SOBOLEV SPACES ON LIE MANIFOLDS AND REGULARITY FOR POLYHEDRAL DOMAINS

BERND AMMANN, ALEXANDRU D. IONESCU, AND VICTOR NISTOR

Abstract. We study some basic analytic questions related to differential operators on Lie manifolds, which are manifolds whose large scale geometry can be described by a a Lie algebra of vector fields on a compactification. We extend to Lie manifolds several classical results on Sobolev spaces, elliptic regularity, and mapping properties of pseudodifferential operators. A tubular neighborhood theorem for Lie submanifolds allows us also to extend to regular open subsets of Lie manifolds the classical results on traces of functions in suitable Sobolev spaces. Our main application is a regularity result on polyhedral domains ℙ ⊂ ℝ^{3} using the weighted Sobolev spaces K_{a}^{m}(ℙ). In particular, we show that there is no loss of K_{a}^{m}–regularity for solutions of strongly elliptic systems with smooth coefficients. For the proof, we identify K_{a}^{m}(ℙ) with the Sobolev spaces on ℙ associated to the metric r_{ℙ}^{−2}g_{E}, where g_{E} is the Euclidean metric and r_{ℙ}(x) is a smoothing of the Euclidean distance from x to the set of singular points of ℙ. A suitable compactification of the interior of ℙ then becomes a regular open subset of a Lie manifold. We also obtain the wellposedness of a non-standard boundary value problem on a smooth, bounded domain with boundary O ⊂ ℝ^{n} using weighted Sobolev spaces, where the weight is the distance to the boundary.

Contents

Introduction 2 1. Lie manifolds 5 2. Submanifolds 11 3. Sobolev spaces 15 4. Sobolev spaces on regular open subsets 21 5. A regularity result 27 6. Polyhedral domains in three dimensions 29 7. A non-standard boundary value problem 31 8. Pseudodifferential operators 33 References 37

MSC:35J40 (Primary) 33J55, 35J70, 35J25, 47G30 (Secondary) Keywords: regularity, polyhedral domains, Lie manifolds, analysis on complete manifolds

Date: June 21, 2006. Ionescu was supported in part by an NSF grant, by an Alfred P. Sloan research fellowship, and

by the Packard Foundation. Nistor was partially supported by the NSF Grant DMS 0200808.

Introduction

We study some basic analytic questions on non-compact manifolds. In order to obtain stronger results, we restrict ourselves to “Lie manifolds,” a class of manifolds whose large scale geometry is determined by a compactification to a manifold with corners and a Lie algebra of vector fields on this compactification (Definition 1.3). One of the motivations for studying Lie manifolds is the loss of (classical Sobolev) regularity of solutions of elliptic equations on non-smooth domains. To explain this loss of regularity, let us recall first that the Poisson problem

(1)  Δu = f ∈ H^{m−1}(Ω), m ∈ ℕ ∪ {0}, Ω ⊂ ℝ^{n} bounded,

has a unique solution u ∈ H^{m+1}(Ω), u = 0 on ∂Ω, provided that ∂Ω is smooth. In particular, u will be smooth up to the boundary if ∂Ω and f are smooth (in the following, when dealing with functions defined on an open set, by “smooth,” we shall mean “smooth up to the boundary”). See the books of Evans [16], or Taylor [58] for a proof of this basic well-posedness result.

This well-posedness result is especially useful in practice for the numerical approximation of the solution u of Equation (1) [8]. However, in practice, it is only rarely the case that Ω is smooth. The lack of smoothness of the domains interesting in applications has motivated important work on Lipschitz domains, see for instance [23, 40] or [65]. These papers have extended to Lipschitz domains some of the classical results on the Poisson problem on smooth, bounded domains, using the classical Sobolev spaces

H^{m}(Ω) := {u, ∂^{α}u ∈ L^{2}(Ω), |α| ≤ m}.

It turns out that, if ∂Ω is not smooth, then the smoothness of f on \overline{Ω} (i. e., up to the boundary) does not imply that the solution u of Equation (1) is smooth as well on \overline{Ω}. This is the loss of regularity for elliptic problems on non-smooth domains mentioned above.

The loss of regularity can be avoided, however, by a conformal blowup of the singular points. This conformal blowup replaces a neighborhood of each connected component of the set of singular boundary points by a complete, but non-compact end. (Here “complete” means complete as a metric space, not geodesically complete.) It can be proved then that the resulting Sobolev spaces are the “Sobolev spaces with weights” considered for instance in [25, 26, 35, 46]. Let f > 0 be a smooth function on a domain Ω, we then define the mth Sobolev space with weight f by

(2) K_{a}^{m}(Ω; f) := {u, f^{|α|−a}∂^{α}u ∈ L^{2}(Ω), |α| ≤ m}, m ∈ ℕ ∪ {0}, a ∈ ℝ.

Indeed, if Ω = ℙ ⊂ ℝ^{2} is a polygon, and if we choose

(3) f(x) = ϑ(x) = the distance to the non-smooth boundary points of ℙ,

then there is no loss of regularity in the spaces K_{a}^{m}(Ω) := K_{a}^{m}(Ω; ϑ) [26, Theorem 6.6.1]. In this paper, we extend this regularity result to polyhedral domains in three dimensions, Theorem 6.1, with the same choice of the weight (in three dimensions the weight is the distance to the edges). The analogous result in arbitrary dimensions leads to topological difficulties [9, 66].

Our regularity result requires us first to study the weighted Sobolev spaces K_{a}^{m}(Ω) := K_{a}^{m}(Ω; ϑ) where ϑ(x) is the distance to the set of singular points on the boundary. Our approach to Sobolev spaces on polyhedral domains is to show first that K_{a}^{m}(Ω) is isomorphic to a Sobolev space on a certain non-compact Riemannian manifold M with smooth boundary. This non-compact manifold M is obtained from our polyhedral domain by replacing the Euclidean metric g_{E} with

(4)  r_{ℙ}^{−2}g_{E}, r_{ℙ} a smoothing of ϑ,

which blows up at the faces of codimension two or higher, that is, at the set of singular boundary points. (The metric r_{ℙ}^{−2}g_{E} is Lipschitz equivalent to ϑ^{−2}g_{E}, but the latter is not smooth.) The resulting non-compact Riemannian manifold turns out to be a regular open subset in a “Lie manifold.” (see Definition 1.3, Subsection 1.6, and Section 6 for the precise definitions). A Lie manifold is a compact manifold with corners M together with a C^{∞}(M)-module V whose elements are vector fields on M. The space V must satisfy a number of axioms, in particular, V is required to be closed under the Lie bracket of vector fields. This property is the origin of the name Lie manifold. The C^{∞}(M)-module V can be identified with the sections of a vector bundle A over M. Choosing a metric on A defines a complete Riemannian metric on the interior of M. See Section 1 or [4] for details.

The framework of Lie manifolds is quite convenient for the study of Sobolev spaces, and in this paper we establish, among other things, that the main results on the classical Sobolev spaces remain true in the framework of Lie manifolds. The regular open sets of Lie manifolds then play in our framework the role played by smooth, bounded domains in the classical theory.

Let ℙ ⊂ ℝ^{n} be a polyhedral domain. We are especially interested in describing the spaces K_{a−1/2}^{m−1/2}(∂ℙ) of restrictions to the boundary of the functions in the weighted Sobolev space K_{a}^{m}(ℙ; ϑ) = K_{a}^{m}(ℙ; r_{ℙ}) on ℙ. Using the conformal change of metric of Equation (4), the study of restrictions to the boundary of functions in K_{a}^{m}(ℙ) is reduced to the analogous problem on a suitable regular open subset Ω_{ℙ} of some Lie manifold. More precisely, K_{a}^{m}(ℙ) = r_{ℙ}^{a−n/2}H^{m}(Ω_{ℙ}). A consequence of this is that

(5)  K_{a−1/2}^{m−1/2}(∂ℙ) = K_{a−1/2}^{m−1/2}(∂ℙ; ϑ) = r_{ℙ}^{a−n/2}H^{m−1/2}(∂Ω_{ℙ}).

(In what follows, we shall usually simply denote K_{a}^{m}(ℙ) := K_{a}^{m}(ℙ; ϑ) = K_{a}^{m}(ℙ; r_{ℙ}) and K_{a}^{m}(∂ℙ) := K_{a}^{m}(∂ℙ; ϑ) = K_{a}^{m}(∂ℙ; r_{ℙ}), where, we recall, ϑ(x) is the distance from x to the set of non-smooth boundary points and r_{ℙ} is a smoothing of ϑ that satisfies r_{ℙ}/ϑ ∈ [c, C], c, C > 0.)

Equation (5) is one of the motivations to study Sobolev spaces on Lie manifolds. 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 appear in the study of the Yamabe problem [32, 48] on compact manifolds, of the Yamabe problem on asymptotically cylindrical manifolds [2], of analysis on locally symmetric spaces, and of the positive mass theorem [49, 50, 67], an analogue of the positive mass theorem on asymptotically hyperbolic manifolds [6]. Lie manifolds also appear in Mathematical Physics and in Numerical Analysis. Classes of Sobolev spaces on non-compact manifolds have been studied in many papers, of which we mention only a few [15, 18, 27, 30, 34, 36, 39, 37, 38, 51, 52, 53, 63, 64] in addition to the works mentioned before. Our work can also be used to unify some of the various approaches found in these papers.

Let us now review in more detail the contents of this paper. 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). If M is a compact manifold with corners, we shall denote by ∂M the union of all boundary faces of M and by M_{0} := M ∖ ∂M the interior of M. 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 a Lie manifold as a pair (M, V), where M is a compact manifold with corners and V is a structural Lie algebra of vector fields that is unrestricted in the interior M_{0} of M. We will explain the above mentioned fact that the interior of M carries a complete metric g. This metric is unique up to Lipschitz equivalence (or quasi-isometry). We also introduce in this section Lie manifolds with (true) boundary and, as an example, we discuss the example of a Lie manifold with true boundary corresponding to curvilinear polygonal domains. In Section 2 we discuss Lie submanifolds, and most importantly, the global tubular neighborhood theorem. The proof of this global tubular neighborhood theorem is based on estimates on the second fundamental form of the boundary, which are obtained from the properties of the structural Lie algebra of vector fields. This property distinguishes Lie manifolds from general manifolds with boundary and bounded geometry, for which a global tubular neighborhood is part of the definition. In Section 3, we define the Sobolev spaces W^{s,p}(M_{0}) on the interior M_{0} of a Lie manifold M, where either s ∈ ℕ ∪ {0} and 1 ≤ p ≤ ∞ or s ∈ ℝ and 1 < p < ∞. We first define the spaces W^{s,p}(M_{0}), s ∈ ℕ ∪ {0} and 1 ≤ p ≤ ∞, by differentiating with respect to vector fields in V. This definition is in the spirit of the standard definition of Sobolev spaces on ℝ^{n}. Then we prove that there are two alternative, but equivalent ways to define these Sobolev spaces, either by using a suitable class of partitions of unity (as in [54, 55, 62] 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 4, we discuss regular open subsets Ω ⊂ M. In the last two sections, several of the classical results on Sobolev spaces on smooth domains were extended to 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 the Hilbert space case (p = 2), and the Rellich-Kondrachov compactness theorem.

In Section 5 we include as an application a regularity result for strongly elliptic boundary value problems, Theorem 5.1. This theorem gives right away the following result, proved in Section 6, which states that there is no loss of regularity for these problems within weighted Sobolev spaces.

Theorem 0.1. Let ℙ ⊂ ℝ^{3} be a polyhedral domain and P be a strongly elliptic, second order differential operator with coefficients in C^{∞}(\overline{ℙ}). Let u ∈ K_{a+1}^{1}(ℙ), u = 0 on ∂ℙ, a ∈ ℝ. If P u ∈ K_{a−1}^{m−1}(ℙ), then u ∈ K_{a+1}^{m+1}(ℙ) and there exists C > 0 independent of u such that

∥u∥_{K_{a+1}^{m+1}(ℙ)} ≤ C(∥P u∥_{K_{a−1}^{m−1}(ℙ)} + ∥u∥_{K_{a+1}^{0}(ℙ)}), m ∈ ℕ ∪ {0}.

