Strichartz Estimate and Nonlinear Klein–Gordon Equation on Nontrapping Scattering Space

We study the nonlinear Klein–Gordon equation on a product space M=R×X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M=\mathbb {R}\times X$$\end{document} with metric g~=dt2-g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{g}=dt^2-g$$\end{document} where g is the scattering metric on X. We establish the global-in-time Strichartz estimate for Klein–Gordon equation without loss of derivative by using the microlocalized spectral measure of Laplacian on scattering manifold showed in Hassell and Zhang (Anal PDE 9:151–192, 2016) and a Littlewood–Paley squarefunction estimate proved in Zhang (Adv Math 271: 91–111, 2015). We prove the global existence and scattering for a family of nonlinear Klein–Gordon equations for small initial data with minimum regularity on this setting.


Introduction and Statement of Main Results
In this paper we consider the evolution of a semilinear Klein-Gordon equations with power-type nonlinearities on a nontrapping scattering manifold. More specifically, we consider the following family of nonlinear Klein-Gordon equation g u + m 2 u = F(u, Du), (t, z) ∈ R × X , u(0) = u 0 (z), ∂ t u(0) = u 1 (z).
Here g = ∂ 2 t − g denotes the d'Alembertian in the metricg = dt 2 − g and g is the Laplacian on the manifold X with scattering metric g introduced by Melrose [34]. We focus on the questions of the minimum regularity for which local well-posedness and small nonlinear scattering hold true. One of the motivations for this study is that in low-dimensional case, we can achieve the level of regularity corresponding to a conserved quantity (e.g., the energy) and thus get global existence for large initial data.
On more general class of physical manifolds, Hintz and Vasy [15][16][17]25] studied the semilinear and quasilinear wave and Klein-Gordon equation on the physically cosmological spacetimes as solutions to Einstein's field equations. In particular they gave a detailed analysis of the long-time behavior of linear and nonlinear waves on Kerr-de Sitter space and nontrapping Lorentzian scattering spaces for large regularity and small initial data. The recent development of [45] allowed them to set up the analysis of the associated linear problem in a framework of Fredholm problem, in which they used Melrose's philosophy [33,34] of studying differential operators P = g on a noncompact space M by compactifying M to a manifold M with boundary or even corners. The concrete choice of compactification is connected to the geometric structure of M near infinity. In our less complicated product setting, we use the same Melrose's idea with P = g on X in the study of the spectral measure of the Laplacian [15,26] and then analyze the propagator e it √ 1− g ; thus, we expect better result on the lowest regularity due to the establishment of the global-in-time Strichartz estimate.
In the simplest flat Euclidean space, there are a large number of literature to study the nonlinear Klein-Gordon equation. In the flat Euclidean space, where X = R n and g jk = δ jk , the dispersive properties of the Klein-Gordon and other dispersive equations have been proved to be powerful in the study of nonlinear problems. The Strichartz estimate for the solution of the homogenous and inhomogeneous Klein-Gordon equation in the form of space time integrability properties gives u(t, z) L q t (I ;L r z (R n )) + u(t, z) C(I ;H s (R n )) u 0 H s (R n ) + u 1 H s−1 (R n ) + F Lq t (I ;Lr z (R n )) , (1.1) where the pairs (q, r ), (q,r ) ∈ [2, ∞] 2 satisfy the admissible condition for 0 θ 1 2 q + n − 1 + θ r n − 1 + θ 2 , (q, r , (n − 1 + θ)/2) = (2, ∞, 1). (1.2) and the gap condition We refer to Brenner [3], Ginibre-Velo [18], and Keel-Tao [29] for more details. In particular when θ = 0, these estimates corresponding to wave equation serve as a tool for existence results about the nonlinear wave equation. For example, Lindblad-Sogge [32] answered the problem of finding minimal regularity conditions on the initial data ensuring local well-posedness for semilinear wave equations. Analogous results for the Klein-Gordon equation can be carried out even though the sharpness of well-posedness result is not known. There are too many references to cite all here, so we refer the reader to [27,36] and the references therein.
In view of the rich Euclidean theory due to the Strichartz estimate, it is natural to consider the corresponding equations on more general manifolds. However it is difficult or impossible to establish the same Strichartz-type estimates as in Euclidean space on the large class of manifold due to the influence of qualitative geometric properties. On asymptotically de Sitter spaces Baskin [4][5][6] established a family of local (in time) weighted Strichartz estimates with derivative losses for the Klein-Gordon equation on asymptotically de Sitter spaces and provided a heuristic argument for the nonexistence of a global dispersive estimate on these spaces. The Strichartz estimates are local-intime or loss of derivatives on the compact manifold with or without boundary, see [8,11,28,39,43] and references therein. On noncompact manifold with nontrapping condition, one can obtain global-in-time Strichartz estimates. For example, the global Strichartz estimates on a exterior manifold in R n to a convex obstacle, for metrics g which agrees with the Euclidean metric outside a compact set with nontrapping assumption, are obtained by Smith-Sogge [40] for odd dimension, and Burq [10] and Metcalfe [35] for even dimension. Blair-Ford-Marzuola [9] established global Strichartz estimates for the wave equation on flat cones C(S 1 ρ ) by using the explicit representation of the fundamental solution. Anker-Pierfelice [2] study the problem on minimal regularity condition on the initial data ensuring well-posedness for wave and Klein-Gordon on hyperbolic space. On the nontrapping scattering manifold, the same setting considered here, Hassell, Tao, and Wunsch first established an L 4 t,z -Strichartz estimate for Schrödinger equation in [21] and then they [22] extended the estimate to full admissible local-in-time Strichartz estimate except endpoint q = 2. More recently, Hassell-Zhang [26] improved the local-in-time one to global-in-time one and fixed the endpoint q = 2 by analyzing the microlocalized spectral measure. Following this Zhang [47] extended the global-in-time result for the wave equation. Bouclet-Mizutani [7] generalized the Schrödinger result to the setting with mild trapping and with more general ends.
In this paper, we will establish the global-in-time Strichartz estimate for the Klein-Gordon and apply it to study the minimal regularity problem for nonlinear Klein-Gordon on the nontrapping scattering manifold (asymptotically conic manifold) which is the same as in [22,26,47] including the asymptotically Euclidean space. The scattering manifold means that X can be compactified to a manifold with boundary X such that g becomes a scattering metric on X ; see more about this next section. For geometric reasons, we expect the same dispersive properties of Klein-Gordon as in the Euclidean setting. The key ingredient is to establish global-in-time Strichartz estimate for Klein-Gordon. It is known that Klein-Gordon behaves like Schrödinger at low frequency and wave equation at high frequency. Similar to the Euclidean space, we introduce a parameter θ for Klein-Gordon admissible pair which is wave admissi-ble at θ = 0 and Schrödinger pair at θ = 1. More precisely, we have the result about Strichartz estimates in the following.
