Schrödinger equations on locally symmetric spaces

Abstract

We prove dispersive and Strichartz estimates for Schrödinger equations on a class of rank one locally symmetric spaces. We present Strichartz estimates applications to the well-posedness and scattering for nonlinear Schrödinger equations.

1 Introduction and statement of the results

Let M be a Riemannian manifold and denote by $\Delta$ its Laplace-Beltrami operator. The nonlinear Schrödinger equation (NLS) on M

$$\begin{aligned} \left\{ \begin{array}{l} i\partial _{t}u\left( t,x\right) +\Delta _{x}u\left( t,x\right) =F\left( u\left( t,x\right) \right) , \\ u\left( 0,x\right) =f\left( x\right) , \end{array} \right. \end{aligned}$$

(1)

has been extensively studied the last thirty years. Its study relies on precise estimates of the kernel $s_{t}$ of the Schrödinger operator $e^{it\Delta }$, the heat kernel of pure imaginary time. The estimates of $s_{t}$ allow us to obtain dispersive estimates of the operator $e^{it\Delta }$ of the form

$$\begin{aligned} \left\| e^{it\Delta }\right\| _{L^{\widetilde{q}^{\prime }}\left( M\right) \rightarrow L^{q}\left( M\right) }\le c\psi \left( t\right) ,\, \quad t\in \mathbb {R}, \end{aligned}$$

(2)

for all $q,\widetilde{q}\in \left( 2,\infty \right]$, where $\psi$ is a positive function and $\widetilde{q}^{\prime }$ is the conjugate of $\widetilde{q}$.

Dispersive estimates of $e^{it\Delta }$ as above, allow us to obtain Strichartz estimates of the solutions $u\left( t,x\right)$ of (1):

$$\begin{aligned} \left\| u\right\| _{L^{p}\left( \mathbb {R};L^{q}\left( M\right) \right) }\le c\{ \left\| f\right\| _{L^{2}\left( M\right) }+\left\| F\right\| _{L^{\widetilde{p}^{\prime }}( \mathbb {R};L^{ \widetilde{q}^{\prime }}\left( M\right) ) }\} , \end{aligned}$$

(3)

for all pairs $( \frac{1}{p},\frac{1}{q})$ and $( \frac{1}{ \widetilde{p}},\frac{1}{\widetilde{q}})$ which lie in a certain triangle.

Strichartz estimates have applications to well-posedness and scattering theory for the NLS equation.

In the case of $\mathbb {R}^{n}$, the first such estimate was obtained by Strichartz himself [41] in a special case. Then, Ginibre and Velo [19] obtained the complete range of estimates except the case of endpoints which were proved by Keel and Tao [30].

In view of the important applications to nonlinear problems, many attempts have been made to study the dispersive properties for the corresponding equations on various Riemannian manifolds (see e.g., [7, 8, 10,11,12,13, 22, 27, 38, 39] and references within). More precisely, dispersive and Strichartz estimates for the Schrödinger equation on real hyperbolic spaces have been stated by Banica [10], Pierfelice [38, 39], Banica and al. [11], Anker and Pierfelice [7], Ionescu and Staffilani [27]. In a recent paper Anker et al. [8] treat NLS in the context of Damek–Ricci spaces, which include all rank one symmetric spaces of noncompact type.

In the present work we treat NLS equations on a class of locally symmetric spaces.

1.1 The Schrödinger class of locally symmetric spaces

In this section we describe the Schrödinger classes of locally symmetric spaces on which we shall treat NLS equations.

For the statement of the results we need to introduce some notation. For more details see Sect. 2. Let G be a semisimple Lie group, connected, noncompact, with finite center and let K be a maximal compact subgroup of G. We denote by X the Riemannian symmetric space G / K. In the sequel we assume that $\dim X=n$.

Denote by $\mathfrak {g}$ and $\mathfrak {k}$ the Lie algebras of G and K. Let also $\mathfrak {p}$ be the subspace of $\mathfrak {g}$ which is orthogonal to $\mathfrak {k}$ with respect to the Killing form. The Killing form induces a K-invariant scalar product on $\mathfrak {p}$ and hence a G-invariant metric on G / K. Denote by $\Delta$ the Laplace–Beltrami operator on X and by $d\left( .,.\right)$ the Riemannian distance and by dx the associated measure on X.

Let $\Gamma$ be a discrete torsion free subgroup of G. Then the locally symmetric space $M=\Gamma \backslash X$, equipped with the projection of the canonical Riemannian structure of X, becomes a Riemannian manifold. We denote also by $\Delta$ the Laplacian on M, by d(., .) the Riemannian distance and by dx the associated measure on M. Note that in general, locally symmetric spaces have not bounded geometry since the injectivity radius of M is not in general strictly positive [18]. In the present work we also assume that $\Gamma$ is non elementary i.e. its limit set is infinite.

Fix $\mathfrak {a}$ a maximal abelian subspace of $\mathfrak {p}$ and denote by $\mathfrak {a}^{*}$ the real dual of $\mathfrak {a}$. If $\dim \mathfrak {a}=1$, we say that X has rank one. Let $\Sigma \subset \mathfrak {a}^{*}$, be the root system of ($\mathfrak {g},\mathfrak {a}$). Denote by W the Weyl group associated to $\Sigma$ and choose a set $\Sigma ^{+}$ of positive roots. Denote by $\rho$ the half sum of positive roots counted with their multiplicities. Let $\mathfrak {a}^{+}\subset \mathfrak {a}$ be the corresponding positive Weyl chamber and let $\overline{ \mathfrak {a}_{+}}$ be its closure.

Denote by $s_{t}$ the fundamental solution of the Schrödinger equation on the symmetric space X:

$$\begin{aligned} -i\partial _{t}s_{t}\left( x,y\right) =\Delta s_{t}\left( x,y\right) ,\quad \, t\in \mathbb {R},\quad \, x,y\in X. \end{aligned}$$

Then $s_{t}$ is a K-bi-invariant function and the Schrödinger operator $S_{t}=e^{it\Delta }$ on X is defined as a convolution operator:

$$\begin{aligned} S_{t}f(x)=\int _{G}f(y)s_{t}(y^{-1}x)dy=\left( f*s_{t}\right) \left( x\right) ,\quad f\in C_{0}^{\infty }(X). \end{aligned}$$

(4)

Using that $s_{t}$ is K-bi-invariant, we deduce that if $f\in C_{0}^{\infty }(M)$, then $S_{t}f$ is right K-invariant and left $\Gamma$-invariant i.e. a function on the locally symmetric space M. Thus the Schrödinger operator $\widehat{S}_{t}$ on M is also defined by formula (4).

The first ingredient for the proof of the dispersive estimate (2 ) are precise estimates of the kernel $s_{t}$. In the context of rank one symmetric spaces we deal with in the present work they are obtained in [8, Section 3].

The second ingredient is the following analogue of Kunze–Stein phenomenon on locally symmetric spaces, proved in [34], and presented in detail in Sect. 3. Let $C_{\rho }$ the convex body in $\mathfrak {a} ^{*}$ generated by the vectors $\left\{ w\rho ;w\in W\right\} ,$ where W is the Weyl group. Let also $\lambda _{0}$ be the bottom of the $L^{2}$-spectrum of $-\Delta$ on M. Then in [34] it is proved that there exists a vector $\eta _{\Gamma }\in C_{\rho }\cap S( 0,( \rho ^{2}-\lambda _{0}) ^{1/2})$, $S\left( 0,r\right)$ is the Euclidean sphere of $\mathfrak {a}^{*}$, such that for all $p\in \left( 1,\infty \right)$ and for every K-bi-invariant function $\kappa$, the convolution operator $*\left| \kappa \right|$ with kernel $\left| \kappa \right|$ satisfies the estimate

$$\begin{aligned} \left\| *\left| \kappa \right| \right\| _{L^{p}(M)\rightarrow L^{p}(M)}\le \int _{G}\left| \kappa \left( g\right) \right| \varphi _{-i\eta _{\Gamma }}\left( g\right) ^{s\left( p\right) }dg, \end{aligned}$$

(5)

where $\varphi _{\lambda }$ is the spherical function with index $\lambda$ and

$$\begin{aligned} s\left( p\right) =2\min ( \left( 1/p\right) ,( 1/p^{\prime }) ). \end{aligned}$$

The class of locally symmetric spaces where (5) is valid contains the following classes:

(i) G possesses Kazhdan’s property (T).

Note that G has property (T) iff G has no simple factors locally isomorphic to SO(n, 1) or SU(n, 1) [21, ch. 2].

Recall that non-compact rank one symmetric spaces are the real, complex and quaternionic hyperbolic spaces, denoted $H^{n}\left( \mathbb {R}\right)$, $H^{n}\left( \mathbb {C}\right)$, $H^{n}\left( \mathbb {H}\right)$ respectively, and the octonionic hyperbolic plane $H^{2}\left( \mathbb {O} \right)$. They have the following representation as quotients:

$$\begin{aligned} H^{n}\left( \mathbb {R}\right)= & {} SO(n,1)/SO(n),\, H^{n}\left( \mathbb { C}\right) =SU(n,1)/SU(n), \\ H^{n}\left( \mathbb {H}\right)= & {} Sp(n,1)/Sp(n)\text {, and }H^{2}\left( \mathbb {O}\right) =F_{4}^{-20}/Spin\left( 9\right) \mathbf {.} \end{aligned}$$

So, (5) is satisfied in the quotients $\Gamma \backslash H^{n}\left( \mathbb {H}\right)$ and $\Gamma \backslash H^{2}\left( \mathbb {O} \right)$ for all discrete subgroups $\Gamma$ of $Sp\left( n,1\right)$ and $F_{4}^{-20}$ respectively.

(ii) $\Gamma$ is amenable.

In particular, (5) is satisfied in the infinite volume quotients $\Gamma \backslash H^{n}\left( \mathbb {R}\right)$, $\Gamma \backslash H^{n}\left( \mathbb {C}\right)$ with $\Gamma$ amenable.

