PSEUDODIFFERENTIAL OPERATORS ON MANIFOLDS WITH A LIE STRUCTURE AT INFINITY

BERND AMMANN, ROBERT LAUTER, AND VICTOR NISTOR

Version: 3.2; Revised: 18/07/2005; Run: September 26, 2006

Abstract. We define and study an algebra Ψ_{1,0,V}^{∞}(M_{0}) of pseudodifferential operators canonically associated to a non-compact, Riemannian manifold M_{0} whose geometry at infinity is described by a Lie algebra of vector fields V on a compactification M of M_{0} to a compact manifold with corners. We show that the basic properties of the usual algebra of pseudodifferential operators on a compact manifold extend to Ψ_{1,0,V}^{∞}(M_{0}). We also consider the algebra Diff_{V}^{∗}(M_{0}) of differential operators on M_{0} generated by V and C^{∞}(M), and show that Ψ_{1,0,V}^{∞}(M_{0}) is a microlocalization of Diff_{V}^{∗}(M_{0}). Our construction solves a problem posed by Melrose in 1990. Finally, we introduce and study semi-classical and “suspended” versions of the algebra Ψ_{1,0,V}^{∞}(M_{0}).

Contents

Introduction 1 1. Manifolds with a Lie structure at infinity 3 2. Kohn-Nirenberg quantization and pseudodifferential operators 6 3. The product 10 4. Properties of Ψ_{1,0,V}^{∞}(M_{0}) 17 5. Group actions and semi-classical limits 21 References 24

Introduction

Let (M_{0}, g_{0}) be a complete, non-compact Riemannian manifold. It is a fundamental problem to study the geometric operators on M_{0}. As in the compact case, pseudodifferential operators provide a powerful tool for that purpose, provided that the geometry at infinity is taken into account. One needs, however, to restrict to suitable classes of non-compact manifolds.

Let M be a compact manifold with corners such that M_{0} = M ∖∂M, and assume that the geometry at infinity of M_{0} is described by a Lie algebra of vector fields V ⊂ Γ(M; T M), that is, M_{0} is a Riemannian manifold with a Lie structure at infinity, Definition 1.3. In [27], Melrose has formulated a far reaching program to study the analytic properties of geometric differential operators on M_{0}. An important

Ammann was partially supported by the European Contract Human Potential Program, Research Training Networks HPRN-CT-2000-00101 and HPRN-CT-1999-00118, Nistor was partially supported by the NSF Grants DMS-9971951 and DMS-0200808. Manuscripts available from http://www.math.psu.edu/nistor/ .

ingredient in Melrose’s program is to define a suitable pseudodifferential calculus Ψ_{V}^{∞}(M_{0}) on M_{0} adapted in a certain sense to (M, V). This pseudodifferential calculus was called a “microlocalization of Diff_{V}^{∗}(M_{0})” in [27], where Diff_{V}^{∗}(M_{0}) is the algebra of differential operators on M_{0} generated by V and C^{∞}(M). (See Section 2.)

Melrose and his collaborators have constructed the algebras Ψ_{V}^{∞}(M_{0}) in many special cases, see for instance [9, 21, 22, 23, 26, 28, 30, 47], and especially [29]. One of the main reasons for considering the compactification M is that the geometric operators on manifolds with a Lie structure at infinity identify with degenerate differential operators on M. This type of differential operators appear naturally, for example, also in the study of boundary value problems on manifolds with singularities. Numerous important results in this direction were obtained also by Schulze and his collaborators, who typically worked in the framework of the Boutet de Monvel algebras. See [39, 40] and the references therein. Other important cases in which this program was completed can be found in [15, 16, 17, 35, 37]. An earlier important motivation for the construction of these algebras was the method of layer potentials for boundary value problems and questions in analysis on locally symmetric spaces. See for example [4, 5, 6, 8, 18, 19, 24, 32].

An outline of the construction of the algebras Ψ_{V}^{∞}(M_{0}) was given by Melrose in [27], provided certain compact manifolds with corners (blow-ups of M^{2} and M^{3}) can be constructed. In the present paper, we modify the blow-up construction using Lie groupoids, thus completing the construction of the algebras Ψ_{V}^{∞}(M_{0}). Our method relies on recent progress achieved in [2, 7, 35].

The explicit construction of the algebra Ψ_{1,0,V}^{∞}(M_{0}) microlocalizing Diff_{V}^{∗}(M_{0}) in the sense of [27] is, roughly, as follows. First, V defines an extension of T M_{0} to a vector bundle A → M (M_{0} = M ∖ ∂M). Let V_{r} := {d(x, y) < r} ⊂ M_{0}^{2} and (A)_{r} = {v ∈ A, ∥v∥ < r}. Let r > 0 be less than the injectivity radius of M_{0} and V_{r} ∋ (x, y) ↦ (x, τ(x, y)) ∈ (A)_{r} be a local inverse of the Riemannian exponential map T M_{0} ∋ v ↦ exp_{x}(−v) ∈ M_{0} × M_{0}. Let χ be a smooth function on A with support in (A)_{r} and χ = 1 on (A)_{r/2}. For any a ∈ S_{1,0}^{m}(A^{∗}), we define

(1) [a_{i}(D)u](x) = (2π)^{−n} ∫_{M_{0}} (∫_{T_{x}^{∗}M_{0}} e^{iτ(x,y)·η}χ(x, τ(x, y))a(x, η)u(y) dη) dy.

