One smoothing property of the scattering map of the KdV on $\mathbb R$

In this paper we prove that in appropriate weighted Sobolev spaces, in the case of no bound states, the scattering map of the Korteweg-de Vries (KdV) on $\mathbb R$ is a perturbation of the Fourier transform by a regularizing operator. As an application of this result, we show that the difference of the KdV flow and the corresponding Airy flow is 1-smoothing.

The aim of this paper is to show that for the KdV on the line, the scattering map is an analytic perturbation of the Fourier transform by a 1-smoothing nonlinear operator. With the applications we have in mind, we choose a setup for the scattering map so that the spaces considered are left invariant under the KdV flow. Recall that the KdV equation on R ∂ t u(t, x) = −∂ 3 x u(t, x) − 6u(t, x)∂ x u(t, x) , u(0, x) = q(x) , Introduce for q ∈ L 2 M with M ≥ 4 the Schrödinger operator L(q) := −∂ 2 x + q with domain H 2 C , where, for any integer N ∈ Z ≥0 , H N C := H N (R, C). For k ∈ R denote by f 1 (q, x, k) and f 2 (q, x, k) the Jost solutions, i.e. solutions of L(q)f = k 2 f with asymptotics f 1 (q, x, k) ∼ e ikx , x → ∞, f 2 (q, x, k) ∼ e −ikx , x → −∞. As f i (q, ·, k), f i (q, ·, −k), i = 1, 2, are linearly independent for k ∈ R \ {0}, one can find coefficients S(q, k), W (q, k) such that for k ∈ R \ {0} one has (1. 2) It's easy to verify that the functions W (q, ·) and S(q, ·) are given by the wronskian identities which are independent of x ∈ R. The functions S(q, k) and W (q, k) are related to the more often used reflection coefficients r ± (q, k) and transmission coefficient t(q, k) by the formulas r + (q, k) = S(q, −k) W (q, k) , r − (q, k) = S(q, k) W (q, k) , t(q, k) = 2ik W (q, k) ∀ k ∈ R \ {0} . (1.5) It is well known that for q real valued the spectrum of L(q) consists of an absolutely continuous part, given by [0, ∞), and a finite number of eigenvalues referred to as bound states, −λ n < · · · < −λ 1 < 0 (possibly none). Introduce the set Q := q : R → R , q ∈ L 2 4 : W (q, 0) = 0, q without bound states . (1.6) We remark that the property W (q, 0) = 0 is generic. In the sequel we refer to elements in Q as generic potentials without bound states. Finally we define We will see in Lemma 3.8 that for any integers N ≥ 0, M ≥ 4, Q N,M is open in H N ∩ L 2 M . Our main theorem analyzes the properties of the scattering map q → S(q, ·) which is known to linearize the KdV flow [GGKM74]. To formulate our result on the scattering map in more details let S denote the set of all functions σ : R → C satisfying (S1) σ(−k) = σ(k), ∀k ∈ R; (S2) σ(0) > 0. where ζ : R → R is an odd monotone C ∞ function with ζ(k) = k for |k| ≤ 1/2 and ζ(k) = 1 for k ≥ 1 . (1.8) The norm on H M ζ is given by Furthermore they are real analytic maps.
(1.11) By Theorem 1.1, S(q, ·) ∈ S , thus property (S2) implies that lim k→0 I(q, k) exists and equals 0. Furthermore, by (S1), the action I(q, ·) is an odd function in k, and strictly positive for k > 0. Thus we will consider just the case k ∈ [0, +∞). The properties of I(q, ·) for k near 0 and k large are described separately. In [KT86], Kappeler and Trubowitz showed that the map q → S(q, ·) is a real analytic diffeomorphism from Q ∩ H N,N to S ∩ H N −1,N ♯ , N ∈ Z ≥3 . They extend their results to potentials with finitely many bound states in [KT88]. Unfortunately, Q ∩ H N,N is not left invariant under the KdV flow.
Results concerning the 1-smoothing property of the inverse scattering map were obtained previously in [Nov96], where it is shown that for a potential q in the space W n,1 (R, R) of real-valued functions with weak derivatives up to order n in L 1 q(x) − 1 π R e −2ikx χ c (k)2ikr + (q, k)dk ∈ W n+1,1 (R, R) .
Here c is an arbitrary number with c > q L 1 and χ c (k) = 0 for |k| ≤ c , χ c (k) = |k|−c for c ≤ |k| ≤ c+1, and 1 otherwise. The main difference between the result in [Nov96] and ours concerns the function spaces considered. For the application to the KdV we need to choose function spaces such as H N ∩L 2 M for which KdV is well posed. To the best of our knowledge it is not known if KdV is well posed in W n,1 (R, R). Furthermore in [Nov96] the question of analyticity of the map q → r + (q) and its inverse is not addressed.
We remark that Theorem 1.1 treats just the case of regular potentials. In [FHMP09,HMP11] a special class of distributions is considered. In particular the authors study Miura potentials q ∈ H −1 loc (R, R) such that q = u ′ + u 2 for some u ∈ L 1 (R, R) ∩ L 2 (R, R), and prove that the map q → r + is bijective and locally bi-Lipschitz continuous between appropriate spaces. Finally we point out the work of Zhou [Zho98], in which L 2 -Sobolev space bijectivity for the scattering and inverse scattering transforms associated with the ZS-AKNS system are proved.

Jost solutions
In this section we assume that the potential q is complex-valued. Often we will assume that q ∈ L 2 M with M ∈ Z ≥4 . Consider the normalized Jost functions m 1 (q, x, k) := e −ikx f 1 (q, x, k) and m 2 (q, x, k) := e ikx f 2 (q, x, k) which satisfy the following integral equations m 1 (q, x, k) = 1 + +∞ x D k (t − x) q(t) m 1 (q, t, k)dt (2.1) where D k (y) := y 0 e 2iks ds. The purpose of this section is to analyze the solutions of the integral equations (2.1) and (2.2) in spaces needed for our application to KdV. We adapt the corresponding results of [KT86] to these spaces. As (2.1) and (2.2) are analyzed in a similar way we concentrate on (2.1) only. For simplicity we write m(q, x, k) for m 1 (q, x, k).
For 1 ≤ p ≤ ∞, M ≥ 1 and a ∈ R, 1 ≤ α < ∞, 1 ≤ β ≤ ∞ we introduce the spaces where x := (1 + x 2 ) 1/2 , L p is the standard L p space, and We consider also the space C 0 x≥a L β := C 0 [a, +∞), L β with f C 0 x≥a L β := sup x≥a f (x, ·) L β < ∞. We will use also the space L α x≤a L β of functions f : . Moreover given any Banach spaces X and Y we denote by (X, Y ) the Banach space of linear bounded operators from X to Y endowed with the operator norm. If X = Y , we simply write (X).
