Generalized and weighted Strichartz estimates

In this paper, we explore the relations between different kinds of Strichartz estimates and give new estimates in Euclidean space $\mathbb{R}^n$. In particular, we prove the generalized and weighted Strichartz estimates for a large class of dispersive operators including the Schr\"odinger and wave equation. As a sample application of these new estimates, we are able to prove the Strauss conjecture with low regularity for dimension 2 and 3.


Introduction
In this paper, we explore the relations between different kinds of Strichartz estimates and give new estimates in Euclidean space R n . In particular, we prove the generalized and weighted Strichartz estimates for a large class of dispersive operators including the Schrödinger and wave equation. As a sample application of these new estimates, we are able to prove the Strauss conjecture with low regularity for dimension 2 and 3. In some sense, this paper can be viewed as a sequel to the work of Fang and the second author [5].
Let D = √ −∆. Typically, Strichartz estimates for the dispersive operators e itD a , a = 1, 2, are a family of estimates which state (1.1) e itD a f L q t L r x (R×R n ) ≤ C f Ḣs , whereḢ s (s < n/2) denotes the homogenous Sobolev space in R n . These estimates were first established by Strichartz [35] for q = r. They were generalized to nonendpoint admissible (q, r) by Ginibre and Velo [7] [8], Lindblad and Sogge [21]. The end point estimates were proven by Keel and Tao [18]. They are powerful tools in the study of the nonlinear Schrödinger and wave equations. See for example Cazenave [2], Sogge [29] and Tao [37] and references therein.
Let ∆ ω = 1≤i<j≤n Ω 2 ij be the Laplace-Beltrami operator on the unit sphere S n−1 ⊂ R n with Ω ij = x i ∂ j − x j ∂ i , ω ∈ S n−1 , and define Λ ω = √ 1 − ∆ ω . Based on the usual Sobolev spaces H s , we introduce the Sobolev spaces with angular regularity as follows (b ≥ 0) For the homogeneous Sobolev spaceḢ s = D −s L 2 , we can similarly define the spacė H s,b ω = Λ −b ω D −s L 2 . We will also use the homogeneous Besov spaceḂ s p,q (for sp < n or sp = n and q = 1), which is defined to be the completion of C ∞ 0 in S ′ , with respect to the norm f Ḃs p,q = 2 sk S k f ℓ q k L p . Here S k are the Fourier multiplier operators of the homogenous Littlewood-Paley decomposition.
Recently, there have been many works on various generalizations of the Strichartz estimates and their applications. Before stating our results and related works, we would like to list different types of Strichartz estimates by the following table. After tagging different Strichartz estimates, it will be easier to explain the history and give an overview of our work by diagrams and lists afterward. Here, in general, we will be able to consider the operators e itD a with a > 0. Recall that the operators e itD a are related to the Schrödinger equation (a = 2) and the wave equation (a = 1). Also we denote L q T L r x as L q t L r x with domain [0, T ] × R n .
Now let us give a brief history of the generalized Strichartz estimates (G. Strichartz) and weighted Strichartz estimates (W. Strichartz) within our best knowledge. The generalized Strichartz estimates were first studied in the endpoint case of the classical Strichartz estimates for the wave and Schrödinger equations. For the wave equation (a = 1), it is known that the 3-dimensional endpoint L 2 t L ∞ x Strichartz estimate fails ( [19]), however, the corresponding generalized estimates (II) and (III) were proven in [23]. For the Schrödinger equation (a = 2), Montgomery-Smith [26] proved the failure of the L 2 t L ∞ x Strichartz estimate and Tao [36] proved the 2-dimensional endpoint L 2 t L ∞ |x| L 2 ω estimate.
In some sense, the Morawetz-KSS estimates can be viewed as a special case of the weighted Strichartz estimates (VII) and (VIII). The weighted Strichartz estimates (VII) with q = r were proven by Fang and Wang [5] (a > 0) and Hidano, Metcalfe, Smith, Sogge and Zhou [13] (a = 1), with the previous work for a = 1 and radial data in [10].