Let H s (X ) = (1 − g ) − s 2 L 2 (X ) be the inhomogeneous Sobolev space over X . Throughout this paper, pairs of conjugate indices are written as r , r , where 1 r + 1 r = 1 with 1 r ∞. Theorem 1.1 (Global-in-time Strichartz estimate) Let (X , g) be nontrapping scattering manifold of dimension n 3. Suppose that u is the solution to the Cauchy problem for some initial data u 0 ∈ H s , u 1 ∈ H s−1 , and the time interval I ⊆ R, then , (1.5) where the pairs (q, r ), (q,r ) ∈ [2, ∞] 2 satisfy the KG-admissible condition with and the gap condition (1.7) Remark 1. 2 We remark that the estimates here are the same as the Strichartz estimates for Klein-Gordon on Euclidean space which are global-in-time and have no loss of derivatives.
We sketch the proof as follows. Similar to [26,47], our strategy is to use the abstract Strichartz estimate proved in Keel-Tao [29]. Thus, with U (t) denoting the (abstract) propagator, we need to show uniform L 2 → L 2 estimate for U (t), and L 1 → L ∞ type dispersive estimate on the U (t)U (s) * with a bound of the form O((1 + |t − s|) −(n−1+θ)/2 ) with 0 θ 1. In the flat Euclidean setting, the estimates are usually obtained by using stationary phase argument. One point is to write the propagator in the form of oscillatory integral. Since the Laplacian in our general setting is degenerate when it is close to the compactified boundary, the formulate turns out to be more complicated. On the other hand, the conjugate point occurring in this nonflat setting may lead to the failure of the dispersive estimate. For example, [24] showed that the Schrödinger propagator e it g failed to satisfy such a dispersive estimate at any pair of conjugate points (z, z ) ∈ X × X (i.e., pairs (z, z ) where a geodesic emanating from z has a conjugate point at z ). Fortunately, we can localize the propagator to separate the conjugating points and write the propagator in a form of oscillatory integral by using a microlocalized spectral measure. The microlocalized spectral measure [26] not only has a size estimate in decay rate but captures its oscillatory behavior, where Q j (λ) is a member of a partition of the identity operator in L 2 (X ). In the stationary phase argument, the Klein-Gordon multiplier e it √ 1+λ 2 behaves like wave at high frequency and Schrödinger at low frequency. We establish the dispersive estimate with norm O((1 + |t − s|) −n/2 ) at low frequency and O((1 + |t − s|) −(n−1+θ)/2 ) at high frequency. We finally show the Strichartz estimate from a frequency-localized Strichartz estimate by a square function estimate proved in [47]. The inhomogeneous Strichartz estimates follow from the homogeneous estimates and the Christ-Kiselev lemma.
Having the Strichartz estimate, we first consider the well-posedness and nonlinear scattering problem of the Cauchy problem on this setting In the case of flat Euclidean space, there are many results on the understanding of the global existence and scattering. We refer the readers to [32,42] and references therein. We here are mostly interested in the range of exponents p ∈ [p conf , 1 + 4 n−2 ] and the initial data are in H s c (X ) × H s c −1 (X ), where p conf = 1 + 4 n and s c = n 2 − 2 p−1 . The critical power 1 + 4/n, which is different from wave equation's 1 + 4 n−1 in [47], is related to the dispersive estimate decay rate and it appears in the theorem because of the fact that NLKG is conformally invariant only if F(u) = u 1+4/n ; see [38]. The other power 1 + 4/(n − 2) is related to the energy-critical index.
The following result is about the well-posedness and nonlinear scattering with smallest regularity.

