Groups of Asymptotic Diffeomorphisms

We consider classes of diffeomorphisms of Euclidean space with partial asymptotic expansions at infinity; the remainder term lies in a weighted Sobolev space whose properties at infinity fit with the desired application. We show that two such classes of asymptotic diffeomorphisms form topological groups under composition. As such, they can be used in the study of fluid dynamics according to the method of V. Arnold. Specific applications have been obtained for the Camassa-Holm equation and the Euler equations.


Introduction
A modern development in fluid dynamics is to view the motion of an incompressible fluid as a geodesic flow on a group of diffeomorphisms of the underlying physical space. This approach was initiated by V. Arnold [1] and further developed by Ebin & Marsden [9] and Bourguignon and Brezis [5] to obtain well-posedness of initial-value problems associated with the Euler and Navier-Stokes equations. In these papers, the underlying physical space was compact (a compact manifold with or without boundary). Subsequently, Cantor [7] used this approach to study the Euler equations on R d by considering diffeomorphisms φ : R d → R d of the form where Id is the identity map and the function f is in a weighted Sobolev space that requires f to decay rapidly at infinity. However, one would like to consider diffeomorphisms of the form (1) where f is bounded but not required to decay rapidly. Moreover, if the initial condition has asymptotics at infinity, one would like to know that the solution has similar asymptotics at infinity (with coefficients depending on t). To make these improvements, we require additional structure.
In this paper, we study groups of diffeomorphism on R d of the form where u is taken from a function space that we call an asymptotic space: these consist of bounded maps on R d having a partial asymptotic expansion at infinity of the form (3) u(x) = a 0 (θ) + a 1 (θ) r + · · · + a N (θ) r N + f N (x) for r = |x| > R, where θ = x/|x|, the functions a 0 , . . . , a N lie in certain Sobolev spaces on the unit sphere S d−1 , and the remainder function f N belongs to a function space R N which ensures that (4) |f N (x)| = o(|x| −N ) as |x| → ∞.
The remainder space R N will be a weighted Sobolev space, but there are different possibilities: the choice will depend upon the application, since it must be compatible with the equations being studied. In this paper we shall consider as remainder space two different classes of weighted Sobolev spaces. In one class, that we shall denote by H m,p N (R d ), the functions have derivatives up to order m that are in L p N (R d ) := {f ∈ L p ℓoc (R d ) : In the other class of weighted Sobolev spaces, that we shall denote by W m,p N −d/p (R d ), the derivatives of functions satisfy D α f ∈ L p N +|α|−d/p (R d ) for all |α| ≤ m. Neither of these classes of weighted Sobolev spaces is new to the literature, but for convenience we shall define them and summarize their properties in Section 1; our exposition is self-contained, with proofs provided in the Appendix.
In Section 2 we give the formal definition of our asymptotic spaces on R d . If we use as R N the weighted Sobolev space H m,p N (R d ) with m ≥ 1 and 1 < p < ∞, then the functions satisfying (4) define an asymptotic space that we denote by A m,p N (R d ). On the other hand, if we let R N = W m,p N − 1 p (R d ), then we denote the associated asymptotic space by A m,p N (R d ). At times it is useful to consider functions with a 0 = · · · = a n−1 = 0 for an integer n ≤ N ; we denote the corresponding spaces by A m,p n,N (R d ) and A m,p n,N (R d ), and identify A m,p 0,N (R d ) = A m,p N (R d ) and A m,p 0,N (R d ) = A m,p N (R d ). Our primary interest in these asymptotic spaces is to control the behavior of diffeomorphisms at infinity. However, in Section 3, we consider an application of the asymptotic spaces to the Helmholtz decomposition of vector fields on R d ; this requires an analysis of the inverse of the Laplacian, which is important in many applications, including our study [15] of Euler's equation on R d .
In Section 4, we introduce and study the associated spaces of diffeomorphisms AD m,p n,N and AD m,p n,N , i.e. diffeomorphisms φ : R d → R d that are of the form (2) where the components of u are in A m,p n,N or A m,p n,N respectively. The main result of this paper is Theorem 4.1 which states that, provided m > 2 + d/p, both AD m,p n,N and AD m,p n,N are topological groups when composition is used as the group operation. (For compact domains and manifolds, the regularity assumption m > 1 + d/p is typically sufficient to show that the associated Sobolev spaces form topological groups under composition; cf. [9] and [5]. Our assumption of additional regularity stems from the fixed point argument that we use in Section 6 to prove the existence of inverses within the group; this argument avoids some of the technical difficulties caused by the asymptotics.) The fact that AD m,p n,N and AD m,p n,N are topological groups allows us to use them to study the asymptotics of various fluid flows on R d . With d = 1, for example, the Camassa-Holm equation [6] is a completely integrable equation that has attracted considerable attention recently. From the differential geometric point of view, Misiolek [17] showed that the equation can be realized as the geodesic flow for a certain metric on the Bott-Virasaro group. Moreover, Constantin [8] studied initial-value problems for Camassa-Holm on R 1 by using a group of diffeomorphisms of the form (1) that not only do not contain asymptotics but require decay o(|x| −3/2 ) as |x| → ∞. In our companion paper [14], we used groups of asymptotic diffeomorphims on R 1 to show that the initialvalue problem for the Camassa-Holm equation is well-posed with respect to asymptotic spaces. For example, we showed that if the initial condition u 0 is in A m,2 n,N for m ≥ 3 and N ≥ n ≥ 0, then there is a unique solution u of Camassa-Holm in C 0 ([0, T ], A m,2 n,N ) ∩ C 1 ([0, T ], A m−1,2 n,N ). With n = 0 this allows u 0 = O(1) as |x| → ±∞, a great improvement over previous results.
With d ≥ 2, we have also used our groups of asymptotic diffeomorphisms AD m,p n,N to study Euler's equation for a velocity field u and pressure p of an incompressible fluid on R d with external forcing f . In fact, in [15] we show that if m > 2 + d/p, 1 ≤ N ≤ d − 1, and f ∈ C([0, T ], A m+1,p There is strong eveidence that these spaces of asymptotic diffeomorphisms will be equally useful in the study of other fluid equations on R d . With d = 1, for example, we note that there are important equations such as KdV and mKdV that have been studied on R 1 with asymptotic conditions at infinity. For example, Menikoff [18] proved the existence of unbounded solutions for KdV on R 1 which are O(|x|) as |x| → ∞; this was subsequently refined and generalized by Bondareva & Shubin [3], [4] and by Kenig, Ponce, and Vega [12]. Building on these ideas, Kappeler, Perry, Shubin, & Topalov [11] proved the existence and uniqueness of unbounded solutions for mKdV on R 1 with asymptotic expansions at infinity. We believe that spaces of asymptotic diffeomorphisms will be useful in the study of these and other fluid equations on R d .