For the notion of an analytic map between complex Banach spaces we refer to Appendix B. We begin by stating a well known result about the properties of m.
Estimates on the Jost functions.
Proposition 2.2. For any q ∈ L 2 M with M ≥ 2, a ∈ R and 2 ≤ β ≤ +∞, the solution m(q) of (2.1) satisfies m(q) − 1 ∈ C 0 x≥a L β ∩ L 2 x≥a L 2 . The map L 2 M ∋ q → m(q) − 1 ∈ C 0 x≥a L β ∩ L 2 x≥a L 2 is analytic. Moreover there exist constants C 1 , C 2 > 0, only dependent on a, β, such that Remark 2.3. In comparison with [KT86], the novelty of Proposition 2.2 consists in the choice of spaces.
To prove Proposition 2.2 we first need to establish some auxiliary results.
Lemma 2.4. (i) For any q ∈ L 1 1 , a ∈ R and 1 ≤ β ≤ +∞, the linear operator is bounded. Moreover for any n ≥ 1, the n th composition K(q) n satisfies K(q) n , is linear and bounded, and Id − K is invertible. More precisely, To compute the norm of the iteration of the map K(q) it's enough to proceed as above and exploit the fact that the integration in t is over a simplex, yielding K(q) n C 0 x≥a L β ≤ C n q n L 1 1 /n! for any n ≥ 1.
Therefore the Neumann series of the operator Id − K(q) −1 = n≥0 K(q) n converges absolutely in C 0 x≥a L β . Since K(q) is linear and bounded in q, the analyticity and, by item (i), the claimed estimate for (Id − K) −1 follow.
(i) For any q ∈ L 2 3/2 , K(q) defines a bounded linear operator L 2 x≥a L 2 → L 2 x≥a L 2 . Moreover the n th composition K(q) n satisfies where C > 0 depends only on a.
(ii) The map K : L 2 3/2 → L 2 x≥a L 2 , q → K(q) is linear and bounded; the map Proof. Proceeding as in the proof of the previous lemma, one gets for x ≥ a the estimate from which it follows that proving item (i). To estimate the composition K(q) n viewed as an operator on L 2 x≥a L 2 , remark that is then proved as in the previous Lemma.
Note that for f ≡ 1, the expression in (2.
Lemma 2.6. For any 2 ≤ β ≤ +∞ and a ∈ R, the map L 2 where C 1 , C 2 > 0 are constants depending on a and β.
Proof. Since the map q → K(q)[1] is linear in q, it suffices to prove its continuity in q. Moreover, it is enough to prove the result for β = 2 and β = +∞ as the general case then follows by interpolation. For any k ∈ R, the bound |D k (y)| ≤ |y| shows that the map k → D k (y) is in L ∞ . Thus where C > 0 is a constant depending only on a ∈ R. The claimed estimate follows by noting that For any x ≥ a one thus has and hence where the last inequality follows from the Hardy-Littlewood inequality. The continuity in x follows from Lebesgue convergence Theorem.
To prove the second inequality, start from the second term in (2.5) and change the order of integration to obtain Proof of Proposition 2.2. Formally, the solution of equation (2.1) is given by (2.6) By Lemma 2.4, 2.5, 2.6 it follows that the r.h.s. of (2.6) is an element of C 0 x≥a L β ∩ L 2 x≥a L 2 , 2 ≤ β ≤ ∞, and analytic as a function of q, since it is the composition of analytic maps.
Properties of ∂ n k m(q, x, k) for 1 ≤ n ≤ M − 1. In order to study ∂ n k m(q, x, k), we deduce from (2.1) an integral equation for ∂ n k m(q, x, ·) and solve it. Recall that for any The result is summarized in the following Proposition 2.7. Fix M ∈ Z ≥4 and a ∈ R. For any integer 1 ≤ n ≤ M − 1 the following holds: K can be chosen uniformly on bounded subsets of L 2 M .
Remark 2.8. In [CK87b] it is proved that if q ∈ L 1 M−1 then for every x ≥ a fixed the map k → m(q, x, k) is in C M−2 ; note that since L 2 M ⊂ L 1 M−1 , we obtain the same regularity result by Sobolev embedding theorem.
To prove Proposition 2.7 we first need to derive some auxiliary results. Assuming that m(q, x, ·) − 1 has appropriate regularity and decay properties, the n th derivative ∂ n k m(q, x, k) satisfies the following integral equation To write (2.7) in a more convenient form introduce for 1 ≤ j ≤ n and q ∈ L 2 n+1 the operators In order to prove the claimed properties for ∂ n k m(q) we must show in particular that the r.h.s. of (2.9) is in C 0 x≥a L 2 . This is accomplished by the following Lemma 2.9. Fix M ∈ Z ≥4 and a ∈ R. Then there exists a constant C > 0, depending only on a, M , such that the following holds: (ii) For any 1 ≤ n ≤ M − 2, the map L 2 M ∋ q → K n (q) ∈ C 0 x≥a L 2 is analytic. Moreover one has As an application of item (i) and (ii), for any integers 1 ≤ n ≤ M − 1 the map L 2 M ∋ q → K n (q)[m(q) − 1] ∈ C 0 x≥a L 2 is analytic, and where K ′ 0 > 0 can be chosen uniformly on bounded subsets of L 2 M .
Proof. First, remark that all the operators q → K n (q) are linear in q, therefore the continuity in q implies the analyticity in q. We begin proving item (i).
and compute the Fourier transform F + (ϕ(x, ·)) with respect to the k variable for x ≥ a fixed, which we denote byφ( By Parseval's Theorem ϕ(x, ·) L 2 = 1 √ π φ(x, ·) L 2 . By changing the order of integration one has by taking the supremum in the x variable one has K n (q) ∈ L 2 x≥a L 2 , C 0 x≥a L 2 , where the continuity in x follows by Lebesgue's convergence theorem. The map q → K n (q) is linear and continuous, therefore also analytic.
We prove now item (ii).
