On Strichartz estimates for Schr\"{o}dinger operators in compact manifolds with boundary

We prove local Strichartz estimates on compact manifolds with boundary. Our results also apply more generally to compact manifolds with Lipschitz metrics.

where H s denotes the L 2 Sobolev space over M , and 2 ≤ p, q ≤ ∞ satisfies 2 p + n q = n 2 , (n, p, q) = (2, 2, ∞). Such estimates are well established in flat Euclidean space, where M = R n and g ij = δ ij . In that case s = 0, and one can take T = ∞; see for example Strichartz [11], Ginibre and Velo [5], Keel and Tao [6], and references therein. Estimates for the standard flat 2-torus were shown by Bourgain [2] to hold for any s > 0.
There is also considerable interest in developing these estimates for non-flat geometries, and also for compact domains. In the case where M is compact and ∂M = ∅, Burq, Gérard, and Tzvetkov [3] established (1.2) with s = 1 p . Hence there is a loss of derivatives in their estimate when compared to the case of flat geometries.
A simple investigation of the Schrödinger evolution on spherical harmonics where M = S n shows that some loss of derivatives must occur. For instance, with n = 2, by taking the initial data to be a highest weight spherical harmonic on S 2 one concludes that the best possible local L 2 x → L 4 t,x bounds would involve a loss of 1/8 derivatives. This sharp estimate and related ones for Zoll surfaces were obtained in [3]. It is not known, however, whether the weaker estimates involving a loss of 1 p derivatives in [3] for general compact manifolds without boundary can be improved.
In the case where ∂M = ∅, one also considers Dirichlet or Neumann boundary conditions in addition to (1.1) The authors were supported by the National Science Foundation, Grants DMS-0140499, DMS-0354668, DMS-0555162, and DMS-0354???.
where N x denotes the unit normal vector field to ∂M . Here one expects a further loss of derivatives in the estimates. The Rayleigh whispering gallery modes over the unit disk in R 2 provide examples of Dirichlet eigenfunctions which accumulate their energy near the boundary, contributing to high L p norms. Applying the Schrödinger evolution to these eigenfunctions show that s ≥ 1 6 is necessary for the Strichartz estimate with p = q = 4. Recently, Anton [1] showed that the estimates (1.2) hold on general manifolds with boundary provided s > 3 2p . In addition, the arguments of [1] work equally well for a manifold M without boundary equipped with a metric g of Lipschitz regularity.
In this work, we improve on the current results for compact (M, g) where either ∂M = ∅, or ∂M = ∅ and g is Lipschitz, by showing that Strichartz estimates hold with a loss of fewer derivatives.
In the case where (M, g) is a boundaryless manifold with g ∈ C ∞ , the estimate of Burq-Gérard-Tzvetkov (1.2) with s = 1 p , while not known to be sharp, is a natural result by the following heuristic argument. For a general compact manifold, there are no conjugate points for the geodesic flow at distance less than the injectivity radius of the manifold. Given a solution to the Schrödinger equation whose frequencies are concentrated at λ, energy propagates at speed ≈ λ. Hence, a frequency λ solution should possess good dispersive properties at least until time T λ ≈ 1 λ . We thus expect to be able to prove a Strichartz estimate with no loss of derivatives for such a solution over a time interval of size roughly 1 λ . By considering a sum over such intervals we should obtain a Strichartz estimate over a time interval of unit size, only with a constant appearing on the right hand size which is a constant multiple of λ 1 p . This corresponds to s = 1 p in the estimate, and Littlewood-Paley theory yields the estimate for arbitrary solutions.
In the case where ∂M = ∅, the boundary conditions affect the flow of energy near the boundary. A key strategy involves reflecting the metric and the solution across the boundary, to obtain a Schrödinger equation on a manifold without boundary, but with a metric that has Lipschitz singularities along ∂M . Hence matters reduce to considering the Schrödinger evolution for Lipschitz metrics. In this case, when establishing estimates for solutions at frequency λ, one can replace the rough metric by a regularized metric which has conjugate points at distance roughly λ − 1 3 apart. Therefore, the solutions should possess good dispersive properties over a time interval of size roughly λ − 4 3 . This now yields a Strichartz estimate over a time interval of unit size with a loss of 4 3p derivatives. Hence, for manifolds with boundary (1.4) appears to be the natural analog of the aforementioned estimates of [3] for the general boundaryless case.
Our proof of Theorem 1.1 follows the above heuristics. In section 2 the solution is localized spatially and a coordinate chart is used to work on R n ; a Littlewood-Paley decomposition then reduce matters to establishing Strichartz estimates for components of the solutions dyadically localized in frequency. As alluded to above, we then seek to prove Strichartz estimates with no loss of derivatives over time intervals of size λ − 4 3 for components of the solution localized at frequency λ. This involves regularizing the metric by truncating its frequency to a scale dependent on λ. Rescaling the solution then reduces the problem to establishing Strichartz estimates for metrics with 2 bounded derivatives over small time intervals whose size also depends on the frequency. Section 3 uses a phase space transform to construct a parametrix for such Schrödinger operators, and section 4 concludes the paper by showing that the parametrix yields the desired estimates.
Notation. In what follows d will denote the gradient operator which maps scalar functions to vector fields and vector fields to matrix functions in the natural way. The expression X Y means that X ≤ CY for some C depending only on n and on the Lipschitz norm of the metric.