Weighted Sobolev Spaces on R d
Let x = |x| 2 + 1. For 1 < p < ∞, δ ∈ R, and a nonnegative integer m, we define the Banach spaces H m,p δ (R d ) and W m,p δ (R d ) to be the closures of C ∞ 0 (R d ) in the respective norms: Notice that H 0,p δ (R d ) = W 0,p δ (R d ) is just a weighted L p -space that may be denoted by L p δ (R d For mp = d, the same conclusions hold for all q ∈ [p, ∞).
In fact, for all 0 ≤ |α| ≤ k, we have For mp = d, the same conclusions hold for all q ∈ [p, ∞).
In fact, for all 0 ≤ |α| ≤ k, we have For both lemmas, the properties (a) and (b) are obvious; properties (c) and (d) are proved in the Appendix.
Using these lemmas, we can prove the following results about pointwise multiplication: Proof. To begin with, it is easy to check that the following weighted Hölder inequality holds: Consequently, we will know that (f, i.e. the proposition holds for k = 0, provided we can find 1 ≤ q 1 , q 2 ≤ ∞ so that To obtain (8), let us first assume that mp < d, so we also have ℓp < d. According to Lemma 1.1 (c), we have Now it is clear that the function f (q 1 , q 2 ) = q −1 1 + q −1 2 takes on all values between 2/p and d − mp dp so whether f (q 1 , q 2 ) ever equals p −1 is determined by whether The second inequality is trivial but the first holds precisely when (m + ℓ)p ≥ d. How does this result change when mp ≥ d? For mp = d we need to require q 1 < ∞, which translates into a strict inequality in (9), so we require (m + ℓ)p > d. For mp > d, then by Lemma 1(d) we can take q 1 = ∞ and q 2 = p. Thus we always have (8) under the assumption (m+ℓ)p > d. This proves the proposition for k = 0. Now, to prove the proposition for all 0 ≤ k ≤ ℓ, we must show that D α (f g) ∈ L p δ1+δ2 for all |α| ≤ k. But if we use the Leibniz rule to write and we observe that (R d ), then we can use (8) provided m − |β| + ℓ − |α − β| > d/p. But this is guaranteed since |β| + |α − β| = |α| ≤ k.
In fact, there is a constant C = C(d, m, ℓ, k, p) such that Proof. As a special case of (7) we have Using this, we will know that (f, g) → f g defines a continuous map W m,p Let us assume first that mp < d, so we also have ℓp < d. According to Lemma 1.2 (c), we have For the same reasons as in the proof of Proposition 1.1, this is possible when (m + ℓ)p ≥ d. The case mp ≥ d also follows as in the proof of Proposition 1.1. Now, to prove the proposition, we must show that x |α| D α (f g) ∈ L p δ1+δ2−(d/p) for all |α| ≤ k. But if we write and we observe that In the next section, we shall also need to consider Sobolev spaces H m,p (S d−1 ) on the unit sphere S d−1 in R d . The boundedness of multiplication on these spaces can be found in the literature or easily derived using the Hölder inequality and Sobolev embedding on S d−1 as in the proofs above. We record here the result.
In fact, there is a constant C = C(d, m, ℓ, k, p) such that f g H k,p ≤ C f H m,p g H ℓ,p for all f ∈ H m,p and g ∈ H ℓ,p .

Asymptotic Spaces of Functions on R d
We want to consider functions u ∈ H m,p ℓoc (R d ) which are bounded on R n and admit a partial asymptotic expansion as |x| → ∞. To describe this partial asymptotic expansion, let χ(t) be a smooth function satisfying χ(t) = 0 for t ≤ 1, χ(t) = 1 for t ≥ 2, and |χ (k) (t)| ≤ M for 0 ≤ k ≤ m and all t. For a nonegative integer N , the functions that we consider are of the following form: In (11b) and throughout this paper, we use r = |x| and θ = x/|x| ∈ S d−1 .
We refer to a in (11b) as the asymptotic function, the function a k on S d−1 as the asymptotic of order k, and f as the remainder function for u. We want to achieve (11c) by requiring the remainder function f to belong to one of the weighted Sobolev spaces discussed in the previous section. Let us begin with W m,p δ . From Lemma 1.2(d) we see that f ∈ W m,p δ (R d ) satisfies (11c) provided mp > d and δ + d p ≥ N . However, for reasons that will become clear in the next section, we want to avoid values of δ for which δ + d p is an integer. Consequently, let us define (12) γ N = N + γ 0 , where γ 0 has been chosen to satisfy 0 . When the domain R d is understood, we simply write A m,p N instead of A m,p N (R d ); when N is fixed or understood, we may simply write γ instead of γ N . The norm on A m,p N is given by (14) u . This norm is complete, so A m,p N is a Banach space. For an integer n with 0 ≤ n ≤ N , we define closed subspaces (15) A m,p n,N = A m,p n,N (R d ) = {u ∈ A m,p N : a k = 0 for k < n} .
Remark 2.1. That the regularity of the asymptotic a k depends on k, i.e. a k ∈ H m+1+N −k,p (S d−1 ), is an important feature of (11); it will prove essential many times in the analysis below.
Remark 2.2. In the definition (11), the specification that χ(r) ≡ 1 for r > 2 is somewhat arbitrary. In fact, if we introduce χ R (t) = χ(R −1 t), then χ R (r) ≡ 1 for r > 2R and we can write where f differs from f by a function with compact support: But we can estimate where C depends on R, χ, m, p, d, and N , to conclude f W m,p γ N ≤ C u A m,p n,N . Similarly, we can estimate f in terms of the a k and f , so if we use χ R in place of χ in (11), we will get a norm on the Banach space A m,p N that is equivalent to (14). This will be important in subsequent sections. In fact, it is sometimes convenient to consider the restriction of u = a + f to the exterior domain We always have N * ≥ N , but we have N * = N when d = 1, or more generally if d < p. In any case, let us define where a satisfies (16b) and f ∈ H m,p N (R d ) . When the domain R d is understood, we simply write A m,p N . We replace (14) by Under this norm, A m,p N is a Banach space, and we define closed subspaces A m,p n,N by requiring a k = 0 for k < n. Of course, Remarks 2.1 and 2.2 apply as well to (17) and (18).
We next investigate some of the properties of these asymptotic spaces. We begin with an elementary result.
The lemma is easy to prove using integration in spherical coordinates and the simple computation: We will use Lemma 2.1 in confirming that our asymptotic spaces have the following properties: We note that the constants C in (d), and throughout the rest of this paper, may depend on d and parameters associated with the functions spaces but not on the functions themselves.
