Strichartz estimates without loss on manifolds with hyperbolic trapped geodesics

TL;DR

This paper demonstrates that Strichartz and dispersive estimates for the Schrödinger equation can hold without loss on manifolds with hyperbolic trapped geodesics of small fractal dimension, contrasting previous results on local smoothing.

Contribution

It establishes lossless Strichartz estimates on manifolds with hyperbolic trapped geodesics, expanding understanding of dispersive properties in such geometric settings.

Findings

Strichartz estimates hold without loss on hyperbolic trapped manifolds.

Dispersive estimates remain valid despite trapping under certain conditions.

Contrast with previous results showing loss in local smoothing effects.

Abstract

Doi proved that the $L^2_t H^{1/2}_x$ local smoothing effect for Schr\"odinger equation on a Riemannian manifold does not hold if the geodesic flow has one trapped trajectory. We show in contrast that Strichartz estimates and $L^1\to L^\infty$ dispersive estimates still hold without loss for $e^{it\Delta}$ in various situations where the trapped set is hyperbolic and of sufficiently small fractal dimension.

Equations

$$||\chi e^{it\Delta }{u}_0|{|}_{L^2((0,T);H^{1/2}(M))}\le C_T||u_0|{|}_{L^2(M)},\forall u_0\in L^2(M)$$

$$||e^{it\Delta }{u}_0|{|}_{L^p((0,T);L^q({\mathbb{R}}^d))}\le C_T||u_0|{|}_{L^2({\mathbb{R}}^d)}\mathrm{if}p,q\ge 2,\frac{2}{p}+\frac{d}{q}=\frac{d}{2},(p,q)\ne (2,\infty ).$$

$$\begin{array}{c}||e^{it{\Delta }_X}|{|}_{L^1(X)\to L^{\infty }(X)}\le \{\begin{array}{l}C|t{|}^{-(n+1)/2},\text{ for }|t|\le 1\\ C|t{|}^{-3/2},\text{ for }|t|>1,\end{array}\\ ||e^{it{\Delta }_X}{u}_0|{|}_{L^p(\mathbb{R};L^q(X))}\le C||u_0|{|}_{L^2(X)}\end{array}$$

$$T_{n}:=\left\{\Big{(}\frac{1}{p},\frac{1}{q}\Big{)}\in\Big{(}0,\frac{1}{2}\Big{]}\times\Big{(}0,\frac{1}{2}\Big{)};\frac{2}{p}\geq\frac{n+1}{2}-\frac{n+1}{q}\right\}\cup\left\{\Big{(}0,\frac{1}{2}\Big{)}\right\}.$$

$$||e^{it\Delta }{u}_0|{|}_{L^p((0,1),L^q(M))}\le C||u_0|{|}_{L^2(M)}$$

$$||\chi (\Delta -\lambda +i0{)}^{-1}\chi |{|}_{L^2\to L^2}\le C\frac{\log (\lambda )}{{\lambda }^{\frac{1}{2}}},\text{ for }\chi \in C_0^{\infty }(M)$$

$$||e^{it{\Delta }_X}|{|}_{L^1\to L^{\infty }}\le \{\begin{array}{l}C|t{|}^{-(n+1)/2},\text{ for }|t|\le 1\\ C|t{|}^{-3/2},\text{ for }|t|>1.\end{array}$$

$$||e^{it{\Delta }_X}{u}_0|{|}_{L^p(\mathbb{R};L^q(X))}\le C||u_0|{|}_{L^2(X)},$$

$$\begin{array}{c}K(t;\rho (z,z^{\prime }))=c|t{|}^{-\frac{1}{2}}{e}^{-it{n}^2/4}(\sinh (\rho {)}^{-1}{\partial }_{\rho }{)}^{\frac{n}{2}}{e}^{i{\rho }^2/4t},n\text{ even}\\ K(t;\rho (z,z^{\prime }))=c|t{|}^{-\frac{3}{2}}{e}^{-it{n}^2/4}(\sinh (\rho {)}^{-1}{\partial }_{\rho }{)}^{\frac{n-1}{2}}{\int }_{\rho }^{\infty }\frac{e^{i{s}^2/4t}s}{\sqrt{\cosh s-\cosh \rho }}ds,n\text{ odd}\end{array}$$

$$\begin{cases}C|t|^{-(n+1)/2}\Big{(}\frac{\rho}{\sinh\rho}\Big{)}^{\frac{n}{2}},\textrm{ for }|t|\leq 1\\ C|t|^{-3/2}\Big{(}\frac{\rho}{\sinh\rho}\Big{)}^{\frac{n}{2}},\ \textrm{ for }|t|>1\end{cases}$$

$$i{\partial }_{t}K(t;z,z^{\prime })=-{\Delta }_{z}K(t;z,z^{\prime })=-{\Delta }_{z^{\prime }}K(t;z,z^{\prime }),$$

