Nonlinear Schrödinger equation on real hyperbolic spaces

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spaces

Jean-Philippe Anker, Vittoria Pierfelice

To cite this version:

Jean-Philippe Anker, Vittoria Pierfelice. Nonlinear Schrödinger equation on real hyperbolic spaces.

Annales de l’Institut Henri Poincaré (C) Non Linear Analysis, Elsevier, 2009, 26 (5), pp.1853-1869.

�10.1016/j.anihpc.2009.01.009�. �hal-00212336v2�

hal-00212336, version 2 - 5 Mar 2008

JEAN–PHILIPPE ANKER & VITTORIA PIERFELICE

Abstract. We consider the Schr¨odinger equation with no radial assumption on real hyperbolic spaces. We obtain 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 prove L² and H¹ global 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, if F is gauge invariant,L² charge is conserved and hence, as in the Euclidean case, it is possible to extend localL² solutions to global ones. The corresponding argument inH¹ requires 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 [4], for small radialL² data or for large radialH¹ data.

The second important application of our global Strichartz estimates is scattering for NLS both inL²and inH¹, with no radial or gauge invariance assumption. Notice that, in the Euclidean case, this is only possible for the critical powerγ= 1+_n⁴ and can be false for subcritical powers while, on hyperbolic spaces, global existence and scattering of small L² solutions holds for all powers 1< γ≤1 +⁴_n. If we restrict to defocusing nonlinearitiesF, we can extend theH¹ scattering results of [4] 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.

1. Introduction

The nonlinear Schr¨odinger equation (NLS) in Euclidean space Rⁿ (1)

( i ∂tu(t, x) + ∆xu(t, x) =F(u(t, x)) u(0, x) = f(x)

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

An essential tool in the study of (1) is the dispersive estimate ke^it∆fkL^∞(Rⁿ) ≤ C |t|⁻ⁿ² kfkL¹(Rⁿ)

1

for the linear homogeneous Cauchy problem (2)

(i ∂tu(t, x) + ∆xu(t, x) = 0, u(0, x) =f(x).

This estimate is classical and follows directly from the representation formula for the fundamental solution. A well known procedure (introduced by Kato [15], Ginibre &

Velo [9], and perfected by Keel & Tao [16]) then leads to theStrichartz estimates (3) kuk_L^p_(I;L^q₍Rⁿ))≤Ckfk_L²₍Rn)+CkFk_Lp˜′

(I;L^q˜′(Rⁿ))

for the linear inhomogeneous Cauchy problem

( i ∂tu(t, x) + ∆xu(t, x) =F(t, x), u(0, x) = f(x).

The estimates (3) hold for any bounded or unbounded time interval I ⊆ R and for all pairs (p, q),(˜p,q)˜ ∈[2,∞]×[2,∞) satisfying the admissibility condition

(4) _n^{2 1}_p = ¹₂ − ¹_q.

Notice that both endpoints (p, q) = (∞,2) and (p, q) = (2,_n²₋ⁿ₂) are included in dimension n≥3 while only the first one is included in dimension n= 2.

The question of well–posedness for the nonlinear Cauchy problem (1) is well under- stood, at least for a power nonlinearity F(u) = ± |u|^γ or F(u) = ±u|u|^γ−1 and for suitable ranges of the exponent γ >1 . Here is a brief account of the classical theory. In the model caseF(u) =|u|^γ, we have

• local well–posedness in L² in the subcritical case γ <1+⁴_n;

• global well–posedness in L² in the critical case γ= 1+_n⁴ for small data ;

• local well–posedness inH¹ in the subcritical case γ <1+_n⁴₋₂ for small data ;

• global well–posedness in H¹ in the critical case γ= 1+_n−⁴₂ for small data.

Notice that the value of the critical exponent depends on the dimension n. On the other hand, in the model case F(u) =u|u|^γ−1, the equation (1) is gauge invariant and defocusing, which impliesL² andH¹ conservation laws. Thus, in addition to the previous results, we have

• global well–posedness in L² in the subcritical case γ <1+_n⁴;

• global well–posedness in H¹ in the subcritical case γ <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 & Zhang [19], . . . ).

