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Strichartz estimates for the wave equation on a 2d model convex domain

We prove better Strichartz type estimates than expected from the (optimal) dispersion we obtained in our earlier work on a 2d convex model. This follows from taking full advantage of the space-time localization of caustics in the parametrix we obtain, despite their number increasing like the inverse square root of the distance from the source to the boundary. As a consequence, we improve known Strichartz estimates for the wave equation. Several improvements on our previous parametrix construction are obtained along the way and are of independent interest for further applications.


Introduction and main results
Let us consider the wave equation on a domain Ω with boundary ∂Ω , Here, ∆ usually stands for the Laplace-Beltrami operator on Ω. If ∂Ω = ∅, the boundary condition could be either Dirichlet (B is the identity map: u| ∂Ω = 0) or Neumann (B = ∂ ν where ν is the unit normal to the boundary.) Solutions to the wave equation on a smooth manifold without boundaries are known to disperse. In particular, on average the wave amplitude is decaying faster than predicted by Sobolev embedding Theorem from energy estimates, and this yields so-called Strichartz estimates. In turn these estimates have been crucial in dealing with a large range of problems, both linear and nonlinear. They are also closely related to localization and decay of eigenfunctions in the compact case (through square function estimates for the wave equation).
Let us be more specific and introduce notations: let d be the spatial dimension of Ω and β = d 1 2 − 1 r − 1 q , with (q, r) be an admissible pair, Then, when Ω is a Riemannian manifold with empty boundary, the solution to (1) is such that, at least for a suitable T < +∞ depending only on Ω, uniformly for 0 < h < 1, where χ ∈ C ∞ 0 is a smooth truncation in a neighborhood of 1, D = −i∂ and H 1 is the Sobolev space associated with ∆ on Ω. One may often state (3) differently by removing the spectral cut-off χ, removing all h factors and replacing (H 1 , L 2 ) by (H β , H β−1 ) for the data on the righthand side, at the expense of introducing fractional Sobolev spaces on Ω. When (3) holds for T = ∞, it is said to be a global in time Strichartz estimate. For Ω = R d with flat metric, the solution u R d (t, x) to (1) with data (u 0 = δ x 0 , u 1 = 0) is known to be u R d (t, x) = 1 (2π) d cos(t|ξ|)e i(x−x 0 )ξ dξ and by stationary phase the classical dispersion estimate follows: Interpolation between (4) and energy estimates, together with a duality argument, routinely provides (3). On any boundaryless Riemannian manifold (Ω, g) one may follow the same path, replacing the exact formula by a parametrix (which may be constructed locally within a small ball, thanks to finite speed of propagation.) On a manifold with boundary, one may no longer think that light rays are slightly distorted straight lines. There may be rays gliding along a convex part of the boundary, rays grazing a convex obstacle or combinations of both. Strichartz estimates outside a strictly convex obstacle were obtained in [11] and turned out to be similar to the free case (see [7] for the more complicated case of the dispersion). Strichartz estimates with losses were obtained later on general domains, [1], using short time parametrices constructions from [12], which in turn were inspired by works on low regularity metrics [13]. Losses in [1] are induced by considering only time intervals that allow for no more than one reflection of a given wave packet and as such, one does not see the full effect of dispersion in the tangential directions.
In our work [8], a parametrix for the wave equation inside a model of strictly convex domain was constructed that provided optimal decay estimates, uniformly with respect to the distance of the source to the boundary, over a time length of constant size. This involves dealing with an arbitrarily large number of caustics and retain control of their order. Our sharp dispersion estimate immediately yields by the usual argument Strichartz estimates with a range of pairs (q, r) such that where, informally, the new 1/4 factor is related to the 1/4 loss in the dispersion estimate. On the other hand, earlier works [3], [4] proved that Strichartz estimates inside strictly convex domains of dimension d ≥ 2 can hold only if (q, r) are such that This provides a counterexample for r > 4 and restricts the dimension to d ≤ 4. In a recent work [10], we improved the 2D counterexample, with (6) replaced by a stronger requirement: which, for r = +∞, yields q ≥ 5 (to be compared to q ≥ 24/5 in (6).) Our purpose now is to improve upon the positive results, in dimension d = 2. In particular, for suitable micro-localized solutions we close the gap with the recent counterexample from [10], providing a near complete picture when x+|D x D −1 y | 2 ∼ h 1/3 , |D y | ∼ h −1 . Before stating our main result, we start by describing the convex model domain under consideration: the Friedlander model domain is the half-space, for d ≥ 2, Ω d = {(x, y)|x > 0, y ∈ R d−1 } with the metric g F inherited from the following Laplace operator, ∆ F = ∂ 2 x + (1 + x)∆ R d−1 y . Remark 1.1. For the metric g = dx 2 + (1 + x) −1 dy 2 , the Laplace-Beltrami operator is △ g = (1 + x) 1/2 ∂ x (1 + x) −1/2 ∂ x + (1 + x)∆ y which is self adjoint with volume form (1 + x) −1/2 dxdy. Our Friedlander model uses instead the Laplace operator associated to the Dirichlet form |∇ g u| 2 dxdy = (|∂ x u| 2 + (1 + x)|∂ y u| 2 )dxdy, which is self adjoint with volume form dxdy. The Friedlander operator ∆ F allows for explicit computations and the difference ∆ g − ∆ F = −(2(1 + x)) −1 ∂ x is a first order differential operator: as such, as long as we are dealing with local in time Strichartz estimates for data close to the boundary, it may be treated as a lower order perturbative term and proving local in time Strichartz estimates for ∆ F implies the same set of estimates for ∆ g . Moreover, (Ω d , g F ) is easily seen to model a strict convex domain, as a first order approximation of the unit disk D(0, 1) in polar coordinates (r, θ), close to the boundary r = 1: set r = 1 − x/2, θ = y. Theorem 1. Strichartz estimates (3) hold true for the wave equation (1) on the domain (Ω 2 , g F ), with ∆ = ∆ F , for pairs (q, r) such that In particular, for r = +∞, we have q ≥ 5 + 1/7.
