{'fraction_non_alphanumeric': 0.06228332337118948, 'fraction_numerical': 0.016736401673640166, 'mean_word_length': 3.7132394366197183, 'pattern_counts': {'":': 0, '<': 11, '<?xml version=': 0, '>': 3, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Global Strichartz estimates for nontrapping perturbations of the Laplacian

The authors prove global Strichartz estimates for compact perturbations of the wave operator in odd dimensions when a non-trapping assumption is satisfied.


Introduction
The purpose of this paper is to establish estimates of Strichartz type, globally in space and time, for solutions to certain nontrapping, spatially compact perturbations of the Minkowski wave equation. Precisely, we consider the following wave equation on the exterior domain Ω to a compact obstacle, where the spatial dimension is an odd integer n ≥ 3.  (1.1) The operator ∆ g is assumed to be the Laplace-Beltrami operator associated to a smooth, time-independent Riemannian metric g(x), such that g ij (x) = δ ij for |x| ≥ R . The set Ω is assumed to be the complement in R n of a smoothly bounded, compact subset of |x| < R, such that Ω is strictly geodesically concave with respect to g. The case that Ω = R n is also permitted. Finally, the geodesic flow on Ω with respect to g and normal reflection on ∂Ω is assumed to be nontrapping, in the sense that all geodesics exit the set |x| ≤ R within some fixed, finite time.
The proof consists of showing that exponential energy decay bounds for ( Global Minkowski Strichartz estimates. For solutions to (1.1), in the case of Ω = R n and g ij (x) = δ ij , the following holds, The ingredient that allows us to establish Theorem 1. Exponential energy decay. For data f, g supported in |x| ≤ R, and β(x) smooth, supported in |x| ≤ R , there exist C < ∞ and α > 0 such that for solutions to (1.1) where F = 0 the following holds, The decay estimate (1.4), which depends on the nontrapping assumption and requires that n ≥ 3 be odd, was first established in the obstacle framework by Taylor [16]. For a more general discussion of energy decay estimates, we refer to Lax-Philips [8], and Vainberg [17]. It was pointed out to us by M.
Zworski that quasimode constructions show that global Strichartz estimates cannot hold if the metric has an elliptic closed geodesic. See, for example, Colin de Verdière [3] and Ralston [12].
There is a long history to establishing the global estimates (1.3) for the Minkowski case, beginning with the original work by Strichartz in [14,15] for the conformal case p = q = 2n+2 n−1 , r = s = 2n+2 n+3 , and γ = 1 2 . We mention here the subsequent work of Genibre-Velo [4], Pecher [11], Kapitanski [6], Lindblad-Sogge [9], Mockenhaupt-Seeger-Sogge [10], and Keel-Tao [7]. Certain local Strichartz estimates (1.2) for the obstacle problem were established by the authors in [13], for the case F = 0 . It will be shown in Theorem 3.2 that these imply the appropriate estimates in the case F = 0, based on an argument of Christ and Kiselev [2]. M. Beals [1] has obtained fixed time L p −L q estimates for the obstacle problem, globally in t, in the case g ij = δ ij , and for data vanishing near ∂Ω. It is not known whether the Strichartz estimates hold, even locally, if ∂Ω has a point of convexity, but related eigenfunction estimates are known to fail in that case, as observed by Grieser [5], and generalized by the authors in [13]. It also is not known whether (1.3) holds for every set of indices for which (1.2) holds, and we do not address that question in this paper. We do mention that if the following conditions are met, then the indices are admissible in the sense of this paper, provided that q, s ′ < 2(n−1) n−3 , We make some remarks concerning the homogeneous Sobolev spacesḢ γ D (Ω) . We always deal with γ < n 2 , so that smooth cutoffs of functions inḢ γ (R n ) are contained in the inhomogeneous Sobolev space H γ (R n ), and the two norms are comparable on functions supported in a fixed ball. For simplicity, if R is large enough so that g ij (x) = δ ij when |x| > R, and ∂Ω ⊂ {|x| < R} , then we whereΩ is a compact manifold with boundary containing Ω is the usual Dirichlet space satisfying f | ∂Ω = 0 (when this makes sense.) For larger γ, the additional compatibility conditions must be satisfied to insure that solutions to the Cauchy problem remain in H γ D (Ω). The spaces with γ < 0 are defined by duality; for the Strichartz estimates (1.3), always γ −1 ≥ −1 , so that the additional compatibility conditions are irrelevant for negative indices.