The third ingredient are norm estimates of the kernel $\widehat{s}_{t}$ of the Schrödinger kernel on M which is given by

$$\begin{aligned} \widehat{s}_{t}(x,y)=\sum _{\gamma \in \Gamma }s_{t}(x,\gamma y). \end{aligned}$$

(6)

Recall that the critical exponent $\delta \left( \Gamma \right)$ is defined by

$$\begin{aligned} \delta \left( \Gamma \right) =\inf \left\{ \alpha >0:P_{\alpha }\left( x,y\right) <\infty \right\} , \end{aligned}$$

where

$$\begin{aligned} P_{\alpha }\left( x,y\right) =\sum _{\gamma \in \Gamma }e^{-\alpha d\left( x,\gamma y\right) } \end{aligned}$$

are the Poincaré series [43]. Note that $\delta \left( \Gamma \right) \le 2\rho$ and that $P_{\alpha }\left( x,y\right)$ converges for $\alpha >\delta \left( \Gamma \right)$ and diverges for $\alpha <\delta \left( \Gamma \right)$. If $P_{\alpha }\left( x,y\right)$ diverges for $\alpha =\delta \left( \Gamma \right)$, we say that $\Gamma$ is of divergent type.

In Sect. 4, we show that the series (6) converges when $\delta \left( \Gamma \right) <\rho$. In this case the Schrödinger operator $\widehat{S}_{t}$ on M is an integral operator with kernel $\widehat{s}_{t}(x,y)$:

$$\begin{aligned} \widehat{S}_{t}f(x)=\int _{M}f(y)\widehat{s}_{t}(x,y)dy. \end{aligned}$$

(7)

The last ingredient we shall use are uniform asymptotics of the counting function $N_{\Gamma }$ of $\Gamma$ which is given by

$$\begin{aligned} N_{\Gamma }\left( x,y,R\right) =\#\left\{ \gamma \in \Gamma :d\left( x,\gamma y\right) \le R\right\} ,\quad \, x,y\in X,\quad R>0, \end{aligned}$$

where $\#\left( A\right)$ is the cardinal of the set A.

The asymptotic properties of the counting function in various geometric contexts has been a subject of many investigations since Margulis [36], (see [9, 14, 40, 42] and the references within). In [42], they are obtained in the context of Hadamard manifolds with pinched negative sectional curvature and in [40] in the more general context of $CAT\left( -1\right)$ spaces. Note that a rank one symmetric space has pinched negative sectional curvature and consequently it is contained in the above mentioned classes of spaces.

In [40, 42] it is proved, under some precise conditions on $\Gamma$, that $N_{\Gamma }$ satisfies the following uniform asymptotics: there is a constant $C>0$, such that for all $x,y\in X$,

$$\begin{aligned} \lim _{R\rightarrow \infty }\frac{N_{\Gamma }\left( x,y,R\right) }{e^{\delta \left( \Gamma \right) R}}=C. \end{aligned}$$

(8)

More precisely, in [42, Theorem 6.2.5], (8) is proved in the context of Hadamard manifolds with pinched sectional negative curvature and for $\Gamma$ convex co-compact i.e. when the quotient by $\Gamma$ of the convex hall of the limit set $L\left( \Gamma \right)$ of $\Gamma$ is compact. Note that if X is real rank 1 symmetric space, then convex co-compact groups is a rich class. For example, it contains uniform lattices and Schottky groups with no parabolic element (see the review article [28]).

In [40, Chapitre 4, Corollaire 2], (8) is proved, in the context of $CAT\left( -1\right)$ spaces, under the following assumptions:

• (i) the length spectrum of $\Gamma$ is non-arithmetic, i.e. the set of lengths of all closed geodesics of $\Gamma \backslash X$ is not contained in a discrete subgroup of $\mathbb {R}$, and

• (ii) the Bowen–Margulis–Sullivan measure associated to $\Gamma$ is finite.

This class of groups $\Gamma$, apart from convex co-compact groups, it contains also all geometrically finite groups of divergence type with at least one parabolic element. Note that if $\Gamma$ is geometrically finite, then the quotient $\Gamma \backslash X$ is composed of a compact part and of finitely many cusps and ends. Note that a convex co-compact group is always geometrically finite and that a geometrically finite group is convex co-compact iff $\Gamma \backslash X$ is cusp free. In Sect. 2.2 we give a detailed description of the class of locally symmetric spaces where the assumption (8) holds true.

Definition 1

We say that a rank one locally symmetric space $M=\Gamma \backslash G/K$ belongs in the class (S) if

• (i) for every K-bi-invariant function $\kappa$ the estimate (5) is satisfied,

• (ii) $\delta \left( \Gamma \right) <\rho$, and

• (iii) the counting function $N_{\Gamma }\left( x,y,R\right)$ satisfies (8).

Note that if $\delta \left( \Gamma \right) <\rho$, then $\lambda _{0}=\rho ^{2}$ [32, 33]. So, if $M\in \left( S\right)$, then the vector $\eta _{\Gamma }$ appearing in (5) equals 0. Note also that if $\hbox {vol}(M)<\infty$, i.e., if M is a lattice, then $\lambda _{0}=0$. So, condition (ii) of Definition 1 implies that if $M\in \left( S\right)$, then $\hbox {vol}(M)=\infty$.

If $M\in \left( S\right)$, then using the expression (7) and under the condition that $N_{\Gamma }\left( x,y,R\right)$ satisfies (8), we deduce in Sect. 4, estimates of the norm $\Vert \widehat{s} _{t}(x,.)\Vert _{L^{q}(M)}$, $q>2$, from the corresponding ones on the symmetric space $X=G/K$. This is the crucial step for the proof of the dispersive estimate (2) of the operator $\widehat{S}_{t}$ for $M\in \left( S\right)$.

Finally, it is important to note that if $M\in \left( S\right)$, then we are able to prove the same results as in the case of hyperbolic spaces [7].

1.2 Dispersive and Strichartz estimates on locally symmetric spaces

The main result of the present paper is the following dispersive estimate.

Theorem 2

Assume that $M\in (S)$. Then for all $q,\tilde{q}\in \left( 2,\infty \right]$, there is a constant $c>0$ such that

$$\begin{aligned} \Vert \widehat{S}_{t}\Vert _{L^{\tilde{q}^{\prime }}(M)\rightarrow L^{q}(M)}\le c|t|^{-n\max {\{}\left( 1/2\right) -\left( 1/q\right) {,\left( 1/2\right) -\left( 1/\widetilde{q}\right) \}}}, \quad \left| t\right| <1, \end{aligned}$$

(9)

and

$$\begin{aligned} \Vert \widehat{S}_{t}\Vert _{L^{\tilde{q}^{\prime }}(M)\rightarrow L^{q}(M)}\le c|t|^{-3/2},\quad \, \left| t\right| \ge 1. \end{aligned}$$

(10)

Consider the following Cauchy problem for the linear inhomogeneous Schrödinger equation on M:

$$\begin{aligned} \left\{ \begin{array}{l} i\partial _{t}u(t,x)+\Delta u(t,x)=F(t,x), \\ u(0,x)=f(x). \end{array} \right. \end{aligned}$$

(11)

Combining the above dispersive estimate with the classical $TT^{*}$ method developed by Kato [29], Ginibre and Velo [19] and Keel and Tao [30], we obtain Strichartz estimates for the solutions u(t, x) of (11). Consider the triangle

$$\begin{aligned} T_{n}=\left\{ \left( \tfrac{1}{p},\tfrac{1}{q}\right) \in \left( 0,\tfrac{1}{ 2}\right] \times \left( 0,\tfrac{1}{2}\right) :\tfrac{2}{p}+\tfrac{n}{q}\ge \tfrac{n}{2}\right\} \cup \left\{ \left( 0,\tfrac{1}{2}\right) \right\} . \end{aligned}$$

(12)

We say that the pair $\left( p,q\right)$ is admissible if $\left( \tfrac{1}{ p},\tfrac{1}{q}\right) \in T_{n}$.

Theorem 3

Assume that $M\in (S)$. Then the solutions $u\left( t,x\right)$ of the Cauchy problem (11) satisfy the Strichartz estimate

$$\begin{aligned} \Vert u\Vert _{L_{t}^{p}L_{x}^{q}}\le c\left\{ \Vert f\Vert _{L_{x}^{2}}+\Vert F\Vert _{L_{t}^{\tilde{p}^{\prime }}L_{x}^{\tilde{q} ^{\prime }}}\right\} , \end{aligned}$$

(13)

for all admissible pairs $\left( p,q\right)$ and $\left( \tilde{p},\tilde{q} \right)$ corresponding to the triangle $T_{n}$.

As it is noticed in [7, 8] the above set $T_{n}$ of admissible pairs is much wider that the admissible set in the case of $\mathbb {R}^{n}$ which is just the lower edge of the triangle. This phenomenon was already observed for hyperbolic spaces in [11, 27].

The paper is organized as follows. In Sect. 2 we describe the geometric context of symmetric spaces and we present the spherical Fourier transform and the uniform asymptotics of the counting function. In Sect. 3 we present an analogue of Kunze–Stein phenomenon for convolution operators on a class of locally symmetric spaces proved in [34] by Lohoué and Marias. In Sect. 4 we study the Schrödinger operator on M and we prove norm estimates for its kernel. In Sect. 5 we prove dispersive estimates for the Schrödinger operator on M and we give the proofs of Theorems 2 and 3. Finally, we apply Strichartz estimates to study well-posedness and scattering for NLS equations.

2 Preliminaries

In this section we recall some basic facts about symmetric spaces of noncompact type we will use for the proof of our results. For more details see [2, 5, 6, 23, 25, 26, 34].