, which implies the claimed estimate. The analyticity follows from the linearity and continuity of the map q → K n (q). Finally we prove item (iii). By Proposition 2.2, the map L 2 n+1 ∋ q → m(q) − 1 ∈ L 2 x≥a L 2 is analytic. By item (i2) above the bilinear map L 2 n+1 × L 2 x≥a L 2 ∋ (q, f ) → K n (q)[f ] ∈ C 0 x≥a L 2 is analytic; since the composition of analytic maps is analytic, the map L 2 n+1 ∋ q → K n (q)[m(q) − 1] ∈ C 0 x≥a L 2 is analytic. By (i2) and Proposition 2.2 one has where K ′ 0 can be chosen uniformly on bounded subsets of L 2 M . Proof of Proposition (2.7). The proof is carried out by a recursive argument in n. We assume that q → ∂ r k m(q) is analytic as a map from L 2 M to C 0 x≥a L 2 for 0 ≤ r ≤ n − 1, and prove that L 2 M → C 0 x≥a L 2 : q → ∂ n k m(q) is analytic, provided that n ≤ M − 1. The case n = 0 is proved in Proposition 2.2. We begin by showing that for every x ≥ a fixed k → ∂ n−1 k m(q, x, k) is a function in H 1 , therefore it has one more (weak) derivative in the k-variable. We use the following characterization of H 1 function [Bre11]: (2.10) where (τ h f )(k) := f (k + h) is the translation operator. Moreover the constant C above can be chosen to be C = ∂ k u L 2 . Starting from (2.9) (with n − 1 instead of n), an easy computation shows that for every x ≥ a fixed (τ h )∂ n−1 k m(q) ≡ ∂ n−1 k m(q, x, k + h) satisfies the integral equation (2.11) In order to estimate the term in the fourth line on the right hand side of the latter identity, use item (i1) of Lemma 2.9 and the characterization (2.10) of H 1 . To estimate all the remaining lines, use the induction hypothesis, the estimates of Lemma 2.9, the fact that the operator norm of (Id − K(q)) −1 is bounded uniformly in k and the estimate to deduce that for every n ≤ M − 1 which is exactly condition (2.10). This shows that k → ∂ n−1 k m(q, x, k) admits a weak derivative in L 2 . Formula (2.7) is therefore justified. We prove now that the map L 2 M ∋ q → ∂ n k m(q) ∈ C 0 x≥a L 2 is analytic for 1 ≤ n ≤ M − 1. Indeed equation (2.9) and Lemma 2.9 imply that where K ′ can be chosen uniformly on bounded subsets of q in L 2 M . Therefore ∂ n k m(q) ∈ C 0 x≥a L 2 and one gets recursively ∂ n k m(q) C 0 x≥a L 2 ≤ K q L 2 M , where K can be chosen uniformly on bounded subsets of q in L 2 M . The analyticity of the map q → ∂ n k m(q) follows by formula (2.9) and the fact that composition of analytic maps is analytic.
Properties of k∂ n k m(q, x, k) for 1 ≤ n ≤ M . The analysis of the M th k-derivative of m(q, x, k) requires a separate attention. It turns out that the distributional derivative ∂ M k m(q, x, ·) is not necessarily L 2integrable near k = 0 but the product k∂ M k m(q, x, ·) is. This is due to the fact that ∂ M k D k (x)q(x) ∼ x M+1 q(x) which might not be L 2 -integrable. However, by integration by parts, it's easy to see that The main result of this section is the following Proposition 2.10. Fix M ∈ Z ≥4 and a ∈ R. Then for every integer 1 ≤ n ≤ M the following holds: (i) for every q ∈ L 2 M and x ≥ a fixed, the function k → k∂ n k m(q, x, k) is in L 2 ; (ii) the map L 2 M ∋ q → k∂ n k m(q) ∈ C 0 x≥a L 2 is analytic. Moreover k∂ n k m C 0 x≥a L 2 ≤ K 1 q L 2 M where K 1 can be chosen uniformly on bounded subsets of L 2 M .
Formally, multiplying equation (2.7) by k, the function k∂ n k m(q) solves where we have introduced for 0 ≤ j ≤ M and q ∈ L 2 M the operators (2.13) We begin by proving that each term of the r.h.s. of (2.13) is well defined and analytic as a function of q. The following lemma is analogous to Lemma 2.9: Lemma 2.11. Fix M ∈ Z ≥4 and a ∈ R. There exists a constant C > 0 such that the following holds: (iii) As an application of item (i) and (ii) we get (2.14) where K ′ 1 can be chosen uniformly on bounded subsets of L 2 M ; (iii2) for any 1 ≤ j ≤ n − 1, the map L 2 M ∋ q →K j (q)[∂ n−j k m(q)] ∈ C 0 x≥a L 2 is analytic with where K ′ 2 can be chosen uniformly on bounded subsets of L 2 M . Proof. (i) Since the maps q →K n (q), 0 ≤ n ≤ M , are linear, it is enough to prove that these maps are continuous.
where ∂ ξ (ξ n ½ [0,t−x] (ξ)) is to be understood in the distributional sense. By Parseval's Theorem Let C ∞ 0 be the space of smooth, compactly supported functions. Since where the last inequality follows from Cauchy-Schwartz and Hardy inequality, and C > 0 is a constant depending on a and M .
(i2) As |k∂ n k D k (t − x)| ≤ 2 n |t−x| n by integration by parts, it follows that for some constant C > 0 depending only on a and M , Now take the supremum over x ≥ a in the expression above and use Lebesgue's dominated convergence theorem to prove item (i2).
(ii) The claim follows by the estimate (iii) By Propositions 2.2 and 2.7 the maps L 2 M ∋ q → m(q) − 1 ∈ C 0 x≥a L 2 ∩ L 2 x≥a L 2 and L 2 M ∋ q → ∂ n−j k m(q) ∈ C 0 x≥a L 2 are analytic; by item (ii) for any 1 ≤ n ≤ M − 1, the bilinear map (q, f ) →K n (q)[f ] is analytic from L 2 M × C 0 x≥a L 2 to C 0 x≥a L 2 . Since the composition of two analytic maps is again analytic, item (iii) follows. MoreoverK n (q)[m(q) − 1],K j (q)[∂ n−j k m(q)] ∈ C 0 x≥a L 2 since m(q, x, k) and ∂ n k m(q, x, k) are continuous in the x-variable. The estimate (2.14) follows from item (ii) and Proposition 2.2, 2.7.
Proof of Proposition 2.10. One proceeds in the same way as in the proof of Proposition 2.7. Given any 1 ≤ n ≤ M , we assume that q → k∂ r k m(q) is analytic as a map from L 2 M to C 0 x≥a L 2 for 1 ≤ r ≤ n−1, and deduce that q → k∂ n k m(q) is analytic as a map from L 2 M to C 0 x≥a L 2 and satisfies equation (2.12) (with r instead of n). We begin by showing that for every x ≥ a fixed, k → k∂ n−1 k m(q, x, k) is a function in H 1 . Our argument uses again the characterization (2.10) of H 1 . Arguing as for the derivation of (2.11) one gets the integral equation Using the estimates |h|, ∀h ∈ R , obtained by integration by parts, the characterization (2.10) of H 1 , the inductive hypothesis, estimates of Lemma 2.9 and Lemma 2.5 one deduces that for every n ≤ M This shows that k → k∂ n−1 k m(q, x, k) admits a weak derivative in L 2 . Since the estimate above and Proposition 2.7 show that k → k∂ n k m(q, x, k) is an L 2 function. Formula (2.7) is therefore justified. The proof of the analyticity of the map q → k∂ n k m(q) is analogous to the one of Proposition 2.7 and it is omitted.