Remark 1.4
On local well-posedness and small scattering with the minimal regularity result, we have to restrict ourself with p ∈ [1+ 4 n−1 , 1+ 4 n−2 ], that is, s c 1/2 and we can extend the similar result to p ∈ [p conf , 1 We next specially consider the well-posedness of the following Yang-Mills-type equations on this setting with dimension n = 3.
The derivative Du is measured relative to the metric structure; more precisely, D is a first-order scattering differential operator. When dropping the linear term u, this equation has the same scaling as cubic NLW, but is more difficult technically because of the derivative term u Du. In the Euclidean space, this Yang-Mills-type wave equation was proved to be local well-posedness when s > 1 in [37] and was showed to be ill-posedness when s 1 in [31].

Theorem 1.5 Let (X , g) be a nontrapping scattering manifold of dimension n = 3 and let
This paper is organized as follows. In Sect. 2 we review the background of scattering manifold, the results of the microlocalized spectral measure for the Laplacian, and the square function inequalities on this setting. Section 3 is devoted to the proofs of the microlocalized dispersive estimates and L 2 -estimates. In Sect. 4, we prove the homogeneous and inhomogeneous Strichartz estimates. Finally, we apply the Strichartz estimates to show Theorems 1.3 and 1.5.

Some Analysis Tools on Scattering Manifold
In this section, we briefly recall the key elements of the microlocalized spectral measure and a fundamental Littlewood-Paley squarefunction estimate. The first one was constructed by Hassell-Zhang [26] to capture both its size and the oscillatory behavior. The second one was proved in [47].

Geometry Setting
Let us recall the manifold with scattering metric introduced by Melrose [34]. There are many works to analyze the Laplacian operator on the scattering manifold, that is, asymptotically conic geometric setting; see [19,20,22,24,26]. Let (X , g) be a complete noncompact Riemannian manifold of dimension n 2 with one end, diffeomorphic to (0, ∞) × Y where Y is a smooth compact connected manifold without boundary. Moreover, we assume (X , g) is scattering manifold which means that X allows a compactification X with boundary, with ∂ X = Y , such that the metric g becomes an asymptotically conic metric on X . In details, the metric g in a collar neighborhood is a boundary defining function for ∂ X and h is a smooth family of metrics on Y . Here we use y = (y 1 , . . . , y n−1 ) for local coordinates on Y = ∂ M, and the local coordinates (x, y) on X near ∂ X . Away from ∂ X , we use z = (z 1 , . . . , z n ) to denote the local coordinates. If h jk (x, y) = h jk (y) is independent of x, we say X is perfectly conic near infinity. Moreover if every geodesic z(s) in X reaches Y as s → ±∞, we say X is nontrapping. The function r := 1/x near x = 0 can be thought of as a "radial" variable near infinity and y can be regarded as the n − 1 "angular" variables; the metric is asymptotic to the exact conic metric ((0, ∞) r ×Y , dr 2 +r 2 h(0)) as r → ∞. The Euclidean space X = R n is an example of an asymptotically conic manifold with Y = S n−1 and the standard metric.

The Laplacian on Scattering Manifold
Our setting is on the scattering manifold, and we turn to the concepts of "scattering geometry." For a full discussion of scattering geometry, we refer the reader to Melrose [34]. The space of sc-vector fields is defined as is the Lie algebra of all smooth vector fields on X X which are tangent to the boundary. The sc-vector field also forms a Lie algebra. These sc-vector fields can be realized as the sections of a vector bundle sc T X , called the sc-tangent bundle. That means is a space of sections of sc T X the sc-tangent bundle over X . Using above notation in which x is the boundary defining function of X and y is the coordinate in ∂ X , we have We denote by Diff * sc (X ) the 'enveloping algebra' of V sc (X ), meaning the ring of differential operator on C ∞ (X ) generated by V sc (X ) and C ∞ (X ). In particular, near the boundary ∂ X , the k-order scattering differential operator is given by If α ∈ R, the α b-density bundle, denoted by sc α , is defined by In particular when α = 1/2, it is convenient to regard such operators as acting on schalf densities, that is, multiples of a half-density taking the form dx . Correspondingly, the Schwartz kernels of such operators can be written as a distribution tensored with a scattering half-density in each of the left and right variables.
Define sc T * X , the scattering cotangent bundle over X , to be the dual vector bundle to sc T X . Locally near the boundary, in the coordinate (x, y), we have Thus for any α ∈ sc T * X it can be written as and this gives linear coordinates (τ, μ) ∈ R × R n−1 on each fiber of sc T * X . Thus this also gives linear coordinates (x, y; τ, μ) on sc T * X near the boundary ∂ X . On the other hand, if (ξ, η) is the dual cotangent variables to (x, y), then We say (τ, μ) as rescaled cotangent variables. Hence this space of operators can be microlocalized by introducing scattering pseudodifferential operators which are formally objects given by b(x, y, x 2 ∂ x , x∂ y ) with b(x, y, τ, μ) a Kohn-Nirenberg symbol on the bundle sc T * X .
In the above coordinates, the Laplacian can be written as where |g| is the determinant of the metric g jk . To compare with the Euclidean space near the boundary, we write the metric near the boundary in the form dr 2 + r 2 h(x, y, dy, r −2 dr) with respect to the local coordinates r = 1/x and y.
Then the metric components satisfy whereh is the induced metric on the boundary. Note that the cross term, with j = 0 and k = 0 or j = 0 and k = 0, vanishes as x 3 when expressed in terms of x∂ x and ∂ y (the components in V b ). Hence near the boundary we write where Diff 2 b is the second-order differential b-operator. In this sense, the Laplacian on this setting is a sc-differential operator. To see more results about its resolvent and calculus, we refer to [25,33].