Reductions
We will establish Theorem 1.1 more generally for operators on M which take the following form in local coordinates Such an operator is self-adjoint in the measure dµ = ρ(x) dx. Neumann conditions and the boundary normal are defined with respect to the metric g ij .
We start by reducing the case of a manifold M with boundary and P smooth, to the case of a compact manifold M without boundary, with P having coefficients of Lipschitz regularity. For this, letM denote the double of M , identified along ∂M . We define a differentiable structure onM near ∂M using geodesic normal coordinates in g ij , so x n > 0 and x n < 0 define the two copies of M . In these coordinates, g ni = 0 for i = n, hence P contains no cross terms between ∂ n and ∂ i . The operatorP with coefficients g ij (x ′ , |x n |) and ρ(x ′ , |x n |) is thus symmetric under x n → −x n , and extends the lift of P toM across ∂M to one with Lipschitz coefficients. Eigenspaces forP decompose into symmetric and antisymmetric functions; these correspond to extensions of eigenfunctions for P satisfying Dirichlet (resp. Neumann) conditions, and each eigenfunction is of regularity C 1,1 across the boundary. The Schrödinger flow forP is thus easily seen to extend that for P , and Strichartz estimates for P follow by establishing such estimates forP onM .
We assume henceforth that M is a compact manifold with smooth differentiable structure, on which an operator P of the form (2.1) is given, with coefficients of Lipschitz regularity. Define L q -Sobolev spaces on M using the spectral resolution of P , When q = 2 we denote W s,q by H s . By elliptic regularity (e.g. [4, Theorem 8.10] for q = 2, and [4,Theorem 9.11] or [14, §2.2] for other q) the spaces W s,q for 1 < q < ∞ coincide with the Sobolev spaces defined using local coordinates, provided 0 ≤ s ≤ 2.
Suppose that u(t, x) = (e itP f )(x). Then we need to establish , hence it suffices to show, uniformly over k, that . By taking a finite partition of unity, it suffices to prove that for each smooth cutoff ψ supported in a suitably chosen coordinate chart. We will choose coordinate charts such that the image contains the unit ball, and for c 0 to be taken suitably small. (This may require multiplying ρ by a harmless constant). We take ψ supported in the unit ball, and assume g ij and ρ are extended so that the above holds globally on R n .
Let {β j (D)} j≥0 be a Littlewood-Paley partition of unity on R n , and v j = β j (D)ψu k . We will prove that, for each j, x , with all norms taken over [−T, T ] × R n , and D = (1 − ∆) , hence the second term on the right of (2.3) is bounded by a geometric series. For the first term, note that Setting λ = 2 j , and denoting v j by v λ , the estimate (2.3) is equivalent to

This, in turn, follows by showing that for any interval
We now regularize the coefficients of P by setting where S λ 2/3 denotes a truncation of a function to frequencies less than λ 2 3 , and let P λ denote the operator with coefficients g ij λ and ρ λ . Since |g ij λ − g ij | λ − 2 3 , and similarly for ρ, it follows that and we may thus replace P by P λ on the right hand side of (2.4) without changing the estimate.
Finally, we rescale the problem. Let µ = λ 2 3 , and define The function u µ (t, ·) is localized to frequencies of size µ, and the coefficients of Q µ are localized to frequencies of size less than µ 1 2 . This implies the following estimates on the coefficients of Q µ The interval I λ scales to an interval of length µ −1 . We have thus reduced the proof of Theorem 1.1 to the following.
Theorem 2.1. Suppose that u µ (t, x) is localized to frequencies |ξ| ∈ [ 1 4 µ, 4µ] and solves Assume also that the metric satisfies . Then the following estimate holds