Proof. (a) Write u ∈ A m,p n1,N1 as = χ a n (θ) r n + · · · a N (θ) r N + g N where a k = 0 for n ≤ k < n 1 and We clearly have a k ∈ H m+1+N −k (S d−1 ) and (using Lemma 2.1) g N ∈ W m,p γN , so u ∈ A m,p n,N . Similarly for A m,p n1,N1 ⊂ A m,p n,N . To prove (b) let us first consider u ∈ A m,p n,N with asymptotics a k ∈ H m+1+N −k,p (S d−1 ) for k = n, . . . , N . We use (19) with |β| = 1 to compute . The term b k,j r −k−1 for k = n, . . . , N is of the form of an asymptotic function in A m−1,p n+1,N +1 while χ ′ (r) θ j a k (θ) r −k has compact support so certainly belongs to the remainder space W m−1,p N +1 . Since we also know that ∂ j : W m,p γ → W m−1,p γ+1 , we have ∂ j : A m,p n,N → A m−1,p n+1,N +1 is bounded. Next consider u ∈ A m,p n,N , with asymptotics a k ∈ H m+1+N * −k,p (S d−1 ) for k = n, . . . , N * .
is bounded, we see that ∂ j : A m,p n,N → A m−1,p n+1,N is bounded. The proof of (c) is immediate. To prove (d), first consider u = χ(r) a k (θ) r −k with n ≤ k ≤ N and a k ∈ H m+1+N −k (S d−1 ). We generalize (22) to conclude and g ∈ H m+2+N −k−|α| (R d ) has support in the annulus A = {x : 1 < |x| < 2} with and we can apply the Sobolev embedding theorem on A to conclude Combining these inequalities, we have Thus, for an asymptotic function a as in (11b), we have and the same holds for the asymptotic function in (16b) provided we replace N by N * . Now let us consider the remainder function f . If f ∈ W m,p γN (R d ), then we use Lemma 1.2 (d) with δ = N to conclude sup On the other hand, if f ∈ H m,p γN (R d ), then we use Lemma 1.1 (d) with δ = N to conclude sup Since n ≤ N , these estimates imply (20) and (21).
What about multiplication? The product of two partial asymptotic expansions involves a number of terms. The product of the remainder functions is covered by Propositions 1.1 and 1.2; for convenience, we record here the following special case of those results:  for f ∈ H k,p δ1 and g ∈ H m,p δ2 .
The product of an asymptotic term like a k (θ)/|x| k and a remainder function is covered by the following (in which we use Lemma 2.1(c) to assume k = 0).
For i > 0, D i a is a sum of products of the form r −i c k (θ)D k θ a where c k is a polynomial in θ and k = 1, . . . , i. Thus we want to estimate For fixed r > 1, let us denote by f (r) the function on S d−1 defined by f (r) (θ) = f (rθ). Now let us use Proposition 1.3 (and s > (d − 1)/p) to estimate .
; this last condition only fails when j = m, which does not occur since we have assumed i > 0. Moreover, for a function g(x) we can use ∂g/∂θ i = r∂g/∂x i to estimate any derivative D k θ g on | denotes the sum of the absolute values of all x derivatives of g of order k. Applying this to g = D j f , we obtain Thus we have shown for i + j = ℓ and k = 1, . . . , i that . Combining the cases i = 0 and i > 0, we have shown (a).
The proof of (b) follows the same outline as for (a). Again we write D ℓ (af ) as a sum of products D i a D j f and treat the case i = 0 by But arguing as above and using (δ − i)p < δp, we can conclude Combining Lemmas 2.2 and 2.3, we obtain the following The analogous statement with W replaced by H and A replaced by A is also true.
We are now able to prove the following result on products for our asymptotic spaces: Proof. We shall prove (27b); the proof of (27a) is analogous. Since p is fixed, we shall drop that notation, but let us introduce Taking the product, we can write To prove c k ∈ H m+1+N −k (S d−1 ), we can use Proposition 1.3 to conclude since the condition which is guaranteed by our assumption m > d/p. This also shows the desired estimate for (27b) To show h ∈ W m γ we have several terms to consider. Let us first consider f g.
As for the other terms, we can useγ ≤ n 2 + γ 1 , n 1 + γ 2 and Lemmas 1.2 and 2.3 to conclude As a special case of Proposition 2.2, we obtain Corollary 2.2. If m > d/p, and 0 ≤ n ≤ N , then A m,p n,N and A m,p n,N are Banach algebras.
Since this paper is mostly concerned with diffeomorphisms of R d , we need to consider asymptotic spaces of vector-valued functions. Here we use bold-face u for a vector-valued function and denote its components by u j . Let us define the Banach spaces As in the scalar-valued case, we will abbreviate A m,p 0,N simply as A m,p N and suppress the notation (R d , R d ) when it is clear from the context that we are considering vector fields on R d .

Application to the Laplacian and Helmholtz Decompositions
The asymptotic spaces A m,p n,N are generally preferable to A m,p n,N in applications involving the In this section we will illustrate this by discussing the mapping properties of ∆ and an application to the Helmholtz decomposition of vector fields.
To begin with, consider the mapping Clearly, (30) is continuous for all δ ∈ R, and in [13] it was shown that (30) is injective for δ > −d/p and an isomorphism (in particular invertible) for it was also shown in [13] that (30) is Fredholm with explicitly specified cokernel. We now observe that for arbitrary where each a k (θ)/r k is harmonic for x = 0 (so a k ∈ C ∞ (S d−1 )) and f ∈ W m,p γN (R d ).
(b) For d = 2, the result also holds, except 1/r 2−d in (31b) is replaced by log r. Of course, this means that the asymptotic space A m+1,p 0,N in (31a) must be replaced by Proof. Let Γ(|x|) denote the fundamental solution for the Laplace operator in R d and K = Γ⋆ denote the convolution operator. As shown in [13], K : is no longer bounded, and we either need to restrict the domain space or expand the range space. Let us first describe what happens for N = d − 2 and then consider the general case. For , let us observe that ∆ (χ(r) Γ(r)) has compact support and we use Green's first identity to calculate Notice that c 0 is finite; in fact, using N = d − 2 and Hölder's inequality, we easily confirm that Then g dx = 0, so g ∈ W m−1,p γN +2 (R d ) and we can let f = K g to find f ∈ W m+1,p γN (R d ). Finally, we define Kg by For d ≥ 3, u = Kg is of the form (31b) for N = d − 2 and satisfies ∆u = g as well as the estimate  [13] that (30) is injective with cokernel equal to the harmonic polynomials of degree less than or equal to k. If we let H k denote the spherical harmonics of degree k, let N (k) = dim H k , and choose an orthonormal basis {φ k,j : j = 1, . . . , N (k)} for H k , then a basis for the space of harmonic polynomials that are homogeneous of degree k is {φ k,j (θ) r k : j = 1, . . . , N (k)}. Consequently, if we define then [13] showed that K : Using Hölder's inequality, we can confirm |c ℓ,j | ≤ C g L p γ N +2 , and in particular that c ℓ,j is finite. Recall that φ ℓ,j (θ) r d−2+ℓ is harmonic for r > 0, so we can use Green's second identity to calculate Then g ∈ W m−1,p γN +2 (R d ) and we can let f = K g ∈ W m+1,p γN . Finally, we define We see that u = Kg is of the form (31b) and satisfies ∆u = g as well as the estimate where C depends on the Sobolev norms of φ ℓ,j on S d−1 , but not on g. This completes the proof.