The algebra Ψ_{1,0,V}^{∞}(M_{0}) is then defined as the linear span of the operators a_{χ}(D) and b_{χ}(D) exp(X_{1}) … exp(X_{k}), a ∈ S^{∞}(A^{∗}), b ∈ S^{−∞}(A^{∗}), and X_{j} ∈ V, and where exp(X_{j}) : C_{c}^{∞}(M_{0}) → C_{c}^{∞}(M_{0}) is defined as the action on functions associated to the flow of the vector field X_{j}.

The operators b_{χ}(D) exp(X_{1}) … exp(X_{k}) are needed to make our space closed under composition. The introduction of these operators is in fact a crucial ingredient in our approach to Melrose’s program. The results of [7, 35] are used to show that Ψ_{1,0,V}^{∞}(M_{0}) is closed under composition, which is the most difficult step in the proof.

A closely related situation is encountered when one considers a product of a manifold with a Lie structure at infinity M_{0} by a Lie group G and operators G invariant on M_{0} × G. We obtain in this way an algebra Ψ_{1,0,V}^{∞}(M_{0}; G) of G– invariant pseudodifferential operators on M_{0} × G with similar properties. The algebra Ψ_{1,0,V}^{∞}(M_{0}; G) arises in the study of the analytic properties of differential geometric operators on some higher dimensional manifolds with a Lie structure at infinity. When G = ℝ^{q}, this algebra is slightly smaller than one of Melrose’s suspended algebras and plays the same role, namely, it appears as a quotient of an algebra of the form Ψ_{1,0,V′}^{∞}(M′_{0}), for a suitable manifold M′_{0}. The quotient map Ψ_{1,0,V′}^{∞}(M′_{0}) → Ψ_{1,0,V}^{∞}(M_{0}; G) is a generalization of Melrose’s indicial map. A convenient approach to indicial maps is provided by groupoids [17].

We also introduce a semi-classical variant of the algebra Ψ_{1,0,V}^{∞}(M_{0}), denoted Ψ_{1,0,V}^{∞}(M_{0}[[h]]), consisting of semi-classical families of operators in Ψ_{1,0,V}^{∞}(M_{0}). For all these algebras we establish the usual mapping properties between appropriate Sobolev spaces.

The article is organized as follows. In Section 1 we recall the definition of manifolds with a Lie structure at infinity and some of their basic properties, including a discussion of compatible Riemannian metrics. In Section 2 we define the spaces Ψ_{1,0,V}^{m}(M_{0}) and the principal symbol maps. Section 3 contains the proof of the crucial fact that Ψ_{1,0,V}^{∞}(M_{0}) is closed under composition, and therefore it is an algebra. We do this by showing that Ψ_{1,0,V}^{∞}(M_{0}) is the homomorphic image of Ψ_{1,0}^{∞}(G), where G is any d-connected Lie groupoid integrating A (d–connected means that the fibers of the domain map d are connected). In Section 4 we establish several other properties of the algebra Ψ_{1,0,V}^{∞}(M_{0}) that are similar and analogous to the properties of the algebra of pseudodifferential operators on a compact manifold. In Section 5 we define the algebras Ψ_{1,0,V}^{∞}(M_{0}[[h]]) and Ψ_{1,0,V}^{∞}(M_{0}; G), which are generalizations of the algebra Ψ_{1,0,V}^{∞}(M_{0}). The first of these two algebras consists of the semi-classical (or adiabatic) families of operators in Ψ_{1,0,V}^{∞}(M_{0}). The second algebra is a subalgebra of the algebra of G–invariant, properly supported pseudodifferential operators on M_{0} × G, where G is a Lie group.

Acknowledgements: We thank Andras Vasy for several interesting discussions and for several contributions to this paper. R. L. is grateful to Richard B. Melrose for numerous stimulating conversations and explanations on pseudodifferential calculi on special examples of manifolds with a Lie structure an infinity. V. N. would like to thank the Institute Erwin Schrödinger in Vienna and University Henri Poincaré in Nancy, where parts of this work were completed.

1. Manifolds with a Lie structure at infinity

For the convenience of the reader, let us recall the definition of a Riemannian manifold with a Lie structure at infinity and some of its basic properties.

1.1. Preliminaries. In the sequel, by a manifold we shall always understand a C^{∞}-manifold possibly with corners, whereas a smooth manifold is a C^{∞}-manifold without corners (and without boundary). By definition, every point p in a manifold with corners M has a coordinate neighborhood diffeomorphic to [0, ∞)^{k} × ℝ^{n−k} such that the transition functions are smooth up to the boundary. If p is mapped by this diffeomorphism to (0, …, 0, x_{k+1}, …, x_{n}), we shall say that p is a point of boundary depth k and write depth(p) = k. The closure of a connected component of points of boundary depth k is called a face of codimension k. Faces of codimension 1 are also called hyperfaces. For simplicity, we always assume that each hyperface H of a manifold with corners M is an embedded submanifold and has a defining function, that is, that there exists a smooth function x_{H} ≥ 0 on M such that

H = {x_{H} = 0} and dx_{H} ≠ 0 on H.