$$|{\Delta }_z^{j}K(t;z,z^{\prime })|+|{\Delta }_{z^{\prime }}^{j}K(t;z,z^{\prime })|\le \{\begin{array}{l}{C}^{\prime }|t{|}^{-(n+1)/2-2j}(1+\rho {)}^{\frac{n}{2}+2j}{e}^{-\frac{n}{2}\rho },\text{ for }|t|\le 1\\ C^{\prime }|t{|}^{-3/2}(1+\rho {)}^{\frac{n}{2}+2j}{e}^{-\frac{n}{2}\rho },\text{ for }|t|>1\end{array}$$

$$P_s(z,z^{\prime }):=\sum _{\gamma \in \Gamma }{e}^{-s\rho (z,\gamma .z^{\prime })},z,z^{\prime }\in {\mathbb{H}}^{n+1}.$$

$$\sum _{\gamma \in \Gamma ,{\ell }_{\gamma }>R}{e}^{-s\rho (z,\gamma z^{\prime })}\le C_{ϵ}(x(z)x(z^{\prime }){)}^s.$$

$$e^{-\rho (z,\gamma z^{\prime })}\le C_2{e}^{-{\ell }_{\gamma }}x(z)x(z^{\prime }).$$

$$K_X(t;z,z^{\prime }):=\sum _{\gamma \in \Gamma }K(t;z,\gamma .z^{\prime })$$

$$|K_{X}(t;z,z^{\prime})|\leq\begin{cases}C_{s}|t|^{-(n+1)/2}\Big{(}\sum_{\gamma\in\Gamma,\ell_{\gamma}\leq R}e^{-s\rho(z,\gamma z^{\prime})}+x(z)^{s}x(z^{\prime})^{s}\Big{)},\textrm{ for }|t|\leq 1\\ C_{s}|t|^{-3/2}\Big{(}\sum_{\gamma\in\Gamma,\ell_{\gamma}\leq R}e^{-s\rho(z,\gamma z^{\prime})}+x(z)^{s}x(z^{\prime})^{s}\Big{)},\textrm{ for }|t|>1\end{cases}$$

$$\begin{array}{c}\underset{z,z^{\prime }\in \mathcal{F}}{\sup }|K_X(t;z,z^{\prime })|\le \{\begin{array}{l}{C}^{\prime }|t{|}^{-(n+1)/2},\text{ for }|t|\le 1\\ C^{\prime }|t{|}^{-3/2},\text{ for }|t|>1\end{array}.\end{array}$$

$$||K_{\gamma}^{1}(t)u_{0}||_{L^{q}(\mathbb{H}^{n+1})}\leq C||\operatorname{1\negthinspace l}_{\mathcal{F}}u_{0}||_{L^{r^{\prime}}(\mathbb{H}^{n+1})}\times\left\{\begin{array}[]{ll}|t|^{-(n+1)\max(\frac{1}{2}-\frac{1}{q},\frac{1}{2}-\frac{1}{r})}&\textrm{ if }|t|\leq 1\\ |t|^{-\frac{3}{2}}&\textrm{ if }|t|>1\end{array}\right.$$

$$|K^2(t;z,z^{\prime })|\le \{\begin{array}{l}{C}_s|t{|}^{-(n+1)/2}x(z{)}^{s}x(z^{\prime }{)}^s,\text{ for }|t|\le 1\\ C_s|t{|}^{-3/2}x(z{)}^{s}x(z^{\prime }{)}^s,\text{ for }|t|>1\end{array}.$$

$$||K^{2}(t)u_{0}||_{L^{q}(\mathcal{F},{\rm dv}_{\mathbb{H}^{n+1}})}\leq C||u_{0}||_{L^{r^{\prime}}(\mathcal{F},{\rm dv}_{\mathbb{H}^{n+1}})}\times\left\{\begin{array}[]{ll}|t|^{-\frac{n+1}{2}}&\textrm{ if }|t|\leq 1\\ |t|^{-\frac{3}{2}}&\textrm{ if }|t|>1\end{array}\right.$$

$$\Vert \chi e^{it{\Delta }_X}{\psi }_{\delta }{\Vert }_{L^q(X)}=\Vert \chi {\psi }_{\delta }{\Vert }_{L^q(X)}\notin L_t^p((0,\infty )),$$

$$\Vert e^{it{\Delta }_Z}{u}_0{\Vert }_{L^p([0,T];L^q(Z))}\le C_T\Vert u_0{\Vert }_{L^2(Z)}$$

$$\Vert \varphi u{\Vert }_{L_t^2([0,T];H^{\frac{1}{2}}(X))}\le C\Vert u_0{\Vert }_{L^2(X)}.$$

$$\Vert \varphi v{\Vert }_{L_t^2([0,T];H^{\frac{1}{2}}(C^2))}\le C\Vert {\tilde{v}}_0{\Vert }_{L^2(C^2)}.$$