The same result holds for strongly elliptic systems.

Note that the above theorem does not constitute a Fredholm (or normal solvability) result, because the inclusion K_{a+1}^{m+1}(ℙ) → K_{a+1}^{0}(ℙ) is not compact. See also [25, 26, 35, 46] and the references therein for similar results.

In Section 7, we obtain a “non-standard boundary value problem” on a smooth domain O in weighted Sobolev spaces with weight given by the distance to the boundary. The boundary conditions are thus replaced by growth conditions. Finally, in the last section, Section 8, we obtain mapping properties for the pseudodifferential calculus Ψ_{V}^{∞}(M) defined in [3] between our weighted Sobolev spaces ρ^{s}W^{r,p}(M). We also obtain a general elliptic regularity result for elliptic pseudodifferential operators in Ψ_{V}^{∞}(M).

Acknowledgements: We would like to thank Anna Mazzucato and Robert Lauter for useful comments. The first named author wants to thank MSRI, Berkeley, CA for its hospitality.

1. Lie manifolds

As explained in the Introduction, our approach to the study of weighted Sobolev spaces on polyhedral domains is based on their relation to Sobolev spaces on Lie manifolds with true boundary. Before we recall the definition of a Lie manifold and some of their basic properties, we shall first look at the following example, which is one of the main motivations for the theory of Lie manifolds.

Example 1.1. Let us take a closer look at the local structure of the Sobolev space K_{a}^{m}(ℙ) associated to a polygon ℙ (recall (2)). Consider Ω := {(r, θ) | 0 < θ < α}, which models an angle of ℙ. Then the distance to the vertex is simply ϑ(x) = r, and the weighted Sobolev spaces associated to Ω, K_{a}^{m}(Ω), can alternatively be described as

(6) K_{a}^{m}(Ω) = K_{a}^{m}(Ω; ϑ) := {u ∈ L_{loc}^{2}(Ω), r^{−a}(r∂_{r})^{i}∂_{θ}^{j}u ∈ L^{2}(Ω), i + j ≤ m}.

The point of the definition of the spaces K_{a}^{m}(Ω) was the replacement of the local basis {r∂_{x}, r∂_{y}} with the local basis {r∂_{r}, ∂_{θ}} that is easier to work with on the desingularization Σ(Ω) := [0, ∞)×[0, α] ∋ (r, θ) of Ω. By further writing r = e^{t}, the vector field r∂_{r} becomes ∂_{t}. Since dt = r^{−1}dr, the space K_{1}^{m}(Ω) then identifies with H^{m}(ℝ_{t} × (0, α)). The weighted Sobolev space K_{1}^{m}(Ω) has thus become a classical Sobolev space on the cylinder ℝ × (0, α), as in [25].

The aim of the following definitions is to define such a desingularisation in general. The desingularisation will carry the structure of a Lie manifold, defined in the next subsection.

We shall introduce a further, related definition, namely the definition of a “Lie submanifolds of a Lie manifold” in Section 4.

1.1. Definition of Lie manifolds. At first, we want to recall the definition of manifolds with corners. A manifold with corners is a closed subset M of a differentiable manifold such that every point p ∈ M lies in a coordinate chart whose restriction to M is a diffeomorphism to [0, ∞)^{k} × ℝ^{n−k}, for some k = 0, 1, …, n depending on p. Obviously, this definition includes the property that the transition map of two different charts are smooth up to the boundary. If k = 0 for all p ∈ M, we shall say that M is a smooth manifold. If k ∈ {0, 1}, we shall say that M is a smooth manifold with smooth boundary.

Let M be a compact manifold with corners. We shall denote by ∂M the union of all boundary faces of M, that is, ∂M is the union of all points not having a neighborhood diffeomorphic to ℝ^{n}. Furthermore, we shall write M_{0} := M ∖ ∂M for the interior of M. In order to avoid confusion, we shall use this notation and terminology only when M is compact. Note that our definition allows ∂M to be a smooth manifold, possibly empty.

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 1.2. 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 boundary 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}

in the sense of C^{∞}(\overline{U}_{p})-modules.

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

(7)  Y = ∑_{j=1}^{k} f_{j}X_{j} on U_{p}.

We now have defined the preliminaries for the following important definition.

Definition 1.3. A Lie structure at infinity on a smooth manifold M_{0} is a pair (M, V), where M is a compact manifold with interior M_{0} and V ⊂ Γ(M; T M) is a structural Lie algebra of vector fields on M with the following property: If p ∈ M_{0}, then any local basis of V in a neighborhood of p is also a local basis of the tangent space to M_{0}.

It follows from the above definition that the constant k of Equation (7) equals to the dimension n 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. (In [4], only the term “manifolds with a Lie structure at infinity” was used.)

Example 1.4. 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. Using the Frobenius theorem it is clear that such vector bundles are exactly the tangent bundles of k-dimensional foliations on M, k = rank F. However, V_{F} does not define a Lie structure at infinity, unless F = T M.

Remark 1.5. We observe that Conditions (iii) and (iv) of Definition 1.2 are equivalent to the condition that V be a projective C^{∞}(M)-module. Thus, by the Serre-Swan theorem [24], 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_{M} : A → T M, called the anchor map. The Condition (ii) of Definition 1.3 is then equivalent to the fact that ϱ_{M} 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_{0} on M_{0} such that, for any p ∈ M, we can choose the basis X_{1}, …, X_{k} in Definition 1.2 (iv’) (7) to be orthonormal with respect to this metric everywhere on U_{p} ∩ M_{0}. (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 other.

Remark 1.6. Using the language of Remark 1.5, g_{0} is a compatible metric on M_{0} if, and only if, there exists a metric g on the vector bundle A → M which restricts to g_{0} 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 [4]. For instance, (M_{0}, g_{0}) is necessarily complete and, if ∂M ≠ ∅, it is of infinite volume. 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 [54] and references therein).

On a Riemannian Lie manifold (M_{0}, M, V, g_{0}), the exponential map exp : T M_{0} → M_{0} is well-defined for all X ∈ T M_{0} and extends to a differentiable map exp : A → M. A convenient way to introduce the exponential map is via the geodesic spray, as done in [4]. Similarly, any vector field X ∈ V = Γ(M; A) is integrable and will map any (connected) boundary face of M to itself. The resulting diffeomorphism of M_{0} will be denoted ψ_{X}.

1.3. Examples. We include here two examples of Lie manifolds together with compatible Riemannian metrics. The reader can find more examples in [4, 31].

Examples 1.7.

(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. This generalizes Example 1.1. See also Subsection 1.6 and Section 6. Let r ≥ 0 to be a smooth function on M that is equal to the distance to the boundary in a neighborhood of ∂M, and is > 0 outside ∂M (i. e., on M_{0}). Let h be a smooth metric on M, then g_{0} = h + (r^{−1}dr)^{2} is a compatible metric on M_{0}.

(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. If ∂M is a smooth manifold (i. e., if M is a smooth manifold with boundary), then V_{0} = rΓ(M; T M), where r is as in (a).

1.4. 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

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

where M_{N}(Diff_{V}^{∗}(M)) is the algebra of N ×N-matrices over the ring Diff_{V}^{∗}(M), and where e_{E}, e_{F} ∈ M_{N}(C^{∞}(M)) are the projections onto E and, respectively, onto F. It follows that Diff_{V}^{∗}(M; E) := Diff_{V}^{∗}(M; E, 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_{0} on M_{0}.

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

One of the main goals of this paper is to prove the results on weighted Sobolev spaces on polyhedral domains that are needed for regularity theorems. We shall do that by reducing the study of weighted Sobolev spaces to the study of Sobolev spaces on “regular open subsets” of Lie manifolds, a class of open sets that plays in the framework of Lie manifolds the role played by domains with smooth boundaries in the framework of bounded, open subsets of ℝ^{n}. Regular open subsets are defined below in this subsection.

Let N ⊂ M be a submanifold of codimension one of the Lie manifold (M, V). Note that this implies that N is a closed subset of M. We shall say that N is a regular submanifold of (M, V) if we can choose a neighborhood V of N in M and a compatible metric g_{0} on M_{0} that restricts to a product-type metric on V ∩ M_{0} ≃ (∂N_{0}) × (−ε_{0}, ε_{0}), N_{0} = N ∖ ∂N = N ∩ M_{0}. Such neighborhoods will be called tubular neighborhoods.

In Section 2, we shall show that a codimension one manifold is regular if, and only if, it is a tame submanifold of M; this gives an easy, geometric, necessary and sufficient condition for the regularity of a codimension one submanifold of M. This is relevant, since the study of manifolds with boundary and bounded geometry presents some unexpected difficulties [47].

In the following, it will be important to distinguish properly between the boundary of a topological subset, denoted by ∂_{top}, and the boundary in the sense of manifolds with corners, denoted simply by ∂.

Definition 1.8. Let (M, V) be a Lie manifold and Ω ⊂ M be an open subset. We shall say that Ω is a regular open subset in M if, and only if, Ω is connected, Ω and \overline{Ω} have the same boundary, ∂_{top}Ω (in the sense of subsets of the topological space M), and ∂_{top}Ω is a regular submanifold of M.

Let Ω ⊂ M be a regular open subset. Then \overline{Ω} is a compact manifold with corners. The reader should be aware of the important fact that ∂_{top}Ω = ∂_{top}\overline{Ω} is contained in ∂\overline{Ω}, but in general ∂\overline{Ω} and ∂_{top}Ω are not equal. The set ∂_{top}Ω will be called the true boundary of \overline{Ω}. Furthermore, we introduce ∂_{∞}Ω := ∂\overline{Ω} ∩ ∂M, and call it the boundary at infinity of \overline{Ω}. Obviously, one has ∂\overline{Ω} = ∂_{top}\overline{Ω}∪∂_{∞}\overline{Ω}. The true boundary and the boundary at infinity intersect in a (possibly empty) set of codimension ≥ 2. See Figure 1. We will also use the notation ∂Ω_{0} := ∂_{top}Ω ∩ M_{0} = ∂\overline{Ω} ∩ M_{0}.

M_{0} ∂_{∞}Ω Ω  ∂_{top}\overline{Ω}

Figure 1. A regular open set Ω. Note that the interior of ∂_{∞}\overline{Ω} is contained in Ω, but the true boundary ∂_{top}Ω = ∂_{top}\overline{Ω} is not contained in Ω

The space of restrictions to Ω or \overline{Ω} of order m differential operators in Diff_{V}^{∗}(M) will be denoted Diff_{V}^{m}(Ω), respectively Diff_{V}^{m}(\overline{Ω}). Similarly, we shall denote by V(Ω) the space of restrictions to \overline{Ω} of vector fields in V, the structural Lie algebra of vector fields on M.

Let F ⊂ ∂Ω be any boundary hyperface of \overline{Ω} of codimension 1. Such a face is either contained in ∂_{top}\overline{Ω} or in ∂_{∞}\overline{Ω}. If F ⊂ ∂_{∞}Ω, then the restrictions of all vector fields in V to F are tangent to F. However, if F ⊂ ∂_{top}\overline{Ω} the regularity of the boundary implies that there are vector fields in V whose restriction to F is not tangent to F. In particular, the true boundary ∂_{top}\overline{Ω} of \overline{Ω} is uniquely determined by (\overline{Ω}, V(Ω)), and hence so is Ω = \overline{Ω} ∖ ∂_{top}\overline{Ω}. We therefore obtain a one-to-one correspondence between Lie manifolds with true boundary and regular open subsets (of some Lie manifold M).