As was clear from [5] and [13], these estimates are intimately related with each other. A starting point can be the homogenous trace lemma (H.T.L., see (1.3) of [5]), i.e., . Using this and interpolation, we can conclude the case q = r in (VII) as follows, Similarly, for the wave equation a = 1, by using the inhomogenous trace lemma (I.T.L., see (1.7) in [5]), i.e., we can conclude a couple of estimates in (VI), (II) and (VIII) (see [33] for the Rodnianski's argument to deduce (II) from (VI.a)). Figure 2. Some consequences of I.T.L..
As long as the estimates in the above diagrams were built, we can get our results as follows.
• Use q = r in (VI) and Hardy's inequality, to get estimates in the range of q ≥ r in (VII), i.e. Theorem 1.7.
We now state our results precisely. First we would like to give an angular generalization of the classical Strichartz estimates.
then we have The technical restriction r < ∞ can essentially be removed (except the endpoint (q, r) = (∞, ∞), (2, ∞)), if we use the real interpolation argument as in Section 5.2.
Remark 1.2. In the case of the wave equation (a = 1), recall that we have the classical Strichartz estimates (see e.g. [4], [18]) The result in Theorem 1.1 extends the Strichartz estimates to the case of by requiring some additional angular regularity on the data. Remark 1.3. The requirement for (q, r) are sharp for a = 1, see Remark 1.7. When a = 1, the sharpness may be different. It will be interesting to determine the sharp range for (q, r), at least for the Schrödinger equation (a = 2). Recall that for the Schrödinger equation (a = 2), the Strichartz estimates can be stated as follows (see e.g. [18], (1.26) of [5]) We note here that for n = a = 2, our estimate is worse than the standard one.
In fact, for the wave equation (a = 1), we can improve the required angular regularity to be almost optimal for the non-admissible (q, r), by interpolating with the classical Strichartz estimates, which recover the results in [33] for n ≥ 3 and [5] for the full range n ≥ 2.
Then we have the estimates On the other hand, if we localize the domain in finite time interval [0, T ], by making use of KSS estimates, we will get the following localized Strichartz estimates for the wave equation.
Then we have For the endpoint case (q, r, n) = (2, ∞, 2) and any ǫ > 0, we have In the recent work of Smith, Sogge and Wang [27] (see also Fang and Wang [6]), we see that when n = 2, we can in fact improve further the generalized Strichartz estimates for the wave equation to the L q t L r |x| L 2 ω estimates, in which case, the angular regularity is not required. Here L r |x| denotes the Lebesgue space for the variable |x| with respect to the measure |x| n−1 d|x|. Inspired by their work, we generalize the results to the general spatial dimensions.
Next, we are interested in exploiting the weighted Strichartz estimates for any q, r ≥ 2. The fist result is for the wave equation and radial initial data, which will serve as a guideline for the general estimates. Theorem 1.5 (Weighted Strichartz estimates for radial initial data). Let q, r ≥ 2, and u be a radial function on R n+1 such that u = (∂ 2 t − ∆)u = 0. Then the following estimates hold with s = α + n( Remark 1.7. The requirement on α is essentially optimal. In fact, since the decay estimates for the wave equation are sharp in general even for radial functions (see e.g. Lemma 4.1 of [12]). By those estimates, it is easy to see that to bound the left hand side of (1.18), we must have 1 q − n − 1 2 + n − 1 r < α < n r , q, r < ∞, The second result is the weighted Strichartz estimates for general a > 0 and general data.
Then we have the following weighted Strichartz estimates, In Theorem 1.6, we have an additional restriction q ≤ r for q, r, compared with (1.18). In general, we can relax this restriction. Theorem 1.7. Let 2 ≤ r ≤ q ≤ ∞, and 1 q − n−1 2 + n−1 r < α < n r . Then we have the following weighted Strichartz estimates, The results in Theorem 1.6 and Theorem 1.7 are generalizations of the weighted Strichartz estimates in Fang and Wang [5](see also [13] for a = 1), i.e.
In particular, when r = 2, we have the generalized Morawetz estimates x , for any b ∈ (1, n) and a > 0.
Lastly, we present local in time weighted Strichartz estimates for the wave equation.
