Huy Quang

Nguyen

A sharp Cauchy theory for the 2D gravity-capillary waves

come     A sharp Cauchy theory for the 2D gravity-capillary waves

1 Introduction

The equations

We consider an incompressible, inviscid fluid with unit density moving in a time-dependent domain Ω = {(t, x, y) ∈ [0, T ] × R × R : (x, y) ∈ Ω t }

where each Ω t is a domain located underneath a free surface

Σ t = {(x, y) × R × R : y = η(t, x)}
and above a fixed bottom Γ = ∂Ω t \Σ t . We make the following assumption on the domain: Ω t is the intersection of the haft space

Ω 1,t = {(x, y) × R × R : y = η(t, x)}
and an open connected set Ω 2 containing a fixed strip around Σ t , i.e., there exists h > 0 such that

{(x, y) ∈ R × R : η(x) -h ≤ y ≤ η(t, x)} ⊂ Ω 2 .
This important assumption prevents the bottom from emerging, or even from coming arbitrarily close to the free surface. The study of water waves without it is an open problem.

Assume that the velocity field v admits a potential φ : Ω → R, i.e, v = ∇φ. Using the Zakharov formulation, we introduce the trace of φ on the free surface ψ(t, x) = φ(t, x, η(t, x)).

Then φ(t, x, y) is the unique variational solution of (1.1) ∆φ = 0 in Ω t , φ(t, x, η(t, x)) = ψ(t, x).

The Dirichlet-Neumann operator is then defined by

G(η)ψ = 1 + |∂ x η| 2 ∂φ ∂n    Σ = (∂ y φ)(t,
x, η(t, x)) -∂ x η(t, x)(∂ x φ)(t, x, η(t, x)).

The gravity water wave problem with surface tension consists in solving the following system of η, ψ:

(1.2)

     ∂ t η = G(η)ψ, ∂ t ψ + gη -H(η) + 1 2 |∂ x ψ| 2 - 1 2 (∂ x η∂ x ψ + G(η)ψ) 2 1 + |∂ x η| 2 = 0
where H(η) is the mean curvature of the free surface:

H(η) = ∂ x ∂ x η 1 + |∂ x η| 2 .
It is important to introduce the vertical and horizontal components of the velocity, which can be expressed in terms of η and ψ:

(1.3) B = (v y )| Σ = ∂ x η∂ x ψ + G(η)ψ 1 + |∂ x η| 2 , V = (v x )| Σ = ∂ x ψ -B∂ x η.

The problem

Our purpose is to study the Cauchy problem for system (1.2) with sharp Sobolev regularity for initial data. For previous results on the Cauchy problem, we refer to the works of Yosihara [START_REF] Yosihara | Gravity waves on the free surface of an incompressible perfect fluid of finite depth[END_REF], Coutand-Shkoller [START_REF] Coutand | Well-posedness of the free-surface incompressible Euler equations with or without surface tension[END_REF], Shatah-Zeng [START_REF] Shatah | Geometry and a priori estimates for free boundary problems of the Euler equation[END_REF][START_REF] Shatah | A priori estimates for fluid interface problems[END_REF][START_REF] Shatah | Local well-posedness for fluid interface problems[END_REF], Ming-Zhang [START_REF] Ming | Well-posedness of the water-wave problem with surface tension[END_REF] for sufficiently smooth solutions; see also the works of Wu [START_REF] Wu | Well-posedness in Sobolev spaces of the full water wave problem in 3-D[END_REF][START_REF] Wu | Well-posedness in Sobolev spaces of the full water wave problem in 2-D. Invent[END_REF], Lannes [START_REF] Lannes | Well-posedness of the water-waves equations[END_REF] for gravity waves without surface tension. In term of regularity of initial data, the work of Alazard-Burq-Zuily [START_REF] Alazard | On the water waves equations with surface tension[END_REF] reached an important threshold: local wellposedness as long as the velocity field is Lipschitz (in term of Sobolev embeddings) up to the free surface. More precisely, this corresponds to data (in view of the formula (1.3))

(η 0 , ψ 0 ) ∈ H s+ 1 2 (R d ) × H s (R d ), s > 2 + d 2 .
This is achieved by the energy method after reducing the system to a single quasilinear equation using a paradifferential calculus approach. However, observe that the linearized of (1.2) around the rest state (0, 0) reads

∂ t Φ + i |D| 3 2 Φ = 0, Φ = |D| 1 2 η + iψ
which is dispersive and enjoys the following Strichartz estimate with a gain of 3 8 derivatives (1.4) Φ

L 4 t W σ-1 8 ,∞ x ≤ C σ Φ| t=0 H σ x , ∀σ ∈ R.
Therefore, one may hope that the fully nonlinear system (1.2) is also dispersive and enjoys similar Strichartz estimates. Indeed, this is true and was first proved by Alazard-Burq-Zuily [START_REF] Alazard | Strichartz estimates for water waves[END_REF]: any solution

(1.5) (η, ψ) ∈ C 0 ([0, T ]; H s+ 1 2 (R) × H s (R)), s > 2 + 1 2 satisfies (1.6) (η, ψ) ∈ L 4 ([0, T ]; W s+ 1 4 ,∞ (R) × W s-1 4 ,∞ (R)).
Comparing to the classical (full) Strichartz estimate (1.4), the estimate (1.6) exhibits a loss of 1 8 derivatives and is called the semi-classical Strichartz estimate. This terminology comes from the work [START_REF] Burq | Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds[END_REF] for Schrodinger equations on manifolds. In fact, slightly earlier in [START_REF] Christianson | Strichartz estimates for the water-wave problem with surface tension[END_REF] the same Strichartz estimate was obtained for the 2D gravity-capillary water waves under another formulation. We also refer to [START_REF] Nguyen | Sharp Strichartz estimates for water waves systems[END_REF] for another proof of (1.6) and the semiclassical Strichartz estimate for 3D waves.

It is known, for instance from the works of Bahouri-Chemin [START_REF] Bahouri | Équations d'ondes quasilinéaires et estimations de Strichartz[END_REF] and Tataru [START_REF] Tataru | Strichartz estimates for operators with nonsmooth coefficients and the nonlinear wave equation[END_REF], that for dispersive PDEs, Strichartz estimates can be used to improve the Cauchy theory for data that are less regular than the one obtained merely via the energy method. We refer to [START_REF] Bahouri | Fourier analysis and nonlinear partial differential equations[END_REF], Chapter 9 for an expository presentation of quasilinear wave equations. Our aim is to proceed such a program for the gravity-capillary water waves system (1.2). For pure gravity water waves, this was considered by Alazard-Burq-Zuily [START_REF] Alazard | Strichartz estimate and the Cauchy problem for the gravity water waves equations[END_REF]. Coming back to our system (1.2), from the semi-classical Strichartz estimate (1.6) for s > 2 + 1 2 it is natural to ask Q: Does the Cauchy problem for (1.2) have a unique solution for data

(η 0 , ψ 0 ) ∈ H s+ 1 2 (R d ) × H s (R d ), s > 2 + 1 2 - 1 4 = 9 4
?.

In the previous joined work [START_REF] Nguyen | Strichartz estimates and local existence for the capillary water waves with non-Lipschitz initial velocity[END_REF], we proved an "intermediate" result for s > 2 + 1/2 -3/20 in 2D case (together with a result for 3D case), which asserts that water waves can still propagate starting from non-Lipschitz velocity (up to the free surface) (see [START_REF] Alazard | Strichartz estimate and the Cauchy problem for the gravity water waves equations[END_REF] for the corresponding result for vanishing surface tension). Our contribution in this work is to prove an affirmative answer for question Q.

Let us give an outline of the proof. In [START_REF] Nguyen | A paradifferential reduction for the gravity-capillary waves system at low regularity and applications[END_REF], using a paradifferential approach we reduced the system (1.2) to a single dispersive equation as follows: assume that for some s > r > 2

(1.7) (η, ψ) ∈ C 0 ([0, T ]; H s+ 1 2 (R) × H s (R)) ∩ L 4 ([0, T ]; W r+ 1 2 ,∞ (R) × W r,∞ (R))
then after paralinearization and symmetrization, (1.2) is reduced to the following equation of a complexed-valued unknown Φ

(1.8)

∂ t Φ + T V ∂ x Φ + iT γ Φ = f
for some paradifferential symbol γ ∈ Σ 3/2 and f (t) satisfies the tame estimate

f (t) H s ≤ F η(t) H s+ 1 2 , ψ(t) H s 1 + η(t) W r+ 1 2 ,∞ + ψ(t) W r,∞ .
Such a reduction was first obtained in [START_REF] Alazard | On the water waves equations with surface tension[END_REF] for solution at the energy threshold (1.5).

Observe that the relation s > r > 2 exhibits a gap of 1 2 derivatives in view of the Sobolev embedding from H s to C s- 1 2 * (see Definition A.1). Having in hand the blow-up criterion and the contraction estimate in [START_REF] Nguyen | A paradifferential reduction for the gravity-capillary waves system at low regularity and applications[END_REF] at the regularity (1.7), the main difficulty in answering question Q is to prove the semi-classical Strichartz estimate for solution Φ to (1.8). Comparing to the Strichartz estimates in [START_REF] Nguyen | Strichartz estimates and local existence for the capillary water waves with non-Lipschitz initial velocity[END_REF] we remark that the semi-classical gain in [START_REF] Alazard | Strichartz estimates for water waves[END_REF] (when s > 2 + 1 2 ) was achieved owing to the fact that when d = 1 one can further reduce (1.8) to an equation where the highest order term T γ becomes the Fourier multiplier |D x | (1.9)

∂ t Φ + T V ∂ x Φ + i|D x | 3 2 Φ = f .
This reduction is proceeded by means of the paracomposition of Alinhac [START_REF] Alinhac | Paracomposition et opérateurs paradifférentiels[END_REF]. Here, we shall see that in our case we need a more precise paracomposition result for 2 purposes:

(1) deal with rougher functions and (2) obtain quantitative estimates. This will be the content of section 3. After having (1.9) we show in section 4 that the method in [START_REF] Alazard | Strichartz estimates for water waves[END_REF] can be adapted to our lower regularity level to derive the semi-classical Strichartz estimate with an arbitrarily small ε loss.

Main results

Let us introduce the Sobolev norm and the Strichartz norm for solution (η, ψ) to the gravity-capillary system (1.2):

M σ (T ) = (η, ψ) L ∞ ([0,T ];H σ+ 1 2 (R)×H σ (R)) , M σ (0) = (η, ψ)| t=0 H σ+ 1 2 (R)×H σ (R) , N σ (T ) = (η, ψ) L 4 ([0,T ];W σ+ 1 2 ,∞ (R)×W σ,∞ (R))
.

Our first result concerns the semi-classical Strichartz estimate for system (1.2).

Theorem 1.1. Assume that (η, ψ) is a solution to (1.2) with (1.10)    (η, ψ) ∈ C 0 ([0, T ]; H s+ 1 2 (R) × H s (R)) ∩ L 4 ([0, T ]; W r+ 1 2 ,∞ (R) × W r,∞ (R)), s > r > 3 2 + 1 2 .
and

(1.11) inf t∈[0,T ] dist(η(t), Γ) ≥ h > 0.
Then, for any µ < 1 4 there exists a non-decreasing function F independent of (η, ψ) such that (1.12)

N s-1 2 +µ (T ) ≤ F (M s (T ) + N r (T )). As a consequence of Theorem 1.1 and the energy estimate in [START_REF] Nguyen | A paradifferential reduction for the gravity-capillary waves system at low regularity and applications[END_REF] we obtain a closed a priori estimate for the mixed norm M s (T ) + N r (T ).

Theorem 1.2. Assume that (η, ψ) is a solution to (1.2) and satisfies conditions (1.10), (1.11)

with 2 < r < s - 1 2 + µ, µ < 1 4 , h > 0.
Then there exists a non-decreasing function F independent of (η, ψ) such that

M s (T ) + N r (T ) ≤ F F (M s (0)) + T F (M s (T ) + N r (T )) .
Finally, we obtain a Cauchy theory for the gravity-capillary system (1.2) with initial data 1 4 derivatives less regular than the energy threshold in [START_REF] Alazard | On the water waves equations with surface tension[END_REF].

Theorem 1.3. Let µ < 1 4 and 2 < r < s -1 2 + µ. Then for any (η 0 , ψ 0 ) ∈ H s+ 1 2 (R) × H s (R) satisfying dist(η 0 , Γ) ≥ h > 0,
there exists T > 0 such that the gravity-capillary waves system (1.2) has a unique solution (η, ψ) in

L ∞ ([0, T ]; H s+ 1 2 (R) × H s (R)) ∩ L 4 ([0, T ]; W r+ 1 2 ,∞ (R) × W r,∞ (R)).
Moreover, we have

(η, ψ) ∈ C 0 [0, T ]; H s0+ 1 2 × H s0 , ∀s 0 < s and inf t∈[0,T ] dist(η(t), Γ) > h 2 .
Remark 1.4. The proof of Theorem 1.3 shows that for each µ < 1 4 the existence time T can be chosen uniformly for data (η 0 , ψ 0 ) lying in a bounded set of H s+ 1 2 (R) × H s (R) and the fluid depth h lying in a bounded set of (0, +∞).

Remark 1.5. We do not know yet if the semi-classical gain is optimal for solutions at the regularity (1.10). However, some remarks can be made as follows. On the one hand, if one proves Strichartz estimate for (1.8) then there is a nontrivial geometry of the symbol γ, for which trapping may occur. According to [START_REF] Burq | Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds[END_REF] (see Section 4), at least in the case of spheres, the semi-classical Strichartz estimates are optimal. On the other hand, if one wishes to eliminate the geometry by making changes of variables, then as we shall see in Proposition 4.3 and Remark 4.4, there will appear a loss of 1 2 derivatives in the source term, which turns out to be optimal for the semi-classical Strichartz estimate (see the proof of Theorem 4.14).

Remark 1.6. The linearized system of (1.2) in dimension 2 (η, ψ : R 2 → R) enjoys the semi-classical Strichartz estimate with a gain 1 2 derivatives (see [START_REF] Nguyen | Sharp Strichartz estimates for water waves systems[END_REF]). It was proved in [START_REF] Nguyen | Sharp Strichartz estimates for water waves systems[END_REF] that the same estimate holds for the nonlinear system (1.2) when

(η, ψ) ∈ C 0 ([0, T ]; H s+ 1 2 (R 2 ) × H s (R 2 )), s > 5 2 + 1.
If the preceding regularity could be improved to (

1 2 derivative    (η, ψ) ∈ C 0 ([0, T ]; H s+ 1 2 × H s ) ∩ L 2 ([0, T ]; W r+ 1 2 ,∞ × W r,∞ ), s - 1 2 > r > 2,
the results in [START_REF] Nguyen | A paradifferential reduction for the gravity-capillary waves system at low regularity and applications[END_REF] would imply a Cauchy theory (see the proof of Theorem 1.3) with initial surface

η 0 ∈ H s+ 1 2 (R 2 ), s > 2 + 1 2 ,
which has the lowest Sobolev regularity to ensure that the initial surface has bounded curvature (see the Introduction of [START_REF] Nguyen | Strichartz estimates and local existence for the capillary water waves with non-Lipschitz initial velocity[END_REF]).

Acknowledgment

This work was partially supported by the labex LMH through the grant no ANR-11-LABX-0056-LMH in the "Programme des Investissements d'Avenir". I would like to sincerely thank my advisor, Nicolas Burq for suggesting me the problem and sharing many fruitful discussions together with Claude Zuily, whom I warmly thank. Finally, I thank Thomas Alazard and Thibault de Poyferré for direct and indirect discussions.