This result improves on known results for strictly convex domains in 2D: for d = 2, [1] obtained γ(2) = 1/6 (but for any boundary), while [8] only provide the weaker γ(2) = 1/4. Note that for d ≥ 3, [8] had a uniform γ(d) = 1/4, which already improved on [1], where γ(3) = 2/3 and γ(d) = (d − 3)/2 for d ≥ 4 (but again, without restrictions on the boundary). We deliberately chose to restrict to 2D, in order to avoid another layer of technicalities. However, Theorem 1 generalizes to d ≥ 3 with at least γ(d) ≤ 1/6. This will be dealt with elsewhere (preliminary results appeared in [9]). Moreover, Theorem 1 will extend to a generic strictly convex domain, building upon [6] and using the present work as a blueprint; recently [6] obtained γ(d) > (1 − 1/d)/4 for any strictly convex domain, using a weaker version of our model 2D argument. Finally, we expect that the present parametrix construction and its counterpart from [6] to be building blocks for sharper versions of propagation of singularities theorems in the presence of boundaries. This in turn would have important applications to control theory that, to our knowledge, have remained out of reach for a long time.
The proof of Theorem 1 will rely on a complete make-over of the parametrix construction of [8]: resulting bounds on the Green function will be improved in several directions together with refinements of estimates on gallery modes from [3], all of which are of independent interest: • After introducing a localization such that x + D −2 y D 2 x ∼ γ ≪ 1, one may further restrict the wave operator to this region of phase space. The worst possible case regarding dispersion will then be x ∼ γ and |D x | √ γ|D y |, for γ h 1/3 ; • At such fixed γ, the parametrix construction from [8] may be extended to initial data δ (x=a,y=0) with h 2/3−ε < γ, for any ε > 0 and any a > 0, improving on the previous requirement a > h 4/7 ; • the degenerate stationary phase estimates in [8] may be refined to isolate precisely the space-time location of the worst case scenario of a swallowtail singularity. It turns out that such singularities only happen at an exceptional, discrete set of times; we then suitably average over such exceptional times, at fixed γ, before recombining the resulting Strichartz estimates. The singular points in the Green function originating at x = a in the (x, t) plane are (a, 4N √ a √ 1 + a), for |N| ≤ 1/ √ a, which creates difficulties in averaging over a > 0, as the usual argument is blind to improvements outside a small neighborhood of this set.
• gallery modes satisfy the usual Strichartz estimates (as already proved in [3]) but with uniform constant with respect to the order of the mode: this allows to deal with the γ < h 2/3−ε region. These improvements proved to be crucial in recent works [5] and [2]: refinements over the space-time localization of the degenerate stationary phase are indeed a key point to obtain semi-classical dispersion estimates in the Schrödinger case, where the methods of [8] fail to yield improvements over previously known results.
In the remaining of the paper, A B means that there exists a constant C such that A ≤ CB and this constant may change from line to line but is independent of all parameters. It will be explicit when (very occasionally) needed. Similarly, A ∼ B means both A B and B A.

The half-wave propagator: spectral analysis and parametrix construction
Let Ai denote the standard Airy function, we have Ai(−z) = A + (z) + A − (z), where (9) A ± (z) = e ∓iπ/3 Ai(e ∓iπ/3 z) , for z ∈ C , By definition, a function f (w) admits an asymptotic expansion for w → 0 when there exists a (unique) sequence (c n ) n such that, for any n, lim w→0 w −(n+1) (f (w) − n 0 c n w n ) = c n+1 . We will denote f (w) ∼ w n c n w n . Then The following lemma (see [10] for a proof) will be crucial in the analysis of reflected phases : A + (ω) , then L is real analytic and strictly increasing. We also have Finally, one may check that where here and thereafter, {−ω k } k≥1 denote the zeros of the Airy function in decreasing order.
In a classical way, for a > 0, the Dirac distribution δ x=a on R + may be decomposed in terms of eigenfunctions {e k } k≥1 : δ x=a = k≥1 e k (x, θ)e k (a, θ). If we consider a data at time t = s such that u 0 (x, y) = ψ(hD y )δ x=a,y=b , where h ∈ (0, 1] is a small parameter and ψ ∈ C ∞ 0 ([ 1 2 , 3 2 ]), we can write the (localized in θ) Green function associated to the half-wave operator on Ω 2 as Notice that, in addition to the cut-off ψ(hθ), which localizes the Fourier variable dual to y, we may add a spectral cut-off ψ 1 (h λ k (θ)) under the θ integral, where ψ 1 is also such that . Indeed, −∆ F (ψ(hθ)e iθy e k (x, θ)) = λ k (θ)ψ(hθ)e iθy e k (x, θ) .