This paper is arranged as follows. In section 2 we prove the estimates stated in Theorem 1.1 and Theorem 1.3; In section 3 we prove Theorem 1.4; In section 4 we prove Theorem 1.5; In section 5 we prove Theorem 1.6 and 1.7; Lastly we provide an application of the Strichartz estimates in Section 6.

Generalized Strichartz Estimates
In this section we prove Theorem 1.1, from which we see how the generalized Strichartz estimates can be obtained from the weighted Strichartz estimates. We also prove Theorem 1.3 which illustrates that local in time generalized Strichatz estimates can be obtained from the Morawetz-KSS estimates.
Let f 1,N be a unit frequency function of angular frequency N and u 1,N = e itD a f 1,N with a > 0. Denote the norm where {Q α } is a partition of R n into cubes Q α of side length 1.
First, since q ≥ 2, by using the Sobolev embedding N − n−1 q L q ω ⊂ L ∞ ω on the unit sphere S n−1 for angular frequency localized functions (see Lemma 7.2 in Appendix 7), we have the following estimate for any tiling of R n by cubes {Q α } of side length 1: This means that we have Interpolating this with the trivial estimate x , we arrive at the following estimate for r ≥ q: . By Sobolev embedding on the sphere S n−1 (see (7.4) in Appendix 7), we have Combining (2.2) and (2.4), we have We can see that (2.5) allows a wider range of (q, r) than that in the usual Strichartz estimates, i.e., we have To see this, we compute, using the Sobolev embedding in Q α , that for any q ≤ r: for some real-valued radially symmetric bump function ϕ(ξ) adapted to {ξ ∈ R n : |ξ| ≤ 2} which equals 1 on the unit ball. Furtherly we make a spherical decomposition of each f j and let f j = N ∈2 N ∪0 f jN , where f jN has angular frequency N . By using Littlewood-Paley-Stein theorem (see Theorem 2 [34]) and applying (2.6), we get for r < ∞ This completes the proof of Theorem 1.1 for 2 ≤ q ≤ r and r < ∞. The case when q ≥ r comes from interpolation with the energy estimate with (q, r) = (∞, 2).

Local in Time Strichartz Estimates for the Wave Equation.
In this subsection, we prove Theorem 1.3.

When considering the wave equation, if we denote
then the Morawetz-KSS estimates can be stated as For the sake of completeness, we present the proof of the Morawetz-KSS estimates in Appendix 7.2.
We consider now the remaining case 1/q ≥ (n − 1)(1/2 − 1/r) and q, r ≥ 2. We will apply the Morawetz-KSS estimates for the wave equation to conclude some local in time generalized Strichartz estimates for a = 1.
Interpolating with the energy estimates, we get that Again by rescaling and the Littlewood-Paley inequality we get (1.12).
In this section, we give the proof of Theorem 1.4, inspired by the recent work of Smith, Sogge and Wang [27] and Fang and Wang [6].
We shall show that We shall also prove that By scaling, Littlewood-Paley theory and interpolation, we get from (3.1) that if we remove the support assumptions on the Fourier transform, then for (q, r, n) satisfying At first, we use scaling and Littlewood-Paley decomposition to conclude from (3.1) that we have Recall that we have the interpolation between spaces of vector-valued functions (see 5 , and the interpolation of the homogeneous Besov spaces (see Theorem 6.4.5 of [1] page 152) Based on (3.4), (3.5) and (3.6) for fixed r ∈ (2, ∞], we get for q with 0 < 1/q < (n − 1)(1/2 − 1/r) and q > 2, This gives us the result (3.3) for 2 < q < ∞. For the case q = ∞ and r < ∞, the result is just the consequence of energy estimates and Sobolev embedding. To prove the remaining case with q = 2 and r < ∞, we need only to use the fact that This concludes the proof of (3.3) for (q, r) = (2, ∞), (∞, ∞) with 1/q < (n − 1)(1/2 − 1/r) and q ≥ 2. Also we can get (1.14) and (1.15) from (3.2) by the same argument.
Remark 3.1. All of the requirements for (q, r) are necessary for the estimates except the requirement (q, r) = (2, ∞). In general, we expect that this restriction can be relaxed. In particular, when n = 3, the estimate with (q, r) = (2, ∞) are proven to be true in Machihara, Nakamura, Nakanishi, and Ozawa [23]. However, we will not exploit this issue.
