Asymptotics in shallow water waves

In this paper we consider the initial value problem for a family of shallow water equations on the line $\R$ with various asymptotic conditions at infinity. In particular we construct solutions with prescribed asymptotic expansion as $x\to\pm\infty$ and prove their invariance with respect to the solution map.


Introduction
In this paper we study the initial value problem for a family of shallow water equations on the line R that contains the Camassa-Holm equation (CH), m t + um x + 2mu x = 0, m := u − u xx u| t=0 = u 0 (1) with initial data u 0 that has an asymptotic expansion of order N ∈ Z ≥0 , where c ± k are real constants for 0 ≤ k ≤ N . (Note that the asymptotic expansion of u 0 as x → +∞ is not necessarily the same as the one when x → −∞.) We are interested to study whether (1) possesses a solution with the prescribed initial data and if the corresponding solution map (provided it exists) preserves the asymptotic expansion (2) in the sense described below. We postpone for the moment the questions related to the regularity of the initial data u 0 and the solutions.
Note that similar questions were studied for the Korteweg-de Vries equation (KdV) in a series of papers [2,3,4] as well as in [16] for the modified KdV equation. In the present paper we develop a different approach to this problem. The new approach is influenced by the seminal paper of Arnold [1] (cf. also the related papers [11,21,19,6]) and is based on the introduction of a group of asymptotic diffeomorphisms on the line. The group of asymptotic diffeomorphisms is an infinite dimensional topological group modeled on a Banach space of functions on the line with prescribed asymptotic expansions at infinity. This provides a general framework for studying the asymptotics of solutions for a relatively large class of nonlinear equations. As a model we consider equation (1) (as well as its generalization (3) below). The approach is not restricted to equations with one spacial dimension (cf. [18]).
Since the time when equation (1) was derived (in [12] using algebraic principles and in [5] as a model for shallow water waves), it has been attracting a lot of attention. One of the reasons is that it is completely integrable and, unlike the classical Korteweg-de Vries equation, admits non-smooth solitary traveling waves u(x, t) = ce −|x−ct| called peakons. 1 Equation (1) is a particular case of a 1-parameter family of shallow water equations ( [13]), where b is a real parameter. When b = 2 we get (1); the case b = 3 is the Degasperis-Procesi equation (DP). Both equations are known to be completely integrable and to admit solitary traveling waves. It is worth noting that, although the cases when b = 2, 3 in (3) are not known to be completely integrable, they still admit solitary traveling waves ( [14]). The main result of this paper states that, under some technical assumptions on the remainder term o(|x| −N ) in (2), for any b ∈ R and for any m ≥ 3 and N ≥ 0 there exists T > 0 and a solution u ∈ C 0 ([0, T ], H m loc (R))∩C 1 ([0, T ], H m−1 loc (R)) of (3) so that for any t ∈ [0, T ], u(t) has the asymptotic expansion (2) with coefficients c ± k for 0 ≤ k ≤ N that may depend on t ∈ [0, T ] -see Theorem 1.1 and Theorem 1.2 below for the precise statement of the result. Here H m loc (R) denotes the space of measurable real valued functions on R whose weak derivatives up to order m are locally square integrable.
Analytic set-up: In order to perform analysis on functions satisfying (2) we assume that the remainder term o(|x| −N ) lies in a suitable weighted Sobolev space. The main idea is to incorporate the asymptotic terms in (2) into a new functional space that we call an asymptotic space. It turns out that the asymptotic spaces defined this way are Banach spaces that enjoy the Banach algebra property in the sense that the (pointwise) product of functions is continuous. Another important feature of these spaces is that they admit a natural (graded) Lie algebra structure that corresponds to the group structure of a special class of diffeomorphisms of R (see below). In this paper we restrict our attention to the following asymptotic spaces: (1) The asymptotic space A m n,N . Take m ≥ 1 and N ≥ 0. In order to define the asymptotic space A m n,N (R), we first define the weighted Sobolev space supplied with the norm , where x := √ 1 + x 2 and f (j) = ∂ j x f denotes the j-th weak derivative of f . Note that f ∈ W m N (R) implies (Lemma 6.3) One easily sees that (2) is equivalent to where the constants a k , b k ∈ R, 0 ≤ k ≤ m, are uniquely determined from c ± k , 0 ≤ k ≤ N . In view of this we define the asymptotic space By (5), the elements of A m N (R) satisfy the asymptotic expansion (6) (and equivalently, (2)). More generally, for 0 ≤ n ≤ N , define We set, A m n,N (R) := W m N (R) for n ≥ N + 1 .
