Strichartz estimates without loss on manifolds with hyperbolic trapped geodesics
aa r X i v : . [ m a t h . A P ] M a r STRICHARTZ ESTIMATES WITHOUT LOSS ON MANIFOLDS WITHHYPERBOLIC TRAPPED GEODESICS.
NICOLAS BURQ, COLIN GUILLARMOU, AND ANDREW HASSELL
Abstract.
In [21], Doi proved that the L t H x local smoothing effect for Schr¨odinger equationon a Riemannian manifold does not hold if the geodesic flow has one trapped trajectory. Weshow in contrast that Strichartz estimates and L → L ∞ dispersive estimates still hold withoutloss for e it ∆ in various situations where the trapped set is hyperbolic and of sufficiently smallfractal dimension. The influence of the geometry on the behaviour of solutions of linear or non linear par-tial differential equations has been widely studied recently, and especially in the context ofwave or Schr¨odinger equations. In particular, the understanding of the smoothing effect for theSchr¨odinger flow and Strichartz type estimates has been related to the global behaviour of thegeodesic flow on the manifold (see for example the works by Doi [21] and Burq [11]). Let us recallthat for the Laplacian ∆ on a d -dimensional non-compact Riemannian manifold ( M, g ), the localsmoothing effect for bounded time t ∈ [0 , T ] and Schr¨odinger waves u = e it ∆ u : M × R → C isthe estimate || χe it ∆ u || L ((0 ,T ); H / ( M )) ≤ C T || u || L ( M ) , ∀ u ∈ L ( M )where C T > T and χ is a compactly supported smoothfunction (the assumption on χ can of course be weakened in many cases, e.g for M = R d ) [19].In other words, although the solution is only L in space uniformly in time, it is actually half aderivative better (locally) in an L -in-time sense. For its description in geometric settings, thepicture now is fairly complete: the so called “nontrapping condition” stating roughly that everygeodesic 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 Schr¨odinger equations is the family of so-called Strichartzestimates introduced by [40]: for Schr¨odinger waves on Euclidean space R d with initial data u ,(0.1) || e it ∆ u || L p ((0 ,T ); L q ( R d )) ≤ C T || u || L ( R d ) if p, q ≥ , p + dq = d , ( p, q ) = (2 , ∞ ) . If sup T ∈ (0 , ∞ ) C T < ∞ , we will say that a global-in-time Strichartz estimate holds. Such aglobal-in-time estimate has been proved by Strichartz for the flat Laplacian on R d while thelocal-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 clearthat such a global-in-time estimate cannot hold on compact manifolds, for it suffices to considerthe function u = 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, thequasi-modes constructed by Babiˇc [4] and Pyˇskina [36] (see also the work by Ralston [37]) showthat for Schr¨odinger solutions, some loss must occur as far as Strichartz (or smoothing) estimatesare concerned; and moreover, that no Strichartz estimates can be true globally in time in thepresence of such orbits. On the other hand Burq-G´erard-Tzvetkov [12] proved that (0.1) holdson compact manifolds for finite time if one replaces || u || L ( M ) by || u || H /p ( M ) , meaning that aStrichartz estimate is satisfied if one accepts some loss of derivatives. It is however certainly notoptimal in general since Bourgain [10] proved that for the flat torus ( R / π Z ) , the Strichartz Mathematics Subject Classification.
Primary 58Jxx, Secondary 35Q41.
Key words and phrases.
Strichartz estimates, Schr¨odinger equation, hyperbolic trapped set. estimate for p = q = 4 holds with ǫ loss of derivatives for any ǫ >
0. Another striking examplehas been given by Takaoka and Tzvetkov [42] by adapting the ideas of Bourgain, namely thecase of the two dimensional infinite flat cylinder S × R where (0.1) holds (with no loss ofderivatives) if p = q = 4; note that this manifold is trapping. An example with a repulsivepotential V ( x , x ) = x − x 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 understoodwhen (i.e. under what geometric conditions) a loss in Strichartz estimates must occur, and if itdoes, how large that loss must be.The purpose of this article is precisely to give some examples of Riemannian manifolds wheretrapping does occur (and consequently loss is unavoidable for the smoothing effect), but neverthe-less, since the dynamics are hyperbolic near the trapped set, we are able to prove (local-in-time)Strichartz estimates without loss for Schr¨odinger solutions.The first example, which we treat in Section 1, is the case of a convex co-compact hyperbolicmanifold of dimension d = n + 1, with a limit set of Hausdorff dimension δ < n/
2. The simplestexample of such a manifold is the two dimensional infinite hyperbolic cylinder with one singletrapped geodesic. In this case, the calculations are quite explicit, representing the Schr¨odingerkernel as an average over the group of the Schr¨odinger kernel on the hyperbolic space H n +1 , andwe are able to prove that not only Strichartz estimates, but the stronger L → L ∞ dispersiveestimates hold for the Schr¨odinger group. Theorem 0.1.
Let X be an ( n + 1) -dimensional convex co-compact hyperbolic manifold suchthat its limit set has Hausdorff dimension δ < n/ . Then the following dispersive and Strichartzestimates without loss hold: || e it ∆ X || L ( X ) → L ∞ ( X ) ≤ ( C | t | − ( n +1) / , for | t | ≤ C | t | − / , for | t | > , || e it ∆ X u || L p ( R ; L q ( X )) ≤ C || u || L ( X ) for all ( p, q ) such that (1 /p, /q ) ∈ T n where (0.2) T n := (cid:26)(cid:16) p , q (cid:17) ∈ (cid:16) , i × (cid:16) , (cid:17) ; 2 p ≥ n + 12 − n + 1 q (cid:27) ∪ (cid:26)(cid:16) , (cid:17)(cid:27) . These manifolds are non-compact, infinite volume, with finitely many ends of funnel type,they have constant curvature − infinitely many closed geodesics; it is remarkablethat despite this last fact, a sharp dispersive estimate holds for all time. We also remark thatStrichartz estimates for the same range of ( p, q ) have been recently shown by Anker-Pierfelice[3] for the model non-trapping case 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 expo-nential decay of the integral kernel of the Schr¨odinger operator at infinity. Notice that in theasymptotically hyperbolic setting, J-M. Bouclet [8] proved Strichartz estimates without loss ofderivatives 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 tendingto − Z given by the connected sum of two copies of Eu-clidean R . This we provide with a Riemannian metric g by gluing two copies of the Euclideanmetric on R 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 transitionregion, but since this region is disjoint from the single trapped orbit, there are no losses in suchlocal smoothing estimates. This example is given in Section 2; the main result is Theorem 2.1.
TRICHARTZ ESTIMATES WITHOUT LOSS 3
Our last family of examples, in Section 3, is a generalization of that in Section 2 to higherdimensions and more complicated trapped sets. It is similar to the class of manifolds studiedrecently by Nonnenmacher and Zworski [32]: we consider asymptotically Euclidean (or moregenerally asymptotically conic) manifolds, the curvature of which is assumed to be negative ina geodesically convex compact part that includes the (projection of) the trapped set, and suchthat the trapped set is small enough in the sense that the topological pressure P (1 /
2) of thetrapped set evaluated at 1 / δ < n/ d = 2) this means that the trapped set (as a subset of the cospherebundle S ∗ M ) has Hausdorff dimension less than 2. More precisely our result (which includesthe example in Section 2 as a special case) is Theorem 0.2.
Let ( M, g ) satisfy assumptions (A1) — (A4) defined in (3.4) . Then Strichartzestimates without loss hold for M : there exists C > such that (0.3) || e it ∆ u || L p ((0 , ,L q ( M )) ≤ C || u || L ( M ) for all u ∈ L ( M ) and ( p, q ) satisfying (0.1) and p > . Note that Christianson [18] and Datchev [20] showed that Strichartz estimates hold with an ǫ loss of derivatives for all ǫ > || χ (∆ − λ + i − χ || L → L ≤ C log( λ ) λ , for χ ∈ C ∞ ( M )together with a sharp dispersive estimate on the logarithmically extended time interval t ∈ (0 , h log( h )) for the frequency localized operator e it ∆ ϕ ( h ∆) where ϕ ∈ C ∞ ((0 , ∞ )) and h ∈ (0 , h ) is small. Roughly speaking this logarithmic extension of the time interval of validity ofthe dispersive estimate allows one to recover the log loss in the local smoothing estimate. This dis-persive estimate is inspired by the works of Anantharaman [1], Anantharaman-Nonnenmacher [2]and Nonnenmacher-Zworski [32]. In particular, the technique for proving the dispersive estimatefor logarithmically extended time originates in [1, Theorem 1.3.3], while the idea of combiningthe 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 forhelpful discussions and references. N.B. is supported by ANR grant ANR-07-BLAN-0250. C.G. issupported by ANR grant ANR-09-JCJC-0099-01 and thanks the Mathematical Sciences Instituteof ANU Canberra where part of this work was done. A.H. is supported by Australian ResearchCouncil Discovery Grant DP0771826 and thanks the mathematics department at Universit´eParis 11 for its hospitality. We are finally grateful to the referee for his careful reading.1.