Preliminaries on dyadic analysis

Dyadic partitions

Our analysis below is sensitive with respect to the underlying dyadic partition of R d . These partitions are constructed by using the cut-off functions given in the following lemma.

Lemma 2.1. For every n ∈ N, there exists φ

(n) ∈ C ∞ (R d ) satisfying (2.1) φ (n) (ξ) = 1, if |ξ| ≤ 2 -n , 0, if |ξ| > 2 n+1 , (2.2) ∀(α, β) ∈ N d × N d , ∃C α,β > 0, ∀n ∈ N, x β ∂ α φ (n) (x) L 1 (R d ) ≤ C α,β .
We leave the proofs of the results in this paragraph to Appendix 2. In fact, to guarantee condition (2.2) we choose φ (n) with support in a ball of size 2 -n + c for some c > 0. We shall skip the subscript (n) and denote φ ≡ φ (n) for simplicity. Setting

φ k (ξ) = φ( ξ 2 k ), k ∈ Z, ϕ 0 = φ = φ 0 , ϕ = χ -χ -1 , ϕ k = φ k -φ k-1 = ϕ( • 2 k ), k ≥ 1, we see that (2.3) supp ϕ 0 ⊂ C 0 (n) := {ξ ∈ R d : |ξ| ≤ 2 n+1 } supp ϕ ⊂ C(n) := {ξ ∈ R d : 2 -(n+1) < |ξ| ≤ 2 n+1 } supp ϕ k ⊂ C k (n) := {ξ ∈ R d : 2 k-(n+1) < |ξ| ≤ 2 k+(n+1) }, ∀k ≥ 1.
Observing also that with

N 0 := 2(n + 1)
we have

C j (n) ∩ C k (n) = ∅ if |j -k| ≥ N 0 .
Definition 2.2. For every φ ≡ φ (n) , defining the following Fourier multipliers

S k u(ξ) = φ k (ξ)û(ξ), k ∈ Z, ∆ k u(ξ) = ϕ k (ξ)û(ξ), k ≥ 0.
Denoting u k = ∆ k u we obtain a dyadic partition

(2.4) u = ∞ p=0 u p ,
where n shall be called the size of this partition. Remark that with the notations above, there hold

∆ 0 = S 0 , q p=0
∆ p = S q , S q+1 -S q = ∆ q+1 .

Throughout this article, whenever R d is equipped with a fixed dyadic partition, we always define the Zygmund-norm (see Definition A.1) of distributions on R d by means of this partition. To prove our paracomposition results we need to choose a particular size n = n 0 , tailored to the diffeomorphism, in Proposition 2.9 below, whose proof requires uniform bounds for the norms of the operators S j , ∆ j in Lebesgue spaces and Hölder spaces, with respect to the size n. This fact in turn stems from property (2.2) of φ (n) .

Lemma 2.3. 1. For every α ∈ N d , there exists C α > 0 independent of n such that ∀j, ∀1 ≤ p ≤ q ≤ ∞, ∂ α S j u L q (R d ) + ∂ α ∆ j u L q (R d ) ≤ C α 2 j(|α|+ d p -d q ) u L p (R d ) .
2. For every µ ∈ (0, ∞), there exists M > 0 independent of n such that

∀j ∈ N, ∀u ∈ W µ,∞ (R d ) : ∆ j u L ∞ (R d ) ≤ M 2 -jµ u W µ,∞ (R d ) .
As a consequence of this lemma, one can examine the proof of Proposition 4.1.16, [START_REF] Métivier | Para-differential calculus and applications to the Cauchy problem for nonlinear systems[END_REF] to have Lemma 2.4. Let µ > 0, µ / ∈ N. Then there exists a constant C µ independent of n, such that for any u ∈ W µ,∞ (R d ) we have

1 C µ u W µ,∞ (R d ) ≤ u C µ * ≤ C µ u W µ,∞ (R d ) .
Moreover, when µ ∈ N the second inequality still holds.

By virtue of Lemma 2.4, we shall identify

W µ,∞ (R d ) with C µ * (R d ) whenever µ > 0, µ /
∈ N, regardless of the size n. For very j ≥ 1, the reverse estimates for ∆ j in Lemma 2.3 1. hold (see Lemma 2.1, [8]) Lemma 2.5. Let α ∈ N d . Then there exists C α (n) > 0 such that for every 1 ≤ p ≤ ∞ and every j ≥ 1, we have

∆ j u L p (R d ) ≤ C α (n)2 -j|α| ∂ α ∆ j u L p (R d ) .
Applying the previous lemmas yields Lemma 2.6. 1. Let µ > 0. Then for every α ∈ N d there exists C α > 0 such that

(2.5) ∀v ∈ C µ * (R d ), ∀p ≥ 0, ∂ α (S p v) L ∞ ≤      C α 2 p(|α|-µ) ∂ α v C µ-|α| * , if |α| > µ C α ∂ α v L ∞ , if |α| < µ C α p v C µ * , if |α| = µ.
2. Let µ < 0. Then for every α ∈ N d there exists C α > 0 such that

(2.6) ∀v ∈ C µ * (R d ), ∀p ≥ 0, ∂ α (S p v) L ∞ ≤ C α 2 p(|α|-µ) v C µ * . 3. Let µ > 0. Then there exists C(n) > 0 such that for any v ∈ S ′ with ∇v ∈ C µ-1 * (R d ) we have (2.7) v -S p v L ∞ ≤ C(n)2 -pµ ∇v C µ-1 * .

On the para-differential operators

In this paragraph we clarify the choice of two cutoff functions χ and ψ appearing in the definition of paradifferential operators A.3 in accordance with the dyadic partitions above. Given a dyadic system of size n on R d , define

(2.8) χ(η, ξ) = ∞ p=0 φ p-N (η)ϕ p (ξ)
with N = N (n) ≫ n large enough. It is easy to check that the so defined χ satisfies (A.3) and (A.4). Plugging (2.8) into (A.2) gives

T a u(x) = ∞ p=0 e i(θ+η)x φ p-N (θ)â(θ, η)ϕ p (η)ψ(η)û(η)dηdθ = ∞ p=0 S p-N (a)(x, D)(ψϕ p )(D)u(x).
Notice that for any p ≥ 1 and η ∈ supp

ϕ p we have |η| ≥ 2 -n . Choosing ψ (depending on n) verifying ψ(η) = 1 if |η| ≥ 2 -n , ψ(η) = 0 if |η| ≤ 2 -n-1 gives (2.9) T a u(x) = ∞ p=1 S p-N a(x, D)∆ p u(x) + S -N (a)(ψϕ 0 )(D)u(x).
Defining the "truncated paradifferential operator" by

(2.10) Ṫa u = ∞ p=1 S p-N a∆ p u.
then the difference T a -Ṫa is a smoothing operator in the following sense: if for some

α ∈ N d , ∂ α u ∈ H -∞ then (T a -Ṫa )u ∈ H ∞ since ψϕ 0 is supported away from 0.
We thus can utilize the symbolic calculus Theorem A.5 for the truncated paradifferential operator Ṫa u when working on distributions u as above. The same remark applies to the paraproduct T P a defined in (A.11). In general, smoothing remainders can be ignored in applications. However, the be precise in constructing the abstract theory we decide to distinguish these objects.

Definition 2.7. For v, w ∈ S ′ define the truncated remainder

Ṙ(v, w) = Ṫv w -Ṫw v.
Comparing to the Bony's remainder R(v, w) defined in (A.12), Ṙ(v, w) satisfies

(2.11) Ṙ(v, w) = R(w, w) + N k=1 (S k-N v∆ k w + S k-N w∆ k v) .
Remark 2.8. The relation (2.11) shows that the estimates (A.13), (A.14), (A.15) are valid for Ṙ.

Choice of dyadic partitions

Let κ : R d 1 → R d 2 be a diffeomorphism satisfying ∃ρ > 0, ∂ x κ ∈ C ρ * (R d 1 ), ∃m 0 > 0, ∀x ∈ R d 1 , |det κ ′ (x)| ≥ m 0 .
We equip on R d 2 a dyadic partition (2.4) with n = 0 and on R d 1 the one with n = n 0 large enough as given the next proposition. Proposition 2.9. Let p, q, j ≥ 0. For ε 0 > 0 arbitrarily small, there exist F 1 , F 2 nonnegative such that with

n 0 = F 1 (m 0 , κ ′ L ∞ ) ∈ N, p 0 = F 2 (m 0 , κ ′ C ε 0 * ) ∈ N, and N 0 = 2(n 0 + 1), we have |S p κ ′ (y)η -ξ| ≥ 1, if either (ξ, η) ∈ C j (n) × C q (1), p ≥ 0, j ≥ q + N 0 + 1 or |ξ| ≤ 2 j+(n+1) , η ∈ C q (1), p ≥ p 0 , 0 ≤ j ≤ q -N 0 -1.
Proof. We consider 2 cases: (i) p ≥ 0, j ≥ q + N 0 + 1. Using Lemma 2.3 we get for some constant

M 1 = M 1 (d) |S p κ ′ (y)η -ξ| ≥ |ξ| -|S p κ ′ (y)η| ≥ 2 q+1 (2 j-q-1-(n+1) -M 1 κ ′ L ∞ ) ≥ 2 N0-(n+1) -M 1 κ ′ L ∞ ≥ 2 n+1 -M 1 κ ′ L ∞ . We choose n ≥ [log 2 (M 1 κ ′ L ∞ + 1)] to have |S p κ ′ (y)η -ξ| ≥ 1. (ii) j ≤ q -N 0 -1.
Note that for any ε 0 > 0, owing to the estimate (2.7), there is a constant

M 2 = M 2 (d, ε 0 ) such that |κ ′ -S p κ ′ | ≤ M 2 2 -pε0 κ ′ C ε 0 *
and consequently, for some increasing function F (2.12)

|det S p κ ′ | ≥ |det κ ′ | -M 2 2 -pε0 F ( κ ′ C ε 0 * ) ≥ m 0 2 if we choose (2.13) p ≥ p 0 := 1 ε 0 ln 2M 2 m 0 F ( κ ′ C ε 0 * ) + 1.
We then use the inverse formula with adjugate matrix

(S p κ ′ ) -1 = 1 det Spκ ′ adj(S p κ ′ ) when d ≥ 2 to get for all d ≥ 1, (S p κ ′ ) -1 ≤ 2 m 0 1 + C(d) κ ′ d-1 L ∞ := K.
It follows that

|S p κ ′ (y)η -ξ| ≥ 1 K |η| -|ξ| ≥ 2 j+n+1 ( 1 K 2 q-2-j-(n+1) 2 -1) ≥ 1 K 2 N0-1-(n+1) -1 ≥ 1 K 2 n -1. Choosing n ≥ [1 + ln K] + 1 lead to |S p κ ′ (y)η -ξ| ≥ 1.
The Proposition then follows with p 0 as in (2.13) and

n 0 = [log 2 (M 1 κ ′ L ∞ + 1)] + [1 + ln K] + 1.
3 Quantitative and global paracomposition results

Motivations

The semi-classical Strichartz estimate for solutions to (1.8) proved in [START_REF] Alazard | Strichartz estimates for water waves[END_REF] relies crucially on the fact that one can make a para-change of variable to convert the highest order term T γ to the simple Fourier multiplier |D x | 3 2 . This is achieved by using the theory of paracomposition of Alinhac [START_REF] Alinhac | Paracomposition et opérateurs paradifférentiels[END_REF]. Let us recall here the main features of this theory: Theorem 3.1. Let Ω 1 , Ω 2 be two open sets in R d and κ : Ω 1 → Ω 2 be a diffeomorphism of class C ρ+1 , ρ > 0. Then, there exists a linear operator κ * A : D ′ (Ω 2 ) → D ′ (Ω 1 ) having the following properties:

1. κ * A applies H s loc (Ω 2 ) to H s loc (Ω 1 ) for all s ∈ R. 2. Assume that κ ∈ H r+1 loc with r > d 2 . Let u ∈ H s loc (Ω 2 ) with s > 1 + d 2 .
Then we have

(3.1) κ * A u = u • κ -T u ′ •κ κ + R with R ∈ H r+1+ε loc (Ω 1 ), ε = min(s -1 -d 2 , r + 1 -d 2 ). 3. Let h ∈ Σ m τ . There exists h * ∈ Σ m ε with ε = min(τ, ρ) such that (3.2) κ * A T h u = T h * κ * A u + R u
where R applies H s loc (Ω 2 ) to H s-m+ε loc (Ω 2 ) for all s ∈ R. Moreover, the symbol h * can be computed explicitly as in the classical pseudo-differential calculus (see Theorem 3.6 below).

Let u ∈ E ′ (Ω 2 ), supp u = K, ψ ∈ C ∞ 0 (Ω 1 ), ψ = 1 near κ -1 (K).
The exact definition of κ *

A in [START_REF] Alinhac | Paracomposition et opérateurs paradifférentiels[END_REF] is given by

(3.3) κ * A u = ∞ p=0 p+N0 j=p-N0 ∆ j (ψ∆ p u • κ)
for some N 0 ∈ N and some dyadic partition 1 = ∆ j depending on κ, K. This local theory was applied successfully by Alinhac in studying the existence and interaction of simple waves for nonlinear PDEs. The equation we have in hand is (1.8). More generally, let us consider the paradifferential equation

(3.4) ∂ t u + N u + iT h u = f, (t, x) ∈ (0, T ) × R,
where u is the unknown, T h is a paradifferential operator of order m > 0 and N u is the lower order part. Assume further that h(x, ξ) = a(x)|ξ| m , a(x) > 0. We seek for a change of variables to convert T h to the Fourier multiplier |D x | m . Set

χ(x) = x 0 a -1 m (y) d y
and let κ be the inverse map of χ. Suppose that a global version of Theorem 3.1 were constructed then part 3. would yield

κ * A T h u = T h * κ * A u + R
u and the principle symbol of h * (as in the case of classical pseudo-differential calculus) would be indeed |ξ| m . However, to be rigorous we have to study the following points

Question 1: A global version of Theorem 1, that is, in all statements H s loc (R) is replaced by H s (R). Question 2: If the symbol h is elliptic: a(x) ≥ c > 0 then the regularity condition κ ∈ C ρ+1 (R) is violated for κ ′ (x) = 1 χ ′ (κ(x)) = a 1 m (κ(x)) ≥ c 1 m .
So, we need a result without any regularity assumption on κ but only on its derivatives; in other words, only on the high frequency part of κ.

Assume now that equation (3.4) is quasilinear: a(t, x) = F (u)(t, x). We then have to consider for each t, the diffeomorphism

χ t (x) = x 0 F (u) -m (t, y) d y
and this gives rise to the following problem Question 3: When one conjugates (3.4)

with κ * A it is requisite to compute (3.5) ∂ t (κ * A u) = κ * A (∂ t u) + R .
This would be complicated in view of the original definition (3.3). In [START_REF] Alazard | Strichartz estimates for water waves[END_REF] the authors overcame this by using Theorem 3.1 2. as a new definition of the paracomposition:

κ * u = u • κ -T u ′ •κ κ.
For this purpose, we need to make use of part 2. of Theorem 3.1 to estimate the remainder

k * A (T h u) -k * (T h u). This in turn requires T h u ∈ H s with s > 1 + d 2 or u ∈ H s with s > m + 1 + d 2 ,
which is not the case if one wishes to study the optimal Cauchy theory for (3.4) since we are alway 1-derivative above the "critical index" µ = m + d 2 . Question 4: If a linearization result as in part 2. of Theorem 3.1 for u ∈ H s (R) with s < 1 + d 2 holds?. Let's suppose that all the above questions can be answered properly. After conjugating (3.4) with κ * the equation satisfied by u * := κ * u reads

(3.6) ∂ t u * + M u * + |D x | m u * = κ * f + g
where g contains all the remainders in Theorem 3.1 2., 3. and in (3.5).