Assume we are given Ω, \overline{Ω} (the closure in M), and V(Ω), with Ω a regular open subset of some Lie manifold (M, V). In the cases of interest, for example if ∂_{top}\overline{Ω} is a tame submanifold of M (see Subsection 2.3 for the definition of tame submanifolds), we can replace the Lie manifold (M, V) in which Ω is a regular open set with a Lie manifold (N, W) canonically associated to (Ω, \overline{Ω}, V(Ω)) as follows. Let N be obtained by gluing two copies of \overline{Ω} along ∂_{top}\overline{Ω}, the so-called double of \overline{Ω}, also denoted \overline{Ω}^{db} = N. A smooth vector field X on \overline{Ω}^{db} will be in W, the structural Lie algebra of vector fields W on \overline{Ω}^{db} if, and only if, its restriction to each copy of \overline{Ω} is in V(Ω). Then Ω will be a regular open set of the Lie manifold (N, W). For this reason, the pair (\overline{Ω}, V(Ω)) will be called a Lie manifold with true boundary. In particular, the true boundary of a Lie manifold with true boundary is a tame submanifold of the double. The fact that the double is a Lie manifold is justified in Remark 2.10.

1.6. Curvilinear polygonal domains. We conclude this section with a discussion of a curvilinear polygonal domain ℙ, an example that generalizes Example 1.1 and is one of the main motivations for considering Lie manifolds. To study function spaces on ℙ, we shall introduce a “desingularization” (Σ(ℙ), κ) of ℙ (or, rather, of \overline{ℙ}), where Σ(ℙ) is a compact manifold with corners and κ : Σ(ℙ) → \overline{ℙ} is a continuous map that is a diffeomorphism from the interior of Σ(ℙ) to ℙ and maps the boundary of Σ(ℙ) onto the boundary of ℙ.

Let us denote by B^{k} the open unit ball in ℝ^{k}.

Definition 1.9. An open, connected subset ℙ ⊂ M of a two dimensional manifold M will be called a curvilinear polygonal domain if, by definition, \overline{ℙ} is compact and for every point p ∈ ∂ℙ there exists a diffeomorphism ϕ_{p} : V_{p} → B^{2}, ϕ_{p}(p) = 0, defined on a neighborhood V_{p} ⊂ M such that

(9) ϕ_{j}(V_{p} ∩ ℙ) = {(r cos θ, r sin θ), 0 < r < 1, 0 < θ < α_{p}} , α_{p} ∈ (0, 2π).

A point p ∈ ∂ℙ for which α_{p} ≠ π will be called a vertex of ℙ. The other points of ∂ℙ will be called smooth boundary points. It follows that every curvilinear polygonal domain has finitely many vertices and its boundary consists of a finite union of smooth curves γ_{j} (called the edges of ℙ) which have no other common points except the vertices. Moreover, every vertex belongs to exactly two edges.

Let \{P_{1}, P_{2}, …, P_{k}\} ⊂ \overline{ℙ} be the vertices of ℙ. The cases k = 0 and k = 1 are also allowed. Let V_{j} := V_{P_{j}} and ϕ_{j} := ϕ_{P_{j}} : V_{j} → B^{2} be the diffeomorphisms defined by Equation (9). Let (r, θ) : ℝ^{2} ∖ {(0, 0)} → (0, ∞) × [0, 2π) be the polar coordinates. We can assume that the sets V_{j} are disjoint and define r_{j}(x) = r(ϕ_{j}(x)) and θ_{j}(x) = θ(ϕ_{j}(x)).

The desingularization Σ(ℙ) of ℙ will replace each of the vertices P_{j}, j = 1, …, k of ℙ with a segment of length α_{j} = α_{P_{j}} > 0. Assume that ℙ ⊂ ℝ^{2}. We can realize Σ(ℙ) in ℝ^{3} as follows. Let ψ_{j} be smooth functions supported on V_{j} with ψ_{j} = 1 in a neighborhood of P_{j}.

Φ : \overline{ℙ} ∖ \{P_{1}, P_{2}, …, P_{k}\} → ℝ^{2} × ℝ, Φ(p) = ( p , ∑_{j} ψ_{j}(p)θ_{j}(p) ).

Then Σ(ℙ) is (up to a diffeomorphism) the closure of Φ(ℙ) in ℝ^{3}. The desingularization map is κ(p, z) = p.

The structural Lie algebra of vector fields V(ℙ) on Σ(ℙ) is given by (the lifts of) the smooth vector fields X on \overline{ℙ} ∖ \{P_{1}, P_{2}, …, P_{k}\} that, on V_{j}, can be written as

X = a_{r}(r_{j}, θ_{j})r_{j}∂_{r_{j}} + a_{θ}(r_{j}, θ_{j})∂_{θ_{j}},

with a_{r} and a_{θ} smooth functions of (r_{j}, θ_{j}), r_{j} ≥ 0. Then (Σ(ℙ), V(ℙ)) is a Lie manifold with true boundary.

To define the structural Lie algebra of vector fields on Σ(ℙ), we now choose a smooth function r_{ℙ} : ℙ → [0, ∞) with the following properties

(i) r_{ℙ} is continuous on \overline{ℙ}, (ii) r_{ℙ} is smooth on ℙ, (iii) r_{ℙ}(x) > 0 on \overline{ℙ} ∖ \{P_{1}, P_{2}, …, P_{k}\}, (iv) r_{ℙ}(x) = r_{j}(x) if x ∈ V_{j}.

Note that r_{ℙ} lifts to a smooth positive function on Σ(ℙ). Of course, r_{ℙ} is determined only up to a smooth positive function ψ on Σ(ℙ) that equals to 1 in a neighborhood of the vertices.

Definition 1.10. A function of the form ψr_{ℙ}, with ψ ∈ C^{∞}(Σ(ℙ)), ψ > 0 will be called a canonical weight function of ℙ.

In what follows, we can replace r_{ℙ} with any canonical weight function. Canonical weight functions will play an important role again in Section 6. Canonical weights are example of “admissible weights,” which will be used to define weighted Sobolev spaces.

Then an alternative definition of V(ℙ) is

(10)  V(ℙ) := { r_{ℙ} (ψ_{1}∂_{1} + ψ_{2}∂_{2}) }, ψ_{1}, ψ_{2} ∈ C^{∞}(Σ(ℙ)).

Here ∂_{1} denotes the vector field corresponding to the derivative with respect to the first component. The vector field ∂_{2} is defined analogously. In particular,

(11)  r_{ℙ}(∂_{j}r_{ℙ}) = r_{ℙ} \frac{∂r_{ℙ}}{∂x_{j}} ∈ C^{∞}(Σ(ℙ)),

which is useful in establishing that V(ℙ) is a Lie algebra. Also, let us notice that both {r_{ℙ}∂_{1}, r_{ℙ}∂_{2}} and {r_{ℙ}∂_{r_{ℙ}}, ∂_{θ}} are local bases for V(ℙ) on V_{j}. The transition functions lift to smooth functions on Σ(ℙ) defined in a neighborhood of κ^{−1}(P_{j}), but cannot be extended to smooth functions defined in a neighborhood of P_{j} in \overline{ℙ}.

Then ∂_{top}Σ(ℙ), the true boundary of Σ(ℙ), consists of the disjoint union of the edges of ℙ (note that the interiors of these edges have disjoint closures in Σ(ℙ)). Anticipating the definition of a Lie submanifold in Section 2, let us notice that ∂_{top}Σ(ℙ) is a Lie submanifold, where the Lie structure consists of the vector fields on the edges that vanish at the end points of the edges.

The function ϑ used to define the Sobolev spaces K_{a}^{m}(ℙ) := K_{a}^{m}(ℙ; ϑ) in Equation (2) is closely related to the function r_{ℙ}. Indeed, ϑ(x) is the distance from x to the vertices of ℙ. Therefore ϑ/r_{ℙ} will extend to a continuous, nowhere vanishing function on Σ(ℙ), which shows that

(12)  K_{a}^{m}(ℙ; ϑ) = K_{a}^{m}(ℙ; r_{ℙ}).

If P is an order m differential operator with smooth coefficients on ℝ^{2} and ℙ ⊂ ℝ^{2} is a polygonal domain, then r_{ℙ}^{m}P ∈ Diff_{V}^{m}(Σ(ℙ)), by Equation (10). However, in general, r_{ℙ}^{m}P will not define a smooth differential operator on \overline{ℙ}.

2. Submanifolds

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

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

We first fix some notation. Let (M_{0}, M, V) and (N_{0}, N, W) 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 1.5. We can assume that V = Γ(M; A) and W = Γ(N; B) and write (M, A) and (N, B) instead of (M_{0}, M, V) and (N_{0}, N, W).

Definition 2.1. Let (M, A) be a Lie manifold with anchor map ϱ_{M} : A → T M. A Lie manifold (N, B) is called a Lie submanifold of (M, A) if

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

(ii) ∂N = N ∩ ∂M (that is, N_{0} ⊂ M_{0}, ∂N ⊂ ∂M), and (iii) B is a sub vector bundle of A|_{N}, and (iv) the restriction of ϱ_{M} to B is the anchor map of B → N.

Remark 2.2. An alternative form of Condition (iv) of the above definition is

(13) W = Γ(N; B) = {X|_{N} | X ∈ Γ(M; A) and X|_{N} tangent to N}

= {X ∈ Γ(N; A|_{N}) | ϱ_{M} ∘ X ∈ Γ(N; T N)}.

We have the following simple corollary that justifies Condition (iv) of Definition 2.1.

Corollary 2.3. Let g_{0} be a metric on M_{0} compatible with the Lie structure at infinity on M_{0}. Then the restriction of g_{0} 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_{0}. 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_{0} 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.

2.2. Second fundamental form. We define the A-normal bundle of the Lie submanifold (N, B) of the Lie manifold (M, A) as ν^{A} = (A|_{N})/B which is a bundle over N. Then the anchor map ϱ_{M} defines a map ν^{A} → (T M|_{N})/T N, called the anchor map of ν^{A}, 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} [4]. Let X, Y, Z ∈ W = Γ(N; B) and X̃, Ỹ , Z̃ ∈ V = Γ(M; A) be such that X = X̃|_{N}, Y = Ỹ |_{N}, Z = Z̃|_{N}. Then ∇A ; X̃ Ỹ |_{N} depends only on X, Y ∈ W = Γ(N; B) and will be denoted ∇_{X}^{A}Y in what follows. Furthermore, the Koszul formula gives

2g(Z̃, ∇A ; ỸX̃) =∂_{ϱ_{M} (X̃)}g(Ỹ , Z̃) + ∂_{ϱ_{M} (Ỹ)}g(Z̃, X̃) − ∂_{ϱ_{M} (Z̃)}g(X̃, Ỹ)
− g([X̃, Z̃], Ỹ) − g([Ỹ , Z̃], X̃) − g([X̃, Ỹ], Z̃),
2g(Z, ∇_{Y}^{B}X) = ∂_{ϱ_{M} (X)}g(Y, Z) + ∂_{ϱ_{M} (Y)}g(Z, X) − ∂_{ϱ_{M} (Z)}g(X, Y)
− g([X, Z], Y) − g([Y, Z], X) − g([X, Y], Z).

As this holds for arbitrary sections Z of Γ(N; B) with extensions Z̃ on Γ(M; A), we see that ∇_{X}^{B}Y is the tangential part of ∇_{X}^{A}Y |_{N}.

The normal part of ∇^{A} then gives rise to the second fundamental form II defined as

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

The Levi-Civita connections ∇^{A} and ∇^{B} are torsion free, and hence II is symmetric because

II(X, Y) − II(Y, X) = [X̃, Ỹ]|_{N} − [X, Y] = 0.

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) = II(X, fY), as usual.) Therefore the second fundamental form is a vector bundle morphism II : B ⊗ B → ν^{A}, and the endomorphism at p ∈ M is denoted by II_{p} : B_{p} ⊗ B_{p} → A_{p}. 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 2.4. Let (N, B) be a submanifold of (M, A) with a compatible metric. Then the (classical) second fundamental form of N_{0} in M_{0} is uniformly bounded.

2.3. Tame submanifolds. We now introduce tame manifolds. Our main interest in tame manifolds is the global tubular neighborhood theorem, Theorem 2.7, 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 may be difficult to check directly.