Hyperbolic manifolds A convex co-compact subgroup Γ ⊂ SO( n + 1 ,
1) is a discrete group of orientation preserv-ing isometries of hyperbolic space H n +1 , consisting of hyperbolic isometries and such that thequotient X := Γ \ H n +1 has finite geometry and infinite volume. If one considers the ball model B n +1 of H n +1 , a hyperbolic isometry is an isometry of H n +1 which fixes exactly two points on B n +1 , and these points are on the boundary S n = ∂ H n +1 . The manifold X := Γ \ H n +1 is saidto be convex co-compact hyperbolic ; it is a smooth complete hyperbolic manifold which admits anatural conformal compactification ¯ X and the hyperbolic metric g on X is of the form g = ¯ g/x where x is a smooth boundary defining function of ¯ X and ¯ g a smooth metric on ¯ X . The set ofclosed geodesics is in correspondence with the classes of conjugacy of the group Γ. The limitset of Γ is the set of accumulation points on the sphere S n = ∂ H n +1 of the orbit Γ .m where NICOLAS BURQ, COLIN GUILLARMOU, AND ANDREW HASSELL m ∈ H n +1 is any point. It has a Hausdorff dimension given by δ ∈ [0 , n ), and the trapped set ofthe geodesic flow on the unit tangent bundle SX has Hausdorff dimension 2 δ + 1; see [41, 44].The simplest example is Γ = Z acting by powers of a fixed dilation D on the upper half spacemodel of H n +1 . Then the limit set consists of two points { , ∞} , δ = 0, and H n +1 / Γ 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 suchthat its limit set has Hausdorff dimension δ < n/ . Then e it ∆ X has a smooth Schwartz kernelfor t = 0 , and there is a constant C such that the following dispersive estimate holds for all t = 0 : (1.1) || e it ∆ X || L → L ∞ ≤ ( C | t | − ( n +1) / , for | t | ≤ C | t | − / , for | t | > . Moreover the following global-in-time Strichartz estimates hold: (1.2) || e it ∆ X u || L p ( R ; L q ( X )) ≤ C || u || L ( X ) , for all ( p, q ) such that (1 /p, /q ) ∈ T n where T n is given by (0.2) .Proof. The integral kernel of the Schr¨odinger operator e it ∆ H n +1 on hyperbolic space has beencomputed by V. Banica [5]. It is a function of the hyperbolic distance(1.3) K ( t ; ρ ( z, z ′ )) = c | t | − e − itn / (sinh( ρ ) − ∂ ρ ) n e iρ / t , n even K ( t ; ρ ( z, z ′ )) = c | t | − e − itn / (sinh( ρ ) − ∂ ρ ) n − Z ∞ ρ e is / t s √ cosh s − cosh ρ ds, n oddwhere ρ = ρ ( z, z ′ ) := d H n +1 ( z, z ′ ). In both cases we remark, like for the heat kernel, that thekernel K ( t ; ., . ) is smooth on H n +1 × H n +1 for t = 0; this is clear when n + 1 is odd, and needsa bit more analysis when n + 1 is even. From this expression we obtain an upper bound for | K ( t ; z, z ′ ) | (see [5, Prop 4.1 and Sec. 4.2]) for t = 0 of the form(1.4) C | t | − ( n +1) / (cid:16) ρ sinh ρ (cid:17) n , for | t | ≤ C | t | − / (cid:16) ρ sinh ρ (cid:17) n , for | t | > C >
0. Using the inequality ( ρ/ sinh ρ ) ≤ (1 + ρ ) e − ρ , and since i∂ t K ( t ; z, z ′ ) = − ∆ z K ( t ; z, z ′ ) = − ∆ z ′ K ( t ; z, z ′ ) , one can deduce that for t = 0 bounded(1.5) | ∆ jz K ( t ; z, z ′ ) | + | ∆ jz ′ K ( t ; z, z ′ ) | ≤ ( C ′ | t | − ( n +1) / − j (1 + ρ ) n +2 j e − n ρ , for | t | ≤ C ′ | t | − / (1 + ρ ) n +2 j e − n ρ , for | t | > K ( t, z, z ′ ) is smooth in z, z ′ .To proceed we use the celebrated result of Patterson and Sullivan [34, 41] that the dimensionof the limit set δ is the exponent of convergence of the Poincar´e series P s ( z, z ′ ) := X γ ∈ Γ e − sρ ( z,γ.z ′ ) , z, z ′ ∈ H n +1 . Lemma 1.2.
Let F ∈ H n +1 be a fundamental domain of the convex co-compact group Γ and let x be a boundary defining function of the compactification ¯ X of X = Γ \ H d +1 , which we also viewas a function on F . For each γ ∈ Γ , define by ℓ γ the translation length of γ . Then there exists R > such that for all ǫ > , there is C ǫ > such that for s > δ + ǫ and all z, z ′ ∈ F , (1.6) X γ ∈ Γ ,ℓ γ >R e − sρ ( z,γz ′ ) ≤ C ǫ ( x ( z ) x ( z ′ )) s . TRICHARTZ ESTIMATES WITHOUT LOSS 5
Proof.
In [23, Lemma 5.2], it is shown that there exist constants C , C > γ ∈ Γ such that ℓ γ > C e − ρ ( z,γz ′ ) ≤ C e − ℓ γ x ( z ) x ( z ′ ) . Now it suffices to sum after raising to the power s and to use the fact that P γ ∈ Γ e − sℓ γ < C ǫ forsome C ǫ if s > δ + ǫ . (cid:3) Combining (1.6) and (1.4), we deduce that for z, z ′ ∈ F , the series K X ( t ; z, z ′ ) := X γ ∈ Γ K ( t ; z, γ.z ′ )converges uniformly and for all s < n/ C s > z, z ′ ∈ F (1.7) | K X ( t ; z, z ′ ) | ≤ C s | t | − ( n +1) / (cid:16) P γ ∈ Γ ,ℓ γ ≤ R e − sρ ( z,γz ′ ) + x ( z ) s x ( z ′ ) s (cid:17) , for | t | ≤ C s | t | − / (cid:16) P γ ∈ Γ ,ℓ γ ≤ R e − sρ ( z,γz ′ ) + x ( z ) s x ( z ′ ) s (cid:17) , for | t | > R is the constant in (1.6). This leads directly to the dispersive estimate(1.8) sup z,z ′ ∈ F | K X ( t ; z, z ′ ) | ≤ ( C ′ | t | − ( n +1) / , for | t | ≤ C ′ | t | − / , for | t | > . for some constants C ′ . Moreover, using (1.5), the same argument shows that the series K X ( t ; z, z ′ )is smooth in z, z ′ for t = 0. Let F be a fundamental domain of Γ. For any u ∈ C ∞ ( X ) thefunction u ( t ) := R F K X ( t ; z, z ′ ) u ( z ′ ) dz ′ is smooth on H n +1 and satisfies u ( t, γz ) = u ( t, z ) forany γ ∈ Γ, thus u ( t ) is smooth on X . Moreover it solves the Schr¨odinger equation on X withinitial data u (0) = u , so u ( t ) = e it ∆ X u . This implies that K X is the Schwartz kernel of e it ∆ 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 fromKeel-Tao [28], following the method of Anker-Pierfelice [3]). Let γ ∈ Γ be such that ℓ γ ≤ R where R is the constant in (1.6), then define K γ ( t ) to be the operator acting on F with L ∞ kernel1l F ( z ) K ( t ; z, γz ′ ) 1l F ( z ′ ). Since γ is an isometry of H n +1 , this operator can also be written as f → F e it ∆ H n +1 γ ∗ (1l F f ). Then from Theorem 3.4 of Anker-Pierfelice [3] and the fact thatpush-forward γ ∗ is an isometry on any L r ′ ( H n +1 ), we get the estimate(1.9) || K γ ( t ) u || L q ( H n +1 ) ≤ C || F u || L r ′ ( H n +1 ) × ( | t | − ( n +1) max( − q , − r ) if | t | ≤ | t | − if | t | > < q, r ≤ ∞ and 1 /r ′ +1 /r = 1, so the same estimate holds for K ( t ) := P γ ∈ Γ ,ℓ γ ≤ R K γ ( t ).Now consider the operator K ( t ) acting on F whose L ∞ kernel is K ( t ; z, z ′ ) := K X ( t ; z, z ′ ) − K ( t ; z, z ′ ). From (1.6) and (1.4), this kernel is bounded (for all s < n/
2) by | K ( t ; z, z ′ ) | ≤ ( C s | t | − ( n +1) / x ( z ) s x ( z ′ ) s , for | t | ≤ C s | t | − / x ( z ) s x ( z ′ ) s , for | t | > . Since the hyperbolic metric on F induces a measure of the form x − n − µ for some boundedmeasure µ on F we see that the function x s is in L α ( F , dv H n +1 ) for all α > n/s , and hencededuce directly that K ( t ) satisfies || K ( t ) u || L q ( F , dv H n +1 ) ≤ C || u || L r ′ ( F , dv H n +1 ) × ( | t | − n +12 if | t | ≤ | t | − if | t | > q, r ∈ (2 , ∞ ] and r ′ the conjugate exponent of r , so the same estimate holds for K X ( t ) whencombining with (1.9). Then it suffices to conclude using the standard T T ∗ argument exactlyas in the proof of Theorem 3.6 of Anker-Pierfelice [3] and we obtained the claimed Strichartzestimate. (cid:3) NICOLAS BURQ, COLIN GUILLARMOU, AND ANDREW HASSELL
Remark 1.3.
For hyperbolic quotients with dimension of limit set δ > n/ , the positive number δ ( n − δ ) is an L eigenvalue with multiplicity one and smooth eigenvector ψ δ . It follows easilyfrom Section 2 of [35] (or the general result of Mazzeo-Melrose [30] about the structure of theresolvent) that ψ δ ∈ L p ( X ) for all p > n/δ , thus in particular for all p ≥ . This implies thatfor q ≥ , < p < ∞ and all χ ∈ L ∞ ( X ) k χe it ∆ X ψ δ k L q ( X ) = k χψ δ k L q ( X ) / ∈ L pt ((0 , ∞ )) , so global-in-time Strichartz estimates cannot hold when δ > n/ , even with a space cut-off. Connected sum of two copies of R In this section, we give an example of a Riemannian manifold (
Z, g ) which is topologically theconnected sum of two copies of R , and is geometrically Euclidean near infinity, and hyperbolicnear the ‘waist’ (and hence with a single trapped ray), for which Strichartz estimates withoutloss are valid. The idea is simple; since Strichartz estimates without loss are valid on flat R , andon the hyperbolic cylinder (thanks to Theorem 1.1), then they should also be valid on a spaceobtained by gluing pieces of these manifolds together, provided that no additional trapping iscreated by the gluing procedure.Let us consider an asymptotically Euclidean manifold ( Z, g ) which is the connected sum oftwo copies of R , joined by a neck which has a neighbourhood U isometric to a neighbourhood U ′ of the short closed geodesic, or ‘waist’, on the hyperbolic two-cylinder C . We denote thisshort closed geodesic by γ , whether on Z or on C . We can write down an explicit metric g forsuch a manifold, on R × S , in the form dr + f ( r ) dθ , where dθ is the metric on S of length2 π , and where f ( r ) = cosh r for small r , say r ≤ η for some small η >
0, and is equal to | r | + a for large | r | , say | r | ≥ R (where a is a constant). We also choose f so that f ′ ( r ) has the samesign 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 ¨ r = 2 f ′ ( r ) f ( r ) ˙ θ ,which has the same sign as r , and it is straightforward to deduce from this that there can be notrapped geodesic other than the waist γ 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 (2.1) k e it ∆ Z u k L p ([0 ,T ]; L q ( Z )) ≤ C T k u k L ( Z ) for all ( p, q ) satisfying (0.1) with d = 2 and all u ∈ L ( Z ) . Before giving the proof we introduce some further notation and definitions. We will compare Z to the hyperbolic cylinder C and to the auxiliary Riemannian manifold ( ˜ Z = R , ˜ g ), givenin standard polar coordinates ( r, θ ) on R by ˜ g = dr + ˜ f ( r ) dθ , where ˜ f ( r ) = f ( r ) for r ≥ η ,˜ f ′ ( r ) >
0, and is equal to r for small r . Reasoning as above, we see that the metric ˜ g isnontrapping. We will take a Schr¨odinger wave u on Z and decompose it so that one piece liveson C and the other lives on ˜ Z , and we will deduce Strichartz without loss on Z from the factthat Strichartz without loss holds for both C and ˜ Z . Proof.
Let U ⊂ Z be a neighbourhood of { r = 0 } , say {| r | < η } , thus containing the projectionof the trapped set of Z . By construction, the metric is exactly hyperbolic in a neighbourhood of U . We decompose u = u i + u e , where u i = χu is supported in U and u e = (1 − χ ) u is supportedwhere the metric g is identical to ˜ g . (Thus ∇ χ is supported where η ≤ | r | ≤ η .) We prove theestimate (2.1) separately for u i and u e . As stated above, the idea is to regard u e as solving aPDE on ˜ Z and to regard u i as solving a PDE on C .We first prove a local smoothing result for Z , ˜ Z and C . This is essentially standard, butwe give the details for the reader’s convenience (and in keeping with the expository character ofthis 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 withtrapped set K ⊂ T ∗ X . Suppose that u solves the Schr¨odinger equation on ( X, g ) with initial TRICHARTZ ESTIMATES WITHOUT LOSS 7 condition u , and suppose that φ ∈ C ∞ c ( M ) is supported away from the projection of the trappedset π ( K ) . Then k φu k L t ([0 ,T ]; H ( X )) ≤ C k u k L ( X ) . (ii) Suppose that v solves the Schr¨odinger equation on ( C , g hyp ) with initial condition v , andsuppose that φ ∈ C ∞ c ( C ) is supported away from the closed geodesic γ . Then k φv k L t ([0 ,T ]; H ( C )) ≤ C k ˜ v k L ( C ) . Proof.