Let A be the analytic subgroup of G with Lie algebra $\mathfrak {a}$. Let $\mathfrak {a}^{+}\subset \mathfrak {a}$ be a positive Weyl chamber and let $\overline{\mathfrak {a}_{+}}$ be its closure. Put $A^{+}=\exp \mathfrak {a} _{+}$. Its closure in G is$\overline{A^{+}}=\exp \overline{\mathfrak {a} _{+}}$. We have the Cartan decomposition

$$\begin{aligned} G=K(\overline{A^{+}})K=K(\exp \overline{\mathfrak {a}_{+}})K. \end{aligned}$$

Let $k_{1}$, $k_{2}$ and $\exp H$ be the components of $g\in G$ in K and $\exp \mathfrak {a}$ according to the Cartan decomposition. Then g is written as $g=k_{1}\left( \exp H\right) k_{2}$. According to Cartan decomposition, the Haar measure on G is written as

$$\begin{aligned} \int _{G}f\left( g\right) dg=c\int _{K}dk_{1}\int _{\mathfrak {a}^{+}}\delta \left( H\right) dH\int _{K}f\left( k_{1}\left( \exp H\right) k_{2}\right) dk_{2}, \end{aligned}$$

(14)

where

$$\begin{aligned} \delta \left( H\right) =\prod _{\alpha \in \Sigma ^{+}}\sinh ^{m_{\alpha }}\alpha \left( H\right) , \end{aligned}$$

with $m_{\alpha }=\dim \mathfrak {g}_{\alpha }$, and

$$\begin{aligned} \mathfrak {g}_{\alpha }=\left\{ X\in \mathfrak {g}:\left[ H,X\right] =\alpha \left( H\right) \quad X\text { for all }H\in \mathfrak {a}\right\} , \end{aligned}$$

is the root space associated to the root $\alpha \in \Sigma ^{+}$. Note that

$$\begin{aligned} \delta \left( H\right) \le ce^{2\rho \left( H\right) },\quad \, H\in \mathfrak {a}_{+}. \end{aligned}$$

(15)

If G has real rank one, then it is well known [25] that the root system $\Sigma$ is either of the form $\left\{ -\alpha ,\alpha \right\}$ or of the form $\left\{ -\alpha ,-2\alpha ,\alpha ,2\alpha \right\}$. Thus $\rho =\left( m_{\alpha }/2\right) \alpha$ or$\rho =\left( m_{\alpha }/2+m_{2\alpha }\right) \alpha$.

Let $H_{0}$ be the unique element of $\mathfrak {a}$ with the property that $\alpha (H_{0})=1$. Set$a(s)=\exp (sH_{0})$, $s\in \mathbb {R}.$ Then $a: \mathbb {R}\longrightarrow A$ is a diffeomorphism and we identify $A=\exp \mathfrak {a}$ with $\mathbb {R}$ via a. We also normalize the Killing form on $\mathfrak {g}$ such that

$$\begin{aligned} d(a(s).\mathbf {0},\mathbf {0})=|s|, \quad \text { for all }s\in \mathbb {R}, \end{aligned}$$

where $\mathbf {0}=\left\{ K\right\}$ is the origin of X.

In this case we have that $A_{+}=\left\{ a(s):s\ge 0\right\} \simeq \mathbb { R}^{+}$ and any K-bi-invariant function $\kappa$ is identified with the function $\kappa :\mathbb {R}_{+}\longrightarrow \mathbb {C}$, given by $\kappa (s)=\kappa (a(s).\mathbf {0})$. So, if $\kappa$ is K-bi-invariant, then from (14) one has

$$\begin{aligned} \int _{G}\kappa \left( g\right) dg\le c\int _{\mathbb {R}^{+}}\kappa \left( s\right) e^{2\rho s}ds. \end{aligned}$$

(16)

2.1 The spherical Fourier transform

Denote by $S\left( K\backslash G/K\right)$ the Schwartz space of K -bi-invariant functions on G. The spherical Fourier transform $\mathcal {H}$ is defined by

$$\begin{aligned} \mathcal {H}f\left( \lambda \right) =\int _{G}f\left( x\right) \varphi _{\lambda }\left( x\right) dx,\quad \, \lambda \in \mathfrak {a}^{*},\quad \ \ f\in S\left( K\backslash G/K\right) , \end{aligned}$$

where $\varphi _{\lambda }\left( x\right)$ are the elementary spherical functions on G, which by a theorem of Harish–Chandra [23, p. 418], are given by

$$\begin{aligned} \varphi _{\lambda }\left( x\right) =\int _{K}e^{\left( \pm i\lambda -\rho \right) \left( H\left( x^{\pm 1}k\right) \right) }dk,\quad \, x\in G,\quad \, \lambda \in \mathfrak {a}_{\mathbb {C}}^{*}. \end{aligned}$$

(17)

Note that for $H\in \overline{\mathfrak {a}_{+}}$,

$$\begin{aligned} e^{-\rho \left( H\right) }\le \varphi _{0}\left( \exp H\right) \le c\left( 1+\left| H\right| \right) ^{a}e^{-\rho \left( H\right) }, \end{aligned}$$

(18)

for some constants c, $a>0$, [23], (see also [3] for a more precise estimate of $\varphi _{0}$). Note also that for $H\in \overline{ \mathfrak {a}_{+}}$ and $\lambda \in \overline{\mathfrak {a}_{+}^{*}}$,

$$\begin{aligned} 0<\varphi _{-i\lambda }\left( \exp H\right) \le e^{\lambda \left( H\right) }\varphi _{0}\left( \exp H\right) . \end{aligned}$$

(19)

In the rank one case the upper estimate in (18) is written as

$$\begin{aligned} \varphi _{0}(r)\le c\left( 1+r\right) e^{-\rho r},\quad \, r>0. \end{aligned}$$

(20)

Let $S\left( \mathfrak {a}^{*}\right)$ be the usual Schwartz space on $\mathfrak {a}^{*}$, and let us denote by $S\left( \mathfrak {a}^{*}\right) ^{W}$ the subspace of W-invariants in $S\left( \mathfrak {a}^{*}\right)$. Then, by a celebrated theorem of Harish–Chandra, $\mathcal {H}$ is an isomorphism between $S\left( K\backslash G/K\right)$ and $S\left( \mathfrak {a}^{*}\right) ^{W}$ and its inverse is given by

$$\begin{aligned} ( \mathcal {H}^{-1}f) \left( x\right) =c\int _{\mathfrak {a}^{*}}f\left( \lambda \right) \varphi _{-\lambda }\left( x\right) \frac{d\lambda }{\left| \mathbf {c}\left( \lambda \right) \right| ^{2}},\ \ x\in G,\ f\in S( \mathfrak {a}^{*}) ^{W}, \end{aligned}$$

(21)

where $\mathbf {c}\left( \lambda \right)$ is the Harish–Chandra function.

Recall that the Schrödinger kernel $s_{t}$ on X is given by

$$\begin{aligned} s_{t}(\exp H)=( \mathcal {H}^{-1}w_{t}) \left( \exp H\right) ,\quad \, H\in \overline{\mathfrak {a}_{+}}, \end{aligned}$$

where

$$\begin{aligned} w_{t}\left( \lambda \right) =e^{it( \left| \rho \right| ^{2}+\left| \lambda \right| ^{2}) }, \lambda \in \mathfrak {p}^{*}. \end{aligned}$$

So, to obtain pointwise estimates of the kernel $s_{t}$ which are crucial for our proofs, we need a manipulable expression of the inverse spherical Fourier transform $\mathcal {H}^{-1}$. This happens in the rank one case we deal with in the present work. See for example [4] for the explicit expression of $\mathcal {H}^{-1}$ in this setting. The estimates of the kernel $s_{t}$ in rank one case are obtained in [8].

2.2 Uniform asymptotics of the counting function

In this section we present uniform asymptotics of the counting function $N_{\Gamma }$ associate to the group $\Gamma$. For that we need to introduce some definitions and notations. For details see [40, 42].

Denote by $L\left( \Gamma \right)$ the limit set of $\Gamma$ and by $C\left( L\left( \Gamma \right) \right)$ its convex hull in X. The group $\Gamma$ is said to be convex co-compact if $\Gamma \backslash C\left( L\left( \Gamma \right) \right)$ is compact. Let $p\in L( \Gamma )$ be a parabolic point of $\Gamma$ and denote by $\Gamma _{p}$ its stabiliser. Then $L\left( \Gamma _{p}\right) =\left\{ p\right\}$ and if $\Gamma \backslash \left( L\left( \Gamma \right) -\left\{ p\right\} \right)$ is compact we say that p is a bounded parabolic point. The group $\Gamma$ is called geometrically finite if the manifold $\Gamma \backslash X$ has finitely many cusps with all of them associated to a bounded parabolic point. Finally, recall that the length spectrum of $\Gamma$ is non-arithmetic if the set of lengths of all closed geodesics of $\Gamma \backslash X$ is not contained in a discrete subgroup of $\mathbb {R}$.

Next, denote by $\mu _{x}$, $x\in X$, a $\Gamma$-invariant Patterson–Sullivan density: that is a family of finite and mutually absolutely continuous measures on $\partial X$ satisfying the following conditions:

(i) for any $x,y\in X$,

$$\begin{aligned} \frac{d\mu _{y}}{d\mu _{x}}\left( \xi \right) =e^{-\delta \left( \Gamma \right) \beta _{\xi }\left( x,y\right) },\quad \, \xi \in \partial X. \end{aligned}$$

Here $\beta _{\xi }\left( x,y\right)$ is the Busemann function:

$$\begin{aligned} \beta _{\xi }\left( x,y\right) =\lim _{t\rightarrow \infty }\left( d\left( \xi _{t},x\right) -d\left( \xi _{t},y\right) \right) , \end{aligned}$$

where $\xi _{t}$ is a geodesic ray tending to $\xi$ as $t\rightarrow \infty$.

   (ii) for any $\gamma \in \Gamma$ and $x\in X$, $\gamma ^{*}\mu _{x}=\mu _{\gamma x}$.

Note that for any $x\in X$, $\mu _{x}$ is supported on the limit set $L\left( \Gamma \right)$.

Fix $x_{0}\in X$. If $\mu _{x_{0}}$ is normalised to be a probability measure, then from (ii) and the fact that $L\left( \Gamma \right)$ is a $\Gamma$-invariant set, it follows that all of $\mu _{x}$, $x\in X$, are also probability measures.

Note that the celebrated Patterson–Sullivan construction insures the existence of such conformal densities in various geometric contexts as Hadamard manifolds and $CAT\left( -1\right)$ spaces, [14, 42].