As observed in [8], the significant part of the sum over k in (14) becomes then a finite sum over k h −1 , considering the asymptotic expansion of ω k ∼ k 2/3 . We now go further and localize with respect to ( As such, introducing a new parameter γ 1, we add a factor ψ 2 ((ω k θ −2/3 )/γ) which, considering the asymptotic expansion of the Airy zeros and the θ localization, is essentially ψ 2 ((kh , this allows to reduce the sum over k in the definition of the Green function to k's such that k γ 3/2 /h. In the following we actually choose ψ 2 ∈ C ∞ 0 ([ 1 2 , 3 2 ]) (later on, it will become clear that for γ ≪ a, the corresponding part of G ± h is irrelevant since the factor e k (a, θ) in this case is exponentially decreasing) and set (rescaling the θ variable for later convenience) Note that from an operator point of view, abusing notations, if G ± is the half-wave propagator, we are in fact considering If γ is taken to be in (2 −m ) m∈N and ψ 2 chosen accordingly (e.g.
h,γ : this sum over γ is finite and restricted to h 2/3 γ 1; the lower bound is induced by the knowledge of −ω 1 < 0 and the upper bound follows from the restriction on k placed by the η and spectral localizations. Before proceeding we remark that we have Indeed, from (15) we write, using (13), using the asymptotics of the Airy function to evaluate the sup z (see [8], Lemma 3.5).
Successive spectral localizations restrict G ± to directions of propagation where the tangential component dominates. As already observed in [8], other directions do see at most one reflection (on a suitable fixed time interval) and may be dealt with using already available arguments (e.g. [1]). We will therefore ignore them from now on. Moreover, we will not only need γ 1 but γ < γ 0 ≪ 1; We conservatively set γ 0 = 1/100 and this may be related to how we sort out directions. Since we are after fongible local in time estimates, we may restrict ourselves to a fixed small size neighborhood in space-time without loss of generality.
We briefly recall a variant of the Poisson summation formula that will be crucial to analyze the spectral sum defining G ± h,γ , see again [10] for a short proof.
N ∈Z

The parametrix regime in 2D
3.1. Localizing waves for γ > h 2/3−ε . In [8] and for a ≫ h 4/7 , a parametrix was carefully constructed by gluing together waves that were mostly located between two consecutive reflections. With a finite number of waves overlapping each other, the supremum of the sum became the supremum of each wave on its own support. In our setting, it is convenient to replace a by the localization parameter γ. Moreover, we replace the parametrix from [8] by the (exact) sum we introduced in the previous section. Let Φ N,a (t, x, y, σ, s, α, η) := y +Φ N (t, x, a, σ, s, α, η) withΦ N defined in (22), We obtain where we have set Observe that χ 1 and ψ 2 induce 2/3 α ∼ γ, which we assume from now on.
We focus on the variables α, s and σ. Using the asymptotic expansion for L(ω), we find where the B ′ term is small compared to 1, provided α 3/2 / is sufficiently large (> 2 is already enough). On the other hand, we have ∂ s Φ N,a = s 2 + a − α and ∂ σ Φ N,a = σ 2 + x − α. If either |s| ≥ 3N ε γ 1/2 or |σ| ≥ 3N ε γ 1/2 , non-stationary phase in one of these variables provides both enough decay to sum in N and an O(h ∞ ) contribution. If |s|, |σ| < 3N ε γ 1/2 , then, for |N| ≥ 1, Φ N,a will not be stationary in α if |t| ≥ (3N + 6N ε )γ 1/2 and non-stationary phase in α provides enough decay to sum in N and an O(h ∞ ) contribution. Hence, the only non trivial contribution comes from |N| |t|γ −1/2 . Now, if a − α > 0, the phase Φ N,a cannot be stationary in s. As α < 2γ, we get the desired result. Observe that, furthermore, by the same reasoning in σ, the only significant contribution of G + h,γ is restricted to x < 2γ. At fixed t ≥ Cγ 1/2 , we can further bound the cardinal of those N that contribute significantly among the C|t|γ −1/2 which are left. Observe that our previous computation tells us that for t = 0, only the N = 0 term may contribute, and from Lemma 4, we can and will restrict ourselves to |t| ≥ Cγ 1/2 .