This result can be deduced from the resolvent estimate of Cardoso-Vodev [15], but forcompleteness we give a proof via a positive commutator argument. We construct a zeroth orderpseudodifferential operator A on X such that i [∆ , A ] has a nonnegative principal symbol whichis elliptic on the support of φ . Then we use the identity(2.2) h Au ( · , T ) , u ( · , T ) i − h Au ( · , , u ( · , i = Z T h i [∆ , A ] u, u i dt valid for any Schr¨odinger wave u . Since i [∆ , A ] is order one and elliptic on the support of φ , theright hand side is equal to c k φu k L ([0 ,T ]; H / ) plus terms which are essentially positive, while theleft hand side is bounded by C k u k L , giving the estimate.Let us set A = A + A , where A is supported in the region where X is Euclidean and A is properly supported. We take R > X has a neighbourhoodisometric to R d \ B (0 , R ) and use Euclidean coordinates x = ( x , x , . . . x d ) on this neighbourhoodwith dual cotangent coordinates ξ . We write r = | x | and take A to have principal symbol(2.3) a = ζ ( r ) h ξ i − r − x · ξ (cid:16) − r − ǫ (cid:17) . Here ζ ( t ) is chosen to be 0 for t < R and 1 for t ≥ R and to be nondecreasing, where R issufficiently large that 1 − R − ǫ > , say. We understand this to mean that a is defined as above on each end of X . Explicitly, we could take A = ζ ( r ) r − (1 + ∆) − / ( x · D x + D x · x ) ζ ( r )(1 − r − ǫ ),where here ∆ denotes the flat Laplacian on R d ; notice that (1 + ∆) − / makes sense since it isboth pre- and post-multiplied by ζ ( r ) which is supported where the metric is Euclidean.Then the derivative a along the Hamilton vector field of σ (∆ X ), namely the geodesic flow2 ξ · ∂ x , is 2 ζ ( r ) h ξ i − r − (cid:0) − r − ǫ (cid:1)(cid:16) r | ξ | − ( x · ξ ) (cid:17) +4 ζ ( r ) ζ ′ ( r ) h ξ i − r − (cid:0) − r − ǫ (cid:1) ( x · ξ ) +2 ζ ( r ) h ξ i − r − ǫr − ǫ ( x · ξ ) . We see that this is nonnegative everywhere, and bounded below by C h ξ i r − − ǫ for r ≥ R .Now we define a symbol a which will be supported in the region r ≤ R . First we introducesome notation: for ˜ R ≥ R , let E ˜ R denote the union of ends R d \ B (0 , ˜ R ), and let U ˜ R ⊂ T ∗ X denote π − ( X \ E ˜ R ). We choose conic neighbourhoods U < and U of K such that U < ⊂ U and π ( U ) is disjoint from supp φ . For any p ∈ π − (supp φ ) we let β = β ( p ) denote the maximallyextended geodesic through p , and we denote by β + , resp. β − , the forward, resp. backwardgeodesic ray starting at p . Standard topological arguments show that one can choose U < sothat for every p ∈ π − (supp φ ), at least one of β + or β − does not meet U < , which we will nowassume.Now choose an arbitrary p ∈ π − (supp φ ) and consider the geodesic β ( p ). Because of the waywe chose U < , either β − or β + does not intersect U < . Suppose, for the sake of definiteness that β + does not meet U < (the argument for β − ∩ U < = ∅ is similar). Let V be a conic neighbourhoodof β + . We may construct a symbol of order 0 that is • supported on V , • non-decreasing with respect to geodesic flow on U R , • strictly increasing with respect to geodesic flow on V < ∩ ( U R \ U ) ∩ {| ξ | ≥ / } , where V < is a conic neighbourhood of β + such that V < ⊂ V , and NICOLAS BURQ, COLIN GUILLARMOU, AND ANDREW HASSELL • vanishing outside U R .To do this, we let t be an arc-length parameter along β with β (0) = p and β + = { β ( t ) | t ≥ } ,and let t = sup { t | β ( t ) ∈ U < } , which is negative by assumption. Also let t = sup { t | β ( t ) ∈ U R } , and t = sup { t | β ( t ) ∈ U R } , both of which are positive by assumption. We choose afunction along β that is 0 for t ≤ t , strictly increasing for t < t < t and zero for t ≥ t . Thiscan be extended to a symbol of order 0 supported in V .Using compactness, we can select a finite number of conic neighbourhoods V < as above,covering π − supp φ \{ } . Summing the corresponding symbols defined above, we obtain a symbol a supported in U R such that the Hamilton vector field of ∆ X applied to a is positive andelliptic on π − supp φ . Let A be a properly supported pseudodifferential operator with symbol a . Let A be the sum of A and a sufficiently large multiple of A . Then i [∆ , A ] has nonnegativesymbol, and (if the symbol a is specified appropriately, i.e. so that { σ (∆) , a } is a sum of squaresof symbols, which is always possible) may be expressed in the form P i B ∗ i B i + B , where the B i are order 1 / P i B i is elliptic on π − supp φ and B is order 0. Then substituting i [∆ , A ] = P i B ∗ i B i + B , and using the sharp G˚arding inequality in the form C P i k B i u k ≥k φu k H / ( X ) − C ′ k u k L ( X ) which is valid for sufficiently large C , we deduce that Z T k φu k H ( M ) dt ≤ C k u k L ( M ) , proving (i).The proof of (ii) is very similar in spirit. Again we construct a pseudodifferential operator A with the property that i [∆ , A ] has a nonnegative principal symbol which is elliptic on thesupport of φ . We construct A as A + A , where A is constructed exactly as above, but A ismodified to reflect the hyperbolic rather than Euclidean structure at infinity. We shall take A to be a zeroth order pseudodifferential operator in the 0-calculus of Mazzeo-Melrose [30]. Recallthat the 0-calculus of pseudodifferential operators on a manifold with boundary is the naturalclass of pseudodifferential operators associated to differential operators generated by vector fieldsthat vanish at the boundary. This calculus is appropriate here since, if we compactify C byadding circles at r = ±∞ , with boundary defining functions e ∓ r , then the Laplacian ∆ C on C is an elliptic combination of such vector fields.Using coordinates ( ρ, ω ) dual to ( r, θ ), we define A to be a zeroth order 0-pseudodifferentialoperator with symbol ζ ( r )(1 − e − ǫr ) ρ · (1 + σ (∆ C )) − / . In these coordinates, the symbol of∆ C is σ (∆ C ) = ρ + (sech r ) ω and the Hamilton vector field is2 (cid:16) ρ ∂∂r + tanh r (sech r ) ω ∂∂ρ + sech r ω ∂∂θ (cid:17) . Applying this to the symbol of A gives the positive term2 ζ ( r )(1 + σ (∆ C )) − / (cid:16) ζ ′ ( r )(1 − e − ǫr ) ρ + ǫζ ( r ) e − ǫr ρ + tanh r (sech r ) ζ ( r )(1 − e − ǫr ) ω (cid:17) which is nonnegative everywhere and bounded below by a multiple of σ (∆ C ) / on the supportof φ . The rest of the proof is the same as in part (i), using the fact that zeroth order operatorsin the 0-calculus are bounded on L ( C ). (cid:3) Remark 2.3.
Exactly the same result holds if X is replaced by an asymptotically conic manifold,with the same proof. We only have to replace r − h ξ i − x · ξ in (2.3) by the cotangent variabledual to dr . We shall use this remark in the next section. Remark 2.4.