The Microlocalized Spectral Measure
In the free Euclidean space, the Klein-Gordon propagator can be written in an explicit formula by using the Fourier transform, but in our setting it turns out to be quite complicated. From the results of [19,23], we have known that the Schwartz kernel of the spectral measure can be described as a Legendrian distribution on the compactification of the space X × X uniformly with respect to the spectral parameter λ. As pointed out in introduction, we really need to choose an operator partition of unity to microlocalize the spectral measure such that the spectral measure can be expressed in a formula capturing not only the size but also the oscillatory behavior. This was constructed and proved in [26]. For convenience, we recall it here.
with N independent of λ, such that for each 1 j N we can write Here d(·, ·) is the Riemannian distance on X .
From this proposition, we can exploit the oscillations both in the multiplier e i(t−s) √ 1+λ 2 and in e ±iλd(z,z ) to obtain the required dispersive estimate for the T T * version of the microlocalized propagator.

The Littlewood-Paley Squarefunction Estimate
In [47], we showed the Gaussian upper bounds on the heat kernel by using the localin-time heat kernel bounds in Cheng-Li-Yau [12], and Guillarmou-Hassell-Sikora's [20] restriction estimate for low frequency. Hence we finally proved the Littlewood-Paley squarefunction estimate on this setting by using a spectral multiplier estimate in Alexopoulos [1] and Stein's [41] classical argument involving Rademacher functions. Now we recall the result here for convenience.
Let φ ∈ C ∞ 0 (R \ {0}) take values in [0, 1] and be supported in [1/2, 2] such that One important application of the traditional Littlewood-Paley theory is the proof of Leibniz (=product) and chain rules for differential operators of noninteger order. For example, if 1 < p, p j < ∞ with j = 1, . . . 4 and s > 0, then For a textbook presentation of these theorems and original references, see [44]. The Leibniz chain rules is a basic tool in the proof of well-posedness. Since we have heat kernel estimate with Gaussian upper bounds and the Littlewood-Paley squarefunction estimate, the Leibniz chain rules can be obtained by similar argument in Euclidean space and it also was proved in [14,Theorem 27]. We record here

L 2 -Estimates and Dispersive Estimates
In this section, we prove the L 2 -estimates for U j,k (t) and dispersive estimates for is a microlocalized Klein-Gordon propagator. The L 2estimate is showed by the spectral theory on Hilbert space. The conjugate points are separated in the microlocalized propagators, and hence we can prove the T T * version dispersive estimates. Since the abstract Klein-Gordon propagator U (t) = e it √ 1− g behaves like the Schrödinger at low frequency and wave at high frequency, we need to establish dispersive estimate by using different arguments at different frequencies.

Microlocalized Propagator
We start by dividing the Klein-Gordon propagator into a low-energy piece and a high-energy piece. Using the dyadic partition of unity 1 = k∈Z φ(2 −k λ) we further define Further using scattering pseudodifferential operator partition of identity operator in Proposition 2.1, we define We divide the microlocalized Klein-Gordon propagator into low frequency and high frequency