Recall that γ < 1/100, |N| > 0 and |t| γ 1/2 . the parameter h is intended to be small and we define a large parameter λ γ = γ 3/2 /h (hence restricting γ > h 2/3−ε .) Set and, in the new variables, the phase function in (26) becomes λ γ Ψ N,a,γ . When γ ∼ a we write Ψ N,a,a and λ γ = λ = a 3/2 /h. The critical points of Ψ N,a,γ with respect to Σ, S, A, η are such that Introducing the term 2N √ A(1 − 3 4 B ′ (ηλ γ A 3/2 )) from (31) in (32) provides a relation between Y and T that doesn't involve N nor B ′ as follows: We first estimate the cardinal of N 1 (t, x, y), with t sufficiently large: let j ∈ {1, 2} and N j ∈ N 1 (t, x, y) be any two elements of N 1 (t, x, y). Then there exist (t j , x j , y j ) such that N j ∈ N (t j , x j , y j ); writing t j = √ γT j , x j = γX j , y j + t j √ 1 + γ = γ 3/2 Y j and rescaling (t, x, y) as in (28), we have |T j − T | ≤ 1, |X j − X| ≤ 1, |Y j − Y | ≤ 1. We now prove (27). From N j ∈ N (t j , x j , y j ), there exist Σ j , A j , η j , S j such that (30), (31), (32) holds with T ,X,Y ,Σ,S,A,η replaced by T j ,X j ,Y j ,Σ j ,S j ,A j ,η j , respectively, and we also have S 2 j + a γ = A j . We re-write (31) as follows Multiplying (34) by A j ′ , where j ′ = j, taking the difference and dividing by We need to estimate |N 1 − N 2 |. Using that Σ j , S j < 3, A j ∼ 1, it follows that for j ∈ {1, 2}. The first difference, involving B ′ , in the right hand side of (35) behaves like We cannot take advantage of the difference itself: each term N j B ′ (η j λ γ A 3/2 j ) corresponds to some η j , A j (close to 1) and the difference 1 is bounded but has no reason to be very small (the difference between A j turns out to be O(1/T ), but we don't have any information about the difference between η j which is simply bounded by a small constant on the support of ψ). Therefore the bound (N 1 + N 2 )/λ 2 γ for the terms involving B ′ in (35) is sharp. Since N j ∼ T j , and |T j − T | ≤ 1, it follows that this contribution is ∼ |T |/λ 2 γ . We are reduced to proving that the difference in the second line of (35) is O(1). Write for some absolute constant C and we only used |T 2 − T 1 | ≤ 2 and A j ∼ 1. For T bounded we can conclude since |T 2 −T | ≤ 1. We are therefore reduced to bound T 2 |A 2 −A 1 | when T 2 is sufficiently large. In order to do so, we need the Y variable. We use (33) with T, X, Y, Σ, S, A, η replaced by T j , X j , Y j , Σ j , S j , A j , η j , j ∈ {1, 2} to eliminate the terms containing N and B ′ as follows: For |T | sufficiently large we also have |T j | large and we divide by T j in order to estimate the difference Taking the difference of (38) with itself for j = 1, 2 gives One may check that, as 1/4 < A < 4 and γ ≪ 1, F γ is a decreasing function of A and , and replacing the last expression in the last line of (36) yields (38)). This ends the proof of (27).
We next proceed with proving the contribution outside of N 1 (t, x, y) to be O(h ∞ ). Consider first 2a γ, then critical points in S are such that S 2 + a/γ = A, with A near 1, therefore On the other hand, if Σ ± , A ± , η ± is a critical point for Ψ ± N,a,γ , then (Σ ± , ± A ± − a/γ, A ± , η ± ) are critical point for Ψ N,a,γ . We have to prove that where W ± N,γ (T, X, Y ) have phase functions Ψ ± N,a,γ and symbols are obtained from the symbol of W N,γ multiplied by the symbol of A ± ((ηλ γ ) 2/3 (A − a γ )). Go back to the system (30), (31), (32) and define the integral curves corresponding to Ψ ε N,a,γ as follows It follows that, for all (T ′ , X ′ , Y ′ ) and all (Σ, A, η) on the support of the symbol, we have . Using (43) and the first inequality in (44), for all (T ′ , X ′ , Y ′ ) ∈ B 1 (T, X, Y ) and all (Σ, A, η), (45)). In other words, (45) implies that for all (Σ, A, η), and using the second inequality in (44) with (T ′ , X ′ , Y ′ ) = (T, X, Y ), for all (Σ, A, η), Therefore non-stationary phase always applies in at least one variable among Σ, A, η and each W ε N,γ with N / ∈ N ε 1 provides a O(λ −M γ ) contribution for any M ≥ 1 (where M corresponds to the number of integrations by parts.) We conclude using that the sum over N in G + h,γ restricts to |N| γ −1/2 from Lemma 4.
Let us now deal with the remaining case: a ∼ γ. We re-scale slightly differently for convenience, with λ = a 3/2 /h and t = √ aT , The phase Ψ N,a,γ∼a =: Ψ N,a,a has the same form as in (29) where γ is now replaced by a and λ γ by λ = λ a . We rewrite the integral in S as an Airy function: in the new variables, We proceed with the first sum in (48). By design, on the support of (1 − χ 0 )((ηλ) 2/3 (1 − A)) we have A ≥ 1. If we set A = 1 + µ 2 , we can perform the standard stationary phase in S with critical points S = ±µ (alternatively, we may rewrite the Airy function as A + + A − and use the associated oscillatory integrals to recover the new phases, indexed by ±). From there, we proceed as we did with the case 2a γ. The only difference is that we now have a symbol of order 2/3 in A (coming from 1 − χ 0 ), so one integration by parts with respect to A provides a factor λ −1 λ 2/3 = λ −1/3 ; an integration by parts with respect to Σ yields a factor λ −1 and with respect to η a factor λ −1 λ 2/3 . Therefore eachW N,a with N / ∈ N 1 has at least an O(λ −M/3 γ ) contribution for any M ≥ 1 (where M corresponds to the number of integrations by parts) and we are done, as the sum is over |N| γ −1/2 . Consider now the second sum in (48). Since the phase function of W 0 N,a does not depend on S, we always have ∂ S Ψ 0 N,a = 0. We proceed exactly like in the case 2a ≤ γ, the only difference being that we may take ε = 0.

3.2.