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

As a byproduct, this method shows that solutionsu(t, x) to (1) are small in a suitable L^q_x sense as t → ±∞. Hence, asymptotically, the contribution of the nonlinearity is dominated by the linear part and thenonlinear equation (1) becomes close to the linear equation (2). This basic observation is at the origin of scattering theory for NLS. By

L²scattering we mean that, for every global solution u(t, x)∈C(R, L²(Hⁿ)), there exist scattering data u_±∈L²(Hⁿ) such that

ku(t, x)−e^it∆xu_±(x)kL²_x → 0 as t→ ±∞. The definition of H¹scattering is analogous.

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

• scattering in L² holds in the critical case γ= 1+_n⁴ for small data ;

• scattering in H¹ holds for 1+⁴_n< γ <1+_n−⁴₂ ;

• scattering in H¹ fails for 1< γ≤1+_n².

This paper is a contribution to the study of Strichartz estimates and NLS on a manifold M. Several results have been obtained for this problem and quite general classes of manifolds. The geometry of M plays 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, G´erard & Tzvetkov [6] after earlier results by Bourgain [5] on the torus. In general one obtains Strichartz estimates

ke^it∆fkL^p(I;L^q(M)) ≤C(I)kfk_H^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 on Rⁿ. Let us mention in particular the local well–

posedness theory in H^s(Tⁿ) developed by Bourgain in the early nineties, extended ten years later to general compact manifolds by Burq, G´erard & Tzvetkov, and improved in some special cases such as spheres Sⁿ [6] or 4–dimensional compact manifolds [8].

In this paper we shall restrict our attention to real hyperbolic spaces M=Hⁿ of di- mension n≥2. Actually our results extend straightforwardly to all hyperbolic spaces i.e. Riemannian symmetric spaces of noncompact type and rank one (they extend fur- thermore to Damek–Ricci spaces and this will be the subject of a forthcoming work).

Consider the following linear Cauchy problem on Hⁿ: (5)

( i ∂tu(t, x) + ∆xu(t, x) =F(t, x), u(0, x) = f(x).

On one hand, Banica [3] (see also [17]) obtained the following weighted dispersive esti- mate, for radial solutions to the homogeneous equation (5) in dimension n≥3 :

w(x)|u(t, x)| ≤ C

|t|⁻ⁿ²+|t|⁻²³ Z

Hⁿ

|f(y)|w(y)⁻¹dy .

Here w(x) = ^sinhr_r , where r denotes the geodesic distance from x 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 dimension n≥3 : kw(x)^21−1q u(t, x)k

L^p_tL^qx ≤ C kf(x)k

L²_x + Ckw(x)^1q˜−12 F(t, x)k_Lp˜′ tL^qx^˜′ . Here w(r) = ^sinh_r ^rn−1

is the jacobian of the exponential map and (¹_p,¹_q), (¹_p˜,¹_˜q) belong to the interval In =

(¹_p,¹_q) ∈ 0,¹₂

× 0,¹₂ ²_n¹_p = ¹₂−¹_q . 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 & Staffilani [4].

Our first main result is the following dispersive estimate (Theorem 3.4), which holds for general functions (no radial assumption) in dimension n≥2 .

Dispersive estimate. Let q,q˜∈(2,∞]. Then, for 0<|t|<1, we have ku(t, x)k

L^qx ≤ C |t|^{−max{12−1q,12−1q˜}n}kf(x)k_Lq˜′ x

while, for |t| ≥1, we have ku(t, x)k

L^qx ≤ C |t|⁻³² kf(x)k_Lq˜′ x .

If q= ˜q= 2, we have of course L² conservation ku(t, x)k_L²_x=kf(x)k_L²_x for all t∈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 (¹_p,¹_q)and (¹_p˜,¹_q˜) belong to the trian- gle Tn =

(¹_p,¹_q)∈ 0,¹₂

× 0,¹_{2 n}²¹_p≥¹₂−¹_q ∪

(0,¹₂) . Then ku(t, x)k

L^p_tL^qx ≤ C kf(x)k

L²_x + CkF(t, x)k_Lp˜′ tL^˜qx^′ .

Notice that the set Tn of admissible pairs for Hⁿ is much wider than the corresponding setInforRⁿ(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

(6)

(i ∂tu(t, x) + ∆xu(t, x) =F(u(t, x)), u(0, x) = f(x).