Notice that (30) generalizes to Then we will use Lemma 3.1 to find a remainder function f so that u = a + f ∈ A m+1,p N (R d ) is an exact solution of ∆u = v. To solve (38), we can use separation of variables. In fact, using where h is the induced metric on S d−1 , we find that a k must satisfy If k(d − 2 − k) > 0, then we can uniquely solve (39) to find a k . However, for k = 0 or k = d − 2, we have a simple solvability condition, namely S d−1 b k+2 ds = 0, and the solution a k is only unique up to an additive constant; this is expected since c 0 and c d−2 r d−2 are harmonic for r > 0. Let us consider two closed subspaces of A m−1,p 2,N +2 (R d ): Of course, if d > N + 2, then the solvability condition b d ds = 0 is vacuous.
. This operator is also bounded (b) For d = 2 and m ≥ 1 the operator is bounded Proof.
As indicated above, N ; in fact, the a k are unique except for k = 0, d − 2. Of course, the same analysis applies for d = 2. However, now let us assume d > 2 and v ∈ A m−1,p 2,N +2 (R d ) with b 2 (θ) ds = 0. Then the necessary solvability condition does not hold in order to be able to solve (39) for k = 0. Instead, let us replace a 0 (θ) by a * 0 log r + a 0 (θ) and instead of (38) try to solve We need to use Lemma 3.1 to find the remainder function f so that ∆u = v. We compute ∆a = χ b + ∆χ (a * 0 log r + a 0 + · · · + r −N a N ) + ∇χ · ∇(a * 0 log r + a 0 + · · · + r −N a N ). So we want f to satisfy ∆f = h := g − ∆χ(a * 0 log r + a 0 + · · · + r −N a N ) − ∇χ · ∇(a * 0 log r + a 0 + · · · + r −N a N ). But h and g differ by a function in H m+1,p (R d ) with compact support, and g ∈ W m−1,p γN +2 , so h ∈ W m−1,p γN +2 . Consequently, we can apply Lemma 3.
We see that u = a + f satisfies ∆u = v and the mapping K : v → u is continuous between the appropriate spaces. Now we turn to the application to Helmholtz decompositions. It is well-known that a C 1 -vector field u in R 3 satisfying D k u = O(|x| −1−k−ε ) as |x| → ∞ for k = 0, 1 and some ε > 0 can be decomposed into the sum of a unique divergence-free vector field with the same decay property and a gradient field: Moreover, v and ∇w are orthogonal in that v · ∇w dx = 0. This is called the Helmholtz decomposition in R 3 . We now show that (43)   Proof of Theorem 3.1. When the 1st-order derivatives of u are O(|x| −2−ε ) as |x| → ∞, the decomposition (43) can be found by letting K div u, where K is defined by convolution with the fundamental solution. However, for a vector field u ∈ A m,p 1,N , we will replace K by the operator discussed in Proposition 3.1. However, we first need to use separation of variables to study div u.
Let ω 1 , . . . , ω d−1 be local coordinates on S d−1 , considered as functions of Euclidean coordinates which, in this proof, we index by superscripts, i.e. x 1 , . . . , x d . Let h = h αβ dω α dω β denote the Riemannian metric on S d−1 induced by the Euclidean metric g = dx 2 = dr 2 + r 2 h. Recall that the divergence of a vector field v may be computed in general coordinatesx 1 , . . . ,x d by To compute the divergence of v in the coordinates (x 0 , . . . ,x d−1 ) = (r, ω 1 , . . . , ω d−1 ), we first compute its components in these coordinates bȳ ∂ ω α ∂x j ∈ S d−1 , and then compute the divergence to find We can use (44) to compute the divergence of a vector field of the form v = r −k a k where k ≥ 1 and a k is a vector field with components a j k (ω) ∈ H m+N +1−k,p (S d−1 ) for j = 1, . . . , d. We conclude where we have used the abbreviation √ h for det h αβ . Now we consider u ∈ A m,p 1,N and claim that div u ∈ A m−1,p 2,N +1 . In fact, using (45) we see that div u = χ(r −2 c 2 + · · · + r −N −1 c N +1 ) + g where c d is given by Since c d is a divergence on S d−1 , we conclude S d−1 b d ds = 0 and so div u ∈ A m−1,p 2,N +1 . By Proposition 3.1 we have w = K(div u) ∈ A m+1,p 0 * ,N −1 , and hence ∇w ∈ A m,p 1,N . In fact, the map Then v ∈ A m,p 1,N and if we compute the divergence, we get div v = div u − ∆w = 0.
The restriction N ≤ d − 1 is necessary to avoid log r terms in the Helmholtz decomposition. To see this, let us consider an example. In fact, let x = r cos φ, y = r sin φ, and simply consider a computation shows that b 2 must satisfy But to solve this, we must have the right hand side orthogonal to Ker(∂ 2 φ + 1) = {cos φ, sin φ}, which need not be the case. Consequently, we must modify our solution: replace r −1 b 2 (φ) by r −1 (b 2 (φ) + c 1 cos φ + c 2 sin φ) where the constants c 1 , c 2 are chosen so that the right hand side of is orthogonal to {cos φ, sin φ}, and hence we can find b 2 . This means that w contains the following asymptotic: Consequently, the terms ∇w and v in (43) will both contain asymptotics of the order r −2 log r as r → ∞, and hence will not be in our asymptotic function space A m,p 1,2 (R 2 , R 2 ).

Groups of Asymptotic Diffeomorphisms on R d
In this section we state the main results for diffeomorphisms of R d whose asymptotic behavior can be described in terms of the asymptotic spaces A m n,N and A m n,N ; proofs of all results will be given in the next two sections. Denote by Diff 1 + (R d , R d ) the group of orientation-preserving C 1 -diffeomorphisms on R d . For simplicity of notation, we will no longer use bold face for vectorvalued functions as we did in the previous two sections.