For the basic facts on the analysis on manifolds with corners we refer to the forthcoming book [25]. We shall denote by ∂M the union of all non-trivial faces of M and by M_{0} the interior of M, i.e., M_{0} := M ∖ ∂M. Recall that a map f : M → N is a submersion of manifolds with corners if df is surjective at any point and df_{p}(v) is an inward pointing vector if, and only if, v is an inward pointing vector. In particular, the sets f^{−1}(q) are smooth manifolds (no boundary or corners).

To fix notation, we shall denote the sections of a vector bundle V → X by Γ(X, V), unless X is understood, in which case we shall write simply Γ(V). A Lie subalgebra 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 provided it is a finitely generated, projective C^{∞}(M)-module and each V ∈ V is tangent to all hyperfaces of M.

Definition 1.1. A Lie structure at infinity on a smooth manifold M_{0} is a pair (M, V), where M is a compact manifold, possibly with corners, and V ⊂ Γ(M, T M) is a structural Lie algebra of vector fields on M with the following properties:

(a) M_{0} is diffeomorphic to the interior M ∖ ∂M of M,

(b) for any vector field X on M_{0} and any p ∈ M_{0}, there is a neighborhood V of p in M_{0} and a vector field Y ∈ V, such that Y = X on V.

A manifold with a Lie structure at infinity will also be called a Lie manifold.

Here are some examples.

Examples 1.2.

(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 manifold with a Lie structure at infinity.

(b) Take V_{0} to be the set of all vector fields vanishing on all faces of a manifold with corners M. Then (M, V_{0}) is a Lie manifold. If ∂M is a smooth manifold (i.e., if M is a manifold with boundary), then V_{0} = rΓ(M; T M), where r is the distance to the boundary.

(c) As another example consider a manifold with smooth boundary and consider the vector fields V_{sc} = rV_{b}, where r and V_{b} are as in the previous examples.

These three examples are, respectively, the “b-calculus”, the “0-calculus,” and the “scattering calculus” from [29]. These examples are typical and will be referred to again below. Some interesting and highly non-trivial examples of Lie structures at infinity on ℝ^{n} are obtained from the N-body problem [45] and from strictly pseudoconvex domains [31]. Further examples of Lie structures at infinity were discussed in [2].

If M_{0} is compact without boundary, then it follows from the above definition that M = M_{0} and V = Γ(M, T M), so a Lie structure at infinity on M_{0} gives no additional information on M_{0}. The interesting cases are thus the ones when M_{0} is non-compact.

Elements in the enveloping algebra Diff_{V}^{∗}(M) of V are called V-differential operators on M. The order of differential operators induces a filtration Diff_{V}^{m}(M), m ∈ ℕ_{0}, on the algebra Diff_{V}^{∗}(M). Since Diff_{V}^{∗}(M) is a C^{∞}(M)-module, we can introduce V-differential operators acting between sections of smooth vector bundles E, F → M, E, F ⊂ M × ℂ^{N} by

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

where e_{E}, e_{F} ∈ M_{N}(C^{∞}(M)) are the projections onto E and, respectively, F. It follows that Diff_{V}^{∗}(M; E, E) =: Diff_{V}^{∗}(M; E) is an algebra that is closed under adjoints.

Let A → M be a vector bundle and ϱ : A → T M a vector bundle map. We shall also denote by ϱ the induced map Γ(M, A) → Γ(M, T M) between the smooth section of these bundles. Suppose a Lie algebra structure on Γ(M, A) is given. Then the pair (A, ϱ) together with this Lie algebra structure on Γ(A) is called a Lie algebroid if ϱ([X, Y]) = [ϱ(X), ϱ(Y)] and [X, fY] = f[X, Y] + (ϱ(X)f)Y for any smooth sections X and Y of A and any smooth function f on M. The map ϱ : A → T M is called the anchor of A. We have also denoted by ϱ the induced map Γ(M, A) → Γ(M, T M). We shall also write Xf := ϱ(X)f.

If V is a structural Lie algebra of vector fields, then V is projective, and hence the Serre-Swan theorem [13] shows that there exists a smooth vector bundle A_{V} → M together with a natural map

(3)  \begin{aligned}ϱ_{V} : A_{V} & & ⟶ & & T M \\ & ↘ & & ↙ \\ & & M \end{aligned}

such that V = ϱ(Γ(M, A_{V})). The vector bundle A_{V} turns out to be a Lie algebroid over M.

We thus see that there exists an equivalence between structural Lie algebras of vector fields V = Γ(A_{V}) and Lie algebroids ϱ : A → T M such that the induced map Γ(M, A) → Γ(M, T M) is injective and has range in the Lie algebra V_{b}(M) of all vector fields that are tangent to all hyperfaces of M. Because A and V determine each other up to isomorphism, we sometimes specify a Lie structure at infinity on M_{0} by the pair (M, A). The definition of a manifold with a Lie structure at infinity allows us to identify M_{0} with M ∖ ∂M and A|_{M_{0}} with T M_{0}.

