SOBOLEV SPACES ON LIE MANIFOLDS AND POLYHEDRAL DOMAINS

BERND AMMANN, ALEXANDRU D. IONESCU, AND VICTOR NISTOR

Abstract. We study Sobolev spaces on Lie manifolds, which we define as a class of manifolds described by vector fields (see Definition 1.2). The class of Lie manifolds includes the Euclidean spaces ℝ^{n}, asymptotically flat manifolds, conformally compact manifolds, and manifolds with cylindrical and polycylindrical ends. As in the classical case of ℝ^{n}, we define Sobolev spaces using derivatives, powers of the Laplacian, or a suitable class of partitions of unity. We extend the basic results about Sobolev spaces on Euclidean spaces to the setting of Lie manifolds. These results include the definition of the trace map, a characterization of its range, the extension theorem, the density of smooth functions, and interpolation properties. One of the main motivations is that, in the examples we have studied so far, the totally-characteristic Sobolev spaces on polyhedral domains identify with Sobolev spaces on suitable Lie manifolds with boundary. The analysis we develop may be useful for solving certain types of non-linear partial differential equations on noncompact manifolds that appear, for instance, in Einstein’s constraint equations. We also sketch two applications, one to the Yamabe functional and one to the regularity of boundary value problems on polyhedral domains.

Contents

Introduction 1 1. Lie manifolds 4 2. Sobolev spaces 9 3. Manifolds with boundary 15 4. The Yamabe functional on Lie manifolds 22 5. Submanifolds 25 6. Pseudodifferential operators 27 References 33

Introduction

Function spaces play a central role in Analysis and are often used in practical applications of mathematics. In many of these applications the domains are not smooth, which has lead to work under Lipschitz-type conditions, as in [35, 54, 57, 77] etc. These papers have extended many classical results on function spaces to Lipschitz

Ammann was partially supported by MSRI’s NSF grant DMS-9810361. Ionescu was supported in part by NSF Grant No. 0100021. Nistor was partially supported by NSF Grants DMS 0200808 and DMS 0209497. Manuscripts available from http://www.math.psu.edu/nistor/ .

domains, but have also revealed the limitations of the standard Sobolev spaces. For example, the usual regularity theorems for solutions of elliptic differential equations on smooth domains [22, 73] do not hold on Lipschitz domains. See [20, 21, 25, 26] in addition to the papers quoted above.

To explain the question of regularity, denote by Δ the Laplace operator and consider the following example. Let ℙ be a polygon in the plane and let u ∈ H^{1}(ℙ) be a solution of the Poisson problem Δu = f ∈ C^{∞}(ℙ) with Dirichlet boundary conditions (i.e., u = 0 on ∂ℙ). One can show [25, 26] that there exists a constant s_{0}, explicitly determined in terms of the angles of ℙ, such that u ∈ H^{s}(ℙ) for any f ∈ C^{∞}(ℙ) and any s < s_{0}, but not better. That is, u ∉ H^{s_{0}}(ℙ) for a suitable choice of f ∈ C^{∞}(ℙ). This is in sharp contrast with the case of smooth boundary, in which case u is smooth whenever f is smooth. A deep study of these issues in the setting of Lipschitz domains can be found in the papers of Jerison and Kenig [35] and Mitrea and Taylor [54], and in the other papers quoted there.

The loss of regularity in the Poisson problem mentioned above can be avoided, however, if one considers a different class of Sobolev spaces on the polygon ℙ [12, 11, 53, 56]. These Sobolev spaces, sometimes denoted H_{b}^{m}(ℙ) are the so called totally characteristic Sobolev spaces and were used by many researchers, see [27, 10, 20, 21, 38, 45, 50, 52, 56, 65] and the references therein. The definition of the spaces H_{b}^{m}(ℙ) uses the distance function ρ(x) from x ∈ ℙ to the vertices of ℙ:

(1)  H_{b}^{m}(ℙ) := {u ∈ L_{loc}^{2}(ℙ), ρ^{|α|−1}D^{α}u ∈ L^{2}(ℙ), |α| ≤ m}.

It is one of the purposes of this paper to study totally characteristic Sobolev spaces on polyhedral domains and to extend some of the main results in the theory of classical Sobolev spaces to this setting. This will help clarify the role of the totally characteristic Sobolev spaces in the study of boundary value problems on polyhedral domains, in particular. More specific results in three dimension will be included in [11].

Our approach to Sobolev spaces on polyhedral domains is to reduce their study to that of Sobolev spaces on certain non-compact manifolds with boundary. These noncompact manifolds are obtained from our polyhedral domain by conformally changing the metric with a factor that blows up at the faces of codimension ≥ 2. The resulting non-compact manifolds are “Lie manifolds with boundary,” (See Definition 1.2 and Subsection 1.5 for definitions.)

Lie manifolds were first introduced informally in [49, 51]. Their definition was formalized in [5], where several simple but basic properties of these manifolds were proved in a general setting. (Lie manifolds were called “manifolds with a Lie structure at infinity” in that paper.) In addition to the non-compact manifolds that arise from polyhedral domains, other examples of Lie manifolds include the Euclidean spaces ℝ^{n}, manifolds that are Euclidean at infinity, conformally compact manifolds, manifolds with cylindrical and polycylindrical ends, and asymptotically hyperbolic manifolds. These classes of non-compact manifolds are relevant in many problems in Mathematical Physics and Computational Sciences, such as domain decomposition methods and the Finite Element Method, quasi-linear parabolic equations, Yamabe’s problem, Einstein’s equations, and the positive mass theorem. Classes of Sobolev spaces on non-compact manifolds have been studied in many papers, of which we mention only a few [7, 2, 34, 37, 39, 40, 43, 51, 47, 46, 63, 64, 71] in addition to the works mentioned before.

A large part of the technical material in this paper is devoted to the study of Sobolev spaces on Lie manifolds, with or without boundary. The first three sections of the paper are more elementary and we have attempted to make them essentially self-contained. We begin in Section 1 with a review of the definition of a structural Lie algebra of vector fields V on a manifold with corners M. This Lie algebra of vector fields will provide the derivatives appearing in the definition of the Sobolev spaces. Then we define Lie manifolds. The interior M_{0} of M is by definition a Lie manifold. It turns out that M_{0} carries a complete metric g, unique up to Lipschitz equivalence (some authors use the term “quasi-isometric” to describe two Lipschitz equivalent metrics).

In Section 1 we define the Sobolev spaces W^{s,p}(M_{0}) on a Lie manifold M_{0}, where the superscripts have the following possible ranges: either s ∈ ℤ_{+} and 1 ≤ p ≤ ∞ or s ∈ ℝ and 1 < p < ∞. The main goal of this paper is to study the spaces W^{s,p}(M_{0}). We first define the spaces W^{s,p}(M_{0}), s ∈ ℤ_{+} and 1 ≤ p ≤ ∞, by differentiating with respect to vector fields in V. This definition is in the spirit of the definition of Sobolev spaces on ℝ^{n}. Then we prove two alternative definitions of these Sobolev spaces, either using a suitable class of partitions of unity (as in [66, 68] for example), or as the domains of the powers of the Laplace operator (for p = 2). We also consider these spaces on open subsets Ω_{0} ⊂ M_{0}. The spaces W^{s,p}(M_{0}), for s ∈ ℝ, 1 < p < ∞ are defined by interpolation and duality or, alternatively, using partitions of unity. In Section 3, we discuss domains Ω_{0} whose boundary ∂Ω_{0} is a (smooth) Lie submanifold of M_{0}.

We extend several of the classical results on Sobolev spaces to the setting of the spaces W^{s,p}(M_{0}). These results include the density of smooth, compactly supported functions, the Gagliardo-Nirenberg-Sobolev inequalities, the extension theorem, the trace theorem, the characterization of the range of the trace map in Hilbert space case (p = 2), and the Rellich-Kondrachov compactness theorem. Some of these results follow from the analogous results for manifolds with bounded geometry. We conclude the first three, more elementary sections, with an application to the regularity of boundary value problems on polyhedral domains, Theorem 3.8.

The last three sections are slightly less elementary. In particular, we no longer attempt to make them self-contained. Some applications to geometry (the Yamabe problem) are given in Section 4. In Section 5 we discuss the geometric results needed on Lie submanifolds, most importantly, the global tubular neighborhood theorem. Finally, in Section 6 we discuss the continuity of pseudodifferential operators acting on the spaces W^{s,p}(M_{0}). The class of pseudodifferential operators that we consider is Ψ_{1,0,V}^{∞}(M_{0}). This algebra was introduced in [6] as a quantization of the algebra of differential operators Diff_{V}^{∗}(M) generated by the structural Lie algebra of vector fields V defining the Lie structure on M_{0}, thus solving a conjecture from [49]. In particular, we obtain a description of the spaces W^{s,p}(M_{0}), s ≥ 0, 1 < p < ∞, as the domain of any elliptic operator P ∈ Ψ_{1,0,V}^{s}(M_{0}).

We include few concrete examples of manifolds with a Lie structure at infinity besides those needed to treat polyhedral domains. The reader can find more examples in [41] or in [5], for example.

Acknowledgements: We would like to thank W. Dahmen, R. Lauter, R. Strichartz, and A. Vasy for useful discussions. The first named author wants to thank MSRI, Berkeley, CA for its hospitality.

1. Lie manifolds

As mentioned already in the Introduction, our approach to the study of totallycharacteristic Sobolev spaces on polyhedral domains is based on their relation to Sobolev spaces on Lie manifolds with boundary. For the convenience of the reader, we recall the definition of a Lie manifold and some basic results. This Section is to a large extent based on [5]. (Note that what we are calling here “Lie manifolds” were called “manifolds with a Lie structure at infinity” in [5].)

We shall treat Lie manifolds as well as Lie submanifolds (of Lie manifolds) in Section 3.

1.1. Definition. We need to recall first manifolds with corners. By definition, every point p in a manifold with corners M has a coordinate neighborhood diffeomorphic to [0, ∞)^{k} × ℝ^{n−k} such that the transition functions are smooth up to the boundary.

We write M_{0} for the interior of M, and ∂M = M ∖ M_{0} for the boundary, i.e., ∂M is the union of all boundary faces of dimension 0 to n − 1. In the sequel, by a manifold we shall always understand a C^{∞}-manifold possibly with corners, whereas a smooth manifold is a C^{∞}-manifold without corners.

As we shall see below, a Lie manifold is described by a Lie algebra of vector fields satisfying certain conditions. We now discuss some of these conditions.

Definition 1.1. A subspace V ⊆ Γ(M, T M) of the Lie algebra of all smooth vector fields on M is said to be a structural Lie algebra of vector fields on M provided that the following conditions are satisfied:

(i) V is closed under the Lie bracket of vector fields; (ii) every V ∈ V is tangent to all hyperfaces of M; (iii) C^{∞}(M)V = V; and (iv) each point p ∈ M has a neighborhood U_{p} such that

V_{U_{p}} := \{X|_{\overline{U}_{p}} |X ∈ V\} ≃ C^{∞}(\overline{U}_{p})^{k},

that is, V_{U_{p}} is a free C^{∞}(\overline{U}_{p})-module of dimension k, for some k.

The condition (iv) in the definition above can be reformulated as follows: (iv’) For every p ∈ M there exist a neighborhood U_{p} ⊂ M of p and vector fields

X_{1}, X_{2}, …, X_{k} ∈ V with the property that for any Y ∈ V, there exist functions

f_{1}, …, f_{k} ∈ C^{∞}(M), uniquely determined on U_{p}, such that
(2)  Y = ∑_{j=1}^{k} f_{j}X_{j} on U_{p}.

Here are some examples of structural Lie algebras of vector fields. If F ⊂ T M is a sub-bundle of the tangent bundle of a smooth manifold (so M has no boundary) such that V_{F} := Γ(M, F) is closed under the Lie bracket, then V_{F} is a structural Lie algebra of vector fields. Another example arises from manifolds with boundary and is related to the totally characteristic Sobolev spaces defined on an angle, as in the introduction. More precisely, let M be a manifold with boundary and let V_{b} be the space of vector fields on M tangent to the boundary of M. Then V_{b} is a structural Lie algebra of vector fields. See [48, 50] and Subsection 1.5.

Definition 1.2. A Lie structure at infinity on a smooth manifold M_{0} is a pair (M, V), where M is a compact manifold, possibly with corners, and V ⊂ Γ(M, T M) is a structural Lie algebra of vector fields on M with the following properties:

(i) M_{0} = M ∖ ∂M, the interior of M, and (ii) If p ∈ M_{0}, then any local basis of V in a neigborhood of p is also a local basis of

the tangent space to M_{0}. (In particular, the constant k of Equation (2) equals the dimension of M_{0}.)

A manifold with a Lie structure at infinity (or, simply, a Lie manifold) is a manifold M_{0} together with a Lie structure at infinity (M, V) on M_{0}. We shall sometimes denote a Lie manifold as above by (M_{0}, M, V), or, simply, by (M, V), because M_{0} is determined as the interior of M.

We include only a few examples of Lie manifolds. The reader can find more examples in [51], from where these examples were borrowed or in [41, 5].

Examples 1.3.

(a) Take V_{b} to be the set of all vector fields tangent to all faces of a manifold with corners M. Then (M, V_{b}) is a Lie manifold. We shall say following Melrose’s terminology that M_{0} = M ∖ ∂M is endowed with the b-structure at infinity.

(b) Take V_{0} to be the set of all vector fields vanishing on all faces of a manifold with corners M. Then (M, V_{0}) is a Lie manifold. We shall say following Melrose’s terminology that M_{0} = M ∖ ∂M is endowed with the zero-structure at infinity.

Remark 1.4. Let us observe, that Conditions (iii) and (iv) of Definition 1.1 are equivalent to the condition that V be a projective C^{∞}(M)-module. Thus, by the Serre-Swan theorem [36], there exists a vector bundle A → M, unique up to isomorphism, such that V = Γ(M, A). Since V consists of vector fields, that is V ⊂ Γ(M, T M), we also obtain a natural vector bundle morphism ϱ : A → M, called the anchor map. The Condition (ii) of Definition 1.2 is then equivalent to the fact that ϱ is an isomorphism A|_{M_{0}} ≃ T M_{0} on M_{0}. We will take this isomorphism to be an identification, and thus we can say that A is an extension of T M_{0} to M (that is, T M_{0} ⊂ A).