Similar to Section 2, we will abbreviate these collections by AD m,p n,N and AD m,p n,N when it is clear that we are considering diffeomorphisms of R d ; we also let AD m,p N and AD m,p N denote AD m,p 0,N and AD m,p 0,N respectively. Now we list some important properties of these spaces of asymptotic diffeomorphisms; as stated before, proofs will be given in the next section. First, we want to show that AD m,p N and AD m,p N are topological groups under composition of functions. Since φ = Id + u means φ(ψ) = ψ + u(ψ), we see that continuity of (φ, ψ) → φ(ψ) in φ reduces to continuity of (u, ψ) → u(ψ). Consequently, we need the following: Next we need to know that inverses of asymptotic diffeomorphisms are asymptotic diffeomorphisms. Due to the complexity of the asymptotics, this is simplest to prove for one degree of regularity greater than that required for the continuity of composition. These two propositions together suggest that AD m,p n,N is a topological group for m > 2+d/p, but we have not shown that φ → φ −1 is continuous AD m,p n,N → AD m,p n,N . However, since the topology in AD m,p N is just a translation of the Banach space topology of A m,p N , this follows from the result of Montgomery [18]. Analogous statements can be made about AD m,p N . Consequently, once we have proved the two propositions above, we will have shown:
Now we have assumed u ∈ A m,p 1,N (B c R ), so by Proposition 2.2 we know that u 2 , . . . , u ℓ ∈ A m,p 1,N (B c R ). Consequently, we will have completed our proof provided we can show (50) To prove (50), we need to consider derivatives of R ℓ (u) up to order m; for notational simplicity, at this point let us assume d = 1. If we calculate the first few derivatives ℓ (u) P k j (u ′ , u ′′ , . . . , u (k) ), where P k j (t 1 , . . . , t k ) is a homogeneous polynomial of degree j and the total number of derivatives is k.
In fact, we can easily prove (51) by induction. It is certainly true for k = 1 (in which case P 1 1 (u ′ ) = u ′ ). Now assume that (51) is true for k. To prove (51) for k + 1, we calculate But u ′ P k j (u ′ , . . . , u (k) ) is a homogeneous polynomial of degree j + 1 with total number of derivatives k + 1, and R (j) ) is a homogeneous polynomial of degree j with total number of derivatives k + 1. Relabeling, we have (51) for k + 1, completing the induction step. Now we want to use the representation (51) to estimate D k x (R ℓ (u)). Using u ∈ A m,p 1,N , we have |u(x)| ≤ C x −1 , and so (48) implies . . , u (k) ), every occurrence of u contributes x −1 and each derivative of u contributes an additional x −1 , so we obtain (52) and (53) with (51), we obtain the estimate , this proves (50) and hence the Lemma when u ∈ A m,p 1,N (B c R ). If we instead assume u ∈ A m,p 1,N (B c R ) and perform the same calculations, then |u (k) (x)| ≤ C x −1 for k = 0, . . . , m − 1, so (52) still holds, but in place of (53) we obtain (55) |P k j (u ′ , . . . , u (k) )| ≤ C x −j , and in place of (54) we have showing that in place of (50) we have and proves the Lemma when u ∈ A m,p 1,N (B c R ). We will need upper and lower bounds on φ(x) when φ = Id + u is an asymptotic diffeomorphism. But these estimates do not require that φ be a diffeomorphism, so we formulate them simply in terms of the vector function u.
Similarly, we use 2x · u ≤ |x| 2 + |u| 2 to conclude These three estimates complete the proof for u ∈ A m,p 0 . Since Lemma 1.2 (d) also implies u ∞ ≤ C u A m,p 0 , the same proof establishes (58) for u ∈ A m,p 0 . Our next estimates concern the Jacobian matrix Dφ of an asymptotic diffeomorphism φ. . Moreover, since we assumed that our diffeomorphisms are orientation-preserving, we know that det(Dφ(x)) > 0, but we need to confirm a lower bound at infinity, so that we have (60) 0 < ε ≤ det(Dφ(x)) for all x ∈ R d .
Given an asymptotic diffeomorphism φ in either AD m,p 0 or AD m,p 0 with m > 1 + d/p, the inverse function φ −1 is a diffeomorphism, but we want to obtain estimates at infinity. Letting x = φ −1 (y) in (58) yields in particular that where C 1 = 1/c 2 and C 2 = 1/c 1 . But since c 1 and c 2 were uniform for bounded u A m,p 0 (R d ) , we find the same is true of (61).
Similarly, for φ = Id + u, we want to show both holding locally uniformly for bounded u A m,p 0 (and similarly for A m,p 0 ).
where m > 1 + d/p. Then (62) and (63) both hold locally uniformly for u ∈ A m,p 0 . The same is true with A replaced by A. Proof. Let us fix φ * = Id + u * and consider φ = Id + u = φ * + u with u A m,p N small. Let ψ = φ −1 and let y = φ(x). Since Dφ(x) = I + Du(x), we can use Lemma 5.3 to conclude where C and ε are uniform for u A m,p 0 ≤ δ. If we differentiate φ(ψ(y)) = ψ(y) + u(ψ(y)) = y, we obtain Dψ + (Du • ψ)Dψ = I, or (I + Du • ψ)Dψ = I. However, (65) shows that I + Du • ψ is invertible, so we can write Using (64) and (65), we conclude where ε 1 = 1/C 0 and C 1 = 1/ε are uniform for u A m,p 0 ≤ δ. In particular, this confirms (63). To prove (62) we want to bound |Dψ| uniformly for small u A m,p 0 . But if we use the adjoint formula for the inverse of a matrix, we see that (62) follows from (64), (65), and (66). The proof for the result with A replaced by A is strictly analogous.
We now consider properties of the composition f • φ when f is in the remainder space and φ is an asymptotic diffeomorphism. In our first result, we allow f to be less regular than φ since this will be useful for later application. We may assume that f is scalar-valued, and we denote its gradient by ∇f .
where C may be taken locally uniformly in φ ∈ AD m,p 0 . The analogous result with A replacing A (and H δ replacing W δ ) is also true.
Proof. We prove (67) by induction. For k = 0, we simply use the change of variables x = ψ(y) = φ −1 (y): where C can be taken locally uniformly by Lemma 5.4. Now we assume (67) holds for k < m and prove it for k + 1. It suffices to assume f ∈ W k+1,p δ and show where C can be chosen uniformly for u A m,p 0 ≤ M . Now we can apply (67) to ∇f (with δ + 1 in place of δ) to conclude where C may be taken locally uniformly for φ ∈ AD m,p 0 . Putting these two inequalities together yields (68), where C may be taken locally uniformly for φ ∈ AD m,p 0 . The proof for f ∈ H m,p δ and φ ∈ AD m,p 0 leads to ∇f ∈ H m−1,p δ and Du ∈ H m−1,p 0 but is otherwise analogous.
The following result will play an important role in proving the continuity of f • φ with respect to φ. Proof. Since m > d/p, we know that φ k and φ are continuous functions with φ k → φ uniformly on compact sets in R d . Moreover, since φ k (x) = x + u k (x) and φ(x) = x + u(x) where u k and u are bounded functions while f has compact support, there is a compact set K such that along with (69) and the fact that φ k → φ uniformly on K to conclude that Next we assume the result for 0 ≤ ℓ < m and prove it for ℓ + 1. Since we may assume f is scalar-valued, this means showing Moreover, we know Dφ k → Dφ in AD m−1,p 1,1 and (by the induction hypothesis) (∇f ) • φ k → (∇f )•φ in W ℓ,p δ+1 . Hence, by Corollary 2.1 concerning products, we find that The previous two lemmas may be used to obtain the following continuity result.