Definition 2.5. Let (N, B) be a Lie submanifold of the Lie manifold (M, A) with anchor map ϱ_{M} : A → T M. Then (N, B) is called a tame submanifold of M if T_{p}N and ϱ_{M}(A_{p}) span T_{p}M for all p ∈ ∂N.

Let (N, B) be a tame submanifold of the Lie manifold (M, A). Then the anchor map ϱ_{M} : A → T M 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_{M} maps B^{⊥}, the orthogonal complement of B in A, injectively into ϱ_{M}(A) ⊂ T M. For any boundary face F and p ∈ F we have ϱ_{M}(A_{p}) ⊂ T_{p}F. Hence, 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 2.6.

(1) Let M be any compact manifold (without boundary). Fix a p ∈ M. Let (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 such that N is transverse to all faces of M. We endow these manifolds with the b-structure at infinity V_{b} (see Example 1.7 (i)). Then (N, V_{b}) is a tame Lie submanifold of (M, V_{b}).

(4) A regular submanifold (see section 1) is a also a tame submanifold.

We now prove the main theorem of this section. Note that this theorem is not true for a general manifold of bounded geometry with boundary (for a manifold with bounded geometry and boundary, the existence of a global tubular neighborhood of the boundary is part of the definition, see [47]).

Theorem 2.7 (Global tubular neighborhood theorem). Let (N, B) be a tame submanifold of the Lie manifold (M, A). For ϵ > 0, let (ν^{A})_{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^{A})_{ϵ} to an open neighborhood V_{ϵ} of N in M. Moreover, dist(exp^{ν}(X), N) = |X| for |X| < ϵ.

Proof. Recall from [4] 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^{A})_{ϵ} → M.

The differential d exp^{ν} at 0_{p} ∈ ν_{p}^{A}, p ∈ N is the restriction of the anchor map to B^{⊥} ≅ ν^{A}, hence any point p ∈ N has a neighborhood U(p) and τ_{p} > 0 such that

(14)  exp^{ν} : (ν^{A})_{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^{A})_{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 theorem holds. Then there are sequences X_{i}, Y_{i} ∈ ν^{A}, 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^{A})_{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 2.8. Let (N, B) be a tame submanifold of codimension one of (M, A). We fix a unit length section X of ν^{A}. Theorem 2.7 states that

exp^{ν} : (ν^{A})_{ϵ} ≅ N × (−ϵ, ϵ) → {x | d(x, N) < ϵ} =: V_{ϵ}
(p, t) ↦ exp(tX(p))

is a diffeomorphism for small ϵ > 0. Then M_{0} carries a compatible metric g_{0} such that (exp^{ν})^{∗}g_{0} is a product metric, i. e., (exp^{ν})^{∗}g_{0} = 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_{0} = (χ ∘ d) g_{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). □

The above definition shows that any tame submanifold of codimension 1 is a regular submanifold. Hence, the concept of a tame submanifold of codimension 1 is the same as that of a regular submanifolds. We hence obtain a new criterion for deciding that a given domain in a Lie manifold is regular.

Proposition 2.9. Assume the same conditions as the previous proposition. Then d exp^{ν} (\frac{∂}{∂t}) defines a smooth vector field on V_{ϵ/2}. This vector field can be extended smoothly to a vector field Y in V. The restriction of A to V_{ϵ/2} splits in the sense of smooth vector bundles as A = A_{1} ⊕ A_{2} where A_{1}|_{N} = ν^{A} and A_{2}|_{N} = B. This splitting is parallel in the direction of Y with respect to the Levi-Civita connection of the product metric g_{0}, i.e. if Z is a section of A_{i}, then ∇_{Y} Z is a section of A_{i} as well.

Proof. Because of the injectivity of the normal exponential map, the vector field Y_{1} := d exp^{ν} (\frac{∂}{∂t}) is well-defined, and the diffeomorphism property implies smoothness on V_{ϵ}. At first, we want to argue that Y_{1} ∈ V(V_{ϵ}). Let π : S(A) → M be the bundle of unit length vectors in A. Recall from [4], section 1.2 that S(A) is naturally a Lie manifold, whose Lie structure is given by the thick pullback π^{#}(A) of A. Now the flow lines of Y_{1} are geodesics, which yield in coordinates solutions to a second order ODE in t. In [4], section 3.4 this ODE was studied on Lie manifolds. The solutions are integral lines of the geodesic spray σ : S(A) → f^{#}(A). As the integral lines of this flow stay in S(A) ⊂ A and as they depend smoothly on the initial data and on t, we see that Y_{1} is a smooth section of constant length one of A|_{V_{ϵ}}.

Multiplying with a suitable cutoff-function with support in V_{ϵ} one sees that we obtain the desired extension Y ∈ V. Using parallel transport in the direction of Y, the splitting A|_{N} = ν^{A} ⊕ T N extends to a small neighborhood of N. This splitting is clearly parallel in the direction of Y. □

Remark 2.10. Let N ⊂ M be a tame submanifold of the Lie manifold (M, V) and Y ∈ V as above. If Y has length one in a neighborhood of N and is orthogonal to N, then V := ⋃_{|t|<ϵ} ϕ_{t}(N) will be a tubular neighborhood of N. According to the previous proposition the restriction of A → M to V has a natural product type decomposition. This justifies, in particular, that the double of a Lie manifold with boundary is again a Lie manifold, and that the Lie structure defined on the double satisfies the natural compatibility conditions with the Lie structure on a Lie manifold with boundary.

3. Sobolev spaces

In this section we study Sobolev spaces on Lie manifolds without boundary. These results will then be used to study Sobolev spaces on Lie manifolds with true boundary, which in turn, will be used to study weighted Sobolev spaces on polyhedral domains. The goal is to extend to these classes of Sobolev spaces the main results on Sobolev spaces on smooth domains.

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. To simplify notation we denote the compatible metric by g instead of the previously used g_{0}. 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}).

3.1. Definition of Sobolev spaces using vector fields and connections. We shall define the Sobolev spaces W^{s,p}(Ω_{0}) in the following two cases:

• s ∈ ℕ ∪ {0}, 1 ≤ p ≤ ∞, and arbitrary open sets Ω_{0} or
• s ∈ ℝ, 1 < p < ∞, and Ω_{0} = M_{0}.

We shall denote W^{s,p}(Ω) = W^{s,p}(Ω_{0}) and W^{s,p}(M) = W^{s,p}(M_{0}). If Ω is a regular open set, then W^{s,p}(\overline{Ω}) = W^{s,p}(Ω_{0}). In the case p = 2, we shall often write H^{s} instead of W^{s,2}. We shall give several definitions for the spaces W^{s,p}(Ω_{0}) and show their equivalence. This will be crucial in establishing the equivalence of various definitions of weighted Sobolev spaces on polyhedral domains. 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 3.1 (∇-definition of Sobolev spaces). The Sobolev space W^{k,p}(Ω_{0}), k ∈ ℕ ∪ {0}, is defined as the space of distributions u on Ω_{0} ⊂ M_{0} such that

(15)  ∥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,

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

We introduce an alternative definition of Sobolev spaces.

Definition 3.2 (vector fields definition of Sobolev spaces). Let again k ∈ ℕ ∪ {0}. 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

(17)  ∥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:

(18)  ∥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,

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

Sometimes, when we want to stress the Lie structure V on M, we shall write W^{k,p}(Ω_{0}; M, V) := W^{k,p}(Ω_{0}).

Example 3.3. Let ℙ be a curvilinear polygonal domain in the plane and let Σ(ℙ)^{db} be the “double” of Σ(ℙ), which is a Lie manifold without boundary (see Subsection 1.6). Then ℙ identifies with a regular open subset of Σ(ℙ)^{db}, and we have

K_{1}^{m}(ℙ) = W^{m,2}(ℙ) = W^{m,2}(ℙ; Σ(ℙ)^{db}, V(ℙ)).

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

Proposition 3.4. 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, H^{k}(Ω_{0}) := 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 [4]. 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(M; 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}} ∈ ℕ ∪ {0}. 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 [16, 60]. □

We shall also need the following simple observation.

Lemma 3.5. Let Ω′ ⊂ Ω ⊂ M be open subsets, Ω_{0} = Ω ∩ M_{0}, and Ω′_{0} = Ω′ ∩ M_{0}, Ω′ ≠ ∅. 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.

3.2. 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 [54, Lemma 1.3], whose definition we now recall. See also [13, 18, 51, 52, 55, 62].

Lemma 3.6. 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}), such that, for some N large enough depending only on the dimension of M_{0}), we have

(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/N) 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} (defined using the exponential map exp_{x_{j}} : T_{x_{j}}M_{0} → M_{0}). The uniform bounds on the Riemann tensor R and its derivatives ∇^{k}R imply uniform bounds on ∇^{k}d exp_{x_{j}}, which directly implies that all derivatives of ψ_{j} are uniformly bounded.

Proposition 3.7. 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

(20)  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 3.6 (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 is similar to the ones in [54, 55, 62, 61] 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 [56, Chapter V] or [60, Section 13.6].

Definition 3.8 (Partition of unity definition of Sobolev spaces). Let s ∈ ℝ, and 1 < p < ∞. Then we define

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

By Proposition 3.7, this norm is equivalent to our previous norm on W^{s,p}(M_{0}) when s is a nonnegative integer.

Proposition 3.9. The space C_{c}^{∞}(M_{0}) is dense in W^{s,p}(M_{0}), for 1 < p < ∞ and s ∈ ℝ, or 1 ≤ p < ∞ and s ∈ ℕ ∪ {0}.

Proof. For s ∈ ℕ ∪ {0}, the result is true for any manifold with bounded geometry, see [7, Theorem 2] or [19, Theorem 2.8], or [20]. 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 (21). (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, s ∈ ℝ. Let k ∈ ℕ ∪ {0} and 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 3.9. That is, let 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 [12] or [58, Chapter 4] for the definition of the complex interpolation spaces.)

The following proposition is an analogue of Proposition 3.7. 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 3.10. 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 isomorphism 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 [60][Equation (6.5), page 23]. In particular, W̃^{s,p}(ℝ^{n}) = W^{s,p}(ℝ^{n}). As in the proof of Proposition 3.7 one shows that the quantity

(22)  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 3.11. Let ϕ ∈ W^{k,∞}(M_{0}), k ∈ ℕ ∪ {0}, p ∈ (1, ∞), and s ∈ ℝ with k ≥ |s|. 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})}. 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 ∈ ℕ ∪ {0}, 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 statement 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 defined to be a diffeomorphism such that ϕ_{∗}(V) = V′. We then have the following invariance property of the Sobolev spaces that we have introduced.

Corollary 3.12. 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 ∈ ℕ ∪ {0}, and p ∈ (1, ∞) if s ∉ ℕ ∪ {0}. Then f → f ∘ ϕ extends to an isomorphism ϕ^{∗} : W^{s,p}(Ω′) → W^{s,p}(Ω) of Banach spaces.

Proof. For s ∈ ℕ∪{0}, this follows right away from definitions and Proposition 3.4. For −s ∈ ℕ ∪ {0}, this follows by duality, Proposition (3.10). For the other values of s, the result follows from the same proposition, by interpolation. □

Recall now that M_{0} is complete [4]. Hence the Laplace operator Δ = ∇^{∗}∇ is essentially self-adjoint on C_{c}^{∞}(M_{0}) by [17, 45]. We shall define then (1+Δ)^{s/2} using the spectral theorem.

Proposition 3.13. 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 ∈ ℕ ∪ {0}, the result is true for any manifold of bounded geometry, by [7, Proposition 3]. For s ∈ ℝ, the result follows from interpolation, because the interpolation spaces are compatible with powers of operators (see, for example, the chapter on Sobolev spaces in Taylor’s book [58]). □

The well known Gagliardo–Nirenberg–Sobolev inequality [7, 16, 19] holds also in our setting.