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

• F is power–like if there exist constants γ >1 and C≥0 such that (7)

( |F(u)| ≤C|u|^γ,

|F(u)−F(v)| ≤C|u−v|(|u|^γ−1+ |v|^γ−1),

• F is gauge invariant if

(8) Im{uF(u)}= 0,

• F is defocusing if there exists a C¹ function G=G(v)≥0 such that

(9) F(u) = u G^′(|u|²).

Notice that gauge invariance implies L² conservation of charge ormass: Z

Hⁿ

|u(t, x)|²dx = Z

Hⁿ

|f(x)|²dx ∀t , while the defocusing assumption implies H¹ conservation of energy:

Z

Hⁿ

|∇u(t, x)|²dx + Z

Hⁿ

G(|u(t, x)|²)dx = constant.

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

F =±u|u|^γ−1 and the defocusing ones

F = +u|u|^γ−1.

Let us first summarize our well–posedness results (Theorems 4.2 & 4.4).

Well–posedness for NLS. Consider the Cauchy problem (6) with a power--like nonlinearity F of order γ.

• Assume γ≤1+⁴_n. Then the problem is globally well–posed for small L²data. For arbitrary L²data, it is locally well–posed if γ <1+_n⁴ .

• Assume γ≤1+_n−⁴₂. Then the problem is globally well–posed for small H¹data. For arbitrary H¹data, it is locally well–posed if γ <1+_n−⁴₂.

• If F is gauge invariant and γ <1+_n⁴ , the problem is globally well–

posed for arbitrary L²data. If F is defocusing and γ <1+_n−⁴₂ , the problem is globally well–posed for arbitrary H¹data.

Similar results were obtained in [4] for radial functions and nonlinearities F=u|u|^γ−1. As expected, they are better for hyperbolic spaces than for Euclidean spaces. For in- stance, onHⁿwe have global well–posedness for smallL²data, for any power 1< γ≤1+_n⁴, while on Rⁿ we must assume in addition gauge invariance. Of course, under this con- dition, 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 nonlinearity F of order γ.

• Assume γ≤1+_n⁴. Then, for all small data f∈L², the unique global solution u(t, x) has the scattering property : there exist u_±∈L² such that

ku(t, x)−e^it∆xu_±(x)k_L²_x →0 as t→ ±∞.

• Assume γ≤1+_n−⁴₂. Then, for all small dataf∈H¹, the unique global solution u(t, x) has the scattering property : there exist u_±∈H¹ such that

ku(t, x)−e^it∆xu_±(x)kH_x¹ →0 as t→ ±∞.

• Assume γ <1+_n−⁴₂ and the defocusing condition. Then, for all data f∈H¹at t=±∞, the NLS has a unique global solution u(t, x) with the following scattering property :

ku(t, x)−e^it∆xf(x)k_Hx¹ →0 as t→ ±∞.

Notice that onHⁿwe have small data scattering for all powers 1< γ≤1+_n⁴ (respectively 1< γ≤1+_n−⁴₂) . This is in sharp contrast with Rⁿ, where scattering is known to fail for the range 1< γ≤1+_n² .

The results in this paper were presented by the second author at the Convegno 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)Inte- gral Geometry, Harmonic Analysis and Representation Theory(Reykjavik, 15–18 August 2007) and at the DFG–JSPS Joint SeminarInfinite Dimensional Harmonic Analysis IV (Tokyo, 10–14 September 2007).

Last minute news. Ionescu and Staffilani [14] have just obtained closely related results. While we are mostly interested in sharp dispersive and Strichartz estimates, with applications to general nonlinearities and scattering, their main aim is scattering in H¹ and 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 non- compact type, namely real hyperbolic spaces Hⁿ of dimension n≥2. We refer to Helga- son’s books ([10], [11], [12]) for their structure, geometric properties, and for harmonic analysis on these spaces. Recall that Hⁿ can be realized as the upper sheet

(x²₀−x²₁− · · · −x²_n= 1, x0 ≥1,

of hyperboloid in R¹⁺ⁿ, equipped with the Riemannian metric dℓ² =−dx²₀ +dx²₁+ . . . +dx²_n,

or as the homogeneous space G/K, where G = SO(1, n)⁰ and K = SO(n). In geodesic polar coordinates, the Riemannian metric is given by

dℓ² =dr²+ (sinhr)²dℓS²ⁿ⁻¹, the Riemannian volume by

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

∆ = ∆Hⁿ=∂_r²+ (n−1) cothr ∂r+ (sinhr)⁻²∆Sⁿ⁻¹.