Using (67), we have for sufficiently large j. Now let us use the density of Again we can use (67) to make The proof is exactly the same as for a).
The following estimates provide a stronger description of the continuity of f • φ when f has an additional degree of regularity. sufficiently close to φ * we have and φ ∈ AD m,p 0 sufficiently close to φ * we have In both (70a) and (70b), C can be taken uniformly for all φ in a fixed neighborhood of φ * .
Proof. For φ = Id + u ∈ AD m,p 0 sufficiently close to φ * = Id + u * , let u = φ − φ * = u − u * so that φ * + t u ∈ AD m,p 0 for 0 ≤ t ≤ 1. Now m > 1 + d/p implies f ∈ C 1 (in fact C 2 ), so we can write By Corollary 5.1, we know that t → ∇f (φ * +t u) is continuous [0, 1] → H m,p δ (R d ), so this mapping is Riemann integrable. Thus we can conclude that where we have also used Corollary 2.1. But now we can apply Lemma 5.5 to ∇f ∈ H m,p δ (R d ): where C can be taken uniform in a neighborhood of φ * . Putting this together yields (70a).
The proof of (70b) is analogous, except now ∇f ∈ W m,p δ+1 so the estimates become where C can be taken uniform in a neighborhood of φ * .
Before considering the continuity of a • φ when a is an asymptotic function and φ is an asymptotic diffeomorphism, we need a refinement of Lemma 5.1. For u ∈ A m,p N (B c 1 ) with m > d/p, let us introduce the scalar-valued function ρ(u) defined by Using Propositions 2.1 and 2.2, we see that ρ : A m,p N (B c 1 ) → A m,p 1,N +1 (B c 1 ) is continuous and we can calculate the asymptotics of ρ(u) in terms of the asymptotics of u. In fact, since we also know by Proposition 2.1 that u is bounded on R d , we see that ρ(u(x)) → 0 as |x| → ∞, and hence we have 1 + ρ(u) ≥ ε > 0 for |x| > R with R sufficiently large. Note that R depends on u, but we can take it uniformly on a bounded neighborhood U of a fixed u * ∈ A m,p N (B c 1 ). Using Lemma 5.1, the scalar-valued function σ(u) defined by and we can calculate its asymptotics in terms of the asymptotics of u, so we have σ : U → A m,p 1,N +1 (B c R ). We now want to show that we can choose U so that σ : U → A m,p 1,N +1 (B c R ) is real-analytic; in particular, this map is continuous.
for m > d/p and N ≥ 0, then there is a neighborhood U of u * and R sufficiently large that σ : U → A m,p 1,N +1 (B c R ) is real analytic. The same is true with A replaced by A.
But we can take η sufficiently small that, for all u A m,p If we let U = {u = u * + u : u A m,p N (B c 1 ) < η} and use the power series (1 + t) −1/2 = 1 − 1 2 t + · · · , we find that Consequently, the same is true of (1 + ρ(u)) −1/2 , from which the result follows.
The proof for the result with A replaced by A is strictly analogous.
Another lemma will be useful in controlling the remainder term in a•φ when a is an asymptotic function and φ is an asymptotic diffeomorphism. In this lemma we consider a function b on S d−1 and extend it to R d \{0} as a function of some degree of homogeneity; however, the specific degree of homogeneity does not matter since we will only be using the behavior of b near |x| = 1.
Then for R sufficiently large we have where C is locally uniform in v ∈ A m,p 0 . The analogous estimate for v ∈ A m,p Proof. Writing γ = γ N , we shall prove by induction that For ℓ = 0 we easily obtain We first want to show that, for R sufficiently large depending locally uniformly on v, I(r) can be estimated by C b p L p (S d−1 ) . To do this we consider the surface Ξ in R d \{0} parameterized by θ ∈ S d−1 : We compute the Jacobian: But ∇v ∈ A m−1,p 1,N +1 with m > 1+d/p implies by Proposition 2.
/|x|. So, for R sufficiently large, we conclude that for r > R the Jacobian is nonsingular and we have which gives us (75) for ℓ = 0. Now we assume (75) for ℓ = m − 1 and prove it for ℓ = m. It suffices to show We can use Lemma 2.3 to estimate where we have used the induction hypothesis for ℓ = m − 1 applied to ∂ j b ∈ H m−1,p (S d−1 ). We can also apply the induction hypothesis to estimate Putting these together proves (76), which completes the induction.
For v ∈ A m,p 0 we follow the same outline, using N − N * − 1 < −d/p to conclude convergence of the radial integral.
We now consider compositions u • φ when u = a + f as in (11a). We may assume that u is scalar-valued but the diffeomorphism φ = Id + u is necessarily vector-valued. We start with generalizing Lemma 5.5. where C may be taken locally uniformly in φ ∈ AD m,p N . The analogous result with A replacing A is also true. Proof. To simplify notation, we assume n = 0. Using the form (14) of the A-norm and Lemma 5.5, it suffices to consider , we need a partial asymptotic expansion for a • φ.
Let us consider a k (x) as a homogeneous of degree 0 function on R d \{0}. In particular, where the remainder R N,k (y, y * ) can be expressed in integral form as This approximation holds for y in a neighborhood of y * , and more generally provided 0 ∈ {y * + t(y − y * ) : 0 ≤ t ≤ 1}. But we now want to take both y and y * on S d−1 . In fact, we shall replace y by φ(x)/|φ(x)| and y * by θ = x/|x|: where v is bounded, means that φ(x)/|φ(x)| → x/|x| as |x| → ∞, so for |x| > R with R sufficiently large we can arrange But we need to investigate the difference φ(x)/|φ(x)| − x/|x| in more detail. Notice that we can write where ρ(v) and σ(v) are defined in (72) and (73) respectively. But by Lemma 5.8 we know that . It is easy to confirm that Note that w ∈ A m,p N (B c R ) and that its asymptotics can be computed in terms of v; in particular, We plug (83) and (82) into (81) to conclude Although this is not quite the partial asymptotic expansion for a • φ, it can be used to estimate a • φ A m,p N . In fact, using Proposition 2.2 and Lemma 5.8, we know a (1 + σ(v)) k we need only estimate the A m,p N (B c R )-norm of the two terms in the brackets in (84). First of all, we claim where C is locally uniform in v ∈ A m,p N (B c 1 ). To see this, we use the algebra property to estimate , then we can consider C in (85) as being locally uniform in v. Secondly, we claim where C may be taken locally uniform in v ∈ A m,p N (B c 1 ). To see this, notice from (80b) that We can apply Lemma 5.9 with b = D α a k ∈ H m,p (S d−1 ) to conclude Using Corollary 2.1 and the above remarks regarding v → w, we obtain (86). Putting this together with (84) and (85), we obtain (79), as desired.