For any R > 0 denote by B A m n,N (R) the ball of radius R in A m n,N centered at the origin. The following theorem is proved in Section 3.
) of (3) that depends continuously on the initial data u 0 ∈ B A m n,N (R) in the sense that the data-to-solution map, is continuous. Moreover, the coefficients a k , b k , n ≤ k ≤ min{2n, N }, in the asymptotic expansion of the solution u(t) are independent of t.
(2) The asymptotic space A m n,N . Although the choice of the remainder space W m N is natural as it mimics the properties of the asymptotic terms 1/ x k and x/ x k+1 with respect to the differentiation in the x-direction, this choice is not unique: one can choose other Banach spaces of functions that satisfy the remainder condition (5). For any m ≥ 1 and N ≥ 0 consider the weighted Sobolev space, One has the following continuous inclusions: . Moreover, the elements of H m N (R) satisfy the remainder condition (5) (see Lemma 6.4). Hence, for 0 ≤ n ≤ N , we can define in a similar way as above the asymptotic space, We set, A m n,N (R) := H m N (R) for n ≥ N + 1 .
For any R > 0 denote by B A m n,N (R) the ball of radius R in A m n,N centered at the origin. The following theorem is proved in Section 4. Theorem 1.2. For any b ∈ R, m ≥ 3, N ≥ 0, n ≥ 0 and R > 0, there exists T > 0 such that for any u 0 ∈ B A m n,N (R), there exists a unique solution u ∈ C 0 ([0, T ], A m n,N (R)) ∩ C 1 ([0, T ], A m−1 n,N (R)) of (3) that depends continuously on the initial data u 0 ∈ B A m n,N (R) in the sense that the data-to-solution map, is continuous. Moreover, the coefficients a k , b k , n ≤ k ≤ min{2n, N }, in the asymptotic expansion of the solution u(t) are independent of t.
Remark 1.1. Note that, unlike in Theorem 1.1, the case n = 0 is not excluded in Theorem 1.2. Hence, the class of solutions obtained in Theorem 1.2 contains, in particular, solutions u(t) such that for any t ∈ [0, T ], u(t, x) → c ± 0 as x → ±∞. The constants c ± 0 are not necessarily zero, or equal to each other. This implies that such solutions are not necessarily summable or square integrable. Remark 1.2. In view of Theorem 1.1 and Theorem 1.2 the coefficients c ± k , n ≤ k ≤ n * := max{2n, N } in the asymptotic expansion of the solution u are conservation laws of equation (3) for any b ∈ R. Hence we get 2(n * − n + 1) functionally independent integrals of motion. This is in contrast to the KdV and the modified KdV equation where only the leading coefficients c ± n are preserved ( [16]). Remark 1.3. By taking n = N + 1 in Theorem 1.1 and Theorem 1.2 we obtain, in view of conventions (7) and (10) [7]. Remark 1.4. As the goal of this work is to study the spacial asymptotics of the solutions we do not attempt to lower the regularity exponent m in our spaces. Note however that the method allows for analogs of Theorem 1.1 and Theorem 1.2 to be proved with m > 3/2. We refer to [9,10,20] where low-regularity results are proved for the CH equation. In order to keep the paper as nontechnical as possible, we also do not consider the case of the asymptotic L p -spaces A m,p n,N and A m,p n,N , p ≥ 1 (see [18] for the definitions and the main properties of these spaces). Since we always assume p = 2, we simplify the notation by omitting reference to p.
Remark 1.5. The definition of the asymptotic group in Section 2 can be extended by allowing linear terms c ± −1 x, c ± −1 = −1 in the asymptotic expansion (2) and the corresponding asymptotic spaces. Note that similarly to KdV, the CH equation (1) possesses the unbounded solution u(t, x) := x/[3(t − 1)] that, at t = 1, blows-up at all points x ∈ R except x = 0.
Organization of the paper: The paper is organized as follows. In Section 2 we define the group of asymptotic diffeomorphisms on the line and discuss its main properties. At the end of the section we formulate Proposition 2.1 that establishes a relation between the solutions of (3) and the solutions of a dynamical system formulated in terms of the asymptotic group. Section 3 is devoted to the proof of Theorem 1.1. Theorem 1.2 is proved in Section 4. Proposition 2.1 is proved in Appendix A. In Appendix B we study the properties of the operator 1 − ∂ 2 x acting on weighted spaces and, for the convenience of the reader, prove in much detail several auxiliary results needed in the main body of the paper.