We can rephrase this result as follows: the operators T X = φe − it ∆ X and T C = φe − it ∆ C , for φ ∈ C ∞ c ( M ) supported away from the trapped set, and T ˜ X = ˜ φe − it ∆ ˜ X , for ˜ φ ∈ C ∞ c ( ˜ X ) are bounded from L ( X ) to L ([0 , T ]); H ( X ) , resp. L ( C ) to L ([0 , T ]); H ( C ) ,resp. L ( ˜ X ) to L ([0 , T ]); H ( ˜ X ) . TRICHARTZ ESTIMATES WITHOUT LOSS 9
We return to the proof of Theorem 2.1. Consider the function u e = (1 − χ ) u . We can regardit as a function on ˜ Z , and as such it satisfies on the time interval [0 , T ](2.4) ( i∂ t − ∆ ˜ Z ) u e = w ≡ − ∇ χ · ∇ u + (∆ ˜ Z χ ) u ; u e (cid:12)(cid:12) t =0 = (1 − χ ) u ∈ L ( ˜ Z ) . By Lemma 2.2, w ∈ L ([0 , T ]; H − ( ˜ Z )). Let us write u e = u ′ e + u ′′ e , where u ′ e solves the PDEabove with zero initial condition, and u ′′ e solves the homogeneous equation ( i∂ t − ∆ ˜ Z ) u ′′ e = 0with initial condition (1 − χ ) u . By [39], the Strichartz estimate (2.1) holds for u ′′ e . The function u ′ e is given by Duhamel’s formula u ′ e ( · , t ) = Z t e − i ( t − s )∆ ˜ Z w ( · , s ) ds. We want to show that this is in L pt L qx for Strichartz pairs ( p, q ). Since we are dimension d = 2,we have p >
2, and hence we can apply the Christ-Kiselev Lemma [17], which tells us that it issufficient to show boundedness of the operator w Z e − i ( t − s )∆ ˜ Z w ( · , s ) ds from L ([0 , T ]; H − ( ˜ Z )) to L pt L qx . But, defining T ˜ Z as above, this is e it ∆ ˜ Z T ∗ ˜ Z w (for any ˜ φ equalto 1 on the support of ∇ χ ). By Lemma 2.2 and duality, T ∗ ˜ Z maps L ([0 , T ]; H − ( ˜ Z )) to L ( ˜ Z ),while by [39], e it ∆ ˜ Z maps L to L pt L qx . This shows that (2.1) holds for the function u e .It remains to consider u i = χu . We regard u i as a function on the hyperbolic cylinder C since it is supported in the region where the metric is hyperbolic, and as such it satisfies on thetime interval [0 , T ](2.5) ( i∂ t − ∆ C ) u i = w ≡ − ∇ χ · ∇ u + (∆ C χ ) u ; u i (cid:12)(cid:12) t =0 = χu ∈ L ( C ) . As we have seen, w ∈ L ([0 , T ]; H − ( C )). Let us write u i = u ′ i + u ′′ i , where u ′ i solves the PDEabove with zero initial condition, and u ′′ i solves the homogeneous equation ( i∂ t − ∆ C ) u ′′ i = 0with initial condition χu . By Theorem 1.1, the Strichartz estimate (2.1) holds for u ′′ i . Thefunction u ′ i is given by Duhamel’s formula u ′ e ( · , t ) = Z t e − i ( t − s )∆ C w ( · , s ) ds. We want to show that this is in L pt L qx for Strichartz pairs ( p, q ). We use the Christ-Kiselev trickagain and show that the operator w Z e − i ( t − s )∆ C w ( · , s ) ds is bounded from L ([0 , T ]; H − ( ˜ Z )) to L pt L qx . But, defining T C as above, this is e it ∆ C T ∗ C w (for any ˜ φ equal to 1 on the support of ∇ χ ). By (ii) of Lemma 2.2, T ∗ C by duality maps L ([0 , T ]; H − ( C )) to L ( C ), while by Theorem 1.1, e it ∆ ˜ Z maps L to L pt L qx . This shows that(2.1) holds for the function u i , and completes the proof of Theorem 2.1. (cid:3) Asymptotically Euclidean (or conic) manifolds with filamentary hyperbolictrapped set
In the previous section, we used the dispersive estimate from section 1 for constant negativecurvature manifolds to prove Strichartz estimates without loss. It is natural to ask if thisresult can be generalized to a variable negative curvature setting. In this section, we shall showthat a more general class of manifolds with hyperbolic trapped set has this ‘Strichartz withoutloss’ property. The class of manifolds we will consider are asymptotically Euclidean (and moregenerally asymptotically conic) but the projection of their trapped set is contained in an openset where the metric has (variable) negative curvature, so that the flow is hyperbolic there, and we will assume, as in [32], that the topological pressure P ( s ) of the unstable Jacobian on thetrapped set satisfies P (1 / < filamentary , although it may contain an infinite number of closed geodesics.An asymptotically conic manifold (or scattering manifold in the sense of [31]) is a completenon-compact Riemannian manifold ( M, g ) which is the interior of a smooth compact manifoldwith boundary M and such that a collar neighbourhood of the boundary is isometric to (cid:16) [0 , ǫ ) x × ∂M , dx x + h ( x ) x (cid:17) where h ( x ) is a one-parameter family of metrics on ∂M depending smoothly on x ∈ [0 , ǫ ). Here ∂M has really to be considered as the ‘points at infinity’ of ( M, g ). The function x can beextended to a nonnegative smooth function on M and the function r = 1 /x is analogous to theradial function on Euclidean space.The geodesic flow Φ t , t ∈ R , is the flow of the Hamiltonian vector field V H associated to H ∈ C ∞ ( T ∗ M ) defined by H ( m, ξ ) := | ξ | g . The trapped set K is defined by K := Γ + ∩ Γ − where(3.1) Γ ± := { ( m, ξ ) ∈ T ∗ M | Φ t ( m, ξ )
6→ ∞ , t → ∓∞} ⊂ T ∗ M. Let us denote by π : T ∗ M → M the projection on the base, and let d = dim M .The geodesic flow is said to be hyperbolic on U ⊂ S ∗ M if for all m ∈ U , the tangent space at m splits into flow, unstable and stable subspaces such that(3.2) i ) T m S ∗ U = R V H ( m ) ⊕ E + m ⊕ E − m , dim E ± m = d − ii ) d Φ tm ( E ± m ) = E ± m , ∀ t ∈ R iii ) ∃ λ > , || d Φ tm ( v ) || ≤ Ce − λ | t | || v || , ∀ v ∈ E ∓ m , ± t ≥ . for some uniform λ >
0; here the norm can be taken with respect to the Sasaki metric on thecotangent bundle (see [33], Definition 1.17). This is true in particular for U = S ∗ M if M isa complete manifold with negative sectional curvatures contained in an interval [ − k , − k ] forsome k i >
0, see for instance [29, Th. 3.9.1].We define the unstable Jacobian J ut ( m ) and the weak unstable Jacobian J wut ( m ) for the flowΦ t at the point m to be(3.3) J ut ( m ) = det (cid:0) d Φ − t (Φ t ( m )) | E +Φ t ( m ) (cid:1) ,J wut ( m ) = det (cid:0) d Φ − t (Φ t ( m )) | E +Φ t ( m ) ⊕ R V H ( m ) (cid:1) where the volume form on d dimensional subspaces of T ( T ∗ M ) is induced by the Sasaki metric.It follows from (iii) of (3.2) that J ut ( m ) , J wut ( m ) ≤ e − λt for t > K is hyperbolic, and s : K → R is a continuousfunction, then the topological pressure of the unstable Jacobian at s is a real number P ( s ),whose definition is given by (3.7). The topological pressure of the unstable Jacobian can beviewed as a real function P of s . The quantity P (0) is known as the topological entropy of theflow. For positive s , P ( s ) in a sense measures two competing effects of the flow: the densityof K (the denser K , the longer points near K stay close by under the flow) and the instabilityof the flow (the more unstable, the more quickly points near K move away from K under theflow). In our analysis we encounter products of square roots of the unstable Jacobian, whichin view of (3.6) and (3.7) make it natural to consider the topological pressure at s = 1 /
2; if P (1 / < M, g ) and on the trapped set K in this section are(A1) ( M, g ) is asymptotically conic.
TRICHARTZ ESTIMATES WITHOUT LOSS 11 (A2) There is an open set X − ⊂ X containing π ( K ) which can be extended to a completemanifold ˜ M with sectional curvatures bounded above by a negative constant (in partic-ular, M − has sectional curvatures bounded above by a negative constant).(A3) M − is geodesically convex in M : i.e. any geodesic entering π − ( M \ M − ) from π − M − remains in this region thereafter.(A4) The topological pressure P ( s ) of K evaluated at s = 1 / P (cid:0) (cid:1) < . For examples, see Section 3.2.
Remark 3.1.
With these assumptions, the geodesic flow is hyperbolic on S ∗ ˜ M and K is thetrapped set on both S ∗ f M and S ∗ M . On S ∗ M , the splitting satisfying (3.2) only makes sense atpoints of K , but we can still consider the splitting T S ∗ M − = R V H ( m ) ⊕ E + m ⊕ E − m coming from theinclusion S ∗ M − ⊂ S ∗ f M : in particular, for all m ∈ S ∗ M − and all t such that Φ t ( m ) ∈ π − ( M − ) we have d Φ tm ( E ± m ) = E ± m , ∃ λ > , || d Φ tm ( v ) || ≤ Ce − λ | t | || v || , ∀ v ∈ E ∓ m , if ± t ≥ . Remark 3.2.
It seems likely that (A2) actually follows from (A1) and (A3); that is, that if M − is negatively curved and geodesically convex, then we expect that it can always be extended toa complete manifold with negative sectional curvature. We do not pursue this question furtherhere as it is a purely differential-geometric question. Our main result in this section is
Theorem 3.3.
Let ( M, g ) satisfy assumptions (A1) — (A4) above. Then local-in-time Strichartzestimates without loss hold for M : there exists C > such that (3.5) || e it ∆ u || L p ((0 , ,L q ( M )) ≤ C || u || L ( M ) for all u ∈ L ( M ) and ( p, q ) satisfying (0.1) . Remark 3.4.
In dimension d = 2 , the condition P (1 / < is equivalent to the Hausdorffdimension of the trapped set satisfying d H ( K ) < , or equivalently d H ( K ∩ S ∗ M ) < . Note thatthis is the natural generalization of the condition δ < n/ in our hyperbolic quotients examplesabove, since d H ( K ∩ S ∗ M ) = 2 δ + 1 in that case (recall d = n + 1 ). Topological pressure.
We now define the topological pressure of the flow P ( s ) on thetrapped set, following [32] (which follows from Def. 20.2.1 of [24]): a set E ⊂ K ∩ S ∗ M is said tobe ( ǫ, T ) separated if given ( x , v ) = ( x , v ) in E , there exists t ∈ [0 , T ] for which the distancebetween Φ t ( x , v ) and Φ t ( x , v ) is at least ǫ >
0. For any s ∈ R we define (3.6) Z T ( ǫ, s ) := sup E X m ∈ E ( J wuT ( m )) s where the sup is taken over all sets E which are ( ǫ, T )-separated. The pressure of s is(3.7) P ( s ) = lim ǫ → lim sup T →∞ T log Z T ( ǫ, s )For instance, if the metric has constant curvature, one has Lemma 3.5.
Let ( M, g ) be a convex cocompact hyperbolic manifold of dimension n + 1 withlimit set of dimension δ . Then the topological pressure at s = 1 / is given by P (1 /
2) = δ − n/ . For later convenience we define the topological pressure using the weak unstable Jacobian rather than theunstable Jacobian which is more standard. However, making this change only changes log Z T ( ǫ, s ) by O (1),uniformly in ǫ , and thus leads to the same value of P ( s ). Proof.
For a constant curvature − J ut ( m ) = e − tn . It follows from (3.6) and (3.7)(and the footnote) that P ( s ) = P (0) − ns . But P (0), which is the topological entropy, is equalto δ by a result of Sullivan [41]. (cid:3) Finally, we recall the alternate definition of topological pressure given in [32, Sec. 5.2], whichturns out to be easier to use. If V = ( V b ) b ∈ B is an open finite cover of K ∩ S ∗ M , let V T ( T ∈ N )be the refined cover made of T -fold intersections V β := T − \ k =0 Φ − k ( V b k ) , β := b b . . . b T − ∈ B T , and consider the set B ′ T ⊂ B T of β such that V β ∩ K = ∅ . For any W ⊂ S ∗ M with W ∩ K = ∅ ,define the coarse-grained unstable Jacobian(3.8) S KT ( W ) := sup m ∈ W ∩ K log J uT ( m ) = − inf m ∈ W ∩ K log det( d Φ T ( m ) | E + m ⊕ R V H ( m ) ) . The topological pressure is defined by P ( s ) := lim diam V → lim T →∞ T log inf n X β ∈ B T exp( sS KT ( V β )); B T ⊂ B ′ T , K ∩ S ∗ M ⊂ [ β ∈ B T V β o . In particular, for all ǫ > ǫ >
0, such that for all ǫ < ǫ and all covers V of K ∩ S ∗ M as above with diameter smaller than ǫ , there is a T ∈ N , a set B T ⊂ B ′ T such that { V β , β ∈ B T } is an open cover of K ∩ S ∗ M and(3.9) X β ∈ B T exp( sS KT ( V β )) ≤ exp( T ( P ( s ) + ǫ / . Moreover V β are all included in π − ( M − ) since they are ǫ close to K . Since by the chain ruleone has J uT ( m ) = T Y j =1 J u (Φ j − ( m )) , and since the unstable foliation is γ -H¨older [38], we deduce that for any V β with β ∈ B T | S KT ( V β ) − S T ( V β ) | ≤ exp( CT ǫ γ ) , where S T ( V β ) := sup m ∈ V β log J uT ( m )for some constant C depending only on ǫ . Therefore, renaming the family ( V β ) β ∈ B T by( W a ) a ∈ A , we get, by taking for instance Cǫ γ ≤ ǫ / X a ∈ A exp( sS T ( W a )) ≤ exp( T ( P ( s ) + ǫ )) . Some examples.