Inhomogeneous Sobolev spaces on Hⁿ (and on more general manifolds) are defined by H^s,q(Hⁿ) = (I−∆)^−s2L^q(Hⁿ) ( 1< q <∞, s∈R).

Using L^q spectral analysis (see for instance [1]), they can be also defined as well by H^s,q(Hⁿ) = (−∆)^−s2 L^q(Hⁿ).

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

W^N,q(Hⁿ) = {f∈L^q(Hⁿ)| |∇^jf| ∈L^q(Hⁿ) ∀0≤j≤N},

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

H^s,q(Hⁿ)⊂H^˜s,˜q(Hⁿ) if s−s˜≥n(¹_q−¹_q˜)>0. 3. Dispersive and Strichartz estimates on Hⁿ

Consider first the homogeneous linear Schr¨odinger equation on the hyperbolic space Hⁿ of dimension n≥2 :

(i ∂tu(t, x) + ∆xu(t, x) = 0, u(0, x) =f(x),

whose solution is given by

u(t, x) = e^it∆f(x) =f∗st(x) = Z

Hn

st(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−21)2t} Z +∞

−∞

e^−itλ2ϕ_λ(r) dλ

|c(λ)|² .

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

(10) st(r) = const.(it)⁻¹² e^{−i(n−21)2t} −_sinh¹ _r∂r^∂n−₂¹ e^4irt2 .

Here (it)⁻¹² =e^{−iπ4sign(t)}|t|⁻²¹ and, in the even dimensional case, the fractional derivative reads

(11) −_sinh¹ _r∂r^∂n−₂¹

e^i4rt2 = ^√1_π Z +∞

|r|

−_sinh¹ _s∂s^∂n₂

e^{4ist2 √}_cosh^sinh_s^{s ds}

−coshr.

Proposition 3.1. There exists a constant C >0 such that the following pointwise kernel estimate holds, for every t∈R^∗ and r≥0:

(12) |s_t(r)| ≤ C

(|t|^−3/2(1 +r)e^−n−21r if |t| ≥1+r ,

|t|^−n/2(1 +r)ⁿ⁻²¹ e^−n−21r if |t| ≤1+r .

Remark 3.2. In dimension n= 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 :

|st(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 expo- nential map j(r) = ^sinh_r ^rn−1

≍ _1+r^er n−1

.

Proof. We shall assume t >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. Set m=ⁿ⁻₂¹ and let us expand

(13) −_sinh¹ _r∂r^∂ m

e^4irt2 = e^4irt2 Pm

j=1 t^−jfj(r).

The functionsfj(r) involved are linear combinations of products ϕℓ1(r)· · · ϕℓj(r) , where ϕℓ(r) = _sinh¹ _r∂r^∂ℓ

r²

and ℓ1, . . . , ℓj∈N∗ are such that ℓ1+· · ·+ℓj =m. Using the elementary global estimate ϕℓ(r) = O (1+r)e^−ℓr

, we are lead to the conclusion :

|s_t(r)| . t⁻¹² Pm j=1 1+r

t

j

e^−mr ≍ t⁻¹² _1+r

t + ^1+r_t m

e^−mr.

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

(14) |s_t(r)| . t⁻¹² Z +∞

r

_1+s

t + ^1+s_t m

e^−ms√_cosh^sinh_s^{s ds}

−coshr.

Here and throughout the proof, we make repeated use of the following elementary esti- mates :

(15) sinhs≍ _1+s^s e^s,

and

(16)

coshs−coshr = 2 sinh^s−₂^r sinh^s+r₂

≍ _1+s^s−₋^r_re^s−2r_1+s+r^s+r e^s+r2

≍ _1+s^s−₋^r_{r 1+s}^s e^s or

( s²−r²

1+r e^r if r≤s≤r+1, e^s if s≥r+1. Thus (14) becomes

(17) |st(r)| . t⁻¹² Z +∞

r

_1+s

t + ^1+s_t m

e^{−(m−12)s√√1+s}_s−⁻_r^{r √√}_1+s^s ds .