Tangential waves for γ > h 2/3(1−ε) . We start with the (most difficult) case γ/4 ≤ a ≤ 2γ and this restricts a ≥ h 2/3(1−ε) due to the χ 1 cut-off. We re-scale variables: Define λ = a 3/2 /h to be our large parameter, then we write where Ψ N,a,a has been introduced in (29) and γ ∼ a. From the compact support of ψ 2 we have A ≤ 6; since critical points are such that S 2 = A − 1, Υ 2 = A − X, we can restrict ourselves to |S|, |Υ| < 3 without changing the contribution modulo O(h ∞ ); we therefore insert a suitable cut-off χ 2 , and obtain (modulo O(h ∞ )) a slightly modified operator, Since for X > A there are no real critical points with respect to Υ, we may restrict to X 1. As the Green function G + h,γ is symmetric with respect to x and a, we may even restrict ourselves to x < a, and we will obtain slightly better bounds in this case.
Given that ∂ S Ψ N,a,a = η(S 2 + 1 −A), by integration by parts in S we may restrict ourselves to A > 9/10 and therefore where ψ 3 has support in [9/10, 6] (and actually includes dependence on an harmless factor a/γ, which we are hiding since it will be irrelevant from now on.) Remark 3.3. When N λ, the phase factor e iN B(ηλA 3/2 ) does not oscillate: we can move it into the symbol. Indeed, we have |NB( The remaining phase is linear in η, stationary phase in η does not produce decay but localizes the wave front in physical space. When N ≥ λ, the term NB(ηλA 3/2 ) produces oscillations, but it will allow to perform stationary phase in both A and η, also providing decay with respect to η (useful especially for N ≥ λ 2 .) (52). Then the stationary phase theorem applies in A and yields where χ 3 has compact support in (S, Υ) and harmless dependency on the parameters a, h, and the phase function φ N,a is given by Proof. Using Remark 3.3, we immediately move the factor e iN B(ηλA 3/2 ) into the symbol. Therefore the phase of W N,a becomes Ψ N,a,a (T, X, Y, Υ, S, A, η) − N λ B(ηλA 3/2 ) which is linear in η and stationary in A where its derivative in A vanishes, which is nothing but (54). It immediately follows from (54) that T /(4N) is bounded when the phase is stationary in A, and even that T /(4N) ∼ √ A c . The second derivative of the phase with respect to A equals , the second derivative does not vanish and has size comparable to |N|; from this we may apply stationary phase to get the desired expansion.
Setting K := T 4N and w := S+Υ 2N , (54) for the critical point A c becomes and where E is a smooth function, uniformly bounded and defined as follows : , so that we can apply the implicit function theorem to get a unique solution A 1/2 . Taking the derivative of (56) with respect to w yields and using again (56) to replace Proposition 3. Let |N| ≥ λ and W N,a (T, X, Y ) be defined in (52), then the stationary phase applies in both A and η and yields where χ 3 has compact support in (S, Υ) and harmless dependency on the parameters a, h, 1/N.
The second order derivatives are given by at the critical points, we have det Hess Ψ N,a,a ∼ N 2 λ 2 , for N ≥ λ, and we may apply stationary phase in (A, η).
We now study critical points in (A, η) for |N| > λ. Applying the implicit function theorem to the system ∂ A Ψ N,a,a = 0, ∂ η Ψ N,a,a = 0 around (S = 0, Υ = 0) yields at most a pair of critical points (A c , η c ) belonging to the support of the symbol and depending on all variables. In the following we will only need the derivatives of A c with respect to the two remaining variables S, Υ. For that, we take the derivatives of (59), (60) with respect to S, Υ. This gives From this system we obtain the following Lemma : belongs to a neighborhood of 1 and at S = Υ = 0, We are left with an integral over (S, Υ). In the remaining part of this section we prove the following dispersive estimates that will be crucial in order to obtain better Strichartz estimates than those implied by the usual duality argument and dispersion (recall t = √ aT ): .
The corollary follows at once from the last bound in Proposition 7 and Proposition 1: We will prove the three propositions in reverse order.

3.2.2.
Proof of Propositions 6 and 5. As 1 ≤ N < λ 1/3 ≪ λ, we move exp(iNB(λA 3/2 )) (from the phase Ψ N,a,a of W N ) into the symbol; the critical point A c is given in Lemma 5, the critical value for the phase is ηφ N,a and φ N,a (defined in (55)) does not depend on η. The following bound is proved in [8] (for a phase that was constructed differently), uniformly for η ∈ supp(ψ), Informally, the decay should be understood as resulting from a non degenerate stationary phase in one variable, followed by an application of Van der Corput lemma (with a non vanishing fourth derivative in the remaining variable). This accounts for the 1/2 + 1/4 = 3/4 exponent on the large parameter. We now obtain better bounds, either because in the remaining variable the phase has non vanishing derivative of order three (generating 1/2 + 1/3 = 5/6 decay) or two. In doing so, we uncover the geometry of the curves on which the phase may degenerate. Set Λ = λ/N 3 to be the new (large) parameter. Re-scale again variables with S = q/N and Υ = p/N and set ΛΦ N,a (T, X, p, q) = λφ N,a (T, X, p/N, q/N). On the support of χ we then have |(q, p)| N. We are reduced to proving We have ∇ (q,p)ΦN,a = q 2 + N 2 (1 − A c ), p 2 + N 2 (X − A c ) , where, using Lemma 5, We define P = (K 2 ∞ (a) − X)N 2 and Q = (K 2 ∞ (a) − 1)N 2 . With these notations, Remark 3.6. For N sufficiently small even the terms with 1 N may provide important contributions. At this stage and given that all variables were properly rescaled with respect to a, the reader may, at first, set a = 0 (and even N = 1 !) to make all subsequent computations more straightforward while capturing the correct asymptotics.