To prove the corresponding result for A, we replace N by N * in (80a) and (80b) and replace (86) by The details are straight-forward.
Proof. As in the proof of Lemma 5.10, we assume n = 0. Since we can write φ j (x) = x + v j (x) and φ(x) = x + v(x) where v j and v are uniformly bounded functions, we can take R large enough that χ(|φ j (x)|) = χ(|φ(x)|) = 1 for all |x| > R, so we want to estimate in A m,p Using the scalar function σ defined in (73), let us introduce (as we did in (83)) the vector functions Since v j → v in A m,p N , we see by Lemma 5.8 that w j → w in A m,p N . Now if we apply (84) to both u • φ j and u • φ, we find for |x| > R that Using Lemma 5.8 again, for each fixed α we know As observed in the proof of Lemma 5.10, |x| k+|α| D α a k (θ) ∈ A m,p N (B c R ), and so To handle the remainder terms in (88), it suffices to show But, using (87), this quantity is given by , so the remainder term in (88) also tends to zero in A m,p N (B c N ) as j → ∞, which is what we needed to show. The proof for A is identical.
Similar to Corollary 5.1, we can use Lemmas 5.10 and 5.11 to show that composition on our asymptotic spaces is continuous; we shall not repeat the argument.
In both cases, the constant C is locally uniform in φ.
Proof. As in the proof of Lemma 5.7, let φ = Id+v and φ * = Id+v * , and let Use the fact that u ∈ C 1 to write We first prove (89a). Assuming u ∈ A m+1,p N , we know ∇u ∈ A m,p N . By Corollary 5.2 the function F : t → (∇u)(φ * + t v) · v is continuous as a map F : [0, 1] → A m,p N , hence Riemann integrable. Consequently, we can apply the algebra property and then Lemma 5.10 to (90) to obtain the desired estimate: Now we consider (89b). Assuming u ∈ A m+1,p N −1 , we know ∇u ∈ A m,p N , so the above steps show Proof of Proposition 4.1. The continuity of (46a) is contained in Corollary 5.2. For the proof that (46b) is C 1 , we shall abbreviate our notation. Let X m represent A m,p n,N (or A m,p n,N ) and XD m represent AD m,p n,N (or AD m,p n,N ); the norm in X m will be denoted simply by · m . We can also assume that u is scalar-valued (although we will not distinguish notation between the m-norms of vector and scalar-valued functions).
Applying this identity with y replaced by φ, we find where Then, replacing u by δu, we find Putting these together, we obtain (91) where Clearly L u,φ is linear in δu and δφ and bounded as desired, so we need to show R u,φ (δu, δφ) m = o( δu m+1 + δφ m ) as δu m+1 + δφ m → 0. But, applying Lemma 5.10 and the algebra property, we can estimate the first term in R u,φ : We can also use Lemma 5.10 and the algebra property to estimate the third term in R u,φ , namely R 2 : Using δu m+1 δφ m ≤ δu 2 m+1 + δφ 2 m in both estimates above, we see that the X m -norms of the first and third terms in R u,φ are actually O( δu 2 m+1 + δφ 2 m ) as δu m+1 + δφ m → 0. To estimate R 1 in the X m -norm as δφ m → 0, we use the continuity of φ → ∇u • φ in X m from the first part of Proposition 4.1 to conclude that ∇u•(φ+tδφ)−∇u•φ m = o(1) uniformly for 0 < t < 1 as δφ m → 0. Using this in the definition of R 1 (u, φ, δφ) we find that This completes the proof.

Proof of Invertibility (Proposition 4.2)
Before we begin the proof of Proposition 4.2 we prove several lemmas that will be useful. The first shows invertibility near the identity; but we require one additional order of differentiability. : v m < η}, where we have chosen η small enough that Id + v ∈ AD m,p N for all v ∈ X. Now let us write φ = Id + u with u m+1 < ε, where ε > 0 will be specified below, and let ψ := φ −1 . Then ψ is a diffeomorphism and φ • ψ = Id implies ψ = Id − u • ψ. But we know that u ∈ A m+1,p N is bounded, so we can write ψ = Id + v, where v := −u • ψ is a bounded function; we want to show that v ∈ A m,p N . However, we know that v satisfies Id + v = Id − u • (Id + v). Consequently, let us define (95) F u (w) := −u • (Id + w).
If we can show that F u : X → X and F u has a fixed point w * , then we will have φ•(Id+w * ) = Id. But we also know φ • (Id + v) = Id. Applying φ −1 to both sides of φ • (Id + w * ) = φ • (Id + v), we find Id + w * = Id + v, showing v = w * ∈ X and hence φ −1 ∈ AD m,p N . First let us confirm that F u : X → X. But we can use Lemma 5.10 to conclude F u (w) m ≤ C 1 u m where C 1 is locally uniform in Id + w ∈ AD m,p N , so can be taken uniform for w ∈ X. Taking ε > 0 sufficiently small that ε C 1 ≤ η, we conclude that u m ≤ u m+1 < ε implies F u : X → X. Next we show that F u : X → X is a contraction. But we can use (89b) to conclude where C 2 can be taken uniform for v ∈ X. Taking ε > 0 small enough that ε C 2 < 1, we have F u : X → X is a contraction. Consequently, F u this map has a fixed point which must be v.
At this point we know We want to use this equation and u ∈ A m+1,p N to show that v ∈ A m+1,p N . If we differentiate (96), we obtain Dv • φ · (I + Du) = −Du. Now sup |Du| ≤ C Du m−1 ≤ C u m < ǫ C, so we can take ε small enough that I + Du is an invertible matrix. Using the adjoint form of the inverse matrix, we have Now Du is a matrix, all of whose elements are in the Banach algebra A m,p 1,N ; since Adj(I + Du) is comprised of products of these elements, it too is a matrix with elements in A m,p N . Similarly, det(I + Du) is a product of elements of A m,p N , and det(I + Du) = 1 + w with w ∈ A m,p 1,N . Using Lemma 5.1, we know (det(I + Du)) −1 ∈ A m,p N . We conclude that the elements of (I + Du) −1 are in A m,p N , so the elements of Dv • φ = −Du · (I + Du) −1 are also in A m,p N . But we know φ −1 ∈ AD m,p N , so we can compose Dv • φ on the right by φ −1 to conclude The second lemma shows that any asymptotic diffeomorphism can be continuously deformed to one whose difference from the identity has compact support. Proof. Write φ = Id + u where u ∈ A m,p N . Since φ is an orientation-preserving diffeomorphism, we know that By continuity, we see that det (D(φ + v)) > 0 provided v is chosen so that sup |Dv| < ε with ε > 0 sufficiently small. But recall that we can write u = χ R (r) (a 0 (θ) + · · · r −N * a N * (θ)) + f where N * satisfies (16a) and f ∈ H m,p N , so we can choose R sufficiently large that D χ R (r) (a 0 (θ) + · · · r −N * a N * (θ)) + χ R (r) f (x) < ǫ.