Groups of asymptotic diffeomorphisms
The topology on AD m n,N (resp. AD m n,N ) is inherited in a natural way from the Banach structure of A m n,N (resp. A m n,N ). The following theorem follows from the analysis in [18].
Theorem 2.1. 1) For any m ≥ 2, N ≥ 0, n ≥ 0, the composition of maps and 3) The same statements in 1) and 2) are true if A is replaced by A. We call AD m n,N and AD m n,N groups of asymptotic diffeomorphisms. Note that AD m n,N ⊆ AD m n,N ⊆ Diff 1 + (R).
Let us also mention another group of diffeomorphisms that was studied in [15]: Observe that the inclusion AD m n,N ⊆ D m (R) holds for n ≥ 1 but D m (R) ⊆ AD m 0,0 ; in fact, by convention (7), D m (R) = AD m 1,0 .
Remark 2.1. For simplicity we have omitted R in the notation for AD m n,N and AD m n,N . Henceforth in this paper we shall also omit R in the notation for other function spaces.
The significance of these groups of asymptotic diffeomorphisms stems from their use for introducing Lagrangian coordinates. This is summarized in the following two results (in which we use˙to denote the t-derivative).
The same statement is true if A is replaced by A.
Remark 2.2. This is an important lemma. It shows in particular that although the asymptotic group AD m n,N is not a Lie group in classical sense (as the composition (11) is not C ∞ -smooth 2 ) one can still define the Lie-group exponential map, exp AD m n,N : A m n,N → AD m n,N , u → ϕ(1), where ϕ ∈ C 1 ([0, ∞), AD m n,N ) is the solution of (12) and u ∈ A m n,N is independent of t. In this sense, the asymptotic space A m n,N can be thought as the "Lie algebra" of AD m n,N . The following Proposition establishes a relation between the solutions of (3) and the solutions of a dynamical system formulated in terms of the asymptotic group.
Proposition 2.1. Assume that m ≥ 3, N ≥ 0, n ≥ 1, and u 0 ∈ A m n,N . Then there exists a bijective correspondence between solutions of equation ( The same statements are true if A is replaced by A; we then allow n ≥ 0. Lemma 2.1 and Proposition 2.1 are proved in Appendix A. They will be used in Section 3 to prove Theorem 1.1 and in Section 4 to prove Theorem 1.2. 3 Analysis on the asymptotic space A m n,N In this section we prove Theorem 1.1. The proof is based on Proposition 2.1 that establishes a correspondence between solutions of (3) and the dynamical system (13) on AD m n,N × A m n,N . Our first goal is to prove that the right hand side of the first equation in (13) defines a C ∞ -smooth vector field on an open neighborhood of (id, 0) in AD m n,N × A m n,N .

Mapping Properties of
Let m ≥ 1, N ≥ 0, and n ≥ 0. In order to study the mapping properties of the operator Λ := 1 − ∂ 2 x and its inverse on A m N we introduce the modified weighted Sobolev space Accordingly, for n ≥ 0 we define the modified asymptotic spacẽ where the inclusions are continuous. There is a natural splitting where the finite-dimensional space is supplied with the norm N k=n (|a k | + |b k |). First, we prove the following Lemma.
Proof. Take f ∈ W m+2 N . For any 0 ≤ j ≤ m, By the definition of W m+2 N , for any 0 ≤ j ≤ m, and for any 0 In addition, for any m ≥ 1, and Combining together (20)-(24) we see that Λ(f ) ∈ W m N and (19) is continuous. As Λ : H m+2 → H m is a linear isomorphism and as W m+2 where the operators Q ± are defined in (83). This together with Lemma 6.1 (i) then gives that Further, in view of (84) and (85) we have Hence, .
As x N +m g (m) ∈ L 2 we obtain from Lemma 6.1 (i) that It follows from (25) that As f, g ∈ W m N we get that Combining (26), (28), and (29) we conclude that f ∈ W m+2 N , and therefore (19) is onto. Finally, the Lemma follows from the open mapping theorem.