Proposition 3.14. 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, [7, Theorem 7] or [19, Corollary 3.1.9]. By duality (see Proposition 3.10), 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 3.10. □

The Rellich-Kondrachov’s theorem on the compactness of the embeddings of Proposition 3.14 for 1/p > 1/q − m/n is true if M_{0} is compact [7, 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 this compactness cannot be true for M_{0} non-compact. We will nevertheless obtain compactness in the next section by using Sobolev spaces with weights, see Theorem 4.6.

4. Sobolev spaces on regular open subsets

Let Ω ⊂ M be an open subset. Recall that Ω is a regular open subset in M if, and only if, Ω and \overline{Ω} have the same boundary in M, denoted ∂_{top}\overline{Ω},, and if ∂_{top}\overline{Ω} is a regular submanifold of M. Let Ω_{0} = Ω∩M_{0}. Then ∂Ω_{0} := (∂Ω)∩M_{0} = ∂_{top}\overline{Ω}∩M_{0} is a smooth submanifold of codimension one of M_{0} (see Figure 1). We shall denote W^{s,p}(\overline{Ω}) = W^{s,p}(Ω) = W^{s,p}(Ω_{0}). Throughout this section Ω will denote a regular open subset of M.

We have the following analogue of the classical extension theorem.

Theorem 4.1. Let Ω ⊂ M be a regular open subset. Then 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. We then set η̃_{i} := ( ∑_{j∈I} η_{j}^{2})^{−1/2}η_{i}. Then ∑_{i∈I} η̃_{i}^{2} = 1, η̃_{i} equals 1 on B(x_{i}, ϵ) and is supported in B(x_{i}, 3ϵ).

Following [56, 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 [56, Chapter 6]. For any u ∈ W^{k,p}(Ω_{0}) and i ∈ J_{1} the function (η̃_{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, (η̃_{i}u) ∘ g_{i} ∈ W^{k,p}(ℝ_{+}^{n}). We define the extension Eu by the formula

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

Notice that for all i ∈ J_{2}, the function η̃_{i} vanishes on the support of Λ_{+}, and hence

(24)  ∑_{i∈J_{1}} η̃_{i}^{2}(x) = ∑_{i∈I} η̃_{i}^{2}(x) = 1 in supp Λ_{+}.

This formula implies 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 [56] using (24), the fact that the extension E_{ℝ^{n}} satisfies the same bound, and the definition of the Sobolev spaces using partitions of unity (Proposition 3.7). □

Let Ω be a regular open subset of M and Ω_{0} = Ω ∩ M, as before. We shall denote by \overline{Ω}_{0} the closure of Ω_{0} in M_{0}.

Theorem 4.2. The space C_{c}^{∞}(\overline{Ω}_{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 4.1, Eu ∈ W^{k,p}(M_{0}). By Proposition 3.9, 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 4.3. The restriction map C_{c}^{∞}(\overline{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. The case p = ∞ is obvious. In the case 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}. As the curvature of M_{0} and the second fundamental form of ∂Ω_{0} in M_{0} are bounded (see Corollary 2.4), there is an ϵ_{1} > 0 such that, in normal coordinates, the hypersurface ∂Ω_{0} is the graph of a function on balls of radius ≤ ϵ_{1}.

We use the definitions of the Sobolev spaces using partitions of unity, Proposition 3.7 and Lemma 3.6 with ε = min(ϵ_{1}, ϵ_{0}, r_{inj}(M_{0}))/10. Let B(x_{j}, 2ε) denote the balls in the cover of M_{0} in Lemma 3.6, let ψ_{j} : B(ϵ, x_{j}) → B(ϵ, 0) denote normal coordinates based in x_{j}, and let 1 = ∑_{j} ϕ_{j} be a corresponding partition of unity. Then ϕ̃_{j} = ϕ_{j}|_{∂Ω_{0}} form a partition of unity on ∂Ω_{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 graph 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 [16, 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, T u is well-defined and 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 that 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 4.7).

In the following, ∂_{ν} denotes derivative in the normal direction of the hypersurface ∂Ω_{0} ⊂ M_{0}.

Theorem 4.4. The closure of C_{c}^{∞}(Ω_{0}) in W^{k,p}(Ω_{0}) is the intersection of the kernels of T ∘ ∂_{ν}^{j} : W^{k,p}(Ω_{0}) → W^{k−j−1,p}(Ω_{0}), 0 ≤ j ≤ k − 1, 1 ≤ p < ∞.

Proof. The proof is reduced to the Euclidean case [1, 16, 33, 58] following the same pattern of reasoning as in the previous theorem. □

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

Theorem 4.5. Denote by n the dimension of M and let Ω ⊂ M be a regular open subset 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 3.14 and Theorem 4.1. Indeed, denote by

j : W^{k,q}(M_{0}) → W^{k−m,p}(M_{0})

the continuous inclusion of Proposition 3.14. 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 Ω ⊂ M be a regular open subset. Let a_{H} ∈ ℝ be a parameter associated to each boundary hyperface (i. e., face of codimension one) of the manifold with corners \overline{Ω}. Fix for any boundary hyperface H ⊂ \overline{Ω} 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

(25)  ρ = ∏ ρ_{H}^{a_{H}} ,

the product being taken over all boundary hyperfaces of \overline{Ω}. A function of the form ψρ, with ψ > 0, ψ smooth on \overline{Ω}, and ρ as in Equation (25) will be called an admissible weight of \overline{Ω} (or simply an admissible weight when Ω is understood). We define then the weighted Sobolev space W^{k,p}(Ω_{0}) by

(26)  ρW^{k,p}(Ω_{0}) := {ρu, 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})}.

Note that in the definition of an admissible weight of \overline{Ω}, for a regular open subset Ω ⊂ M of the Lie manifold (M, V), we allow also powers of the defining functions of the boundary hyperfaces contained in ∂Ω = ∂_{top}\overline{Ω}, the true boundary of \overline{Ω}. In the next compactness theorem, however, we shall allow only the powers of the defining functions of M, or, which is the same thing, only powers of the defining functions of the boundary hyperfaces of \overline{Ω} whose union is ∂_{∞}\overline{Ω} (see Figure 1).

Theorem 4.6. Denote by n the dimension of M and let Ω ⊂ M be a regular open subset, Ω_{0} = Ω ∩ M_{0}. Assume that 1/p > 1/q − m/n > 0, 1 ≤ q < ∞, where m ∈ {1, …, k} is an integer, and that s > s′ are real parameters. Then ρ^{s}W^{k,q}(Ω_{0}) is compactly embedded in ρ^{s′}W^{k−m,p}(Ω_{0}) for any admissible weight ρ := ∏_{H} ρ_{H}^{a_{H}} of M such that a_{H} > 0 for any boundary hyperface H of M.

Proof. The same argument as that in the proof of Theorem 4.5 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. 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 nonnegative 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 3.11 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 [7, 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}).

Theorem 4.7. 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 2 for further explanation of this notation.) The existence of tubular neighborhoods, Theorem 2.7, 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 obtain the product metric on M_{0}.

Let S^{1} be the unit circle in the plane. Let us denote by Δ_{M}, Δ_{N}, and Δ_{S^{1}} the Laplace operators on M_{0}, N_{0}, and S^{1}, respectively. Then Δ_{M} = Δ_{N} + Δ_{S^{1}} and Δ_{S^{1}} = −∂^{2}/∂θ^{2} has spectrum {4π^{2}n^{2} | n ∈ ℕ ∪ {0}}. 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} ≃ L^{2}(N_{0}) is obtained by restricting to N_{0} = N_{0} × {1}, 1 ∈ S^{1}. We use this isomorphism to identify the above spaces in what follows.

Let ξ ∈ L^{2}(N_{0} × S^{1}). Then ξ identifies with a sequence (ξ_{n}) under the above isomorphism. By Proposition 3.13, we have that ξ ∈ H^{s}(N_{0} × S^{1}) if, and only if, (1 + Δ_{M})^{s/2}ξ = ∑_{n} ((1 + n^{2} + Δ_{N})^{s/2}ξ_{n}) ∈ ⨁_{n∈ℤ} L^{2}(N_{0}) ≃ L^{2}(N_{0} × S^{1}). The restriction of ξ to N_{0} is then given by ∑_{n} ξ_{n}. We want to show that ∑_{n} ξ_{n} ∈ H^{s−1/2}(N_{0}), which is equivalent to (1 + Δ_{N})^{s/2−1/4}( ∑ ξ_{n}) ∈ L^{2}(N_{0}).

The spectral spaces of Δ_{N} corresponding to [m, m + 1) ⊂ ℝ, m ∈ ℕ ∪ {0} give an orthogonal direct sum decomposition of L^{2}(N_{0}).

We decompose ξ_{n} = ∑_{m} ξ_{mn}, with ξ_{mn} in the spectral space corresponding to [m, m + 1) of Δ_{N}. Note that ξ_{mn} is orthogonal to ξ_{m′n} for m ≠ m′. Denote h = (1 + m^{2})^{−1/2}, f(t) = (1 + t^{2})^{−s}, and C = 1 + ∫_{ℝ} f(t)dt. Then an application of the Cauchy–Schwartz inequality gives

(27) (1 + m^{2})^{s−1/2}(∑_{n} ∥ξ_{mn}∥)^{2}

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

≤ h(∑_{n} f(nh)) ∑_{n} ∥(1 + n^{2} + m^{2})^{s/2}ξ_{mn}∥^{2} ≤ C_{s} ∑_{n} ∥(1 + n^{2} + m^{2})^{s/2}ξ_{mn}∥^{2}.

The constant C_{s} is independent of m (but depends on s). We sum over m and obtain

(28) ∥ ∑_{n}(1 + Δ_{N})^{s/2−1/4}ξ_{n}∥^{2} = ∑_{m} ∥ ∑_{n}(1 + Δ_{N})^{s/2−1/4}ξ_{nm}∥^{2}

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

≤ 2^{s}C_{s} ∑_{n,m} ∥(1 + n^{2} + m^{2})^{s/2}ξ_{nm}∥^{2} ≤ 2^{s}C_{s} ∑_{n,m} ∥(1 + n^{2} + Δ_{N})^{s/2}ξ_{nm}∥^{2}

= 2^{s}C_{s} ∑_{n} ∥(1 + n^{2} + Δ_{N})^{s/2}ξ_{n}∥^{2},

with the same constant C_{s} as in Equation (27). This shows that ζ := ∑_{n} ξ_{n} ∈ H^{s−1/2}(N_{0}) if ξ = (ξ_{n}) ∈ ⨁_{n} L^{2}(N_{0}) ≃ L^{2}(N_{0} × S^{1}) is a finite sequence such that ∥ξ∥_{H^{s}} := ∑_{n} ∥(1 + n^{2} + Δ_{N})^{s/2}ξ_{n}∥_{L^{2}(N_{0})}^{2} < ∞, and that ζ depends continuously on ξ ∈ H^{s}(N_{0} × S^{1}). This completes the proof. □

We finally obtain the following consequences for a curvilinear polygonal domain ℙ (see Subsection 1.6). First, recall that the distance ϑ(x) from x to the vertices of a curvilinear polygon ℙ and r_{ℙ} have bounded quotients, and hence define the same weighted Sobolev spaces (Equation (12)). Moreover, the function r_{ℙ} is an admissible weight. Recall that ℙ has a compactification Σ(ℙ) that is a Lie manifold with boundary (that is, the closure of a regular open subset of a Lie manifold M). Let us write W^{m,p}(Σ(ℙ)) := W^{m,p}(ℙ) the Sobolev spaces defined by the structural Lie algebra of vector fields on Σ(ℙ). Then

(29)  K_{a}^{m}(ℙ; ϑ) = r_{Ω}^{a−1}K_{1}^{m}(ℙ; r_{ℙ}) = r_{ℙ}^{a−1}W^{m,2}(Σ(ℙ)).

This identifies the weighted Sobolev spaces on ℙ with a weighted Sobolev space of the form ρW^{k,p}(Ω_{0}).