Let us now recall the definition of the Bowen–Margulis–Sullivan (BMS) measure m on X associated to a $\Gamma$-invariant Patterson–Sullivan density $\mu _{x}$, $x\in X$. For simplicity we assume that X is a rank one symmetric space. Denote by SX the unit tangent bundle of X and fix $x_{0}\in X$. For $u=\left( \xi ,\eta ,s\right) \in \left( SX\right) ^{2}\times \mathbb {R}$, the BMS measure m on X is given by

$$\begin{aligned} dm\left( u\right) =e^{\delta \left( \Gamma \right) \left( \beta _{\xi }\left( x_{0},u\right) +\beta _{\eta }\left( x_{0},u\right) \right) }d\mu _{x_{0}}\left( \xi \right) d\mu _{x_{0}}\left( \eta \right) ds, \end{aligned}$$

[40, Chapitre 1C]. The quotient of m by $\Gamma$ is the BMS measure $m_{\Gamma }$ on $\Gamma \backslash X$. The measure $m_{\Gamma }$ is supported on the set $\Gamma \backslash G\left( \Gamma \right)$, where $G\left( \Gamma \right) =\left( L\left( \Gamma \right) \times L\left( \Gamma \right) \backslash diagonal\right) \times \mathbb {R}$, i.e. the union of all geodesics $\phi$ in X with $\phi \left( -\infty \right)$ and $\phi \left( +\infty \right)$ in $L\left( \Gamma \right)$. Note that if $\Gamma$ is discrete and torsion free, then $G\left( \Gamma \right)$ is not convex but is always a subset of the convex hull $C\left( L\left( \Gamma \right) \right)$ of $L\left( \Gamma \right)$.

As it is already mentioned in the Introduction, in [42, Theorem 6.2.4] it is shown, in the context of Hadamard manifolds with pinched negative sectional curvature, that if $\Gamma$ is convex co-compact, then there is a constant $C>1$, such that for all $x,y\in X$,

$$\begin{aligned} \lim _{R\rightarrow \infty }\frac{N_{\Gamma }\left( x,y,R\right) }{e^{R\delta \left( \Gamma \right) }}=C\mu _{x}\left( L\left( \Gamma \right) \right) \mu _{y}\left( L\left( \Gamma \right) \right) . \end{aligned}$$

(22)

But, for all $x\in X$, $\mu _{x}$ is a probability measure supported on $L\left( \Gamma \right)$. So, $\mu _{x}\left( L\left( \Gamma \right) \right) =1$ and (22) is in fact uniform in $x,y\in X$.

Denote by $\left\| m_{\Gamma }\right\|$ the total mass of the BMS measure $m_{\Gamma }$. In [40, Chapitre 4, Corollaire 2], it is proved, in the general context of $CAT\left( -1\right)$ spaces, that

$$\begin{aligned} \lim _{R\rightarrow \infty }\frac{N_{\Gamma }\left( x,y,R\right) }{e^{R\delta \left( \Gamma \right) }}=\frac{\mu _{x}\left( L\left( \Gamma \right) \right) \mu _{y}\left( L\left( \Gamma \right) \right) }{\delta \left( \Gamma \right) \left\| m_{\Gamma }\right\| }=\frac{1}{\delta \left( \Gamma \right) \left\| m_{\Gamma }\right\| }, \end{aligned}$$

(23)

provided that

• (i) the length spectrum of $\Gamma$ is non-arithmetic and

• (ii) the BMS measure $m_{\Gamma }$ is finite.

Let us make some comments about the two assumptions above.

Note first that the support of $m_{\Gamma }$ is the set $\Gamma \backslash G\left( \Gamma \right)$. But, $G\left( \Gamma \right) \subseteq C\left( L\left( \Gamma \right) \right)$,[42, Section 6], and $\Gamma \backslash C\left( L\left( \Gamma \right) \right)$ is compact if $\Gamma$ is convex co-compact. So, $\Gamma \backslash G\left( \Gamma \right)$ is also compact and consequently $m_{\Gamma }$ is finite. In fact, as it is noticed in [40, Chapitre 1F], convex co-compact groups are precisely those admitting a BMS measure with compact support.

A larger class of discrete groups with finite BMS measure is given by the finiteness criterion proved in [17, Theorem B]. Let $L_{bp}\left( \Gamma \right)$ be the set of all bounded parabolic points of $L\left( \Gamma \right)$. The finiteness criterion states that if $\Gamma$ is geometrically finite and of divergence type, then the BMS measure $m_{\Gamma }$ is finite iff for all $p\in L_{bp}\left( \Gamma \right)$,

$$\begin{aligned} \sum _{\gamma \in \Gamma _{p}}d\left( x,\gamma x\right) e^{-\delta \left( \Gamma \right) d\left( x,\gamma x\right) }<\infty . \end{aligned}$$

(24)

Next, as it is noticed in [16], the conditions that ensure that the length spectrum of a non elementary group $\Gamma$ is non-arithmetic are not yet known in their full generality. Nevertheless, it is known, [16] , that the length spectrum of $\Gamma$ is non-arithmetic in the following cases:

• (1) X is a hyperbolic space $\mathbb {H}^{n}$,

• (2) $L\left( \Gamma \right)$ possesses a connected component with at least two elements,

• (3) $\Gamma$ contains a parabolic element,

• (4) $\dim X=2$.

From (3) it follows that (23) holds true if $\Gamma$ is geometrically finite, of divergence type, with at least one parabolic element and satisfying (24).

3 An analogue of Kunze–Stein phenomenon on locally symmetric spaces

Let us recall that a central result in the theory of convolution operators on semisimple Lie groups is the Kunze–Stein phenomenon, which states that if $p\in [1,2)$, $f\in L^{2}(G)$ and $\kappa \in L^{p}(G)$, then

$$\begin{aligned} ||f*\kappa ||_{L^{2}(G)}\le C\left( p\right) ||f||_{L^{2}(G)}||\kappa ||_{L^{p}(G)}, \end{aligned}$$

(25)

(see [26, p. 3361]). This inequality was proved by Kunze and Stein [31] in the case when $G=SL(2,\mathbb {R})$ and by Cowling [15] in the general case. In [24] Herz noticed that the inequality (25) can be sharpened if the kernel $\kappa$ is K-bi-invariant. Indeed, Herz’s criterion [24] asserts that if $p\ge 1$ and $\kappa$ is a K -bi-invariant kernel, then

$$\begin{aligned} ||*\left| \kappa \right| ||_{L^{p}(G)\rightarrow L^{p}(G)}= & {} C\int _{G}\left| \kappa (g)\right| \varphi _{-i\rho _{p}}(g)dg \nonumber \\= & {} C\int _{\mathfrak {a}_{+}}\left| \kappa (\exp H)\right| \varphi _{-i\rho _{p}}(\exp H)\delta (H)dH, \end{aligned}$$

(26)

where $\rho _{p}=|2/p-1|\rho$, $p\ge 1$.

Note that for $p=2$, the best we can obtain in the Euclidean case is the inequality

$$\begin{aligned} ||*\left| \kappa \right| ||_{L^{2}(\mathbb {R}^{n})\rightarrow L^{2}(\mathbb {R}^{n})}\le ||\kappa ||_{L^{1}(\mathbb {R}^{n})}, \end{aligned}$$

while in the semisimple case we have that

$$\begin{aligned} ||*\left| \kappa \right| ||_{L^{2}(G)\rightarrow L^{2}(G)}=C\int _{\mathfrak {a}_{+}}\left| \kappa (\exp H)\right| \varphi _{0}(\exp H)\delta (H)dH. \end{aligned}$$

(27)

Bearing in mind that

$$\begin{aligned} ||\kappa ||_{L^{1}(G)}=\int _{\mathfrak {a}_{+}}\left| \kappa (\exp H)\right| \delta (H)dH, \end{aligned}$$

we deduce from the above norm estimates that for $p=2$, the non-trivial gain over the Euclidean case is the factor $\varphi _{0}(\exp H)$.

Kunze–Stein’s phenomenon is no more valid on locally symmetric spaces $M=\Gamma \backslash G/K$. In [34] Lohoué and Marias proved an analogue of this phenomenon for a class of locally symmetric spaces. More precisely, let $\lambda _{0}$ be the bottom of the$L^{2}$-spectrum of $-\Delta$ on M. We say that M possesses property (KS) if there exists a vector $\eta _{\Gamma }\in C_{\rho }\cap S( 0,( \rho ^{2}-\lambda _{0}) ^{1/2})$, such that for all $p\in \left( 1,\infty \right)$,

$$\begin{aligned} \left\| *\left| \kappa \right| \right\| _{L^{p}(M)\rightarrow L^{p}(M)}\le \int _{G}\left| \kappa \left( g\right) \right| \varphi _{-i\eta _{\Gamma }}\left( g\right) ^{s\left( p\right) }dg, \end{aligned}$$

(28)

where

$$\begin{aligned} s\left( p\right) =2\min ( \left( 1/p\right) ,( 1/p^{\prime }) ). \end{aligned}$$

(29)

As it is already mentioned in the Introduction, in [34] it is shown that M possesses property (KS) if it is contained in the following three classes:

• (i) $\Gamma$ is a lattice i.e. $\hbox {vol}\left( \Gamma \backslash G\right) <\infty$,

• (ii) G possesses Kazhdan’s property (T). Recall that G has property (T) iff G has no simple factors locally isomorphic to SO(n, 1) or SU(n, 1), [21, chap. 2]. In this case $\Gamma \backslash G/K$ possesses property (KS) for all discrete subgroups $\Gamma$ of G. Recall that $H^{n}\left( \mathbb {H}\right) =Sp\left( n,1\right) /Sp\left( n\right)$ and $H^{2}\left( \mathbb {O}\right) =F_{4}^{-20}/Spin\left( 9\right)$. So, $\Gamma \backslash H^{n}\left( \mathbb {H}\right)$ and $\Gamma \backslash H^{2}\left( \mathbb {O}\right)$ have property (KS) for all discrete subgroups $\Gamma$ of $Sp\left( n,1\right)$ and $F_{4}^{-20}$ respectively.

Thus, from cases (i) and (ii) we deduce that all locally symmetric spaces $\Gamma \backslash H^{n}\left( \mathbb {H}\right)$ and $\Gamma \backslash H^{2}\left( \mathbb {O}\right)$ have property (KS).

On the contrary, the isometry groups SO(n, 1) and SU(n, 1) of real and complex hyperbolic spaces do not have property (T) and consequently the quotients $\Gamma \backslash H^{n}\left( \mathbb {R}\right)$ and $\Gamma \backslash H^{n}\left( \mathbb {C}\right)$ of infinite volume do not in general belong in the class (ii). The class (iii) below covers also this case.