We start with |(P, Q)| ≥ r 0 for some large, fixed r 0 , in which case we can follow the same approach as in the previous case. Set again P = r cos θ and Q = r sin θ. If |(p, q)| < r 0 /2, then the corresponding integral is non stationary and we get decay by integration by parts. We change variables (p, q) = r 1/2 (p ′ , q ′ ) with r 0 ≤ r N 2 and aim at proving (87) r R 2 e ir 3/2 ΛηΦ N,a χ(r 1/2 q ′ /N, r 1/2 p ′ /N, a, 1/N, h, η) dp ′ dq ′ r −1/4 Λ −5/6 , where χ is compactly supported and Φ N,a (T, X, p ′ , q ′ ) := r −3/2Φ N,a (T, X, r 1/2 p ′ , r 1/2 q ′ ). Compute As in the previous case, cos θ and sin θ play symmetrical parts: hence we set P ≥ Q, cos θ ≥ sin θ. If |(p ′ , q ′ )| ≥ M for some large M ≥ 1, then, for critical points, p ′2 c ≥ q ′2 c and if M is sufficiently large non-stationary phase applies in p ′ . Therefore we are reduced, again, to bounded |(p ′ , q ′ )|. We deal with three cases, depending upon Q = r sin θ: if sin θ < − C √ r for some sufficiently large constant C > 0, then As |(p ′ , q ′ )| is bounded, E is bounded, N is sufficiently large (from N > √ r ≥ √ r 0 ) and 1 √ r ≥ 1 N , it follows that non-stationary phase applies: the sum of the last three terms in the previous inequality is greater than C/(2r 1/2 ) for C large enough. If | sin θ| ≤ C √ r then, again, θ ∈ (− C √ r 0 , π 4 ) and cos θ ≥ √ 2 2 . We have |Q| = |r sin θ| ≤ C √ r; if |Q| < C, then 1 + |Q| r 1/2 , while |P | ∼ r. As in the previous case the stationary phase applies in p ′ with non-degenerate critical points p ′ ± and yields a factor (r 3/2 Λ) −1/2 ; the critical values Φ ±,N,a of the phase function at these critical points are such that |∂ 3 r 0 ) and therefore the integral with respect to q ′ is bounded by (r 3/2 Λ) −1/3 by Van der Corput lemma. We obtain (87). If we are interested solely in X < 1, we may bound r −1/4 (1 + |Q| 1/2 ) −1 . Finally, if sin θ > C √ r , then Q = r sin θ > C √ r and therefore N 2 |K 2 ∞ (a) − 1| > Cr 1/2 . We directly perform stationary phase with large parameter r 3/2 Λ as the determinant of the Hessian matrix at the critical point is at least C √ cos θ sin θ: this yields the following bound for the left hand side term in (87), where the last inequality holds only for X < 1 cr Considering we may exchange P and Q, we have just proved Proposition 6: for N < λ 1/3 and |T − 4N| 1/N, In the special case X < 1, we may replace N 1/4 |T − 4N| 1/4 by N 1/2 |T − 4N| 1/2 . We now move to the most delicate case |(P, Q)| ≤ r 0 . For |(p, q)| large, the phase is non stationary and integrations by parts provide O(Λ −∞ ) decay. So we may replace χ by a cutoff, that we still call χ, that is compactly supported in |(p, q)| < R. We will improve the estimates from [8] that were sharp only at p = q = 0 on this limiting case and prove (88) I := R 2 e iΛηΦ N,a χ(q/N, p/N, a, 1/N, h, η) dpdq Λ −5/6 Λ −1/12 + |Q| 1/6 .
In [8], we proved a general lemma covering the most degenerate cases. Here, as explained earlier, we just proceed by identifying one variable where the usual stationary phase may be performed, and then evaluate the remaining 1D oscillating integral using Van der Corput lemma with different decay rates depending on the lower bounds on derivatives of order at most 4. Replacing A c using (86) in ∂ qΦN,a = q 2 + N 2 (1 − A c w= (p+q) 2N 2 ) and ∂ pΦN,a = Define H N such that −pP − qQ + H N (q, p, X, T ) =Φ N,a (q, p, X, T ), then Moreover, the second order derivatives of H N follow directly from those ofΦ N,a : and we recall from (57) that the derivative of A c with respect to w is given by .
We need to prove that, when ∂ ξ 1 (h N (ξ 1 , ξ 2,c )) = ∂ 2 ξ 1 (h N (ξ 1 , ξ 2,c )) = 0, the third order derivative doesn't vanish. Taking the derivative of (100) with respect to ξ 1 yields where we setȆ : . For all |N| ≥ 2, the bracket in the right hand side term of (102) is strictly positive and we may apply Van der Corput lemma with nonvanishing third order derivative. We are left with |N| = 1 where the third order derivative may vanish when M 2 is very small : but in this case, the second derivative given in (100) doesn't vanish and we can apply the usual stationary phase. Indeed, recall that (ξ 1 , ξ 2 ) is bounded (from (p, q) bounded). Therefore, when N 2 = 1, the coefficient of ξ 1 in (100) is O(a) and for M 2 such that |M 2 | ≤ 4c 2 we have | | ≤ c for every |ξ 1 | ≥ c. We conclude using that K ∞ (a) ∼ 1. As such, the contribution of the set |ξ 1 | > c (88) is at most C(c)×1/Λ 1/2 ×1/Λ 1/3 , where the first factor is the non degenerate stationary phase in ξ 2 and the second one is coming from Van der Corput in ξ 1 , irrespective of the value of N. We stress that we made no use of the value of the Hessian here.