$$⟨Au(\cdot ,T),u(\cdot ,T)⟩-⟨Au(\cdot ,0),u(\cdot ,0)⟩={\int }_0^T⟨i[\Delta ,A]u,u⟩dt$$

$$a_{1}=\zeta^{2}(r)\langle\mathbf{\xi}\rangle^{-1}r^{-1}\mathbf{x}\cdot\mathbf{\xi}\Big{(}1-r^{-\epsilon}\Big{)}.$$

$$\begin{gathered}2\zeta^{2}(r)\langle\mathbf{\xi}\rangle^{-1}r^{-3}\big{(}1-r^{-\epsilon}\big{)}\Big{(}r^{2}|\mathbf{\xi}|^{2}-(\mathbf{x}\cdot\mathbf{\xi})^{2}\Big{)}\\ +4\zeta(r)\zeta^{\prime}(r)\langle\mathbf{\xi}\rangle^{-1}r^{-2}\big{(}1-r^{-\epsilon}\big{)}(\mathbf{x}\cdot\mathbf{\xi})^{2}\\ +2\zeta^{2}(r)\langle\mathbf{\xi}\rangle^{-1}r^{-3}\epsilon r^{-\epsilon}(\mathbf{x}\cdot\mathbf{\xi})^{2}.\end{gathered}$$

$${\int }_0^T\Vert \varphi u{\Vert }_{H^{\frac{1}{2}}(M)}^{2}dt\le C\Vert u_0{\Vert }_{L^2(M)}^2,$$

$$\sigma ({\Delta }_{C^2})={\rho }^2+(\mathrm{sech}r{)}^2{\omega }^2$$

Full text

The influence of the geometry on the behaviour of solutions of linear or non linear partial differential equations has been widely studied recently, and especially in the context of wave or Schrödinger equations. In particular, the understanding of the smoothing effect for the Schrödinger flow and Strichartz type estimates has been related to the global behaviour of the geodesic flow on the manifold (see for example the works by Doi [21] and Burq [11]). Let us recall that for the Laplacian $\Delta$ on a $d$-dimensional non-compact Riemannian manifold $(M,g)$, the local smoothing effect for bounded time $t\in[0,T]$ and Schrödinger waves $u=e^{it\Delta}u_{0}:M\times\mathbb{R}\to\mathbb{C}$ is the estimate

[TABLE]

where $C_{T}>0$ is a constant depending a priori on $T$ and $\chi$ is a compactly supported smooth function (the assumption on $\chi$ can of course be weakened in many cases, e.g for $M=\mathbb{R}^{d}$) [19]. In other words, although the solution is only $L^{2}$ in space uniformly in time, it is actually half a derivative better (locally) in an $L^{2}$-in-time sense. For its description in geometric settings, the picture now is fairly complete: the so called “nontrapping condition” stating roughly that every geodesic maximally extended goes to infinity, is known to be essentially necessary and sufficient (modulo reasonable conditions near infinity) [11].

Another tool for analyzing non-linear Schrödinger equations is the family of so-called Strichartz estimates introduced by [40]: for Schrödinger waves on Euclidean space $\mathbb{R}^{d}$ with initial data $u_{0}$,

[TABLE]

If $\sup_{T\in(0,\infty)}C_{T}<\infty$, we will say that a global-in-time Strichartz estimate holds. Such a global-in-time estimate has been proved by Strichartz for the flat Laplacian on $\mathbb{R}^{d}$ while the local-in-time estimate is known in several geometric situations where the manifold is non-trapping (asymptotically Euclidean, conic or hyperbolic); see [9, 8, 25, 39]. On the other hand it is clear that such a global-in-time estimate cannot hold on compact manifolds, for it suffices to consider the function $u_{0}=1$. The situation is similar for the non-compact case in the presence of elliptic (stable) non-degenerate periodic orbits of the geodesic flow: as remarked by M. Zworski, the quasi-modes constructed by Babič [4] and Pyškina [36] (see also the work by Ralston [37]) show that for Schrödinger solutions, some loss must occur as far as Strichartz (or smoothing) estimates are concerned; and moreover, that no Strichartz estimates can be true globally in time in the presence of such orbits. On the other hand Burq-Gérard-Tzvetkov [12] proved that (0.1) holds on compact manifolds for finite time if one replaces $||u_{0}||_{L^{2}(M)}$ by $||u_{0}||_{H^{1/p}(M)}$, meaning that a Strichartz estimate is satisfied if one accepts some loss of derivatives. It is however certainly not optimal in general since Bourgain [10] proved that for the flat torus $(\mathbb{R}/2\pi\mathbb{Z})^{2}$, the Strichartz estimate for $p=q=4$ holds with $\epsilon$ loss of derivatives for any $\epsilon>0$. Another striking example has been given by Takaoka and Tzvetkov [42] by adapting the ideas of Bourgain, namely the case of the two dimensional infinite flat cylinder $S^{1}\times\mathbb{R}$ where (0.1) holds (with no loss of derivatives) if $p=q=4$; note that this manifold is trapping. An example with a repulsive potential $V(x_{1},x_{2})=x_{1}^{2}-x_{2}^{2}$ has also been studied by Carles [16], who proved that global-in-time Strichartz estimates with no loss hold in this case. To summarize, it is not really understood when (i.e. under what geometric conditions) a loss in Strichartz estimates must occur, and if it does, how large that loss must be.