1.2. Riemannian metric. Let (M_{0}, M, V) be a Lie manifold. By definition, a Riemannian metric on M_{0} compatible with the Lie structure at infinity (M, V) is a metric g such that for any p ∈ M, we can choose the basis X_{1}, …, X_{k} in Definition 1.1, (iv’) and (2) to be orthonormal with respect to this metric everywhere on U_{p}. (Note that this condition is a restriction only for p ∈ ∂M := M ∖ M_{0}.) Alternatively, we will also say that (M_{0}, g_{0}) is a Riemannian Lie manifold. Any Lie manifold carries a compatible Riemannian metric, and any two compatible metrics are bi-Lipschitz to each others.

Remark 1.5. Using the language of Remark 1.4, g is a compatible metric on M_{0} if, and only if, there exists a metric on the vector bundle A → M which restricts to g on T M_{0} ⊂ A.

The geometry of a Riemannian manifold (M_{0}, g_{0}) with a Lie structure (M, V) at infinity has been studied in [5]. For instance, (M_{0}, g_{0}) is necessarily of infinite volume and complete. Moreover, all the covariant derivatives of the Riemannian curvature tensor are bounded. Under additional mild assumptions, we also know that the injectivity radius is bounded from below by a positive constant, i.e., (M_{0}, g_{0}) is of bounded geometry. (A manifold with bounded geometry is a Riemannian manifold with positive injectivity radius and with bounded covariant derivatives of the curvature tensor, see [66] and references therein).

On a Riemannian Lie manifold (M_{0}, M, V), the exponential map exp_{p} : T M_{0} → M_{0} is well-defined for all p ∈ M_{0} and extends to a differentiable map exp_{p} : A_{p} → M depending smoothly on p ∈ M. A convenient way to introduce the exponential map is via the geodesic spray, as done in [5]. Similarly, any vector field X ∈ V = Γ(M, A) is integrable and will map any (connected) face of M to itself. The resulting diffeomorphism of M_{0} will be denoted ψ_{X}.

We assume from now on that r_{inj}(M_{0}), the injectivity radius of (M_{0}, g_{0}), is positive.

1.3. V-differential operators. We are especially interested in the analysis of the differential operators generated using only derivatives in V. Let Diff_{V}^{∗}(M) be the algebra of differential operators on M generated by multiplication with functions in C^{∞}(M) and by differentiation with vector fields X ∈ V. The space of order m differential operators in Diff_{V}^{∗}(M) will be denoted Diff_{V}^{m}(M). A differential operator in Diff_{V}^{∗}(M) will be called a V-differential operator.

We can define V-differential operators acting between sections of smooth vector bundles E, F → M, E, F ⊂ M × ℂ^{N} by

(3)  Diff_{V}^{∗}(M; E, F) := e_{F} M_{N}(Diff_{V}^{∗}(M))e_{E} ,

where e_{E}, e_{F} ∈ M_{N}(C^{∞}(M)) are the projections onto E and, respectively, F. It follows that Diff_{V}^{∗}(M; E, E) =: Diff_{V}^{∗}(M; E) is an algebra. It is also closed under taking adjoints of operators in L^{2}(M_{0}), where the volume form is defined using a compatible metric g on M_{0}.

1.4. Lie manifolds with boundary. One of the main motivation for this work is to study Sobolev spaces on polyhedral domains. We shall do that by reducing their study to that of Sobolev spaces on “Lie manifolds with boundary,” a class of manifolds with boundary that we introduce below.

To understand the following constructions, let us take a closer look at the the local structure of the Sobolev space H_{b}^{m}(ℙ) associated to a polygon ℙ (recall (1)). Consider Ω := {(r, θ) | 0 < r < r_{0}, 0 < θ < α}, which models an angle of ℙ. Then the totally characteristic Sobolev spaces associated to Ω, H_{b}^{m}(Ω), can alternatively be described as

(4)  H_{b}^{m}(Ω) := {u ∈ L_{loc}^{2}(Ω), r^{−1}(r∂_{r})^{i}∂_{θ}^{j}u ∈ L^{2}(Ω), i + j ≤ m}.

An important point of the above definition is that first the angle Ω was desingularized and then a different basis of vector fields on the desingularization was used instead of the standard basis in the definition of the usual Sobolev spaces. For example, in the case above of the angle Ω, the basis r∂_{r} and ∂_{θ} was used instead of the usual basis ∂_{x} and ∂_{y}. This underscores the importance of vector fields in our approach, which owes to the work of several authors. See [17, 19, 18, 49, 51, 45, 46, 56, 63, 65, 66, 67, 68, 69] and the references therein.

Let N ⊂ M be a submanifold with corners of codimension one of M (see Section 5). Recall that this implies that N is transverse to all faces of M. We shall say that N is a regular submanifold of (M, V) if we can choose a tubular neighborhood V of N_{0} := N ∖ ∂N = N ∩ M_{0} in M_{0} and a compatible metric g on M_{0} that restricts to a product-type metric on V ≃ (∂N_{0}) × (−ε_{0}, ε_{0}). (In Section 5, we shall show that every tame submanifold of codimension one is regular; in turn, this will give an easy, geometric criterion to decide when a codimension one submanifold of M is regular.)

Let Ω ⊂ M be an open subset. We say that Ω is a Lie domain in M if and only if ∂Ω = ∂\overline{Ω} and ∂Ω is a regular submanifold of M. Let Ω_{0} = Ω ∩ M_{0}. Then ∂Ω_{0} = (∂Ω) ∩ M_{0} is a smooth submanifold of codimension one of M_{0}.

Definition 1.6. A Lie manifold with boundary is a triple (Ω_{0}, Ω′, V′), where Ω_{0} is a smooth manifold with boundary, Ω′ is a compact manifold with corners containing Ω_{0} as an open subset, and V′ is a Lie algebra of vector fields on Ω′ with the property that there exists a Lie manifold (M_{0}, M, V), a Lie domain Ω ⊂ M in M and a diffeomorphism ϕ : Ω′ → Ω such that ϕ(Ω_{0}) = Ω ∩ M_{0} and ϕ^{∗}(V|_{Ω}) = V′.

Note that if (Ω_{0}, Ω, V) is a Lie manifold with boundary, then Ω_{0} is determined by (Ω, V), so we can write (Ω, V) instead of (Ω_{0}, Ω, V).

1.5. Polyhedral domains. We now discuss in an example the relation between polyhedral domains and Lie manifolds with boundary. Let us consider

Δ_{n} := {(x_{1}, …, x_{n}) ∈ ℝ^{n}, x_{j} ≥ 0, ∑ x_{j} ≤ 1},

the unit simplex in ℝ^{n}. To Δ_{n} we now associate a Lie manifold with boundary (Σ(Δ_{n}), V(Δ_{n})), together with a “desingularization” map

κ_{n} : Σ(Δ_{n}) → Δ_{n},

satisfying κ_{n}(∂Σ(Δ_{n})) ⊂ ∂Δ_{n}. Let S ⊂ Δ_{n} be the set of points not belonging to a face of codimension ≥ 2. The map κ_{n} will turn out to be a bijection between κ_{n}^{−1}(S) and S. We shall proceed by induction as follows.

Let ℙ be the angle {0 ≤ θ ≤ α}, a closed subset of ℝ^{2}, where (r, θ) are the polar coordinates in the plane. We define its canonical desingularization by Σ(ℙ) := [0, ∞) × [0, α], which maps surjectively to ℙ by κ(r, θ) = (r cos θ, r sin θ). The Lie algebra of vector fields on Σ(ℙ) is given by

(5)  {X ∈ Γ(T Σ(ℙ)), X(0, θ) tangent to {0} × [0, α] }

We can realize Σ(ℙ) as a Lie domain in the manifold with boundary [0, ∞) × S^{1}, by realizing [0, α] as a subset of the unit circle S^{1}. On [0, ∞) × S^{1} we then consider all vector fields tangent to the boundary.

The desingularization Σ(Δ_{2}) of the unit triangle Δ_{2} is obtained as follows. First we desingularize each angle with the opposite face removed. Then we glue these desingularizations using the desingularization map κ. This yields, up to a diffeomorphism a hexagon. Let e_{1}, e_{2}, e_{3} be the three non-intersecting edges of this hexagon corresponding to the vertices of Δ_{2}. These faces are the ones that, under the desingularization map will go to the three vertices of the triangle. Then V(Δ_{2}) consists of vector fields tangent to the faces e_{1}, e_{2}, e_{3}.

Assume now that (Σ(Δ_{n}), V(Δ_{n}), κ_{n}) were constructed, and let us construct (Σ(Δ_{n+1}), V(Δ_{n+1}), κ_{n+1}). We first construct an analogous desingularization of

CΔ_{n} := {(x_{0}, x_{1}, …, x_{n}) ∈ ℝ^{n+1}, x_{j} ≥ 0}.

The space CΔ_{n} is the “cone” over Δ_{n}. We shall use next the alternative description of Δ_{n} as

Δ_{n} := {(y_{0}, y_{1}, …, y_{n}) ∈ ℝ^{n+1}, y_{j} ≥ 0, ∑ y_{j} = 1}.

The desingularization of CΔ_{n} is, by definition, [0, ∞)×Σ(Δ_{n}), with desingularization map κ(r, z) = rκ_{n}(z). The Lie algebra of vector fields on this desingularization is

V = {(X, Y) ∈ Γ(T [0, ∞)) × Γ(T Σ(Δ_{n})), X(0) = 0 and Y ∈ V(Δ_{n})}.

In other words, if we split the tangent space to [0, ∞) × Σ(Δ_{n}) into the direct sum of the vector bundles tangent to the two factors, then on the first component we get vector fields tangent to the boundary (“tangent to the boundary” means “vanishing at the boundary” in this case of [0, ∞)) and on the second component we simply get vectors in the structural Lie algebra of vector fields V(Σ(Δ_{n})) corresponding to the desingularization of Δ_{n}.

The desingularization Σ(Δ_{n+1}) of Δ_{n+1} is obtained by gluing the desingularizations of the cones corresponding to each of the vertices, these cones being obtained by removing the opposite face to the given vertex, one vertex at a time.

This rather complicated construction is justified by the following simple proposition.

Proposition 1.7. Let h be the standard euclidean metric on Δ_{n}. Let ρ(x) be the distance from the point x to the set of points belonging to a face of codimension ≥ 2. Then ρ^{−2}h is Lipschitz equivalent to any compatible metric g on the interior of Σ(Δ_{n}), that is, there exists C > 0 such that

C^{−1}g(ξ) ≤ ρ^{−2}h(ξ) ≤ Cg(ξ)

for any tangent vector ξ ∈ T Δ_{n} tangent to an interior point of Δ_{n}. In particular,

C^{−n/2}d vol_{g} ≤ ρ^{−n}d vol_{h} ≤ C^{n/2}d vol_{g} .

Proof. By induction. For n = 2, this follows right away from the definition of totally characteristic Sobolev spaces on a triangle. For the induction step, denote by h_{n} the metric on Δ_{n}. We can cover Δ_{n} with open sets V_{j}, such that the closure \overline{V}_{j} contains exactly one vertex of Δ_{n}. It is enough to prove our statement on \overline{V}_{j}, for all j. This will change the function ρ on \overline{V}_{j}, but the metric ρ^{−2}h_{n} will be in the same Lipschitz equivalence class, because the function ρ changes only by a factor that is bounded from above and bounded from zero (i.e., it takes values in a compact interval of (0, ∞)).

We then consider the same construction on CΔ_{n−1}, the cone over the simplex Δ_{n−1}. Note that the function ρ does not change by considering the bigger set CΔ_{n−1} instead of \overline{V}_{j}. Let r denote the distance to the vertex of the cone CΔ_{n−1}. After a suitable translation, we can (and will) assume that the vertex of this cone is the origin. The metric r^{−2}h_{n} makes the interior of CΔ_{n−1} isometric to the interior of Δ_{n−1} × ℝ with the product metric h̃_{n−1} × (dt)^{2}, t ∈ ℝ, where h̃_{n−1} is the metric on Δ_{n−1} obtained by mapping Δ_{n−1} to the unit sphere via the map x → x/∥x∥. This metric is Lipschitz equivalent to h_{n−1}.

Let ρ̃(x) be the distance from x ∈ CΔ_{n−1} to the subset of points belonging to a face of codimension ≥ 2 in the metric h_{n−1} × (dt)^{2}. The induction hypothesis is that ρ̃^{−2}(h_{n−1} × (dt)^{2}) is Lipschitz equivalent to g. Hence

(rρ̃)^{−2}h_{n} = ρ̃^{−2}h̃_{n−1} × (dt)^{2}

is Lipschitz equivalent to g. To prove our result, it is enough then to show that f(x) := r(x)ρ̃(x)/ρ(x) is bounded from above and bounded from zero. Let us observe that ρ(tx) = tρ(x), r(tx) = tr(x), and ρ̃(tx) = ρ̃(x), for all t ∈ (0, ∞) for which this makes sense. Consequently, f(tx) = f(x), whenever both sides are defined. To prove that f takes on values in a compact interval contained in (0, ∞), it is therefore enough to do that for ∥x∥ constant. The result hence follows from the fact that f is continuous ≠ 0 on the set {∥x∥ = c > 0} ∩ CΔ_{n−1}. □

For further reference, let us record there a consequence of the proof of the above proposition.

Corollary 1.8. We use the notation of Proposition 1.7. There exists f ∈ C(Δ_{n}), f smooth and ≠ 0 on the interior of Δ_{n} such that f/ρ takes values in a compact interval of (0, ∞). Moreover, f^{−2}h is a compatible metric on the interior of Δ_{n} for any such f.

2. Sobolev spaces

In this section we discuss the Sobolev spaces on M_{0} from an elementary point of view, that is, without using pseudodifferential operators. Our treatment is standard, following [22, 28, 29]. See also [8]. Some of these elementary results simply follow from the fact that M_{0} has bounded geometry whenever its injectivity radius is positive. These results include the density of smooth, compactly supported functions, the identification of the L^{2}–Sobolev spaces with the domains of suitable powers of 1 + Δ, and the Gagliardo–Nirenberg–Sobolev Embedding theorem. We include these results here for completeness and further references.