As a corollary of Lemma 3.1 we obtain Proposition 3.1. For any m ≥ 1, N ≥ 0, and n ≥ 0, the mapping, is a linear isomorphism.
Proof. For the sake of further reference we first record that for any k ≥ 0, and As a consequence, for any k ≥ N + 1 and for any r ≥ 1, This implies that for any k ≥ 0 and for any r ≥ 1, Take u ∈Ã m+2 n,N , In view of the splitting (18), (31), and (32), we have, + Λ(f ) .
Here and below we use the convention that a sum vanishes if its upper limit is strictly less than its lower one. It follows from (35)-(37) and Lemma 3.1 that (30) is continuous. Moreover, (30) is injective by Lemma 6.1 (iv): by (89), For any k ≥ 0 and l ≥ 0 we have and, similarly, Taking l ≥ 0 such that k + (2l + 2) ≥ N + 1 we get from (31)-(33) that Hence, in view (39), (40), (41) and Lemma 3.1, and (34), for any k ≥ 0 there exists u k , w k ∈Ã m+2 n,N such that 1 x k = Λ(u k ) and This together with Lemma 3.1 and (38) implies that there exists u ∈Ã m+2 n,N such Finally, the Proposition follows from the open mapping theorem.

Smoothness of the conjugate map
Assume that m ≥ 3, N ≥ 0, and n ≥ 0. In this section we study the conjugate map Hence, Now we claim that and are both C ∞ -smooth. In fact, let us first verify that is C ∞ -smooth. This follows from Lemma 6.10 as ϕ ′ = 1 + u ′ > 0, u ∈ A m n,N , and as the linear map u → u ′ , A m n,N → A m−1 n+1,N +1 , is continuous by Lemma 6.6. Note that any continuous linear (or multilinear) map is C ∞ -smooth. In particular, by the continuity of the pointwise product in A m−1 0,N +1 we conclude from (47) that is smooth. Then the smoothness of (45) follows from the smoothness of the maps (48), (49), and the continuity of the pointwise product (Lemma 6.6), Finally, in view of the continuity of the inclusionÃ m n,N +2 ⊆ A m−2 n,N +2 , one gets from (44), (45), and (46) that (42) is C ∞ -smooth.
The main result of this subsection is the following Proposition.
is a C ∞ -diffeomorphism.
Proof. By Lemma 3.2 the map (42) is C ∞ -smooth. The differential of σ at the point (id, 0), d (id,0) σ : A m n,N ×Ã m n,N +2 → A m n,N × A m−2 n,N +2 , is given by Then, it follows from Proposition 3.1 that (51) is a C ∞ -diffeomorphism onto its image. Here we used that σ(ϕ, 0) = (ϕ, 0) for any ϕ ∈ AD m n,N . Note also that for any (ϕ, v) ∈ AD m n,N ×Ã m n,N +2 , is a linear isomorphism of Banach spaces. In particular, the mapping is bijective. Finally, by computing the differential of C at an arbitrary point (ϕ, v) ∈ V ×Ã m n,N +2 we obtain that where id A m n,N : A m n,N → A m n,N is the identity in A m n,N and * denotes a bounded linear map A m n,N ×Ã m n,N +2 → A m−2 n,N +2 . As (53) is a linear isomorphism we conclude from (55) that is a linear isomorphism. Applying the inverse function theorem we get that C is a local C ∞ -diffeomorphism. As C is bijective we conclude that it is a C ∞diffeomorphism.

Smoothness of the vector field
Take b ∈ R, m ≥ 3, and N ≥ 0, and n ≥ 0. Here we consider the mapping, First we prove the following Proposition. Proof. We follow the lines of the proof of Lemma 3.2. For any (ϕ, v) ∈ AD m n,N × A m n,N one has in view of Lemma 6.2, Using Lemma 6.6 and Lemma 6.10 we get, v ∈ A m n,N , Hence, in view of Lemma 6.6, v · 1 ϕ ′ ∈ A m−1 n,N , and therefore, where we used that n ≥ 1. This combined with the continuity of the pointwise product (Lemma 6.6) implies that is C ∞ -smooth. The other terms in (57) are treated similarly.
Remark 3.1. The proof of Proposition 3.3 shows that Finally, combining Proposition 3.2, Proposition 3.3, and the fact that the inclusion,Ã m 2n+1,N +2 ⊆ A m 2n+1,N , is continuous, we see that for n ≥ 1 the mapping, where U is the open neighborhood given by Proposition 3.2 is C ∞ -smooth. In particular, we see that the mapping, is C ∞ -smooth. Hence, we proved the following Theorem.