The purpose of this article is precisely to give some examples of Riemannian manifolds where trapping does occur (and consequently loss is unavoidable for the smoothing effect), but nevertheless, since the dynamics are hyperbolic near the trapped set, we are able to prove (local-in-time) Strichartz estimates without loss for Schrödinger solutions.

The first example, which we treat in Section 1, is the case of a convex co-compact hyperbolic manifold of dimension $d=n+1$, with a limit set of Hausdorff dimension $\delta<n/2$. The simplest example of such a manifold is the two dimensional infinite hyperbolic cylinder with one single trapped geodesic. In this case, the calculations are quite explicit, representing the Schrödinger kernel as an average over the group of the Schrödinger kernel on the hyperbolic space $\mathbb{H}^{n+1}$, and we are able to prove that not only Strichartz estimates, but the stronger $L^{1}\to L^{\infty}$ dispersive estimates hold for the Schrödinger group.

[TABLE]

for all $(p,q)$ such that $(1/p,1/q)\in T_{n}$ where

[TABLE]

These manifolds are non-compact, infinite volume, with finitely many ends of funnel type, they have constant curvature $-1$ and possess infinitely many closed geodesics; it is remarkable that despite this last fact, a sharp dispersive estimate holds for all time. We also remark that Strichartz estimates for the same range of $(p,q)$ have been recently shown by Anker-Pierfelice [3] for the model non-trapping case $\mathbb{H}^{n+1}$ (see also [5, 27] for the estimate (0.1) in that setting). The triangle of admissibility for the Strichartz exponent $(p,q)$ is a consequence of the exponential decay of the integral kernel of the Schrödinger operator at infinity. Notice that in the asymptotically hyperbolic setting, J-M. Bouclet [8] proved Strichartz estimates without loss of derivatives for bounded times and with admissibility exponents satisfying (0.1) for non-trapping such manifolds (in this case the sectional curvature is not assumed constant, but rather tending to $-1$ at infinity).

The second example is on the manifold $Z$ given by the connected sum of two copies of Euclidean $\mathbb{R}^{2}$. This we provide with a Riemannian metric $g$ by gluing two copies of the Euclidean metric on $\mathbb{R}^{2}$ with the metric on the 2-dimensional hyperbolic cylinder. Essentially because $Z$ is formed from pieces all of which satisfy Strichartz estimates without loss, the same is true for $(Z,g)$. Actually we need to use local smoothing estimates to control error terms in the transition region, but since this region is disjoint from the single trapped orbit, there are no losses in such local smoothing estimates. This example is given in Section 2; the main result is Theorem 2.1.

Our last family of examples, in Section 3, is a generalization of that in Section 2 to higher dimensions and more complicated trapped sets. It is similar to the class of manifolds studied recently by Nonnenmacher and Zworski [32]: we consider asymptotically Euclidean (or more generally asymptotically conic) manifolds, the curvature of which is assumed to be negative in a geodesically convex compact part that includes the (projection of) the trapped set, and such that the trapped set is small enough in the sense that the topological pressure $P(1/2)$ of the trapped set evaluated at $1/2$ is negative (see the definition in Section 3.1). This is a dynamical condition which generalizes the condition $\delta<n/2$ above to this more general setting and roughly speaking means that the trapped set is filamentary with small fractal dimension. For instance, for surfaces (dimension $d=2$) this means that the trapped set (as a subset of the cosphere bundle $S^{*}M$) has Hausdorff dimension less than $2$. More precisely our result (which includes the example in Section 2 as a special case) is

[TABLE]

for all $u_{0}\in L^{2}(M)$ and $(p,q)$ satisfying (0.1) and $p>2$.