The next lemma concerns the differential of an asymptotic diffeomorphism φ ∈ AD m+1,p N for m > 1 + d/p as a C 1 -map AD m,p N → AD m,p N defined by composition. Of course, we must take the differential of φ at a particular diffeomorphism ψ, which we take as the identity ψ = Id and denote the resultant differential d ψ φ by d 0 φ; this will be a linear map d 0 φ : A m,p N → A m,p N . If we write φ = Id + u, then we have d 0 φ = I + Du, since for v ∈ A m,p N we can calculate pointwise We now show that this linear map d 0 φ : A m,p N → A m,p N is invertible. Proof. Write φ = Id + u, where u ∈ A m+1,p N , so d 0 φ = I + Du. But since φ is a diffeomorphism, considered as a matrix, I + Du is invertible and its inverse is given by the adjoint formula (97). Now Du is a matrix, all of whose elements are in A m,p 1,N ; since Adj(I + Du) is comprised of products of these elements, it too is a matrix with elements in A m,p N . Since m > d/p, A m,p N is an algebra and Adj(I + Du) maps A m,p N → A m,p N . Also, det(I + Du) is a product of elements of A m,p N , so it too is in A m,p N . Since det(I + Du)(x) > 0 for all x ∈ R d and Du(x) → 0 as |x| → ∞, we have det(I + Du)(x) ≥ ε > 0; therefore, we can use Lemma 5.1 to conclude that det(I + Du) −1 ∈ A m,p N . Consequently, (I + Du) −1 is a matrix with all elements in A m,p N , so it is bounded as a map A m,p N → A m,p N , completing the proof. The proof for AD is strictly analogous.
Next we want to show that left-translation by a fixed asymptotic diffeomorphism is an open map in a neighborhood of the identity; the next lemma shows that this is true provided we have one additional order of differentiability in the fixed diffeomorphism. Proof. Let φ * = Id + u * and for φ = Id + u near Id in AD m,p N , we can write Hence there is an open neighborhood U 0 of 0 in A m,p N such that F * : U 0 → A m,p N and F * (0) = 0; in fact, since u * ∈ A m+1,p N , by Proposition 4.1, F * : U 0 → A m,p N is C 1 . If we compute the differential of F * at 0, which we also denote by d 0 F * , we find d 0 F * (v) = Du * · v + v = (I + Du * ) · v = d 0 φ * (v) for any v ∈ A m,p N . But, using Lemma 6.3 (with m in place of m − 1), we know that d 0 φ * = d 0 F * is invertible. By the inverse function theorem, we conclude that F * admits a continuous inverse near 0 ∈ A m,p N , which translates as the desired conclusion for φ * and U = Id + U 0 .
Finally, we are ready to prove Proposition 4.2.
Proof of Proposition 4.2. We give the proof for AD, the case of AD being analogous. Using Lemma 6.2, there exists a continuous map φ t : [0, 1] → AD m+1,p N such that φ 1 = φ and φ 0 − Id has compact support. But then φ −1 0 is the identity outside a compact set, so trivially we have φ −1 0 ∈ AD m+1,p N . We want to use the continuity method to show that this property can gradually be extended to φ 1 = φ.
Let us denote by U m+1 the neighborhood of Id in AD m+1,p N which Lemma 6.1 guarantees consists of diffeomorphisms that are invertible in AD m+1,p N . Now let U m+1 t = φ t (U m+1 ) = {φ t • ψ : ψ ∈ U m+1 }. We first want to show that every φ * in U m+1 . Now by compactness we can cover the path φ t for 0 ≤ t ≤ 1 by a finite number of these translated neighborhoods, i.e. , which is both invertible in AD m+1,p N and of the formφ = φ t1 • ψ for some ψ ∈ U m+1 . However, we can compose with ψ −1 to conclude φ t1 =φ • ψ −1 and hence φ −1 t1 = ψ •φ −1 ∈ AD m+1,p N . Clearly this process can be continued to show every φ * ∈ U m+1 1 is invertible in AD m+1,p N . In particular, φ 1 = φ is invertible in AD m+1,p N , as desired. Finally, we want to show that φ → φ −1 is C 1 as a map AD m+1,p N → AD m,p N . We shall do this using the implicit function theorem; this is valid since a neighborhood of a fixed φ * in AD m+1,p N may be parameterized by a neighborhood of 0 in the Banach space A m+1,p N . In fact, let F : AD m+1,p N × AD m,p N → AD m,p N represent composition, i.e. F (φ, ψ) = φ • ψ, which we know from Proposition 4.1 is C 1 . Let us fix φ * ∈ AD m+1,p N and consider the differential of the map ψ → F (φ * , ψ) at the point ψ = φ −1 * ∈ AD m,p N , which is given by Since dφ * ∈ A m,p N , by Proposition 4.1 we know that dφ * •φ −1 * ∈ A m,p N , and hence (using Proposition 2.2) the linear operator T is bounded A m,p N → A m,p N . In fact, T is invertible on A m,p N , since its inverse is just T −1 (h) = d(φ * ) −1 • φ * · h, which is also bounded A m,p N → A m,p N . Consequently, the implicit function theorem implies that there is a neighborhood U of φ * in AD m+1,p N and a unique C 1 map G : U → AD m,p N such that F (φ, G(φ)) = Id holds for all φ near φ * . But uniqueness of the inverse of φ shows G(φ) = φ −1 , and hence φ → φ −1 is C 1 . For g ∈ H m,p (Q) and p ≤ q ≤ pd/(d − mp), the Sobolev inequality states g L q (Q) ≤ C(d, m, p, q) g H m,p (Q) .
Apply this to g = x δ f and for |α| ≤ m use Now let Q 0 denote the d-box of side length 1 centered at the origin in R d , and Q ℓ for ℓ = 0, 1, . . . be an enumeration of all d-boxes of side length 1 and centers at integral coordinates, so Then we use the inequality (98) and then the elementary inequality a q j 1/q ≤ a p j 1/p to estimate where C = C(d, m, p, q, δ). But the last term is equivalent to C f H m,p δ , which establishes the inequality in (c). The same argument works for d = mp, provided we assume p ≤ q < ∞. Now suppose d < mp and k < m − (d/p). For g ∈ H m−k,p (Q), Morrey's inequality implies g ∈ C(Q) and sup x∈Q |g(x)| ≤ C(m, p, k) g H m−k,p (Q) .