We first give examples of Riemannian manifolds satisfying assumptions(A1) — (A4). Consider any convex co-compact hyperbolic manifold (
M, g ). Near infinity, itis conformally compact. That is, it admits a compactification to a compact manifold M witha boundary defining function x , such that x g is a smooth nondegenerate metric up to theboundary of M . In other words, near infinity g takes the form g = h/x for some smooth metric h on M . We shall now modify the metric g near infinity to a metric that is asymptotically conic,in such a way that the trapped set is left unchanged. (We recall that both asymptotically conicand conformally compact metrics are nontrapping near infinity.) This is straightforward: wehave near infinity, in suitable coordinates ( x, y ) where x is a boundary defining function for M and y is a local coordinate on Y = ∂M g = dx + h ( x ) x . Notice that E + m makes sense since π ( V β ) is in the negative curved part M − . TRICHARTZ ESTIMATES WITHOUT LOSS 13
Here h ( x ) is a smooth family of metrics on Y , i.e. is smooth in all its arguments. Changingvariables to the ‘geometric’ coordinate r = log(1 /x ), this reads g = dr + e r h ( e − r ) . Assume that this is valid for r ≥ R . Then, for some R ′ ≥ R we choose a function f ( r ) such that f ( r ) = e r for r ≤ R ′ and f ( r ) = cr for r ≥ R ′ , and such that f ′ ( r ) > f ( r ) / r for all r . Thisis possible for all R ′ ≥ R and some c depending on R ′ . Define the metric(3.11) g ac = dr + f ( r ) h ( e − r ) , r ≥ R ; g ac ≡ g for r ≤ R. This is an asymptotically conic metric on M and satisfies assumption (A2). The symbol of theLaplacian with respect to this metric is ρ + f ( r ) − ( h − ( e − r )) ij η i η j and along geodesics we have¨ r = 2 (cid:16) f ′ ( r ) f ( r ) | η | h − ( e − r ) + e − r f ( r ) | η | h − ( e − r ) (cid:17) . Here ˙ h − ( e − r ) means d/ds ( h − ( s )) | s = e − r . The metric d/ds ( h − ( s )) is bounded above by aconstant times h − ( s ) uniformly for s ∈ [0 , log R − ]. Also, using f ′ ( r ) > f ( r ) / r , we see that f ′ ( r ) /f ( r ) ≫ e − r /f ( r ) for large r . It follows that for R ′ sufficiently large, and r ≥ R ′ , we have¨ r ≥ r ≥ R ′ for the metric g ac ; moreover, the set { r ≤ R ′ } is geodesically convex. Hence the metric satisfies condition (A3).Finally to verify assumption (A4) it suffices to use Lemma 3.5.3.3. Strategy of the proof.
The proof of Theorem 3.3 is much more involved than that ofTheorem 2.1. We will need to localize both in frequency and in time. To explain the idea, wefirst show how the Strichartz estimates on an asymptotically conic nontrapping manifold M maybe deduced via frequency and time localization. Here we focus on estimating the solution on acompact set contained in M . Thus, we consider χu , where χ ∈ C ∞ ( M ) vanishes for small x .We introduce the semiclassical parameter h , where h − will be (up to a constant) the frequencyof our frequency-localized wave u . Let ψ ∈ C ∞ (1 / , c ,(3.12) k ψ ( h ∆ M ) e − it ∆ M k L ( M ) → L ∞ ( M ) ≤ Ct − n/ for t ∈ [0 , ch ] . Let us assume that u is localized near frequencies ≈ h − in the sense that ψ ( h ∆ M ) u = u (thiswill then be true for all times t ). This assumption is harmless as a Littlewood-Paley argument(see [12], Section 2.3.2 using the Littlewood-Paley estimate from [6]) shows that if Strichartz istrue for frequency-localized u , then it holds for all u . It follows from (3.12) and Keel-Tao thatStrichartz holds for u on a time interval of length ch :(3.13) k e − it ∆ M u k L p [0 ,ch ]; L q ( M ) ≤ C k u k L ( M ) for ( p, q ) satisfying (0.1). Here, c depends on the injectivity radius of M and the support of ψ ;for simplicity, below we assume that c = 1.To extend this to a fixed length time interval, we use time cutoffs and local smoothing esti-mates. To define the time cutoffs, let ϕ ( s ) ∈ C ∞ [ − ,
1] satisfy ϕ (0) = 1 and P j ∈ Z ϕ ( s − j ) = 1.Then we can write, for any Schr¨odinger wave u ( · , t ) = e − it ∆ M u , χu = X j ∈ Z ϕ ( t/h − j ) χu ≡ X j u j , where each u j is supported on a time-interval of length 2 h . We work on the time interval [0 , h − = N ∈ N ; thus, we need to consider u j for j = 0 , , . . . , N . The functions u and u N are dealt with from the semiclassical Strichartz estimate (3.13). So consider u j for1 ≤ j ≤ N −
1. These functions satisfy the equation(3.14) ( i∂ t − ∆ M ) u j = h − ϕ ′ ( t/h − j ) χu + 2 ϕ ( t/h − j ) (cid:0) ∇ χ · ∇ u − ∆ M χu (cid:1) ≡ w j . Since M is nontrapping, we have from Lemma 2.2 and Remark 2.3 the local smoothing estimate(3.15) k ˜ χu k L [0 , H / ( M ) ∼ h − / k ˜ χu k L [0 , L ( M ) ≤ C k u k L ( M ) for any ˜ χ ∈ C ∞ ( M ). Choose ˜ χ to be 1 on the support of χ . It follows that(3.16) N X j =1 k w j k L t ; L ( M ) ≤ Ch − k u k L ( M ) . We can express u j in terms of w j using Duhamel’s formula:(3.17) u j ( t ) = ˜ χ Z t ( j − h e − i ( t − s )∆ M ˜ χw j ( s ) ds. By the Christ-Kiselev lemma, if p >
2, in order to estimate the L pt norm of u j in terms of the L t norm of w j it is sufficient to estimate the L p norm of ˜ u j defined by(3.18) ˜ u j ( t ) = Z ( j +1) h ( j − h e − i ( t − s )∆ M w j ( s ) ds = e − it ∆ M Z ( j +1) h ( j − h e is ∆ M w j ( s ) ds. Now we can use the semiclassical Strichartz estimate since the time interval is O ( h ). The dualestimate to (3.15) gives(3.19) k Z ( j +1) h ( j − h e is ∆ M w j ( s ) ds k L ( M ) ≤ Ch / k w j k L t ; L ( M ) . Then (3.13) applied to this L function shows that(3.20) k ˜ u j k L pt ; L q ( M ) ≤ Ch / k w j k L t ; L ( M ) and the same estimate holds for u j by Christ-Kiselev. Squaring this inequality, summing over j and using (3.16) shows that(3.21) N − X j =1 k u j k L pt ; L q ( M ) ≤ C k u k L ( M ) . Using the continuous embedding from l ( N ) to l p ( N ) if p ≥
2, we obtain(3.22) (cid:16) N − X j =1 k u j k pL pt ; L q ( M ) (cid:17) /p ≤ C k u k L ( M ) and this gives(3.23) k u k L p [0 , L q ( M ) ≤ C k u k L ( M ) with C independent of h .Now suppose that M is trapping, but obeys assumptions (A1) – (A4) above. In that case, thelocal smoothing estimate definitely fails [11], [21], and then the argument only gives Strichartzestimate with a loss (i.e. with additional negative powers of h on the right hand side) arising fromthe loss in the local smoothing estimate. If the trapped set has negative topological pressure,the local smoothing loss is | log h | / as follows from work of Nonnenmacher-Zworski [32] andDatchev [20], Theorem 3.7 below (see also [11, 14]). In combination with the argument above,this gives Strichartz with logarithmic loss, as shown for example in the case of the exterior ofseveral convex obstacles by Burq [11].However, when the topological pressure is negative then more is true: essentially from thework of Anantharaman [1] and Nonnenmacher-Zworski [32], it follows that the semiclassicalStrichartz estimate can be improved by a logarithm: it is valid not just on a time interval oflength O ( h ), but actually on an interval of length h | log h | — see Theorem 3.8 below. Then itturns out that this logarithmic improvement exactly compensates for the logarithmic loss in thelocal smoothing estimate and we recover the Strichartz estimate on a fixed finite time interval TRICHARTZ ESTIMATES WITHOUT LOSS 15 without loss. This is achieved by localizing in time on intervals of length h | log h | rather than h for the part of u localized near the trapped set.More precisely, we proceed as above but with ϕ ( t/h − j ) replaced with ϕ ( t/h | log h | − j ); thatis, we localize to time intervals of length h | log h | which is the maximum for which we can applythe semiclassical Strichartz estimate. We then write w j = w ′ j + w ′′ j , where(3.24) w ′ j = 1 h | log h | ϕ ′ ( t/h | log h | − j ) u j , w ′′ j = 2 ϕ ( t/h | log h | − j ) (cid:0) ∇ χ · ∇ u − ∆ M χu (cid:1) . Then w ′′ j is of size ∼ h − (since a derivative of u costs h − ). On the other hand, it is supportedin the nontrapping region and may be dealt with as above, as the local smoothing estimate isvalid without loss in the nontrapping region. The other term, w ′ j , is supported in the trappedregion but is of size O (( h | log h | ) − ). On this term we apply the local smoothing estimate,losing | log h | / as compared to the argument above. When we apply the dual estimate at thestep (3.19) we lose a further | log h | / , and then applying semiclassical Strichartz completes theargument with no overall loss. The details are given in Section 3.7.In summary, the key ingredients of the proof will be the following three results: Theorem 3.6 (Strichartz in the nontrapping region) . Let M be an asymptotically conic man-ifold, and suppose that χ ∈ C ∞ ( M ) ∩ L ∞ ( M ) vanishes in a neighbourhood of π ( K ) where K ⊂ T ∗ M is the trapped set. Then we have Strichartz estimates without loss for χe − it ∆ M u : (3.25) k χe − it ∆ M u k L p [0 , L q ( M ) ≤ C k u k L ( M ) for all ( p, q ) satisfying (0.1) , p > . Theorem 3.7 (Local smoothing with logarithmic loss) . Suppose that M satisfies assumptions(A1) — (A4). Then for any χ ∈ C ∞ ( M ) and ψ ∈ C ∞ c (1 / , , we have (3.26) k χe − it ∆ M ψ ( h ∆) u k L [0 , L ( M ) ≤ C ( h | log h | ) / k u k L ( M ) . Moreover, if χ is supported outside the trapping region, then the estimate holds without thelogarithmic loss in h on the right hand side. Theorem 3.8 (Semiclassical Strichartz on a logarithmic interval) . Suppose that M satisfiesassumptions (A1) — (A4). Then for any χ supported in M − , we have on a time interval oflength h | log h | k χe − it ∆ M ψ ( h ∆) u k L p [0 ,h | log h | ]; L q ( M ) ≤ C k u k L ( M ) for all ( p, q ) satisfying (0.1) . Strichartz in the nontrapping region.