Conventions. Throughout the rest of this paper, (M_{0}, M, V) will be a fixed Lie manifold. We also fix a compatible metric g on M_{0} (i.e., , a metric compatible with the Lie structure at infinity on M_{0}, see Subsection 1.2). By Ω we shall denote an open subset of M and Ω_{0} = Ω ∩ M_{0}. The letters C and c will be used to denote possibly different constants that may depend only on (M_{0}, g) and its Lie structure at infinity (M, V).

We shall denote the volume form (or measure) on M_{0} associated to g by d vol_{g}(x) or simply by dx, when there is no danger of confusion. Also, we shall denote by L^{p}(Ω_{0}) the resulting L^{p}-space on Ω_{0} (i.e., defined with respect to the volume form dx). These spaces are independent of the choice of the compatible metric g on M_{0}, but their norms, denoted by ∥ · ∥_{L^{p}}, do depend upon this choice, although this is not reflected in the notation. Also, we shall use the fixed metric g on M_{0} to trivialize all density bundles. Then the space D′(Ω_{0}) of distributions on Ω_{0} is defined, as usual, as the dual of C_{c}^{∞}(Ω_{0}). The spaces L^{p}(Ω_{0}) identify with spaces of distributions on Ω_{0} via the pairing

⟨u, ϕ⟩ = ∫_{Ω_{0}} u(x)ϕ(x)dx, where ϕ ∈ C_{c}^{∞}(Ω_{0}) and u ∈ L^{p}(Ω_{0}).

2.1. Definition of Sobolev spaces using vector fields and connections. Anticipating, let us mention that we will define the Sobolev spaces W^{s,p}(Ω_{0}) in the following two cases: s ∈ ℤ_{+}, 1 ≤ p ≤ ∞, and arbitrary open sets Ω_{0} or s ∈ ℝ, 1 < p < ∞, and Ω_{0} = M_{0}. In fact, we will give several definitions and then show their equivalence. The first definition is in terms of the Levi-Civita connection ∇ on T M_{0}. We shall denote also by ∇ the induced connections on tensors (i.e., on tensor products of T M_{0} and T^{∗}M_{0}).

Definition 2.1 (∇-definition of Sobolev spaces). Let k ∈ ℤ_{+}, then the Sobolev space W^{k,p}(Ω_{0}) is the space of distributions u on Ω_{0} ⊂ M_{0} such that

(6)  ∥u∥_{∇,W^{k,p}}^{p} := ∑_{l=1}^{k} ∫_{Ω_{0}} |∇^{l}u(x)|^{p}dx < ∞ , 1 ≤ p < ∞.

For p = ∞ we change this definition in the obvious way, namely we require that,

(7)  ∥u∥_{∇,W^{k,∞}} := sup |∇^{l}u(x)| < ∞ , 0 ≤ l ≤ k.

Let p = 2. When \overline{Ω} = Σ(Δ_{n}), see Subsection 1.5, we can regard \overline{Ω} as a subset of its double, which is a Lie manifold. This gives then the totally characteristic Sobolev spaces on Δ_{n}, denoted H_{b}^{k}(Δ_{n}). Let ρ(x) be the distance to the set of points belonging to a face of codimension ≥ 2 of Δ_{n}, as in Proposition 1.7. Then

(8) H_{b}^{k}(Δ_{n}) = W^{k,2}(Σ(Δ_{n})) = {u, ∫_{Δ_{n}} |∂^{α}u|^{2}ρ^{−n+2|α|}d vol_{h} < ∞, |α| ≤ k}

= {u, ρ^{|α|−n/2}∂^{α}u ∈ L^{2}(Δ_{n}) = L^{2}(Δ_{n}, dx), |α| ≤ k}.

We introduce an alternative definition of Sobolev spaces.

Definition 2.2 (vector fields definition of Sobolev spaces). Let again k ∈ ℤ_{+}. Choose a finite set of vector fields X such that C^{∞}(M)X = V. This condition is equivalent to the fact that the set {X(p), X ∈ X } generates A_{p} linearly, for any p ∈ M. Then the system X provides us with the norm

(9)  ∥u∥_{X ,W^{k,p}}^{p} := ∑ ∥X_{1}X_{2} … X_{l}u∥_{L^{p}}^{p} , 1 ≤ p < ∞,

the sum being over all possible choices of 0 ≤ l ≤ k and all possible choices of not necessarily distinct vector fields X_{1}, X_{2}, …, X_{l} ∈ X. For p = ∞, we change this definition in the obvious way:

(10)  ∥u∥_{X ,W^{k,∞}} := max ∥X_{1}X_{2} … X_{l}u∥_{L^{∞}} ,

the maximum being taken over the same family of vector fields.

In particular,

(11)  W^{k,p}(Ω_{0}) = {u ∈ L^{p}(Ω_{0}), P u ∈ L^{p}(Ω_{0}), for all P ∈ Diff_{V}^{k}(M)}

The following proposition shows that the second definition yields equivalent norms.

Proposition 2.3. The norms ∥ · ∥_{X ,W^{k,p}} and ∥ · ∥_{∇,W^{k,p}} are equivalent for any choice of the compatible metric g on M_{0} and any choice of a system of the finite set X such that C^{∞}(M)X = V. The spaces W^{k,p}(Ω_{0}) are complete Banach spaces in the resulting topology. Moreover, W^{k,2}(Ω_{0}) is a Hilbert space.

Proof. As all compatible metrics g are bi-Lipschitz to each others, the equivalence classes of the ∥ · ∥_{X ,W^{k,p}}-norms are independent of the choice of g. We will show that for any choice X and g, ∥ · ∥_{X ,W^{k,p}} and ∥ · ∥_{∇,W^{k,p}} are equivalent. It is clear that then the equivalence class of ∥ · ∥_{X ,W^{k,p}} is independent of the choice of X, and the equivalence class of ∥ · ∥_{∇,W^{k,p}} is independent of the choice of g.

We argue by induction in k. The equivalence is clear for k = 0. We assume now that the W^{l,p}-norms are already equivalent for l = 0, …, k − 1. Observe that if X, Y ∈ V, then the Koszul formula implies ∇_{X}Y ∈ V [5]. To simplify notation, we define inductively X^{0} := X, and X^{i+1} = X^{i} ∪ {∇_{X}Y | X, Y ∈ X^{i}}.

By definition any V ∈ Γ(T^{∗}M^{⊗k}) satisfies (∇∇V)(X, Y) = ∇_{X}∇_{Y} V − ∇_{∇_{X}Y} V. This implies for X_{1}, …, X_{k} ∈ X

(\underbrace{∇ … ∇f}_{k-times})(X_{1}, …, X_{k}) = X_{1} … X_{k}f + ∑_{l=0}^{k−1} ∑_{Y_{j}∈X^{k−l}} a_{Y_{1},...,Y_{l}}Y_{1} … Y_{l} f,

for appropriate choices of a_{Y_{1},...,Y_{l}} ∈ ℤ_{+}. Hence,

∥(\underbrace{∇ … ∇f}_{k-times})∥_{L^{p}} ≤ C ∑ ∥∇ … ∇f(X_{1}, …, X_{k})∥_{L^{p}} ≤ C∥f∥_{X ,W^{k,p}}.

By induction, we know that ∥Y_{1}, …, Y_{l}f∥_{L^{p}} ≤ C∥f∥_{∇,W^{l,p}} for Y_{i} ∈ X^{k−l}, 0 ≤ l ≤ k − 1, and hence

∥X_{1} … X_{k}f∥_{L^{p}} ≤ \underbrace{∥∇ … ∇f∥_{L^{p}}∥X_{1}∥_{L^{∞}} ⋯ ∥X_{k}∥_{L^{∞}}}_{≤C∥f∥_{∇,W^{k,p}}}

+ \underbrace{∑_{l=0}^{k−1} ∑_{Y_{1},...,Y_{l}∈X^{k−l}} a_{Y_{1},...,Y_{l}}Y_{1} … Y_{l} f}_{≤C∥f∥_{∇,W^{k−1},p}} .

This implies the equivalence of the norms.

The proof of completeness is standard, see for example [22, 75]. □

We shall also need the following simple observation.

Lemma 2.4. Let Ω′ ⊂ Ω ⊂ M be open subsets, Ω_{0} = Ω∩M_{0}, and Ω′_{0} = Ω′∩M_{0}. The restriction then defines continuous operators W^{s,p}(Ω_{0}) → W^{s,p}(Ω′_{0}). If the various choices (X , g, x_{j}) are done in the same way on Ω and Ω′, then the restriction operator has norm 1.

2.2. Definition of Sobolev spaces using partitions of unity. Yet another description of the spaces W^{k,p}(Ω_{0}) can be obtained by using suitable partitions of unity as in [66, Lemma 1.3], whose definition we now recall. See also [15, 68, 58].

Lemma 2.5. For any 0 < ϵ < r_{inj}(M_{0})/6 there is a sequence of points {x_{j}} ⊂ M_{0}, and a partition of unity ϕ_{j} ∈ C_{c}^{∞}(M_{0}) with the following properties:

(i) supp(ϕ_{j}) ⊂ B(x_{j}, 2ϵ); (ii) ∥∇^{k}ϕ_{j}∥_{L^{∞}(M_{0})} ≤ C_{k,ϵ}, with C_{k,ϵ} independent of j; and (iii) the sets B(x_{j}, ϵ/2) are disjoint, the sets B(x_{j}, ϵ) form a covering of M_{0}, and the

sets B(x_{j}, 4ϵ) form a covering of M_{0} of finite multiplicity, i.e.,

sup_{y∈M_{0}} #{x_{j} | y ∈ B(x_{j}, 4ϵ)} < ∞.

Fix ϵ ∈ (0, r_{inj}(M_{0})/6). Let ψ_{j} : B(x_{j}, 4ϵ) → B_{ℝ^{n}}(0, 4ϵ) normal coordinates around x_{j}, i.e., a composition of the exponential maps exp_{x_{j}} : T_{x_{j}}M_{0} → M_{0} and by some isometries T_{x_{j}}M_{0} ≃ ℝ^{n}. The uniform bounds on the Riemann tensor R and its derivatives ∇^{k}R imply uniform bounds on

∇^{k}d exp_{x_{j}} : B_{T_{x_{j}} M_{0}}(0, 4ϵ) → ℝ^{n(k+1)} ⊗ T M,

which simply means that all derivatives of ψ_{j} are uniformly bounded.

Proposition 2.6. Let ϕ_{i} and ψ_{i} be as in the two paragraphs above. Let U_{j} = ψ_{j}(Ω_{0} ∩ B(x_{j}, 2ϵ)) ⊂ ℝ^{n}. We define

ν_{k,∞}(u) := sup_{j} ∥(ϕ_{j}u) ∘ ψ_{j}^{−1}∥_{W^{k,∞}(U_{j})}

and, for 1 ≤ p < ∞,

ν_{k,p}(u)^{p} := ∑_{j} ∥(ϕ_{j}u) ∘ ψ_{j}^{−1}∥_{W^{k,p}(U_{j})}^{p}.

Then u ∈ W^{k,p}(Ω_{0}) if, and only if, ν_{k,p}(u) < ∞. Moreover, ν_{k,p}(u) defines an equivalent norm on W^{k,p}(Ω_{0}).

Proof. We shall assume p < ∞, for simplicity of notation. The case p = ∞ is completely similar. Consider then μ(u)^{p} = ∑_{j} ∥ϕ_{j}u∥_{W^{k,p}(Ω_{0})}^{p}. Then there exists C_{k,ε} > 0 such that

(12)  C_{k,ε}^{−1}∥u∥_{W^{k,p}(Ω_{0})} ≤ μ(u) ≤ C_{k,ε}∥u∥_{W^{k,p}(Ω_{0})},

for all u ∈ W^{k,p}(Ω_{0}), by Lemma 2.5 (i.e., the norms are equivalent). The fact that all derivatives of exp_{x_{j}} are bounded uniformly in j further shows that μ and ν_{k,p} are also equivalent. □

The proposition gives rise to a third, equivalent definition of Sobolev spaces. This definition was inspired from [66, 68], and can be used to define the spaces W^{s,p}(Ω_{0}), for any s ∈ ℝ, 1 < p < ∞, and Ω_{0} = M_{0}. The cases p = 1 or p = ∞ are more delicate and we shall not discuss them here.

Recall that the spaces W^{s,p}(ℝ^{n}), s ∈ ℝ, 1 < p < ∞ are defined using the powers of 1 + Δ, see [67, Chapter V] or [75, Section 13.6].

Definition 2.7 (Partition of unity definition of Sobolev spaces). Let s ∈ ℝ, and 1 < p < ∞. Then we define

(13)  ∥u∥_{W^{s,p}(M_{0})}^{p} := ∑_{j} ∥(ϕ_{j}u) ∘ ψ_{j}^{−1}∥_{W^{s,p}(ℝ^{n})}^{p}, 1 < p < ∞.

By Proposition 2.6 this norm is equivalent to our previous norm on W^{k,p}(M_{0}) when k is a nonnegative integer.

Proposition 2.8. The space C_{c}^{∞}(M_{0}) is dense in W^{k,p}(M_{0}), for 1 < p < ∞ and s ∈ ℝ, or 1 ≤ p < ∞ and s = k ∈ ℤ_{+}.

Proof. For k ∈ ℤ_{+}, the result is true for any manifold with bounded geometry, see [8, Theorem 2] or [28, Theorem 2.8], or [29]. For Ω_{0} = M_{0}, s ∈ ℝ, and 1 < p < ∞, the definition of the norm on W^{s,p}(M_{0}) allows us to reduce right away the proof to the case of ℝ^{n}, by ignoring enough terms in the sum defining the norm (13). (We also use a cut-off function 0 ≤ χ ≤ 1, χ ∈ C_{c}^{∞}(B_{ℝ^{n}}(0, 4ϵ)), χ = 1 on B_{ℝ^{n}}(0, 4ϵ).) □

We now give a characterization of the spaces W^{s,p}(M_{0}) using interpolation. Let W̃^{−k,p}(M_{0}) be the set of distributions on M_{0} that extend by continuity to linear functionals on W^{k,q}(M_{0}), p^{−1}+q^{−1} = 1, using Proposition 2.8. That is, W̃^{−k,p}(M_{0}) be the set of distributions on M_{0} that define continuous linear functionals on W^{k,q}(M_{0}), p^{−1} + q^{−1} = 1. We let

W̃^{θk,k,p}(M_{0}) := [W̃^{0,p}(M_{0}), W^{k,p}(M_{0})]_{θ} , 0 ≤ θ ≤ 1 ,

be the complex interpolation spaces. Similarly, we define

W̃^{−θk,k,p}(M_{0}) = [W̃^{0,p}(M_{0}), W^{−k,p}(M_{0})]_{θ}.