To prove Strichartz estimates for (3.6), we need to control g in L p t L q x norms, which in turns requires tame estimates for g. It is then crucial to have quantitative estimates for the remainders appearing in g and hence quantitative results for the paracomposition.

Statement of main results

Let κ : R d 1 → R d 2 be a diffeomorphism. We equip on R d 2 and R d 1 two dyadic partitions as in (2.4) with n = 0 and n = n 0 , respectively, where n 0 is given in Proposition 2.9. Notation 3.2. 1. For a fixed integer N sufficiently large (larger than N given in (2.8) and N 0 = 2(n 0 + 1) and to be chose appropriately in the proof of Theorem 3.6), we set for any v ∈ S ′ (R d 1 )

(3.7) thepiece[v] p = |j-p|≤ N ∆ j v.

For any positive real numer µ we set µ

-= µ if µ / ∈ N and µ -= µ -ε if s ∈ N with ε > 0 arbitrarily small so that µ -ε / ∈ N.
Henceforth, we always assume the following assumptions on κ:

Assumption I (3.8) ∃ρ > 0, ∂ x κ ∈ C ρ * (R d 1 ), ∃α ∈ N d , r > -1, ∂ α0 x κ ∈ H r+1-|α0| (R d 1 )
.

Assumption II

(3.9) ∃m 0 > 0, ∀x ∈ R d 1 , |det κ ′ (x)| ≥ m 0 . Definition 3.3. (Global paracomposition) For any u ∈ S ′ (R d 2 )
we define formally

κ * g u = ∞ p=0 [u p • κ] p .
We state now our precise results concerning the paracomposition operator κ * g . Theorem 3.4. (Operation) For every s ∈ R there exists F independent of κ such that

∀u ∈ C s * (R d 2 ), κ * g u C s * ≤ F (m 0 , κ ′ L ∞ ) u C s * , ∀u ∈ H s (R d 2 ), κ * g u H s ≤ F (m 0 , κ ′ L ∞ ) u H s . Theorem 3.5. (Linearization) Let s ∈ R. For all u ∈ S ′ (R d 2 ) we define (3.10) R line u = u • κ -κ * g u + Ṫu ′ •κ κ . (i) If 0 < σ < 1, ρ + σ > 1 and r + σ > 0 then there exists F independent of κ, u such that R line u H s ≤ F (m 0 , κ ′ C ρ * ) ∂ α0 x κ H r+1-|α 0 | 1 + u ′ H s-1 + u C σ * where s = min(s + ρ, r + σ). (ii) If σ > 1, set ε = min(σ -1, ρ + 1) -then there exists F independent of κ, u such that R line u H s ≤ F (m 0 , κ C ρ * ) ∂ α0 x κ H r+1-|α 0 | 1 + u ′ H s-1 + u C σ *
where s = min(s + ρ, r + 1 + ε).

Theorem 3.6. (Conjugation) Let m, s ∈ R and τ > 0. Set ε = min(τ, ρ). Then for every h(x, ξ) ∈ Γ m τ , homogeneous in ξ there exist • h * ∈ Σ m ε , • F nonnegative, independent of κ, h, • k 0 = k 0 (d, τ ) ∈ N such that we have for all u ∈ H s (R d 2 ), κ * g T h u = T h * κ * g u + R conj u, (3.11) R conj u H s-m+ε ≤ F (m 0 , κ ′ C ρ * )M m τ (h; k 0 ) (1 + ∂ α0 κ H +1-α 0 ) u H s . (3.12) (the semi-norm M m τ (h; k 0 ) is defined in (A.1)).
Moreover, h * is computed by the formula

(3.13) h * (x, ξ) = [ρ] j=0 h * j := [ρ] j=0 1 j! ∂ j ξ D j y h κ(x), R(x, y) -1 ξ |det ∂ y κ(y)| |det R(x, y)| | y=x , R(x, y) = t 1 0 ∂ x κ(tx + (1 -t)y) d t.
Remark 3.7.

• The definition (3.10) of R line involves Ṫu•κ ′ κ which does not require the regularity on the low frequency part of the diffeomorphism κ.

• Part (i) of Theorem 3.6 gives an estimate for the remainder of the linearization of κ * g u where u is allowed to be non C 1 . • In part (ii) of Theorem 3.6, the possible loss of arbitrarily small regularity in ε = min(σ -1, ρ + 1) -is imposed to avoid the technical issue in the composition of two functions in Zygmund spaces (see the proof of Lemma 3.10). On the other hand, there is no loss in part (i) when σ ∈ (0, 1).

• In the estimate (3.12), u is assumed to have Sobolev regularity. Therefore, in the conjugation formula (3.11) the paradifferential operators T h , T h * can be replaced by their truncated operators Ṫh , Ṫh * , modulo a remainder bounded by the right-hand side of (3.12).

Proof of the main results

Notation 3.8. To simplify notations, we denote throughout this section C µ = C µ * (R d ).

Technical lemmas

First, for every u ∈ S ′ (R d 2 ) we define formally

(3.14) R g u = κ * g u - p≥0 [u p • S p κ] p .
The remainder R g is ρ-regularized as to be shown in the following lemma.

Lemma 3.9. For every µ ∈ R there exists F independent of κ such that:

∀v ∈ H µ (R d 2 ), R g v H µ+ρ ≤ F (m 0 , κ ′ C ρ ) v ′ H µ-1 (1 + ∂ α0 κ H r+1-α 0 ) .
Proof. By definition, we have

R g v = - p≥0 [v p • S p κ -v p • κ] p = - p≥0 [A p ] p .
Each term A p can be written using Taylor's formula:

A p (x) = 1 0 v ′ p (κ(x) + t(S p κ(x) -κ(x))) d t(S p κ(x) -κ(x)). 1. Case 1: p ≥ p 0 . Setting y(x) = κ(x) + t(S p κ(x) -κ(x)), one has as in (2.12) |det(y ′ )| ≥ m0 2 , hence (3.15) 1 0 v ′ p (κ(x) + t(S p κ(x) -κ(x))) d t L 2 ≤ F (m 0 , κ ′ C ρ ) v ′ p L 2 .
Then by virtue of the estimate (2.7) we obtain (3.16)

∀p ≥ p 0 , A p L 2 ≤ 2 -p(ρ+1) 2 -p(µ-1) F (m 0 , κ ′ C ρ )e p = 2 -p(ρ+µ) F (m 0 , κ ′ C ρ )e p with ∞ p=p0 e 2 p ≤ v ′ 2 H µ-1 .
2. Case 2: 0 ≤ p < p 0 . We have by the Sobolev embedding

H d/2+1 ֒→ L ∞ 1 0 v ′ p (κ(x) + t(S p κ(x) -κ(x))) d t L ∞ ≤ v ′ p L ∞ ≤ 2 p( d 2 -s+2) v ′ H s-1 .
Applying Lemma 2.5 we may estimate with p≥0

f 2 p ≤ F (m 0 , κ ′ C ρ ) ∂ α0 κ 2 H r+1-|α 0 | κ -S p κ L 2 ≤ ∞ j=p+1 ∆ j κ L 2 ≤ ∞ j=p+1 2 -j|α0| ∆ j ∂ α0 κ L 2 ≤ ∞ j=p+1 2 -j|α0| 2 -j(r+1-|α0|) f p ≤ F (m 0 , κ ′ C ρ ) ∂ α0 κ H r+1-α 0 ,
where we have used the assumption that r + 1 > 0. Therefore, (3.17)

∀p < p 0 , A p L 2 ≤ F (m 0 , κ ′ C ρ ) v ′ H µ-1 ∂ α0 κ H r+1-α 0 .
3. Finally, noticing that the spectrum of [A p ] p is contained in an annulus {M -1 2 p ≤ |ξ| ≤ 2 p M } with M depending on n 0 , the lemma then follows from (3.16), (3.17). Lemma 3.10. Let µ > 0 and ε = min(µ, ρ + 1) -. For every v ∈ C µ (R d 2 ), set

r p := S p (v • κ) -(S p v) • (S p κ).
Then for every α ∈ N there exists a non-decreasing function F α independent of κ and v such that

∂ α x r p L ∞ ≤ 2 p(|α|-ε) F α ( κ ′ C ρ )(1 + v C µ ). Proof.
We first remark that by interpolation, it suffices to prove the estimate for α = 0 and all |α| large enough. By definition of ε we have v

• κ ∈ C ε with norm bounded by F ( κ ′ C ρ )(1 + v C µ ). 1. α = 0. One writes r p = (S p (v • κ) -v • κ) + (v -S p v) • κ + (S p v • κ -S p v • S p κ)
and use (2.7) to estimate the first two terms. For the last term, by Taylor's formula and (2.7) (consider µ > 1, = 1 or < 1) we have

S p v • κ -S p v • S p κ L ∞ ≤ S p v ′ L ∞ κ -S p κ L ∞ ≤ C2 -pε v C µ κ ′ C ρ . Therefore, r p L ∞ ≤ C2 -pε ( v • κ C ε + v C ε + v C µ κ ′ C ρ ) . 2. |α| > ρ + 1. The estimate (2.5) implies S p (v • κ) (α) L ∞ ≤ C α 2 p(|α|-ε) v • κ C ε .
On the other hand, part 2. of the proof of Lemma 2.1.1, [START_REF] Alinhac | Paracomposition et opérateurs paradifférentiels[END_REF] gives

(S p v • S p κ) (α) L ∞ ≤ C α 2 p(|α|-ε) (1 + κ ′ C ρ ) |α| v C µ .
Consequently, we get the desired estimate for all |α| > 1+ρ, which completes the proof.

Lemma 3.11. Let v ∈ C ∞ (R d 2 ), supp v ∈ C q (0). Recall that N 0 = 2(n 0 + 1)
with n 0 given by Proposition 2.9. (i) For p ≥ 0, j ≥ q + N 0 + 1 and k ∈ N there exists F k independent of κ, v such that

(v • S p κ) j L 2 (R d ) ≤ 2 -jk 2 p(k-ρ)+ v L 2 F k (m 0 , κ ′ C ρ ). (ii) For p ≥ p 0 , 0 ≤ ℓ ≤ ℓ ′ ≤ q -N 0 -1 and k ∈ N there exists F k independent of κ, v such that ℓ ′ j=ℓ (v • S p κ) j L 2 (R d ) ≤ 2 -qk 2 p(k-ρ)+ v L 2 F k (m 0 , κ ′ C ρ ). (iii) Set R p u = [u p • S p κ] p -(u p • S p κ).
For any p ≥ p 0 , there exists

F k independent of κ, u such that R p u L 2 ≤ 2 -pρ u p L 2 F k (m 0 , κ ′ C ρ ). Proof. First, it is clear that (iii) is a consequence of (i) and (ii) both applied with k > ρ.
The proof of (i) and (ii) follows mutadis mutandis that of Lemma 2.1.2, [START_REF] Alinhac | Paracomposition et opérateurs paradifférentiels[END_REF], using the technique of integration by parts with non-stationary phase. We only explain how to obtain the non-stationariness here. Let ϕ = 1 on C(0) and supp ϕ ⊂ C(1). The phase of the integral (with respect to y) appearing in the expression of (v•S p κ) j and

ℓ ′ j=ℓ (v•S p κ) j is S p κ(y)η -yξ
where,

• in case (i), (η, ξ) ∈ supp( ϕ(2 -q •) × supp ϕ(2 -j •), • in case (ii), (η, ξ) ∈ supp( ϕ(2 -q •) × supp φ(2 -ι •) with ι = ℓ or ℓ ′ + 1, which comes from the fact that ℓ j=ℓ ′ ϕ(2 -j ξ) = φ(2 -ℓ ξ) -φ(2 -ℓ ′ -1 ).
In both cases,

|∂ y (S p κ(y)η -yξ)| = |S p κ ′ (y)η -ξ| ≥ 1
by virtue of Proposition 2.9.

Proof of Theorem 3.4

By definition 3.3 of the global paracomposition κ

* g u = ∞ p=0 [u p •κ] p . Since each [u p •κ]
p is spectrally localized in a dyadic cell depending on n 0 = F (m, κ ′ L ∞ ), the theorem follows from Lemma 2.3 after making the change of variables y = κ(x).

Proof of Theorem 3.5

Using the dyadic partition 1 = p≥0 u p and the fact that S p → Id in S ′ we have in

D ′ (R d 1 ) u • κ = p≥0 u p • κ = p≥0 q≥0 (u p • S q+1 κ -u p • S q κ) + p≥0 u p • S 0 κ.
Denoting by S the first right-hand side term, one has by Fubini,

S = q≥0 0≤p≤q (u p • S q+1 χ -u p • S q κ) + q≥0 p≥q+1 (u p • S q+1 κ -u p • S q κ) =: (I) + (II).
For (I) we take the sum in p first and notice that S 0 = ∆ 0 to get

(I) = q≥0 (S q u • S q+1 κ -S q u • S q κ).
For (II) we write

(II) = p≥1 0≤q≤p-1 (u p • S q+1 κ -u p • S q κ) = p≥1 (u p • S p κ -u p • S 0 κ).
Summing up, we derive

(3.18) u • κ = p≥0 u p • S p κ + q≥0 (S q u • S q+1 κ -S q u • S q κ) =: A + B.
Thanks to lemma 3.9, there hold

(3.19) A = p≥0 u p • S p κ = κ * g u + R g u, with (3.20) R g u H s+ρ ≤ F (m 0 , κ ′ C ρ ) u ′ H s-1 (1 + ∂ α0 κ H r+1-α 0 ) . On the other hand, B = q≥0 B q with B q := S q u • S q+1 κ -S q u • S q κ = r q+1 κ q+1 + S q-N +1 (u ′ • κ)κ q+1 where r q+1 = 1 0 (S q u ′ )(tS q+1 κ + (1 -t)S q κ) d t -S q-N +1 (u ′ • κ).
By definition of truncated paradifferential operators

(3.21) q≥0 S q-N +1 (u ′ • κ)κ q+1 = p≥1 S p-N (u ′ • κ)κ p = Ṫu ′ •κ κ. Thus, it remains to estimate q≥0 r q+1 κ q+1 = q≥1 r q κ q . (i) Case 1: 0 < σ < 1, σ + ρ > 1 In this case, we see that u • κ ∈ C σ , hence (u • κ) ′ ∈ C σ-1 with norm bounded by F (m 0 , κ ′ C ρ ) u C σ . Then, using (A.22) with α = 1 -σ, β = ρ -yields u ′ • κ C σ-1 = (κ ′ ) -1 (u • κ) ′ C σ-1 ≤ F (m 0 , κ ′ C ρ ) (κ ′ ) -1 C ρ -(u • κ) ′ C σ-1 .
By writing

(κ ′ ) -1 = 1 det(κ ′ ) adj(κ ′ ) we get easily that (κ ′ ) -1 C ρ -≤ F (m 0 , κ ′ C ρ ) and hence u ′ • κ C σ-1 ≤ F (m 0 , κ ′ C ρ ) u C σ . Now, we claim that (3.22) ∀q ≥ 1, ∀α ∈ N d , ∂ α x r p L ∞ ≤ 2 q(|α|+1-σ) F α (m 0 , κ ′ C ρ ) u C σ .
Since σ -1 < 0 it follows from (2.6) that

∂ α x S q-N (u ′ • κ) L ∞ ≤ C α 2 q(|α|+1-σ) u ′ • κ C σ-1 ≤ 2 q(|α|+1-σ) F α (m 0 , κ ′ C ρ ) u C σ .
Thus, to obtain (3.22) it remains to prove

