrate by parts:

I₂(t, r)+I₃(t, r)=e^4is

2 t

s sinhs

m−1

√ 1

coshs−coshr^s=r+1

s=√

r²+t+^s=+∞

s=r+1

+ r+1

√r²+t

+ +∞

r+1

e^4is

2 t

(m−1)

s sinhs

m−2

scoths−1

sinhs (coshs−coshr)⁻¹² +1

2 s

sinhs m−1

(coshs−coshr)⁻³²sinhs

ds.

The boundary terms are estimated asI₁(t, r):

s sinhs

m−1

√ 1

coshs−coshr

s=√

r²+t

t⁻¹²(1+r)^m−12e^−(m−12)r. The integral terms are bounded by

(1+r)^m−2e^−(m−12)r

r+1

√r²+t

1+r s²−r²

¹

2 +

1+r s²−r²

³

2

s ds+ +∞

r+1

(1+s)^m−1e^−(m−12)sds. (20)

Here we have used (15), (16) and the elementary estimate^scoths_sinhs⁻¹se^−s. In the expression between braces, the first factor is dominated by the second one. Thus the first integral in (20) is bounded by

(1+r)³²

r+1

√r²+t

s²−r²₋³₂

s ds=(1+r)³²

−

s²−r²₋¹₂^s=r+1

s=√

r²+t t⁻¹²(1+r)³². The second integral in (20) is estimated as (17):

+∞

r+1

(1+s)^m−1e^−(m−12)sds= +∞

1

(1+r+u)^m−1e^{−(m−12)(r+u)}du(1+r)^m−1e^−(m−12)r.

As a conclusion, we obtain

I (t, r)t⁻¹²(1+r)^m−12e^−(m−12)r. Thus we have shown that

st(r)t^−m(1+r)^m−12e^−(m−12)r when 0< t1+r. 2

Corollary 3.3. Let2< q <∞and 1α∞. Then there exists a constantC >0such that the following kernel estimate holds, with respect to Lorentz norms:

s_tL^q,αC

|t|^−n/2 if0<|t|1,

|t|^−3/2 if|t|1. (21)

Proof. Recall that Lorentz spacesL^q,α(Hⁿ)are variants of the classical Lebesgue spaces, whose norms are defined by

fL^q,α= [_+∞

0 {s^1/qf^∗(s)}^{α ds}_s ]^1/α if 1α <∞, sup_s>0s^1/qf^∗(s) ifα= ∞,

wheref^∗denotes the decreasing rearrangement off. In particular, iff is a positive radial decreasing function onHⁿ, thenf^∗=f ◦V⁻¹, where

V (r)=C r

0

(sinhs)ⁿ⁻¹ds

rⁿ asr→0, e^(n−1)r asr→ +∞

is the volume of a ball of radiusr >0 inHⁿ. Hence fL^q,α=

_+∞

0

V (r)^1/qf (r) ^αV(r) V (r) dr

1/α

1

0

f (r)^αr^αnq−1dr 1/α

+ _+∞

1

f (r)^αe^α(nq−1)rdr 1/α

if 1α <∞and fL^q,∞=sup

r>0

V (r)^1/qf (r) sup

0<r<1

r^nqf (r)+sup

r1

e^n−q1rf (r).

The Lorentz norm estimate (21) follows from these considerations and from the pointwise estimate (12). 2 Let us turn toL^qmapping properties of the Schrödinger propagatore^itonHⁿ. Recall thate^itis a one parameter group of unitary operators onL²(Hⁿ).

Theorem 3.4.Let2< q,q˜∞. Then there exists a constantC >0such that the following dispersive estimates hold:

e^it

L^q˜→L^qC

|t|^{−max{12−1q,12−1q˜}n} if0<|t|<1,

|t|⁻³² if|t|1.

Remark 3.5.In the Euclidean setting, small time estimates are similar forq= ˜q, other small time estimates do not hold, and large time estimates are drastically different.

Proof of Theorem 3.4. These estimates are obtained by interpolation and by using Corollary 3.3. Specifically, small time estimates follow from⎧

⎪⎪

⎨

⎪⎪

⎩ e^it

L¹→L^q= stL^qCq|t|⁻ⁿ² ∀q >2, e^it

L^q→L^∞= s_tL^qC_q|t|⁻ⁿ² ∀q >2, e^it

L²→L²=1, and large time estimates from

⎧⎪

⎪⎨

⎪⎪

⎩ e^it

L¹→L^q= stL^qCq|t|⁻³² ∀q >2, e^it

L^q→L^∞= s_tL^qC_q|t|⁻³² ∀q >2, e^it

L^q→L^qC_qs_tL^q,1C_q|t|⁻³² ∀q >2.

Fig. 1. Admissible set forHⁿin dimensionn3.

The key ingredient here is a sharp version of the Kunze–Stein phenomenon, due to Cowling, Meda and Setti (see [7]) and improved by Ionescu [13], which yields in particular

L^q(K\G)∗L^q(G/K)⊂L^q,∞(K\G/K) ∀q >2.

By such an inclusion, we mean that there exists a constantC_q>0 such that f ∗g_L^q,∞C_qf_L^qg_L^q ∀f∈L^p(K\G),∀g∈L^p(G/K).

Hence by duality

L^q(G/K)∗L^q,1(K\G/K)⊂L^q(G/K) ∀q >2. 2

Consider next the inhomogeneous linear Schrödinger equation (5) onHⁿ: i∂_tu(t, x)+u(t, x) =F (t, x),

u(0, x)=f (x),

whose solution is given by Duhamel’s formula:

u(t, x)=e^itf (x)−i t 0

e^i(t−s)F (s, x) ds. (22)

Strichartz estimates onHⁿinvolveadmissible pairsof indices(p, q)corresponding to the triangle Tn=

1 p,1

q

∈

0,1 2

×

0,1 2 2

p+n q n

2

∪

0,1 2

(23) (see Fig. 1).

Theorem 3.6.Assume that(p, q)and(p,˜ q)˜ are admissible pairs as above. Then there exists a constantC >0such that the following Strichartz estimate holds for solutions to the Cauchy problem(5):

uL^p_tL^q_x C

fL²_x + F_Lp˜ t L^qx^˜

. (24)

Remark 3.7.This result was obtained previously for radial functions in [4], using sharp weighted Strichartz estimates [18] in dimension n4, the elementary kernel expression (10) in dimension n=3, and specific kernel estimates in dimension n=2. Notice that the admissible set for Hⁿ is much larger than the admissible set for Rⁿ (which corresponds to the lower edge of the triangleTn). This is due to large scale dispersive effects in negative curvature.

Actually it could be even larger if the region_p² +ⁿ_q <ⁿ₂was not excluded for purely local reasons. This happens for dispersive equations on homogeneous trees and will be discussed in another paper.