Motivated by Equation (29), we now define

(30)  K_{a}^{m}(∂ℙ) = K_{a}^{m}(∂ℙ; ϑ) = K_{a}^{m}(∂ℙ; r_{ℙ}) = r_{ℙ}^{a−1/2}W^{m,2}(∂ℙ).

More precisely, let us notice that we can identify each edge with [0, 1]. Then K_{a}^{m}(∂ℙ) consists of the functions f : ∂ℙ → ℂ that, on each edge, are such that t^{k}(1 − t)^{k}f^{(k)} ∈ L^{2}([0, 1]), 0 ≤ k ≤ m (here we identify that edge with [0, 1]). This last condition is equivalent to [t(1 − t)∂_{t}]^{k}f ∈ L^{2}([0, 1]), 0 ≤ k ≤ m.

Proposition 4.8. Let ℙ ⊂ ℝ^{2} be a curvilinear polygonal domain and P be a differential operator of order m with coefficients in C^{∞}(\overline{ℙ}). Then P_{λ} := r_{ℙ}^{λ}P r_{ℙ}^{−λ} defines a continuous family of bounded maps P_{λ} : K_{a}^{s}(ℙ) → K_{a−m}^{s−m}(ℙ), for any s, a ∈ ℝ. Let ℙ′ be \overline{ℙ} with the vertices removed. Then C_{c}^{∞}(ℙ′) is dense in K_{a}^{m}(ℙ).

Also, the restriction to the boundary extends to a continuous, surjective trace map K_{a}^{s}(ℙ) → K_{a−1/2}^{s−1/2}(∂ℙ). If s = 1, then the kernel of the trace map is the closure of C_{c}^{∞}(ℙ) in K_{a}^{1}(ℙ).

The above proposition, except maybe for the description of the restrictions to the boundary, is well known in two dimensions. It will serve as a model for the results in three dimensions that we present in the last section.

5. A regularity result

We include in this section an application to the regularity of boundary value problems, Theorem 5.1. Its proof is reduced to the Euclidean case using a partition of unity argument and the tubular neighborhood theorem 2.7, both of which require some non-trivial input from differential geometry.

Let us introduce some notation first that will be also useful in the following. 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 admissible weight function ρ defined in Equation (25) and the weighted Sobolev spaces ρ^{s}W^{k,p}(Ω_{0}) := {ρ^{s}u, u ∈ W^{k,p}(Ω_{0})} introduced in Equation 26.

Recall [58], Chapter 5, Equation (11.79), that a differential operator P of order m is called strongly elliptic if there exists C > 0 such that Re(σ^{(m)}(P)(ξ)) ≥ C∥ξ∥^{m} for all ξ.

Theorem 5.1. Let Ω ⊂ M be a regular open subset of the Lie manifold (M, V). Let P ∈ Diff_{V}^{2}(M) be an order 2 strongly elliptic operator on M_{0} generated by V and s ∈ ℝ, t ∈ ℤ, 1 < p < ∞. Then there exists C > 0 such that, for any u ∈ ρ^{s}W^{1,p}(Ω_{0}), u|_{∂Ω_{0}} = 0, we have

∥u∥_{ρ^{s}W^{t+2,p}(Ω_{0})} ≤ C(∥P u∥_{ρ^{s}W^{t,p}(Ω_{0})} + ∥u∥_{ρ^{s}L^{p}(Ω_{0})}).

In particular, let u ∈ ρ^{s}W^{1,p}(Ω_{0}) be such that P u ∈ ρ^{s}W^{t,p}(Ω_{0}), 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

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

for any u ∈ W_{loc}^{t+2,p}(Ω_{0}). Here, of course, ∥u∥_{ρ^{s}L^{p}(M_{0})} = ∥ρ^{−s}u∥_{L^{p}(M_{0})} (see Equation (26)).

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 ≤ −2, so we will assume t ≥ −1 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 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

(32)  ∥u∥_{W^{t+2,p}(T_{x}M_{0})}^{p} ≤ C(∥P_{x}u∥_{W^{t,p}(T_{x}M_{0})}^{p} + ∥u∥_{L^{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}. (For p = 2, a complete proof can be found in [58], Propositions 11.10 and 11.16. For general p, the result can be proved as [16], Theorem 1 in subsection 5.8.1, page 275.) 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 1.5 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 upper semi-continuous (i. e., the set {C_{x} < η} is open for any x). Since M is compact, C_{x} will attain its maximum, which therefore must be positive. Let C be that maximum value.

Let now ϕ_{j} be the partition of unity and ψ_{j} be the diffeomorphisms appearing in Equation (22), for some 0 < ϵ < r/6. In particular, the partition of unity ϕ_{j} satisfies the conditions of Lemma 3.6, 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∥_{L^{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∥_{L^{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∥_{L^{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 3.7 and 3.10) 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 ≥ −1. This is known to imply

(33)  ∥u∥_{W^{t+2,p}(M_{0})} ≤ C(∥P u∥_{W^{t,p}(M_{0})} + ∥u∥_{L^{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 of arbitrary domains Ω_{0} follows in exactly the same way, but using a product type metric in a neighborhood of ∂_{top}Ω_{0} and the analogue of Equation (32) for a half-space, which shows that Equation (31) continues to hold for M_{0} replaced with Ω_{0}.

The case of arbitrary s ∈ ℝ is obtained by applying Equation (33) 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 (31) right away. □

For p = 2, by combining the above theorem with Theorem 4.7, we obtain the following corollary.

Corollary 5.2. We keep the assumptions of Theorem 5.1. 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

(34) ∥u∥_{ρ^{s}H^{t+2}(Ω_{0})} ≤ C(∥P u∥_{ρ^{s}H^{t}(Ω_{0})} + ∥u∥_{ρ^{s}L^{2}(Ω_{0})} + ∥u|_{∂Ω_{0}}∥_{ρ^{s}H^{t+3/2}(Ω_{0})}).

Proof. For u|_{∂Ω_{0}} = 0, the result follows from Theorem 5.1. In general, choose a suitable v ∈ H^{t+2}(Ω_{0}) such that v|_{∂Ω_{0}} = u|_{∂Ω_{0}}, which is possible by Theorem 4.7. Then we use our result for u − v. □

6. Polyhedral domains in three dimensions

We now include an application of our results to polyhedral domains ℙ ⊂ ℝ^{3}. A polyhedral domain in ℙ ⊂ ℝ^{3} is a bounded, connected open set such that ∂ℙ = ∂\overline{ℙ} = ⋃ \overline{D}_{j} where

• each D_{j} is a polygonal domain with straight edges contained in an affine 2-dimensional subspace of ℝ^{3}

• each edge is contained in exactly two closures of polygonal domains \overline{D}_{j}. (See Subsection 1.6 for the definition of a polygonal domain.)

The vertices of the polygonal domains D_{j} will form the vertices of ℙ. The edges of the polygonal domains D_{j} will form the edges of ℙ. For each vertex P of ℙ, we choose a small open ball V_{P} centered in P. We assume that the neighborhoods V_{P} are chosen to be disjoint. For each vertex P, there exists a unique closed polyhedral cone C_{P} with vertex at P, such that \overline{ℙ} ∩ V_{P} = C_{P} ∩ V_{P}. Then ℙ ⊂ ⋃ C_{P}.

We now proceed to define canonical weight functions of ℙ in analogy with the definition of canonical weights of curvilinear polygonal domains, Definition 1.10. We want to define first a continuous function r_{ℙ} : \overline{Ω} → [0, ∞) that is positive and differentiable outside the edges. Let ϑ(x) be the distance from x to the edges of ℙ, as before. We want r_{ℙ}(x) = ϑ(x) close to the edges but far from the vertices and we want the quotients r_{ℙ}(x)/ϑ(x) and ϑ(x)/r_{ℙ}(x) to extend to continuous functions on \overline{Ω}. Using a smooth partition of unity, in order to define r_{ℙ}, we need to define it close to the vertices.

Let us then denote by {P_{k}} the set of vertices of ℙ. Choose a continuous function r : \overline{ℙ} → [0, ∞) such that r(x) is the distance from x to the vertex P if x ∈ V_{P} ∩ ℙ, and such that r(x) is differentiable and positive on \overline{ℙ} ∖ \{P_{k}\}. Let S^{2} be the unit sphere centered at P and let r_{P} be a canonical weight associated to the curvilinear polygon C_{P} ∩S^{2} (see Definition 1.10). We extend this function to C_{P} to be constant along the rays, except at P, where r_{P} (P) = 0. Finally, we let r_{ℙ}(x) = r(x)r_{P} (x), for x close to P. Then a canonical weight of ℙ is any function of the form ψr_{ℙ}, where ψ is a smooth, nowhere vanishing function on \overline{ℙ}.

For any canonical weight r_{ℙ}, we then we have the following analogue of Equation (12)

(35)  K_{a}^{m}(ℙ) := K_{a}^{m}(ℙ; ϑ) = K_{a}^{m}(ℙ; r_{ℙ}).

Let us define, for every vertex P of ℙ, a spherical coordinate map Θ_{P} : ℙ∖{P } → S^{2} by Θ_{P} (x) = |x − P |^{−1}(x − P). Then, for each edge e = [AB] of ℙ joining the vertices A and B, we define a generalized cylindrical coordinate system (r_{e}, θ_{e}, z_{e}) to satisfy the following properties:

(i) r_{e}(x) be the distance from x to the line containing e. (ii) A as the origin (i. e., r_{e}(A) = z_{e}(A) = 0), (iii) θ_{e} = 0 on one of the two faces containing e, and (iv) z_{e} ≥ 0 on the edge e. Let ψ : S^{2} → [0, 1] be a smooth function on the unit sphere that is equal to 1 in a neighborhood of (0, 0, 1) = {ϕ = 0} ∩ S^{2} and is equal to 0 in a neighborhood of (0, 0, −1) = {ϕ = π} ∩ S^{2}. Then we let

θ̃_{e}(x) = θ_{e}(x)ψ(Θ_{A}(x))ψ(−Θ_{B}(x))

where θ_{e}(x) is the θ coordinate of x in a cylindrical coordinate system (r, θ, z) in which the point A corresponds to the origin (i. e., r = 0 and z = 0) and the edge AB points in the positive direction of the z axis (i. e., B corresponds to r = 0 and z > 0). By choosing ψ to have support small enough in S^{2} we may assume that the function θ̃_{e} is defined everywhere on ℙ ∖ e. (This is why we need the cut-off function ψ.)

We then consider the function

Φ : ℙ → ℝ^{N}, Φ(x) = (x, Θ_{P} (x), r_{e}(x), θ̃_{e}(x)),

with N = 3 + 3n_{v} + 2n_{e}, n_{v} being the number of vertices of ℙ and n_{e} being the number of edges of ℙ. Finally, we define Σ(ℙ) to be the closure of Φ(ℙ) in ℝ^{N}. Then Σ(ℙ) is a manifold with corners that can be endowed with the structure of a Lie manifold with true boundary as follows. (Recall that a Lie manifold with boundary Σ is the closure \overline{Ω} of a regular open subset Ω in a Lie manifold M and the true boundary of Σ is the topological boundary ∂_{top}Ω.) The true boundary ∂_{top}Σ(Ω) of Σ(Ω) is defined as the union of the closures of the faces D_{j} of ℙ in Σ(ℙ). (Note that the closures of D_{j} in Σ(ℙ) are disjoint.) We can then take M to be the union of two copies of Σ(ℙ) with the true boundaries identified (i. e., the double of Σ(ℙ)) and Ω = Σ(ℙ) ∖ ∂_{top}Σ(ℙ). In particular, Ω_{0} := Ω ∩ M_{0} identifies with ℙ.

To complete the definition of the Lie manifold with true boundary on Σ(ℙ), we now define the structural Lie algebra of vector fields V(ℙ) of Σ(ℙ) by

(36)  V(ℙ) := {r_{ℙ}(ϕ_{1}∂_{1} + ϕ_{2}∂_{2} + ϕ_{3}∂_{3}), ϕ_{j} ∈ C^{∞}(Σ(ℙ))}.