In doing so, we may possibly reduce the size of the constant c without loss of generality. Knowing the explicit formula for a = 0 yields F (·, 0) = F 0 .
Therefore G + h,γ (t, x, y, a, 0, 0 Here we dropped the χ 1 cut-off that is irrelevant since we consider λ γ to be large. We will prove the following propositions (recall t = √ γT , therefore tγ/h = T λ γ ): Proposition 11. For |T | λ 2 γ , we have We start with W N,γ . As a/γ < 1/4 and A > 1/2, |S ± | = A − a γ > 1/2 and the usual stationary phase applies in this variable; the critical values of the phase at S ± are {Ψ ± N,a,γ (T, X, Y, Υ, A, η)}, where Ψ ± N,a,γ (T, X, Y, Υ, A, η) have been defined in (40) and accordingly we set W N,γ = W + N,γ + W − N,γ . The phases Ψ ± N,a,γ are stationary in A when The second derivative with respect to A is If N = 0, the middle term dominates the first one (as γ ≪ 1). If |N| ≥ 1, the last term overcomes the next to last: this is obvious unless |N| = 1, in which case √ A ∼ 2 A − a/γ would imply 3A ∼ a/γ < 1/4 which is excluded by the support condition on A. Therefore |∂ 2 A Ψ ± N,a,γ | 1. The stationary phase in A applies so we are left to deal with the remaining variables Υ, η. When |T | is not too large, we will only integrate in Υ and then simply discard the integral in η which is bounded as it has compact support; when |T | > λ 2 γ the stationary phase in η turns out to be particularly useful. The critical value of the phase functions Ψ ± N,a,γ (T, X, Y, Υ, A c , η) satisfies For a given |N| ≥ 2 and a given point (T, X, Y ) with T ∼ 4N, the phase Ψ ± N,a,γ|Ac has at most one degenerate critical point of order two with respect to Υ.
Notice that for every given T, N we obtain an unique solution Υ c (T, N, γ). Asking Υ c to be a critical point gives also Υ 2 c = A c − X, which provides the relation between T and X at those points where the phase is degenerate of order at least two. To check the order of degeneracy we consider higher order derivatives. We have 3 which means that for |N| ≥ 2 the critical point Υ c is degenerate of order exactly two and the absolute value of the third derivative is bounded from below by a constant.
and we obtain, after replacing Υ with its expression from (126) which ends the proof.
Proof. Let N = 1 : for Ψ − 1,a,γ the proof follows the one of Lemma 3.3, except for the lower bound on T that is a direct consequence of (126), with ± = −. For Ψ + 1,a,γ we proceed as in the proof of Lemma 10; notice that in that case, (126), with ± = + forces |T | 1. Proof. The derivatives with respect to A, η of Ψ ± N,a,γ are given by The second order derivatives are and when ∇ A,η Ψ ± N,a,γ = 0, the determinant of the Hessian matrix of Ψ ± N,a,γ is It remains to deal with the integral in Υ. We proceed exactly like in the proof of Lemma 3.3: the critical point A c is given by the same formula while the critical point η c doesn't interfere here, as and at the stationary point Υ 2 + X − A c = 0, we get the same formula as in (129) (with a factor η c near 1). In the same way, the third derivative is hence when the first and second derivative vanish, the third one behaves exactly like (131).
We now consider several cases, depending on T . If |T | 1, we can easily see that Ψ ± N,a,γ can be stationary only for N ∈ {0, ±1}, since for |N| ≥ 2 (126) has no solution for A: , which is outside the support of A for λ γ large enough.
Remark 3.9. Notice that this last bound is the same as on a domain without boundary. More precisely, it matches the boundary-less case for a frequency localized Dirac data, with η ∼ 1/h and ξ ∼ √ γ/h, where dispersion takes over Sobolev embedding for t > h/γ.
Let N = ±1, then, according to Lemma 11, ∂ 2 Υ Ψ ± ±1,a,γ ∼ T ; we conclude as for N = 0 by splitting according to whether tγ/h M 2 or tγ/h > M 2 . On the other hand, the phases Ψ ∓ ±1,a,γ may have critical points degenerate of order (exactly) two, but they only contribute if |T | ≥ 1 from Lemma 11. As such, for small |T |, the significant contribution to G + h,γ is W 0,γ + W + 1,γ + W − −1,γ and we obtain the second term in (122), completing the proof of Proposition 8. Now, let 1 ≤ |T | ≤ 9; equation (126) has solutions for N ∈ {0, ±1, ±2} and stationary phase applies in A with large parameter λ γ . When N = 0, the usual stationary phase applies in Υ since |T | ≥ 1; when N = ±1, stationary phase would apply for W ± ±1,γ but for |T | 1 they are non stationary and do not contribute significantly; according to Lemma , and this completes the proof of Proposition 9.