Analysis of ∂ x m(q, x, k). Introduce a odd smooth monotone function ζ : R → R with ζ(k) = k for |k| ≤ 1/2 and ζ(k) = 1 for k ≥ 1. We prove the following Proposition 2.12. Fix M ∈ Z ≥4 and a ∈ R. Then the following holds: where K 2 can be chosen uniformly on bounded subsets of L 2 M .
where K 3 can be chosen uniformly on bounded subsets of L 2 M .
The integral equation for ∂ x m(q, x, k) is obtained by taking the derivative in the x-variable of (2.1): Taking the derivative with respect to the k-variable one obtains, for 0 ≤ n ≤ M − 1, and rewrite (2.17) in the more compact form Proposition 2.12 (i) follows from Lemma 2.13 below. The M th derivative requires a separate treatment, as ∂ M k m might not be well defined at k = 0. Indeed To deal with this issue we use the function ζ described above. Multiplying (2.19) with n = M by ζ we formally obtain Proposition 2.12 (ii) follows from item (iii) of Lemma 2.13 and the fact that ζ ∈ L ∞ : Lemma 2.13. Fix M ∈ Z ≥4 and a ∈ R. There exists a constant C > 0 such that (i) for any integer 0 ≤ n ≤ M the following holds: (iii) For any 1 ≤ n ≤ M − 1, 0 ≤ j ≤ n − 1 and ζ : R → R odd smooth monotone function with ζ(k) = k for |k| ≤ 1/2 and ζ(k) = 1 for k ≥ 1, the following holds: can be chosen uniformly on bounded subsets of L 2 M .
Proof. As before it's enough to prove the continuity in q of the maps considered to conclude that they are analytic.
The claim follows by taking the supremum over x ≥ a in the inequality above.
(iii1) By Proposition 2.7 one has that for any x≥a L 2 is analytic. Since composition of analytic maps is again an analytic map, the claim regarding the analyticity follows. The first estimate follows from item (ii). A similar argument can be used to prove the second estimate.
The following corollary follows from the results obtained so far: Corollary 2.14. Fix M ∈ Z ≥4 . Then the normalized Jost functions m j (q, x, k), j = 1, 2, satisfy: where K 2 > 0 can be chosen uniformly on bounded subsets of L 2 M . Proof. The Corollary follows by evaluating formulas (2.1), (2.7), (2.17) at x = 0 and using the results of Proposition 2.2, 2.7, 2.10 and 2.12.
3 One smoothing properties of the scattering map.
The aim of this section is to prove the part of Theorem 1.1 related to the direct problem. To begin, note that by Theorem 2.1, for q ∈ L 2 4 real valued one has m 1 (q, x, k) = m 1 (q, x, −k) and m 2 (q, which by continuity holds for k = 0 as well. In the case where q ∈ Q, the latter identity implies that S(q, 0) = 0.
Recall that for q ∈ L 2 4 the Jost solutions f 1 (q, x, k) and f 2 (q, x, k) satisfy the following integral equations Note that the integrals above are well defined thanks to the estimate in item (ii) of Theorem 2.1.
Inserting formula (3.3) into (3.5), one gets that The main result of this section is an estimate of saying that A is 1-smoothing. To formulate the result in a precise way, we need to introduce the following Banach spaces.
* is a real Banach space. We will use also the complexification of the Banach spaces H M * and H M ζ (this last defined in (1.7)), in which the reality condition f (k) = f (−k) is dropped: Note that for any M ≥ 2 We can now state the main theorem of this section. Let L 2 M,R := f ∈ L 2 M | f real valued .
Theorem 3.1. Let N ∈ Z ≥0 and M ∈ Z ≥4 . Then one has: where the constant C A > 0 can be chosen uniformly on bounded subsets of are real analytic. The following corollary follows immediately from identity (3.7), item (ii) of Theorem 3.1 and the properties of the Fourier transform: where the constant C S > 0 can be chosen uniformly on bounded subsets of H N C ∩ L 2 4 .
In [KST13], it is shown that in the periodic setup, the Birkhoff map of KdV is 1-smoothing. As the map q → S(q, ·) on the spaces considered can be viewed as a version of the Birkhoff map in the scattering setup of KdV, Theorem 3.1 confirms that a result analogous to the one on the circle holds also on the line.
The proof of Theorem 3.1 consists of several steps. We begin by proving item (i). Since F − : L 2 M → H M C is bounded, item (i) will follow from the following proposition: As by Corollary 2.14, m j (q, 0, ·) − 1 ∈ H M * ,C and ∂ x m j (q, 0, ·) ∈ H M ζ,C , j = 1, 2, the identity (3.9) yields follows by Corollary 2.14.
Proof of Theorem 3.1 (i). The claim is a direct consequence of Proposition 3.3 and the fact that for any real valued potential q, In order to prove the second item of Theorem 3.1, we expand the map q → A(q) as a power series of q. More precisely, iterate formula (3.3) and insert the formal expansion obtained in this way in the integral term of (3.5), to get where, with dt = dt 0 · · · dt n , is a polynomial of degree n + 1 in q (cf Appendix B) and ∆ n+1 is given by Since by Proposition 3.3 S(q, ·) is in L 2 , it remains to control the decay of A(q, ·) in k at infinity. Introduce a cut off function χ with χ(k) = 0 for |k| ≤ 1 and χ(k) = 1 for |k| > 2 and consider the series Item (ii) of Theorem 3.1 follows once we show that each term χ(k)sn(q,k) k n of the series is bounded as a map from H N C ∩ L 2 4 into L 2 N +1 and the series has an infinite radius of convergence in L 2 N +1 . Indeed the analyticity of the map then follows from general properties of analytic maps in complex Banach spaces, see Remark B.4. In order to estimate the terms of the series, we need estimates on the maps k → s n (q, k). A first trivial bound is given by However, in order to prove convergence of (3.12), one needs more refined estimates of the norm of k → s n (q, k) in L 2 N . In order to derive such estimates, we begin with a preliminary lemma about oscillatory integrals: Then g ∈ L 2 and for any component α i = 0 one has (3.14) Proof. The lemma is a variant of Parseval's theorem for the Fourier transform; indeed Integrating first in the k variable and using the distributional identity R e ikx dk = 1 2π δ 0 , where δ 0 denotes the Dirac delta function, one gets Denoting dσ i = dt 1 · · · dt i · · · dt n and dσ i = ds 1 · · · ds i · · · ds n , one has, integrating first in the variables s i and t i , where in the second line we have used the Cauchy-Schwarz inequality and the invariance of the integral To get bounds on the norm of the polynomials k → s n (q, k) in