L 2 -Estimate for U j,k (t).
In this subsection we show this definition is well defined and prove U j,k (t) is a bounded operator on L 2 (X ). Essentially this has been proved in [47,Proposition 3.2]. For convenience, we sketch it here. Indeed it suffices to show the above integrals are well defined over any compact interval in (0, ∞). Let A(λ) = e it √ 1+λ 2 φ(2 −k )Q j (λ). Then A(λ) is a family of bounded operators on L 2 (X ), compactly supported in [2 k−1 , 2 k+1 ] and C 1 in λ ∈ (0, ∞). Integrating by parts, the integral of 2 k+1 is given by Hence the operators U j,k (t) are well defined by using the following lemma which is the consequence of [26, Lemmas 2.3, 3.1].
We observe that this is independent of t and we also note that the integrand is a bounded operator on L 2 , with an operator bound of the form C/λ where C is uniform, as we see from Lemma 3.1 and the support property of φ. The integral is therefore uniformly bounded, as we are integrating over a dyadic interval in λ. Hence we have shown that Proposition 3.2 (L 2 -estimates) Let U j,k (t) be defined in (3.2). Then there exists a constant C independent of t, z, z such that U j,k (t) L 2 →L 2 C for all j 1, k ∈ Z.
Since there is no difference between e itλ 2 and e it √ 1+λ 2 in the proof [26, Proposition 5.1] (using a almost orthogonal property in the summation of k), we have Proposition 3.3 (L 2 -estimates) Let U low j (t) be defined in (3.3). Then there exists a constant C independent of t, z, z such that U low j (t) L 2 →L 2 C for all j 1.

Dispersive Estimates
In this subsection, we use stationary phase argument and Proposition 2.1 to establish the microlocalized dispersive estimates. Before doing this, we prove a fundamental result on decay estimate.

Proposition 3.4 (Microlocalized dispersive estimates for low frequency) Let Q j (λ) be in Proposition 2.1. Then for all integers j 1, the kernel estimate
holds for a constant C independent of points z, z ∈ X.

Proof
The key things in the proof are to use the property of spectral measure in Proposition 2.1 and stationary phase argument. When |t| 1, it is easy to show it due to the compact support of φ 0 . From now on, we only need to consider the case t 1 by symmetry. Let r = d(z, z ) andr = rt − 1 2 . In this case, we write the kernel using where a ± satisfies estimates First, we show the contribution of the above term with b(λ, z, z ). We can use the estimate (2.7) to obtain d dλ Let δ be a small constant to be chosen later. Recall that we We use integration by parts N times to obtain, using (3.9) Choosing δ = |t| − 1 2 , we have thus proved (3.10) Now we consider first term in RHS of (3.7). We divide it into two pieces using the partition of unity above. It suffices to prove that there exists a constant C independent ofr such that The estimate for I ± is obvious, since λ 1. For I I + , we use integration by parts. Notice that Note that if 0 < λ < √ t, we have for k 0 by induction and integrating by parts, we gain a factor of λ −2N thanks to (3.16) and (3.11). Thus I I + can be estimated by To treat I I − , we introduce a further decomposition, based on the size ofr λ. We write I I − = I I − 1 + I I − 2 , where (dropping the-superscripts and subscripts from here on) Let (λ,r ) = √ t 2 + tλ 2 −r λ. We first consider I I 1 . Since the integral for I I 1 is supported where λ (8r ) −1 and λ 1/2, the integrand is only nonzero when By (3.16) and using integration by parts, we obtain for N > n/2 Finally we consider I I 2 . Here, we replace the decomposition m φ(2 −m λ) with a different decomposition, based on the size of ∂ λ .
Ifr 10, note λ < √ t again, then for the integrand of I I 1 2 to be nonzero we must have λ 100, due to the second φ 0 factor in I I 1 2 . Then it is easy to see that I I 1 2 is uniformly bounded. Ifr 10, by | tλ √ t 2 +tλ 2 −r | 1 and λ < √ t, we haver ∼ λ. Hence, using (3.16) with α = 0, Now we consider the second term. We write then we have the rough estimate, due to the support of b |∂ α λ b| C α λ n−1 (1 +r λ) −(n−1)/2 .
Hence we obtain Therefore we obtain by using integrating by parts and (3.16) Ifr 2 m+1 , then λ 2 m+2 on the support of the integrand.
Ifr 2 m+1 , we have λ ∼r , thus which is summable for N > 1. Therefore we have completed the proof of Proposition 3.4.
holds for 0 θ 1 and a constant C independent of k and points z, z ∈ X.
The key to the proof is to use the estimates in Proposition 2.1. If |t| h, it is easy to see (3.13) due to From now on, we only consider |t| h = 2 −k . By the scaling, this is a directly consequence of (3.14) Indeed if we have done this, we have for 0 θ 1 which implies (3.13). Now we prove (3.14).
where a ± satisfies estimates and therefore Consider the terms with the 'b' term, then we can use the estimate (2.7) to obtain Let δ be a small constant to be chosen later. Recall that we We use integration by parts N times to obtain, using (3.9), Choosing δ = (|t|/h) − 1 2 and noting |t| h, we have thus proved Next we consider the terms with a ± . Without loss of generality, we consider t h. Let ± (λ, h, r , t) = √ h 2 + λ 2 ± λr t , it suffices to show there exists a constant C independent of r , t, and h such that where If r < t/4 or r > 2t, a simpler computation gives It is not difficult to use the Leibniz rule to prove Lemma 3.6 Let L = ( it h ∂ λ ) −1 ∂ λ and let L * be its adjoint operator. Suppose that b(λ) satisfies Then we have for any N 0 By integrating by parts and using this lemma, we obtain for r < t 4 or r > 2t which implies (3.19) since t h. Therefore we only need consider the case t ∼ r . A rough estimate gives and by using the same stationary phase argument again, we obtain To estimate I − h (t, r ), we need the following Van der Corput lemma, see [41]  It is easy to check for h 1 and λ ∼ 1 By using the Van der Corput lemma with λ = th, we show (3.26) This together with (3.23), we prove (3.19).
As two consequences of Propositions 3.4 and 3.5, respectively, we immediately have Proposition 3.8 Let U low j (t) be defined in (3.3). Then there exists a constant C independent of t, z, z for all j 1, such that Proposition 3.9 Let U j,k (t) be defined in (3.2). Then there exists a constant C independent of t, z, z for all j 1, k ∈ Z + such that