The Parametrix
We will establish Theorem 2.1 using a short-time wave packet parametrix for the equation (2.5). Wave packet parametrices have been used to establish Strichartz estimates for Schrödinger equations in the work of Staffilani-Tataru [10] and Koch-Tataru [7]; see Tataru [13] for an overview of the methods. The result we need, in fact, is included as a special case in Theorem 2.5 of [7]. The proof of the short time estimate Theorem 2.1 is comparatively simple, though, and therefore we include a self-contained proof here for the reader's benefit.
In this section, then, we use a wave packet transform to construct a parametrix for the operator in (2.5) that will yield the Strichartz estimates. For convenience, we suppress the µ from both the operator and the solution. Let g be a radial Schwartz function over R n such that supp( g) ⊂ B 1 (0) and g L 2 = (2π) − n 2 . For µ ≥ 1, we define the operator T µ : T µ enjoys the property that its adjoint as a map from L 2 x,ξ (R 2n ) → L 2 z (R n ) also serves as a left inverse for T µ , that is, T * µ T µ = I. This implies that T µ is an isometry We conjugate A(x, D) by T µ and take a suitable approximation to the resulting operator. Specifically, define the following differential operator over (x, ξ) By a standard argument from wave packet methods (see for example [12] or [13] where g is Gaussian, or Lemmas 3.1-3.3 in [8] for g as above) we have that if β µ is a Littlewood-Paley cutoff truncating to frequencies |ξ| ≈ µ then This yields that, ifũ(t, x, ξ) = (T µ u µ (t, ·))(x, ξ), thenũ solves the equation Given an integral curve γ(r) ∈ R 2n x,ξ of the vector field This allows us to writẽ which expressesũ as an integrable superposition over r of functions invariant under the flow ofÃ, truncated to t > r.
Since u(t, x) = T * µũ (t, x, ξ) it thus suffices to obtain estimates where W t acts on functions f (x, ξ) by the formula . By a standard duality argument and an application of the endpoint estimates of  this results from establishing The inequality (3.3) follows from the fact that T µ is an isometry and χ 0,t (x, ξ) is a symplectomorphism, hence preserves the measure dx dξ. The inequality (3.2) is the focus of the next section.

The dispersive estimate
In this section, we establish the inequality (3.2). We write the kernel K(t, y, s, x) Recall that supp(ĝ) ⊂ B 1 (0). Since we are concerned with W t W * s β µ , we can insert a cutoff S µ (ζ) into the integrand which is supported in a set |ζ| ≈ µ. Note that the Hamiltonian vector field is independent of time, and hence χ t,s = χ t−s,0 . We drop the zero and abbreviate the latter transformation as χ t−s (z, ζ) = (z t−s , ζ t−s ). It then suffices to consider s = 0, and we write the kernel K(t, x, 0, y) as µ n 2 e −i ζ,x−z −iψ(t,z,ζ)+i ζt,y−zt g(µ 1 2 (y − z t ))g(µ 1 2 (x − z))S µ (ξ) dz dζ .
We need to establish uniform bounds over x and y, |K(t, x, 0, y)| t − n 2 . A straightforward estimate shows that |K(t, x, 0, y)| µ n meaning that the dispersive estimate holds for t ≤ µ −2 . We thus assume t ≥ µ −2 for the remainder of the section. Lastly we suppose that, in addition, t ≤ εµ −1 with ε chosen sufficiently small and independent of µ.
Estimate (4.3) follows by differentiating Hamilton's equations as above and applying the bounds (4.2).
Estimate (4.3), and the fact that d 2 ξ a(x, ξ) − 2I = 2 a ij − δ ij ≪ 1 yields for l, m ∈ Z n and t ≤ εµ −1 |x t (x, ξ m ) − x t (x, ξ l )| ≈ t |ξ m − ξ l | = t  Since the sum on the right converges for N large this establishes the dispersive estimate.
The expression vanishes at t = 0 since d ζ z 0 = 0, and Hamilton's equations show that the derivative of the expression with respect to t vanishes.
As in Theorem 5.4 of Smith-Sogge [9], we now proceed by defining the differential operator L = 1 + it −1 (x − z − d ζ ζ t · (y − z t )) · d ζ 1 + t −1 |x − z − d ζ ζ t · (y − z t )| 2 . By the observation above, L preserves the phase function in the definition of K m . The estimates (4.2) and (4.4) show that, if p is any one of the functions φ m (ζ), t − 1 2 z t , µ