3.1. Proof of (3.1). To begin, let us recall some basic knowledge about the spherical harmonics (for detailed discussion, see e.g. Stein and Weiss [32]). Let n ≥ 2. For any k ≥ 0, we denote by H k the space of spherical harmonics of degree k on By orthogonality, we observe that F (t, r·) L 2 ω = a k,l (t, r) l 2 k,l . Due to the support assumptions for the Fourier inversion of f (denoted byf ) we have that If we expand the angular part off using spherical harmonics, we find that if ξ = ρω with ω ∈ S n−1 , then there are generalized Fourier coefficients c k (ρ) which vanish when So, by (3.9) and Plancherel's theorem for S n−1 and R we have ifč k,l denotes the one-dimensional Fourier inversion of c k,l (ρ).

3.2.
The estimates for ψ ik (m, r). Now we present the proof of the key estimates (3.21)-(3.23) for ψ ik (m, r), to conclude the proof of (3.1).
Lemma 3.1. Let n ≥ 3 be odd, β(ρ, r, u) = α(ρ)e −ρr sinh u and r > 1. Then there is a uniform constant C, which is independent of k and r > 1 so that the following inequality hold Proof. Notice that β ∈ S(R) with respect to all variables (together with the support in [1/4, 2] for ρ). If we use Hölder's inequality, Plancherel Theorem and the facts r > 1 and k + n−2 2 ≥ 1 2 , then   Proof. First, observe that the case n = 2 is trivial since α ∈ S. For the case n ≥ 4 and even, then n−2 2 ∈ N. The estimate (3.26) follows immediately, if we use the Taylor expansion ofα up to order n/2, in terms of r cos θ, and recall that we have the orthogonality relation  Proof. Notice that r < 1 and γ(ρ) = α(ρ)ρ k+ n−2 , and we see that the last quantity is less than or equal to 1. For any k < n+1 2 , the last quantity is bounded, and this concludes the proof of (3.23) with a constant independent of k.

Radial Weighted Strichartz Estimates, A Motivation
In this section we study the weighted Strichartz estimates for the wave equation with radial initial data and prove Theorem 1.5. The argument of [33] in proving Strichartz estimates of the wave equation with radial initial data can be adapted for our purpose.
Here the function m 3 is smooth, and the remaining m i have asymptotic expansions: as y → ∞. In other words, the functions m 1 (rρ) and m 2 (rρ) are smooth with derivatives in ρ uniformly bounded for all 1 2 ρ 2 and r 2. Substituting the asymptotic formula (4.2) into the integral formula, we may assume without loss of generality that we are trying to bound integrals of the form: where m ± is a smooth function with derivatives in ρ uniformly bounded for all r 0, and χ (1/4,4) is a smooth bump function on the interval ( 1 4 , 4). It is now apparent that the integrals in (4.3) are essentially time translated inverse Fourier transforms of a one dimensional unit frequency function. Therefore, we can localize these integrals in physical space (i.e. the t ± r variable) on a O(1) scale. Since the function f 1 is compactly supported in the interval (0, 4), we may take its Fourier series development: An important thing to notice here is that we can recover the L 2 norm of f 1 as a function on R n in terms of {c k }: Sticking the series (4.4) into the the integrals (4.3) yields: Integrating by parts as many times as necessary in the above formula, we see that we have the asymptotic bound: Using the expansion (4.6) and the asymptotic bound (4.7) we can directly compute that The manipulation to get the last line above follows from Hölder's inequality. Note that to make the function integrable in L p , we must have α be the number such that n−1 p − α > − 1 p (or −α ≥ 0 when p = ∞), i.e., If we also choose M large enough, then by integrating each expression in this line term by term, we arrive at the bound: Testing this last expression for L q in time, and using the inclusion ℓ min{p,q} ⊆ ℓ p , we see that: In conclusion, using the characterization (4.5), we know that if we have q, p ≥ 2 and (we can take the first inequality with equality when q = ∞ and the second inequality with equality when p = ∞), then we have Now we use the Littlewood-Paley decomposition f = j∈Z f j , and apply (4.10) to get Next we use real interpolation to prove the estimate (1.18).