L 2 N it is convenient to study the multilinear maps associated with them: The boundedness of these multilinear maps is given by the following Lemma 3.5. For each n ≥ 1 and N ∈ Z ≥0 ,s n : (H N C ∩ L 1 ) n+1 → L 2 N is bounded. In particular there exist constants C n,N > 0 such that For the proof, introduce the operators I j : L 1 → L ∞ , j = 1, 2, defined by n j=1 sin k(t j − t j−1 ) e iktn as a sum of complex exponentials. Note that the arguments of the exponentials are obtained by taking all the possible combinations of ± in the expression t 0 ± (t 1 − t 0 ) ± . . . ± (t n − t n−1 ) + t n . To handle this combinations, define the set and introduce For any σ ∈ Λ n , define α σ = (α j ) 0≤j≤n as Note that for any t = (t 0 , . . . , t n ), one has α σ · t = t 0 + n j=1 σ j (t j − t j−1 ) + t n . For every σ ∈ Λ n , α σ satisfies the following properties: To prove the induction step n n + 1, let α 0 = 1 − σ 1 , . . . , α n = σ n − σ n+1 , α n+1 = 1 + σ n+1 , and defineα n := 1 + σ n ∈ {0, 2}. By the induction hypothesis the vectorα σ = (α 0 , . . . , α n−1 ,α n ) has an odd number of elements non zero. Caseα n = 0: in this case the vector (α 0 , . . . , α n−1 ) has an odd number of non zero elements. Then, since α n = σ n − σ n+1 =α n − α n+1 = −α n+1 , one has that (α n , α n+1 ) ∈ {(0, 0), (−2, 2)}. Therefore the vector α σ has an odd number of non zero elements. Caseα n = 2: in this case the vector (α 0 , . . . , α n−1 ) has an even number of non zero elements. As α n = 2 − α n+1 , it follows that (α n , α n+1 ) ∈ {(2, 0), (0, 2)}. Therefore the vector α σ has an odd number of non zero elements. This proves (3.21). As s n can be written as a sum of complex exponentials,s n (f 0 , . . . , f n )(k) = σ∈Λn The case N = 0 follows directly from Lemma 3.4, since for each σ ∈ Λ n one has by (3.21) that there We now prove by induction thats n : (H N C ∩ L 1 ) n+1 → L 2 N for any N ≥ 1. We start with n = 1. Since we have already proved thats 1 is a bounded map from (L 2 ∩ L 1 ) 2 to L 2 , it is enough to establish the stated decay at ∞. One verifies that Moreover We prove the induction step n n + 1 with n ≥ 1 for any N ≥ 1 (the case N = 0 has been already treated). The terms n+1 where we multiplied and divided by the factor e iktn . Writing Integrating by parts N -times in the integral expression displayed above one has Inserting the formula above in the expression fors n+1 , and using the multilinearity ofs n+1 one gets We analyze the first term in the r.h.s. of (3.23).
where χ is chosen as in (3.12). For the second term in (3.23) it is enough to note that f n I 1 (f n+1 ) ∈ H N C ∩ L 1 and by the inductive assumption it follows thats n (f 0 , . . . , f n I 1 (f n+1 )) ∈ L 2 N . We are left with (3.24). Due to the factor (2ik) N in the denominator, we need just to prove that the integral term is L 2 integrable in the k-variable. Since the oscillatory factor e 2iktn+1 doesn't get canceled when we express the sine functions with exponentials, we can apply Lemma 3.4, integrating first in L 2 w.r. to the variable t n+1 , getting Putting all together, it follows thats n+1 is bounded as a map from (H N C ∩ L 1 ) n+2 to L 2 N for each N ∈ Z ≥0 and the estimate (3.18) holds.
By evaluating the multilinear maps n on the diagonal, Lemma 3.5 says that for any N ≥ 0, ∀n ≥ 1. We now show that for any ρ > 0,Ã(q, ·) is an absolutely and uniformly convergent series in L 2 N +1 for q in B ρ (0), where B ρ (0) is the ball in H N C ∩ L 1 with center 0 and radius ρ. By (3.25) the map q → N +1 n=1 χ(k)sn(q,k) k n is analytic as a map from H N C ∩ L 1 to L 2 N +1 , being a finite sum of polynomialscf. Remark B.4. It remains to estimate the sum It is absolutely convergent since by the L ∞ estimate (3.13) for an absolute constant C > 0. Therefore the series in (3.26) converges absolutely and uniformly in B ρ (0) for every ρ > 0. The absolute and uniform convergence implies that for any N ≥ 0, q →Ã(q, ·) is analytic as a map from H N C ∩ L 1 to L 2 N +1 . It remains to show that identity (3.12) holds, i.e., for every q ∈ H N C ∩ L 1 one has χA(q, ·) =Ã(q, ·) in L 2 N +1 . Indeed, fix q ∈ H N C ∩ L 1 and choose ρ such that q H N C ∩L 1 ≤ ρ. Iterate formula (3.3) N ′ ≥ 1 times and insert the result in (3.5) to get for any k ∈ R \ {0}, By the definition (3.7) of A(q, k) and the expression of S N ′ +1 displayed above Let now N ′ ≥ N , then by Theorem 2.1 (ii) there exists a constant K ρ , which can be chosen uniformly on B ρ (0) such that where for the last inequality we used that q L 1 1 ≤ C q L 2 2 for some absolute constant C > 0. Since For later use we study regularity and decay properties of the map k → W (q, k). For q ∈ L 2 4 real valued with no bound states it follows that W (q, k) = 0, ∀ Im k ≥ 0 by classical results in scattering theory. We define (3.28) We will prove in Lemma 3.8 below that The properties of the map W are summarized in the following Proposition: Proposition 3.6. For M ∈ Z ≥4 the following holds: where the constant C W > 0 can be chosen uniformly on bounded subsets of L 2 M .
(ii) The map Q 0,M are analytic. Here ζ is a function as in (1.8).
We are now able to prove the direct scattering part of Theorem 1.1.
We conclude this section with a lemma about the openness of Q N,M and S M,N .
Proof. The proof can be found in [KT88]; we sketch it here for the reader's convenience. By a classical result in scattering theory [DT79], W (q, k) admits an analytic extension to the upper plane Im k ≥ 0. By definition (3.28) one has Q C = {q ∈ L 2 4 : W (q, k) = 0 ∀ Im k ≥ 0}. Using that (q, k) → W (q, k) is continuous on L 2 4 × R and that by Proposition 3.6, W (q, ·) − 2ik L ∞ is bounded locally uniformly in q ∈ L 2 4 one sees that Q C is open in L  0)) > 0 and σ ∈ H M ζ,C ∩ L 2 N . In the following we denote by C n,γ (R, C), with n ∈ Z ≥0 and 0 < γ ≤ 1, the space of complex-valued functions with n continuous derivatives such that the n th derivative is Hölder continuous with exponent γ. Proof. Clearly H 4 ζ,C ⊆ H 3 C , and by the Sobolev embedding theorem H 3 C ֒→ C 2,γ (R, C) for any 0 < γ < 1/2. It follows that σ → σ(0) is a continuous functional on H 4 ζ,C . In view of the definition of S M,N , the claimed statement follows.