(See [13] or [73, Chapter 4] for the definition of the complex interpolation spaces.)

The following proposition is an analogue of Proposition 2.6. Its main role is to give an intrinsic definition of the spaces W^{s,p}(M_{0}), a definition that is independent of choices.

Proposition 2.9. Let 1 < p < ∞ and k > |s|. Then we have a topological equality W̃^{s,k,p}(M_{0}) = W^{s,p}(M_{0}). In particular, the spaces W^{s,p}(M_{0}), s ∈ ℝ, do not depend on the choice of the covering B(x_{j}, ϵ) and of the subordinated partition of unity and we have

[W^{s,p}(M_{0}), W^{0,p}(M_{0})]_{θ} = W^{θs,p}(M_{0}) , 0 ≤ θ ≤ 1 .

Moreover, the pairing between functions and distributions defines an isomorpism W^{s,p}(M_{0})^{∗} ≃ W^{−s,q}(M_{0}), where 1/p + 1/q = 1.

Proof. This proposition is known if M_{0} = ℝ^{n} with the usual metric [75][Equation (6.5), page 23]. In particular, W̃^{s,p}(ℝ^{n}) = W^{s,p}(ℝ^{n}). As in the proof of Proposition 2.6 one shows that the quantity

(14)  ν_{s,p}(u)^{p} := ∑_{j} ∥(ϕ_{j}u) ∘ ψ_{j}^{−1}∥_{W̃^{s,p}(ℝ^{n})}^{p},

is equivalent to the norm on W̃^{s,p}(M_{0}). This implies W̃^{s,p}(M_{0}) = W^{s,p}(M_{0}). Choose k large. Then we have

[W^{s,p}(M_{0}), W^{0,p}(M_{0})]_{θ} = [W^{s,k,p}(M_{0}), W^{0,k,p}(M_{0})]_{θ}

= W^{θs,k,p}(M_{0}) = W^{θs,p}(M_{0}).

The last part follows from the compatibility of interpolation with taking duals. This completes the proof. □

The above proposition provides us with several corollaries. First, from the interpolation properties of the spaces W^{s,p}(M_{0}), we obtain the following corollary.

Corollary 2.10. Let ϕ ∈ W^{k,∞}(M_{0}) and p ∈ (1, ∞). Then multiplication by ϕ defines a bounded operator on W^{s,p}(M_{0}) of norm at most C_{k}∥ϕ∥_{W^{k,∞}(M_{0})} where k ≥ |s|, k ∈ ℤ_{+}. Similarly, any differential operator P ∈ Diff_{V}^{m}(M) defines continuous maps P : W^{s,p}(M_{0}) → W^{s−m,p}(M_{0}).

Proof. For s ∈ ℤ_{+}, this follows from the definition of the norm on W^{k,∞}(M_{0}) and from the definition of Diff_{V}^{m}(M) as the linear span of differential operators of the form fX_{1} … X_{k}, (f ∈ C^{∞}(M) ⊂ W^{k,∞}, X_{j} ∈ V, and 0 ≤ k ≤ m), and from the definition of the spaces W^{k,p}(Ω_{0}).

For s ≤ m, the follows by duality. For the other values of s, the result follows by interpolation. □

Next, recall that an isomorphism ϕ : M → M of the Lie manifolds (M_{0}, M, V) and (M′_{0}, M′, V′) is a diffeomorphism such that ϕ_{∗}(V) = V′. We then have the following invariance property of the Sobolev spaces that we have introduced.

Corollary 2.11. Let ϕ : M → M′ be an isomorphism of Lie manifolds, Ω_{0} ⊂ M_{0} be an open subset and Ω′ = ϕ(Ω). Let p ∈ [1, ∞] if s ∈ ℤ_{+}, and p ∈ (1, ∞) if s ∉ ℤ_{+}. Then f → f ∘ ϕ extends to an isomorphism ϕ^{∗} : W^{s,p}(Ω′) → W^{s,p}(Ω) of Banach spaces.

Proof. For k ∈ ℤ_{+}, this follows right away from definitions and Proposition 2.3. For −k ∈ ℤ_{+}, this follows by duality, Proposition (2.9). For the other values of s, the result follows from the same proposition, by interpolation. □

Recall now that M_{0} is complete [5]. Hence the Laplace operator Δ = ∇^{∗}∇ is essentially self-adjoint on C_{c}^{∞}(M_{0}) by [23, 59, 70]. We shall define then (1 + Δ)^{s/2} using the spectral theorem.

Proposition 2.12. The space H^{s}(M_{0}) := W^{s,2}(M_{0}), s ≥ 0, identifies with the domain of (1 + Δ)^{s/2}, if we endow the latter with the graph topology.

Proof. For s ∈ ℤ_{+}, the result is true for any manifold of bounded geometry, by [8, Proposition 3]. For s ∈ ℝ, the result follows from interpolation, because the interpolation spaces are compatible with powers of operators, see the chapter on Sobolev spaces in Taylor’s book [73]. □

The well known Gagliardo–Nirenberg–Sobolev inequality [8, 22, 24, 28] holds also in our setting.

Proposition 2.13. Denote by n the dimension of M_{0}. Assume that 1/p = 1/q − m/n, 1 < q ≤ p < ∞, where m ≥ 0. Then W^{s,q}(M_{0}) is continuously embedded in W^{s−m,p}(M_{0}).

Proof. If s and m are integers, s ≥ m ≥ 0, the statement of the proposition is true for manifolds with bounded geometry, [8, Theorem 7] or [28, Corollary 3.1.9]. By duality (see Proposition 2.9), we obtain the same result when s ≤ 0, s ∈ ℤ. Then, for integer s, m, 0 < s < m we obtain the corresponding embedding by composition W^{s,q}(M_{0}) → W^{0,r}(M_{0}) → W^{s−m,p}(M_{0}), with 1/r = 1/q − s/n. This proves the result for integral values of s. For non-integral values of s, the result follows by interpolation using again Proposition 2.9. □

The Rellich-Kondrachov’s theorem on the compactness of the embeddings of Proposition 2.13 for 1/p > 1/q−m/n is true if M_{0} is compact [8, Theorem 9]. This happens precisely when M = M_{0}, which is a trivial case of a manifold with a Lie structure at infinity. On the other hand, it is easily seen (and well known) that these compactness cannot be true for M_{0} non-compact. We will nevertheless restore this by using Sobolev spaces with weights in the next section, see Theorem 3.6.

3. Manifolds with boundary

We continue to assume that (M_{0}, M, V) is Lie manifold. Let N ⊂ M be a submanifold with corners of codimension one of M. Recall that this implies that N is transverse to all faces of M. Also, recall that N is a regular submanifold of (M, V) if we can choose a tubular neighborhood V of N_{0} := N ∖ ∂N = N ∩ M_{0} in M_{0} and a compatible metric g on M_{0} that restricts to a product-type metric on V ≃ (∂N_{0}) × (−ε_{0}, ε_{0}) (see Proposition 5.9). In Section 5, we shall show that every tame manifold is regular; in turn, this will give an easy, geometric criterion to decide when a codimension one submanifold of M is regular.

Let Ω ⊂ M be an open subset. Recall that Ω is a Lie domain in M if, and only if, ∂Ω = ∂\overline{Ω} and ∂Ω is a regular submanifold of M. Let Ω_{0} = Ω ∩ M_{0}. Then ∂Ω_{0} := (∂Ω) ∩ M_{0} is a smooth submanifold of codimension one of M_{0}. We have the following analogue of the classical extension theorem.

Theorem 3.1. There exists a linear operator E mapping measurable functions on Ω_{0} to measurable functions on M_{0} with the properties:

(i) E maps W^{k,p}(Ω_{0}) continuously into W^{k,p}(M_{0}) for every p ∈ [1, ∞] and every integer k ≥ 0, and

(ii) Eu|_{Ω_{0}} = u.

Proof. Since ∂Ω_{0} is a regular submanifold we can fix a compatible metric g on M_{0} and a tubular neighborhood V_{0} of ∂Ω_{0} such that V_{0} ≃ (∂Ω_{0}) × (−ε_{0}, ε_{0}), ε_{0} > 0. Let ε = min(ε_{0}, r_{inj}(M_{0}))/20, where r_{inj}(M_{0}) > 0 is the injectivity radius of M_{0}. By Zorn’s lemma and the fact that M_{0} has bounded geometry we can choose a maximal, countable set of disjoint balls B(x_{i}, ε), i ∈ I. Since this family of balls is maximal we have M_{0} = ∪_{i}B(x_{i}, 2ε). For each i we fix a smooth function η_{i} supported in B(x_{i}, 3ε) and equal to 1 in B(x_{i}, 2ε). This can be done easily in local coordinates around the point x_{i}; since the metric g is induced by a metric g on A we may also assume that all derivatives of order up to k of η_{i} are bounded by a constant C_{k,ε} independent of i. By replacing η_{i} with η_{i}/\sqrt{∑_{j} η_{j}^{2}}, we can further assume that ∑_{i} η_{i}^{2} = 1.

Following [67, Ch. 6] we also define two smooth cutoff functions adapted to the set Ω_{0}. We start with a function ψ : ℝ → [0, 1] which is equal to 1 on [−3, 3] and which has support in [−6, 6]

Let φ = (φ_{1}, φ_{2}) denote the isomorphism between V_{0} and ∂Ω_{0} × (−ε_{0}, ε_{0}), where φ_{1} : V_{0} → ∂Ω_{0} and φ_{2} : V_{0} → (−ε_{0}, ε_{0}). We define

Λ_{+}(x) := \{ \begin{matrix}0 & if x ∈ M_{0} \ V_{0} \\ ψ(φ_{2}(x)/ε) & if x ∈ V_{0}, \end{matrix}

and Λ_{−}(x) := 1 − Λ_{+}(x). Clearly Λ_{+} and Λ_{−} are smooth functions on M_{0} and Λ_{+}(x) + Λ_{−}(x) = 1. Obviously, Λ_{+} is supported in a neighborhood of ∂Ω_{0} and Λ_{−} is supported in the complement of a neighborhood of ∂Ω_{0}.

Let ∂Ω_{0} = A_{1} ∪ A_{2} ∪ … denote the decomposition of ∂Ω_{0} into connected components. Let V_{0} = B_{1} ∪ B_{2} ∪ … denote the corresponding decomposition of V_{0} into connected components, namely, B_{j} = φ^{−1}(A_{j} ×(−ε_{0}, ε_{0})). Since ∂Ω_{0} = ∂\overline{Ω}_{0}, we have φ(Ω_{0} ∩ B_{j}) = A_{j} × (−ε_{0}, 0) or φ(Ω_{0} ∩ B_{j}) = A_{j} × (0, ε_{0}). Thus, if necessary, we may change the sign of φ on some of the connected components of V_{0} in such a way that

φ(Ω_{0} ∩ V_{0}) = ∂Ω_{0} × (0, ε_{0}).

Let ψ_{0} denote a fixed smooth function, ψ_{0} : ℝ → [0, 1], ψ_{0}(t) = 1 if t ≥ −ε and ψ_{0}(t) = 0 if t ≤ −2ε, and let

Λ_{0}(x) = \{ \begin{matrix}1 & if x ∈ Ω_{0} \ V_{0} \\ 0 & if x ∈ M_{0} \ (Ω_{0} ∪ V_{0}) \\ ψ_{0}(φ_{2}(x)) & if x ∈ V_{0}. \end{matrix}

We look now at the points x_{i} defined in the first paragraph of the proof. Let J_{1} = {i ∈ I : d(x_{i}, ∂Ω_{0}) ≤ 10ε} and J_{2} = {i ∈ I : d(x_{i}∂Ω_{0}) > 10ε}. For every point x_{i}, i ∈ J_{1}, there is a point y_{i} ∈ ∂Ω_{0} with the property that B(x_{i}, 4ε) ⊂ B(y_{i}, 15ε). Let B_{∂Ω_{0}}(y_{i}, 15ε) denote the ball in ∂Ω_{0} of center y_{i} and radius 15ε (with respect to the induced metric on ∂Ω_{0}). Let h_{i} : B_{∂Ω_{0}}(y_{i}, 15ε) → B_{ℝ^{n−1}}(0, 15ε) denote the normal system of coordinates around the point y_{i}. Finally let g_{i} : B_{ℝ^{n−1}}(0, 15ε) × (−15ε, 15ε) → V_{0} denote the map g_{i}(v, t) = φ^{−1}(h_{i}^{−1}(v), t).

Let E_{ℝ^{n}} denote the extension operator that maps W^{k,p}(ℝ_{+}^{n}) to W^{k,p}(ℝ^{n}) continuously, where ℝ_{+}^{n} denotes the half-space {x : x_{n} > 0}. Clearly, E_{ℝ^{n}}u|_{ℝ_{+}^{n}} = u. The existence of this extension operator is a classical fact, for instance, see [67, Chapter 6]. For any u ∈ W^{k,p}(Ω_{0}) and i ∈ J_{1} the function (η_{i}u) ∘ g_{i} is well defined on ℝ_{+}^{n} simply by setting it equal to 0 outside the set B_{ℝ^{n−1}}(0, 15ε) × (0, 15ε). Clearly, (η_{i}u) ∘ g_{i} ∈ W^{k,p}(ℝ_{+}^{n}). We define the extension Eu by the formula

(15)  Eu(x) = Λ_{0}(x)Λ_{−}(x)u(x) + Λ_{0}(x)Λ_{+}(x) ∑_{i∈J_{1}} η_{i}(x)E_{ℝ^{n}}[(η_{i}u) ∘ g_{i}](g_{i}^{−1}x) .

Notice that

(16)  ∑_{i∈J_{1}} η_{i}^{2}(x) = 1 in supp Λ_{+}.

Indeed, since M_{0} = ∪_{i∈I}B(x_{i}, 2ε), we have ∑_{i∈I} η_{i}^{2}(x) ≥ 1 for any x ∈ M_{0}. Also, η_{i}(x) ≡ 0 in supp Λ_{+} if i ∈ J_{2}, thus (16) follows. This shows that Eu in (15) is well-defined. Clearly, by the formula, Eu|_{Ω_{0}} = u. It remains to verify that

∥Eu∥_{W^{k,p}(M_{0})} ≤ C_{k}∥u∥_{W^{k,p}(Ω_{0})}.