(3.23) ∀q ≥ 1, ∀α ∈ N d , ∂ α x (S q u ′ (S q κ)) L ∞ ≤ 2 q(|α|+1-σ) F α (m 0 , κ ′ C ρ ) u C σ .
By interpolation, this will follow from the corresponding estimates for α = 0 and |α| > 1 + ρ. Again, since σ -1 < 0 we have (3.23) for α = 0. Now, consider |α| > 1 + ρ. By the Faa-di-Bruno formula ((S q u ′ ) • (S q κ)) (α) is a finite sum of terms of the following form

A = (S q u ′ ) (m) t j=1 [(S q κ) (γj ) ] sj , where 1 ≤ |m| ≤ |α|, |γ j | ≥ 1, |s j | ≥ 1, t j=1 |s j |γ j = α, t j=1 s j = m.
By virtue of (2.5), one gets

(S q κ) (γj ) L ∞ = (S q κ ′ ) (γj -1) L ∞ ≤      C2 q(|γj |-1-ρ) κ ′ C ρ , if |γ j | -1 > ρ C κ ′ C ρ , if |γ j | -1 < ρ C η 2 qη κ ′ C ρ , ∀η > 0 if |γ j | -1 = ρ. ≤ C α 2 q(|γj |-1)(1-ρ |α|-1 ) κ ′ C ρ .
Consequently,

(3.24) t j=1 [(S q κ) (γj ) ] sj L ∞ ≤ C α 2 q(|α|-|m|)(1-ρ |α|-1 ) κ ′ |m| C ρ .
Combining (3.24) with the estimate (applying (2.5) since m + 1 > σ)

(S q u ′ ) (m) L ∞ ≤ C m 2 q(m+1-σ) u C σ yields ∂ α x (S q u ′ (S q κ)) L ∞ ≤ 2 qM F α ( κ ′ C ρ ) u C σ with M = (m + 1 -σ) + (|α| -m)(1 - ρ |α| -1 ) ≤ |α| + 1 -σ,
which concludes the proof the claim (3.23). On the other hand, according to Lemma 2.5 (ii) for any q ≥ 1, α ∈ N there holds

(3.25) ∂ α x κ q L 2 ≤ C α 2 q|α| κ q L 2 ≤ C α 2 q(|α|-|α0|) ∂ α0 κ q L 2 ≤ C α 2 q(|α|-|α0|) 2 -q(r+1-|α0|) a p = C α 2 q(|α|-r-1) a p , with q≥1 a 2 q ≤ F (m 0 , κ ′ C ρ ) ∂ α0 x κ 2 H r+1-α 0 .
We deduce from (3.23) and (3.25) that

∀α ∈ N d , ∀q ≥ 1, ∂ α x (r q κ q ) L 2 ≤ 2 q(|α|-r-σ) F α ( κ C ρ ) u C σ a q .
By the assumption r + σ > 0 we conclude

(3.26) q≥1 r q κ q H r+σ ≤ u C σ -F ( κ C ρ ) ∂ α0 x κ H r+1-α 0 .
Combining (3.19), (3.20), (3.21), (3.26) we obtain the assertion (i) of Theorem 3.5.

(ii) Case 2: σ > 1. This case was studied in [START_REF] Alinhac | Paracomposition et opérateurs paradifférentiels[END_REF]. One writes

S q-N (u ′ • κ) = t q + S q-1 u ′ • S q-1 κ + s q . with t q = S q-N (u ′ • κ) -S q-1 (u ′ • κ), s q = S q-1 (u ′ • κ) -S q-1 u ′ • S q-1 κ.
Plugging this into r q gives r q = z qt qs q with z q = 1 0

(S q-1 u ′ )(tS q κ + (1 -t)S q-1 κ) d t -S q-1 u ′ • S q-1 κ = κ q 1 0 t 1 0 ((S q-1 u ′ )) ′ (S q-1 κ + stκ q ) d s d t.
Now we estimate the L ∞ -norm of derivatives of r q . Since

t q = - q-1 j=q-N +1 (u ′ • κ) j we get with ε = min(σ -1, ρ + 1) - (3.27) ∀q ≥ 1, ∀α ∈ N, ∂ α x t q L ∞ ≤ 2 p(|α|-ε) F α ( κ ′ C ρ )(1 + u C σ ).
Applying Lemma 3.10 we have the same estimate as (3.27) for s q . Finally, following exactly part d) of the proof of Lemma 3.1, [START_REF] Alinhac | Paracomposition et opérateurs paradifférentiels[END_REF] with the use of Lemma 2.6 one obtains the same bounds for z q . We conclude by using (3.25) that

q≥1 r q κ q H r+1+ε ≤ (1 + u C σ ) ∂ α0 x κ H r+1-α 0 F ( κ ′ C ρ ),
which combines with (3.20) gives the assertion (ii) of Theorem 3.5.

Proof of Theorem 3.6

We recall first the following lemma in [START_REF] Alinhac | Paracomposition et opérateurs paradifférentiels[END_REF].

Lemma 3.12 ([6, Lemme d), page 111]). Let K ⊂ R d be a compact set. Let a(x, y, η) be a bounded function; C ∞ in η and its support w.r.t η is contained in K; its derivatives w.r.t η are bounded. For every p ∈ N, define the associated pseudo-differential operator

A p v(x) = e i(x-y)ξ a(x, y, 2 -pξ )v(y) d y d ξ.
Then, there exist a constant C > 0 independent of a, p and an integer

k 1 = k 1 (d) such that with M = sup |α|≤k1 ∂ α η a L ∞ (R d ×R d ×R d ) we have ∀v ∈ L 2 (R d ), A p v L 2 ≤ CM v L 2 .
Now we quantify the proof of Lemma 3.3 in [START_REF] Alinhac | Paracomposition et opérateurs paradifférentiels[END_REF]. Let m, s ∈ R, τ > 0, ε = min(τ, ρ) and h(x, ξ) ∈ Γ m τ , homogeneous in ξ. We say that a quantity Q is controllable if Q H s-m+ε is bounded by the right-hand side of (3.12) and therefore can be neglected. Also, by A ∼ B we mean that A -B is controllable.

Step 1. First, by Lemma 3.9 we have

(3.28) κ * g (T h u) ∼ p≥0 [∆ p T h u • S p κ] p .
Then with v q = (S q-N h)(x, D)u q , it holds that

κ * g (T h u) ∼ p≥0 q≥1 [∆ p v q • S p κ] p .
One can see easily that if N is chosen larger than n enough then the spectrum of v q is contained in the annulus

ξ ∈ R d : 2 p-M1 ≤ |ξ| ≤ 2 p+M1
with M 1 = M 1 (N, n 0 ). This implies

∆ p v q = 0 if |p -q| > M := M 1 + n 0 + 1 = M (N, n 0 )
and thus,

κ * g (T h u) = |p-q|≤M [∆ p v q • S p κ] p . Set S 1 = |p-q|≤M [∆ p v q • S p κ] p -[∆ p v q • S p κ] q , S 2 = |p-q|≤M [∆ p v q • S p κ] q -[∆ p v q • κ] q .
We shall prove that S 1 , S 2 are controllable so that

(3.29) κ * g (T h u) ∼ p,q≥0 [∆ p v q • κ] q = q≥0 [v q • κ] q = p≥0 [(S p-N h)∆ p u • κ] p .
The estimate for S 2 is proved along the same lines as in the proof of Lemma 3.9. We now consider S 1 . If we choose

N ≫ M + N 0 in the definition of [•] p then S 1 = p,q |p-q|≤M j N0<|j-p|≤ N ∆ j (∆ p v q •S p κ)- p,q |p-q|≤M j N0<|j-q|≤ N ∆ j (∆ p v q •S p κ) = S 1,1 -S 1,2 .
Each of S 1,1 and S 1,2 is treated in the same way. Let us consider S 1,1 = j a 1 j + j a 2 j ,

a 1 j = p p<j-N0 |p-j|≤ N q |q-p|≤M ∆ j (∆ p v q • S p κ), a 2 j = p p>j+N0 |p-j|≤ N q |q-p|≤M ∆ j (∆ p v q • S p κ).
Since for all q v q L 2 ≤ M m 0 (h) ∆ q u H m , by virtue of Lemma 3.11 (i) (applied with k > ρ) one has

a 1 j L 2 ≤ p<j-N0,|p-j|≤ N ,|q-p|≤M C k 2 -jk 2 p(k-ρ) M m 0 (h) ∆ q u H m F k (m 0 , κ ′ C ρ ) ≤ |q-j|≤M+ N C k 2 -jρ+mq M m 0 (h) ∆ q u L 2 F k (m 0 , κ ′ C ρ ) ≤ C k 2 -j(ρ-m+s) M m 0 (h)F k (m 0 , κ ′ C ρ ) |q-j|≤M+ N b q .
with b ℓ 2 ≤ C u H s . Then, thanks to the spectral localization of a 1 j we conclude that

j a 1 j H s-m+ρ ≤ M m 0 (h) u H s F k (m 0 , κ ′ C ρ ).
For the second sum a 2 j we apply Lemma 3.11 (ii).

Step 2. Recall from (3.29) that

κ * g T h u = p≥0 [A p ] p , A p = (S p-N h)(x, D)u p • κ.
One writes

A p (y) = e i(κ(y)-y ′ )ξ ϕ(2 -p ξ)(S p-N h)(κ(y), ξ)u p (y ′ ) d y ′ d ξ
where ϕ is a cutt-off function analogous to ϕ and equal to 1 on the support of ϕ.

In the expression of A p we make two changes of variables

y ′ = κ(z), ξ = t R -1 η, R = R(y, z) := 1 0 κ ′ (ty + (1 -t)z) d t to derive A p (y) = e i(y-z)η ϕ(2 -p . t R -1 )(S p-N h)(κ(y), t R -1 η)u p (κ(z)) |κ ′ (z)| |det R| d z d η.
The rest of the proof follows the same method as in the classical case classical pseudodifferential calculus except that we shall regularize first the symbol a p (y, z, η) of A p : set

b p (y, z, η) = ϕ(2 -p . t R -1 p )(S p-N h)(S p κ(y), t R -1 p η) |S p κ ′ (z)| |det R p | with R p = R p (y, z) = 1 0 S p κ ′ (ty + (1 -t)z) d t.
Thanks to the homogeneity of S p-N h we write a p (y, z, η) = 2 pm a p (y, z, 2 -p η) and similarly for b p . Then due to the presence of the cut-off function ϕ one can prove without any difficulty that

∀k ∈ N, sup |α|≤k ∂ α η ( a p -b p )(y, z, η) ≤ C k 2 -pρ F k (m 0 , κ ′ C ρ )M m 0 (h; k + 1).
Therefore, in view of Lemma 3.12 we see that in κ * g T h u the replacement of a p by b p gives rise to a controllable remainder.

Step 3. Next, we expand b p (y, z, η) by Taylor's formula w.r.t z up to order ℓ = [ρ], at z = y to have

b p (y, z, η) = b 0 p (y, η) + b 1 p (y, η)(z -y) + ... + b ℓ p (y, η)(z -y) ℓ + r ℓ+1 p (y, z, η)(z -y) ℓ+1
where b j is the j th -derivative of b p with respect to z, taken at z = y and

r ℓ+1 p (y, z, η) = C 1 0 b ℓ+1 p (y, y + t(z -y), η) d t(z -y) ℓ+1 .
In the pseudo-differential operator R ℓ+1 p with symbol r ℓ+1 p we integrate by parts w.r.t η ℓ + 1 times to obtain a sum of symbols of the form 2 p(m-ℓ-1) r p (y, z, 2 -p η),

r p (y, z, η) = C 1 0 ∂ α z ∂ β η b p (y, y + t(z -y), η) d t, |α| = |β| = ℓ + 1. For |α| = ℓ + 1, |γ| = ℓ + 1 + k, k ∈ N it holds ∂ α z ∂ γ η b p (y, z, η) ≤ C k 2 p(ℓ+1-ρ) M m 0 (h, [ρ] + 1 + k)F k (m 0 , κ ′ C ρ ).
Lemma 3.12 then gives for some

k 1 = k 1 (d) ∈ N, R ℓ+1 p L 2 ≤ 2 p(ℓ+1-ρ) 2 p(m-ℓ-1) u p L 2 M m 0 (h, [ρ] + 1 + k 1 )F k (m 0 , κ ′ C ρ ).
Therefore, the remainder p [R ℓ+1 p ] p is controllable.

Step 3. We write

B j p u(y) = e i(y-z)η b j p (y, η)(z -y) j u p (κ(z)) d z d η
and integrate by parts j times w.r.t η to get

B j p u = e i(y-z)η c j p (y, η)u p (κ(z)) d z d η.
The key point here is: in the expression above we shall replace u p • κ, p ≥ 0 by its "recoupe" [u p • κ] p which will enter T h * κ * g u. Therefore, one has to estimate the L 2 -norm of the difference

W p := u p • κ -[u p • κ] p as W p L 2 ≤ 2 -pρ F (m 0 , κ ′ C ρ ) u p L 2 .
For 0 ≤ p < p 0 . We treat separately each term in W p by making the change of variables x → κ(x) to have

W p L 2 ≤ F (m 0 , κ ′ C ρ ) u p L 2 .
For p ≥ p 0 we write

W p = (u p • κ -u p • S p κ) + (u p • S p κ -[u p • S p κ] p ) + ([u p • S p κ] p -[u p • κ] p ).
The second term is estimated using directly Lemma 3.11 (iii). The first and the last term are treated exactly as in the first part (case 1.) of the proof of Lemma 3.9 (see (3.16)). Again, by virtue of Lemmma 3.12 we conclude that: in κ g T h u the replacement of u p • κ by [u p • κ] p is (ρ + jm)-regularized and controllable.

Step 4. Set

C j p u(y) = e i(y-z)η c j p (y, η)[u p • κ] p (z) d z d η.
We observe that if the cut-off function ϕ is chosen appropriately then all the terms in c j p relating to ∂ α ϕ is 1 if α = 0 and is 0 if α = 0, on the spectrum of [u p • κ] p . Therefore, comparing to the classical calculus (3.13) for S p-N h we can prove that

sup |α|≤k 0<c1≤|η|≤c2 ∂ α η c j p -S p-N h * j (y, η) ≤ C k 2 -pεj M m τ (h; k + j + 1)F k (m 0 , κ ′ C ρ )
with ε j = min(τ, ρj).

Then, Lemma 3.12 implies that in κ * g T h u our replacement of C j p u by

D j u(y) = e i(y-z)η (S p-N h * j )(y, η)[u p • κ] p (z) d z d η
leaves a controllable remainder of order mjε j ≤ mε.