In this section we sketch how to prove Theo-rem 3.6. The argument is quite related to similar ideas in [39]. Here, we follow fairly straight-forward modifications of the argument in [25] for nontrapping metrics. Notice that none ofthe results about ‘Local Schr¨odinger integral operators’ in section 3 of [25] use the nontrappingproperty, which only enters when the local smoothing estimate is used. To adapt the results of[25] to prove (3.25), we modify the definition of the Banach space X in (4.6) of that paper toinclude a cutoff function χ + , supported in the nontrapping region and equal to 1 on the supportof χ in (3.25) in the k f k H / , − / − ρ/ ( M ) term. This cutoff function then needs to be includedin Lemma 4.4 and Lemma 5.5 of [25]. The proof in section 6 then goes through provided that ψ (0) α is supported in the nontrapping region.3.5. Local smoothing effect.
Theorem 3.7 follows fairly directly from the resolvent estimatefrom [20] (which generalizes [32, Theorem 5] to scattering manifolds):(3.27) k χ ( h ∆ − (1 ± iǫ )) − χ k L → L ≤ C | log h | h , < h < h << , with C independent of ǫ (and h ). From this, we deduce (3.26), following [13]. Indeed, denote by T the operator T = χe it ∆ ψ ( h ∆) . The boundedness of T from L to L (( −∞ , + ∞ ); L ( M )) is equivalent to the boundedness of itsadjoint T ∗ from L (( −∞ , + ∞ ); L ( M )) to L (with same norm), which in turn is equivalent tothe boundedness of T T ∗ from L (( −∞ , + ∞ ); L ( M )) to itself (with same norm squared). But T T ∗ f ( t ) = Z + ∞−∞ χe i ( t − s )∆ ψ ( h ∆) χf ( s ) ds = χ Z t −∞ + χ Z + ∞ t ≡ χA f ( t ) + χA f ( t )and it is enough to estimate for example χA f . For this we can assume that f has compactsupport, and consider u ǫ = e − ǫt A f ( t ) and f ǫ = e − ǫt f (notice that u ǫ is supported in the set { t ≥ C } ) which satisfy ( i∂ t + ∆ + iǫ ) u ǫ = ψ ( h ∆) χf ǫ Taking Fourier transform with respect to the variable t , we get χ b u ǫ ( τ ) = χ (∆ − ( τ − iǫ )) − ψ ( h ∆) χ b f ǫ and according to the Plancherel formula (recall that the Plancherel formula is true for functionstaking values in any (separable) Hilbert space), and using (3.27) we obtain k χu ǫ k L (( −∞ , + ∞ ); L ( M )) ≤ Ch log(1 /h ) k χf ǫ k L (( −∞ , + ∞ ); L ( M )) . Letting ǫ > χA to T T ∗ satisfies the requiredestimate. The other contribution χA is dealt with similarly.The improved estimate when the support of χ does not meet the trapped set is a consequenceof Lemma 2.2 and Remark 2.3.3.6. Semiclassical Strichartz on a logarithmic interval.
Using the work of Nonnenmacher-Zworski [32] (which follows techniques of Anantharaman [1]), we shall obtain a sharp dispersiveestimate for the propagator e − it ∆ M for time t ∈ (0 , h | log h | ) and in frequency localization inwindows of size h . We want to prove the following: Proposition 3.9.
There exists δ > and C > such that for all ψ ∈ C ∞ ((1 − δ/ , δ/ ,all t ∈ (0 , h | log h | ) with h ∈ (0 , h ) small, we have for every χ ∈ C ∞ ( M ) supported in M − || χe it ∆ ψ ( h ∆) χ || L → L ∞ ≤ Ct − n/ . Then Theorem 3.8 follows immediately from this by applying the main result of Keel-Tao[28].The proof of Proposition 3.9 decomposes in several parts. Let us first introduce the objectstaken from [32] that we need to use for the proof.Without loss of generality and to simplify notation, we assume as in [1] that the injectivityradius of M is larger than 1. For t ∈ (0 , h ), the result is essentially contained in [12, 9] since weare localized in a compact set of M . Now take s ∈ [0 ,
1] and an integer L , with 1 ≤ L ≤ log(1 /h ).We want to obtain a dispersive estimate for U ( L + s ), where U ( t ) := e ith ∆ , h ∈ (0 , h )by following [1, 32]. We consider, as in Section 6.3 of [32], a microlocal partition of unity (Π a ) a ∈ A of the energy layer E δ := { ( m, ξ ) ∈ T ∗ M, | ξ | ∈ (1 − δ, δ ) } for some δ > a ) is defined. The operators Π a are associated to an open covering ( W a ) a ∈ A of E δ in the sense that the semi-classical wavefront set WF h (Π a ) ⊂ W a and P a ∈ A Π a = I microlocally near E δ/ , i.e. WF h (Π ∞ ) ∩ E δ/ = ∅ if Π ∞ is defined by Π ∞ := I − P a ∈ A Π a .Following Section 5.2 and 5.3 in [32], the set A is decomposed in 3 parts, A = A ⊔ A ⊔ { } .The open set W is defined by W := E δ ∩ π − ( M \ M − ) . TRICHARTZ ESTIMATES WITHOUT LOSS 17 ( W a ) a ∈ A is chosen so that W a ⊂ M − if a ∈ A and, as a ranges over A , these sets covers K ∩ E δ in such a way that for any ǫ > δ, ǫ > T ∈ N with W a ⊂ { m ∈ T ∗ M, d ( W a , K ) ≤ ǫ } and(3.28) X a ∈ A exp (cid:16) sS T ( W a ) (cid:17) ≤ exp (cid:16) T ( P ( s ) + ǫ ) (cid:17) , where S T ( W a ) is defined by (3.8). This is possible as explained in Section 3.1 and using thehomogeneity of the Hamiltonian on T ∗ M to deal with E δ instead of S ∗ M . Finally, the W a for a ∈ A are defined so that there exists d > d ( W a , Γ + ∩ E δ ) + d ( W a , Γ − ∩ E δ ) > d where Γ ± are the forward/backward trapped sets defined in (3.1). By [32, Lem. 5.1], there exists L ∈ N such that for all a ∈ A (3.29) Φ t ( W a ) ⊂ W for t ≥ L or t ≤ − L . For notational simplicity we replace both T and L by max( T , L ); hence we have(3.30) Φ t ( W a ) ⊂ W for t ≥ T or t ≤ − T . To summarize, the energy layer E δ is decomposed into the part W covering the (spatial)infinity of E δ , the part ∪ a ∈ A W a covering the trapped set K ∩ E δ , and finally the part coveringthe complementary, whose flowout by Φ t lies in W after some large (positive or negative) time.We write U ( t ) = e ith ∆ and we shall prove(3.31) || χψ ( h ∆) U ( T ) u || L ∞ ≤ C ( T h ) − n/ || u || L for T ∈ (2 T , | log h | ). First we need a technical lemma. Lemma 3.10. (i) Let ( M, g ) be a scattering manifold and ∆ M its Laplacian. Then for any ψ ∈ C ∞ ( R ) , ψ ( h ∆ M ) is a semiclassical scattering operator of order ( −∞ , , in the sense ofWunsch-Zworski [43] .(ii) Let ψ be as above, then for each fixed t , the operator ψ ( h ∆ M ) e − ith ∆ M is a semiclassicalFourier Integral operator associated to the canonical relation { (cid:0) ( z, ζ ) , ( z ′ , ζ ′ ) (cid:1) ∈ T ∗ M × T ∗ M | ( z, tζ ) = exp( z ′ , tζ ′ ) } Proof. (i) This follows from the argument in [26] for scattering pseudodifferential operators.(We also remark that a similar, weaker result proved under more general assumptions about thenature of the ends of the manifold by Bouclet [6] would also suffice for our purposes.)(ii) It is shown in [32] that e ith ∆ h φ ( h ∆) is a semiclassical FIO for each fixed t and all φ ∈ C ∞ ( R ). Thus, using the result of (i), ψ ( h ∆) e ith ∆ φ ( h ∆) is a semiclassical FIO. If φ = 1 on thesupport of ψ then this is precisely ψ ( h ∆) e ith ∆ h by functional calculus, proving the result. (cid:3) We decompose
T > T in the form T = L − s = (2 + N ) T + t , where L, N ∈ N and s ∈ (0 , t ∈ (0 , T ]. Choosing ψ + ∈ C ∞ c ( R ) to be 1 on the support of ψ , we have ψ ( h ∆) U ( T ) = ψ ( h ∆) ψ L + ( h ∆) U ( T ) = ψ ( h ∆) U ( s ) (cid:16) ψ + ( h ∆) U (1) (cid:17) L − ψ + ( h ∆)and we can decompose(3.32) ψ + ( h ∆) U (1) = X a ∈ A ∪∞ U a , U a := ψ + ( h ∆) U (1)Π a . Hence we may write(3.33) ψ ( h ∆) U ( T ) = X α ∈ A L ψ ( h ∆) U ( s )Π α L U α L − . . . U α ψ + ( h ∆) + R T where R T term is the sum over all sequences α containing at least one index α j = ∞ . The firstestimate one obtains corresponds to Lemma 6.5 in [32]: Lemma 3.11. If ψ, ψ + ( x, hD ) are chosen as above, one has k χR T χ k L → L ∞ = O ( h ∞ ) for ≤ T ≤ | log h | , with implied constants independent of T .Proof. Since both Π ∞ and ψ + ( h ∆) are order zero pseudodifferential operators and they havedisjoint operator wavefront set, the composition Π ∞ ψ + ( h ∆) is O ( h ∞ ) as an operator from L to L . The other factors in (3.33) are all bounded from L to L , and there are at most Ce CT terms in the sum. As T ≤ | log h | this contributes at most a factor of a fixed power of h . Hence k R T k L → L = O ( h ∞ ).To get an L → L ∞ estimate from this, we compose on the left with ψ ( h ∆ M ) and observethat we still obtain an O ( h ∞ ) estimate if we pre- and post-multiply by (1 + ∆ M ) m for any m ,since this has the effect of increasing the operator norm by at most Ch − m . This is equivalent toan O ( h ∞ ) estimate from Sobolev spaces H − m ( M ) to H m ( M ), from which we obtain L → L ∞ by Sobolev embedding for m larger than half the dimension of M . (cid:3) The second estimate needed is similar to Lemma 6.6 of [32], but it is even better since we cuton the left on a compact set. Let A L ⊂ ( A \ { } ) L defined by(3.34) α ∈ A L ⇐⇒ (cid:26) Φ ( W α j ) ∩ W α j +1 = ∅ , j = 1 , . . . , L − α j ∈ A for all j = T , . . . , L − T . Lemma 3.12. If α ∈ A L \ A L and χ, ψ, ψ + ( x, hD ) are chosen as above, then (3.35) || χψ ( h ∆) U ( s )Π α L U α L − . . . U α ψ ( x, hD ) χ || L → L ∞ = O ( h ∞ ) for T < T ≤ | log h | , with T = L − s and s ∈ [0 , .Proof. This is proved in the same way as the previous lemma. We only need to show that eachterm in (3.33) corresponding to a multi-index α ∈ A L \ A L has a factor which is O ( h ∞ ) asa map from L to L . This follows directly from Egorov’s theorem if there is a j such thatΦ ( W α j ) ∩ W α j +1 = ∅ . Indeed, referring back to Lemma 3.10 we can write U α j +1 ◦ U α j = ψ + ( h ∆) U (1)Π α j +1 ψ + ( h ∆) e ih ∆ ψ ++ ( h ∆) Π α j where ψ ++ ∈ C ∞ c ( R ) is 1 on the support of ψ + . By Egorov, we have e ih ∆ ψ ++ ( h ∆) Π α j = Qe ih ∆ ψ ++ ( h ∆) for some pseudodifferential operator Q with wavefront set given by Φ − ( W F ′ (Π α j )).Since this is disjoint from the operator wavefront set of Π α j +1 by hypothesis, this factor is O ( h ∞ )as a map from L to L .If either α = 0 or α L = 0 then the O ( h ∞ ) estimate is immediate because Π is microsupportedin M \ M − and χ is supported in M − . If any of the other α j = 0 then the O ( h ∞ ) estimatefollows because of assumption (A3), which implies that either a = 0 or a L = 0, or else thecondition Φ ( W α j ) ∩ W α j +1 = ∅ has to hold for some intermediate j , showing that we are backin the situation considered above. Similarly, if α j ∈ A for some T ≤ j ≤ L − T then (3.30)shows that we are again back in the situation considered above. (cid:3) This Lemma clearly implies the bound(3.36) X α/ ∈ A L || χψ ( h ∆) U ( s )Π α L U α L − . . . U α χ || L → L ∞ = O ( h ∞ )since | A | L = O ( h − log | A | ).It remains to deal with the elements α ∈ A L . We can obtain, again essentially from theanalysis of [1] (and in a comparable way to [32, Prop 6.3]), the following bounds: Lemma 3.13.