(iii) $\Gamma \backslash G$ is non-amenable. Note that since G isnon-amenable, then $\Gamma \backslash G$ is non-amenable if $\Gamma$ is amenable. So, if $\Gamma$ is amenable, then the quotients $\Gamma \backslash H^{n}\left( \mathbb {R}\right)$ and $\Gamma \backslash H^{n}\left( \mathbb {C}\right)$ possesses property (KS) even if they have infinite volume. Note that if $\Gamma$ is finitely generated and has subexponential growth, then $\Gamma$ is amenable [1, 20].

Let us now explain when a finitely generated group $\Gamma$ has subexponential growth. Let $A=\{a_{1},a_{2},\ldots ,a_{m}\}$ be a system of generators of $\Gamma$. The length $|g|_{A}$ of $g\in \Gamma$ with respect to A is the length n of the shortest representation of g in the form $g=a_{i_{1}}^{\pm }a_{i_{2}}^{\pm }\cdots a_{i_{n}}^{\pm }$, $a_{i_{j}}\in A$ . This depends on the set A but, for any two systems of generators A and B, $|g|_{A}$ and $|g|_{B}$ are equivalent. The growth function of $\Gamma$ with respect to the set A is the function

$$\begin{aligned} \gamma _{\Gamma }^{A}\left( n\right) =\#\left\{ g\in G:|g|_{A}\le n\right\} . \end{aligned}$$

We say that $\Gamma$ has subexponential growth if $\gamma _{\Gamma }^{A}\left( n\right)$ grows more slowly than any exponential function.

4 Norm estimates of the Schrödinger kernel on M

As it is already mentioned, the Schrödinger kernel $s_{t}$ on X is given by

$$\begin{aligned} s_{t}(\exp H)=( \mathcal {H}^{-1}w_{t}) \left( \exp H\right) ,\quad \, H\in \overline{\mathfrak {a}_{+}}, \end{aligned}$$

(30)

where

$$\begin{aligned} w_{t}\left( \lambda \right) =e^{it\left( \left| \rho \right| ^{2}+\left| \lambda \right| ^{2}\right) },\quad \, \lambda \in \mathfrak {p}^{*}. \end{aligned}$$

Using (30) and the expression of the inverse spherical Fourier transform $\mathcal {H}^{-1}$ in the case of rank one symmetric spaces, Anker et al. [8] obtained the following estimates of $s_{t}$:

$$\begin{aligned} \left| s_{t}(r)\right| \le c\psi _{1}\left( t,r\right) e^{-\rho r}, \end{aligned}$$

(31)

where

$$\begin{aligned} \psi _{1}\left( t,r\right) =\left\{ \begin{array}{l} |t|^{-n/2}(1+r)^{\left( n-1\right) /2}, \quad \text { if }\left| t\right| \le 1+r, \\ |t|^{-3/2}(1+r), \quad \text { if }\left| t\right| >1+r, \end{array} \right. \end{aligned}$$

(32)

and $n=\dim X$, (see also [7, 35] for the case of the real hyperbolic space).

Recall now that the Schrödinger operator $\widehat{S}_{t}$ on M is initially defined as aconvolution operator

$$\begin{aligned} \widehat{S}_{t}f(x)=\int _{G}s_{t}(y^{-1}x)f(y)dy,\quad \, f\in C_{0}^{\infty }(M). \end{aligned}$$

(33)

Set $s_{t}(x,y)=s_{t}(y^{-1}x)$ and

$$\begin{aligned} \widehat{s}_{t}\left( x,y\right) =\sum _{\gamma \in \Gamma }s_{t}\left( x,\gamma y\right) =\sum _{\gamma \in \Gamma }s_{t}( \left( \gamma y\right) ^{-1}x) . \end{aligned}$$

(34)

Proposition 4

For all groups $\Gamma$ with $\delta \left( \Gamma \right) <\rho$, the series (34) converges and the Schrödinger operator $\widehat{S}_{t}$ on M is given by

$$\begin{aligned} \widehat{S}_{t}f\left( x\right) =\int _{M}\widehat{s}_{t}\left( x,y\right) f\left( y\right) dy. \end{aligned}$$

(35)

Proof

Use the Cartan decomposition and write $\left( \gamma y\right) ^{-1}x=k_{\gamma }\exp H_{\gamma }k_{\gamma }^{\prime }$. Then, since $s_{t}$ is K-bi-invariant, $s_{t}( \left( \gamma y\right) ^{-1}x) =s_{t}( \exp H_{\gamma })$ and the estimate (31) implies that

$$\begin{aligned} \left| \widehat{s}_{t}\left( x,y\right) \right|\le & {} \sum _{\gamma \in \Gamma }\vert s_{t}( ( \gamma y) ^{-1}x) \vert \le \sum _{\gamma \in \Gamma }\vert s_{t}( \exp H_{\gamma }) \vert \\\le & {} c\sum _{\gamma \in \Gamma }\psi ( t,\vert H_{\gamma }\vert ) e^{-\rho \left| H_{\gamma }\right| }. \end{aligned}$$

By (32) we have that for every $\varepsilon >0$

$$\begin{aligned} \left| \widehat{s}_{t}\left( x,y\right) \right|&\le \sum _{\gamma \in \Gamma }\psi ( t,\vert H_{\gamma }\vert ) e^{-\rho \left| H_{\gamma }\right| } \nonumber \\&\le c|t|^{-n/2}\sum _{\{\gamma \in \Gamma :\left| t\right| \le 1+\left| H_{\gamma }\right| \}}e^{(\varepsilon -\rho )\left| H_{\gamma }\right| } \nonumber \\&+c|t|^{-3/2}\sum _{\{\gamma \in \Gamma :\left| t\right| >1+\left| H_{\gamma }\right| \}}e^{(\varepsilon -\rho )\left| H_{\gamma }\right| } \nonumber \\&\le c( |t|^{-n/2}+|t|^{-3/2}) \sum _{\gamma \in \Gamma }e^{(\varepsilon -\rho )d\left( 0,\left( \gamma y\right) ^{-1}x\right) } \nonumber \\&\le c(|t|^{-3/2}+|t|^{-n/2})\sum _{\gamma \in \Gamma }e^{(\varepsilon -\rho )d\left( x,\gamma y\right) } \nonumber \\&\le c(|t|^{-3/2}+|t|^{-n/2})P_{\rho -\varepsilon }\left( x,y\right) <\infty \end{aligned}$$

(36)

provided that $\rho >\delta \left( \Gamma \right)$.

Let us now prove (35). Since $s_{t}$ and f are right K -invariant, from (33) we get that

$$\begin{aligned} \widehat{S}_{t}f(x)=\int _{X}s_{t}(x,y)f(y)dy. \end{aligned}$$

Now, since f is left $\Gamma$-invariant, by Weyl’s formula we find that

$$\begin{aligned} \widehat{S}_{t}(f)(x)= & {} \int _{X}f(y)s_{t}(x,y)dy=\int _{\Gamma \backslash X}\left( \sum _{\gamma \in \Gamma }f(\gamma y)s_{t}(x,\gamma y)\right) dy \\= & {} \int _{M}f(y)\widehat{s}_{t}(x,y)dy. \end{aligned}$$

$\square$

Next, we prove norm estimates for the Schrödinger kernel on M which are crucial for the proofs of our results.

Proposition 5

If $M\in \left( S\right)$, then for any $q>2$ there is a constant $c=c\left( q,\Gamma \right) >0$, such that for all $x\in X$,

$$\begin{aligned} \Vert \widehat{s}_{t}(x,.)\Vert _{L^{q}(M)}\le c\Psi \left( t\right) ,\quad \, t\in \mathbb {R}, \end{aligned}$$

(37)

where

$$\begin{aligned} \Psi \left( t\right) =\left\{ \begin{array}{l} \left| t\right| ^{-n/2},\quad \text { if }|t|\le 1, \\ |t|^{-3/2},\quad \text { if }|t|>1. \end{array} \right. \end{aligned}$$

For the proof of the proposition we need the following lemma.

Set $\Gamma _{R}\left( x,y\right) =\{\gamma \in \Gamma :d(x,\gamma y)\le R\},$ $x,y\in X,$ $R>0$, and recall that the counting function of $\Gamma$ is given by $N_{\Gamma }\left( x,y,R\right) =\#\Gamma _{R}\left( x,y\right)$ .

Lemma 6

Assume that $M\in \left( S\right)$. If $s>\delta (\Gamma )$, then there are positive constants $R_{0}$ and $c_{0}$ such that for $R\ge R_{0}$ and for all $x,y\in X$,

$$\begin{aligned} \sum _{\gamma \in \Gamma }e^{-sd\left( x,\gamma y\right) }\le c_{0}\sum _{\gamma \in \Gamma _{R}\left( x,y\right) }e^{-sd\left( x,\gamma y\right) }. \end{aligned}$$

(38)

Proof

By our assumption, if $M\in \left( S\right)$ then $N_{\Gamma }$ satisfies the following uniform asymptotics: there exists $c>0$ such that for all $x,y\in X,$

$$\begin{aligned} \frac{N_{\Gamma }\left( x,y,R\right) }{e^{R\delta \left( \Gamma \right) }} \underset{R\rightarrow \infty }{\rightarrow }c. \end{aligned}$$

(39)

Recall also that $N_{\Gamma }\left( x,y,R\right)$ satisfies the following formula:

$$\begin{aligned} \sum _{\gamma \in \Gamma _{R}\left( x,y\right) }e^{-sd\left( x,\gamma y\right) }=N_{\Gamma }\left( x,y,R\right) e^{-sR}+s\int _{0}^{R}N_{\Gamma }\left( x,y,t\right) e^{-st}dt, \end{aligned}$$

(40)

(see [37, p. 46] and [33, p. 131]).