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

√r+u

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

(18) |s_t(r)| . t⁻¹² _1+r

t + ^1+r_t m

e^−(m−12)r. This allows us to conclude that

(19) |st(r)| ≤ C t^{−12 1+r}_t e^−(m−12)r

when t≥1+r. If t≤1+r, the polynomial power m in (18) must be brought down to m− ¹₂. For this purpose, let us rewrite more carefully the expansion

−_sinh¹ _s∂s^∂m

e^i4st2 = P

0<j<m

t^−jfj(s)e^i4st2 + t^−(m−1) −₂ⁱ_sinhs^s m−1

−_sinh¹ _s∂s^∂ e^4ist2. The contribution of the sum (which doesn’t occur in dimension n= 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) = Z +∞

r s sinhs

m−1 ∂

∂se^4ist2 _ds

√coshs−coshr

is O t⁻¹²(1+r)^m−12e^−(m−12)r

when t≤1+r. Let us split I(t, r) = I1(t, r) + I2(t, r) + I3(t, r) according to

Z +∞ r

=

Z ^√r²+t r

+ Z r+1

√r²+t

+ Z +∞

r+1

.

The first integral is easy to handle. We simply differentiate _∂s^∂ e^i4st2 = ₂^is_te^4ist2 and use the elementary estimates (15), (16) together with the fact that s∈[r, r+1 ] . As a result,

|I1(t, r)| . t⁻¹(1+r)^m−12 e^−(m−12)r

Z ^√r²+t r

√s ds

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

I2(t, r) +I3(t, r) = e^4ist2 _sinhs^s m−1√ 1 coshs−coshr

n

s=r+1 s=√

r²+t+

s=+∞ s=r+1

o + n Z ^r+1

√r²+t

+ Z +∞

r+1

oe^4ist2 ×

×

(m−1) _sinh^s _sm−2 scoths−1

sinhs (coshs−coshr)⁻¹² + + ¹_{2 sinhs}^s m−1

(coshs−coshr)⁻³² sinhs ds . The boundary terms are estimated as I1(t, r) :

s sinhs

m−1√ 1 coshs−coshr

s=√

r²+t ≍ t⁻¹² (1+r)^m−12 e^−(m−12)r.

The integral terms are bounded by

(20)

(1+r)^m−2e^−(m−12)r Z r+1

√r²+t

_1+r

s²−r²

¹₂

+ _s^1+r2−r²

³₂ s ds +

Z +∞ r+1

(1+s)^m−1e^−(m−12)sds .

Here we have used (15), (16) and the elementary estimate ^scoth_sinh^s_s⁻¹ ≍ s e^−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 Z r+1

√r²+t

(s²−r²)⁻³² s ds = (1+r)³²n

−(s²−r²)^−12s=r+1

s=√ r²+t

o . t⁻¹²(1+r)²³.

The second integral in (20) is estimated as (17) : Z +∞

r+1

(1+s)^m−1e^−(m−12)sds = Z +∞

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−21e^−(m−12)r. Thus we have shown that

|s_t(r)| . t^−m(1 +r)^m−12e^−(m−12)r

when 0< t≤1+r.

Corollary 3.3. Let 2< q <∞ and 1≤α≤ ∞. Then there exists a constant C >0 such that the following kernel estimate holds, with respect to Lorentz norms :

(21) kstkL^q,α ≤ C

(|t|^−n/2 if 0<|t| ≤1,

|t|^−3/2 if |t| ≥1.