We now turn our attention to Riemannian structures on M_{0}. Any metric on A induces a metric on T M_{0} = A|_{M_{0}}. This suggests the following definition.

Definition 1.3. A manifold M_{0} with a Lie structure at infinity (M, V), V = Γ(M, A), and with metric g_{0} on T M_{0} obtained from the restriction of a metric g on A is called a Riemannian manifold with a Lie structure at infinity.

The geometry of a Riemannian manifold (M_{0}, g_{0}) with a Lie structure (M, V) at infinity has been studied in [2]. For instance, (M_{0}, g_{0}) is necessarily of infinite volume and complete. Moreover, all the covariant derivatives of the Riemannian curvature tensor are bounded. Under additional mild assumptions, we also know that the injectivity radius is bounded from below by a positive constant, i.e., (M_{0}, g_{0}) is of bounded geometry. (A manifold with bounded geometry is a Riemannian manifold with positive injectivity radius and with bounded covariant derivatives of the curvature tensor, see [41] and references therein). A useful property is that all geometric operators on M_{0} that are associated to a metric on A are V-differential operator (i.e., in Diff_{V}^{m}(M) [2]).

On a Riemannian manifold M_{0} with a Lie structure at infinity (M, V), V = Γ(M, A), the exponential map exp_{p} : T_{p}M_{0} → M_{0} is well-defined for all p ∈ M_{0} and extends to a differentiable map exp_{p} : A_{p} → M depending smoothly on p ∈ M. A convenient way to introduce the exponential map is via the geodesic spray, as done in [2]. A related phenomenon is that any vector field X ∈ Γ(A) is integrable, which is a consequence of the compactness of M. The resulting diffeomorphism of M_{0} will be denoted ψ_{X}.

Proposition 1.4. Let F_{0} be an open face of M and X ∈ Γ(M; A). Then the diffeomorphism ψ_{X} maps F_{0} to itself.

Proof. This follows right away from the assumption that all vector fields in V are tangent to all faces [2]. □

2. Kohn-Nirenberg quantization and pseudodifferential operators

Throughout this section M_{0} will be a fixed manifold with Lie structure at infinity (M, V) and V := Γ(A). We shall also fix a metric g on A → M, which induces a metric g_{0} on M_{0}. We are going to introduce a pseudodifferential calculus on M_{0} that microlocalizes the algebra of V-differential operators Diff_{V}^{∗}(M_{0}) on M given by the Lie structure at infinity.

2.1. Riemann-Weyl fibration. Fix a Riemannian metric g on the bundle A, and let g_{0} = g|_{M_{0}} be its restriction to the interior M_{0} of M. We shall use this metric to trivialize all density bundles on M. Denote by π : T M_{0} → M_{0} the natural projection. Define

(4)  Φ : T M_{0} ⟶ M_{0} × M_{0}, Φ(v) := (x, exp_{x}(−v)), x = π(v).

Recall that for v ∈ T_{x}M we have exp_{x}(v) = γ_{v}(1) where γ_{v} is the unique geodesic with γ_{v}(0) = π(v) = x and γ′_{v}(0) = v. It is known that there is an open neighborhood U of the zero-section M_{0} in T M_{0} such that Φ|_{U} is a diffeomorphism onto an open neighborhood V of the diagonal M_{0} = Δ_{M_{0}} ⊆ M_{0} × M_{0}.

To fix notation, let E be a real vector space together with a metric or a vector bundle with a metric. We shall denote by (E)_{r} the set of all vectors v of E with |v| < r.

We shall also assume from now on that r_{0}, the injectivity radius of (M_{0}, g_{0}), is positive. We know that this is true under some additional mild assumptions and we conjectured that the injectivity radius is always positive [2]. Thus, for each 0 < r ≤ r_{0}, the restriction Φ|_{(T M_{0})_{r}} is a diffeomorphism onto an open neighborhood V_{r} of the diagonal Δ_{M_{0}}. It is for this reason that we need the positive injectivity radius assumption.

We continue, by slight abuse of notation, to write Φ for that restriction. Following Melrose, we shall call Φ the Riemann-Weyl fibration. The inverse of Φ is given by

M_{0} × M_{0} ⊇ V_{r} ∋ (x, y) ⟼ (x, τ(x, y)) ∈ (T M_{0})_{r} ,

where −τ(x, y) ∈ T_{x}M_{0} is the tangent vector at x to the shortest geodesic γ : [0, 1] → M such that γ(0) = x and γ(1) = y.

2.2. Symbols and conormal distributions. Let π : E → M be a smooth vector bundle with orthogonal metric g. Let

(5)  ⟨ξ⟩ := \sqrt{1 + g(ξ, ξ)}.