Combining with (39), we get that there are positive constants $R_{0}$, $c_{1}$ such that for $R\ge R_{0}$ and for all $x,y\in X$,

$$\begin{aligned} \sum _{\gamma \in \Gamma _{R}\left( x,y\right) }e^{-sd\left( x,\gamma y\right) }\ge N_{\Gamma }\left( x,y,R\right) e^{-sR}\ge c_{1}e^{-R\left( s-\delta \left( \Gamma \right) \right) }. \end{aligned}$$

(41)

Next, using again (39), we get that for $R\ge R_{0}$ and for all $x,y\in X$,

$$\begin{aligned} \sum _{\gamma \in \Gamma _{R}\left( x,y\right) ^{c}}e^{-sd\left( x,\gamma y\right) }= & {} \sum _{k\ge 0}\sum _{\left\{ \gamma :R+k<d\left( x,\gamma y\right) \le R+k+1\right\} }e^{-sd\left( x,\gamma y\right) } \nonumber \\\le & {} \sum _{k\ge 0}e^{-s\left( R+k\right) }N\left( x,y,R+k+1\right) \nonumber \\\le & {} c\sum _{k\ge 0}e^{-s\left( R+k\right) }e^{\left( R+k+1\right) \delta \left( \Gamma \right) } \nonumber \\\le & {} ce^{-R\left( s-\delta \left( \Gamma \right) \right) }\sum _{k\ge 0}e^{-k\left( s-\delta \left( \Gamma \right) \right) } \nonumber \\\le & {} c_{2}e^{-R\left( s-\delta \left( \Gamma \right) \right) }, \end{aligned}$$

(42)

since $s>\delta (\Gamma )$.

From (41) and (42) it follows that for $R\ge R_{0}$ and all $x,y\in X$

$$\begin{aligned} \sum _{\gamma \in \Gamma _{R}\left( x,y\right) ^{c}}e^{-sd\left( x,\gamma y\right) }\le c_{2}e^{-R\left( s-\delta \left( \Gamma \right) \right) }\le \frac{c_{2}}{c_{1}}\sum _{\gamma \in \Gamma _{R}\left( x,y\right) }e^{-sd\left( x,\gamma y\right) }, \end{aligned}$$

and (38) follows by taking $c_{0}=1+\left( c_{2}/c_{1}\right)$. $\square$

Proof of Proposition 5

From (36) we have that for every $\varepsilon >0$,

$$\begin{aligned} \left| \widehat{s}_{t}\left( x,y\right) \right| \le c\Psi \left( t\right) \sum _{\gamma \in \Gamma }e^{(\varepsilon -\rho )d\left( x,\gamma y\right) }. \end{aligned}$$

From (38), (39) and the fact that $\Gamma _{R}\left( x,y\right)$ is a finite set, we get that for $R>R_{0}$ and $q>2$

$$\begin{aligned} \left( \sum _{\gamma \in \Gamma }e^{(\varepsilon -\rho )d\left( x,\gamma y\right) }\right) ^{q}\le & {} \left( c_{0}\sum _{\gamma \in \Gamma _{R}\left( x,y\right) }e^{(\varepsilon -\rho )d\left( x,\gamma y\right) }\right) ^{q} \nonumber \\\le & {} c_{0}^{q}\left( \#\Gamma _{R}\left( x,y\right) \right) ^{q-1}\sum _{\gamma \in \Gamma _{R}\left( x,y\right) }e^{q(\varepsilon -\rho )d\left( x,\gamma y\right) } \nonumber \\\le & {} cN_{\Gamma }\left( x,y,R\right) ^{q-1}\sum _{\gamma \in \Gamma }e^{q(\varepsilon -\rho )d\left( x,\gamma y\right) } \nonumber \\\le & {} ce^{\left( q-1\right) R\delta \left( \Gamma \right) }\sum _{\gamma \in \Gamma }e^{q(\varepsilon -\rho )d\left( x,\gamma y\right) }. \end{aligned}$$

(43)

Using Weyl’s formula, we have that

$$\begin{aligned} \int _{M}\sum _{\gamma \in \Gamma }e^{q(\varepsilon -\rho )d\left( x,\gamma y\right) }dy= & {} \int _{X}e^{q(\varepsilon -\rho )d\left( x,y\right) }dy \nonumber \\= & {} c\int _{0}^{\infty }e^{-r\left( \left( q-2\right) \rho -q\varepsilon \right) }dr<\infty , \end{aligned}$$

(44)

if we choose $\varepsilon <\frac{\left( q-2\right) \rho }{q}$.

Combining (43) and (44) we get that

$$\begin{aligned} \int _{M}\left( \sum _{\gamma \in \Gamma }e^{(\varepsilon -\rho )d\left( x,\gamma y\right) }\right) ^{q}dy\le c. \end{aligned}$$

Consequently, we have that

$$\begin{aligned} \begin{aligned} \Vert \widehat{s}_{t}(x,.)\Vert _{L^{q}(M)}^{q}&=\int _{M}\left| \widehat{s}_{t}\left( x,y\right) \right| ^{q}dy \\&\le c\Psi (t)^{q}\int _{M}\left( \sum _{\gamma \in \Gamma }e^{(\varepsilon -\rho )d\left( x,\gamma y\right) }\right) ^{q}dy \\&\le c\Psi (t)^{q}, \end{aligned} \end{aligned}$$

and the proof is complete. $\square$

5 Dispersive estimates for the Schrödinger operator

In this section we prove dispersive estimates for the Schrödinger operator $\widehat{S}_{t}$ on locally symmetric spaces in the class (S). To begin with, we make use of the estimates of the $L^{q}$-norm of the kernel $\widehat{s}_{t}\left( x,y\right)$ obtained in the previous section, in order to estimate the operator norms of $\widehat{S}_{t}$.

Lemma 7

If $M\in \left( S\right)$, then for any $q>2$

$$\begin{aligned} \Vert \widehat{S}_{t}\Vert _{L^{1}\left( M\right) \rightarrow L^{q}\left( M\right) }\le c\Psi \left( t\right) ,\quad \, t\in \mathbb {R}, \end{aligned}$$

(45)

and

$$\begin{aligned} \Vert \widehat{S}_{t}\Vert _{L^{q^{\prime }}\left( M\right) \rightarrow L^{\infty }\left( M\right) }\le c\Psi \left( t\right) ,\quad \, t\in \mathbb {R}, \end{aligned}$$

(46)

where $\Psi \left( t\right)$ is defined in Proposition 5.

Proof

If $f\in L^{1}(M)$ and $q>2$, then Minkowski’s inequality and Proposition 5 imply that

$$\begin{aligned} \Vert \widehat{S}_{t}f\Vert _{L^{q}(M)}= & {} \left( \int _{M}\left| \int _{M} \widehat{s}_{t}(x,y)f(y)dy\right| ^{q}dx\right) ^{1/q} \\\le & {} \int _{M}\left( \int _{M}\left| \widehat{s}_{t}(x,y)f(y)\right| ^{q}dx\right) ^{1/q}dy \\= & {} \int _{M}|f(y)|\left( \int _{M}\left| \widehat{s}_{t}(x,y)\right| ^{q}dx\right) ^{1/q}dy \\\le & {} \Vert f\Vert _{L^{1}(M)}\sup _{x}\Vert \widehat{s}_{t}(x,.)\Vert _{L^{q}(M)} \\\le & {} c\Psi \left( t\right) \Vert f\Vert _{L^{1}(M)}. \end{aligned}$$

The estimate (46) follows from (45) by duality. $\square$

Next, we make use of the analogue of Kunze–Stein phenomenon for locallysymmetric spaces proved in [34] and presented in Sect. 3 , in order to obtain the estimate of the norm $\Vert *s_{t}\Vert _{L^{q^{\prime }}(M)\rightarrow L^{q}(M)}$.

Fix a base point $x_{0}\in X$ and consider the K-bi-invariant weight

$$\begin{aligned} \omega \left( x\right) =e^{\left( \delta \left( \Gamma \right) +\varepsilon \right) d\left( x,x_{0}\right) },\quad \, x\in X, \end{aligned}$$

where $0<\varepsilon <\rho -\delta \left( \Gamma \right)$.

Proposition 8

Assume that $M\in \left( S\right)$. If $\psi$ is a bounded K -bi-invariant function on G, then for every $q\ge 2$

$$\begin{aligned} \left\| *( \omega ^{-1}\psi ) \right\| _{L^{q^{\prime }}\left( M\right) \rightarrow L^{q}\left( M\right) }\le c_{q}\left( \int _{G}\omega ^{-1}(g)\left| \psi \left( g\right) \right| ^{q/2}\varphi _{0}(g)dg\right) ^{2/q}. \end{aligned}$$

(47)

Proof

We shall prove (47) for $q=2$ and $q=\infty$ and then interpolate to get the general case.

The estimate (47) for $q=2$ is a direct consequence of Kunze and Stein’sphenomenon. In fact, if $M\in \left( S\right)$, then the vector $\eta _{\Gamma }$ appearing in (28) is equal to 0. Thus for $q=2$, (28) is written as

$$\begin{aligned} \left\| *( \omega ^{-1}\psi ) \right\| _{L^{2}(M)\rightarrow L^{2}(M)}\le c\int _{G}\omega ^{-1}\left( g\right) \left| \psi \right| \left( g\right) \varphi _{0}\left( g\right) dg. \end{aligned}$$

(48)

Next, we prove (47) for $q=\infty$. Consider a bounded and K-bi-invariant function $\kappa$ on G such that $\left\| \kappa \omega \right\| _{\infty }<\infty$. Since $\kappa$ is K-bi-invariant, then by Weyl’s formula, we get that

$$\begin{aligned} \left( f*\kappa \right) \left( x\right) =\int _{M}\widehat{\kappa }\left( x,y\right) f\left( y\right) dy,\quad \, f\in C_{0}^{\infty }\left( M\right) , \end{aligned}$$

where

$$\begin{aligned} \widehat{\kappa }\left( x,y\right) =\sum _{\gamma \in \Gamma }\kappa ( ( \gamma y) ^{-1}x),\quad x,y\in X. \end{aligned}$$

For $R\ge 0$, denote by $\kappa _{R}$ and $\omega _{R}$ the restrictions of $\kappa$ and $\omega$ on the ball $B\left( x_{0},R\right)$ and write

$$\begin{aligned} \left| \widehat{\kappa }_{R}\left( x,y\right) \right|\le & {} \sum _{\gamma \in \Gamma }\vert \kappa _{R}( ( \gamma y) ^{-1}x)\vert \omega _{R}\left( \gamma x\right) \omega _{R}\left( \gamma x\right) ^{-1} \nonumber \\\le & {} \sup _{\gamma \in \Gamma }\left( \left| \kappa _{R}\left( \left( \gamma y\right) ^{-1}x\right) \right| \omega _{R}\left( \gamma x\right) \right) \sum _{\gamma \in \Gamma }e^{-\left( \delta \left( \Gamma \right) +\varepsilon \right) d\left( \gamma x,x_{0}\right) } \nonumber \\\le & {} \left\| \kappa \omega \right\| _{\infty }P_{\delta \left( \Gamma \right) +\varepsilon }\left( x,x_{0}\right) . \end{aligned}$$