(Here ∂_{j} are the standard unit vector fields. Also, the vector fields in V(ℙ) are determined by their restrictions to ℙ.) This is consistent with the fact that ∂_{top}Σ(ℙ), the true boundary of Σ(ℙ), is defined as the union of the boundary hyperfaces of Σ(ℙ) to which not all vector fields are tangent. This completes the definition of the structure of Lie manifold with boundary on Σ(ℙ).

The function r_{ℙ} is easily seen to be an admissible weight on Σ(ℙ). It hence satisfies

r_{ℙ}(∂_{j}r_{ℙ}) = r_{ℙ} \frac{∂r_{ℙ}}{∂x_{j}} ∈ C^{∞}(Σ(ℙ)),

which is equivalent to the fact that V(ℙ) is a Lie algebra. This is the analogue of Equation (11).

To check that Σ(ℙ) is a Lie manifold, let us notice first that g = r_{ℙ}^{−2}g_{E} is a compatible metric on Σ(ℙ), where g_{E} is the Euclidean metric on ℙ. Then, let us denote by ν the outer unit normal to ℙ (where it is defined), then r_{ℙ}∂_{ν} is the restriction to ∂_{top}Σ(Ω) of a vector field in V(ℙ). Moreover r_{ℙ}∂_{ν} is of length one and orthogonal to the true boundary in the compatible metric g = r_{ℙ}^{−2}g_{E}.

The definition of V(ℙ) together with our definition of Sobolev spaces on Lie manifolds using vector fields shows that

(37)  K_{a}^{m}(ℙ) = r_{ℙ}^{a−3/2}W^{m,2}(Σ(ℙ)) = r_{ℙ}^{a−3/2}H^{m}(Σ(ℙ)).

The induced Lie manifold structure on Σ(ℙ) consists of the vector fields on the faces D_{j} that vanish on the boundary of D_{j}. The Soblev spaces on the boundary are

(38)  K_{a}^{m}(∂ℙ) = r_{ℙ}^{a−1}W^{m,2}(∂_{top}Σ(ℙ)) = r_{ℙ}^{a−1}H^{m}(∂_{top}Σ(ℙ)).

The factors −3/2 and −1 in the powers of r_{ℙ} appearing in the above two equations are due to the fact that the volume elements on ℙ and Σ(ℙ) differ by these factors.

If P is an order m differential operator with smooth coefficients on ℝ^{3} and ℙ ⊂ ℝ^{3} is a polyhedral domain, then r_{ℙ}^{m}P ∈ Diff_{V}^{m}(Σ(ℙ)), by Equation (10). However, in general, r_{ℙ}^{m}P will not define a smooth differential operator on \overline{ℙ}.

In particular, we have the following theorem, which is a direct analog of Proposition 4.8, if we replace “vertices” with “edges:”

Theorem 6.1. Let ℙ ⊂ ℝ^{3} be a polyhedral domain and P be a differential operator of order m with coefficients in C^{∞}(\overline{ℙ}). Then P_{λ} := r_{ℙ}^{λ}P r_{ℙ}^{−λ} defines a continuous family of bounded maps P_{λ} : K_{a}^{s}(ℙ) → K_{a−m}^{s−m}(ℙ), for any s, a ∈ ℝ. Let ℙ′ be \overline{ℙ} with the edges removed. Then C_{c}^{∞}(ℙ′) is dense in K_{a}^{m}(ℙ). Also, the restriction to the boundary extends to a continuous, surjective trace map K_{a}^{s}(ℙ) → K_{a−1/2}^{s−1/2}(∂ℙ). If s = 1, then the kernel of the trace map is the closure of C_{c}^{∞}(ℙ) in K_{a}^{1}(ℙ).

See [11] for applications of these results, especially of the above theorem.

Theorem 5.1 and the results of this section immediately lead to the proof of Theorem 0.1 formulated in the Introduction.

7. A non-standard boundary value problem

We present in this section a non-standard boundary value problem on a smooth manifold with boundary. Let O be a smooth manifold with boundary. We shall assume that O is connected and that the boundary is not empty.

Let r : \overline{O} → [0, ∞) be a smooth function that close to the boundary is equal to the distance to the boundary and is > 0 on O. Then we recall [14] that there exists a constant depending only on O such that

(39)  ∫_{O} r^{−2}|u(x)|^{2}dx ≤ C ∫_{O} |∇u(x)|^{2}dx

for any u ∈ H^{1}(O) that vanishes at the boundary. If we denote, as in Equation (2),

K_{a}^{m}(O; r) := {u ∈ L_{loc}^{2}(O), r^{|α|−a}∂^{α}u ∈ L^{2}(O), |α| ≤ m}, m ∈ ℕ ∪ {0}, a ∈ ℝ,

with norm ∥ · ∥_{K_{a}^{m}}, the Equation (39) implies that ∥u∥_{K_{1}^{1}} ≤ C∥∇u∥_{L^{2}}. Let M = \overline{O} with the structural Lie algebra of vector fields

V = V_{0} := {X, X = 0 at ∂O} = rΓ(M; T M),

(see Example 1.7). Recall from Subsection 1.4 that Diff_{V}^{m}(M) is the space of order m differential operators on M generated by multiplication with functions in C^{∞}(M) and by differentiation with vector fields X ∈ V. It follows that

(40)  r^{m}P ∈ Diff_{V}^{m}(M)

for any differential operator P of order m with smooth coefficients on M.

Lemma 7.1. The pair (M, V) is a Lie manifold with M_{0} = O satisfying

(41)  K_{a}^{m}(O; r) = r^{a−n/2}H^{m}(M).

If P is a differential operator with smooth coefficients on M, then r^{m}P is a differential operator generated by V, and hence P_{λ} := r^{λ}P r^{−λ} gives rise to a continuous family of bounded maps P_{λ} : K_{a}^{s}(O; r) → K_{a−m}^{s−m}(O; r).

Because of the above lemma, it makes sense to define K_{a}^{s}(O; r) = r^{a−n/2}H^{s}(M), for all s, a ∈ ℝ, with norm denoted ∥ · ∥_{K_{a}^{s}}. The regularity result (Theorem 5.1) then gives

Lemma 7.2. Let P be an order m elliptic differential operator with smooth coefficients defined in a neighborhood of M = \overline{O}. Then, for any s, t ∈ ℝ, there exists C = C_{st} > 0 such that

∥u∥_{K_{a}^{s}} ≤ C(∥P u∥_{K_{a−m}^{s−m}} + ∥u∥_{K_{a}^{t}}).

In particular, let u ∈ K_{a}^{t}(O; r) be such that P u ∈ K_{a−m}^{s−m}(O; r), then u ∈ K_{a}^{s}(O; r). The same result holds for elliptic systems.

Proof. We first notice that r^{m}P ∈ Diff_{V}^{m}(M) is an elliptic operator in the usual sense (that is, its principal symbol σ^{(m)}(r^{m}P) does not vanish outside the zero section of A^{∗}). For this we use that σ^{(m)}(r^{m}P) = r^{m}σ^{(m)}(P) and that A^{∗} is defined such that multiplication by r^{m} defines an isomorphism C^{∞}(T^{∗}M) → C^{∞}(A^{∗}) that maps order m elliptic symbols to elliptic symbols. Then the proof is exactly the same as that of Theorem 5.1, except that we do not need strong ellipticity, because we do not have boundary conditions (and hence we have no condition of the form u = 0 on the boundary). □

An alternative proof of our lemma is obtained using pseudodifferential operators generated by V [3] and their L^{p}–continuity.

Theorem 7.3. There exists η > 0 such that Δ : K_{s+1}^{a+1}(O; r) → K_{s−1}^{a−1}(O; r) is an isomorphism for all s ∈ ℝ and all |a| < η.

Proof. The proof is similar to that of Theorem 2.1 in [10], so we will be brief. Consider

B : K_{1}^{1}(O; r) × K_{1}^{1}(O; r) → ℂ, B(u, v) = ∫_{O} ∇u · ∇\overline{v}dx.

Then |B(u, v)| ≤ ∥u∥_{K_{1}^{1}}∥v∥_{K_{1}^{1}}, so B is continuous.

On the other hand, by Equation (39), B(u, u) ≥ θ∥u∥_{K_{1}^{1}}^{2}, for all u with compact support on O and for some θ > 0 independent of u. Since C_{c}^{∞}(O) is dense in K_{1}^{1}(O; r), by Theorem 4.2, the Lax-Milgram Lemma can be used to conclude that

Δ : K_{1}^{1}(O; r) → K_{−1}^{−1}(O; r) := K_{1}^{1}(O; r)^{∗}

is an isomorphism. Since multiplication by r^{a} : K_{1}^{1}(O; r) → K_{a+1}^{1}(O; r) is an isomorphism and the family r^{a}Δr^{−a} depends continuously on a by Lemma 7.1, we obtain that Δ : K_{a+1}^{1}(O; r) → K_{a−1}^{−1}(O; r) is an isomorphism for |a| < η, for some η > 0 small enough.

Fix now a, |a| < η. We obtain that Δ : K_{a+1}^{s+1}(O; r) → K_{a−1}^{s−1}(O; r) is a continuous, injective map, for all s ≥ 0. The first part of the proof (for a = 0) together with the regularity result of Lemma 7.2 show that this map is also surjective. The Open Mapping Theorem therefore completes the proof for s ≥ 0. For s ≤ 0, the result follows by considering duals. □

It can be shown as in [10] that η is the least value for which Δ : K_{η+1}^{1}(O; r) → K_{η−1}^{−1}(O; r) is not Fredholm. This, in principle, can be decided by using the Fredholm conditions in [43] that involve looking at the L^{2} invertibility of the same differential operators when M is the half-space {x_{n+1} ≥ 0}. See also [5] for some non-standard boundary value problems on exterior domains in weighted Sobolev spaces.

8. 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}.

8.1. 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 [4]. Also, recall that A → M is a vector bundle such that V = Γ(A).

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

(42)  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 (42) 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 [21, 42, 57, 59]. See [3] 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 [3], 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} [22]. 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}) [3].

We now enlarge the class of order −∞ operators that we consider. Any X ∈ V = 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

(43)  ψ_{X} := Ψ_{X}(1, ·) : M → M.

We now define the pseudodifferential calculus on M_{0} that we will consider following [3]. See [28, 29, 41, 44] for the connections between this calculus and groupoids.

Definition 8.1. 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).

8.2. 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 [3]. 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 [22]. 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

(44)  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 8.2. The spaces Ψ_{1,0,V}^{∞}(M_{0}) and Ψ_{cl,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_{0}) ⊆ 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 8.3. The principal symbol establishes isomorphisms

(45)  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

(46)  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 8.4. Let ρ 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.

8.3. 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 5.1, which the reader may find convenient to review first. Let us recall from Equation (25) that an admissible weight ρ of M is a function of the form ρ := ∏_{H} ρ_{H}^{a_{H}}, where a_{H} ∈ ℝ and ρ_{H} is a defining function of H.

Theorem 8.5. Let ρ be an admissible weight of M and 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 8.4 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

(47)  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} [60, XIII, §5] or [56].

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 3.6. 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 [13, 54].

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 3.10 and Equation (21). It is enough to prove that

(48) 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 κ such that at any given point x, at most κ 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 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. □

We obtain the following standard description of Sobolev spaces.

Theorem 8.6. 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 map T : L^{p}(M_{0}) ⊕ L^{p}(M_{0}) ∋ (u, v) ↦ u + P v ∈ W^{−s,p}(M_{0}) is surjective and identifies W^{−s,p}(M_{0}) with the quotient (L^{p}(M_{0}) ⊕ L^{p}(M_{0}))/ ker(T).

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 also the Hahn-Banach theorem. □

We conclude our paper with the sketch of a regularity results for solutions of elliptic equations. Recall the Sobolev spaces with weights ρ^{s}W^{s,p}(Ω_{0}) introduced in Equation (26).