Note that Christianson [18] and Datchev [20] showed that Strichartz estimates hold with an $\epsilon$ loss of derivatives for all $\epsilon>0$ in that setting. Our method is based on the use of the local smoothing effect with log loss, which follows from the resolvent estimate of [32] (see also [20])

[TABLE]

together with a sharp dispersive estimate on the logarithmically extended time interval $t\in(0,h\log(h))$ for the frequency localized operator $e^{it\Delta}\varphi(h^{2}\Delta)$ where $\varphi\in C_{0}^{\infty}((0,\infty))$ and $h\in(0,h_{0})$ is small. Roughly speaking this logarithmic extension of the time interval of validity of the dispersive estimate allows one to recover the $\log$ loss in the local smoothing estimate. This dispersive estimate is inspired by the works of Anantharaman [1], Anantharaman-Nonnenmacher [2] and Nonnenmacher-Zworski [32]. In particular, the technique for proving the dispersive estimate for logarithmically extended time originates in [1, Theorem 1.3.3], while the idea of combining the exponential decay provided by this theorem with the topological pressure assumption (see (A4) in Section 3) is due to [32].

Acknowledgement. We thank S. Nonnenmacher, N. Anantharaman and F. Planchon for helpful discussions and references. N.B. is supported by ANR grant ANR-07-BLAN-0250. C.G. is supported by ANR grant ANR-09-JCJC-0099-01 and thanks the Mathematical Sciences Institute of ANU Canberra where part of this work was done. A.H. is supported by Australian Research Council Discovery Grant DP0771826 and thanks the mathematics department at Université Paris 11 for its hospitality. We are finally grateful to the referee for his careful reading.

1. Hyperbolic manifolds

A convex co-compact subgroup $\Gamma\subset{\rm SO}(n+1,1)$ is a discrete group of orientation preserving isometries of hyperbolic space $\mathbb{H}^{n+1}$, consisting of hyperbolic isometries and such that the quotient $X:=\Gamma\backslash\mathbb{H}^{n+1}$ has finite geometry and infinite volume. If one considers the ball model $B^{n+1}$ of $\mathbb{H}^{n+1}$, a hyperbolic isometry is an isometry of $\mathbb{H}^{n+1}$ which fixes exactly two points on $\overline{B^{n+1}}$, and these points are on the boundary $S^{n}=\partial\overline{\mathbb{H}^{n+1}}$. The manifold $X:=\Gamma\backslash\mathbb{H}^{n+1}$ is said to be convex co-compact hyperbolic; it is a smooth complete hyperbolic manifold which admits a natural conformal compactification $\bar{X}$ and the hyperbolic metric $g$ on $X$ is of the form $g=\bar{g}/x^{2}$ where $x$ is a smooth boundary defining function of $\bar{X}$ and $\bar{g}$ a smooth metric on $\bar{X}$. The set of closed geodesics is in correspondence with the classes of conjugacy of the group $\Gamma$. The limit set of $\Gamma$ is the set of accumulation points on the sphere $S^{n}=\partial\overline{\mathbb{H}^{n+1}}$ of the orbit $\Gamma.m$ where $m\in\mathbb{H}^{n+1}$ is any point. It has a Hausdorff dimension given by $\delta\in[0,n)$, and the trapped set of the geodesic flow on the unit tangent bundle $SX$ has Hausdorff dimension $2\delta+1$; see [41, 44].

The simplest example is $\Gamma=\mathbb{Z}$ acting by powers of a fixed dilation $D$ on the upper half space model of $\mathbb{H}^{n+1}$. Then the limit set consists of two points $\{0,\infty\}$, $\delta=0$, and $\mathbb{H}^{n+1}/\Gamma$ is the $(n+1)$-dimensional hyperbolic cylinder.

We prove the following.

Theorem 1.1**.**

Let $X$ be an $(n+1)$-dimensional convex co-compact hyperbolic manifold such that its limit set has Hausdorff dimension $\delta<n/2$. Then $e^{it\Delta_{X}}$ has a smooth Schwartz kernel for $t\neq 0$, and there is a constant $C$ such that the following dispersive estimate holds for all $t\neq 0$:

[TABLE]

Moreover the following global-in-time Strichartz estimates hold:

[TABLE]

for all $(p,q)$ such that $(1/p,1/q)\in T_{n}$ where $T_{n}$ is given by (0.2).

Proof.

The integral kernel of the Schrödinger operator $e^{it\Delta_{\mathbb{H}^{n+1}}}$ on hyperbolic space has been computed by V. Banica [5]. It is a function of the hyperbolic distance

[TABLE]

where $\rho=\rho(z,z^{\prime}):=d_{\mathbb{H}^{n+1}}(z,z^{\prime})$. In both cases we remark, like for the heat kernel, that the kernel $K(t;.,.)$ is smooth on $\mathbb{H}^{n+1}\times\mathbb{H}^{n+1}$ for $t\not=0$; this is clear when $n+1$ is odd, and needs a bit more analysis when $n+1$ is even. From this expression we obtain an upper bound for $|K(t;z,z^{\prime})|$ (see [5, Prop 4.1 and Sec. 4.2]) for $t\not=0$ of the form

[TABLE]

for some constant $C>0$. Using the inequality $(\rho/\sinh\rho)\leq(1+\rho)e^{-\rho}$, and since

[TABLE]

one can deduce that for $t\not=0$ bounded

[TABLE]

and in particular $K(t,z,z^{\prime})$ is smooth in $z,z^{\prime}$.