We shall denote by S_{1,0}^{m}(E) the symbols of type (1, 0) in Hörmander’s sense [12]. Recall that they are defined, in local coordinates, by the standard estimates

|∂_{x}^{α}∂_{ξ}^{β}a(ξ)| ≤ C_{K,α,β}⟨ξ⟩^{m−|β|}, π(ξ) ∈ K,

where K is a compact trivializing subset (i.e., π^{−1}(K) ≃ K × ℝ^{n}) and α and β are multi-indices. If a ∈ S_{1,0}^{m}(E), then its image in S_{1,0}^{m}(E)/S_{1,0}^{m−1}(E) is called the principal symbol of a and denoted σ^{(m)}(a). A symbol a will be called homogeneous of degree μ if a(x, λξ) = λ^{μ}a(x, ξ) for λ > 0 and |ξ| and |λξ| large. A symbol a ∈ S_{1,0}^{m}(E) will be called classical if there exist symbols a_{k} ∈ S_{1,0}^{m−k}(E), homogeneous of degree m−k, such that a−∑_{j=0}^{N−1} a_{k} ∈ S_{1,0}^{m−N}(E). Then we identify σ^{(m)}(a) with a_{0}. (See any book on pseudodifferential operators or the corresponding discussion in [3].)

We now specialize to the case E = A^{∗}, where A → M is the vector bundle such that V = Γ(M, A). Recall that we have fixed a metric g on A. Let π : A → M and \overline{π} : A^{∗} → M be the canonical projections. Then the inverse of the Fourier transform F_{fiber}^{−1}, along the fibers of A^{∗} gives a map

(6) F_{fiber}^{−1} : S_{1,0}^{m}(A^{∗}) ⟶ C^{−∞}(A) := C_{c}^{∞}(A)′ , ⟨F_{fiber}^{−1}a, φ⟩ := ⟨a, F_{fiber}^{−1}φ⟩,

where a ∈ S_{1,0}^{m}(A^{∗}), φ is a smooth, compactly supported function, and

(7)  F_{fiber}^{−1}(φ)(ξ) := (2π)^{−n} ∫_{\overline{π}(ζ)=π(ξ)} e^{i⟨ξ,ζ⟩}φ(ζ) dζ.

Then I^{m}(A, M) is defined as the image of S_{1,0}^{m}(A^{∗}) through the above map. We shall call this space the space of distributions on A conormal to M. The spaces I^{m}(T M_{0}, M_{0}) and I^{m}(M_{0}^{2}, Δ_{M_{0}}) = I^{m}(M_{0}^{2}, M_{0}) are defined similarly. In fact, these definitions are special cases of the following more general definition. Let X ⊂ Y be an embedded submanifold of a manifold with corners Y. On a small neighborhood V of X in Y we define a structure of a vector bundle over X, such that X is the zero section of V. As a bundle V is isomorphic to the normal bundle of X in Y. Then we define the space of distributions on Y that are conormal of order m to X, denoted I^{m}(Y, X), to be the space of distributions on M that are smooth on Y ∖X and, that are, in a tubular neighborhood V → X of X in Y, the inverse Fourier transforms of elements in S^{m}(V^{∗}) along the fibers of V → X. For simplicity, we have ignored the density factor. For more details on conormal distributions we refer to [11, 12, 42] and the forthcoming book [25] (for manifolds with corners).

The main use of spaces of conormal distributions is in relation to pseudodifferential operators. For example, since we have

I^{m}(M_{0}^{2}, M_{0}) ⊆ C^{−∞}(M_{0}^{2}) := C_{c}^{∞}(M_{0}^{2})′ ,

we can associate to a distribution in K ∈ I^{m}(M_{0}^{2}, M_{0}) a continuous linear map T_{K} : C_{c}^{∞}(M_{0}) → C^{−∞}(M_{0}) := C_{c}^{∞}(M_{0})′, by the Schwartz kernel theorem. Then a well known result of Hörmander [11, 12] states that T_{K} is a pseudodifferential operator on M_{0} and that all pseudodifferential operators on M_{0} are obtained in this way, for various values of m. This defines a map

(8)  T : I^{m}(M_{0}^{2}, M_{0}) → Hom(C_{c}^{∞}(M_{0}), C^{−∞}(M_{0})).

Recall now that (A)_{r} denotes the set of vectors of norm < r of the vector bundle A. We agree to write I_{(r)}^{m}(A, M) for all k ∈ I^{m}(A, M) with supp k ⊆ (A)_{r}. The space I_{(r)}^{m}(T M_{0}, M_{0}) is defined in an analogous way. Then restriction defines a map

(9)  R : I_{(r)}^{m}(A, M) ⟶ I_{(r)}^{m}(T M_{0}, M_{0}).

Recall that r_{0} denotes the injectivity radius of M_{0} and that we assume r_{0} > 0. Similarly, the Riemann–Weyl fibration Φ of Equation (4) defines, for any 0 < r ≤ r_{0}, a map

(10)  Φ_{∗} : I_{(r)}^{m}(T M_{0}, M_{0}) → I^{m}(M_{0}^{2}, M_{0}).

We shall also need various subspaces of conormal distributions, which we shall denote by including a subscript as follows:

• “cl” to designate the distributions that are “classical,” in the sense that they correspond to classical pseudodifferential operators.

• “c” to denote distributions that have compact support.