Let χ be as above and let ǫ be the small parameter in (3.28) . Then for all small ǫ > , there exists C > such that (3.37) X α ∈ A L || χψ ( h ∆) U ( s )Π α L U α L − . . . U α χ || L → L ∞ ≤ Ch − d/ e T ( P (1 / ǫ + ǫ )TRICHARTZ ESTIMATES WITHOUT LOSS 19 for all h ∈ (0 , h ) and T ≤ T ≤ | log h | , where T = L − s with s ∈ [0 , . Proof : We start by proceeding as in [1, Sec. 3]. If the cover is taken thin enough, we may usecoordinates ( z, ξ ) in each W a , a ∈ A , where z ∈ π ( W a ) and ξ ∈ T ∗ z M are cotangent variables.We can write for u ∈ L ( M ) and z ∈ π ( W α )Π α χu ( z ) = Z π ( W α ) δ y ( z ) u ( y ) dy + O ( h ∞ ) , with δ y ( z ) := 1(2 πh ) d Z ( z,ξ ) ∈ W α e i ( z − y ) ξh σ ( z, ξ ) dξ. where σ ( x, ξ ) is the local symbol of Π α χ in W α . An upper bound for the left hand side of(3.37) is then the sum over all α ∈ A L of(3.38) sup y,z (cid:12)(cid:12)(cid:12)(cid:12)(cid:16)(cid:0) ψ ( h ∆) U ( t )Π α J +1 (cid:1) U α J . . . U α e ih ∆ δ y (cid:17) ( z ) (cid:12)(cid:12)(cid:12)(cid:12) . where we shall choose t = s and J = L −
1. Thus we take Π α χu and evolve it through e ih ∆ then microlocally cutoff in W α , evolve again, microlocally cut off again, and so on. For Anosovflows, it is shown in [1, Sec. 3] that, for any J, K ∈ N fixed (independently of L ), there existsa function S J ( ., t ) ∈ C ∞ ( π ( W α J )) and b J ( ., h, t ) ∈ C ∞ ( π ( W α J )) with b J smooth in h ∈ [0 , h )such that for t ∈ [0 , (cid:16)(cid:0) ψ ( h ∆) U ( t )Π α J +1 (cid:1) U α J . . . U α e ih ∆ δ y (cid:17) ( z ) = (2 πh ) − d/ e iSJ ( z,t ) h b J,K ( z, h, t ) + R J,K ( h, t ) b J,K ( z, h, t ) = K X k =0 h k b J ; k ( z, t )with || R J,K ( h, t ) || L = C K Jh K for some C K > t, J (this estimate is shown in[1, Lemma 3.2.2]). The function S J ( z, t ) generates a smooth Lagrangian submanifold L J + t = L J ( t ) = { ( z, d z S J ( z, t )) ∈ T ∗ M ; z ∈ π ( W α J ) } which is part of the graph of the canonicaltransformation Φ J + t , namely that part with first coordinate lying in the Lagrangian { ( y, ξ ) | − ǫ < | ξ | < ǫ } . Remark 3.14.
The key to the proof of Lemma 3.13 which we owe to [1] is the following fact:as J → ∞ and since we only consider α ∈ A L , the geodesics generating L J ( t ) lie entirely within π − ( M − ) which has sectional curvatures bounded above by a negative constant, these Lagrangians L J + t converge uniformly (indeed, exponentially) to the weak unstable foliation as J → ∞ . Thiswill allow us to compare the size of b J with the weak unstable Jacobians J wut ( m ) , as we shortlyshow. We now take J = L − t = s and K large. We obviously have || χR L,K ( h, s ) || L ≤ C K h K | log h | so using Sobolev embedding arguments as in the proof of Lemma 3.11, we canreplace the L norm by the L ∞ norm, up to a loss of h − d/ − , so || χR L,K ( h, s ) || L ∞ ≤ Ch K − d/ − | log h | . Taking K large enough and summing over the α ∈ A L , the number of which is bounded by h − log | A | , we conclude that these terms do not contribute.It thus remains to study the L ∞ norm of elements of the form(2 πh ) − d/ χe iSL − z,s h b L − ,K ( z, h, s );that is, the L ∞ norm of b L − ,K . We can essentially use the estimates in [1] but first we need tomake some remarks on the different partitions of unity used here as compared to [1]. There, thequantum partition of unity is implemented by multiplication operators that cut off at the scale h − κ , 0 < κ < /
2, while here we use semiclassical pseudodifferential operators with symbols smooth in h . There are two main differences: in [1], the multiplication operators are triviallybounded on L ( M ) with operator norm 1, while in our case, the operator norm of our microlocalcutoffs is 1 + O ( h ) since the principal symbols are bounded by 1. This is inessential sinceit contributes at most a factor (1 + Ch ) | log h | to each term, which is bounded uniformly as h →
0. Second, we need to replace the estimate on the derivatives of the microlocal cutoffs from | D m A a | ≤ Ch − mκ in [1] to k ad m ( D, Π a ) k L → L ≤ C , where D indicates differentiation and ad m indicates the m th iterated commutator (which is even better than in [1], as we do not get anynegative powers of h in our case).With these remarks made, we can follow the analysis of Section 3.2 of [1]. Let us de-fine J t L L + s ( z ) to be the Jacobian of the map Φ t , restricted to L L + s , and evaluated at z =( z, dS L ( s )( z )) ∈ L L + s . Then the construction of [1] shows that b L − k ( z, s ) is only nonzero ifΦ − j ( z, dS L − ( z, ∈ π ( W α L − j ) for all j = 1 , . . . , L and k = 0 , . . . , K , in which case(3.40) | b L − k ( z, s ) | ≤ C k L k ( J L − s L − ( L − s ( z )) / . Notice that this is the analogue of [1, Lemma 3.2.1]. Let us write T = T + N T + T , where T = T + t ∈ [ T , T ]. Then we can decompose(3.41) J − T L L − s ( z ) = J − T L L − s ( z ) × J − T L ( N +1) T (cid:0) Φ − T ( z ) (cid:1) × J − T L NT (cid:0) Φ − T − T ( z ) (cid:1) . . .. . . × J − T L T (cid:0) Φ − T + T ( z ) (cid:1) . The first and last Jacobian factors are uniformly bounded with respect to L ; they only dependon T since they can be written as a supremum of the Jacobian of the flow at some time boundedby 2 T on some set independent of L . Now using Remark 3.14, by assuming that T is largeenough, the Lagrangians L jT , j ≥ ǫ > T sufficiently). Thus J − T L jT (Φ − T + jT ( z )) ≤ J wuT (Φ − T + jT ( z ))(1 + ǫ )where J wut ( m ) is defined in (3.3). But the right hand side is uniformly bounded byexp (cid:16) S T ( W α j ′ ) (cid:17) (1 + ǫ ) , with j ′ := T − jT . Consequently, using (3.39), (3.40) and (3.41), we find that (3.38) is bounded uniformly by C (1 + ǫ ) N exp (cid:16) N X j =1 S T ( W α j ′ ) (cid:17) for some subsequence ( α j ′ ) j ′ ∈ A N and some C > T . Now summing overall α in A L , we clearly obtain the bound X α ∈ A L || χU ( s )Π α L U α L − . . . U α χ || L → L ∞ ≤ Ch − d/ (1 + ǫ ) N X α ∈ A L N Y j ′ =1 e S T ( W α ′ j ) ≤ Ch − d/ (1 + ǫ ) N (cid:16) X a ∈ A e S T ( W a ) (cid:17) N which from (3.28) proves the Lemma since N T is comparable to T . (cid:3) Completion of the proof of Proposition 3.9 : We first note that for times T ≤
1, the estimate || χψ ( h ∆) e iT h ∆ χ || L → L ∞ ≤ C ( T h ) − n/ follows from the parametrix construction in [12, Section 2.2]. TRICHARTZ ESTIMATES WITHOUT LOSS 21
For times 1 ≤ T ≤ T , the estimate can be obtained essentially as above: if T = N + s with N ∈ N and s ∈ (0 , χU ( T ) ψ ( h ∆) χ is, modulo O ( h ∞ )), a finite sum of termsof the form(3.42) χU ( T ) ψ ( h ∆) χ = χψ ( h ∆) U ( s ) U α N U α N − . . . U α χ where U α j = U (1)Π α j ψ + ( h ∆) and α j ∈ A like above. Using the assumption that the region π ( W ) is geodesically convex and that the support of χ is included in M − , we see that only thecases where all the α j are non-zero is not O ( h ∞ ). But then, since the Π α j are microsupportedin the part of the manifold which has negative curvature, then by the method of Anantharaman[1] as we just explained before, the operators of (3.42) are Lagrangian distributions and enjoythe L → L ∞ estimate || χψ ( h ∆) U ( s ) U α N U α N − . . . U α χ || L → L ∞ ≤ C ( T h ) − n . and we sum those finitely many terms to obtain the desired result.For T ≥ T we can apply Lemmas 3.11, 3.12 with the estimate of Lemma 3.13 where ǫ + ǫ is chosen smaller than − P (1 / || χψ ( h ∆) e iT h ∆ χ || L → L ∞ ≤ Ch − n/ e − βT for some β >
0, and all T ∈ (0 , log(1 /h )). It suffices to set t = T h and we get the desired resultsince e − βT T n/ ≤ C . (cid:3) Proof of Theorem 3.3.