This follows as in [67] using (16), the fact that the extension E_{ℝ^{n}} satisfies the same bound, and the definition of the Sobolev spaces using partitions of unity (Proposition 2.6). □

Theorem 3.2. The space C_{c}^{∞}(\overline{Ω}_{0}), where the closure is defined in M_{0}, is dense in W^{k,p}(Ω_{0}), for 1 ≤ p < ∞.

Proof. For any u ∈ W^{k,p}(Ω_{0}) let Eu denote its extension from Theorem 3.1, Eu ∈ W^{k,p}(M_{0}). By Proposition 2.8, there is a sequence of functions f_{j} ∈ C_{c}^{∞}(M_{0}) with the property that

lim_{j→∞} f_{j} = Eu in W^{k,p}(M_{0}).

Thus lim_{j→∞} f_{j}|_{Ω_{0}} = u in W^{k,p}(Ω_{0}), as desired. □

Theorem 3.3. The restriction map C_{c}^{∞}(Ω_{0}) → C_{c}^{∞}(∂Ω_{0}) extends to a continuous map T : W^{k,p}(Ω_{0}) → W^{k−1,p}(∂Ω_{0}), for 1 ≤ p ≤ ∞.

Proof. To simplify the notation assume 1 ≤ p < ∞. We shall assume that the compatible metric on M_{0} restricts to a product type metric on V_{0}, our distinguished tubular neighborhood of ∂Ω_{0}.

We use the definitions of the Sobolev spaces using partitions of unity, Proposition 2.6 and Lemma 2.5 with ε = min(ϵ_{0}, r_{inj}(M_{0}))/10. Let B(x_{j}, 2ε) denote the balls in the cover of X in Lemma 2.5 and 1 = ∑_{j} ϕ_{j} the corresponding partition of unity. Then ϕ̃_{j} = ϕ_{j}|_{∂Ω_{0}} form a partition of unity on ∂Ω_{0}. Clearly,

r_{inj}(∂Ω_{0}) ≥ r_{inj}(M_{0}) > 0.

Start with a function u ∈ W^{k,p}(Ω_{0}) and let u_{j} = (uϕ_{j}) ∘ ψ_{j}^{−1}, u_{j} ∈ W^{k,p}(ψ_{j}(Ω_{0} ∩ B(x_{j}, 4ε))). In addition u_{j} ≡ 0 outside the set ψ_{j}(Ω_{0} ∩ B(x_{j}, 2ε)). If B(x_{j}, 4ε) ∩ ∂Ω_{0} = ∅ let T̃(u_{j}) = 0. Otherwise notice that B(x_{j}, 4ε) is included in V_{0}, the tubular neighborhood of ∂Ω_{0}, thus the set ψ_{j}(∂Ω_{0} ∩ B(x_{j}, 4ε)) is the intersection of a hyperplane and the ball B_{ℝ^{n}}(0, 4ε). We can then let T̃(u_{j}) denote the Euclidean restriction of u_{j} to ψ_{j}(∂Ω_{0} ∩ B(x_{j}, 4ε)) (see [22, Section 5.5]). Clearly T̃(u_{j}) is supported in ψ_{j}(∂Ω_{0} ∩ B(x_{j}, 2ε)) and

∥T̃(u_{j}) ∘ ψ̃_{j}∥_{W^{k−1,p}(∂Ω_{0})} ≤ C∥u_{j}∥_{W^{k,p}(ψ_{j}(Ω_{0}∩B(x_{j},4ε)))},

where ψ̃_{j} = ψ_{j}|_{Ω_{0}} and the constant C is independent of j (recall that ψ_{j}(∂Ω_{0} ∩ B(x_{j}, 4ε)) is the intersection of a hyperplane and the ball B_{ℝ^{n}}(0, 4ε)). Let

T u = ∑_{j} T̃(u_{j}) ∘ ψ̃_{j}.

Since the sum is uniformly locally finite we have

\begin{aligned}∥T u∥_{W^{k−1,p}(∂Ω_{0})}^{p} & ≤ C ∑_{j} ∥T̃(u_{j}) ∘ ψ̃_{j}∥_{W^{k−1,p}(∂Ω_{0})}^{p} \\ & ≤ C ∑_{j} ∥u_{j}∥_{W^{k,p}(ψ_{j}(Ω_{0}∩B(x_{j},4ε)))}^{p} ≤ C∥u∥_{W^{k,p(Ω_{0})}}^{p}, \end{aligned}

with constants C independent of u. The fact T u|_{C_{c}^{∞}(Ω_{0})} is indeed the restriction operator follows immediately from the definition. □

We shall see that if p = 2, we get a surjective map W^{s,2}(Ω_{0}) → W^{s−1/2,2}(∂Ω_{0}) (Theorem 3.7).

Theorem 3.4. The closure of C_{c}^{∞}(Ω_{0}) in W^{k,p}(Ω_{0}) is the intersection of the kernels of T ∘ ∂_{ν}^{j}, 0 ≤ j ≤ k − 1.

Proof. The proof is reduced to the Euclidean case [1] following the same pattern of reasoning as in the previous theorem. □

The Gagliardo–Nirenberg–Sobolev theorem holds also for manifolds with boundary.

Theorem 3.5. Denote by n the dimension of M and let Ω ⊂ M be a Lie domain in M. Assume that 1/p = 1/q − m/n > 0, 1 ≤ q < ∞, where m ≤ k is an integer. Then W^{k,q}(Ω_{0}) is continuously embedded in W^{k−m,p}(Ω_{0}).

Proof. This can be proved using Proposition 2.13 and Theorem 3.1. Indeed, denote by

j : W^{k,q}(M_{0}) → W^{k−m,p}(M_{0})

the continuous inclusion of Proposition 2.13. Also, denote by r the restriction maps W^{k,p}(M_{0}) → W^{k,p}(Ω_{0}). Then the maps

W^{k,q}(Ω_{0}) ⟶^{E} W^{k,q}(M_{0}) ⟶^{j} W^{k−m,p}(M_{0}) ⟶^{r} W^{k−m,p}(Ω_{0})

are well defined and continuous. Their composition is the inclusion of W^{k,q}(Ω_{0}) into W^{k−m,p}(Ω_{0}). This completes the proof. □

For the proof of a variant of Rellich–Kondrachov’s compactness theorem, we shall need Sobolev spaces with weights. Let a_{H} > 0 be a parameter associated to each hyperface (i.e., face of codimension one) of M. Fix for any hyperface H ⊂ M a defining function ρ_{H}, that is a function ρ_{H} ≥ 0 such that H = {ρ_{H} = 0} and dρ_{H} ≠ 0 on H. Let

(17)  ρ = ∏ ρ_{H}^{a_{H}},

the product being taken over all hyperfaces of M. The function ρ will be called an admissible weight. (The function ρ considered in the beginning of this paper is, in fact, an admissible weight, so there is no conflict in the notation.) We define then

(18)  ρ^{s}W^{k,p}(Ω_{0}) := {ρ^{s}u, u ∈ W^{k,p}(Ω_{0})},

with the norm ∥ρ^{s}u∥_{ρ^{s}W^{k,p}(Ω_{0})} := ∥u∥_{W^{k,p}(Ω_{0})}.

Theorem 3.6. Denote by n the dimension of M and let Ω ⊂ M be a Lie domain, Ω_{0} = Ω ∩ M_{0}. Let s < s′ be real parameters. Assume that 1/p > 1/q − m/n > 0, 1 ≤ q < ∞, where m ≤ k is an integer. Then ρ^{s}W^{k,q}(Ω_{0}) is compactly embedded in ρ^{s′}W^{k−m,p}(Ω_{0}).

Proof. The same argument as that in the proof of Theorem 3.5 allows us to assume that Ω_{0} = M_{0}. The norms are chosen such that W^{k,p}(Ω_{0}) ∋ u → ρ^{s}u ∈ ρ^{s}W^{k,p}(Ω_{0}) is an isometry. Thus, it is enough to prove that ρ^{s} : W^{k,q}(Ω_{0}) → W^{k−m,p}(Ω_{0}), s > 0, is a compact operator.

For any defining function ρ_{H} and any X ∈ V, we have that X(ρ_{H}) vanishes on H, since X is tangent to H. Recall now the function ρ defined in Equation (17). We obtain that X(ρ^{s}) = ρ^{s}f_{X}, for some f_{X} ∈ C^{∞}(M). Then, by induction, X_{1}X_{2} … X_{k}(ρ^{s}) = ρ^{s}g, for some g ∈ C^{∞}(M).

Let χ ∈ C^{∞}([0, ∞) be equal to 0 on [0, 1/2], equal to 1 on [1, ∞), and non-negative everywhere. Define ϕ_{ϵ} = χ(ϵ^{−1}ρ^{s}). Then

∥X_{1}X_{2} … X_{k}(ρ^{s}ϕ_{ϵ} − ρ^{s})∥_{L^{∞}} → 0 , as ϵ → 0,

for any X_{1}, X_{2}, …, X_{k} ∈ V. Corollary 2.10 then shows that ρ^{s}ϕ_{ϵ} → ρ^{s} in the norm of bounded operators on W^{s,p}(Ω_{0}). But multiplication by ρ^{s}ϕ_{ϵ} is a compact operator, by the Rellich-Kondrachov’s theorem for compact manifolds with boundary [8, Theorem 9]. This completes the proof. □

We end with the following generalization of the classical restriction theorem for the Hilbertian Sobolev spaces H^{s}(M_{0}) := W^{s,2}(M_{0}). The tame submanifolds are defined in the next section.

Theorem 3.7. Let N_{0} ⊂ M_{0} be a tame submanifold of codimension k of the Lie manifold (M_{0}, M, V). Restriction of smooth functions extends to a bounded, surjective map

H^{s}(M_{0}) → H^{s−k/2}(N_{0}),

for any s > k/2. In particular, H^{s}(Ω_{0}) → H^{s−1/2}(∂Ω_{0}) is continuous and surjective.

Proof. Let B → N be the vector bundle defining the Lie structure at infinity (N, B) on N_{0} and A → M be the vector bundle defining the Lie structure at infinity (M, A) on M_{0}. (See Section 5 for further explanation of this notation.) The existence of tubular neighborhoods, Theorem 5.8, and a partition of unity argument, allows us to assume that M = N × S^{1} and that A = B × T S^{1} (external product). Since the Sobolev spaces H^{s}(M_{0}) and H^{s−1/2}(N_{0}) do not depend on the metric on A and B, we can assume that the circle S^{1} is given the invariant metric making it of length 2π and that M_{0} is given the product metric. The rest of the proof now is independent of the way we have arrived at the product metric on M_{0}.

Then Δ_{M_{0}} = Δ_{N_{0}} + Δ_{S^{1}} and Δ_{S^{1}} = −∂^{2}/∂θ^{2} has spectrum {4π^{2}n^{2}}, n ∈ ℤ. We can decompose L^{2}(N_{0} × S^{1}) according to the eigenvalues n ∈ ℤ of −\frac{1}{2πı}∂_{θ}:

L^{2}(N_{0} × S^{1}) ≃ ⊕_{n∈ℤ}L^{2}(N_{0} × S^{1})_{n} ≃ ⊕_{n∈ℤ}L^{2}(N_{0}),

where the isomorphism L^{2}(N_{0} × S^{1})_{n} ≃ ⊕_{n∈ℤ}L^{2}(N_{0}) is obtained by restricting to N_{0} = N_{0} × {1}, 1 ∈ S^{1}.

To prove our theorem, it is enough to check that, if ξ_{n} ∈ L^{2}(N_{0}) is a sequence such that

(19)  ∑_{n} ∥(1 + n^{2} + Δ_{N_{0}})^{s/2}ξ_{n}∥^{2} < ∞,

then ∑(1 + Δ_{N_{0}})^{s/2−1/4}ξ_{n} is convergent.

Let C = 1 + ∫_{ℝ}(1 + t^{2})^{−s}dt and assume that each ξ_{n} is in the spectral subspace of Δ_{N_{0}} corresponding to [m, m + 1) ⊂ ℝ_{+}. Then

(1 + m^{2})^{s−1/2}(∑_{n} ∥ξ_{n}∥)^{2} ≤ C ∑_{n} ∥(1 + n^{2} + m^{2})^{s/2}ξ_{n}∥^{2}.

Since the constant C is independent of m and the spectral spaces of Δ_{N_{0}} corresponding to [m, m + 1) ⊂ ℝ give an orthogonal direct sum decomposition of L^{2}(N_{0}), this checks Equation (19) and completes the proof. □

We conclude this section with an application to the regularity of boundary value problems. Applications of this result to the regularity of boundary value problems on polyhedra as well as more details will be included in [11].

Let us introduce some notation first that will be also useful in the following, especially in Section 6.

Let exp : T M_{0} ⟶ M_{0} × M_{0} be given by exp(v) := (x, exp_{x}(v)),, v ∈ T_{x}M_{0}. If E is a real vector bundle with a metric, we shall denote by (E)_{r} the set of all vectors v of E with |v| < r. Let (M_{0}^{2})_{r} := {(x, y), x, y ∈ M_{0}, d(x, y) < r}. Then the exponential map defines a diffeomorphism exp : (T M_{0})_{r} → (M_{0}^{2})_{r}. We shall also need the function ρ defined in Equation (17) and the weighted Sobolev spaces ρ^{s}W^{k,p}(Ω_{0}) := {ρ^{s}u, u ∈ W^{k,p}(Ω_{0})} introduced in Equation 18.

Theorem 3.8. Let Ω_{0} ⊂ M_{0} be a Lie domain in a Lie manifold (M_{0}, M, V). Let P ∈ Diff_{V}^{2}(M) be an order 2 elliptic operator on M_{0} generated by V. Let u ∈ ρ^{s}W^{1,p}(Ω_{0}) be such that P u ∈ ρ^{s}W^{t,p}(Ω_{0}), s ∈ ℝ, t ∈ ℤ, 1 < p < ∞, and u|_{∂Ω_{0}} = 0. Then u ∈ ρ^{s}W^{t+2,p}(Ω_{0}).