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

kfkL^q,α =





h Z ^+∞

0

s^1/qf^∗(s) ^{α ds}_s i1/α

if 1≤α <∞, sup_s>0 s^1/qf^∗(s) if α=∞,

where f^∗ denotes the decreasing rearrangement of f. In particular, if f is a positive radial decreasing function on Hⁿ, then f^∗=f◦V⁻¹, where

V(r) = C Z r

0

(sinhs)ⁿ⁻¹ds ≍

( rⁿ as r→0 e^(n−1)r as r→+∞

is the volume of a ball of radius r >0 in Hⁿ. Hence kfkL^q,α = h Z ^+∞

0

V(r)^1/qf(r) ^{α V}_V^′_(r)^(r)dri1/α

≍ h Z ¹

0

f(r)^αr^{α nq −1}dri1/α

+ h Z ^+∞

1

f(r)^αe^α(n

−1)

q r

dri1/α

if 1≤α <∞ and

kfkL^q,∞ = sup

r>0

V(r)^1/qf(r) ≍ sup

0<r<1

r^nqf(r) + sup

r≥1

e^n−q1rf(r).

The Lorentz norm estimate (21) follows from these considerations and from the pointwise

estimate (12).

Let us turn toL^qmapping properties of the Schr¨odinger propagatore^it∆onHⁿ. Recall that e^it∆ is a one parameter group of unitary operators on L²(Hⁿ).

Theorem 3.4. Let 2< q,q˜≤ ∞. Then there exists a constant C >0 such that the following dispersive estimates hold :

ke^it∆k_Lq˜′

→L^q ≤ C

(|t|^{−max{12−1q,12−1q˜}n} if 0<|t|<1,

|t|⁻³² if |t| ≥1.

Remark 3.5. In the Euclidean setting, small time estimates are similar for q= ˜q, other small time estimates don’t hold, and large time estimates are drastically different.

Proof. These estimates are obtained by interpolation and by using Corollary 3.3. Specif- ically, small time estimates follow from







ke^it∆kL¹→L^q =kstkL^q ≤Cq |t|⁻ⁿ² ∀q >2, ke^it∆k_Lq′

→L^∞=kstkL^q ≤Cq|t|⁻ⁿ² ∀q >2, ke^it∆kL²→L² = 1,

and large time estimates from







ke^it∆k_L¹_→L^q =ks_tk_L^q ≤C_q |t|⁻³² ∀q >2, ke^it∆k_Lq′

→L^∞=kstkL^q ≤Cq |t|⁻³² ∀q >2, ke^it∆k_Lq′

→L^q≤ CqkstkL^q,1 ≤Cq |t|⁻³² ∀q >2.

The key ingredient here is a sharp version of the Kunze–Stein phenomenon, due to Cowling, Meda & 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 constant Cq>0 such that kf∗gk_Lq′,∞≤Cqkfk_Lq′kgk_Lq′ ∀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.

Consider next the inhomogeneous linear Schr¨odinger equation (5) onHⁿ: ( i ∂tu(t, x) + ∆xu(t, x) =F(t, x),

u(0, x) = f(x), whose solution is given by Duhamel’s formula : (22) u(t, x) = e^it∆xf(x)−i

Z t 0

e^i(t−s)∆xF(s, x)ds .

Strichartz estimates onHⁿ involveadmissible pairs of indices (p, q) corresponding to the triangle

(23) Tn = ₁

p,¹_q

∈ 0,¹₂

× 0,¹_{2 n}^{2 1}_p ≥ ¹₂ − ¹_q ∪ 0,¹₂ (see Figure 1).

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

(24) ku(t, x)k

L^p_tL^qx ≤ C

kf(x)k

L²_x+kF(t, x)k_Lp˜′ tL^qx^˜′ .

Remark 3.7. This result was obtained previously for radial functions in [4], using sharp weighted Strichartz estimates [18] in dimension n≥4, the elementary kernel expression (10)in dimensionn= 3, and specific kernel estimates in dimensionn= 2. Notice that the admissible set forHⁿ is much larger than the admissible set forRⁿ (which corresponds to the lower edge of the triangle Tn). This is due to large scale dispersive effects in negative curvature. Actually it could be even larger if the region _n²¹_p<¹₂−¹_q was not excluded for purely local reasons. This happens for dispersive equations on homogeneous trees and will be discussed in another paper.

Proof. We resume the standard strategy developed by Kato [15], Ginibre & Velo [9], and Keel & Tao [16]. Consider the operator

T f(t, x) =e^it∆xf(x) and its formal adjoint

T^∗F(x) = Z +∞

−∞

e^−is∆xF(s, x)ds .