Inverse scattering map
The aim of this section is to prove the inverse scattering part of Theorem 1.1. More precisely we prove the following theorem.
In the next proposition we discuss the properties of the map σ → l(σ, ·). To this aim we introduce, for M ∈ Z ≥2 and ζ as in (1.8), the auxiliary Banach space and its complexification both endowed with the norm f 2 Note that W M ζ differs from H M ζ since we require that f lies just in L ∞ (and not in L 2 as in H M ζ ).
Proposition 4.7. Let N ∈ Z ≥0 and M ∈ Z ≥4 be fixed. Then the following holds true: (i) σ → F ± (σ, ·) are real analytic as maps from S 4,0 to H 1 ∩ L 2 3 . Moreover there exists C > 0 so that where C ′ depends locally uniformly on σ ∈ S M,N .
We are finally able to prove that there exists a potential q ∈ Q with prescribed scattering coefficient σ ∈ S M,N . More precisely the following theorem holds.
Theorem 4.8. Let N ∈ Z ≥0 , M ∈ Z ≥4 and σ ∈ S M,N be fixed. Then there exists a potential q ∈ Q such that S(q, ·) = σ.
It remains to show that q ∈ Q N,M and that the map S −1 : S M,N → Q N,M is real analytic. We take here a different approach then [KT86]. In [KT86] the authors show that the map S is complex differentiable and its differential d q S is bounded invertible. Here instead we reconstruct q by solving the Gelfand-Levitan-Marchenko equations and we show that the inverse map S M,N → Q N,M , σ → q is real analytic. We outline briefly the procedure. Given two reflection coefficients ρ ± satisfying items (i)-(v) of Proposition 4.5 and arbitrary real numbers c + ≤ c − , it is possible to construct a potential q + on [c + , ∞) using ρ + and a potential q − on (−∞, c − ] using ρ − , such that q + and q − coincide on the intersection of their domains, i.e., q + | [c+,c−] = q − | [c+,c−] . Hence q defined on R by q| [c+,+∞) = q + and q| (−∞,c−] = q − is well defined, q ∈ Q and r ± (q, ·) = ρ ± , i.e., ρ + and ρ − are the reflection coefficients of the potential q [Fad64, Mar86,DT79]. We postpone the details of this procedure to the next section.

Gelfand-Levitan-Marchenko equation
In this section we prove how to construct for any σ ∈ S M,N two potentials q + and q − with q + ∈ H N x≥c ∩ L 2 M,x≥c respectively q − ∈ H N x≤c ∩ L 2 M,x≤c , where for any c ∈ R and 1 ≤ p ≤ ∞ where x := (1 + x 2 ) 1/2 . We will write H N C,x≥c for the complexification of H N x≥c . For 1 ≤ α, β ≤ ∞, we define . Analogously one defines the spaces L p x≤c , H N x≤c , L 2 M,x≤c and L α x≤c L β y≤0 , mutatis mutandis. The potentials q + and q − mentioned at the beginning of this section are constructed by solving an integral equation, known in literature as the Gelfand-Levitan-Marchenko equation, which we are now going to described in more detail. Given σ ∈ S , define the functions F ± (σ, ·) as in (4.10). See Proposition 4.7 for the analytical properties of the maps σ → F ± (σ, ·). To have a more compact notation, in the following we will denote F ±,σ := F ± (σ, ·).
Remark 4.9. From the decay properties of F ′ ±,σ one deduces corresponding decay properties of F ±,σ . Indeed one has (4.17) The Gelfand-Levitan-Marchenko equations are the integral equations given by where E ±,σ (x, y) are the unknown functions and F ±,σ are given and uniquely determined by σ through formula (4.10). If (4.18) and (4.19) have solutions with enough regularity, then one defines the potentials q + and q − through the well-known formula - [Fad64] q (4.20) The main purpose of this section is to study the maps R ±,c defined by . As such they are real analytic. In order to prove Theorem 4.10 we look for solutions of (4.18) and (4.19) of the form where B ±,σ (x, y) are to be determined. Inserting the ansatz (4.22) into the Gelfand-Levitan-Marchenko equations (4.18), (4.19), one gets We will prove in Lemma 4.12 below that there exists a solution B +,σ of (4.23) and a solution B −,σ of (4.24) with ∂ x B +,σ (·, 0) ∈ H 1 x≥c respectively ∂ x B −,σ (·, 0) ∈ H 1 x≤c . By (4.20) we get therefore with B ±,σ (x, y) := E ±,σ (x, y) + F ±,σ (x, y) as in (4.22). Now we study analytic properties of the maps B ±,c in case the scattering coefficient σ belongs to S 4,N with arbitrary N ∈ Z ≥0 . Later we will treat the case where σ ∈ S M,0 , M ∈ Z ≥4 .
where K > 0 is a constant which can be chosen locally uniformly in σ ∈ S 4,N .
The main ingredient of the proof of Proposition 4.11 is a detailed analysis of the solutions of the integral equations (4.23)-(4.24), which we rewrite as where for every x ∈ R fixed, the two operators K + x,σ : L 2 y≥0 → L 2 y≥0 and K − x,σ : L 2 y≤0 → L 2 y≤0 are defined by and the functions f ±,σ are defined by As the claimed statements for B +,c and B −,c can be proved in a similar way we consider B +,c only. To simplify notation, in the following we omit the subscript " + ". In particular we write B σ ≡ B +,σ , F σ ≡ F +,σ , f σ ≡ f +,σ and K x,σ ≡ K + x,σ .
We give the following definition: a function h σ : [c, ∞) × [0, ∞) → R, which depends on σ ∈ S 4,N , will be said to satisfy (P ) if the following holds true: (P 2) There exists a constant K c > 0, which depends locally uniformly on σ ∈ H 4 ζ,C ∩ L 2 N , such that (4.31) is real analytic as a map from S 4,N to L 2 x≥c L 2 y≥0 [L 2 x≥c ]. We have the following lemma: Lemma 4.12. Fix N ≥ 0 and c ∈ R. For every σ ∈ S 4,N equation (4.23) has a unique solution . Moreover for all integers n 1 , n 2 ≥ 0 with n 1 + n 2 ≤ N + 1 , the function ∂ n1 x ∂ n2 y B σ satisfies (P ).
Proof. Let N ∈ Z ≥0 and c ∈ R be fixed. The proof is by induction on j 1 + j 2 = n, 0 ≤ n ≤ N . For each n we prove that ∂ j1 x ∂ j2 y B σ and its derivatives ∂ j1+1 x ∂ j2 y B σ , ∂ j1 x ∂ j2+1 y B σ satisfy (P ). Thus the claim follows.