Weighted Strichartz Estimates
In this section, we prove the weighted Strichartz estimates stated in Theorem 1.6, based on Rodnianski's argument, the weighted Strichartz estimates (1.22) and a localized version of the weighted HLS estimates.
In particular, if we choose β = −α, the corresponding estimate on R n is true for −n/q ′ < α < n/r, 1 < q ≤ r < ∞ and s = n/q − n/r.
In this subsection, we are interested in the localized version of the estimate with β = −α. More precisely, if we denote B 1 be the unit ball, and B 2 the ball centered at origin with radius 2, then we aim at the proof of the following lemma.
Lemma 5.1 (Localized Weighted HLS). Let 1 < q ≤ r < ∞, and −n/q ′ < α < n/r, we have the localized version of the weighted Hardy-Littlewood-Sobolev inequality if s is large enough (in fact, we need only to choose s = 2m with m > n/2).
Proof. We will prove the result for s = 2m with n/2 < m ∈ N. At first, we observe that the proof of the estimate (5.1) can be reduced to the proof of the following where φ ∈ C ∞ 0 and Λ 2m = (1 − ∆) m is a differential operator. In fact, if this estimate is true for any f , we can choose φ = 1 in and supp∂ψ ⊂ B 2 \B 1 . We have So it is sufficient to prove (5.2). First we write (5.2) in the equivalent form Recall that Λ −2m f = F −1 (1 + |ξ| 2 ) −mf ) = K * f , and if m > n/2, then K(x) = O( x −N ) for any N . By introducing ψ ∈ C ∞ 0 such that ψφ = φ, if α < n/r, we can control By Hölder's inequality, we can estimate T f L ∞ by f L q , if we have which can be easily seen to hold if we have α > −n/q ′ .

5.2.
Rodnianski's Argument for the Weighted Estimates. Let f 1,N be a unit frequency function of angular frequency N and u 1,N = e itD a f 1,N with a > 0.
To see this, we can compute, using the Sobolev embedding on Q β , that for any r ≥ 2, Then an application of Littlewood-Paley-Stein inequality gives us (5.8) for r < ∞.
This gives the proof of (1.20) in the region |x| ≥ 1, for frequency localized functions.
To prove the homogeneous estimates for |x| ≤ 1, we use the localized weighted HLS estimates proven in Lemma 5.1. The estimate is if 1 < q ≤ r < ∞, −n/q ′ < α < n/r and s is large enough. Then by (5.7) and (5.9) with q = 2, we have Recall that [Ω ij , ∂ k ] = δ jk ∂ i − δ ik ∂ j and f 1 = φ * f for some spectral localized radial bump function φ, we have the following estimates for the even positive numbers s x . By duality and interpolation, the same estimates are true for a general real number s. Applying this estimate in the previous inequalities, we find that Combining (5.8) and (5.10), we get An application of Littlewood-Paley-Stein inequality gives us (5.11) for f 1 since 2 ≤ r < ∞.
Now we use the Littlewood-Paley decomposition f = j∈Z f j , and apply (5.11) to get Next we can use real interpolation as in section 4 to get the case q = 2 < r < ∞ proved. Recall that we have the weighted Strichartz estimates (1.22) for 2 ≤ q = r ≤ ∞, and hence (1.20) with 2 ≤ q = r < ∞ (see e.g. (2.4)), we have the estimate (1.20) for 2 ≤ q ≤ r < ∞, by using real interpolation again.

5.3.
Weighted Estimates for q ≥ r. In this subsection we will prove Theorem 1.7.
Recall that Hardy's inequality gives, , where we have used Littlewood-Paley-Stein decomposition for f in the last inequality. Also we can rewrite the weighted Strichartz estimates (1.22) as , a > 0 and p ∈ [2, ∞]. Now for fixed q ≥ r ≥ 2, if we set θ = 1 − 2( 1 r − 1 q ), p = q − 2q r + 2, by complex interpolation between (5.12) and (5.13) and using Theorem 5.6.3 in [1], we get 6. An Application: Strauss Conjecture when n = 2, 3 As an application of the generalized Strichartz estimates in Theorem 1.1, Theorem 1.3 and Theorem 1.4, we prove the Strauss conjecture with low regularity when n = 2, 3.