Step 5. We have proved in step 4 that

κ * g T h u ∼ [ρ] j=0 p D j p u p = [ρ] j=0 p (S p-N h * j )(x, D)[u p • κ] p p .
Now, notice that if in the definition of κ * g T h u in (3.28) we had chosen instead of [•] p a larger piece [•] ′ p corresponding to N ≫ N (remark that such a replacement is controllable according to Lemma 3.9 and Lemma 3.11) we would have obtained

κ * g T h u = [ρ] j=0 p (S p-N h * j )(x, D)[u p • κ] p ′ p = [ρ] j=0 p |k-p|≤N ∆ k S p-N h * j )(x, D)[u p • κ] p .
Remark that the spectrum of

(S p-N h * j )(x, D)[u p • κ] p is contained in the annulus ξ ∈ R d : 2 p-M ≤ |ξ| ≤ 2 p+M for some M = M ( N , N ) > 0. Therefore, if we choose N ≫ M ( N , N ) then ∆ k S p-N h * j )(x, D)[u p • κ] p = 0 if |k -p| > N and hence (3.30) κ * g T h u = [ρ] j=0 p (S p-N h * j )(x, D)[u p • κ] p .
Finally, we write for 0 ≤ j ≤ [ρ]

T h * j κ * u = p (S p-N h * j )(x, D)∆ p q [u q • κ] q = p (S p-N h * j )(x, D)∆ p q k |k-q|≤ N ∆ k (u q • κ) = p k,q:|k-p|≤N0,|k-q|≤ N (S p-N h * j )(x, D)∆ p ∆ k (u q • κ).
In the sum above, the replacement of (S p-N h * j )(x, D) by (S q-N h * j )(x, D) leaves a controllable remainder, so

T h * j κ * u = k |p-k|≤N0 |q-k|≤ N (S q-N h * j )(x, D)∆ p ∆ k (u q • κ) = k,q |q-k|≤ N (S q-N h * j )(x, D)∆ k (u q • κ) = q (S q-N h * j )(x, D)[u q • κ] q .
Therefore, we conclude in view of (3.30

) that κ * g T h u ∼ [ρ] j=0 T h * j κ * u.
4 The semi-classical Strichartz estimate

Para-change of variable

First of all, let us recall the symmetrization of (1.2) to a paradifferential equation proved in [START_REF] Nguyen | A paradifferential reduction for the gravity-capillary waves system at low regularity and applications[END_REF] for rough solutions. This symmetrization requires the introduction of the following symbols:

• γ = 1 + (∂ x η) 2 -3 4 |ξ| 3 2 , • ω = -i 2 ∂ x ∂ ξ γ, • q = 1 + (∂ x η) 2 -1 2 , • p = 1 + (∂ x η) 2 -5 4 |ξ| 1 2 + p (-1 2 ) , where p (-1 2 ) = F (∂ x η, ξ)∂ 2 x η, F ∈ C ∞ (R × R \ {0}; C) is homogeneous of order -1/2 in ξ.
   (η, ψ) ∈ C 0 ([0, T ]; H s+ 1 2 (R) × H s (R)) ∩ L 4 ([0, T ]; W r+ 1 2 ,∞ (R) × W r,∞ (R)), s > r > 3 2 + 1 2 . Define U := ψ -T B η, Φ = T p η + T q U, then Φ solves the problem (4.2) ∂ t Φ + T V ∂ x Φ + iT γ Φ = f
and there exists a function F : R + × R + → R + , non-decreasing in each argument, independent of (η, ψ) such that for a.e. t ∈ [0, T ],

(4.3) f (t) H s ≤ F η(t) H s+ 1 2 , ψ(t) H s 1 + η(t) W r+ 1 2 ,∞ + ψ(t) W r,∞ .
We assume throughout this section that (η, ψ) is a solution to (1.2) with regularity (4.1).

We shall apply our results on the paracomposition in the preceding section to reduce further equation (4.2) by adapting the method in [START_REF] Alazard | Strichartz estimates for water waves[END_REF]. Define for every

(t, x) ∈ [0, T ] × R χ(t, x) = x 0 1 + (∂ y η(t, y)) 2 d y
then for each t ∈ I := [0, T ], the mapping x → χ(t, x) is a diffeomorphism from R to itself. Introduce then for each t ∈ I the inverse κ(t) of χ(t).

Concerning the underlying dyadic partitions, we shall write

η(t), ψ(t) : R 2 → R 1 , κ(t) : R 1 → R 2 ,
where, R 2 is equipped with the dyadic partition (2.4) of size n = 0 and R 1 is equipped with the one of size n = n 0 determined in Proposition 2.9:

n 0 = F 1 (m 0 , κ ′ L ∞ ). Since κ ′ (x) = 1 (∂ x χ) • κ = 1 1 + (∂ x η) • κ(x)) 2 ,
we get (4.4)

m 0 := 1 + ∂ x η 2 L ∞ t L ∞ x -1/2 ≤ κ ′ (x) ≤ 1, ∀x ∈ R.
Therefore, up to a constant of the form

F ( ∂ x η L ∞ t L ∞
x ) we will not distinguish between R 1 and R 2 in the rest of this article.

As mentioned in the introduction of our paracomposition results, we shall consider the linearized part of κ * g as a new definition for paracomposition. More precisely, we set

(4.5) u = κ * Φ := Φ • κ -Ṫ(∂xΦ)•κ κ,
where, for any function g : I × R 2 → C we have denoted

(g • κ)(t, x) = g(t, κ(t, x)), ∀(t, x) ∈ I × R 1 .
Let us first gather various estimates that will be used frequently in the sequel. To be concise, we denote

N = F ( η L ∞ t H s+ 1 2 x , ψ L ∞ t H s x )
where F is non-decreasing in each argument, independent of η, ψ and F may change from line to line.

Lemma 4.2. The following estimates hold 1.

Φ L ∞ t H s x ≤ N , 2. Φ(t) C r * ,x ≤ N 1 + η(t) W r+ 1 2 ,∞ + ψ(t) W r,∞ , 3. ∂ x χ -1 L ∞ t H s-1 2 x ≤ N , 4. ∂ t χ L ∞ t,x ≤ N , 5. ∂ t κ L ∞ t,x ≤ N , 6. ∂ x κ L ∞ t W (s-1) -,∞ x ≤ N , 7. ∂ x κ -1 L ∞ t H s-1 2 x ≤ N , 8. ∂ t ∂ x χ(t) L ∞ x ≤ N 1 + ψ(t) C r *
Proof. The estimates 1., 2., 3. can be deduced straightforwardly from the definition of Φ and the regularity of (η, ψ) given in ( ≤ N . On the other hand, using the first equation in (1.2) and the fact that s > 2 we get

∂ x ∂ t η(t) L 2 x ≤ G(η)ψ(t) H 1 x ≤ G(η)ψ(t) H s-1 x ≤ N .
5. This follows from 4. by using the formula ∂ t κ = -∂tχ ∂xχ • κ and noticing that

∂ x χ ≥ 1. 6. With F (z) = 1 √ 1+z 2 -1 and G := F • (∂ x η) we have (4.7) ∂ x κ = 1 (∂ x χ) • κ = 1 + F • (∂ x η) • κ = 1 + G • κ From 3. and Sobolev's embedding, ∂ x η ∈ L ∞ t C s-1 * ⊂ L ∞ t W (s-1)-,∞ x
. This together with the fact that

F ∈ C ∞ b (R) implies G ∈ L ∞ t W (s-1)-,∞ x and (4.8) G L ∞ t W (s-1) -,∞ x ≤ N .
Then, bootstrap the recurrence relation (4.7) we deduce that

∂ x κ ∈ L ∞ t W [(s-1)-],∞ x and (4.9) ∂ x κ L ∞ t W [(s-1) -],∞ x ≤ N .
Now, set µ = (s -1) --[(s -1) -] ∈ (0, 1). Again, by (4.7)

(4.10) ∂ [(s-1)-] x (∂ x κ) = ∂ [(s-1)-] x (G • κ)
is a finite combination of terms of the form

(4.11) A = [(∂ q G) • κ] m j=1 ∂ γj x κ, 1 ≤ q ≤ [(s -1) -], γ j ≥ 1, m j=1 γ j = [(s -1) -].
Using (4.9) and (4.8) it follows easily that A belongs to W µ,∞ (R d ) with norm bounded by N and thus 6. is proved. 7. First, the nonlinear estimate (A.23) implies that G = F • ∂ x η defined in the proof of 6. satisfies

(4.12) G L ∞ t H s-1 2 x ≤ N .
Then changing the variable x → χ(x) in (4.7) gives

∂ x κ -1 L ∞ t L 2 x ≤ G L ∞ t L 2 x χ ′ 1 2 L ∞
t,x ≤ N . Now using (4.7), (4.9) and induction we get (4.13)

∂ x κ -1 L ∞ t H [(s-1) -] x ≤ N . Next, set µ = (s -1 2 ) -[(s -1) -] ∈ [ 1 2 , 1 2 + ε], ε arbitrarily small (so that µ ∈ [ 1 2 , 1
)). To obtain 7. we are left with the estimate for ∂

[(s-1)-] x ∂ x κ in H µ -norm. This amounts to estimating (4.14) R 2 |∂ [(s-1)-] x (G • κ)(x) -∂ [(s-1)-] x (G • κ)(y)| 2 |x -y| 1+2µ d x d y
where ∂

[(s-1)-] x (G • κ
) is a finite linear combination of terms of the form A in (4.11). Inserting A into (4.14) one estimates successively the difference of each factor in A under the double integral while the others are estimated in L ∞ -norm. This is done using (4.12), (4.13) for Sobolev-norm estimates and (4.8), (4.9) for Hölder-norm estimates.

8. By definition of χ, it holds with

F 0 (z) = z √ 1+z 2 ∂ t ∂ x χ(t, x) = F 0 (∂ x η)∂ x ∂ t η = F 0 (∂ x η)∂ x G(η)ψ.
Then, applying the Holder estimate for the Dirichlet-Neumann operator in Proposition 2.10, [START_REF] Nguyen | A paradifferential reduction for the gravity-capillary waves system at low regularity and applications[END_REF] we get

∂ x G(η)ψ L ∞ ≤ ∂ x G(η)ψ C r-2 * ≤ N 1 + ψ(t) C r *
and hence the result.

The main task here is to apply Theorem 3.5 and Theorem 3.6 to convert the highest order paradifferential operator T γ to the Fourier multiplier

|D x | 3 2 .
Proposition 4.3. The function u defined by (4.5) satisfies the equation

(4.15) ∂ t + T W ∂ x + i|D x | 3 2 u = f where (4.16) W = (V • κ)(∂ x χ • κ) + ∂ t χ • κ
and for a.e. t ∈ [0, T ],

(4.17)

f (t) H s-1 2 ≤ F ( η L ∞ t H s+ 1 2 x , ψ L ∞ t H s x ) 1 + η(t) W r+ 1 2 ,∞ + ψ(t) W r,∞ .
Proof. We proceed in 4 steps. We shall say that A is controllable if for a.e. t ∈ [0, T ],

A(t) H s-1 2 is bounded by the right-hand side of (4.17) denoted by RHS.

Step 1. Let us first prove that for some controllable remainder R 1 ,

(4.18) κ * (∂ t Φ) = ∂ t + T (∂tχ)•κ ∂ x u + R 1 .
By definition of κ * we have

κ * (∂ t Φ) = ∂ t Φ • κ -Ṫ(∂x∂tΦ)•κ κ = ∂ t (Φ • κ) -(∂ x Φ • k)∂ t κ -Ṫ(∂x∂tΦ)•κ κ. Therefore, (4.19) κ * (∂ t Φ) = ∂ t (κ * Φ) + A 1 + A 2 A 1 = Ṫ(∂ 2 x Φ•κ)∂tκ κ, A 2 = Ṫ(∂xΦ)•κ ∂ t κ -(∂ x Φ • κ)∂ t κ. 1. Since the truncated paradifferential operator Ṫ(∂ 2
x Φ•κ)∂tκ κ involves only the high frequency part of κ we have (4.20)

A 1

H s+ 1 2 x ≤ N (∂ 2 x Φ • κ)∂ t κ L ∞ x ∂ 2 x κ H s-3 2 .
From Lemma 4.2 2., 5. there holds

(∂ 2 x Φ • κ)∂ t κ L ∞ x ≤ N 1 + η(t) W r+ 1 2 ,∞ + ψ(t) W r,∞ .
On the other hand, Lemma 4.2 7. gives

∂ 2 x κ H s-3 2 ≤ N , hence A 1 is controllable. 2. To study A 2 , one uses ∂ t κ = -ab with a = (∂ t χ) • κ, b = ∂ x κ. Set c = (∂ x Φ) • κ then ∂ x (κ * Φ) = bc -Ṫc b -Ṫ∂xc κ, hence A 2 = -Ṫc (ab) + abc = Ṫab c + Ṙ(c, ab) = Ṫa Ṫb c + R 2 + Ṙ(c, ab) = Ṫa (bc -Ṫc b) -Ṫa Ṙ(b, c) + R 2 + Ṙ(c, ab) = Ṫa (∂ x (κ * Φ)) + Ṫa Ṫ∂xc κ -Ṫa Ṙ(b, c) + R 2 + Ṙ(c, ab)
where

R 2 = Ṫab c -Ṫa Ṫb c. (i) The symbolic calculus Theorem A.5 implies for a.e. t ∈ [0, T ] R 2 (t) H s ≤ K a(t) W 1,∞ b(t) L ∞ + a(t) L ∞ b(t) W 1,∞ c(t) H s-1 .
Now, from Lemma 4.2 6. and the fact that s -

1 > 1 one gets b L ∞ t W 1,∞
x ≤ N . On the other hand, Lemma 4.2 4., 8. give, respectively

a(t) L ∞ ≤ N , a(t) W 1,∞ ≤ RHS.
Applying Lemma 3.2 in [START_REF] Alazard | On the water waves equations with surface tension[END_REF] and Lemma 4.2 1., 6. yield

(4.21) c(t) L ∞ t H s-1 x ≤ N .
Therefore, R 2 (t) H s is controllable.

(ii) In views of Lemma 4.2 2., 4., 7. the term Ṫa Ṫ∂xc κ can be estimated by

Ṫa Ṫ∂xc κ (t) H s ≤ N a(t) L ∞ ∂ x c(t) L ∞ ∂ 2 x κ(t) H s-2 ≤ RHS.
(iii) The estimate 7. in Lemma 4.2 and Sobolev's embedding imply that b L ∞ t C s-1 * ≤ N . Then according to (A.14) and the fact that s > 2 we obtain Ṫa Ṙ(b, c)(t)

H s ≤ N a(t) L ∞ b(t) C s-1 * c(t) H s-1 N .
By the same argument, to estimate Ṙ(ab, c)(t) H s it remains to bound (ab)(t) C 1 * which is in turn bounded by (ab)(t) W 1,∞ . From Lemma 4.2 1. and 4. we have

a(t) L ∞ + b(t) L ∞ ≤ N .
On the other hand, the estimate 6. (or 7.) of that lemma gives

∂ x b L ∞ ≤ N . Finally, we write ∂ x a = [(∂ t ∂ x χ) • κ]∂ x κ
κ * T h Φ = T h * κ * Φ -R line (T h Φ) + T h * R line Φ + R conj Φ.
It follows from Lemma 4.2 7. that

∂ x κ -1 L ∞ t C s-1 * ≤ ∂ x κ -1 L ∞ t H s-1 2 x ≤ N .
Therefore, κ satisfies condition (3.8) with

(4.23) ρ = 1, r 1 = s - 1 2 , α 0 = 2
where we have changed the notation in (3.8): ∂ α0 x κ ∈ H r1+1-|α0| to avoid the r used in (4.1) for the Hölder regularity of ψ. On the other hand, we have seen from (4.4) that κ ′ ≥ m 0 and thus the Assumptions I, II on κ are fulfilled. For the transport term, the symbol is h(x, ξ) = iξV (x).