Then, when 9 ≤ |T | 4λ 2 γ , equation (126) has a solution only for |N| ≥ 2 and the third order derivative with respect to A is bounded from below, therefore we get where the first factor λ −1/2 γ comes from the stationary phase in S, the factor (Nλ γ ) −1/2 from the stationary phase with respect to A and λ −1/3 γ from the degenerate critical point of order two in Υ. Using that t/ √ γ ∼ 4N we find as here we consider only 9 T 4λ 2 γ . Finally, let 4λ 2 γ |T | hence λ 2 γ |N|. Since |T | 1/ √ γ, this regime corresponds to γ h 4/7 that hasn't been dealt with in [8]. Here, stationary phase applies in (A, η). Using Lemma 12, we get where the first factor λ −1/2 γ comes from the integration in S, λ −1 γ (N 2 /λ 2 γ ) −1/2 from the stationary phase in A, η and λ

Strichartz estimates
We intend to prove Theorem 1. We may reduce ourselves to half-wave frequency localized operators G ± h , and then, to a sum of operators G ± h,γ . Assuming we get a bound with a constant γ ε , ε > 0, we will have the same bound (with fixed constant) on G ± h . We proceed as usual by duality, hereby reducing ourselves to an inhomogeneous estimate on 4.1. The parametrix regime: γ h 2/3−ε . We search for the smallest q such that, for |t| 1, with C q (γ) 1. It will turn out to be convenient to prove with p = q/2. As the adjoint of G ± h,γ is G ∓ h,γ , we will recover the previous estimate by duality and interpolation: duality yields , and interpolation midway between (135) and (136) provides (134). We may now replace f ∈ L 1 s,a,b in (135) by a Dirac at (s, a, b): then we have v ± (x, y, t) = G ± h,γ (x, y, t, a, b, s), and we are left to prove that, for s ∈ (−1, 1), we have sup a,b,s v ± Notice that s is actually irrelevant, and so is b: set s = b = 0, we would like to prove We recall that, at fixed γ, we know that 0 < a γ and only sees at most one reflection, and as such it satisfies the free case dispersion estimate, as proved in [1] with a generic boundary. In our particular setting we do have Propositions 4 and 8 that provide the correct decay estimate, and we get which is bounded by h 1/2 /h 2 for p = 2 (except for an irrelevant log that may be removed by computing the weak L 2 norm). Next, we record the bound from (138) but for any 2 < p: We then split the operator defined by (133): u ± h,γ = u ±,0 h,γ + w ± h,γ , where the kernel of u ±,0 h,γ is restricted to |t − s| γ. Having the usual dispersion bound for this truncated kernel, we get . Hence on such a short time scale we recover the classical L 4/3 L 1 → L 4 L ∞ bound on the 2D wave equation in R 2 . Going back to an homogeneous estimate and using conservation of energy, we may pile up 1/ √ γ estimates to obtain an homogeneous estimate on a longer time interval, at the expense of a large constant 1/( √ γ) 1/4 , and then convert that estimate to an inhomogeneous estimate again, and it will hold for both u ± h,γ and w ± h,γ (using (140) for w ± h,γ = u ± h,γ − u ±,0 h,γ ): x . Interpolating this with (141) (for u ± h,γ ) yields an L 6 L ∞ bound, which is nothing but the bound from [1].
We now proceed with larger times, and start with γ > h 1/3 , where one may easily check that the condition N < λ 1/3 is always satisfied. Moreover, in this regime, there are no overlaps between waves. We first consider the tangential part, G +,♯ h,γ = N W N,a . From Propositions 5 and 6, set N max ∼ 1/ √ γ λ 1/3 (the maximum number of reflections we may observe on our given time interval of size comparable to one), and write Changing variables, we compute the right-hand side, for 3 ≤ p < 6: where 1 λγ 3/2 = γ 3 /h in our regime. We now do the same computation but for the transverse part, G +,♭ h,γ = N W N,γ , using Propositions 9 and 10, for T > 1: and, for 2 < p the sum is finite. Therefore, summing both tangential and transverse estimates, for all a γ and choosing p = 4 yields Using (142) we get an estimate for w ± h,γ : first from L 1 s,a,b to L 4 t L ∞ x,y and then by duality and interpolation, We notice that h 1/12 γ 1/8 γ 3/8 . We may now interpolate between (143) and (141): pick an interpolating exponent θ = 3/5, we have 1/5 = θ/4 + (1 − θ)/8 and 3(1 − θ)/8 − θ/4 = 0, we get The same bound holds for u ±,0 h,γ , with a much better γ 1/5 replacing the log factor. If in the interpolation step, we pick θ < 3/5, there is an additional positive power of γ left, making the log irrelevant, and we can even sum over γ h 1/3 : denote this sum by G ± h, h 1/3 , we proved Proposition 12. The half-wave operator G ± h, h 1/3 is such that, for any q > 5, x,y h 1/q−1 u 0 2 .

Remark 4.3.
Considering that the counterexample in [10] precludes q < 5, Proposition 12 is optimal up to the endpoint q = 5. In fact, one may refine our argument to get an optimal L q L ∞ estimate for G ± h,γ , depending on h 1/3 γ 1, and connecting q = 5 and q = 4.
These solutions to the wave equation and are the so-called gallery modes. Using Lemma 2, (u k,± ) k is an orthogonal family in L 2 (Ω 2 ). We also set u k,0 (x, y) = u k,± (0, x, y).
As in Section 2.1, the cut-off φ γ reduces the sum over k to k γ 3/2 /h = λ γ .
The Proposition is a simple consequence of the Cauchy-Schwarz inequality: as remarked earlier on, the localization w.r.t. Q x,y restricts the sum over k to (hk) 2/3 γ, and therefore Chose q > 4: by interpolation between our two bounds, we can sum over all h 2/3 γ h 2/3−ε : consider q > 4 and θ = 1 − 4/q, then the summability condition on ε and q reads, with µ > 0 ) .