Proof of Theorem 1.1
In view of Theorem 3.1 the right-hand side of the first equation in (13) is a C ∞ -smooth vector field on U × A m n,N . Hence, by the existence, uniqueness, and continuous (or smooth) dependence on parameters theorem for ODE's in Banach spaces [17], there exists R ′ > 0 and T ′ > 0 so that for any u 0 in the ball B A m n,N (R ′ ) of radius R ′ in A m n,N centered at the origin, there exists a unique solution (ϕ, v) of (13) in C 1 ([0, T ′ ], U × A m n,N ). This solution depends continuously on the initial data u 0 ∈ B A m n,N (R ′ ), in the sense that the data-tosolution map, is continuous. In view of Proposition 2.1, the curve u : N ) and, by item 1) of Theorem 2.1, u depends continuously on the initial data u 0 ∈ B A m n,N (R ′ ), in the sense that the data-to-solution map, is continuous. Next, recall that the solutions of (3) possess the following scaling invariance: if u(t), t ∈ [0, τ ], τ > 0, is a solution of (3) then for any λ > 0, u λ (t) := λu(λt), t ∈ [0, τ /λ], is a solution of the first equation in (3) so that u| t=0 = λu 0 . Now, take an arbitrary R > R ′ and denote T := µT ′ where µ := R ′ /R. Let w 0 ∈ B A m n,N (R). Then, u 0 := µw 0 ∈ B A m n,N (R ′ ) and hence, there exists a unique solution u of equation (3)

By the scaling invariance, w(t) := u(t/µ)/µ is a solution of the first equation in
The solution w is necessarily unique. Otherwise, using scaling invariance, we will obtain that u is not unique. The same argument also shows that the data-to-solution map The second statement can be proved as follows: If n ≥ N + 1 then by convention (7) there are no asymptotic terms and the statement trivially holds. Assume that 0 ≤ n ≤ N and let (ϕ, v) ∈ C 1 ([0, T ], AD m n,N × A m n,N ) be the solution of (13). Then, by Proposition 2.1, . In view of the last statement of Theorem 3.1, the coefficients, in the asymptotic expansion of v are independent of t ∈ [0, T ]. Then, by Lemma By Lemma 6.3 and Lemma 6.11, for any t ∈ [0, T ] and for any n ≤ k ≤ n * , This and Lemma 6.3 also imply that Hence, in view of (61), (62), and (63), for any t ∈ [0, T ], This shows that the first 2(n * − n + 1) coefficients in the asymptotic expansion of u coincide with (60) and therefore they are independent of t ∈ [0, T ].

Smoothness of the conjugate map
Assume that m ≥ 3 and consider the conjugate map where ϕ −1 denotes the inverse of ϕ in AD m n,N . Arguing as in Section 3.2 one gets from Lemma 6.7, Lemma 6.8, and Lemma 6.10, the following Lemma.
is a C ∞ -diffeomorphism.

Smoothness of the vector field
Take b ∈ R and consider the mapping, where As is Section 3.3 one has  Proof. As in the proof of Proposition 3.3 we see that for any (ϕ, v) ∈ AD m n,N × A m n,N , By Lemma 6.8 and Lemma 6.10, v ∈ A m n,N , v ′ ∈ A m−1 n+1,N , and In view of Lemma 6.8, v · 1 ϕ ′ ∈ A m−1 n,N , and hence, This combined with the continuity of the pointwise product (Lemma 6.8) implies that is C ∞ -smooth. The other terms in (69) are treated similarly. In fact, as we see from the continuity of the pointwise product (Lemma 6.8) that and are C ∞ -smooth. In conclusion, we obtain from (69)-(72) that is C ∞ -smooth. Now, arguing as in Section 3.2 we see that the mapping, where U is the open neighborhood given by Proposition 4.2 is C ∞ -smooth. Hence, we proved the following Theorem.

Proof of Theorem 1.2
The proof of Theorem 1.2 follows by the same arguments as in the proof of Theorem 1.1.

Appendix A: Lagrangian description
In this Appendix we give the proofs of Lemma 2.1 and Proposition 2.1.