(i) Now one can apply Theorem 3.6 with τ = ρ = 1 (hence ε = min(τ, ρ) = 1 ) to have

h * (x, ξ) = iV • κ(x) ξ κ ′ (x) = i(V • κ)(∂ x χ • κ)ξ and at a.e. t ∈ [0, T ] R conj Φ H s ≤ F (m 0 , κ ′ C ρ * )M 1 1 (h; k 0 ) 1 + ∂ 2 κ H s-3 2 Φ H s .
On the right-hand side, we estimate

κ ′ C ρ * + u H s + ∂ 2 κ H s-3 2 ≤ N , M 1 1 (h; k 0 ) ≤ RHS hence, R conj Φ(t) H s ≤ RHS.
(ii) The term T h * R line Φ is bounded as

T h * R line Φ(t) H s ≤ M 1 0 (h * ) R line Φ(t) H s+1
where M 1 0 (h * ) ≤ N . Applying Theorem 3.5 (ii) with

Φ(t) ∈ C 2 * , σ = r, ε = min(σ - 1, 1 + ρ) -≥ 1 we have s = min(s + ρ, r 1 + 1 + ε) = min(s + 1, s - 1 2 + 1 + ε) = s + 1, R line Φ(t) H s+1 ≤ F (m 0 , κ ′ C ρ * ) ∂ 2 x κ H s-3 2 1 + Φ ′ (t) H s-1 + Φ(t) C σ * ≤ RHS.
(In the last inequality, we have used Lemma 4.2 1., 2.) Therefore (4.24)

T h * R line Φ(t) H s ≤ RHS.
In (4.22) we are left with the estimate for R line (T h Φ). Notice that since

M 1 0 (h) ≤ N , with v = T h Φ one has v(t) H s-1 ≤ N , v(t) C r-1 * ≤ RHS.
Then, by virtue of Theorem 3.5 (ii) applied to v and σ = r -1, ε = min(r -2, 2) -we have

s = min(s + 1, s - 1 2 + 1 + ε) > s + 1 2 , R line v H s+ 1 2 ≤ RHS.
Summing up, we conclude from (4.22) that

κ * T h Φ = T h * κ * Φ + R 2 , R 2 (t) H s ≤ RHS.
Step 3. We now conjugate the highest order term T γ Φ with κ * . This is the point where we really need Theorem 3.5 (i) for non-C 1 functions. Recall the formula (4.22) and the verifications of Assumptions I, II given by (4.23) and (4.4). With

c 0 = (1 + (∂ x η)) -1/2 , we have that γ = c 0 |ξ| 3/2 satisfies M 3 2
1 (γ) ≤ N . Theorem 3.6 applied with m = 3/2, τ = 1 then yields

h * (x, ξ) = h(κ(x), ξ κ ′ (x) ) = (c 0 • κ)(x) |ξ| 3 2 κ ′ (x) = |ξ| 3 2 for 1/κ ′ (x) = (χ ′ • κ)(x) = (c 0 • κ)(x); and (at a.e. t ∈ [0, T ]) R conj Φ H s-3 2 +1 ≤ F (m 0 , κ ′ C ρ * )M 3 2 1 (h; k 0 ) 1 + ∂ 2 κ H s-3 2 Φ H s ≤ N .
The term T h * R line Φ(t) is estimated exactly as in (4.24) noticing that h * now is of order 3/2 we get

T h * R line Φ(t) H s-1 2 ≤ RHS. Consider the remaining term R line T h Φ(t). Since T h Φ(t) belongs to C r-3 2 *
and r - 3 2 can be smaller than 1, we have to use in this case Theorem 3.5 (i):

σ = 1 2 , ρ + σ = 3 2 > 1, s = min((s - 3 2 ) + 1, (s - 1 2 )
+ 1 2 ) = s - 1 2 , R line T h Φ(t) H s-1 2 ≤ F (m 0 , κ ′ C ρ * ) ∂ 2 x κ H s-3 2 1 + T h Φ(t) H s-3 2 + T h Φ(t) C σ * .
We conclude in this step that

κ * T γ Φ = |D x | 3 2 κ * Φ + R 3 , R 3 (t) H s-1 2 ≤ RHS.
Step 4. Since ω ∈ Γ

1 2
0 with the semi-norms bounded by N , one gets by virtue of Theorem 3.4 and Theorem 3.5 (ii)

κ * T ω Φ(t) H s-1 2 ≤ N . Similarly, f (t) ∈ H s ֒→ C s-1 2 * with s -1 2 > 3 2 we also have κ * f (t) H s-1 2 ≤ RHS.
Putting together the results in the previous steps, we conclude the proof of Proposition 4.3.

Remark 4.4. In fact, in the above proof, we have proved that

κ * (∂ t + T V ∂ x )Φ(t) = (∂ t + T W ∂ x )κ * Φ(t) + f 1 (t) with f 1 (t) H s ≤ N 1 + η(t) W r+ 1 2 ,∞ + ψ(t) W r,∞ .
We loose 1 2 derivative only in Step 3 and Step 4 when conjugating κ * with T γ Φ and also T ω , where in Step 3 we applied Theorem 3.6 with ρ = 1, τ = 3 2 and thus ε = 1. The reason is that we want to keep the right-hand side of (4.17) to be tame. On the other hand, if we apply the mentioned theorem with ρ = 3 2 then it follows that

κ * T γ Φ = |D x | 3 2 κ * Φ + R 3 with R 3 (t) H s ≤ F ( η L ∞ t H s+ 1 2 x , ψ L ∞ t H s x )F 1 1 + η(t) W r+ 1 2 ,∞ + ψ(t) W r,∞ .
If we assume more regularity: s > 2 + 1 2 then by Sobolev's embedding R 3 (t) H s ≤ N and we see again the result proved in [START_REF] Alazard | On the water waves equations with surface tension[END_REF] (cf. Proposition 3.3) (after performing in addition another change of variable to suppress the 1 2 order terms). In the next paragraphs, we shall prove Strichartz estimates for u solution to (4.15). To have an independent result, let us restate the problem as follows. Let I = [0, T ], s 0 ∈ R and

(4.25) W ∈ L ∞ ([0, T ]; L ∞ (R)) ∩ L 4 ([0, T ]; W 1,∞ (R)), f ∈ L 4 (I; H s0-1 2 (R)). If u ∈ L ∞ (I, H s0 (R)) is a solution to the problem (4.26) ∂ t + T W ∂ x + i|D x | 3 2 u = f
we shall derive the semi-classical Strichartz estimate for u (with a gain of 1 4 -ε derivatives). Remark that the same problem was considered in [START_REF] Alazard | Strichartz estimates for water waves[END_REF] at the following regularity level

W ∈ L ∞ ([0, T ]; H s-1 (R)), f ∈ L ∞ (I; H s (R)), s > 2 + 1 2 .
We shall in fact examine the proof in [START_REF] Alazard | Strichartz estimates for water waves[END_REF] to show that our regularity (4.25) is sufficient. It turns out that for the semi-classical Strichartz estimate, the loss of 1 2 derivatives in the source term f is optimal. Remark also that u is defined on R equipped with a dyadic partition of size n 0 . Then as remarked before, up to a constant of the form

F ( ∂ x η L ∞ t L ∞ x
), which will appear in our final Strichartz estimate, we shall work as if n 0 = 0.

Frequency localization

To prove Strichartz estimates for equation (4.26), we will adapt the proof of Theorem 1.1 in [START_REF] Alazard | Strichartz estimates for water waves[END_REF]: microlocalize the solution using Littlewood-Paley theory and establish dispersive estimates for those dyadic pieces. The first step consists in conjugating (4.26) with the dyadic operator ∆ j to get the equation satisfied by ∆ j u: (4.27)

∂ t + 1 2 (T W ∂ x +∂ x T W )+i|D x | 3 2 ∆ j u = ∆ j f + 1 2 ∆ j (T ∂xW u)+ 1 2 [T W , ∆ j ]∂ x u+∂ x [T W , ∆ j ]u .
After localizing u at frequency 2 j one can replace the paradifferential operator T W by the paraproduct with S j-N (W ) as follows Lemma 4.5 ([4, Lemma 4.9]). For all j ≥ 1 and for some integer N , we have

T W ∂ x ∆ j u = S j-N (W )∂ x ∆ j u + R j u ∂ x T W ∆ j u = = ∂ x S j-N (W )∆ j u + R ′ j u
where R j u, R ′ j u have spectrum contained in an annulus {c 1 2 j ≤ |ξ| ≤ c 2 2 j } and satisfies the following estimate for all s 0 ∈ R:

R j u H s 0 (R) + R ′ j u H s 0 (R d ) ≤ C(s 0 ) W W 1,∞ (R d ) u H s 0 (R d ) .
From now on, we always consider the high frequency part of u, that is ∆ j u with j ≥ 1. Combining (4.27) and Lemma 4.5 leads to (4.28)

∂ t + 1 2 (S j-N (W )∂ x + ∂ x S j-N (W )) + i|D x | 3 2 ∆ j u = ∆ j f + 1 2 ∆ j (T ∂xW u) + 1 2 [T W , ∆ j ]∂ x u + ∂ x [T W , ∆ j ] u + R j u + R ′ j u.
Next, as in [START_REF] Bahouri | Équations d'ondes quasilinéaires et estimations de Strichartz[END_REF], [START_REF] Tataru | Strichartz estimates for operators with nonsmooth coefficients and the nonlinear wave equation[END_REF], [START_REF] Alazard | Strichartz estimate and the Cauchy problem for the gravity water waves equations[END_REF] we smooth out the symbols (see for instance Lemma 4.4, [4]) Definition 4.6. Let δ > 0 and U ∈ S ′ (R). For any j ∈ Z, j ≥ -1 we define

S δj (U ) = ψ(2 -δj D x )U. Let χ 0 ∈ C ∞ 0 (R), supp χ ⊂ { 1 4 ≤ |ξ| ≤ 4}, ξ = 1 in { 1 2 ≤ |ξ| ≤ 2}. Define (4.29) a(ξ) = χ 0 (ξ)|ξ| 3 2 , h = 2 -j , L δ = ∂ t + 1 2 (S (j-N )δ (W ) • ∂ x + ∂ x • S δ(j-N ) (W )) + iχ 0 (hξ)|D x | 3 2 .
Using (4.28), we have

(4.30) L δ ∆ j u = F j , where (4.31)
F j = ∆ j f + 1 2 ∆ j (T ∂xW u) + 1 2 [T W , ∆ j ]∂ x u + ∂ x [T W , ∆ j ]u + R j u + R ′ j u+ 1 2 S (j-N )δ (W ) -S (j-N ) (W ) ∂ x ∆ j u + ∂ x S (j-N )δ (W ) -S (j-N ) (W ) ∆ j u .

Semi-classical parametrix and dispersive estimate

Recall that ϕ is the cut-off function employed to defined the dyadic partition of size n = 0 in paragraph 2.1. To simplify the presentation, let us rescale the existence time to T = 1 and set h = 2 -j , j ≥ 1,

E 0 = L ∞ ([0, T ]; L ∞ (R)), E 1 = L 4 ([0, T ]; W 1,∞ (R).
The main result of this paragraph is the following semi-classical dispersive estimate for the operator L δ .

Theorem 4.7. Let δ < 1 2 and t 0 ∈ R. For any u 0 ∈ L 1 (R d ) set u 0,h = ϕ(hD x )u 0 . Denote by S(t, t 0 )u 0,h solution of the problem

L δ u h (t, x) = 0, u h (t 0 , x) = u 0,h (x).
Then there exists F : R + → R + such that

(4.32) S(t, t 0 )u 0,h L ∞ (R d ) ≤ F ( W E0 ) h -1 4 |t -t 0 | -1 2 u 0,h L 1 (R d ) for all 0 < |t -t 0 | ≤ h 1 2
and 0 < h ≤ 1. We make the change of temporal variables t = h

L δ = h∂ σ + h 1 2 W h (h∂ x ) + 1 2 h∂ x W h + ia(hD x ).
For this new differential operator, we shall prove the the corresponding (classical) dispersive estimate:

Theorem 4.8. Let δ < 1 2 and σ 0 ∈ [0, 1]. For any u 0 ∈ L 1 (R d ) and u 0,h = ϕ(hD x )u 0 . Denote by S(σ, σ 0 )u 0,h solution of the problem

L δ U h (σ, x) = 0, U h (σ 0 , x) = u 0,h (x).
Then there exists F : R + → R + such that

(4.35) S(σ, σ 0 )u 0,h L ∞ (R d ) ≤ F ( W E0 ) h -1 2 |σ -σ 0 | -1 2 u 0,h L 1 (R d )
for all σ ∈ [0, 1].

Theorem 4.8 will imply Theorem 4.7. Indeed, the relation

L δ u h (σ, x) = h 3 2 L δ u h (σh 1 2 , x), yields S(σ, σ 0 )u 0,h (x) = S(h 1 2 σ, h 1 2 σ 0 )u 0,h (x).
If Theorem 4.8 were proved then via the relation t = σh

S(t, t 0 )u 0,h L ∞ x = S(σ, σ 0 )u 0,h L ∞ x ≤ F ( W E0 ) h -1 2 |σ -σ 0 | -1 2 u 0,h L 1 (R d ) ≤ F ( W E0 ) h -1 4 |t -t 0 | -1 2 u 0,h L 1 (R d )
which proves Theorem 4.7.

To prove Theorem 4.8, we use the WKB method to construct a parametrix of the following integral form

(σ, x, ξ, h) = b(σ, x, ξ, h)ζ(x -z -σa ′ (ζ)) with ζ ∈ C ∞ 0 (R), ζ(s) = 1 if |s| ≤ 1 and ζ(s) = 0 if |s| ≥ 2.
We shall work with the following class of symbols. Definition 4.9. For small h 0 to be fixed, we set

O = (σ, x, ξ, h) ∈ R 4 : h ∈ (0, h 0 ), |σ| < 1, 1 < |ξ| < 3 .
If m ∈ R and ρ ∈ R + , we denote by S m ρ (O) the set of all functions f defined on O which are C ∞ with respect to (σ, x, ξ) and satisfy

|∂ α x f (σ, x, ξ, h)| ≤ C α h m-αρ , ∀α ∈ N, ∀(σ, x, ξ, h) ∈ O.
Remark 4.10. Recall that

W h (σ, x) = S (j-N )δ (W )(σh 1 2 , x) ≡ φ(2 -(j-N )δ D x )W (σh 1 2 , x).
Hence, for any α ∈ N, there hold

(4.38) |∂ α x W h (σ, x)| ≤ C α h -δα W (σh 1 2 , •) L ∞ , |∂ α+1 x W h (σ, x)| ≤ C α h -δα W (σh 1 2 , •) W 1,∞ .
The following result for transport problems is elementary.

Lemma 4.11. If v is a solution of the problem

(∂ σ + m(ξ)∂ x + if ) v(σ, x, ξ) = g(σ, x, ξ), u| σ=0 = z ∈ C,
where f be real-valued, then v satisfies

|v(σ, x, ξ)| ≤ |z| + σ 0 |g(σ ′ , x + (σ ′ -σ)a ′ (ξ), ξ)| d σ ′ .
The existence of the parametrix is given in the following Proposition.

e i h (ϕ(σ,x,ξ,h)-zξ) r(σ, x, z, ξ, h)u 0,h (z)dzdξ H 1 (Rx) ≤ h N F N ( W E0 + W E1 ) u 0,h L 1 (R) .
Proof. We proceed in several steps.