(49)

But, by the asymptotics (39) of the counting function $N_{\Gamma }$ and the fact that the function $R\rightarrow N_{\Gamma }\left( x,y,R\right)$ is increasing, we get that for $k_{0}$ big enough and for all $x_{0},x\in X,$

$$\begin{aligned}&P_{_{\delta \left( \Gamma \right) +\varepsilon }}\left( x,x_{0}\right) =\sum _{k=0}^{\infty }\sum _{\gamma \in \Gamma _{k-1}\left( x_{0},x\right) \backslash \Gamma _{k}\left( x_{0},x\right) }e^{-\left( \delta \left( \Gamma \right) +\varepsilon \right) d\left( \gamma x,x_{0}\right) } \\&\quad \le \sum _{k=0}^{k_{0}-1}e^{-\left( \delta \left( \Gamma \right) +\varepsilon \right) \left( k-1\right) }N\left( x_{0},x,k\right) +\sum _{k=k_{0}}^{\infty }e^{-\left( \delta \left( \Gamma \right) +\varepsilon \right) \left( k-1\right) }N\left( x_{0},x,k\right) \\&\quad \le cN\left( x_{0},x,k_{0}\right) \sum _{k=0}^{k_{0}}e^{-\left( \delta \left( \Gamma \right) +\varepsilon \right) k}+c\sum _{k=k_{0}}^{\infty }e^{-\left( \delta \left( \Gamma \right) +\varepsilon \right) k}e^{\delta \left( \Gamma \right) k} \\&\quad \le ce^{\delta \left( \Gamma \right) k_{0}}+c\sum _{k=0}^{\infty }e^{-\varepsilon k}=c\left( \varepsilon ,k_{0},\Gamma \right) <\infty . \end{aligned}$$

Combining the estimate above with (49) it follows that for $x,y\in X$

$$\begin{aligned} \left| \widehat{\kappa }\left( x,y\right) \right| \le c\left( \varepsilon ,k_{0},\Gamma \right) \left\| \kappa \omega \right\| _{\infty }<\infty , \end{aligned}$$

(50)

since by our assumption $\left\| \kappa \omega \right\| _{\infty }<\infty$.

Taking $\kappa =\omega ^{-1}\psi$, it follows from (50) that for $f\in C_{0}^{\infty }\left( M\right)$ and any $x\in X$

$$\begin{aligned} \vert f*( \omega ^{-1}\psi ) \left( x\right) \vert\le & {} c\int _{M}\widehat{\left| ( \omega ^{-1}\psi ) \right| }\left( x,y\right) \left| f\left( y\right) \right| dy \\\le & {} c\left( \varepsilon ,k_{0},\Gamma \right) \left\| \psi \right\| _{\infty }\left\| f\right\| _{L^{1}\left( M\right) }, \end{aligned}$$

i.e.

$$\begin{aligned} \left\| *( \omega ^{-1}\psi ) \right\| _{L^{1}\left( M\right) \rightarrow L^{\infty }\left( M\right) }\le c\left( \varepsilon ,k_{0},\Gamma \right) \left\| \psi \right\| _{\infty }. \end{aligned}$$

(51)

Interpolating the inequalities (51) and (48) with $\theta =1-\left( 2/q\right)$, we obtain (47). $\square$

5.1 Proof of the results

Once we have established the necessary ingredients for the proof of dispersiveestimates of the Schrödinger operator $\widehat{S}_{t}$ on $M\in \left( S\right)$, mainly the norm estimates of the kernel $\widehat{s} _{t}(x,y)$ obtained in Proposition 5 and the estimates of the operator norms of $\widehat{S}_{t}$ obtained in Lemma 7, we give the proofs of Theorems 2 and 3 and present their applications to the study of well-posedness and scattering for NLS equations.

Having obtained the ingredients mentioned above, the proofs become standard and they are similar to the proofs of the corresponding results in [7, 8]. Consequently, we shall be brief and present only their main lines or simply we will omit them.

Proof of Theorem 2

Recall first that in Lemma 7 we obtained that for any $q>2$

$$\begin{aligned} \Vert \widehat{S}_{t}\Vert _{L^{1}\left( M\right) \rightarrow L^{q}\left( M\right) }\le c\Psi \left( t\right) ,\quad \, t\in \mathbb {R}, \end{aligned}$$

(52)

and

$$\begin{aligned} \Vert \widehat{S}_{t}\Vert _{L^{q^{\prime }}\left( M\right) \rightarrow L^{\infty }\left( M\right) }\le c\Psi \left( t\right) ,\quad \, t\in \mathbb {R}, \end{aligned}$$

(53)

where

$$\begin{aligned} \Psi \left( t\right) =\left\{ \begin{array}{l} \left| t\right| ^{-n/2},\quad \text { if }|t|\le 1, \\ |t|^{-3/2},\quad \text { if }|t|>1. \end{array} \right. \end{aligned}$$

Next, by Proposition 8 with $\psi =\omega s_{t}$, we obtain that for $q>2$

$$\begin{aligned} \left\| \widehat{S}_{t}\right\| _{L^{q^{\prime }}\left( M\right) \rightarrow L^{q}\left( M\right) }^{q/2}\le c\int _{0}^{\infty }\omega (r)^{-1}\left| \left( \omega s_{t}\right) (r)\right| ^{q/2}\varphi _{0}(r)\delta (r)dr. \end{aligned}$$

(54)

Also, from the estimates (32) of the kernel $s_{t}$ it follows that

$$\begin{aligned} \omega \left( r\right) \left| s_{t}\left( r\right) \right| \le c\left\{ \begin{array}{ll} \left| t\right| ^{-n/2}\left( 1+r\right) ^{\left( n-1\right) /2}e^{-\left( \rho -\delta \left( \Gamma \right) -\varepsilon \right) r}, &{} \quad \left| t\right| \le 1+r, \\ \left| t\right| ^{-3/2}\left( 1+r\right) e^{-\left( \rho -\delta \left( \Gamma \right) -\varepsilon \right) r}, &{}\quad \left| t\right| \le 1+r, \end{array} \right. \end{aligned}$$

(55)

with $\rho -\delta \left( \Gamma \right) -\varepsilon >0$.

Finally, recall that

$$\begin{aligned} \varphi _{0}(r)\le c\left( 1+r\right) e^{-\rho r}\quad \text { and }\quad \delta (r)\le e^{2\rho r},\, r>0. \end{aligned}$$

(56)

Putting (55) and (56) in (54) we get that for $q>2$ and $\left| t\right| \le 1$,

$$\begin{aligned} \left\| \widehat{S}_{t}\right\| _{L^{q^{\prime }}\left( M\right) \rightarrow L^{q}\left( M\right) }^{2/q}\le & {} c\left| t\right| ^{-nq/4}\int _{0}^{\infty }\left( 1+r\right) ^{\frac{n\left( q-1\right) }{4} +1}e^{-\left( \frac{q}{2}-1\right) \varepsilon r}dr \\\le & {} c\left| t\right| ^{-nq/4}. \end{aligned}$$

So,

$$\begin{aligned} \left\| \widehat{S}_{t}\right\| _{L^{q^{\prime }}\left( M\right) \rightarrow L^{q}\left( M\right) }\le c\left| t\right| ^{-n/2},\quad \, \left| t\right| \le 1. \end{aligned}$$

(57)

Similarly, if $\left| t\right| \ge 1$, then

$$\begin{aligned} \left\| \widehat{S}_{t}\right\| _{L^{q^{\prime }}\left( M\right) \rightarrow L^{q}\left( M\right) }^{2/q}\le & {} c\left| t\right| ^{-3q/4}\int _{0}^{\left| t\right| -1}\left( 1+r\right) ^{\frac{q}{2} +1}e^{-\left( \frac{q}{2}-1\right) \varepsilon r}dr \\&+c\left| t\right| ^{-3q/4}\int _{\left| t\right| -1}^{\infty }\left( 1+r\right) ^{\frac{n\left( q-1\right) }{4}+1}e^{-\left( \frac{q}{2}-1\right) \varepsilon r}dr \\\le & {} c\left| t\right| ^{-3q/4}\int _{0}^{\infty }e^{-\left( \frac{q }{2}-1\right) \frac{\varepsilon r}{2}}dr\le c\left| t\right| ^{-3q/4}. \end{aligned}$$

So,

$$\begin{aligned} \left\| \widehat{S}_{t}\right\| _{L^{q^{\prime }}\left( M\right) \rightarrow L^{q}\left( M\right) }\le c\left| t\right| ^{-3/2}, \quad \, \left| t\right| \ge 1. \end{aligned}$$

(58)

Finally, by the spectral theorem, we have that

$$\begin{aligned} \Vert \widehat{S}_{t}\Vert _{L^{2}(M)\rightarrow L^{2}\left( M\right) }=1. \end{aligned}$$

Putting together the above norm estimates of $\widehat{S}_{t}$ we get that: for $\left| t\right| \le 1$,

$$\begin{aligned} \left\{ \begin{array}{l} \Vert \widehat{S}_{t}\Vert _{L^{1}(M)\rightarrow L^{q}\left( M\right) }\le c\left| t\right| ^{-n/2}, \\ \Vert \widehat{S}_{t}\Vert _{L^{q^{\prime }}(M)\rightarrow L^{\infty }\left( M\right) }\le c\left| t\right| ^{-n/2}, \\ \Vert \widehat{S}_{t}\Vert _{L^{2}(M)\rightarrow L^{2}\left( M\right) }=1, \end{array} \right. \end{aligned}$$

and for $\left| t\right| \ge 1$,

$$\begin{aligned} \left\{ \begin{array}{l} \Vert \widehat{S}_{t}\Vert _{L^{1}(M)\rightarrow L^{q}\left( M\right) }\le c\left| t\right| ^{-3/2}, \\ \Vert \widehat{S}_{t}\Vert _{L^{q^{\prime }}(M)\rightarrow L^{\infty }\left( M\right) }\le c\left| t\right| ^{-3/2}, \\ \Vert \widehat{S}_{t}\Vert _{L^{q^{\prime }}(M)\rightarrow L^{q}\left( M\right) }\le c\left| t\right| ^{-3/2}. \end{array} \right. \end{aligned}$$