This together with the existence theorem for ODE's in Banach spaces [17] implies that for any t 0 ∈ [0, T ] there exists an open neighborhood U t0 of (t 0 , id) in [0, T ] × AD m−1 n,N and ε t0 > 0 such that for any (τ, ψ) ∈ U t0 there exists a unique solution n,N we see that there exists ε > 0 such that for any t 0 ∈ [0, T ] there exists a unique solution is a solution of (75) then for any ψ ∈ AD m−1 n,N the curve t → ϕ(t)•ψ, ϕ This solution is unique as F (t, ϕ) = u(t) • ϕ is locally Lipschitz continuous on [0, T ] × AD m−1 n,N . As ε > 0 is independent of the choice of t 0 ∈ [0, T ] and ψ ∈ AD m−1 n,N we conclude that (12) has a unique solution As ∂ x : A m−1 n,N → A m−2 n+1,N +1 is a bounded linear map we see from Lemma 6.
where the t-derivative is understood in A m−2 n+1,N +1 . As ϕ ∈ C 1 [0, T ], AD m−1 n,N we see that ϕ x ∈ C 1 [0, T ], AD m−2 n+1,N +1 . By the inclusion A m−2 n+1,N +1 ⊆ L ∞ one concludes that for any given x ∈ R, ϕ x (·, x) ∈ C 1 ([0, T ], R). This together with ϕ x (t, x) > 0 implies that (log ϕ x (t, x)) · = u x (t, ϕ(t, x)) for any given x ∈ R and t ∈ [0, T ]. Hence, for any x ∈ R and for any t ∈ [0, T ], Note that by assumption u we conclude from Theorem 2.1, Lemma 6.11, and Lemma 6.6 that u as the integrand is a continuous curve in A m−1 n+1,N +1 . As A m−1 n+1,N +1 is a Banach algebra we conclude from (78) that . As in addition ϕ ∈ C 1 ([0, T ], AD m−1 n,N ) we see that This solution is unique in AD m n,N as it is unique as a curve in AD m−1 n,N .
Proof of Proposition 2.1. Let u ∈ C 0 ([0, T ], A m n,N ) ∩ C 1 ([0, T ], A m−1 n,N ) be a solution of (3). Then we have It follows from Lemma 6.6 that u t + uu x ∈ A m−1 n,N and β b (u) ∈ A m−2 n,N . In view of Lemma 6.1 (iv), we get where (1 − ∂ 2 x ) −1 f := Q(f ) and Q(f ) is defined by (84). Next, consider the differential equatioṅ In view of Lemma 2.1 there exists a unique solution ϕ ∈ C 1 ([0, T ], AD m n,N ).
It follows from Sobolev embedding theorem that u(t, x) and ϕ(t, x) are functions in C 1 ([0, T ] × R, R). In particular, by (79), Hence, for any given As u ∈ C 0 ([0, T ], A m n,N ) we obtain from Lemma 6.6 and n ≥ 1 that In view of Corollary 3.1 we get As ϕ ∈ C 1 ([0, T ], AD m n,N ) we obtain from Theorem 2.1 that This implies that the integrand in (81) converges in A m n,N . Hence, is a solution of (13). Denote In view of Theorem 2.1, By inspection one checks that u is a solution of (3). The fact that the described above correspondence between solutions of (3) and (13) is bijective follows from the fact that if u 1 , u 2 ∈ C 0 ([0, T ], A m n,N ), u 1 = u 2 then the corresponding solutions ϕ 1 , ϕ 2 ∈ C 1 ([0, T ], AD m n,N ) of (80) do not coincide.

Appendix B: Auxiliary Results
In this Appendix we collect for the convenience of the reader some auxiliary results needed in the main body of the paper. For any γ ∈ R denote For any g ∈ L 2 * define the integral transforms and Q(g) := 1 2 (Q + (g) + Q − (g)) . (84) Moreover, for any γ ∈ R, (ii) For any g ∈ H 1 loc ∩ L 2 * , Q ± (g) ∈ H 2 loc . If in addition g ′ ∈ L 2 * then The restriction of Q ± to H 1 is a linear isomorphism Q ± | H 1 : (iii) For any g ∈ H 1 loc ∩ L 2 * , Q(g) ∈ H 3 loc , and If in addition g ′ ∈ L 2 * then Q(g) ′ = Q(g ′ ) .