Let n = 2, 3, s c = n 2 − 2 p−1 be the critical index of regularity, s d = 1 2 − 1 p , p conf = 1 + 4 n−1 be the conformal index, p c be the solution of s c = s d and p > 1. Let F p (u) be a function such that (6.1) |F p (u)| ≤ C|u| p , |F ′ p (u)| ≤ C|u| p−1 for some C > 0 and p > 1, consider the following semilinear wave equations Strauss conjecture asserts that for p > p c , the problem (6.2) has a global solution, when the initial data (f, g) is sufficiently small and smooth. This conjecture was verified by Georgiev, Lindblad and Sogge in [9] and Tataru [38] for smooth data. Then the remaining problem is the case of low regularity. There has been many partial results on this field.
Remark 6.1. The lifespan given in this result is essentially sharp. In [42] and [43], Zhou proved that the life span T ǫ of classical solutions to the equation (6.2) has order ǫ 1 sc−s d when n = 2, 3 and 2 < p < p c (see also Lindblad [20] for the case n = 3). Remark 6.3. Theorem 1.8 can also be employed to give an alternative proof of Theorem 6.1 for 2 < p < p c . See Yu [41] for the related argument for n = 3.
In this section, we give a proof of the Strauss conjecture with low regularity when the dimension is 2, 3, i.e., Theorem 6.1, by using essentially only the generalized Strichartz estimates. Moreover, we can also prove the almost global result Theorem 6.2 for the endpoint case p = p c and n = 2, by using local in time generalized Strichartz estimates.
6.1. Global Results for p > p c . When p ∈ (p c , p conf ), we have 1 q = 2 p−1 − (n − 2). If u ∈ S, we define Πu to be the solution of the wave equation with initial data (f, g) ∈ D sc,b . Then it suffices to show that when the initial data is small enough in D sc,b , then Π : S → S and the map is a contraction map on small balls of S. Now we prove this claim. First, note that Πu = v h +v i with v h being the solution to the homogeneous equation with initial data (f, g) ∈ D sc,b , and v i the solution to the inhomogeneous equation with null initial data (0, 0).
Thus by Christ-Kiselev lemma [3], we have for n = 2 rad . Moreover, if n = 3, Sogge (Theorem 4.2 of [29]) proves the same radial inhomogeneous inequality (6.3). Then by the comparison principle for the wave equation with n = 2, 3, we have Since (f, g) ∈ D sc,b with b > 1 pq + n−1 p , an application of Theorem 1.1 and Sobolev embedding on the sphere yields that v h S (f, g) D sc,b .
Thus we know that (6.4) Πu S (f, g) D sc,b + u p S . From this inequality, our claim follows immediately (recall (6.1)).

6.2.
Local Results for p ∈ (2, p c ). In this subsection, we prove the local results in Theorem 6.1 when p ∈ (2, p c ). Define 1 and let S Tǫ = L qp Tǫ L p |x| L ∞ ω be the solution space with T ǫ = cǫ 1 sc−s d , and let ǫ be the norm of the data in D s d ,b . We want to prove that the map Π is a contraction map on small balls of S Tǫ .
If n = 2 and u is radial, let s = s d = 1/2 − 1/p, then (qp, p, s) and (q ′ , ∞, 1 − s) satisfy the condition for (1.14) and (1.13). Thus by Christ-Kiselev lemma [3], we have for n = 2 rad . Moreover, if n = 3, Sogge (Theorem 4.2 in [29]) proves the same radial inhomogeneous inequality (6.5). Then by the comparison principle for wave equation with n = 2, 3, we have Since (f, g) ∈ D s d ,b with b > 1 qp + 1 p and norm ǫ, an application of Theorem 1.3 yields that Thus we know that Moreover, we have (recall (6.1)) Thus if we choose T ǫ = cǫ 1 sc−s d with c sufficiently small, we can see from (6.6) and (6.7) that Π is a contraction map on the complete set which completes the proof of Theorem 6.1.