Case n = 0. Then j 1 = j 2 = 0. We need to prove existence and uniqueness of the solution of equation (4.27). By Lemma D.2 [Proposition 4.7 and Lemma D.1] the function f σ and its derivatives ∂ x f σ , ∂ y f σ [F σ ] satisfy assumption (P ) [(H)-cf Appendix C]. Thus by Lemma C.7 (i) it follows that B σ = (Id + K σ ) −1 f σ and its derivatives ∂ x B σ , ∂ y B σ satisfy (P ). Note that if N = 0 the lemma is proved. Thus in the following we assume N ≥ 1.
Lemma 4.12 implies in a straightforward way Proposition 4.11.
Proof of Proposition 4.11. By Lemma 4.12, ∂ n x B σ satisfies (P ) for every 1 ≤ n ≤ N + 1. In particular for every 1 ≤ n ≤ N +1, σ → ∂ n x B σ (·, 0) is real analytic as a map from S 4,N to L 2 x≥c and ∂ n x B σ (·, 0) L 2 0) is real analytic as a map from S 4,N to H N x≥c . The claimed estimate follows in a straightforward way.
In the next result we study the case σ ∈ S M,0 for arbitrary M ≥ 4. Proof. We prove the result just for R +,c , since for R −,c the proof is analogous. As before, we suppress the subscript "+" from the various objects. Consider the Gelfand-Levitan-Marchenko equation (4.18). Multiply it by x M−3/2 to obtain (4.34) The function satisfies h σ (x, ·) ∈ L 2 y≥0 and one checks that h σ ∈ C 0 x≥c L 2 y≥0 ∩ C 0 x≥c,y≥0 . We show now that h σ ∈ L 2 x≥c L 2 y≥0 . By Lemma A.1 (A3) and Proposition 4.7 for N = 0 it follows that Consider now h σ (x, 0) = − x M−3/2 F σ (x). By (4.17) it follows that h σ (·, 0) ∈ L 2 x≥c . Finally the map σ → h σ [σ → h σ (·, 0)] is real analytic as a map from S M,0 to L 2 x≥c L 2 y≥0 [L 2 M−3/2,x≥c ]. Proceeding as in the proof of Lemma C.5, one shows that there exists a solution E σ of equation (4.18) which 0)] is real analytic as a map from S M,0 to L 2 x≥c L 2 y≥0 [L 2 M−3/2,x≥c ]. Furthermore its derivative ∂ x E σ satisfies the integral equation Multiply the equation above by x M−3/2 , to obtain (Id where h ′ σ (x, y) := F ′ σ (x + y). We claim thath σ ∈ L 2 x≥c L 2 y≥0 and σ →h σ is real analytic as a map S M,0 → L 2 x≥c L 2 y≥0 . By Lemma A.1 (A0) the first term of (4.36) satisfies and by Lemma A.1 (A1) the second term of (4.36) satisfies Moreover σ →h σ is real analytic as a map from S M,0 to L 2 x≥c L 2 y≥0 , being composition of real analytic maps.
Since S is 1-1 it follows that q c ≡ q.

By
We prove the following lemma.
Proof. As a first step we show that Ω t (σ) ∈ S for every t ≥ 0.
Remark 5.2. One can adapt the proof above, putting ζ(k) ≡ 1, to shows that the spaces H N ∩ L 2 M , with integers N ≥ 2M ≥ 2, are invariant by the Airy flow. Indeed the Fourier transform F − conjugates the Airy flow with the linear flow Ω t , i.e., U t Proof of Theorem 1.3. Recall that by [GGKM74] the scattering map S conjugate the KdV flow with the linear flow Ω t (σ)(k) := e −i8k 3 t σ(k), i.e., where N, M are integers with N ≥ 2M ≥ 8. By Theorem 1.1, S(q) ≡ S(q, ·) ∈ S M,N . By Lemma 5.1 the flow Ω t preserves the space S M,N for every t ≥ 0. Thus Ω t • S(q) ∈ S M,N , ∀t ≥ 0. By the bijectivity of S it follows that S −1 • Ω t • S(q) ∈ Q N,M ∀t ≥ 0. Thus item (i) is proved.
We prove now item (ii). Remark that by item (i), U t KdV (q) ∈ L 2 M for any t ≥ 0. Since U t Airy preserves the space H N ∩ L 2 M (N ≥ 2M ≥ 8), it follows that for q ∈ Q N,M the difference U t KdV (q) − U t Airy (q) ∈ H N ∩ L 2 M , ∀t ≥ 0. We prove now the smoothing property of the difference U t KdV (q) − U t Airy (q). Since and since S = F − + A, The 1-smoothing property of the difference U t KdV (q)−U t Airy (q) follows now from the smoothing properties of A and B described in item (ii) of Theorem 1.1. The real analyticity of the map q → U t KdV (q)−U t Airy (q) follows from formula (5.5) and the real analyticity of the maps A, B and S.

A Auxiliary results.
For the convenience of the reader in this appendix we collect various known estimates used throughout the paper.
Lemma A.1. Fix an arbitrary real number c. Then the following holds: (A0) The linear map T 0 : L 2 1/2,x≥c → L 2 x≥c L 2 y≥0 defined by is continuous, and there exists a constant K c > 0, depending on c, such that is continuous, and is continuous, and there exists a constant K c > 0, depending on c, such that is continuous, and there exists a constant K c > 0, depending on c, such that (A4) The bilinear map T 4 : L 1 x≥c × L 2 x≥c L 2 y≥0 → L 2 x≥c L 2 y≥0 defined by is continuous, and there exists a constant K c > 0, depending on c, such that is bounded and satisfies Proof. Inequality (A1), (A4) can be verified in a straightforward way. To prove (A0) make the change of variable ξ = x + y and remark that +∞ c +∞ 0 |g(x + y)| 2 dx dy ≤ K c +∞ 0 |ξ − c| |g(ξ)| 2 dξ .
We prove now (A2): using Cauchy-Schwartz, one gets In order to prove (A3) take a function h ∈ L 2 x≥c and remark that where for the last inequality we used the Hardy-Littlewood inequality.

B Analytic maps in complex Banach spaces
In this appendix we recall the definition of an analytic map from [Muj86]. Let E and F be complex Banach spaces. A mapP k : E k → F is said to be k-multilinear if P k (u 1 , . . . , u k ) is linear in each variable u j ; a multilinear map is said to be bounded if there exist a constant C such that P k (u 1 , · · · , u k ) ≤ C u 1 · · · u k ∀u 1 , . . . , u k ∈ E.
Its norm is defined by P k := sup u j ∈E, u j ≤1 P k (u 1 , . . . , u k ) .
A map P k : E → F is said to be a polynomial of order k if there exists a k-multilinear mapP k : E → F such that P k (u) =P k (u, . . . , u) ∀u ∈ E.
The polynomial is bounded if it has finite norm We denote with P k (E, F ) the vector space of all bounded polynomials of order k from E into F .
