On global inversion of homogeneous maps
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In this note we prove a global inverse function theorem for homogeneous mappings on . The proof is based on an adaptation of the Hadamard’s global inverse theorem which provides conditions for a function to be globally invertible on . For the latter adaptation, we give a short elementary proof assuming a topological result.
KeywordsInverse function theorem Homogeneous mappings Global inverse
Mathematics Subject ClassificationPrimary 26B10 Secondary 26B05 26-01
The purpose of this note is to prove a global inverse function for homogeneous mappings on . The main difficulty is in the fact that such mappings are not smooth at the origin and thus, the known global inverse function theorems on are not readily applicable. The main motivation for studying such inverses are applications to the global invertibility of Hamiltonian flows, and further applications to construction of suitable phase functions of Fourier integral operators, a topic that will be addressed elsewhere.
Thus, our aim here is to establish the existence and the following properties for the global inverses of homogeneous mappings.
We remark that for the conclusion is not always true because, for example, the Jacobian of never vanishes on but is not globally invertible since .
Our proof is based on the application of an adaptation of the Hadamard global inverse theorem. Thus, the second aim of this note is to give a short elementary proof for it assuming a well-known topological result (very likely also known to Hadamard).
This kind of global inverse function theorem is a classical subject and of independent interest, but sometimes it also plays an important role when we discuss the global boundedness of oscillatory integrals. For example, Asada-Fujiwara  established this boundedness based on the global invertibility of maps defined by phase functions which are smooth everywhere. Generalising such results to the case of homogeneous phase as in Theorem 1.1 is also important but it is not straightforward because of the singularity at the origin. The global boundedness of oscillatory integrals with phases as in Theorem 1.1 has been analysed by the authors in  and applied to questions of global smoothing for partial differential equations in . The application of Theorem 1.1 to the global analysis of hyperbolic partial differential equations and the corresponding Hamiltonian flows will appear elsewhere.
2 Global inverses
Let us start with a topological result we assume. A differentiable map between manifolds is called a -diffeomorphism if it is one-to-one and its inverse is also differentiable. A mapping is called proper if is compact whenever is compact.
Let and be connected, oriented, -dimensional -manifolds, without boundary. Let be a proper -map such that the Jacobian never vanishes. Then is surjective. If is simply connected in addition, then is also injective.
This fact was known to Hadamard, but a rigorous proof for surjectivity can be found in . As for the injectivity, a precise proof can be found in [2, Sect. 3]. We remark that it follows in principle from the general fact that a proper submersion between smooth manifolds without boundary is a fibre bundle, meaning in our setting that it is a covering map. Since a simply connected manifold is its own universal covering space, it implies that is a diffeomorphism.
We will mostly discuss the mappings , in which case we denote its Jacobian by . The Hadamard global inverse function theorem states:
A -map is a -diffeomorphism if and only if the Jacobian never vanishes and whenever .
This theorem goes back to Hadamard [3, 4, 5]. In fact, in 1972 W. B. Gordon wrote “This theorem goes back at least to Hadamard, but it does not appear to be ‘well-known’. Indeed, I have found that most people do not believe it when they see it and that the skepticism of some persists until they see two proofs.” The reason behind this is that while we know that the function is locally a -diffeomorphism by the usual local inverse function theorem, the condition that as , guarantees that the function is both injective and, more importantly, surjective on the whole of . And indeed, W. B. Gordon proceeds in  by giving two different proofs for it, for and for mappings.
It turns out that for dimensions the conclusion remains still valid even if we relax the assumption, in some sense rather substantially, almost removing it at a finite number of points. While often this is not the case (the famous one being the ‘hairy ball theorem’, which fails completely if we assume that the vector field may be not differentiable at one point), the theory of covering spaces assures that it is the case here. In fact, here we have the following:
Let . Let . Let be such that is on , with on , and that is continuous at . Let , and assume that and that implies . Then the mapping is a global homeomorphism and its restriction is a global -diffeomorphism.
While general topological considerations as outlined above are possible here, we prefer to also give an elementary proof in Sect. 3 of the fact that Theorem 2.1 implies Theorem 2.3, also noting that exactly the same proof yields the following further version:
Let . Let be a closed set. Let be such that is on , with on , that is continuous and injective on , and that is simply connected. Assume that and that implies . Then the mapping is a global homeomorphism and its restriction is a global -diffeomorphism.
Given Theorem 2.3, we can apply it to derive Theorem 1.1:
Proof of Theorem 1.1
First we observe that in the setting of Theorem 2.3, by translation (by ) in and by subtracting from , we may assume without loss of generality that . To prove Theorem 2.3, we start with preliminary statements.
Let . Then is compact in if and only if it is compact in .
We recall that a mapping is called proper if is compact whenever is compact.
Let be proper and such that and . Then the restriction is proper.
Let be compact in . By Lemma 3.1 it is compact in , and, since is proper, the set is compact in . We notice that if then we would have , which is impossible since . Hence , and by Lemma 3.1 again, the set is compact in . Hence the restriction is proper.
Let be continuous everywhere. Then is proper if and only if implies .
We show the if part. Let be compact. Then it is closed and hence is closed. Suppose is not bounded. Then there is a sequence such that . Hence and also by the assumption on , which yields a contradiction with the boundedness of . The converse implication is clearly also true.
Let be proper, injective, and continuous. Then is an open map.
Let us assume that is not open for an open subset . Then there is a point such that is on the boundary of , and we can construct a sequence such that . Since is proper, there exists a subsequence which converges to some point . Note that . Since is continuous, , but we also have which contradicts to the fact that is injective.
The following result is a straight forward consequence of Theorem 2.1.
Let . Let be proper and such that and and . Moreover, assume that is on , with on . Then the restriction is surjective. If in addition, is also injective.
By Lemma 3.2, the restriction of is a proper map from to . Note that is simply connected if . Then Theorem 2.1 implies the statement.
With all these facts, Theorem 2.3 is immediate:
Proof of Theorem 2.3
By Corollary 3.5 and Lemma 3.3, the map is bijective. Hence it is a global diffeomorphism by the usual local inverse function theorem. Furthermore, the map is also bijective since , hence the global inverse exists and is continuous by Lemma 3.4. Since is also continuous, it is a homeomorphism.
The authors would like to thank Professor Adi Adimurthi for valuable remarks on the first version of our manuscript, leading to its considerable improvement.
- 3.Hadamard, J.: Sur les correspondances ponctuelles. Paris: Oeuvres, Editions du Centre Nationale de la Researche Scientifique, pp. 383–384 (1998)Google Scholar
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