6.3. Almost Global Results for p = p c and n = 2. In this subsection, we prove the almost global results in Theorem 6.2. Recall that when p = p c and n = 2, we have the local in time estimates (1.10).
(ln(2 + T ǫ )) 1/qp u p ST ǫ . By Theorem 1.4 in the radial case and the comparison principle.
Moreover, if u, v ∈ S Tǫ with norm bounded by 2ǫ 1/p , then where we have used the assumption (6.1). Thus the map Π is a contraction map on S Tǫ with norm bounded by 2ǫ 1/p . This completes the proof of Theorem 6.2.
7. Appendix 7.1. Sobolev embedding on the sphere S n−1 . We give a simple proof of the Sobolev inequalities used in of Section 2.1 and 5.2. These inequalities should be true in general. For the sake of completeness, we give a proof here.
Lemma 7.1 (Sobolev embedding I). Let 2 ≤ q < ∞, we have the following Proof. Our proof is based on the spectral cluster estimates on the sphere (see e.g. [28] Lemma 4.2.4 page 129). Recall that if we define the spectral cluster operator where E k is the projection onto the one dimensional eigenspace with eigenvalue λ k , then we have Thus for the Littlewood-Paley projector S λ on the the spectral interval [λ, 2λ], by using the Cauchy-Schwartz inequality in j with j ∈ Z, we have Then by Hölder's inequality, we have Finally, by the Littlewood-Paley-Stein theorem (see Theorem 2 [34]) for the sphere, we have (7.5) f L q ∼ S λ f L q ℓ 2 λ ≤ S λ f ℓ 2 λ L q ≤ C f H σ(q) for any 2 ≤ q < ∞, which completes the proof. Lemma 7.2 (Sobolev embedding II). Let p ≥ 2, then for the Littlewood-Paley projector S λ on the the spectral interval [λ, 2λ] we have the following (7.6) S λ f L ∞ (S n−1 ) ≤ C(1 + λ) (n−1)/p f L p (S n−1 ) .
Proof. Since the estimate for λ 1 is trivial, we assume that λ ≫ 1. Note that it is equivalent to prove the dual estimate of (7.6), which is a consequence of the interpolation between the dual of (7.3), which says S λ f L 2 (S n−1 ) ≤ Cλ (n−1)/2 f L 1 (S n−1 ) and (7.7) S λ f L 1 (S n−1 ) ≤ C f L 1 (S n−1 ) .
In order to prove (7.8) , we shall use the finite propagation speed for solutions to the wave equation. Specifically, if f is supported in a geodesic ball B(x 0 , R) centered at x 0 with radius R, then x → cos tP f vanishes outside of B(x 0 , R + T ) if 0 ≤ t ≤ T . We will prove T λ (P ) satisfies (7.8) by showing T 0 λ (P ) and j≥1 T j λ (P ) both satisfy (7.8). cos(tλ k )β(λ k /λ)e k (x)e k (y)dt}f (y)dy = S n−1 K 0 λ (x, y)f (y)dy The finite propagation speed of the wave equation mentioned before implies that the kernel K 0 λ (x, y) must satisfy K 0 λ (x, y) = 0 if dist(x, y) > 8λ −1 , since cos tP will have a kernel that vanishes on this set when t belongs to the support of the integral defining K 0 λ (x, y). Because of this, in order to prove T 0 λ satisfies (7.8), it suffices to show that for all geodesic balls B λ,0 with radius 8λ −1 one has the bound (7.11) T 0 λ f L 1 (B λ,0 ) ≤ C f L 1 (S n−1 ) , for the L 1 norm over B λ,0 .
Recall also that we have the energy estimate The Morawetz estimates with µ > 1/2 can be proven as follows, Now we deal with the case µ ≤ 1/2. We consider first the case when T 1. In this case, the Morawetz-KSS estimates are in fact weaker than the energy estimate, For the remaining case with T ≥ 2, we use the energy estimate to deal with the region |x| ≥ T , For the remaining region, we use instead (7.14) x −µ e itD f 2 x . This completes the proof of (2.7).