The restriction of Q to H 1 is a linear isomorphism Q| H 1 : Proof. (i) Take g ∈ L 2 * . As g ∈ L 2 γ for some γ ∈ R one sees that for any a ∈ R the function y → g(∓y) e −y = g(∓y) y γ · e −y y −γ is summable on [a, ∞) and This formula implies that Q ± (g) is an absolutely continuous function on any finite interval of R and its derivative 4 belongs to L 2 loc . Differentiating (90) we see that, which implies (85). As Q ± (g) ∈ L 2 loc we get from (91) that Q ± (g) ∈ H 1 loc . Now, consider the second statement in (i). Take γ ≥ 0 and note that for any x, z ∈ R, It follows from (92) and Young's inequality that for any measurable function f such that x γ f ∈ L 2 , x γ Q + (f ) ∈ L 2 and where χ [0,∞) is the characteristic function of [0, ∞) and C > 0 is independent of the choice of f . The case of the transformation Q − is considered in the same way. If γ < 0 one uses the inequality x ± z 2 ≤ 2 x 2 z 2 and argues similarly.
(ii) As g ∈ H 1 loc ∩ L 2 * we see from (i) that Q ± (g) ∈ H 1 loc and Q ± (g) ′ = g − Q ± (g). This implies that Q ± (g) ∈ H 2 loc . Assume in addition that g ′ ∈ L 2 * . Then, for any test function ψ ∈ C ∞ 0 such that supp ψ ⊆ (−R, R), 0 < R < ∞, we get from Fubini's theorem and an integration by parts that, We can apply Fubini's theorem as by the first statement of (i), ∞ 0 |g(x ∓ z)| e −z dz and ∞ 0 |g ′ (x∓z)| e −z dz are locally summable functions in the variable x. Formula (93) then implies that, This together with (91) implies that Now, we prove the last statement in (ii). Assume that g ∈ H 1 . Then g, g ′ ∈ L 2 ⊆ L 2 * , and in view of (94) and the second statement in (i) (with α = 0) we see that Q ± (g) ′ = Q ± (g ′ ) ∈ H 1 . This shows that Q ± (g) ∈ H 2 . Hence, the linear map Q ± | H 1 : H 1 → H 2 is well defined. It follows from (95) that Q ± | H 1 : This together with the open mapping theorem imply that Q ± H 1 : H 1 → H 2 is a linear isomorphism.
Proof. As the both items of the Lemma follow in a similar way from a simple approximation argument, we prove only (i). Assume that f ∈ H 1 loc and ϕ : and Combining (97) with (98) we see that Using a change of variables one easily sees that f • ϕ and f ′ • ϕ are locally square integrable. This and (99) then imply that f • ϕ ∈ H 1 loc . The proof of the last statement of (i) follows from (99) in a similar way. Lemma 6.3. Let m ≥ 1 and N ∈ R. Then there exists C > 0 such that for any f ∈ W m N and for any 0 ≤ j ≤ m − 1, If N ≥ 0 one has the continuous inclusions W m N ⊆ H m ⊆ C m−1 . Proof. First assume that f ∈ W 1 N and N ∈ R. Consider the function A direct differentiation shows that As dy where c ± ≥ 0 are non-negative constants. In particular, Now, assume that c + > 0. Then, there exist α ≥ 0 and ε > 0 such that g(x) ≥ ε 2 > 0 for any x ≥ α. In particular, for any x ≥ α, that contradicts x N f ∈ L 2 . Hence, c + = 0. Arguing in a similar way we see that c − = 0. In particular, and lim |x|→∞ g(x) = 0 .
This shows that lim It follows from (100) and (101) that Hence, there exists C N > 0 such that for any f ∈ W 1 N and for any x ∈ R, Now, assume that f ∈ W m N , m ≥ 1, and N ∈ R. Take 0 ≤ j ≤ m − 1. Then In view of (103) we have that for any x ∈ R, This prove the lemma with C := max 0≤j≤m−1 C N +j . In view of (102) we also get that lim |x|→∞ |f (j) (x)| x N +j+ 1 2 = 0. The last statement of the lemma follows from Sobolev's embedding theorem.
Arguing in a similar way one proves. Lemma 6.4. Let m ≥ 1 and N ∈ R. Then there exists C > 0 such that for any f ∈ H m N and for any 0 ≤ j ≤ m − 1, If N ≥ 0 one has the continuous inclusions H m N ⊆ H m ⊆ C m−1 . Proof. The proof of this statement follows the arguments of the proof of Lemma 6.3 applied to the function g( The following Lemma can be proved in a straightforward way. Lemma 6.5. Assume that m 1 , m 2 ≥ 1, N 1 , N 2 ≥ 0, and k ≥ 0. Then the mappings, , are continuous. In particular, for any m ≥ 1, N ≥ 0, W m N is a ring with continuous (pointwise) product.