Definition B.1. Let E and F be complex Banach spaces. Let U be a open subset of E. A mapping f : U → F is said to be analytic if for each a ∈ U there exists a ball B r (a) ⊂ U with center a and radius r > 0 and a sequence of polynomials P k ∈ P k (E, F ), k ≥ 0, such that is convergent uniformly for u ∈ B r (a); i.e., for any ǫ > 0 there exists K > 0 so that Finally let us recall the notion of real analytic map. Therefore analytic maps inherit the properties of C-differentiable maps; in particular the composition of analytic maps is analytic. For a proof of the equivalence of the two notions see [Muj86], Theorem 14.7.
Remark B.4. Any P k ∈ P k (E, F ) is an analytic map. Let f (u) = ∞ m=0 P m (u) be a power series from E into F with infinite radius of convergence with P m ∈ P m (E, F ). Then f is analytic ([Muj86], example 5.3, 5.4).

C Properties of the solutions of integral equation (4.27)
In this section we discuss some properties of the solution of equation (4.27) which we rewrite as Here σ ∈ S 4,N , N ≥ 0, h σ is a function h σ : [c, +∞) × [0, +∞) → R, with c arbitrary, which satisfies (P ). We denote by is real analytic as a map from S 4,N to H 1 ∩ L 2 3 [L 2 4 ]. Moreover the operators Id ± K x,σ : L 2 y≥0 → L 2 y≥0 with K x,σ defined as are invertible for any x ≥ c, and there exists a constant C σ > 0, depending locally uniformly on σ ∈ H 4 ζ,C ∩ L 2 N , such that sup Finally σ → (Id ± K x,σ ) −1 are real analytic as maps from S 4,N to (L 2 x≥c L 2 y≥0 ).
Remark C.1. The pairing is a bounded bilinear map and hence analytic. Let now σ → h σ be a real analytic map from S 4,0 to L 2 x≥c L 2 y≥0 and let K σ as in (H). Then by Lemma D.1 (iii) it follows that σ → (Id analytic as a map from S 4,0 to L 2 x≥c L 2 y≥0 as well. Remark C.2. By the Sobolev embedding theorem, assumption (H) implies that F σ ∈ C 0,γ (R, C), γ < 1 2 . By assumption (H) the map (c, ∞) → (L 2 y≥0 ), x → K x,σ is differentiable and its derivative is the operator as one verifies using that for x > c and ǫ = 0 sufficiently small and the fact that the translations are continuous in L 1 . Therefore the following lemma holds Lemma C.3. K x,σ and thus (Id + K x,σ ) −1 is a family of operators from L 2 y≥0 to L 2 y≥0 which depends continuously on the parameter x. Moreover the map (c, ∞) → (L 2 y≥0 ), x → K x,σ is differentiable and its derivative is the operator K ′ x,σ defined in (C.5).
(P 3) It follows by Lemma A.1 (A2) and the fact that F σ and F σ (·, 0) are composition of real analytic maps.
We study now the solution of equation (C.1).
The following lemma will be useful in the following: Lemma C.6. (i) Let F σ satisfy (H), and g σ ∈ C 0 x≥c L 2 y≥0 ∩ L 2 x≥c L 2 y≥0 be such that g σ L 2 where K c > 0 depends locally uniformly on σ ∈ H 4 ζ,C ∩ L 2 N .
(iii) Assume that F σ satisfies (H) and that the map S 4,N → H 1 Then the function , and since g σ ∈ L 2 x≥c L 2 y≥0 it follows that , which implies (C.11). We show now that The continuity of the translation in L 1 and the assumption g σ ∈ C 0 x≥c L 2 y≥0 imply that Φ σ (x + ǫ, ·) − Φ σ (x, ·) L 2 y≥0 → 0 as ǫ → 0, thus Φ σ ∈ C 0 x≥c L 2 y≥0 . The real analyticity of σ → Φ σ follows from Lemma A.1 (A4) and the fact that Φ σ is composition of real analytic maps.
(iii) We skip an easy proof.
If the function h σ is more regular one deduces better regularity properties of the corresponding solution of (C.1).

D Proof from Section 4
D.1 Properties of K ± x,σ and f ±,σ .
We begin with proving some properties of K ± x,σ and f ±,σ , defined in (4.28) and (4.30), which will be needed later.

Properties of Id + K ±
x,σ . In order to solve the integral equations (4.27) we need the operator Id + K + x,σ to be invertible on L 2 y≥0 (respectively Id + K − x,σ to be invertible on L 2 y≤0 ). The following result is well known: CK87a]). Let σ ∈ S 4,0 and fix c ∈ R. Then the following holds: (i) For every x ≥ c, K + x,σ : L 2 y ≥ 0 → L 2 y ≥ 0 is a bounded linear operator; moreover sup x≥c K + x,σ (L 2 y≥0 ) < 1, , is linear and bounded. Moreover the operators Id ± K + σ are invertible on L 2 x≥c L 2 y≥0 and there exists a constant K c > 0, which depends locally uniformly on σ ∈ S 4,0 , such that −1 are real analytic as maps from S 4,0 to (L 2 x≥c L 2 y≥0 ). Analogous results hold also for K − x,σ replacing L 2 x≥c L 2 y≥0 by L 2 x≤c L 2 y≤0 .
Properties of f ±,σ . First note that f ±,σ , defined by (4.30), are well defined. Indeed for any σ ∈ S 4,0 , Proposition 4.7 implies that F ±,σ ∈ H 1 ∩ L 2 3 ⊂ L 2 . Hence for any x ≥ c, y ≥ 0 the map given by z → F +,σ (x + y + z)F +,σ (x + z) is in L 1 z≥0 . Similarly, for any x ≥ c, y ≥ 0, the map given by z → F −,σ (x + y + z)F −,σ (x + z) is in L 1 z≤0 . In the following we will use repeatedly the Hardy inequality [HLP88] x m +∞ x g(z)dz The inequality is well known, but for sake of completeness we give a proof of it in Lemma A.1 (A3).
We analyze now the maps σ → f ±,σ . Since the analysis of f +,σ and the one of f −,σ are similar, we will consider f +,σ only. To shorten the notation we will suppress the subscript " + " in what follows.
Lemma D.3. Fix c ∈ R, N ∈ Z ≥0 and let σ ∈ S 4,N . Let F σ be given as in (4.10). Then the following holds true: (i) Let σ → b σ be real analytic as a map from S 4,N to H 1 x≥c , satisfying b σ H 1 where K c > 0 depends locally uniformly on σ ∈ H 4 ζ,C ∩L 2 N . Then for every integer k with 0 ≤ k ≤ N , the function H σ (x, y) := F (k) σ (x + y) b σ (x) (D.10) satisfies (P ).
H σ satisfies (P 3). The real analyticity property follows from Lemma A.1 and Proposition 4.7, since for every 0 ≤ k ≤ N , H σ is product of real analytic maps.