• “pr” to indicate operators that are properly supported or distributions that

give rise to such operators. For instance, I_{c}^{m}(Y, X) denote the space of compactly supported conormal distributions, so that I_{(r)}^{m}(A, M) = I_{c}^{m}((A)_{r}, M). Occasionally, we shall use the double subscripts “cl,pr” and “cl,c.” Note that “c” implies “pr”. .

2.3. Kohn-Nirenberg quantization. For notational simplicity, we shall use the metric g_{0} on M_{0} (obtained from the metric on A) to trivialize the half-density bundle Ω^{1/2}(M_{0}). In particular, we identify C_{c}^{∞}(M_{0}, Ω^{1/2}) with C_{c}^{∞}(M_{0}). Let 0 < r ≤ r_{0} be arbitrary. Each smooth function χ, with χ = 1 close to M ⊆ A and support contained in the set (A)_{r}, induces a map q_{Φ,χ} : S_{1,0}^{m}(A^{∗}) ⟶ I^{m}(M_{0}^{2}, M_{0}),

(11)  q_{Φ,χ}(a) := Φ_{∗} (R (χF_{fiber}^{−1}(a))) .

Let a_{χ}(D) be the operator on M_{0} with distribution kernel q_{Φ,χ}(a), defined using the Schwartz kernel theorem, i.e., a_{χ}(D) := T ∘ q_{Φ,χ}(a) . Following Melrose, we call the map q_{Φ,χ} the Kohn-Nirenberg quantization map. It will play an important role in what follows.

For further reference, let us make the formula for the induced operator a_{χ}(D) : C_{c}^{∞}(M_{0}) → C_{c}^{∞}(M_{0}) more explicit. Neglecting the density factors in the formula, we obtain for u ∈ C_{c}^{∞}(M_{0})

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

Specializing to the case of Euclidean space M_{0} = ℝ^{n} with the standard metric we have τ(x, y) = x − y, and hence

(13)  a_{χ}(D)u(x) = (2π)^{−n} ∫_{ℝ^{n}} ∫_{ℝ^{n}} e^{i(x−y)η}χ(x, x − y)a(x, η)u(y) dη dy ,

i.e., the well-known formula for the Kohn-Nirenberg-quantization on ℝ^{n}, if χ = 1. The following lemma states that, up to regularizing operators, the above quantization formulas do not depend on χ.

Lemma 2.1. Let 0 < r ≤ r_{0}. If χ_{1} and χ_{2} are smooth functions with support (A)_{r} and χ_{j} = 1 in a neighborhood of M ⊆ A, then (χ_{1} − χ_{2})F_{fiber}^{−1}(a) is a smooth function, and hence a_{χ_{1}}(D) − a_{χ_{2}}(D) has a smooth Schwartz kernel. Moreover, the map S_{1,0}^{m}(A^{∗}) → C^{∞}(A) that maps a ∈ S_{1,0}^{m}(A^{∗}) to (χ_{1} −χ_{2})F_{fiber}^{−1}(a) is continuous, where the right hand side is endowed with the topology of uniform C^{∞}-convergence on compact subsets.

Proof. Since the singular supports of χ_{1}F_{fiber}^{−1}(a) and χ_{2}F_{fiber}^{−1}(a) are contained in the diagonal Δ_{M_{0}} and χ_{1} − χ_{2} vanishes there, we have that (χ_{1} − χ_{2})F_{fiber}^{−1}(a) is a smooth function.

To prove the continuity of the map S_{1,0}^{m}(A^{∗}) ∋ a ↦ (χ_{1} − χ_{2})F_{fiber}^{−1}(a) ∈ C^{∞}(A), it is enough, using a partition of unity, to assume that A → M is a trivial bundle. Then our result follows from the standard estimates for oscillatory integrals (i.e., by formally writing |v|^{2} ∫ e^{i⟨v,ξ⟩}a(ξ)dξ = − ∫ (Δ_{ξ}e^{i⟨v,ξ⟩})a(ξ)dξ and then integrating by parts, see [12, 33, 43, 44] for example). □

We now verify that the quantization map q_{Φ,χ}, Equation (11), gives rise to pseudodifferential operators.

Lemma 2.2. Let r ≤ r_{0} be arbitrary. For each a ∈ S_{1,0}^{m}(A^{∗}) and each χ ∈ C_{c}^{∞}((A)_{r}) with χ = 1 close to M ⊆ A, the distribution q_{Φ,χ}(a) is the Schwartzkernel of a pseudodifferential operator a_{χ}(D) on M_{0}, which is properly supported if r < ∞ and has principal symbol σ^{(μ)}(a) ∈ S_{1,0}^{m}(E)/S_{1,0}^{m−1}(E). If a ∈ S_{cl}^{μ}(A^{∗}), then a_{χ}(D) is a classical pseudodifferential operator.

Proof. Denote also by χ : I^{m}(T M_{0}, M_{0}) → I_{(r)}^{m}(T M_{0}, M_{0}) the “multiplication by χ” map. Then

(14)  a_{χ}(D) = T ∘ Φ_{∗} ∘ R ∘ χ ∘ F_{fiber}^{−1}(a) := T_{Φ_{∗}(R(χF_{fiber}^{−1}(a)))} = T ∘ q_{Φ,χ}(a)

where T is defined in Equation (8). Hence a_{χ}(D) is a pseudodifferential operator by the Hörmander’s result mentioned above [11, 12] (stating that the distribution conormal to the diagonal are exactly the Schwartz kernels of pseudodifferential operators. Since χR(a) is properly supported, so will be the operator a_{χ}(D)).