As a consequence from Lemma 6.5 we get Lemma 6.6. Assume that m 1 , m 2 ≥ 1, N 1 , N 2 , n 1 , n 2 , k ≥ 0. Then the mappings, , are continuous. In particular, for any m ≥ 1, N ≥ 0, and n ≥ 0, A m n,N is a ring with continuous (pointwise) product. Remark 6.1. As the mappings considered in Lemma 6.5 and Lemma 6.6 are multilinear and continuous they are C ∞ -smooth.
Left composition with analytic maps: First we introduce some additional notation. Take r ≥ 0 and denote R r := {x ∈ R | |x| ≥ r} .
As n ≥ 1, the product u N +1 ∈ W m N (R r ) (Lemma 6.5) and the claimed inequality (105) then follows from (109) and an induction argument.
As a corollary we obtain the following Proposition 6.1. Assume that m ≥ 1, N ≥ 0, and n ≥ 1. Let α : I → R be a real-analytic function in the open interval I ⊆ R that contains zero, 0 ∈ I. Then, if u ∈ A m n,N and Image(u) ⊆ I then α • u − α(0) ∈ A m n,N . 7 The same statement holds if A is replaced by A.
Proof. Take u ∈ A m n,N so that Image(u) ⊆ I. As 0 ∈ I and as α : I → R is real-analytic there exists δ > 0 such that where the series converges absolutely on (−δ, δ). As n ≥ 1 we see from Lemma 6.3 that there exists r ≥ 0 so that for any x ∈ R r , |u(x)| < δ .
In particular, for any x ∈ R r , where the series converges absolutely. For any x ∈ R r , we have In view of Lemma 6.9, (111) By taking r ≥ 0 so that u A m n,N (Rr) ≤ δ/C we obtain that (110) converges in A m n,N (R r ). Hence, α • u − α(0) | Rr ∈ A m n,N (R r ) . As u ∈ H m loc − (r + 1), r + 1 , Image(u) ⊆ I, and as α is real-analytic in I, one concludes by standard arguments that α • u ∈ H m loc − (r + 1), r + 1 . The statement of the Proposition now follows from Lemma 6.9, 1). The case of the space A m n,N is treated in the same way.
We will use Proposition 6.1 for proving the following important Lemmas.
Lemma 6.10. Assume that ψ := 1 + f > 0 where f ∈ A m n,N , m ≥ 1, N ≥ 0, and n ≥ 1. Then 1 ψ − 1 ∈ A m n,N and there exists an open neighborhood U of zero in A m n,N such that for any g ∈ U, ψ + g > 0, and the mapping, is real-analytic. The same statements are true if A is replaced by A.
Proof. The fact that 1 ψ − 1 ∈ A m n,N follows from Proposition 6.1 applied to the real-analytic function α(x) := 1/(1 + x), α : (−1, ∞) → R. Let us prove the second statement of the Lemma. It follows from the continuity of the pointwise product in A m n,N and the continuity of the inclusion A m n,N ⊆ L ∞ that there exists an open neighborhood U of zero in A m n,N such that g · 1 ψ A m n,N < 1 and g · 1 ψ L ∞ < 1 .
Then for any g ∈ U, where the series converges in A m n,N . The case of the space A m n,N is treated in the same way.
Similar arguments show that the following Lemma hold. Lemma 6.11. Assume that m ≥ 2, N ≥ 0, and n ≥ 0. Then for any ϕ ∈ AD m n,N where 1 · • ϕ stands for 1/ 1 + ϕ(x) 2 . Moreover, the mapping, is real-analytic. The same statements are true if A is replaced by A.
Proof. Take ϕ ∈ AD m n,N . Then ϕ(x) = x + u(x), u ∈ A m n,N , and In view of Lemma 6.6, the expression 2 x x u x + u x 2 ∈ A m n+1,N +1 . As 1 + 2 x x u x + u This together with (112) completes the proof of the first statement of the Lemma. The second statement follows as in the proof of the second statement of Lemma 6.10. The case of the space A m n,N is treated in the same way.