Proof. Note that locally, this is a well known statement. In particular, ϕu ∈ W^{t+2,p}(Ω_{0}), for any ϕ ∈ C_{c}^{∞}(M_{0}). The result will follow then if we prove that

(20)  ∥u∥_{ρ^{s}W^{t+2,p}(M_{0})} ≤ C(∥P u∥_{ρ^{s}W^{t,p}(M_{0})} + ∥u∥_{ρ^{s}W^{1,p}(M_{0})})

for any u ∈ W_{loc}^{t+2,p}(Ω_{0}).

Let r < r_{inj}(M_{0}) and let exp : (T M_{0})_{r} → (M_{0}^{2})_{r} be the exponential map. The statement is trivially true for t ≤ −1, so we will assume t ≥ 0 in what follows. Also, we will assume first that s = 0. The general case will be reduced to this one at the end. Assume first that Ω_{0} = M_{0}.

Let P_{x} be the differential operators on defined on B_{T_{x}M_{0}}(0, r) obtained from P by the local diffeomorphism exp : B_{T_{x}M_{0}}(0, r) → M_{0}. We claim that there exists a constant C > 0, independent of x ∈ M_{0} such that

(21)  ∥u∥_{W^{t+2,p}(T_{x}M_{0})}^{p} ≤ C(∥P_{x}u∥_{W^{t,p}(T_{x}M_{0})}^{p} + ∥u∥_{W^{1,p}(T_{x}M_{0})}^{p}),

for any function u ∈ C_{c}^{∞}(B_{T_{x}M_{0}}(0, r)). This is seen as follows. We can find a constant C_{x} > 0 with this property for any x ∈ M_{0} by the ellipticity of P_{x} using [73]. Choose C_{x} to be the least such constant. Let π : A → M be the extension of the tangent bundle of M_{0}, see Remark 1.4 and let A_{x} = π^{−1}(x). The family P_{x}, x ∈ M_{0}, extends to a family P_{x}, x ∈ M, that is smooth in x. The smoothness of the family P_{x} in x ∈ M shows that C_{x} is lower semi-continuous. Since M is compact, C_{x} will attain its minimum, which therefore must be positive. Let C be that minimum value.

Let now ϕ_{j} be the partition of unity and ψ_{j} be the diffeomorphisms appearing in Equation (14), for some 0 < ϵ < r/6. In particular, the partition of unity ϕ_{j} satisfies the conditions of Lemma 2.5, which implies that supp(ϕ_{j}) ⊂ B(x_{j}, 2ϵ) and the sets B(x_{j}, 4ϵ) form a covering of M_{0} of finite multiplicity. Let η_{j} = 1 on the support of ϕ_{j}, supp(η_{j}) ⊂ B(x_{j}, 4ϵ). We then have

ν_{t+2,p}(u)^{p} := ∑_{j} ∥(ϕ_{j}u) ∘ ψ_{j}^{−1}∥_{W^{t+2,p}(ℝ^{n})}^{p}

≤ C ∑_{j} (∥P_{x}(ϕ_{j}u)∥_{W^{t,p}(T_{x}M_{0})}^{p} + ∥ϕ_{j}u∥_{W^{1,p}(T_{x}M_{0})}^{p})

≤ C ∑_{j} (∥ϕ_{j}P_{x}u∥_{W^{t,p}(T_{x}M_{0})}^{p} + ∥[P_{x}, ϕ_{j}]u∥_{W^{t,p}(T_{x}M_{0})}^{p} + ∥ϕ_{j}u∥_{W^{1,p}(T_{x}M_{0})}^{p})

≤ C ∑_{j} (∥ϕ_{j}P_{x}u∥_{W^{t,p}(T_{x}M_{0})}^{p} + ∥η_{j}u∥_{W^{t+1,p}(T_{x}M_{0})}^{p} + ∥ϕ_{j}u∥_{W^{1,p}(T_{x}M_{0})}^{p})

≤ C(ν_{t,p}(P u)^{p} + ν_{t+1}(u)^{p}).

The equivalence of the norm ν_{s,p} with the standard norm on W^{s,p}(M_{0}) (Propositions 2.6 and 2.9) shows that ∥u∥_{W^{t+2,p}(M_{0})} ≤ C(∥P u∥_{W^{t,p}(M_{0})} + ∥u∥_{W^{t+1,p}(M_{0})}), for any t. This is known to imply

(22)  ∥u∥_{W^{t+2,p}(M_{0})} ≤ C(∥P u∥_{W^{t,p}(M_{0})} + ∥u∥_{W^{1,p}(M_{0})})

by a boot-strap procedure, for any t ≥ −1. This proves our statement if s = 0 and Ω_{0} = M_{0}.

The case Ω_{0} arbitrary follows in exactly the same way, but using a product type metric in a neighborhood of ∂Ω_{0} and the analogue of Equation (21) for a half-space, which shows that Equation (20) continues to hold for M_{0} replaced with Ω_{0}.

The case s arbitrary is obtained by applying Equation (22) to the elliptic operator ρ^{−s}P ρ^{s} ∈ Diff_{V}^{2}(M) and to the function ρ^{−s}u ∈ W^{k,p}(Ω_{0}), which then gives Equation (20) righaway. □

For p = 2, by combining the above theorem (including its proof, especially Equation (20)) with Theorem 3.7, we obtain the following corollary.

Corollary 3.9. We keep the assumptions of Theorem 3.8. Let u ∈ ρ^{s}H^{1}(Ω_{0}) be such that P u ∈ ρ^{s}H^{t}(Ω_{0}) and u|_{∂Ω_{0}} ∈ ρ^{s}H^{t+3/2}(Ω_{0}), s ∈ ℝ, t ∈ ℤ. Then u ∈ ρ^{s}H^{t+2}(Ω_{0}) and

(23)  ∥u∥_{ρ^{s}H^{t+2}(Ω_{0})} ≤ C(∥P u∥_{ρ^{s}H^{t}(Ω_{0})} + ∥u∥_{ρ^{s}H^{1}(Ω_{0})} + ∥u|_{∂Ω_{0}}∥_{ρ^{s}H^{1}(Ω_{0})}).

Proof. For u|_{∂Ω_{0}} = 0, the result follows from Equation (20), with M_{0} replaced by Ω_{0}, which is proved as explained in the proof of Theorem 3.8. In general, choose v ∈ H^{t+2}(Ω_{0}) such that v|_{∂Ω_{0}} = u|_{∂Ω_{0}}, which is possible by Theorem 3.7. Then we use our result for u − v. □

4. The Yamabe functional on Lie manifolds

As an example of the consequences and applications of the analysis developped in the preceeding sections, we will show that the Yamabe functional on a Lie manifold is bounded from below.

As before, let (M_{0}, g) be a Riemannian Lie manifold, with (M, V) the Lie structure at infinity. Recall that the scalar curvature scal_{g}(p) at p ∈ M_{0} is defined as scal_{g}(p) = ∑_{i,j=1}^{n}⟨R(e_{i}, e_{j})e_{j}, e_{i}⟩, where e_{1}, e_{2}, …, e_{n} denotes an orthonormal basis of T_{p}M_{0}.

We define the Yamabe functional for functions u ∈ C_{c}^{∞}(M_{0})

Y (u) = inf \frac{∫_{M}(4 \frac{n−1}{n−2}|du|^{2} + u^{2} scal_{g}) dvol_{g}}{∥u∥_{L^{p}(M_{0},g)}^{2}} ,

where p = 2n/(n − 2). The infimum is a conformal invariant. It is called the Yamabe constant of (M_{0}, g).

The geometric problem of finding a minimizer of Y is called the Yamabe problem. The interest in this problem is due to the following observation. If a minimizer u exists, then |u| is an everywhere strictly positive minimizer of Y. Then, the Euler Lagrange equation in this case is equivalent to the statement that the metric |u|^{4/(n−2)}g is a metric of constant scalar curvature.

Intensive investigations from 1960 to 1987 (see [42] for a nice overview, and [76], [9], [60, 61, 62] for original literature) lead to the celebrated result that a minimizer of Y always exists on compact manifolds.

Remark 4.1. The original motivation for studying the Yamabe functional comes from the following observation (essentially due to Einstein and Hilbert): We view Y as a functional on the space of all metrics g on M, and positive functions u on M; then (g, u) a stationary point of Y (with respect to compactly supported perturbations of g and u) if and only if u^{4/(n−2)}g is an Einstein manifold (see e.g. [14]). Yamabe’s idea [78] was to find such stationary points via a minimax procedure. As a first step one mimimzes u for fixed g, as described above. Then, as a second step one takes the maximum over all metrics g. However, Yamabe’s program did not succeed as Yamabe was not aware of analytical difficulties in his approach. As indicated above, on compact manifolds, the first step in his program (i.e., the solution of the Yamabe problem) could be repaired. However, the second step cannot be repaired, there are obstructions to the existence of an Einstein metric.

For various reasons, one is interested in finding minimizers of Y on a large class of non-compact, but complete manifolds. E.g. to derive glueing formulas that describe the behavior of the Yamabe constant under taking a connected sum, it is helpful to find minimizers of Y on manifolds with cylindrical ends [2].

The following proposition is a preliminary step in the solution of the Yamabe problem on Lie manifolds.

Proposition 4.2. The Yamabe functional extends to a continuous functional on W^{1,2}(M_{0}), and it is bounded from below.

Proof. Because of bounded geometry

∫_{M_{0}} 4 \frac{n − 1}{n − 2}|du|^{2} + scal_{g}u^{2} ≤ C∥u∥_{W^{1,2}(M_{0})}^{2}.

Theorem 3.5 tells us that W^{1,2}(M_{0}) embeds into L^{\frac{2n}{n−2}} (M_{0}), hence the functional extends to W^{1,2}(M_{0}), and the constant from the embedding yields a lower bound for the Yamabe functional. □

It would be interesting to obtain a criterion under which the infimum of he functional is attained on general non-compact Lie manifolds.

Another important case, in which the Yamabe constant on a Lie manifold is of central importance, is general relativity theory. A preliminary step to obtaining solutions to the Einstein equations on a space-time is to obtain solutions to the Einstein constraint equations on a asymptotically euclidean or hyperbolic Riemannian manifold M of dimension 3, embedded as a space-like hypersurface into a space-time. The constraint equation is the compatibility condition at the metric and at the second fundamental form on M in order that it can be extended to a solution of an Einstein metric on a space-time.

The “conformal-cmc method” is a method to obtain solutions to these Einstein constraint equations. After having chosen a divergence free symmetric (0, 2)-tensor σ on M and an additional real constant τ in the space-time, each solution the Lichnerowicz equation

8Δu + scal · u − |σ|^{2}u^{−7} = 0

gives rise to a solution of the Einstein constraint equations with mean curvature τ. Whether the Lichnerowicz equation has solutions depends on the Yamabe constant of M [16, 32, 33, 44].

String theory suggests to search for solutions to the Einstein constraint equations on 9, 10 and 25 dimensional manifolds with other kind of asymptotic at infinity.

Remark 4.3. Most of the results of this paper, in particular Corollary 2.10, Proposition 2.13, and Theorem 3.6, still hold if one replaces functions by sections of a vector bundle, if this vector bundle extends to M in the sense of vector bundles with metric and connection.

For the special case that ∂M = ∅ (i.e., M_{0} compact) these statements are applied to sections of the spinor bundle in [4, 3]. This proves the existence of a maximizer of the functional

F(ϕ) = \frac{∫_{M}⟨Dϕ, ϕ⟩}{∥Dϕ∥_{L^{q}(M)}^{2}} , q = 2n/(n + 1),

where ϕ is a section of the spinor bundle, and D is the Dirac operator. The functional is bounded from above because of the boundedness of the Sobolev embedding

W^{1,2}(M_{0}) → W^{1/2,q}(M_{0}).

Maximizers of F satisfy Dϕ = c|ϕ|^{2/(n+1)}ϕ. If dim M = 2, then the spinorial Weierstrass representation tells us that these solutions represent constant mean curvature surfaces in ℝ^{3} and S^{3}.

Extensions of this functional to non-compact Lie manifolds are the object of current research.

5. Submanifolds

In this section we introduce various classes of submanifolds of a Lie manifold. Some of these classes were already used in the previous sections.

5.1. General submanifolds. We first introduce the most general class of submanifolds of a Lie manifold.

We first fix some notation. Let (M_{0}, M, A) and (N_{0}, N, B) be Lie manifolds. We know that there exist vector bundles A → M and B → N such that V ≃ Γ(M, A) and W ≃ Γ(N, B), see Remark 1.4. We can assume that V = Γ(M, A) and W = Γ(N, B).

Definition 5.1. Let (M_{0}, M, A) be a Lie manifold. Then (N_{0}, N, B) is called a submanifold of (M_{0}, M, A) if

(i) N is a closed submanifold of M (possibly with corners, no transversality at the boundary required),

(ii) N_{0} ⊂ M_{0}, ∂N ⊂ ∂M, and B ⊂ A|_{N}. (iii) (N_{0}, N, B) is a Lie manifold, (iv) the Lie structures at infinity satisfy the compatibility condition

Γ(N; B) = {X ∈ Γ(N; A|_{N}) | ρ ∘ X ∈ Γ(N; T N)}

We now make three simple observations.

Remark 5.2. An alternative form of Condition (iv) of the above definition is

(24)  Γ(N; B) = {X|_{N} | X ∈ Γ(M, A) and X|_{N} tangent to N}.

Remark 5.3. Equation 24 shows that there exists a natural vector bundle morphism f : B → A. Since B_{x} = T_{x}N_{0} ⊂ T_{x}M_{0} = A_{x} for x ∈ N_{0}, the map f is injective above N_{0}. The assumption ii of our definition implies that f is injective everywhere.

We have the following simple corollary that justifies Condition (iv) of Definition 5.1.

Corollary 5.4. Let g_{0} be a metric on M_{0} compatible with the Lie structure at infinity on M_{0}. Then the restriction of g to N_{0} is compatible with the Lie structure at infinity on N_{0}.

Proof. Let g be a metric on A whose restriction to T M_{0} defines the metric g. Then g restricts to a metric h on B, which in turn defines a metric h_{0} on N_{0}. By definition, h_{0} is the restriction of g to N_{0}. □

We thus see that any submanifold (in the sense of the above definition) of a Riemannian Lie manifold is itself a Riemannian Lie manifold.