Then, a standard argument of interpolation gives that

$$\begin{aligned} \Vert \widehat{S}_{t}\Vert _{L^{\tilde{q}^{\prime }}(M)\rightarrow L^{q}(M)}\le c|t|^{-n\max {\{\frac{1}{2}-\frac{1}{q},\frac{1}{2}-\frac{1}{ \tilde{q}}\}}},\quad \text { for }\left| t\right| \le 1, \end{aligned}$$

and

$$\begin{aligned} \Vert \widehat{S}_{t}\Vert _{L^{{\tilde{q}}^{\prime }}(M)\rightarrow L^{q}(M)}\le c|t|^{-3/2},\quad \text { for }\left| t\right| \ge 1. \end{aligned}$$

$\square$

Proof of Theorem 3

As it is already mentioned in the Introduction, to prove the Strichartz estimates

$$\begin{aligned} \Vert u\Vert _{L_{t}^{p}L_{x}^{q}}\le c\{ \Vert f\Vert _{L_{x}^{2}}+\Vert F\Vert _{L_{t}^{\tilde{p}^{\prime }}L_{x}^{\tilde{q} ^{\prime }}}\} , \end{aligned}$$

(59)

for the solutions $u\left( t,x\right)$ of the Cauchy problem

$$\begin{aligned} \left\{ \begin{array}{l} i\partial _{t}u(t,x)+\Delta u(t,x)=F(t,x),\quad \, t\in \mathbb {R},\quad \, x\in M, \\ u(0,x)=f(x), \end{array} \right. \end{aligned}$$

(60)

we shall combine the dispersive estimates of the operator $e^{it\Delta }$ obtained in Theorem 2 with the classical $TT^{*}$ method. This method consists in proving $L_{t}^{p^{\prime }}L_{x}^{q^{\prime }}\rightarrow L_{t}^{p}L_{x}^{q}$ boundedness of the operator

$$\begin{aligned} TT^{*}{F}(t,x)=\int _{-\infty }^{+\infty }{\widehat{S}}_{t-s}{F}(s,x)ds \end{aligned}$$

and of its truncated version

$$\begin{aligned} \widetilde{TT}^{*}{F}(t,x)=\int _{0}^{t}{\widehat{S}}_{t-s}{F}(s,x)ds, \end{aligned}$$

for admissible indices $\left( p,q\right)$. For that we shall make also use of the fact that the solutions of (60) are given by Duhamel’s formula:

$$\begin{aligned} u(t,x)=e^{it\Delta }f\left( x\right) -i\int _{0}^{t}e^{i\left( t-s\right) \Delta }F\left( s,x\right) ds. \end{aligned}$$

(61)

Here we shall treat only the case when the admissible pairs (p, q) correspond to the triangle

$$\begin{aligned} \widetilde{T}_{n}=\left\{ \left( \tfrac{1}{p},\tfrac{1}{q}\right) \in \left( 0,\tfrac{1}{2}\right) \times \left( 0,\tfrac{1}{2}\right) :\tfrac{2}{p}+ \tfrac{n}{q}\ge \tfrac{n}{2}\right\} \cup \left\{ \left( 0,\tfrac{1}{2} \right) \right\} . \end{aligned}$$

The other cases are treated similarly with the analogue cases in the context of $\mathbb {R}^{n}$ or of $\mathbb {H}^{n}$ [7, 30].

So, if $( \frac{1}{p},\frac{1}{q}) \in \widetilde{T}_{n}$, then from (61) it follows that to finish the proof of the theorem, it is enough to show that

$$\begin{aligned} \left\| \int _{-\infty }^{+\infty }{\widehat{S}}_{t-s}{F(s)}ds\right\| _{L_{t}^{p}L_{x}^{q}}\le c\Vert F\Vert _{L_{t}^{\tilde{p}^{\prime }}L_{x}^{ \tilde{q}^{\prime }}} \end{aligned}$$

(62)

and

$$\begin{aligned} \left\| {\widehat{S}}_{t}{f(s)}\right\| _{L_{t}^{p}L_{x}^{q}}\le c\Vert f\Vert _{L_{x}^{2}}. \end{aligned}$$

(63)

We give only the proof of (62). The proof of (63) is similar. We have that

$$\begin{aligned} \left\| \int _{-\infty }^{+\infty }{\widehat{S}}_{t-s}{F(s)}ds\right\| _{L_{x}^{q}}\le & {} \int _{-\infty }^{+\infty }\left\| {\widehat{S}}_{t-s}{ F(s)}\right\| _{L_{x}^{q}}ds \\\le & {} \int _{-\infty }^{+\infty }\left\| {\widehat{S}}_{t-s}\right\| _{L_{x}^{q^{\prime }}\rightarrow L_{x}^{q}}\left\| F\left( s\right) \right\| _{L_{x}^{q^{\prime }}}ds. \end{aligned}$$

Then, from Theorem 2 we get that

$$\begin{aligned} \left\| \int _{-\infty }^{+\infty }{\widehat{S}}_{t-s}{F(s)}ds\right\| _{L_{t}^{p}L_{x}^{q}}\le & {} \left\| \int _{|t-s|\ge 1}|t-s|^{-3/2}\Vert { F}(s)\Vert _{L_{x}^{q^{\prime }}}ds\right\| _{L_{t}^{p}} \\&+\left\| \int _{|t-s|\le 1}|t-s|^{-\left( 1/2-1/q\right) n}\Vert {F} (s)\Vert _{L_{x}^{q^{\prime }}}ds\right\| _{L_{t}^{p}} \\&:=I_{1}+I_{2}. \end{aligned}$$

To estimate $I_{1}$ and $I_{2}$ we consider the operators

$$\begin{aligned} T_{1}(f)(t)=\int _{|t-s|\ge 1}|t-s|^{-3/2}f(s)ds \end{aligned}$$

and

$$\begin{aligned} T_{2}(f)(t)=\int _{|t-s|\le 1}|t-s|^{-\left( 1/2-1/q\right) n}f(s)ds. \end{aligned}$$

Note that the kernel $k_{1}(u)=|u|^{-3/2}\chi _{\{|u|\ge 1\}}$ of $T_{1}$ is bounded on $L^{1}$. So, $T_{1}$ is bounded from $L^{p^{\prime }}\$to $L^{p}$ for every $p\in [2,\infty ]$. Similarly, $k_{2}(u)=|u|^{- \left( 1/2-1/q\right) n}\chi _{\{|u|\le 1\}}$ is bounded on $L^{1}$ if $\tfrac{1}{q}\in \left( \left( \tfrac{{1}}{2}-\tfrac{1}{n}\right) ,\tfrac{1}{2 }\right)$. This implies that and $T_{2}$ is bounded from $L^{p_{1}}$ to $L^{p_{2}}$ for all $p_{1},p_{2}\in \left( 1,\infty \right)$, such that $0\le \frac{1}{p_{1}}-\frac{1}{p_{2}}\le 1-\left( \frac{1}{2}-\frac{1}{q} \right) n$. So,

$$\begin{aligned} I_{1},I_{2}\le c\Vert {F}(s)\Vert _{L_{t}^{\tilde{p}^{\prime }}L_{x}^{ \tilde{q}^{\prime }}}, \end{aligned}$$

for all admissible pairs $\left( p,q\right)$ and $\left( \tilde{p},\tilde{q} \right)$ corresponding to the triangle $\widetilde{T}_{n}$. $\square$

5.2 Applications of Strichartz estimates

In this section we apply Strichartz estimates to study well-posedness and scattering for NLS equations, in the case when $M\in \left( S\right)$.

Consider the Cauchy problem for the inhomogeneous Schrödinger equation on M:

$$\begin{aligned} \left\{ \begin{array}{l} i\partial _{t}u(t,x)+\Delta u(t,x)=F(u(t,x)),\, t\in \mathbb {R} , \, x\in M, \\ u(0,x)=f(x), \end{array} \right. \end{aligned}$$

(64)

and assume that F has a power-like nonlinearity of order $\gamma$, i.e.

$$\begin{aligned} |F(u)|\le c|u|^{\gamma },\,\,\,|F(u)-F(v)|\le c( |u|^{\gamma -1}+|v|^{\gamma -1}) |u-v|. \end{aligned}$$

Recall that the NLS is globally well-posed in $L^{2}(M)$ if, for any bounded subset B of $L^{2}(M),$ there exists a Banach space Y continuously embedded into $C(\mathbb {R};L^{2}(M))$, such that for any $f\in B$, the NLS has a unique solution $u\in Y$ with $u(0,x)=f(x)$ and the map $T:B\rightarrow Y,$ $T(f)=u$ is continuous. Here, as in [7], we take

$$\begin{aligned} Y=Y_{\gamma }=C(\mathbb {R};L^{2}(M))\cap {L^{\gamma +1}(\mathbb {R};L^{\gamma +1}(M))}, \end{aligned}$$

which is a Banach space for the norm

$$\begin{aligned} \Vert u\Vert _{Y_{\gamma }}=\Vert u\Vert _{L_{t}^{\infty }L_{x}^{2}}+\Vert u\Vert _{L_{t}^{\gamma +1}L_{x}^{\gamma +1}}, \end{aligned}$$

and $B=B\left( 0,\varepsilon \right) \subset L^{2}(M)$.

We have the following result.

Theorem 9

Assume that $M\in \left( S\right)$ and that F has a power-like nonlinearity of order $\gamma$. If $\gamma \in (1,1+\frac{4}{n}]$ , then the NLS (64) is globally well-posed for small $L^{2}$ data.

Also, one can apply Strichartz’s estimates in order to show scattering for the NLS in the case of power-like nonlinearity of order $\gamma$ and for small $L^{2}$ data.

Theorem 10

Assume that $M\in \left( S\right)$. Consider the Cauchy problem (64) and assume that F has a power-like nonlinearity of order $\gamma \in (1,1+\frac{4}{n}]$. Then, for every global solution u corresponding to small $L^{2}$ data, there exists $u_{\pm }\in L^{2}(M)$ such that

$$\begin{aligned} \Vert u(t)-e^{it\Delta }u_{\pm }\Vert _{L^{2}(M)}\rightarrow 0,\quad \text { as } t\rightarrow \pm \infty . \end{aligned}$$

The proof of Theorems 9 and 10 are similar to the proofs of the corresponding results in [7] and then omitted.