For the statement about the principal symbol, we use the principal symbol map for conormal distributions [11, 12], and the fact that the restriction of the anchor A → T M to the interior A|_{M_{0}} is the identity. (This also follows from the Equation (13) below.) This proves our lemma. □

Let us denote by Ψ^{m}(M_{0}) the space of pseudodifferential operators of order ≤ m on M_{0} (no support condition). We then have the following simple corollary.

Corollary 2.3. The map σ_{tot} : S_{1,0}^{m}(A^{∗}) → Ψ^{m}(M_{0})/Ψ^{−∞}(M_{0}),

σ_{tot}(a) := a_{χ}(D) + Ψ^{−∞}(M_{0})

is independent of the choice of the function χ ∈ C_{c}^{∞}((A)_{r}) used to define a_{χ}(D) in Lemma 2.2.

Proof. This follows right away from the previous lemma, Lemma 2.2. □

Let us remark that our pseudodifferential calculus depends on more than just the metric.

Remark 2.4. Non-isomorphic Lie structures at infinity can lead to the same metric on M_{0}. An example is provided by ℝ^{n} with the standard metric, which can be obtained either from the radial compactification of ℝ^{n} with the scattering calculus, or from [−1, 1]^{n} with the b-calculus. See Examples 1.2 and the paragraph following it. The pseudodifferential calculi obtained from these Lie algebra structures at infinity will be, however, different.

The above remark readily shows that not all pseudodifferential operators in Ψ^{m}(M_{0}) are of the form a_{χ}(D) for some symbol a ∈ S_{1,0}^{m}(A^{∗}), not even if we assume that they are properly supported, because they do not have the correct behavior at infinity. Moreover, the space T ∘q_{Φ,χ}(S_{1,0}^{∞}(A^{∗})) of all pseudodifferential operators of the form a_{χ}(D) with a ∈ S_{1,0}^{∞}(A^{∗}) is not closed under composition. In order to obtain a suitable space of pseudodifferential operators that is closed under composition, we are going to include more (but not all) operators of order −∞ in our calculus.

Recall that we have fixed a manifold M_{0}, a Lie structure at infinity (M, A) on M_{0}, and a metric g on A with injectivity radius r_{0} > 0. Also, recall that any X ∈ Γ(A) ⊂ V_{b} generates a global flow Ψ_{X} : ℝ × M → M. Evaluation at t = 1 yields a diffeomorphism Ψ_{X}(1, ·) : M → M, whose action on functions is denoted

(15)  ψ_{X} : C^{∞}(M) → C^{∞}(M).

We continue to assume that the injectivity radius r_{0} of our fixed manifold with a Lie structure at infinity (M, V) is strictly positive.

Definition 2.5. Fix 0 < r < r_{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, A) is the linear space of operators C_{c}^{∞}(M_{0}) → C_{c}^{∞}(M_{0}) generated by a_{χ}(D), a ∈ S_{1,0}^{m}(A^{∗}), and b_{χ}(D)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, A) is obtained by using classical symbols a in the construction above.

It is implicit in the above definition that the spaces Ψ_{1,0,V}^{−∞}(M_{0}) and Ψ_{cl,V}^{−∞}(M_{0}) are the same. They will typically be denoted by Ψ_{V}^{−∞}(M_{0}). As usual, we shall denote

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}).

At first sight, the above definition depends on the choice of the metric g on A. However, we shall soon prove that this is not the case.

As for the usual algebras of pseudodifferential operators, we have the following basic property of the principal symbol.

Proposition 2.6. The principal symbol establishes isomorphisms

(16)  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

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

Proof. This follows from the classical case of the spaces Ψ^{m}(M_{0}) using also Lemma 2.2. □

3. The product

We continue to denote by (M, V), V = Γ(A), a fixed manifold with a Lie structure at infinity and with positive injectivity radius. In this section we want to show that the space Ψ_{1,0,V}^{∞}(M_{0}) is an algebra (i.e., it is closed under multiplication) by showing that it is the homomorphic image of the algebra Ψ_{1,0}^{∞}G) of pseudodifferential operators on any d-connected groupoid G integrating A (Theorem 3.2).

First we need to fix the terminology and to recall some definitions and constructions involving groupoids.

3.1. Groupoids. Here is first an abstract definition that will be made more clear below. Recall that a small category is a category whose morphisms form a set. A groupoid is a small category all of whose morphisms are invertible. Let G denote the set of morphisms and M denote the set of objects of a given groupoid. Then each g ∈ G will have a domain d(g) ∈ M and a range r(g) ∈ M such that the product g_{1}g_{2} is defined precisely when d(g_{1}) = r(g_{2}). Moreover, it follows that the multiplication (or composition) is associative and every element in G has an inverse. We shall identify the set of objects M with their identity morphisms via a map ι : M → G. One can think then of a groupoid as being a group, except that the multiplication is only partially defined. By abuse of notation, we shall use the same notation for the groupoid and its set of morphisms (G in this case). An intuitive way of thinking of a groupoid with morphisms G and objects M is to think of the elements of G as being arrows between the points of M. The points of M will be called units, by identifying an object with its identity morphism. There will be structural maps d, r : G → M, domain and range, μ : {(g, h), d(g) = r(h)} → G, multiplication, G ∋ g → g^{−1} ∈ G, inverse, and ι : M → G satisfying the usual identities satisfied by the composition of functions.