To proceed we use the celebrated result of Patterson and Sullivan [34, 41] that the dimension of the limit set $\delta$ is the exponent of convergence of the Poincaré series

[TABLE]

Lemma 1.2**.**

Let $\mathcal{F}\in\mathbb{H}^{n+1}$ be a fundamental domain of the convex co-compact group $\Gamma$ and let $x$ be a boundary defining function of the compactification $\bar{X}$ of $X=\Gamma\backslash\mathbb{H}^{d+1}$, which we also view as a function on $\mathcal{F}$. For each $\gamma\in\Gamma$, define by $\ell_{\gamma}$ the translation length of $\gamma$. Then there exists $R>0$ such that for all $\epsilon>0$, there is $C_{\epsilon}>0$ such that for $s>\delta+\epsilon$ and all $z,z^{\prime}\in\mathcal{F}$,

[TABLE]

Proof.

In [23, Lemma 5.2], it is shown that there exist constants $C_{1},C_{2}>0$ such that for all $\gamma\in\Gamma$ such that $\ell_{\gamma}>C_{1}$

[TABLE]

Now it suffices to sum after raising to the power $s$ and to use the fact that $\sum_{\gamma\in\Gamma}e^{-s\ell_{\gamma}}<C_{\epsilon}$ for some $C_{\epsilon}$ if $s>\delta+\epsilon$. $\square$

Combining (1.6) and (1.4), we deduce that for $z,z^{\prime}\in\mathcal{F}$, the series

[TABLE]

converges uniformly and for all $s<n/2$ there exists $C_{s}>0$ such that for all $z,z^{\prime}\in\mathcal{F}$

[TABLE]

where $R$ is the constant in (1.6). This leads directly to the dispersive estimate

[TABLE]

for some constants $C^{\prime}$. Moreover, using (1.5), the same argument shows that the series $K_{X}(t;z,z^{\prime})$ is smooth in $z,z^{\prime}$ for $t\not=0$. Let $\mathcal{F}$ be a fundamental domain of $\Gamma$. For any $u_{0}\in C_{0}^{\infty}(X)$ the function $u(t):=\int_{\mathcal{F}}K_{X}(t;z,z^{\prime})u_{0}(z^{\prime})dz^{\prime}$ is smooth on $\mathbb{H}^{n+1}$ and satisfies $u(t,\gamma z)=u(t,z)$ for any $\gamma\in\Gamma$, thus $u(t)$ is smooth on $X$. Moreover it solves the Schrödinger equation on $X$ with initial data $u(0)=u_{0}$, so $u(t)=e^{it\Delta_{X}}u_{0}$. This implies that $K_{X}$ is the Schwartz kernel of $e^{it\Delta_{X}}$ on $X$.

We next prove the global-in-time Strichartz estimates (1.2) (notice that these estimates for a finite time interval follow immediately from the small-time dispersive estimate (1.1) and from Keel-Tao [28], following the method of Anker-Pierfelice [3]). Let $\gamma\in\Gamma$ be such that $\ell_{\gamma}\leq R$ where $R$ is the constant in (1.6), then define $K_{\gamma}^{1}(t)$ to be the operator acting on $\mathcal{F}$ with $L^{\infty}$ kernel $\operatorname{1\negthinspace l}_{\mathcal{F}}(z)K(t;z,\gamma z^{\prime})\operatorname{1\negthinspace l}_{\mathcal{F}}(z^{\prime})$. Since $\gamma$ is an isometry of $\mathbb{H}^{n+1}$, this operator can also be written as $f\to\operatorname{1\negthinspace l}_{\mathcal{F}}e^{it\Delta_{\mathbb{H}^{n+1}}}\gamma_{*}(\operatorname{1\negthinspace l}_{\mathcal{F}}f)$. Then from Theorem 3.4 of Anker-Pierfelice [3] and the fact that push-forward $\gamma_{*}$ is an isometry on any $L^{r^{\prime}}(\mathbb{H}^{n+1})$, we get the estimate

[TABLE]

for all $2<q,r\leq\infty$ and $1/r^{\prime}+1/r=1$, so the same estimate holds for $K^{1}(t):=\sum_{\gamma\in\Gamma,\ell_{\gamma}\leq R}K^{1}_{\gamma}(t)$. Now consider the operator $K^{2}(t)$ acting on $\mathcal{F}$ whose $L^{\infty}$ kernel is $K^{2}(t;z,z^{\prime}):=K_{X}(t;z,z^{\prime})-K^{1}(t;z,z^{\prime})$. From (1.6) and (1.4), this kernel is bounded (for all $s<n/2$) by

[TABLE]