The method consists in proving the L^p_t^′L^q_x^′→L^p_tL^q_x boundedness of the operator

(25) T T^∗F(t, x) =

Z +∞

−∞

e^i(t−s)∆xF(s, x)ds and of its truncated version

(26) T Tg^∗F(t, x) =

Z t 0

e^i(t−s)∆xF(s, x)ds ,

for every admissible pair (p, q). The endpoint (¹_p,¹_q) = (0,¹₂) is settled byL² conservation and the endpoint (¹_p,¹_q) = (¹₂,¹₂−_n¹) in dimension n≥3 will be handled at the end. Thus

we are left with the pairs (p, q) such that ¹₂−_n¹<¹_q<¹₂ and (¹₂−¹_q)ⁿ₂≤¹_p≤¹₂. According to the dispersive estimates in Theorem 3.4, the L^p_tL^q_x norms of (25) and (26) are bounded above by

(27) Z

|t−s|≥1

|t−s|⁻³² kF(s, x)k_Lq′ x

L^p_t+

Z

|t−s|≤1

|t−s|^{−(12−1q)n}kF(s, x)k_Lq′ x

L^p_t . On one hand, the convolution kernel |t−s|⁻³² 1l_{{|t−s|≥1}} onRdefines a bounded operator fromL^p_s¹ to L^p_t², for all 1≤p₁≤p₂≤ ∞, in particular from L^p_s^′ toL^p_t, for all 2≤p≤ ∞. On the other hand, the convolution kernel |t−s|^{−(12−1q)n}1l_{{|t−s|≤1}} defines a bounded operator from L^p_s¹ to L^p_t², for¹all 1< p1, p2<∞ such that 0≤_p¹

1−_p¹

2≤1−(¹₂−¹_q)n, in particular fromL^p_s^′ to L^p_t, for all 2≤p <∞ such that ¹_p≥(¹₂−¹_q)ⁿ₂. Consider eventually the endpoint (¹_p,¹_q) = (¹₂,¹₂−¹_n) in dimension n ≥3. The integrals over |t−s| ≥1 are estimated as before. The integrals over |t−s| ≤1 are handled as in [16], except that only small dyadic intervals are involved. Indices are finally decoupled, using the T T^∗

argument.

4. Well–posedness results for NLS on Hⁿ

Strichartz estimates for inhomogeneous linear equations are used to prove local and global well–posedness results for nonlinear perturbations. We present here a few results in this direction for the Schr¨odinger equation (6)

(i ∂tu(t, x) + ∆xu(t, x) =F(u(t, x)), u(0, x) = f(x),

onM=Hⁿ, with a power–like nonlinearity as in (7) :

|F(u)| ≤C|u|^γ, |F(u)−F(v)| ≤C|u−v|(|u|^γ−1+|v|^γ−1). Let us recall the definition of well–posedness.

Definition 4.1. Let s∈R. The NLS equation (6)is locally well–posed in H^s(M)if, for any bounded subsetB ofH^s(M), there exist T >0and a Banach space XT, continuously embedded into C([−T,+T];H^s(M)), such that

• for any Cauchy data f(x)∈B, (6) has a unique solution u(t, x)∈XT ;

• the map f(x)7→u(t, x) is continuous from B to XT.

The equation is globally well–posed if these properties hold with T=∞.

In the Euclidean setting, γ = 1 +_n⁴ is known to be the critical exponent for well–

posedness in L²(Rⁿ). Specifically, the NLS (1) has a unique local solution for arbitrary data f∈L² provided γ <1+_n⁴; in general this solution cannot be extended to a global one ; this is possible under the assumption (8) of gauge invariance. In the critical case γ= 1+⁴_n, the NLS (1) is globally well–posed forL² data satisfying a smallness condition.

On the other hand, γ= 1+_n−⁴₂ is known to be the critical exponent for well–posedness

1Actually for all 1≤p₁, p₂≤ ∞ such that 0≤_p¹

1−_p¹

2≤1−(¹₂−¹_q)n, except for the dual endpoints (_p¹

1, ¹

p₂) = (1,(¹₂−¹_q)ⁿ₂) and (_p¹

1, ¹

p₂) = (1−(₂¹−¹_q)ⁿ₂,0).