5.2. Second fundamental form. We define the A-normal bundle of the sub-manifold (N_{0}, N, B) of (M_{0}, M, A) as ν^{A} = (A|_{N})/B which is a bundle over N. Then the anchor map ϱ defines a map ρ^{ν} : ν^{A} → (T M|_{N})/T N which is an isomorphism over N_{0}.

We denote the Levi-Civita-connection on A by ∇^{A} and the Levi-Civita connection on B by ∇^{B} [5]. Let X, Y ∈ V = Γ(M, A) such that X|_{N_{0}}, Y |_{N_{0}} ∈ T N_{0}. Then ∇_{X}^{A}Y |_{N} depends only on X|_{N}, Y |_{N} ∈ W = Γ(N, B). Furthermore the Koszul formula implies that ∇_{X}^{B}Y is the tangential part of ∇_{X}^{A}Y |_{N}. The normal part gives rise to the second fundamental form

II : B × B → ν^{A}, II(X, Y) = ∇_{X}^{A}Y − ∇_{X}^{B}Y.

The Levi-Civita connections ∇^{A} and ∇^{B} are torsion free, and hence II(X, Y) − II(Y, X) = [X, Y] − [Y, X] = 0 is symmetric. A direct computation reveals also that II(X, Y) is tensorial in X, and hence, because of the symmetry, it is also tensorial in Y. (“Tensorial” here means II(fX, Y) = fII(X, Y).) It then follows from the compactness of N that

∥II_{p}(X_{p}, Y_{p})∥ ≤ C∥X_{p}∥ ∥Y_{p}∥,

with a constant C independent of p ∈ N. Clearly, on the interior N_{0} ⊂ M_{0} the second fundamental form coincides with the classical second fundamental form.

Corollary 5.5. Let (N_{0}, N, B) be a submanifold of (M_{0}, M, A) with a compatible metric. Then the (classical) second fundamental form of N_{0} in M_{0} is uniformly bounded.

5.3. Tame submanifolds. We now introduce tame manifolds. Our main interest in tame manifolds is the tubular neighborhood theorem, Theorem 5.8, which asserts that a tame submanifold of a Lie manifold has a tubular neighborhood in a strong sense. In particular, we will obtain that a tame submanifold of codimension one is regular. This is interesting because being tame is an algebraic condition that can be easily verified by looking at the structural Lie algebras of vector fields. On the other hand, being a regular submanifold is an analytic condition on the metric that is very difficult to check directly.

Definition 5.6. A submanifold (N_{0}, N, B) of a Lie manifold (M_{0}, M, A) is called tame if T_{p}N and ϱ(A_{p}) span T_{p}M for all p ∈ ∂N.

As a consequence of these properties the anchor map ϱ defines an isomorphism from A_{p}/B_{p} to T_{p}M/T_{p}N for any p ∈ N. In particular, the anchor map ρ maps B^{⊥}, the orthogonal complement of B in A, injectively into ϱ(A) ⊂ T M. For any boundary face F and p ∈ F we have ρ(A_{p}) ⊂ T_{p}F. Hence, N is transversal to F, i.e., for any p ∈ N ∩ F, the space T_{p}M is spanned by T_{p}N and T_{p}F. As a consequence, N ∩ F is a submanifold of F of codimension dim M − dim N. The codimension of N ∩F in F is therefore independent of F, in particular independent of the dimension of F.

Examples 5.7.

(1) Let M be any compact manifold (without boundary). Fix a p ∈ M. Let (N_{0}, N, B) be a manifold with a Lie structure at infinity. Then (N_{0} × {p}, N × {p}, B) is a tame submanifold of (N_{0} × M, N × M, B × T M).

(2) If ∂N ≠ ∅, the diagonal N is a submanifold of N ×N, but not a tame submanifold.

(3) Let N be a submanifold with corners of M (so N is transverse to all faces of M). We endow these manifolds with the b-structure at infinity V_{b} (see Example 1.3 (i)). Then (N, V_{b}) is a tame Lie submanifold of (M, V_{b}).

We now prove the main theorem of this section.

Theorem 5.8 (Global tubular neighborhood theorem). Let (N_{0}, N, B) be a tame submanifold of the Lie manifold (M_{0}, M, A). For ϵ > 0, let (ν)_{ϵ} be the set of all vectors normal to N of length smaller than ϵ. If ϵ > 0 is sufficiently small, then the normal exponential map exp^{ν} defines a diffeomorphism from (ν)_{ϵ} to an open neighborhood V_{ϵ} of N in M. Moreover, dist(exp^{ν}(X), N) = |X| for |X| < ϵ.

Proof. Recall from [5] that the exponential map exp : T M_{0} → M_{0} extends to a map exp : A → M. The definition of the normal exponential function exp^{ν} is obtained by identifying the quotient bundle ν^{A} with B^{⊥}, as discussed earlier. This gives

exp^{ν} : (ν)_{ϵ} → M.

The differential d exp^{ν} at 0_{p} ∈ ν_{p}, p ∈ N is the identity, hence any point p ∈ N has a neighborhood U(p) and τ_{p} > 0 such that

(25)  exp^{ν} : (ν)_{τ_{p}}|_{U_{p}} → M

is a diffeomorphism onto its image. By compactness τ_{p} ≥ τ > 0. Hence, exp^{ν} is a local diffeomorphism of (ν)_{τ} to a neighborhood of N in M. It remains to show that it is injective for small ϵ ∈ (0, τ).

Let us assume now that there is no ϵ > 0 such that the proposition holds. Then there are sequences X_{i}, Y_{i} ∈ ν, i ∈ ℕ, X_{i} ≠ Y_{i} such that exp^{ν} X_{i} = exp^{ν} Y_{i} with |X_{i}|, |Y_{i}| → 0 for i → ∞. After taking a subsequence we can assume that the basepoints p_{i} of X_{i} converge to p_{∞} and the basepoints q_{i} of Y_{i} converge to q_{∞}. As the distance in M of p_{i} and q_{i} converges to 0, we conclude that p_{∞} = q_{∞}. However, exp^{ν} is a diffeomorphism from (ν)_{τ}|_{U(p_{∞})} into a neighborhood of U(p_{∞}). Hence, we see that X_{i} = Y_{i} for large i, which contradicts the assumptions. □

We now prove that every tame, codimension one Lie submanifold is regular.

Proposition 5.9. Let (N_{0}, N, B) be a tame submanifold of codimension one of (M_{0}, M, A). Fix a diffeomorphism

exp^{ν} : (ν)_{ϵ} ≅ N × (−ϵ, ϵ) → {x | d(x, N) < ϵ} := U_{ϵ}

as in Theorem 5.8. Then M_{0} carries a compatible metric g such that (exp^{ν})^{∗}g is a product metric, i.e., (exp^{ν})^{∗}g = g_{N} + dt^{2} on N × (−ϵ/2, ϵ/2).

Proof. Choose any compatible metric g_{1} on M_{0}. Let g_{2} be a metric on U_{ϵ} such that (exp^{ν})^{∗}g_{2} = g_{1}|_{N} + dt^{2} on N × (−ϵ, ϵ). Let d(x) := dist(x, N). Then

g = χ ∘ dg_{1} + (1 − χ ∘ d)g_{2},

has the desired properties, where the cut-off function χ : ℝ → [0, 1] is 1 on (−ϵ/2, ϵ/2) and has support in (−ϵ, ϵ), and satisfies χ(−t) = χ(t). □

6. Pseudodifferential operators

We now recall the definition of pseudodifferential operators on M_{0} generated by a Lie structure at infinity (M, V) on M_{0}.

6.1. Definition. We fix in what follows a compatible Riemannian metric g on M_{0} (that is, a metric coming by restriction from a metric on the bundle A → M extending T M_{0}), see Section 1. In order to simplify our discussion below, we shall use the metric g to trivialize all density bundles on M. Recall that M_{0} with the induced metric is complete [5].

Let exp_{x} : T_{x}M_{0} → M_{0} be the exponential map, which is everywhere defined because M_{0} is complete. We let

(26)  Φ : T M_{0} ⟶ M_{0} × M_{0}, Φ(v) := (x, exp_{x}(−v)), v ∈ T_{x}M_{0},

If E is a real vector bundle with a metric, we shall denote by (E)_{r} the set of all vectors v of E with |v| < r. Let (M_{0}^{2})_{r} := {(x, y), x, y ∈ M_{0}, d(x, y) < r}. Then the map Φ of Equation (26) restricts to a diffeomorphism Φ : (T M_{0})_{r} → (M_{0}^{2})_{r}, for any 0 < r < r_{inj}(M_{0}), where r_{inj}(M_{0}) is the injectivity radius of M_{0}, which was assumed to be positive. The inverse of Φ is of the form

(M_{0}^{2})_{r} ∋ (x, y) ⟼ (x, τ(x, y)) ∈ (T M_{0})_{r} .

We shall denote by S_{1,0}^{m}(E) the space of symbols of order m and type (1, 0) on E (in Hörmander’s sense) and by S_{cl}^{m}(E) the space of classical symbols of order m on E [30, 55, 72, 74]. See [6] for a review of these spaces of symbols in our framework.

Let χ ∈ C^{∞}(A^{∗}) be a smooth function that is equal to 1 on (A^{∗})_{r} and is equal to 0 outside (A^{∗})_{2r}, for some r < r_{inj}(M_{0})/3. Then, following [6], we define

q(a)u(x) = (2π)^{−n} ∫_{T^{∗}M_{0}} e^{iτ(x,y)·η}χ(x, τ(x, y))a(x, η)u(y) dη dy .

This integral is an oscillatory integral with respect to the symplectic measure on T^{∗}M_{0} [31]. Alternatively, we consider the measures on M_{0} and on T_{x}^{∗}M_{0} defined by some choice of a metric on A and we integrate first in the fibers T_{x}^{∗}M_{0} and then on M_{0}. The map σ_{tot} : S_{1,0}^{m}(A^{∗}) → Ψ^{m}(M_{0})/Ψ^{−∞}(M_{0}),

σ_{tot}(a) := q(a) + Ψ^{−∞}(M_{0})

is independent of the choice of the function χ ∈ C_{c}^{∞}((A)_{r}) [6].

We now enlarge the class of order −∞ operators that we consider. Any X ∈ Γ(A) generates a global flow Ψ_{X} : ℝ × M → M because X is tangent to all boundary faces of M and M is compact. Evaluation at t = 1 yields a diffeomorphism

(27)  ψ_{X} := Ψ_{X}(1, ·) : M → M.

We now define the pseudodifferential calculus on M_{0} that we will consider following [6].

Definition 6.1. Fix 0 < r < r_{inj}(M_{0}) and χ ∈ C_{c}^{∞}((A)_{r}) such that χ = 1 in a neighborhood of M ⊆ A. For m ∈ ℝ, the space Ψ_{1,0,V}^{m}(M_{0}) of pseudodifferential operators generated by the Lie structure at infinity (M, V) is defined to be the linear space of operators C_{c}^{∞}(M_{0}) → C_{c}^{∞}(M_{0}) generated by q(a), a ∈ S_{1,0}^{m}(A^{∗}), and q(b)ψ_{X_{1}} … ψ_{X_{k}}, b ∈ S^{−∞}(A^{∗}) and X_{j} ∈ Γ(A), ∀j.

Similarly, the space Ψ_{cl,V}^{m}(M_{0}) of classical pseudodifferential operators generated by the Lie structure at infinity (M, V) is obtained by using classical symbols a in the construction above.

We have that Ψ_{cl,V}^{−∞}(M_{0}) = Ψ_{1,0,V}^{−∞}(M_{0}) =: Ψ_{V}^{−∞}(M_{0}) (we dropped some subscripts).

6.2. Properties. We now review some properties of the operators in Ψ_{1,0,V}^{m}(M_{0}) and Ψ_{cl,V}^{m}(M_{0}) from [6]. These properties will be used below. Let Ψ_{1,0,V}^{∞}(M_{0}) = ∪_{m∈ℤ}Ψ_{1,0,V}^{m}(M_{0}) and Ψ_{cl,V}^{∞}(M_{0}) = ∪_{m∈ℤ}Ψ_{cl,V}^{m}(M_{0}).

First of all, each operator P ∈ Ψ_{1,0,V}^{m}(M_{0}) defines continuous maps C_{c}^{∞}(M_{0}) → C^{∞}(M_{0}), and C^{∞}(M) → C^{∞}(M), still denoted by P. An operator P ∈ Ψ_{1,0,V}^{m}(M_{0}) has a distribution kernel k_{P} in the space I^{m}(M_{0} × M_{0}, M_{0}) of distributions on M_{0} × M_{0} that are conormal of order m to the diagonal, by [31]. If P = q(a), then k_{P} has support in (M_{0} × M_{0})_{r}. If we extend the exponential map (T M_{0})_{r} → M_{0} × M_{0} to a map A → M, then the distribution kernel of P = q(a) is the restriction of a distribution, also denoted k_{P} in I^{m}(A, M).

If P denotes the space of polynomial symbols on A^{∗} and Diff(M_{0}) denotes the algebra of differential operators on M_{0}, then

(28)  Ψ_{1,0,V}^{∞}(M_{0}) ∩ Diff(M_{0}) = Diff_{V}^{∞}(M) = q(P).

The spaces Ψ_{1,0,V}^{m}(M_{0}) and Ψ_{1,0,V}^{m}(M_{0}) are independent of the choice of the metric on A and the function χ used to define it, but depend, in general, on the Lie structure at infinity (M, A) on M_{0}. They are also closed under multiplication, which is a quite non-trivial fact.

Theorem 6.2. The spaces Ψ_{1,0,V}^{∞}(M_{0}) and Ψ_{1,0,V}^{∞}(M_{0}) are filtered algebras that are closed under adjoints.

For Ψ_{1,0,V}^{m}(M_{0}), the meaning of the above theorem is that

Ψ_{1,0,V}^{m}(M_{0})Ψ_{1,0,V}^{m′}(M_{V}) ⊆ Ψ_{1,0,V}^{m+m′}(M_{0}) and (Ψ_{1,0,V}^{m}(M_{0}))^{∗} = Ψ_{1,0,V}^{m}(M_{0})

for all m, m′ ∈ ℂ ∪ {−∞}.

The usual properties of the principal symbol remain true.