Since the hyperbolic metric on $\mathcal{F}$ induces a measure of the form $x^{-n-1}\mu$ for some bounded measure $\mu$ on $\mathcal{F}$ we see that the function $x^{s}$ is in $L^{\alpha}(\mathcal{F},{\rm dv}_{\mathbb{H}^{n+1}})$ for all $\alpha>n/s$, and hence deduce directly that $K^{2}(t)$ satisfies

[TABLE]

for all $q,r\in(2,\infty]$ and $r^{\prime}$ the conjugate exponent of $r$, so the same estimate holds for $K_{X}(t)$ when combining with (1.9). Then it suffices to conclude using the standard $TT^{*}$ argument exactly as in the proof of Theorem 3.6 of Anker-Pierfelice [3] and we obtained the claimed Strichartz estimate. $\square$

Remark 1.3**.**

For hyperbolic quotients with dimension of limit set $\delta>n/2$, the positive number $\delta(n-\delta)$ is an $L^{2}$ eigenvalue with multiplicity one and smooth eigenvector $\psi_{\delta}$. It follows easily from Section 2 of [35] (or the general result of Mazzeo-Melrose [30] about the structure of the resolvent) that $\psi_{\delta}\in L^{p}(X)$ for all $p>n/\delta$, thus in particular for all $p\geq 2$. This implies that for $q\geq 2$, $2<p<\infty$ and all $\chi\in L^{\infty}(X)$

[TABLE]

so global-in-time Strichartz estimates cannot hold when $\delta>n/2$, even with a space cut-off.

2. Connected sum of two copies of $\mathbb{R}^{2}$

In this section, we give an example of a Riemannian manifold $(Z,g)$ which is topologically the connected sum of two copies of $\mathbb{R}^{2}$, and is geometrically Euclidean near infinity, and hyperbolic near the ‘waist’ (and hence with a single trapped ray), for which Strichartz estimates without loss are valid. The idea is simple; since Strichartz estimates without loss are valid on flat $\mathbb{R}^{2}$, and on the hyperbolic cylinder (thanks to Theorem 1.1), then they should also be valid on a space obtained by gluing pieces of these manifolds together, provided that no additional trapping is created by the gluing procedure.

Let us consider an asymptotically Euclidean manifold $(Z,g)$ which is the connected sum of two copies of $\mathbb{R}^{2}$, joined by a neck which has a neighbourhood $U$ isometric to a neighbourhood $U^{\prime}$ of the short closed geodesic, or ‘waist’, on the hyperbolic two-cylinder $C^{2}$. We denote this short closed geodesic by $\gamma$, whether on $Z$ or on $C^{2}$. We can write down an explicit metric $g$ for such a manifold, on $\mathbb{R}\times S^{1}$, in the form $dr^{2}+f(r)^{2}d\theta^{2}$, where $d\theta^{2}$ is the metric on $S^{1}$ of length $2\pi$, and where $f(r)=\cosh r$ for small $r$, say $r\leq 3\eta$ for some small $\eta>0$, and is equal to $|r|+a$ for large $|r|$, say $|r|\geq R$ (where $a$ is a constant). We also choose $f$ so that $f^{\prime}(r)$ has the same sign as $r$; it is easy to see that this is compatible with the condition that $f(r)=\cosh r$ for small $|r|$ and $|r|+a$ for large $|r|$. The equations of motion for geodesic flow then give $\ddot{r}=2f^{\prime}(r)f(r)\dot{\theta}^{2}$, which has the same sign as $r$, and it is straightforward to deduce from this that there can be no trapped geodesic other than the waist $\gamma$ at $r=0$. For any such manifold $(Z,g)$ we have

Theorem 2.1**.**

For any finite $T$ there is a constant $C_{T}$ such that

[TABLE]

for all $(p,q)$ satisfying (0.1) with $d=2$ and all $u_{0}\in L^{2}(Z)$.

Before giving the proof we introduce some further notation and definitions. We will compare $Z$ to the hyperbolic cylinder $C^{2}$ and to the auxiliary Riemannian manifold $(\tilde{Z}=\mathbb{R}^{2},\tilde{g})$, given in standard polar coordinates $(r,\theta)$ on $\mathbb{R}^{2}$ by $\tilde{g}=dr^{2}+\tilde{f}(r)^{2}d\theta$, where $\tilde{f}(r)=f(r)$ for $r\geq\eta$, $\tilde{f}^{\prime}(r)>0$, and is equal to $r$ for small $r$. Reasoning as above, we see that the metric $\tilde{g}$ is nontrapping. We will take a Schrödinger wave $u$ on $Z$ and decompose it so that one piece lives on $C^{2}$ and the other lives on $\tilde{Z}$, and we will deduce Strichartz without loss on $Z$ from the fact that Strichartz without loss holds for both $C^{2}$ and $\tilde{Z}$.

Proof.