Theorem 8.7. 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 8.5, 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 5.1. The proof here is much shorter, however, it attests to the power of pseudodifferential operator algebra techniques.

References

[1] R. A. Adams. Sobolev spaces. Pure and Applied Mathematics, Vol. 65. Academic Press, New York-London, 1975.

[2] K. Akutagawa and B. Botvinnik. Yamabe metrics on cylindrical manifolds. Geom. Funct. Anal., 13(2):259–333, 2003.

[3] B. Ammann, R. Lauter, and V. Nistor. Pseudodifferential operators on manifolds with a Lie structure at infinity. To appear in Annals of Math.

[4] B. Ammann, R. Lauter, and V. Nistor. On the geometry of Riemannian manifolds with a Lie structure at infinity. Int. J. Math. Math. Sci., (1-4):161–193, 2004.

[5] C. Amrouche, V. Girault, and J. Giroire. Dirichlet and Neumann exterior problems for the n-dimensional Laplace operator: an approach in weighted Sobolev spaces. J. Math. Pures Appl. (9), 76(1):55–81, 1997.

[6] L. Andersson and M. Dahl. Scalar curvature rigidity for asymptotically locally hyperbolic manifolds. Ann. Global Anal. Geom., 16(1):1–27, 1998.

[7] T. Aubin. Espaces de Sobolev sur les variétés riemanniennes. Bull. Sc. math., 100:149–173, 1970.

[8] I. Babuška and A. K. Aziz. Survey lectures on the mathematical foundations of the finite element method. In The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972), pages 1–359. Academic Press, New York, 1972. With the collaboration of G. Fix and R. B. Kellogg.

[9] C. Bacuta, V. Nistor, and L. Zikatanov. Boundary value problems and regularity on polyhedral domains. IMA preprint #1984, August 2004.

[10] C. Bacuta, V. Nistor, and L. Zikatanov. Improving the rate of convergence of ‘high order finite elements’ on polygons and domains with cusps. Numerische Mathematik, 100:165–184, 2005.

[11] C. Bacuta, V. Nistor, and L. Zikatanov. Improving the rate of convergence of high-order finite elements on polyhedra. I. A priori estimates. Numer. Funct. Anal. Optim., 26(6):613–639, 2005.

[12] J. Bergh and J. Löfström. Interpolation spaces. An introduction. Springer-Verlag, Berlin, 1976. Grundlehren der Mathematischen Wissenschaften, No. 223.

[13] M. Cheeger, J. Gromov and M. Taylor. Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds. J. Differential Geom., 17(1):15–53, 1982.

[14] H. Donnelly. Spectrum of domains in Riemannian manifolds. Illinois J. Math., 31(4):692–698, 1987.

[15] A. K. Erkip and E. Schrohe. Normal solvability of elliptic boundary value problems on asymptotically flat manifolds. J. Funct. Anal., 109:22–51, 1992.

[16] L. C. Evans. Partial differential equations, volume 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1998.

[17] M. P. Gaffney. The harmonic operator for exterior differential forms. Proc. Nat. Acad. Sci. U. S. A., 37:48–50, 1951.

[18] G. Grubb. Functional calculus of pseudodifferential boundary problems, volume 65 of Progress in Mathematics. Birkhäuser Boston Inc., Boston, MA, second edition, 1996.

[19] E. Hebey. Sobolev spaces on Riemannian manifolds, volume 1635 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1996.

[20] E. Hebey. Nonlinear analysis on manifolds: Sobolev spaces and inequalities, volume 5 of Courant Lecture Notes in Mathematics. New York University Courant Institute of Mathematical Sciences, New York, 1999.

[21] L. Hörmander. Pseudo-differential operators and hypoelliptic equations, volume X of Proc. Symp. in Pure Math. – Singular Integrals, pages 138–183. Amer. Math. Soc., Providence, Rhode Island, 1967.

[22] L. Hörmander. The analysis of linear partial differential operators, vol. 3. Pseudo-differential operators, volume 274 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin - Heidelberg - New York, 1985.

[23] D. Jerison and C. E. Kenig. The inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal., 130(1):161–219, 1995.

[24] M. Karoubi. Homologie cyclique et K-theorie. Asterisque, 149:1–147, 1987.

[25] V. A. Kondrat′ev. Boundary value problems for elliptic equations in domains with conical or angular points. Transl. Moscow Math. Soc., 16:227–313, 1967.

[26] V. A. Kozlov, V. G. Maz′ya, and J. Rossmann. Elliptic boundary value problems in domains with point singularities, volume 52 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1997.

[27] R. Lauter. Pseudodifferential analysis on conformally compact spaces. Mem. Amer. Math. Soc., 163(777), 2003.

[28] R. Lauter, B. Monthubert, and V. Nistor. Pseudodifferential analysis on continuous family groupoids. Doc. Math., 5:625–655 (electronic), 2000.

[29] R. Lauter, B. Monthubert, and V. Nistor. Spectral invariance for certain algebras of pseudodifferential operators. Preprint Nr. 1/02, Fachbereich Mathematik, Johannes Gutenberg-Universität Mainz, January 2002.

[30] R. Lauter and S. Moroianu. Homology of pseudodifferential operators on manifolds with fibered cusps. Trans. Amer. Math. Soc., 355(8):3009–3046 (electronic), 2003.

[31] R. Lauter and V. Nistor. Analysis of geometric operators on open manifolds: a groupoid approach. In N. P. Landsman, M. Pflaum, and M. Schlichenmaier, editors, Quantization of Singular Symplectic Quotients, volume 198 of Progress in Mathematics, pages 181–229. Birkhäuser, Basel - Boston - Berlin, 2001.

[32] J. M. Lee and T. H. Parker. The Yamabe problem. Bull. Am. Math. Soc., New Ser., 17:37–91, 1987.

[33] J.-L. Lions and E. Magenes. Non-homogeneous boundary value problems and applications. Vol. I. Springer-Verlag, New York, 1972. Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181.

[34] L. Maniccia and P. Panarese. Eigenvalue asymptotics for a class of md-elliptic ψdo’s on manifolds with cylindrical exits. Ann. Mat. Pura Appl. (4), 181(3):283–308, 2002.

[35] V. Maz’ya and J. Rossmann. Weighted L_{p} estimates of solutions to boundary value problems for second order elliptic systems in polyhedral domains. ZAMM Z. Angew. Math. Mech., 83(7):435–467, 2003.

[36] R. Mazzeo. Elliptic theory of differential edge operators. I. Commun. Partial Differ. Equations, 16(10):1615–1664, 1991.

[37] R. B. Melrose. Pseudodifferential operators, corners and singular limits. In Proceeding of the International Congress of Mathematicians, Kyoto, pages 217–234, Berlin - Heidelberg - New York, 1990. Springer-Verlag.

[38] R. B. Melrose. The Atiyah-Patodi-Singer index theorem. Research Notes in Mathematics (Boston, Mass.). 4. Wellesley, MA: A. K. Peters, Ltd., 1993.

[39] R. B. Melrose. Geometric scattering theory. Stanford Lectures. Cambridge University Press, Cambridge, 1995.

[40] M. Mitrea and M. Taylor. Boundary layer methods for Lipschitz domains in Riemannian manifolds. J. Funct. Anal., 163(2):181–251, 1999.

[41] B. Monthubert. Pseudodifferential calculus on manifolds with corners and groupoids. Proc. Amer. Math. Soc., 127(10):2871–2881, 1999.

[42] A. Nagel and E. Stein. Lectures on pseudodifferential operators: regularity theorems and applications to nonelliptic problems, volume 24 of Mathematical Notes. Princeton University Press, Princeton, N.J., 1979.

[43] V. Nistor. Pseudodifferential opearators on non-compact manifolds and analysis on polyhedral domains. In Workshop on Spectral Geometry of Manifolds with Boundary and Decomposition of Manifolds, Roskilde University, pages 307–328, 2005.

[44] V. Nistor, A. Weinstein, and P. Xu. Pseudodifferential operators on groupoids. Pacific J. Math., 189:117–152, 1999.

[45] W. Roelcke. Über den Laplace-Operator auf Riemannschen Mannigfaltigkeiten mit diskontinuierlichen Gruppen. Math. Nachr., 21:131–149, 1960.

[46] J. Rossmann. On two classes of weighted Sobolev-Slobodetskiĭ spaces in a dihedral angle. In Partial differential equations, Part 1, 2 (Warsaw, 1990), volume 2 of Banach Center Publ., 27, Part 1, pages 399–424. Polish Acad. Sci., Warsaw, 1992.

[47] T. Schick. Manifolds with boundary and of bounded geometry. Math. Nachr., 223:103–120, 2001.

[48] R. Schoen. Conformal deformation of a Riemannian metric to constant scalar curvature. J. Diff. Geom., 20:479–495, 1984.

[49] R. Schoen and S.-T. Yau. On the proof of the positive mass conjecture in general relativity. Comm. Math. Phys., 65:45–76, 1979.

[50] R. Schoen and S.-T. Yau. Proof of the positive mass theorem. II. Comm. Math. Phys., 79:231– 260, 1981.

[51] E. Schrohe. Spectral invariance, ellipticity, and the Fredholm property for pseudodifferential operators on weighted Sobolev spaces. Ann. Global Anal. Geom., 10(3):237–254, 1992.

[52] E. Schrohe. Fréchet algebra techniques for boundary value problems: Fredholm criteria and functional calculus via spectral invariance. Math. Nachr., 199:145–185, 1999.

[53] E. Schrohe and B.-W. Schulze. Boundary value problems in Boutet de Monvel’s algebra for manifolds with conical singularities II. In Boundary value problems, Schrödinger operators, deformation quantization, volume 8 of Math. Top. Akademie Verlag, Berlin, 1995.

[54] M. A. Shubin. Spectral theory of elliptic operators on noncompact manifolds. Astérisque, 207:5, 35–108, 1992. Méthodes semi-classiques, Vol. 1 (Nantes, 1991).

[55] L. Skrzypczak. Mapping properties of pseudodifferential operators on manifolds with bounded geometry. J. London Math. Soc. (2), 57(3):721–738, 1998.

[56] E. Stein. Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N.J., 1970.

[57] M. Taylor. Pseudodifferential operators, volume 34 of Princeton Mathematical Series. Princeton University Press, Princeton, N.J., 1981.

[58] M. Taylor. Partial differential equations I, Basic theory, volume 115 of Applied Mathematical Sciences. Springer-Verlag, New York, 1995.

[59] M. Taylor. Partial differential equations II, Qualitative studies of linear equations, volume 116 of Applied Mathematical Sciences. Springer-Verlag, New York, 1996.

[60] M. Taylor. Partial differential equations III, Nonlinear equations, volume 117 of Applied Mathematical Sciences. Springer-Verlag, New York, 1997.

[61] H. Triebel. Spaces of Besov-Hardy-Sobolev type on complete Riemannian manifolds. Ark. Mat., 24(2):299–337, 1986.

[62] H. Triebel. Characterizations of function spaces on a complete Riemannian manifold with bounded geometry. Math. Nachr., 130:321–346, 1987.

[63] A. Vasy. Propagation of singularities in many-body scattering. Ann. Sci. École Norm. Sup. (4), 34(3):313–402, 2001.

[64] A. Vasy and M. Zworski. Semiclassical estimates in asymptotically Euclidean scattering. Comm. Math. Phys., 212(1):205–217, 2000.

[65] G. C. Verchota. Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains. J. Funct. Anal., 59(3):572–611, 1984.

[66] G. C. Verchota and A. L. Vogel. A multidirectional Dirichlet problem. J. Geom. Anal., 13(3):495–520, 2003.

[67] E. Witten. A new proof of the positive energy theorem. Comm. Math. Phys., 80:381–402, 1981.

Bernd Ammann, Institut Élie Cartan, Université Henri Poincaré Nancy 1, B.P. 239, 54506 Vandoeuvre-Les-Nancy, France, http://www.differentialgeometrie.de/ammann

   E-mail address: bernd.ammann@gmx.net

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