Step 1. Construction of the phase ϕ. We find ϕ under the form

|∂ k ξ ∂ α x ψ(σ, x, ξ, h)| ≤ C kα |σ| 3 4 h -δ(α+k-1) + W L 4 ([0,T ],W 1,∞ x ) ,
for every (α, k) ∈ N 2 , for every (σ, x, ξ, h) ∈ O; where m + = max{m, 0}. Remark that in [START_REF] Alazard | Strichartz estimates for water waves[END_REF] where W ∈ L ∞ ([0, T ], W 1,∞ (R)), one has the better estimate

(4.44) |∂ k ξ ∂ α x ψ(σ, x, ξ, h)| ≤ C kα |σ|h -δ(α+k-1) + W L ∞ ([0,T ],W 1,∞ x ) .
However, (4.43) is enough to get ∂ x ψ ∈ S 0 δ (O). Consequently, the estimates from (4.17) to (4.30) in [START_REF] Alazard | Strichartz estimates for water waves[END_REF] still hold and thus we have by (4.29), [START_REF] Alazard | Strichartz estimates for water waves[END_REF] (4.45)

r = h ∂ σ b + a ′ (ξ)∂ x b + if b + h µ0 M1 l=0 e l (h δ ∂ x ) l b ζ + i 4 j=1 r j with e l ∈ S 0 δ (O), (4.46) µ 0 = 1 2 ( 1 2 -δ) > 0, f = W h ∂ x ψ + a ′′ (ξ)(∂ x ψ) 2 (real valued);
and with ρ(x, y) =

1 0 ∂ x ϕ(σ, λx + (1 -λ)y, ξ, h)dλ,
the remainders r ′ i s are then given by (4.47)

r 1 = ch M-1 1 0 e i h (x-y)η κ 0 (η)(1 -λ) M-1 ∂ M y a (M) (λη + (ρ(x, y)) b(y) dλdydη, (4.48) r 2 = M-1 k=0 c k,M h M+k 1 0 z M κ0 (z)(1 -λ) M-1 ∂ M+k y a (k) ((ρ(x, y)) b(y) y=x-λhz dλdz. (4.49) r 3 = M-1 k=0 k j=1 c ′ j,k h k ∂ k-j y (∂ k ξ a)(ρ(x, y))b(y) | y=x ζ (j) . (4.50) r 4 = 1 i h -a ′ (ξ) + h 1 2 W h bζ ′
where c, c k,M , c ′ jk are constants and κ 0 ∈ C ∞ 0 (R), κ = 1 in a neighborhood of the origin. Now, combining (4.43) with the fact that W h ∈ S 0 δ (O) (by (4.38)) we obtain the following estimate for f

(4.51) |∂ k ξ ∂ α x f (σ, x, ξ, h)| ≤ |σ| 3 4 h -δ(α+k) F kα W L 4 ([0,T ],W 1,∞ x ) W L ∞ ([0,T ],L ∞ x ) , ∀(α, k) ∈ N 2 , ∀(σ, x, ξ, h) ∈ O.
Step 2. Construction of the amplitude b. According to the WKB method, ones find b under the form

(4.52) b = M-1 j=0 h jµ0 b j where b 0 solves ∂ σ b 0 + a ′ (ξ)∂ x b 0 + if b 0 = 0, b 0 | σ=0 = χ 1 (ξ)
and b ′ j s, j ≥ 1 solves

∂ σ b j + a ′ (ξ)∂ x b j + if b j = - M1 l=0 e l (h δ ∂ x ) l b j-1 , b j | σ=0 = 0.
Owing to Lemma 4.11 and the estimate (4.51), one can use induction for the preceding transport problems (see Lemma 4.7,[START_REF] Alazard | Strichartz estimates for water waves[END_REF]) to have (4.53) b j (σ, x, ξ, h) = χ 1 (ξ)c j (σ, x, ξ, h), ∀0 ≤ j ≤ J -1 and the c j satisfies ∀(α, k) ∈ N 2 , ∀(σ, x, ξ, h) ∈ O,

(4.54) |∂ k ξ ∂ α x c j (σ, x, ξ, h)| ≤ h -δ(α+k) F jkα ( W E0 + W E1 ) .
Step 3. Estimate for the remainder r. Plugging (4.52) into (4.45) we obtain r = 5 j=0 r j with r 5 = h Mµ0 b M-1 ζ. We want to prove (4.40), i.e, for a.e. t ∈ [0, T ] and for all j = 1, ..5, (4.55) e i h (ϕ(σ,x,ξ,h)-zξ) r(σ, x, z, ξ, h)u 0,h (z)dzdξ

H 1 (Rx) ≤ h N F N ( W E0 + W E1 ) u 0,h L 1 (R) .
Let us denote the function inside the norm on the left-hand side by F j h . The proofs for F j h H 1

x , j = 1, 2, 3, 5 remain unchanged compared to section those in 4.1.1, [START_REF] Alazard | Strichartz estimates for water waves[END_REF] (using integration by parts). The only point that we need to take care is the estimate for F 4 H 1 since r 4 contains W h which is less regular than it was in [START_REF] Alazard | Strichartz estimates for water waves[END_REF]. Recall that

r 4 = 1 i h -a ′ (ξ) + h 1 2 W h bζ ′ .
On the support of all derivatives of ζ one has |xzσa ′ (ξ)| ≥ 1. Now, by (4.43)

h 1 2 ∂ x ψ ≤ Ch 1 2 |σ| 3 4 ≤ ch 1 2
hence using (4.41) we deduce that

|∂ ξ (ϕ(σ, x, ξ, h) -zξ)| = |x -z -σa ′ (ξ) -h 1 2 ∂ ξ ψ| ≥ 1 2
for h small enough. Therefore, we can integrate by parts N times in the integral defining F 4 using the vector filed

L = h i∂ ξ (ϕ(σ, x, ξ, h) -zξ) ∂ ξ .
Taking into account the fact that for all α ∈ N, on the support of ζ, xzσa ′ (ξ) ≤ C and (due to (4.38), (4.54) and (4.43))

|∂ α ξ r 4 (σ, x, ξ, h)| ≤ C(1 + W h (σ) L ∞ x )h 1-αδ F α ( W E0 + W E1 ) , |∂ α+1 ξ (ϕ(σ, x, ξ, h) -zξ)| ≤ C(1 + W E1 )h -αδ we obtain F 4 h (σ) L 2 x ≤ h 1+N (1-δ) (1 + W h (σ) L ∞ x )F α ( W E0 + W E1 ) × × |u 0,h (z)|dz |χ 1 (ξ)|dξ ≤ h 1+N (1-δ) F N ( W E0 + W E1 ) u 0,h (z) L 1
x . Similarly, one gets

∂ x F 4 h (σ) L 2 x ≤ h 1+N (1-δ) (1 + ∂ x W h (σ) L ∞ x )F N ( W E0 + W E1 ) u 0,h (z) L 1 x ≤ h 1+N (1-δ) (1 + h -δ W (σh 1 2 ) L ∞ x )F N ( W E0 + W E1 ) u 0,h (z) L 1 x ≤ h (N +1)(1-δ) F N ( W E0 + W E1 ) u 0,h (z) L 1
x . Therefore, we end up with

sup σ∈[0,1] F 4 h (σ) H 1 (R) ≤ h N (1-δ) F N ( W E0 + W E1 ) u 0,h (z) L 1 x ,
which concludes the proof.

Having established the previous Proposition, we turn to prove Theorem 4.8. Proof of Theorem 4.8 Without loss of generality, we take σ 0 = 0. By a scaling argument, it suffices to prove the dispersive estimate (4.35) for the operator S for σ = 1. Indeed, let σ 1 ∈ (0, 1], making the following changes of variables

τ = σ σ 1 , x = x σ 1 , h = h σ 1
we see that the operator L δ becomes

Lδ = h∂ τ + h 1 2 Wh ( h∂ x) + 1 2 h 3 2 (∂ x Wh ) + i| hD x| 3 2
where Wh (τ, x) = σ 1 2

1 W h (σ 1 τ, σ 1 x). Observe that there exists C > 0 independent of σ 1 ∈ (0, 1] for which there holds

Wh E0 + Wh E1 ≤ C.
Suppose that the dispersive estimate (4.35) for L δ were proved for σ = 1, it then would imply the same estimate for Lδ for τ = 1. Calling S the propagator of Lδ , we have for all

σ ∈ [0, 1] S(σ, 0)u 0 (x) = ( S( σ σ 1 )ū)( x σ 1 ), ū( x σ 1 ) = u 0 (x).
Taking σ = σ 1 then it would follow that

S(σ 1 )u 0 L ∞ (R) = S( σ 1 σ 1 )ū L ∞ (R) ≤ C h 1 2 ū0 L 1 (R) ≤ Cσ 1 2 1 h 1 2 σ 1 u 0 L 1 (R) ≤ C |hσ 1 | 1 2 u 0 L 1 (R) ,
which is the estimate (4.35) for L δ for σ = σ 1 . Therefore, it suffices to prove (4.35) for σ = 1. Now, combining (4.36) and Proposition 4.12 yields (4.56)

L δ U h (σ, x) = F h (σ, x) with (4.57) sup σ∈[0,1] F h (σ) H 1 x (R)) ≤ C N h N F N ( W E0 + W E1 ) u 0,h L 1 (R) .
Using integration by parts we can show that U h is a good parametrix at the initial time (see (4.53), [START_REF] Alazard | Strichartz estimates for water waves[END_REF]) in the following sense

(4.58) U h (0, •) = u 0,h + v 0,h , v 0,h H 1 (R) ≤ C N h N u 0,h H 1 (R) .
Combining (4.56), (4.58) and the Duhamel formula gives

S(σ, 0)u 0,h = R 1 + R 2 + R 3 where      R 1 = U h (σ, x), R 2 = -S(σ, 0)v 0,h , R 3 = - σ 0 S(σ, r)[F h (r,
x)]dr. We shall successively estimate R i . First, by Sobolev's inequalities and (4.58),

R 2 (σ) L ∞ x ≤ C S(σ, 0)v 0,h H 1 x = C v 0,h H 1 x ≤ C N h N u 0,h L 1 .
Next, for R 3 we estimate

R 3 (σ) L ∞ x ≤ σ 0 S(σ, r)[F h (r, x)] H 1 x dr ≤ σ 0 F h (r, x) H 1 x dr.
Then, by virtue of the estimate (4.57) we deduce that

R 3 (σ) L ∞ x ≤ h N F N ( W E0 + W E1 ) u 0,h L 1 (R) . Finally, from (4.36) we have U h (σ, x) = K(σ, x, z, h)u 0,h (z)dz with K(σ, x, z, h) = 1 2πh e i h (ϕ(σ,x,ξ,h)-zξ) b(σ, x, z, ξ, h)dξ.
Because σ = 1 is fixed, the proof of Proposition 4.8, [START_REF] Alazard | Strichartz estimates for water waves[END_REF] still works and we obtain for some F : R + → R + independent of all parameters

|K(1, x, z, h)| ≤ 1 h 1 2 F ( W E0 + W E1 ) .
This gives

R 1 (1) L ∞ x = U h (1) L ∞ x ≤ h -1 2 F ( W E0 + W E1 ) u 0,h L 1 .
The proof is complete.

The semi-classical Strichartz estimate

Combining the dispersive estimate (4.32) with the usual T T * argument and Duhamel's formula, we derive the Strichartz estimate on small time interval [0, h

.

Corollary 4.13. Let I h = [0, h

] and u be a solution to the problem

Lu(t, x) = f (t, x), u(0, x) = 0 with supp f ⊂ {c 1 h -1 ≤ |ξ| ≤ c 2 h -1 }. Then there exists F : R + → R + (independent of u, f, W, h) such that u L 4 (I h ,L ∞ (R)) ≤ h -1 8 F ( W E0 + W E1 ) f L 1 (I h ,L 2 (R)) .
Finally, we glue these estimates together both in frequency and in time to obtain the semi-classical Strichartz estimate for u on [0, T ].

∈ R. Let W ∈ E 0 ∩ E 1 and f ∈ L 4 (I; H s0-1 2 (R)). If u ∈ L ∞ (I, H s0 (R)) is a solution to the problem ∂ t + T W ∂ x + i|D x | 3 2 u = f, then for every ε > 0, there exists F ε (independent of u, f, W ) such that (4.59) u L 4 (I;C s 0 -1 4 -ε * (R)) ≤ F ε (Ξ) f L 4 (I;H s 0 -1 2 -ε (R)) + u L ∞ (I;H s 0 (R)) , where Ξ = W E0 + W E1 + ∂ x η L ∞ t L ∞ x .
Proof. Recall from (4.16) that W is given by

W = (V • κ)(∂ x χ • κ) + ∂ t χ • κ.
First, by Sobolev's embedding and Lemma 4.2 4.,

W L ∞ t L ∞ x ≤ F (M s (T )). To estimate W E1 we compute ∂ x W = (∂ x V • κ)(∂ x χ • κ)∂ x κ + (V • κ)(∂ 2 x χ • κ)∂ x κ + (∂ t ∂ x χ • κ)∂ x κ.
Using the expression (1.3) for V together with the Hölder estimate for the Dirichlet-Neumann operator proved in Proposition 2.10, [START_REF] Nguyen | A paradifferential reduction for the gravity-capillary waves system at low regularity and applications[END_REF], we obtain for a.e. t ∈ [0, T ]

(5.4)

∂ x V (t) L ∞ x ≤ F ( η(t) H s+ 1 2 , ψ(t) H s ) (1 + ψ(t) W r,∞ ) . On the other hand, Lemma 4.2 3. gives ∂ x χ L ∞ t L ∞ x ≤ F (M s (T )), hence (5.5) (∂ x V • κ)(∂ x χ • κ)∂ x κ L 4 t L ∞ x ≤ F (M s (T ))(1 + N r (T )
). The other two terms in the expression of ∂ x W are treated in the same way.

Corollary 5.3. For every 0 < µ < 1 4 , there exists F : R + → R + such that

(5.6) u L 4 (I;W s-1 2 +µ,∞ (R)) ≤ F (M s (T ) + N r (T )).
Proof. In views of the Strichartz estimate (5.3) and Lemma 5.1, Lemma 5.2, there holds

(5.7) u L 4 (I;W s-1 2 +µ,∞ (R)) ≤ F (M s (T ) + N r (T )) f L 4 (I;H s-1 2 (R)) + 1 .
On the other hand, from the estimate (4.17) we have

f L 4 (I;H s-1 2 (R)) ≤ F (M s (T ))(1 + N r (T )),
which concludes the proof.

Having established the estimate (5.6) for u, we now go back from u to the original unknown (η, ψ). To this end, we proceed in 2 steps:

u = k * Φ -→ Φ -→ (η, ψ). Fix µ ∈ (0, 1 4 ). Step 1. By definition (4.5), Φ • κ = u + Ṫ∂xΦ•κ κ. It is easy to see that Ṫ∂xΦ•κ κ L ∞ t H s+ 1 2 x ≤ F (M s (T ))
and thus by Sobolev's embedding and the estimate (5.6) it holds

Φ • κ L 4 (I;W s-1 2 +µ,∞ ) ≤ F (M s (T ) + N s (T )).
We then may estimate

Φ(t) W s-1 2 +µ,∞ = Φ • κ • χ(t) W s-1 2 +µ,∞ ≤ Φ(t) • κ(t) W s-1 2 +µ,∞ x F ( χ ′ (t) W s-3 2 +µ,∞ ) ≤ Φ(t) • κ(t) W s-1 2 +µ,∞ F (M s (T )), which implies Φ L 4 (I;W s-1 2 +µ,∞ ) ≤ F (M s (T ) + N s (T )).
Step 2. By definition of Φ and the inequality • C σ ≤ C σ • W σ,∞ for any σ > 0, the preceding estimate gives (5.8)

T p η L 4 (I;C s-1 2 +µ * ) + T q (ψ -T B η) L 4 (I;C s-1 2 +µ * ) ≤ F (M s (T ) + N s (T )). 1. Since sup t∈[0,T ] M -1/2 0 (p (-1/2) (t)) + sup t∈[0,T ] M 1/2 1 (p (1/2) (t)) ≤ F (M s (T ))
it follows from (A.6) that

T p (-1/2) η L 4 (I;C s-1 2 +µ * ) ≤ F (M s (T )) η L 4 (I;C s-1+µ * ) ≤ F (M s (T )).
Consequently, we have

T p (1/2) η L 4 (I;C s-1 2 +µ * ) ≤ F (M s (T ) + N s (T )).
Since p (1/2) ∈ Γ In summary, we have proved that for all (η, ψ) solution to (1.2) with (5.10)

   (η, ψ) ∈ C 0 ([0, T ]; H s+ 1 2 (R) × H s (R)) ∩ L 4 ([0, T ]; W r+ 1 2 ,∞ (R) × W r,∞ (R)), s > r > 3 2 + 1 2
there holds for any µ < Of course, (5.11) is meaningful only if r < s -1 2 + µ. Under this constrain, using an interpolation argument (see [START_REF] Alazard | Strichartz estimate and the Cauchy problem for the gravity water waves equations[END_REF], page 88, for instance) we deduce easily that N r (T ) ≤ F T M s (T ) + N r (T ) .