Proposition 6.3. The principal symbol establishes isomorphisms

(29)  σ^{(m)} : Ψ_{1,0,V}^{m}(M_{0})/Ψ_{1,0,V}^{m−1}(M_{0}) → S_{1,0}^{m}(A^{∗})/S_{1,0}^{m−1}(A^{∗})

and

(30)  σ^{(m)} : Ψ_{cl,V}^{m}(M_{0})/Ψ_{cl,V}^{m−1}(M_{0}) → S_{cl}^{m}(A^{∗})/S_{cl}^{m−1}(A^{∗}).

Moreover, σ^{(m)}(q(a)) = a + S_{1,0}^{m−1}(A^{∗}) for any a ∈ S_{1,0}^{m}(A^{∗}) and σ^{(m+m′)}(P Q) = σ^{(m)}(P)σ^{(m′)}(Q), for any P ∈ Ψ_{1,0,V}^{m}(M_{0}) and Q ∈ Ψ_{1,0,V}^{m′}(M_{0}).

We shall need also the following result.

Proposition 6.4. Let x be a defining function of some hyperface of M. Then ρ^{s}Ψ_{1,0,V}^{m}(M_{0})ρ^{−s} = Ψ_{1,0,V}^{m}(M_{0}) and ρ^{s}Ψ_{cl,V}^{m}(M_{0})ρ^{−s} = Ψ_{cl,V}^{m}(M_{0}) for any s ∈ ℂ.

6.3. Continuity on W^{s,p}(M_{0}). The preparations above will allow us to prove the continuity of the operators P ∈ Ψ_{1,0,V}^{m}(M_{0}) between suitable Sobolev spaces. This is the main result of this section. Some of the ideas and constructions in the proof below have already been used in 3.8, which the reader may find convenient to review first.

Theorem 6.5. Let P ∈ Ψ_{1,0,V}^{m}(M_{0}) and p ∈ (0, ∞). Then P maps ρ^{r}W^{s,p}(M_{0}) continuously to ρ^{r}W^{s−m,p}(M_{0}) for any r, s ∈ ℝ.

Proof. We have that P maps ρ^{r}W^{s,p}(M_{0}) continuously to ρ^{r}W^{s−m,p}(M_{0}) if, and only if, ρ^{−r}P ρ^{r} maps W^{s,p}(M_{0}) continuously to W^{s−m,p}(M_{0}). By Proposition 6.4 it is therefore enough to check our result for r = 0.

We shall first prove our result if the Schwartz kernel of P has support close enough to the diagonal. To this end, let us choose ϵ < r_{inj}(M_{0})/9 and assume that the distribution kernel of P is supported in the set (M_{0}^{2})_{ϵ} := {(x, y), d(x, y) < ϵ} ⊂ M_{0}^{2}. This is possible by choosing the function χ used to define the spaces Ψ_{1,0,V}^{m}(M_{0}) to have support in the set (M_{0}^{2})_{ϵ}. There will be no loss of generality then to assume that P = q(a).

Then choose a smooth function η : [0, ∞) → [0, 1], η(t) = 1 if t ≤ 6ϵ, η(t) = 0 if t ≤ 7ϵ. Let ψ_{x} : B(x, 8ϵ) → B_{T_{x}M_{0}}(0, 8ϵ) denote the normal system of coordinates induced by the exponential maps exp_{x} : T_{x}M_{0} → M_{0}. Denote π : A → M be the natural (vector bundle) projection and

(31)  B := A ×_{M} A := {(ξ_{1}, ξ_{2}) ∈ A × A, π(ξ_{1}) = π(ξ_{2})},

which defines a vector bundle B → M. In the language of vector bundles, B := A⊕A. For any x ∈ M_{0}, let η_{x} denote the function η ∘ exp_{x}, and consider the operator η_{x}P η_{x} on B(x, 13ϵ). The diffeomorphism ψ_{x} then will map this operator to an operator P_{x} on B_{T_{x}M_{0}}(0, 8ϵ). Then P_{x} maps continuously W^{s,p}(T_{x}M_{0}) → W^{s−m,p}(T_{x}M_{0}), by the continuity of pseudodifferential operators on ℝ^{n} [75, XIII, §5] or [72, 69].

The distribution kernel k_{x} of P_{x} is a distribution with compact support on

T_{x}M_{0} × T_{x}M_{0} = A_{x} × A_{x} = B_{x}

If P = q(a) ∈ Ψ_{1,0,V}^{m}(M_{0}), then the distributions k_{x} can be determined in terms of the distribution k_{P} ∈ I^{m}(A, M) associated to P. This shows that the distributions k_{x} extend to a smooth family of distributions on the fibers of B → M. From this, it follows that the family of operators P_{x} : W^{s,p}(A_{x}) → W^{s−m,p}(A_{x}), x ∈ M_{0}, extends to a family of operators defined for x ∈ M (recall that A_{x} = T_{x}M_{0} if x ∈ M_{0}). This extension is obtained by extending the distribution kernels. In particular, the resulting family P_{x} will depend smoothly on x ∈ M. Since M is compact, we obtain, in particular, that the norms of the operators P_{x} are uniformly bounded for x ∈ M_{0}.

By abuse of notation, we shall denote by P_{x} : W^{s,p}(M_{0}) → W^{s−m,p}(M_{0}) the induced family of pseudodifferential operators, and we note that it will still be a smooth family that is uniformly bounded in norm. Note that it is possible to extend P_{x} to an operator on M_{0} because its distribution kernel has compact support.

Then choose the sequence of points {x_{j}} ⊂ M_{0} and a partition of unity ϕ_{j} ∈ C_{c}^{∞}(M_{0}) as in Lemma 2.5. In particular, ϕ_{j} will have support in B(x_{j}, 2ϵ). Also, let ψ_{j} : B(x_{j}, 4ϵ) → B_{ℝ^{n}}(0, 4ϵ) denote the normal system of coordinates induced by the exponential maps exp_{x} : T_{x}M_{0} → M_{0} and some fixed isometries T_{x}M_{0} ≃ ℝ^{n}. Then all derivatives of ψ_{j} ∘ ψ_{k}^{−1} are bounded on their domain of definition, with a bound that may depend on ϵ but does not depend on j and k [15, 66].

Let

ν_{s,p}(u)^{p} := ∑_{j} ∥(ϕ_{j}u) ∘ ψ_{j}^{−1}∥_{W^{s,p}(ℝ^{n})}^{p}.

be one of the several equivalent norms defining the topology on W^{s,p}(M_{0}) (see Proposition 2.9 and Equation (13). It is enough to prove that

(32) ν_{s,p}(P u)^{p} := ∑_{j} ∥(ϕ_{j}P u) ∘ ψ_{j}^{−1}∥_{W^{s,p}(ℝ^{n})}^{p}

≤ C ∑_{j} ∥(ϕ_{j}u) ∘ ψ_{j}^{−1}∥_{W^{s,p}(ℝ^{n})}^{p} =: Cν_{s,p}(u)^{p},

for some constant C independent of u.

We now prove this statement. Indeed, for the reasons explained below, we have the following inequalities.

∑_{j} ∥(ϕ_{j}P u) ∘ ψ_{j}^{−1}∥_{W^{s,p}(ℝ^{n})}^{p} ≤ C ∑_{j,k} ∥(ϕ_{j}P ϕ_{k}u) ∘ ψ_{j}^{−1}∥_{W^{s,p}(ℝ^{n})}^{p}

= C ∑_{j,k} ∥(ϕ_{j}P_{x_{j}}ϕ_{k}u) ∘ ψ_{j}^{−1}∥_{W^{s,p}(ℝ^{n})}^{p} ≤ C ∑_{j,k} ∥(ϕ_{j}ϕ_{k}u) ∘ ψ_{j}^{−1}∥_{W^{s,p}(ℝ^{n})}^{p}

≤ C ∑_{j} ∥(ϕ_{j}u) ∘ ψ_{j}^{−1}∥_{W^{s,p}(ℝ^{n})}^{p} = Cν_{s,p}(u)^{p}.

Above, the first and last inequalities are due to the fact that the family ϕ_{j} is uniformly locally finite, that is, there exists a constant N such that at any given point x, at most N of the functions ϕ_{j}(x) are different from zero. The first equality is due to the support assumptions on ϕ_{j}, ϕ_{k}, and P_{x_{j}}. Finally, the second inequality is due to the fact that the operators P_{x_{j}} are continuous, with norms bounded by a constant independent of j, as explained above. We have therefore proved that P = q(a) ∈ Ψ_{1,0,V}^{m}(M_{0}) defines a bounded operator W^{s,p}(M_{0}) → W^{s−m,p}(M_{0}), provided that the Schwartz kernel of P has support in a set of the (M_{0}^{2})_{ϵ}, for ϵ < r_{inj}(M_{0})/9.

Assume now that P ∈ Ψ_{V}^{−∞}(M_{0}). We shall check that P is bounded as a map W^{2k,p}(M_{0}) → W^{−2k,p}(M_{0}). For k = 0, this follows from the fact that the Schwartz kernel of P is given by a smooth function k(x, y) such that ∫_{M_{0}} |k(x, y)|d vol_{g}(x) and ∫_{M_{0}} |k(x, y)|d vol_{g}(y) are uniformly bounded in x and y. For the other values of k, it is enough to prove that the bilinear form

W^{2k,p}(M_{0}) × W^{2k,p}(M_{0}) ∋ (u, v) → ⟨P u, v⟩ ∈ ℂ

is continuous. Choose Q a parametrix of Δ^{k} and let R = 1 − QΔ^{k} be as above. Let R′ = 1 − Δ^{k}Q ∈ Ψ_{V}^{−∞}(M_{0}). Then

⟨P u, v⟩ = ⟨(QP Q)Δ^{k}u, Δ^{k}v⟩ + ⟨(QP R)u, Δ^{k}v⟩ + ⟨(R′P Q)Δ^{k}u, v⟩ + ⟨(R′P R)u, v⟩, which is continuous since QP Q, QP R, R′P Q, and R′P R are in Ψ_{V}^{−∞}(M_{0}) and hence they are continuous on on L^{p}(M_{0}) and because Δ^{k} : W^{2k,p}(M_{0}) → L^{p}(M_{0}) is continuous.

Since any P ∈ Ψ_{1,0,V}^{m}(M_{0}) can be written P = P_{1} + P_{2} with P_{2} ∈ Ψ_{V}^{−∞}(M_{0}) and P_{1} = q(a) ∈ Ψ_{1,0,V}^{m}(M_{0}) with support arbitrarily close to the diagonal in M_{0}, the result follows. □

As in [68][Proposition 1.8], we obtain the following characterization of Sobolev spaces.

Theorem 6.6. Let s ∈ ℝ_{+} and p ∈ (1, ∞). We have that u ∈ W^{s,p}(M_{0}) if, and only if, u ∈ L^{p}(M_{0}) and P u ∈ L^{p}(M_{0}) for any P ∈ Ψ_{1,0,V}^{s}(M_{0}). The norm u → ∥u∥_{L^{p}(M_{0})} + ∥P u∥_{L^{p}(M_{0})} is equivalent to the original norm on W^{s,p}(M_{0}) for any elliptic P ∈ Ψ_{1,0,V}^{s}(M_{0}).

Similarly, the space W^{−s,p}(M_{0}) is the quotient of L^{p}(M_{0}) ⊕ L^{p}(M_{0}) with respect to the map (u, v) → u + P v.

Proof. Clearly, if u ∈ W^{s,p}(M_{0}), then P u, u ∈ L^{p}(M_{0}). Let us prove the converse.

Assume P u, u ∈ L^{p}(M_{0}). Let Q ∈ Ψ_{1,0,V}^{−s}(M_{0}) be a parametrix of P and let R, R′ ∈ Ψ_{V}^{−∞}(M_{0}) be defined by R := 1−QP and R′ = 1−P Q. Then u = QP u+Ru. Since both Q, R : L^{p}(M_{0}) → W^{s,p}(M_{0}) are defined and bounded, u ∈ W^{s,p}(M_{0}) and ∥u∥_{W^{s,p}(M_{0})} ≤ C(∥u∥_{L^{p}(M_{0})} + ∥P u∥_{L^{p}(M_{0})}). This proves the first part.

To prove the second part, we observe that the mapping

W^{s,q}(M_{0}) ∋ u → (u, P u) ∈ L^{q}(M_{0}) ⊕ L^{q}(M_{0}), q^{−1} + p^{−1} = 1,

is an isomorphism onto its image. The result then follows by duality using the Hahn-Banach theorem. □

We conclude our paper with the sketch of two regularity results on solutions of elliptic equations. We formulate the first result only for order two operators with Dirichlet boundary conditions, in order to avoid a discussion of regular boundary conditions [73] in our setting.

The proof of the following result is a standard application of the previous ideas. Recall the Sobolev spaces with weights ρ^{s}W^{s,p}(Ω_{0}) introduced in Equation (18).

Theorem 6.7. Let P ∈ Diff_{V}^{m}(M) be an order m elliptic operator on M_{0} generated by V. Let u ∈ ρ^{s}W^{r,p}(M_{0}) be such that P u ∈ ρ^{s}W^{t,p}(M_{0}), s, r, t ∈ ℝ, 1 < p < ∞. Then u ∈ ρ^{s}W^{t+m,p}(M_{0}).

Proof. Let Q ∈ Ψ_{V}^{−∞}(M_{0}) be a parametrix of P. Then R = I − QP ∈ Ψ_{V}^{−∞}(M_{0}). This gives u = Q(P u) + Ru. But Q(P u) ∈ ρ^{s}W^{t+m,p}(M_{0}), by Theorem 6.5, because P u ∈ ρ^{s}W^{t,p}(M_{0}). Similarly, Ru ∈ ρ^{s}W^{t+m,p}(M_{0}). This completes the proof. □

Note that the above theorem was already proved in the case t ∈ ℤ and m = 2, using more elementary methods, as part of Theorem 3.8. The proof here is much shorter, however, which attests to the power of pseudodifferential operator algebra techniques.

References

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Bernd Ammann, Math. Science Res. Inst., 17 Gauss Way, Berkeley, CA 94720-5070, USA, http://www.math.uni-hamburg.de/home/ammann

Alexandru D. Ionescu, University of Wisconsin, Department of Mathematics, Madison, WI 53705, USA

  E-mail address: ionescu@math.wisc.edu

Victor Nistor, Pennsylvania State University, Math. Dept., University Park, PA 16802, USA

  E-mail address: nistor@math.psu.edu