Let $U\subset Z$ be a neighbourhood of $\{r=0\}$, say $\{|r|<2\eta\}$, thus containing the projection of the trapped set of $Z$. By construction, the metric is exactly hyperbolic in a neighbourhood of $\overline{U}$. We decompose $u=u_{i}+u_{e}$, where $u_{i}=\chi u$ is supported in $U$ and $u_{e}=(1-\chi)u$ is supported where the metric $g$ is identical to $\tilde{g}$. (Thus $\nabla\chi$ is supported where $\eta\leq|r|\leq 2\eta$.) We prove the estimate (2.1) separately for $u_{i}$ and $u_{e}$. As stated above, the idea is to regard $u_{e}$ as solving a PDE on $\tilde{Z}$ and to regard $u_{i}$ as solving a PDE on $C^{2}$.

We first prove a local smoothing result for $Z$, $\tilde{Z}$ and $C^{2}$. This is essentially standard, but we give the details for the reader’s convenience (and in keeping with the expository character of this section). For applications in the following section, we give a result in any dimension.

Lemma 2.2**.**

(i) Suppose that $X$ is a $d$-dimensional manifold with Euclidean ends and with trapped set $K\subset T^{*}X$. Suppose that $u$ solves the Schrödinger equation on $(X,g)$ with initial condition $u_{0}$, and suppose that $\phi\in C_{c}^{\infty}(M)$ is supported away from the projection of the trapped set $\pi(K)$. Then

[TABLE]

(ii) Suppose that $v$ solves the Schrödinger equation on $(C^{2},g_{\operatorname{hyp}})$ with initial condition $v_{0}$, and suppose that $\phi\in C_{c}^{\infty}(C^{2})$ is supported away from the closed geodesic $\gamma$. Then

[TABLE]

Proof.

This result can be deduced from the resolvent estimate of Cardoso-Vodev [15], but for completeness we give a proof via a positive commutator argument. We construct a zeroth order pseudodifferential operator $A$ on $X$ such that $i[\Delta,A]$ has a nonnegative principal symbol which is elliptic on the support of $\phi$. Then we use the identity

[TABLE]

valid for any Schrödinger wave $u$. Since $i[\Delta,A]$ is order one and elliptic on the support of $\phi$, the right hand side is equal to $c\|\phi u\|_{L^{2}([0,T];H^{1/2})}^{2}$ plus terms which are essentially positive, while the left hand side is bounded by $C\|u_{0}\|_{L^{2}}^{2}$, giving the estimate.

Let us set $A=A_{1}+A_{2}$, where $A_{1}$ is supported in the region where $X$ is Euclidean and $A_{2}$ is properly supported. We take $R>0$ large enough so that each end of $X$ has a neighbourhood isometric to $\mathbb{R}^{d}\setminus\overline{B(0,R)}$ and use Euclidean coordinates $\mathbf{x}=(x_{1},x_{2},\dots x_{d})$ on this neighbourhood with dual cotangent coordinates $\mathbf{\xi}$. We write $r=|\mathbf{x}|$ and take $A_{1}$ to have principal symbol

[TABLE]

Here $\zeta(t)$ is chosen to be [math] for $t<R$ and $1$ for $t\geq 2R$ and to be nondecreasing, where $R$ is sufficiently large that $1-R^{-\epsilon}>\frac{1}{2}$, say. We understand this to mean that $a_{1}$ is defined as above on each end of $X$. Explicitly, we could take $A_{1}=\zeta(r)r^{-1}(1+\Delta)^{-1/2}\frac{1}{2}(\mathbf{x}\cdot D_{\mathbf{x}}+D_{\mathbf{x}}\cdot\mathbf{x})\zeta(r)(1-r^{-\epsilon})$, where here $\Delta$ denotes the flat Laplacian on $\mathbb{R}^{d}$; notice that $(1+\Delta)^{-1/2}$ makes sense since it is both pre- and post-multiplied by $\zeta(r)$ which is supported where the metric is Euclidean.

Then the derivative $a_{1}$ along the Hamilton vector field of $\sigma(\Delta_{X})$, namely the geodesic flow $2\mathbf{\xi}\cdot\partial_{\mathbf{x}}$, is

[TABLE]

We see that this is nonnegative everywhere, and bounded below by $C\langle\mathbf{\xi}\rangle r^{-1-\epsilon}$ for $r\geq 2R$.

Now we define a symbol $a_{2}$ which will be supported in the region $r\leq 4R$. First we introduce some notation: for $\tilde{R}\geq R$, let $E_{\tilde{R}}$ denote the union of ends $\mathbb{R}^{d}\setminus B(0,\tilde{R})$, and let $U_{\tilde{R}}\subset T^{*}X$ denote $\pi^{-1}(X\setminus E_{\tilde{R}})$. We choose conic neighbourhoods $U_{<}$ and $U$ of $K$ such that $\overline{U_{<}}\subset U$