Strichartz Estimates
In this section, we show the Strichartz estimates in Theorem 1.1. To obtain the Strichartz estimates for high frequency, we need a variant of Keel-Tao's abstract Strichartz estimate.

Semiclassical Strichartz Estimates
We recall a variety of the abstract Keel-Tao's Strichartz estimates theorem proved in [47], which is an analogue of the semiclassical Strichartz estimates for Schrödinger in [30,46].

Homogeneous Strichartz Estimates
Now we prove the homogeneous Strichartz estimates. Using the Littlewood-Paley frequency cutoff φ m ( √ H), we define Then the frequency-localized solutions {u m } m∈Z solves the family of Cauchy problems where f m = φ m ( √ H)u 0 and g m = φ m ( √ H)u 1 . Then we can write the solution Noting that we can write By the squarefunction estimates (2.9) and Minkowski's inequality, we obtain for q, r 2 To prove the homogeneous estimates in Theorem 1.1, that is F = 0, we need where the K-G admissible pair (q, r ) ∈ [2, ∞] 2 and s satisfy (1.6) and (1.7).

Inhomogeneous Strichartz Estimates
In this subsection, we prove the inhomogeneous Strichartz estimates. Let U (t) = e it √ 1+H : L 2 → L 2 . We have already proved that Then we have by duality . By the Christ-Kiselev lemma [13], we thus obtain for q >q , τ <t Notice that for all (q, r ), (q,r ) satisfy (1.6) and (1.7), we must have q >q . Therefore we have proved all inhomogeneous Strichartz estimates including q = 2.

Well-posedness and Small Nonlinear Scattering
In this section, we prove Theorems 1.3 and 1.5. We prove the results by a contraction mapping argument. The key point is the application of Strichartz estimates.

Proof of Theorem 1.5
For a constant C we define Consider the solution map defined by where F(u) is replaced by F(u) = u Du + |u| 2 u. By Theorem 1.1 with 0 < θ = 2δ 1, we obtain The standard contraction argument on Y completes the proof of Theorem 1.5.