We shall be brief here since the proof was outlined already inSection 3.3. We use the notation from that section. Thus, u j = ϕ ( t/h | log h | − j ) χu satisfies( i∂ t − ∆ M ) u j = w ′ j + w ′′ j where w ′ j , w ′′ j are defined in (3.24). Choose χ − ∈ C ∞ ( M ) supported in M − and identically 1 onthe support of χ , and χ + ∈ C ∞ ( M ) so that 1 − χ + ∈ C ∞ ( M ) is identically 1 on π ( K ) and is 0on the support of ∇ χ . Then u j = χ + u j and w ′ j = χ − w ′ j , w ′′ j = χ + w ′′ j . We define u ′ j by(3.43) u ′ j ( t ) = χ − Z t ( j − h | log h | e − i ( t − s )∆ M χ + w ′ j ( s ) ds with u ′′ j defined analogously. Clearly u ′ j + u ′′ j = u j .To treat w ′′ j , consider ˜ u ′′ j defined by(3.44)˜ u ′′ j ( t ) = χ − Z ( j +1) h | log h | ( j − h | log h | e − i ( t − s )∆ M χ + w ′′ j ( s ) ds = χ − e − it ∆ M Z ( j +1) h | log h | ( j − h | log h | e is ∆ M χ + w ′′ j ( s ) ds. Using Lemma 2.2 and Remark 2.3 we see that k Z ( j +1) h | log h | ( j − h | log h | e is ∆ M χ + w j ( s ) ds k L ( M ) ≤ Ch / k w ′′ j k L t ; L ( M ) . Then Theorem 3.8 applied to this L function shows that(3.45) k ˜ u ′′ j k L pt ; L q ( M ) ≤ Ch / k w ′′ j k L t ; L ( M ) and the same estimate holds for u ′′ j by Christ-Kiselev. To treat w ′ j , consider ˜ u ′ j defined by(3.46)˜ u ′ j ( t ) = χ − Z ( j +1) h | log h | ( j − h | log h | e − i ( t − s )∆ M χ + w ′ j ( s ) ds = χ − e − it ∆ M Z ( j +1) h | log h | ( j − h | log h | e is ∆ M χ + w ′ j ( s ) ds. The dual estimate to Theorem 3.7 implies k Z ( j +1) h | log h | ( j − h | log h | e is ∆ M χ + w ′ j ( s ) ds k L ( M ) ≤ C (cid:0) h | log h | (cid:1) / k w ′ j k L t ; L ( M ) ≤ C k χu k L t ; L ( M ) ( h | log h | ) / . using also ω ′ j = i ( h | log h | ) − ϕ ′ ( t/h | log h | − j ) χu . Then we can use Theorem 3.8 applied to this L function shows that(3.47) k ˜ u ′ j k L pt ; L q ( M ) ≤ C ( h | log h | ) − / k χu k L t ; L ( M ) and the same estimate holds for u ′ j by Christ-Kiselev.Squaring and summing over j gives(3.48) N − X j =1 k u j k L pt ; L q ( M ) ≤ C N − X j =1 (cid:16) h k w ′′ j k L t ; L ( M ) + 1 h | log h | k w ′ j k L t ; L ( M ) (cid:17) and the right hand side is no bigger than C k u k L ( M ) using Lemma 2.2 for w ′′ j and Theorem 3.7for w ′ j . Using the continuous embedding from l ( N ) to l p ( N ) as in Section 3.3 gives k χe − it ∆ M ψ ( h ∆) u k L p [0 , L q ( M ) ≤ C k u k L ( M ) . Together with Theorem 3.6 this gives the Strichartz estimate without the space cutoff χ : k e − it ∆ M ψ ( h ∆) u k L p [0 , L q ( M ) ≤ C k u k L ( M ) . Finally using Bouclet’s Littlewood-Paley estimate (equation (1.4) of [6]) and the argument in[12], we remove the frequency cutoff and obtain (3.5), which completes the proof.
Remark 3.15.
The restriction p > in Theorem 0.2 is only required because we use the Christ-Kiselev lemma. It is likely that this condition could be eliminated (for d > ) with a more carefulanalysis. References [1] N. Anantharaman,
Entropy and the localization of eigenfunctions , Annals of Math. 168 (2008), no 2, 435-475.[2] N. Anantharaman, S. Nonnenmacher,
Half-delocalization of eigenfunctions for the Laplacian on an Anosovmanifold , Ann. Inst. Fourier (Grenoble), 57 (2007), no 7, 1465-2523[3] J-P. Anker, V. Pierfelice,
Nonlinear Schr¨odinger equation on real hyperbolic spaces , Ann. Henri Poincar´eAnalyse Non-Lin´eaire, in press (2009).[4] V. M. Babiˇc,
Eigenfunctions which are concentrated in the neighborhood of a closed geodesic (Russian), Zap.Nauˇcn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 9 (1968), 15–63.[5] V. Banica,
The nonlinear Schr¨odinger equation on the hyperbolic space , Comm. PDE. (2007), no 10,1643-1677.[6] J-M. Bouclet, Semi-classical functional calculus on manifolds with ends and weighted L p estimates ,Arxiv:0711.358.[7] J-M. Bouclet, Littlewood-Paley decompositions on manifolds with ends , Bull. Soc. Math. Fr., , fascicule1 (2010), 1-37.[8] J-M. Bouclet,
Strichartz estimates for asymptotically hyperbolic manifolds , Analysis and PDE, in press.[9] J-M. Bouclet, N. Tzvetkov,
Strichartz estimates for long range perturbations , Amer. J. Math. (2007),no 6, 1665-1609.[10] J. Bourgain,
Fourier transform restriction phenomena for certain lattice subsets and application to nonlinearevolution equations. I. Schr¨odinger equations . Geom. Funct. Anal. (1993), 107-156.[11] N. Burq, Smoothing effect for Schr¨odinger boundary value problems . Duke Math. J. (2004), no. 2,403–427.[12] N. Burq, P. Gerard, N. Tzvetkov,
Strichartz inequalities and the non-linear Schr¨odinger equation on compactmanifolds , Amer. J. Math, (2004), 569-605.[13] N. Burq, P. Gerard, N. Tzvetkov,
On Nonlinear Schr¨odinger equations in exterior domains , Ann. I.H.P.Ana. non lin., (2004), 295-318[14] N. Burq and M. Zworski, Geometric control in the presence of a black box. , J. Amer. Math. Soc. 17 (2004),no. 2, 443-471[15] F. Cardoso, G. Vodev,
Uniform estimates of the Laplace-Beltrami operator on infinite volume Riemannianmanifolds II , Ann. H. Poincar´e (2002) 673-691.[16] R. Carles, Global existence results for nonlinear Schrdinger equations with quadratic potentials.
DiscreteContin. Dyn. Syst. (2005), no. 2, 385–398.[17] M. Christ, A. Kiselev, Maximal functions associated to filtrations , J. Funct. Anal. (2001) 409-425.[18] H. Christianson,
Cutoff resolvent estimates and the semilinear Schr¨odinger equation , Proc. Amer. Math.Soc. (2008), no. 10, 3513-3520.[19] P. Constantin, J.-C. Saut,
Effets r´egularisants locaux pour des ´equations dispersives g´en´erales , C. R. Acad.Sci. Paris S´er. I Math. 304 (1987), no. 14, 407–410.
TRICHARTZ ESTIMATES WITHOUT LOSS 23 [20] K. Datchev,
Local smoothing for scattering manifolds with hyperbolic trapped sets , Comm. Math. Phys. ,no. 3, 837–850.[21] S.-I. Doi,
Smoothing effects for Schr¨odinger evolution equation and global behaviour of geodesic flow , Math.Ann. (2000), 355 – 389.[22] P. Gaspard and S. A. Rice,
Semiclassical quantization of the scattering from a classically chaotic repellor ,J. Chem. Phys. (1989), 2242-2254.[23] C. Guillarmou, S. Moroianu, J. Park, Eta invariant and Selberg Zeta function of odd type over convexco-compact hyperbolic manifolds. , to appear Advances in Math. Arxiv 0901.4082.[24] B. Hasselblatt, A. Katok,
Introduction to the modern theory of dynamical systems , Cambridge UniversityPress, 1997.[25] A. Hassell, T. Tao, J. Wunsch,
Sharp Strichartz estimates on nontrapping asymptotically conic manifolds ,Amer. J. Math. (2006), no. 4, 963–1024.[26] A. Hassell, A. Vasy,
Symbolic functional calculus and N -body resolvent estimates , J. Funct. Anal. (2000),no. 2, 257–283.[27] A. Ionescu, G. Staffilani, Semilinear Schr¨odinger flow on hyperbolic space: scattering in H , Math. Ann. (2009), no. 1, 133–158.[28] M. Keel, T. Tao, Endpoint Strichartz estimates , Amer. J. Math. (1998), 955-980.[29] W. Klingenberg, Riemannian geometry , Studies in Mathematics, de Gruyter. Berlin, New-York 1982.[30] R. Mazzeo, R. Melrose,
Meromorphic extension of the resolvent on complete spaces with asymptoticallyconstant negative curvature , J.Funct.Anal. (1987), 260-310.[31] R.B. Melrose, Spectral and scattering theory for the Laplacian on asymptotically Euclidian spaces, in Spectraland Scattering Theory , M. Ikawa, ed., Marcel Dekker, 1994.[32] S. Nonnenmacher, M. Zworski,
Quantum decay rates in chaotic scattering , Acta Math. (2009), no. 2,149–233[33] G.P. Paternain,
Geodesic flows . Volume 180,
Progress in Mathematics . Birkh¨auser Boston Inc., Boston MA,1999.[34] S.J. Patterson,
The limit set of a Fuchsian group . Acta Math. (1976), 241-273.[35] P. Perry,
Asymptotics of the length spectrum for hyperbolic manifolds of infinite volume , GAFA (2001),132-141.[36] M. F. Pyˇskina, The asymptotic behavior of eigenfunctions of the Helmholtz equation that are concentratednear a closed geodesic (Russian), Zap. Nauˇcn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 15 (1969),154–160.[37] J.V. Ralston,
On the construction of quasimodes associated with stable periodic orbits , Comm. Math. Phys.,51-3, (1976), 219–242.[38] J. Schmeling, Ra. Siegmund-Schultze,
H¨older continuity of the holonomy maps for hyperbolic basic sets. I. ,Ergodic theory and related topics, III (Gstrow, 1990), 174–191, Lecture Notes in Math., 1514, Springer,Berlin, 1992.[39] G. Staffilani, D. Tataru,
Strichartz estimates for a Schr¨odinger operator with nonsmooth coefficients , Comm.Part. Diff. Eq. (2002), no. 7-8, 1337–1372.[40] R. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equa-tions , Duke Math. J. 44 (1977), no. 3, 705–714.[41] D. Sullivan,
The density at infinity of a discrete group of hyperbolic motions.
Publ. IHES (1979), 172-202.[42] H. Takaoka, N. Tzvetkov, On 2D nonlinear Schr¨odinger equation on R × T , J. Funct. Anal. (2001), no.2, 427-442.[43] J. Wunsch, M. Zworski, Distribution of resonances for asymptotically Euclidean manifolds , J. Diff. Geom. (2000), 43-82.[44] M. Zworski, Dimension of the limit set and the density of resonances for convex co-compact hyperbolicsurfaces.
Invent. Math.136