Global Strichartz Estimates
In this section we provide the proof of Theorem 1.1, based on the assumptions (1.2), (1.3), and (1.4). By dilating, we will take R = 1 2 .
where the data f, g are supported in { |x| ≤ 1 }, and F, G are supported in Then, for any ρ < α, and any admissible p, q, r, s, γ, there exists C < ∞ such that the following holds.
Proof. We establish the estimate on t ≥ 0. Observe that, by (1.2) and Duhamel's principle, the inequality holds for the L p t L q x norm of u over [0, 1]×Ω . Also, by (1.2), .
Next consider (1 − β) u . Since ∆ g = ∆ on the support of (1 − β)u, we have and by Duhamel's principle we have , which leads to the desired estimate for u . Then the following holds Proof. By Plancherel's theorem over t, x , the left hand side can be written as We next apply the Schwarz inequality over η to bound this by From the fact that Let u solve the Cauchy problem for the Minkowski wave equation, with data f, g, F. Then the following holds, Proof. If F = 0, this is a direct consequence of Lemma 2.2 above. If f = g = 0 , then the Strichartz estimate (1.3), duality, and Huygen's principle imply the Since r, s ≤ 2 , the following holds Proof of Theorem 1.1. By Lemma 2.1, we may without loss of generality assume that f and g vanish for |x| ≤ 1 . We write where u 0 solves the Cauchy problem for the Minkowski wave equation, with data f, g, F , where we set F = 0 on R n \Ω . By (1.3), we can restrict attention to βu 0 − v . We write by Corrollary 2.3. Note that the initial data of βu 0 − v vanishes. Let F j , G j denote the restrictions of F, G to the set t ∈ [j, j + 1] , and write (for t > 0) where u j (t, x) is the forward solution to ∂ 2 t u j − ∆ g u j = βF j + G j .
By Lemma 2.1, the following holds where we use the fact that p, q ≥ 2 ≥ r, s .

Homogeneous estimates imply inhomogeneous estimates
In this section, we provide an elementary proof that mixed-norm estimates for the homogeneous Cauchy problem imply the appropriate estimates for the inhomogeneous Cauchy problem. The proof is quite general, but we present it here in the context of the obstacle problem. The main ingredient is a special case of a lemma of Christ and Kiselev [2], the proof of which we present here for completeness. We thank T. Tao for pointing out the relevance of this lemma to inhomogeneous estimates for the wave equation. Assume that As remarked in [2], if K(s, t) = 1/(t − s), then Lemma 3.1 does not hold in the case p = q ∈ (1, ∞).
Using this lemma, we establish the following.
. Then the following estimate holds for solutions to the inhomogeneous Cauchy problem (1.1), provided that f, g and F are supported in the set |x| ≤ R .
Proof of Theorem 3.2. By finite propagation velocity, we may replace Ω by a compact manifoldΩ with boundary. Let Λ = −∆ g , so that the spectrum of Λ is bounded below. By the assumptions of the theorem and duality, the following hold.
Let K(t, s) = Λ −1 sin (t − s)Λ . An application of the addition formula to sin (t − s)Λ yields the following. Proof of Lemma 3.1. Since the argument for q = ∞ is similar to the case q < ∞, we assume for simplicity that q is finite.
We can normalize f so that We may also assume without loss of generality that f (s) is a continuous function (with values in X) and that if Next consider the set of all dyadic subintervals of [0, 1] . If I and J are two such subintervals, we say that I ∼ J if the following hold. First, I and J must have the same length, and I must lie to the left of J. We also require that I and J be non-adjacent but have adjacent parents (i.e., I ⊂ I 0 and J ⊂ J 0 where I 0 and J 0 are adjacent dyadic intervals of twice the length).
Note then that if J is fixed there are only two intervals with I ∼ J. Moreover, for almost every (x, y) ∈ [0, 1] 2 with x < y there is a unique pair I, J with I ∼ J and x ∈ I and y ∈ J.
If we apply this fact using the variables x = F (s) and y = F (t) then we conclude that the following identity holds almost everywhere: The lemma follows by (3.3) after summing over j, where we use the fact that p < q.