On the other hand, in Theorem 1.1 [START_REF] Nguyen | A paradifferential reduction for the gravity-capillary waves system at low regularity and applications[END_REF] it was proved the following energy estimate at the regularity (5.10) M s (T ) ≤ F F (M s (0)) + T F (M s (T ) + N r (T )) .

Consequently, one gets a closed a priori estimate for the mixed norm M s (T ) + N r (T ) as in Theorem 1.2:

(5.12)

M s (T ) + N r (T ) ≤ F F (M s (0)) + T F (M s (T ) + N r (T )) .

Finally, by virtue of the contraction estimate for two solution (η j , ψ j ) j = 1, 2 in the norm M s-1,T + N r-1,T established in Theorem 5.9, [START_REF] Nguyen | A paradifferential reduction for the gravity-capillary waves system at low regularity and applications[END_REF] (whose proof makes use of Theorem 4.14) one can use the standard regularized argument (see section 6, [START_REF] Nguyen | A paradifferential reduction for the gravity-capillary waves system at low regularity and applications[END_REF]) to solve uniquely the Cauchy problem for system (1.2) with initial data (η 0 , ψ 0 ) ∈ H s+ 1 2 (R) × H s (R) with s > 2 + 1 2µ for any µ < 1 4 . The proof of Theorem 1.3 is complete.

A Appendix 1: Paradifferential calculus Let us review notations and results about Bony's paradifferential calculus (see [START_REF] Bony | Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires[END_REF][START_REF] Métivier | Para-differential calculus and applications to the Cauchy problem for nonlinear systems[END_REF]).

Here we follow the presentation by Métivier in [START_REF] Métivier | Para-differential calculus and applications to the Cauchy problem for nonlinear systems[END_REF] and [START_REF] Alazard | On the Cauchy problem for gravity water waves[END_REF]. a (m-j) where each a (m-j) ∈ Γ m-j ρ-j is homogeneous of degree mj with respect to ξ. Definition A.3. (Paradifferential operators) Given a symbol a, we define the paradifferential operator T a by

(A.2)
T a u(ξ) = (2π) -d χ(ξη, η) a(ξη, η)ψ(η) u(η) dη, where a(θ, ξ) = e -ix•θ a(x, ξ) dx is the Fourier transform of a with respect to the first variable; χ and ψ are two fixed C ∞ functions such that: (i) ψ is identical to 0 near the origin and identical to 1 away from the origin, (ii) there exists 0 < ε 1 < ε 2 < 1 such that

(A.3) χ(η, ξ) = 1 if |η| ≤ ε 1 (1 + |ξ|), 0 if |η| ≥ ε 2 (1 + |ξ|)
and for any (α, β) ∈ N 2 there exists C α,β > 0 such that

(A.4) ∀(η, ξ) ∈ R d × R d , ∂ α η ∂ β ξ χ(η, ξ) ≤ C α,β (1 + |ξ|) -α-β .
Definition A.4. An operator T is said to be of order m ∈ R (or equivalently, -mregularized) if, for all µ ∈ R, it is bounded from H µ to H µ-m and from C µ * to C µ-m * .

Symbolic calculus for paradifferential operators is summarized in the following theorem. B.2 Proof of Lemma 2.3

1. Let 1 ≤ p ≤ q ≤ ∞. Remark first that the estimates for ∆ j follows immediately from those of S j since ∆ 0 = S 0 and ∆ j = S j -S j-1 , ∀j ≥ 1. By definition 2.2 we have for each n ∈ N, S j u = f j * u where f j is the inverse Fourier transform of φ j , where φ ≡ φ (n) . With r satisfying 1 p + 1 r = 1 + 1 q we get by Young's inequality

∂ α S j L p →L q ≤ ∂ α f j L r .
The problem then reduces to showing that

∂ α f j L r ≤ C α 2 j(|α|+ d p -d q )
which in turn reduces to

∂ α F -1 (φ (n) )(x) L r ≤ C α ,
which is true by virtue of (2.2).

2. The boundedness of the operators 2 jµ ∆ j , j ≥ 1 from W µ,∞ (R d ) to L ∞ (R d ) is proved in Lemma 4.1.8, [START_REF] Métivier | Para-differential calculus and applications to the Cauchy problem for nonlinear systems[END_REF]. Following that proof we see that

2 jµ ∆ j W µ,∞ →L ∞ ≤ 2 jµ

Theorem 4 . 1 ([ 19 ,

 4119 Proposition 4.1]). Assume that (η, ψ) is a solution to (1.2) and satisfies(4.1)

  σ,x,ξ,h)-zξ) b(σ, x, z, ξ, h)u 0,h (z)dzdξ where (i) the phase ϕ satisfies ϕ(σ = 0) = xξ, (ii) the amplitude b has the form (4.37) b

Proposition 4 . 12 . 1 2

 4121 There exists a phase ϕ of the formϕ(σ, x, ξ, h) = xξσa(ξ) + h ψ(σ, x, , ξ, h) with ∂ x ψ ∈ S 0 δ (O)and there exists a symbol b ∈ S 0 δ (O) such that with the amplitude b defined by (4.37), we have (4.39) L δ e i h φ b = e i h φ r h , where for any N ∈ N there holds (4.40) sup σ∈[0,1]

( 4 . 1 2

 41 41) ϕ(t, x, ξ, h) = xξσa(ξ) + h ψ(σ, x, , ξ, h) where ψ solves the following transport problem (4.42) ∂ σ ψ + a ′ (ξ)∂ x ψ = -ξW h , ψ| σ=0 = 0. Differentiating (4.42) with respect to x and ξ then using Lemma 4.11 together with (4.38) and Hölder's inequality we derive (4.43)

Theorem 4 . 14 .

 414 Let I = [0, T ] and s 0

L 4 (L 4 (

 44 is elliptic, applying(A.8) yields η = T 1/p (1/2) T p (1/2) η + Rη, where R is of order -1 and for any σ ∈ R sup t∈[0,T ] R(t) C σ * →C σ+1 * ≤ F (M s (T )).Thus, (5.9) η L 4 (I;C s+µ * ) ≤ F (M s (T ) + N r (T )).Likewise, we deduce from (5.8) thatψ -T B η M s (T ) + N r (T )).Owing to (5.9) and the fact thatB L ∞ t L ∞ x ≤ F (M s (T )), we obtain ψ M s (T ) + N r (T )).

  M s (T ) + N r (T ))and thus (since µ < 1 4 is arbitrary)(5.11) N s-1 2 +µ (T ) ≤ F (M s (T ) + N r (T )), where M σ (T ), N σ (T ) are respectively the Sobolev-norm and the Strichartz norm defined in (5.1).(5.11) is the semi-classical Strichartz estimate announced in Theorem 1.1.

  -Paley partition. For any real number s, we define the Zygmund class C s * (R d ) as the space of tempered distributions u such thatu C s * := sup q 2 qs ∆ q u L ∞ < +∞.2. (Hölder spaces) For k ∈ N, we denote by W k,∞ (R d ) the usual Sobolev spaces. For ρ = k + σ, k ∈ N, σ ∈ (0, 1) denote by W ρ,∞ (R d ) the space of functions whose derivatives up to order k are bounded and uniformly Hölder continuous with exponent σ.

Definition A. 2 . 2 ( 1 +

 221 1. (Symbols) Given ρ ∈ [0, ∞) and m ∈ R, Γ m ρ (R d ) denotes the space of locally bounded functions a(x, ξ) on R d × (R d \ 0), which are C ∞ with respect to ξ for ξ = 0 and such that, for all α ∈ N d and all ξ = 0, the functionx → ∂ α ξ a(x, ξ) belongs to W ρ,∞ (R d ) and there exists a constant C α such that, •, ξ) W ρ,∞ (R d ) ≤ C α (1 + |ξ|) m-|α| . Let a ∈ Γ m ρ (R d ),we define for every n ∈ N the semi-norm(A.1) M m ρ (a; n) = sup |α|≤n sup |ξ|≥1/|ξ|) |α|-m ∂ α ξ a(•, ξ) W ρ,∞ (R d ).When n = [d/2] + 1 we denote M m ρ (a; n) = M m ρ (a).2. (Classical symbols) For any m ∈ R and ρ > 0 we denote by Σ m ρ (R d ) the class of classical symbols a(x, ξ) such that a(x, ξ) = 0≤j≤[ρ]

Theorem A. 5 .

 5 (Symbolic calculus,[START_REF] Métivier | Para-differential calculus and applications to the Cauchy problem for nonlinear systems[END_REF]) Let m ∈ R and ρ ∈ [0, ∞). Denote by ρ the smallest integer that is not smaller than ρ andn 1 = [d/2] + ρ + 1. (i) If a ∈ Γ m 0 (R d ),then T a is of order m. Moreover, for all µ ∈ R there exists a constant K such thatT a H µ →H µ-m ≤ KM m 0 (a), (A.5) T a C µ * →C µ-m * ≤ KM m 0 (a). (A.6) (ii) If a ∈ Γ m ρ (R d ), b ∈ Γ m ′ ρ (R d ) with ρ > 0. Then T a T b -T a♯b is of order m + m ′ρ where a♯b := |α|<ρ (-i) α α! ∂ α ξ a(x, ξ)∂ α x b(x, ξ).Moreover, for all µ ∈ R there exists a constant K such thatT a T b -T a♯b H µ →H µ-m-m ′ +ρ ≤ KM m ρ (a; n 1 )M m ′ 0 (b) + KM m 0 (a)M m ′ ρ (b; n 1 ), (A.7) T a T b -T a♯b C µ * →C µ-m-m ′ +ρ * ≤ KM m ρ (a; n 1 )M m ′ 0 (b) + KM m 0 (a)M m ′ ρ (b; n 1 ). (A.8) (iii) Let a ∈ Γ m ρ (R d ) with ρ > 0.Denote by (T a ) * the adjoint operator of T a and by a the complex conjugate of a. Then (T a ) * -T a * is of order mρ wherea * = |α|<ρ 1 i |α| α! ∂ α ξ ∂ α x a.

  R d|x| µ |g j (x)|dx := I, where g j is the inverse Fourier transform of ϕ j = φ jφ j-1 . Owing to (2.2) it holds that∀α ∈ N d , ∃C α > 0, ∀(j, n) ∈ N * × N, |x α g j (x)| d x ≤ C α 2 -j|α| .Thus, if µ ∈ N we have the result. If µ = δn + (1δ)(n + 1) for some δ ∈ (0, 1), n ∈ N we use Hölder's inequality to estimateI ≤ 2 jµ |x| n |g j (x)| d x δ |x| n+1 |g j (x)| d x 1-δ ≤ C µ 2 jµ 2 -jnδ-j(n+1)(1-δ) = C µ ,which concludes the proof.

  and use Lemma 4.2 8. to get ∂ x a L ∞ ≤ RHS. We have proved that modulo a controllable remainder, A 2 = Ṫ∂tχ•κ u. Consequently, modulo a controllable remainder, A 2 = T ∂tχ•κ u. Then putting together this and (4.[START_REF] Nguyen | A paradifferential reduction for the gravity-capillary waves system at low regularity and applications[END_REF]), (4.20) we end up with the claim (4.18).Step 2. With the definitions of R line and R conj in Theorem 3.5 and Theorem 3.6 we write for any h ∈ Γ m

τ

(4.22)

The author was supported in part by Agence Nationale de la Recherche project ANAÉ ANR-13-BS01-0010-03.

Proof. Throughout this proof, we denote F = F ( W E0 + W E1 ) and RHS the righthand side of (4.59). Remark first that by (4.30) we have L δ u h = F h , where F h is given by (4.31).

Step 1. Let χ ∈ C ∞ 0 (0, 2) equal to one on [ 1 2 , 3 2 ]. For 0 ≤ k ≤ [T h -1 ] -2 define

, u h,k = χ h,k (t)u h .

Then

Applying Corollary 4.13 to each u h,k on the interval I h,k we obtain, since χ h,k (t) = 1 for

Raising to the power 4 both sides of the preceding estimate, summing in k from 0 to [T h -1 2 ] -2 and then taking the power 1/4 we get (4.60)

Set ν = 1 2δ. Multiplying both sides of the above inequality by h -s0+ 1 4 +ν and taking into account the fact that u h and F h are spectrally supported in annulus of size h -1 , it follows that (4.61)

Step 2. We now estimate F h L 4 (I;H s 0 -1+δ (R)) , where recall from (4.31) that (4.62)

, we can apply the symbolic calculus Theorem A.5 (i), (ii) to have

Next, remark that the spectrum of Λ j := S (j-N )δ (W ) -S j-N (W ) ∂ x ∆ j u is contained in a ball of radius C 2 j we can write for fixed t

According to the convolution formula,

where φ is the inverse Fourier transform of the Littlewood-Paley function φ. It follows that

Therefore, we obtain

Similarly, it also holds that (4.65)

Now, combining (4.63), (4.64), (4.65) and Lemma 4.5 and the fact that 0 < δ < 1 2 we conclude (4.66)

Now, combining this estimate with (4.61) we derive (4.67)

Finally, for every given

5 Proof of Theorems 1.1, 1.2, 1.3

Throughout this section, we assume that (η, ψ) is a solution to the gravity-capillary water waves system (1.2) having the regularity given by (4.1). For any real number σ, let us define the Sobolev-norm and the Strichartz-norm of the solution:

From the Strichartz estimate (4.59) we have for any ε > 0 (5.3)

We shall estimate the norms of W and u appearing on the right-hand side of (5.3) in terms of of M s and N s .

Proof. By definition (4.5), u is given by

Lemma 5.1 then follows from Theorem 3.4 and Theorem 3.5 (ii).

Lemma 5.2. We have

Moreover, for all µ there exists a constant K such that We shall use frequently various estimates about paraproducts (see Chapter 2, [8] and [START_REF] Alazard | On the Cauchy problem for gravity water waves[END_REF]) which are recalled here.

Theorem A.7.

3. Let m > 0 and s ∈ R. Then

Theorem A.9. 1. Let s ≥ 0 and consider F ∈ C ∞ (C N ) such that F (0) = 0. Then there exists a non-decreasing function F : R + → R + such that, for any

Then there exists a non-decreasing function F : R + → R + such that, for any

Theorem A.10. [8, Theorem 2.92](Paralinearization) Let r, ρ be positive real numbers and F be a C ∞ function on R such that F (0) = 0. Assume that ρ is not an integer. For

We then define φ (n) = f n * g. It is easy to see that φ (n) ≥ 0 and satisfies condition (2.1).

To verify condition (2.2) we use ∂ α φ (n) = f n * ∂ α g to have

Each term on the right-hand side is estimated by

where (•) β2 ∂ α g L 1 is independent of n. It remains to have a uniformly bound with respect to n for (•) β1 f n L 1 . To this end, one can choose the following piecewise affine functions