A Lie groupoid is a groupoid G such that the space of arrows G and the space of units M are manifolds with corners, all its structural maps (i.e., multiplication, inverse, domain, range, ι) are differentiable, the domain and range maps (i.e., d and r) are submersions. By the definition of a submersion of manifolds with corners, the submanifolds G_{x} := d^{−1}(x) and G^{x} := r^{−1}(x) are smooth (so they have no corners or boundary), for any x ∈ M. Also, it follows that that M is an embedded submanifold of G.

The d–vertical tangent space to G, denoted T_{vert}G, is the union of the tangent spaces to the fibers of d : G → M, that is

(18)  T_{vert}G := ∪_{x∈M}T G_{x} = ker d_{∗},

the union being a disjoint union, with topology induced from the inclusion T_{vert}G ⊂ T G. The Lie algebroid of G, denoted A(G) is defined to be the restriction of the d–vertical tangent space to the set of units M, that is, A(G) = ∪_{x∈M}T_{x}G_{x}, a vector bundle over M. The space of sections of A(G) identifies canonically with the space of sections of the d-vertical tangent bundle (= d-vertical vector fields) that are right invariant with respect to the action of G. It also implies a canonical isomorphism between the vertical tangent bundle and the pull-back of A(G) via the range map r : G → M:

(19)  r^{∗}A(G) ≃ T_{vert}G.

The structure of Lie algebroid on A(G) is induced by the Lie brackets on the spaces Γ(T G_{x}), G_{x} := d^{−1}(x). This is possible since the Lie bracket of two right invariant vector fields is again right invariant. The anchor map in this case is given by the differential of r, r_{∗} : A(G) → T M.

Let G be a Lie groupoid with units M, then there is associated to it a pseudodifferential calculus (or algebra of pseudodifferential operators) Ψ_{1,0}^{∞}(G), whose operators of order m form a linear space denoted Ψ_{1,0}^{m}(G), m ∈ ℝ, such that Ψ_{1,0}^{m}(G)Ψ_{1,0}^{m′}(G) ⊂ Ψ_{1,0}^{m+m′}(G). This calculus is defined as follows: Ψ_{1,0}^{m}(G) consists of smooth families of pseudodifferential operators (P_{x}), P_{x} ∈ Ψ_{1,0}^{m}(G_{x}), x ∈ M, that are right invariant with respect to multiplication by elements of G and are “uniformly supported.” To define what uniformly supported means, let us observe that the right invariance of the operators P_{x} implies that their distribution kernels K_{P_{x}} descend to a distribution k_{P} ∈ I^{m}(G, M). Then the family P = (P_{x}) is called uniformly supported if, by definition, k_{P} has compact support. If P is uniformly supported, then each P_{x} is properly supported. The right invariance condition means, for P = (P_{x}) ∈ Ψ_{1,0}^{∞}(G), that right multiplication G_{x} ∋ g′ ↦ g′g ∈ G_{y} maps P_{y} to P_{x}, whenever d(g) = y and r(g) = x. By definition, the evaluation map

(20)  Ψ_{1,0}^{∞}(G) ∋ P = (P_{x}) ↦ e_{z}(P) := P_{z} ∈ Ψ_{1,0}^{∞}(G_{z})

is an algebra morphism for any z ∈ M. If we require that the operators P_{x} be classical of order μ ∈ ℂ, we obtain spaces Ψ_{cl}^{μ}(G) having similar properties. These spaces were considered in [35].

All results and constructions above remain true for classical pseudodifferential operators. This gives the algebra Ψ_{cl}^{∞}(G) consisting of families P = (P_{x}) of classical pseudodifferential operators satisfying all the previous conditions.

Assume that the interior M_{0} of M is an invariant subset. Recall that the so called vector representation π_{M} : Ψ_{1,0}^{∞}(G) → End(C_{c}^{∞}(M_{0})) associates to a pseudodifferential operator P on G a pseudodifferential operator π_{M}(P) : C_{c}^{∞}(M_{0}) → C_{c}^{∞}(M_{0}) [17]. This representation π_{M} is defined as follows. If φ ∈ C_{c}^{∞}(M_{0}), φ∘r is a smooth function on G, and we can let the family (P_{x}) act along each G_{x} to obtain the function P (φ ∘ r) on G defined by P (φ ∘ r)|_{G_{x}} = P_{x}(φ ∘ r|_{G_{x}}). The fact that P_{x} is a smooth family guarantees that P (φ ∘ r) is also smooth. Using then the fact that r is a submersion, so locally it is a product map, we obtain that P (φ ∘ r) = φ_{0} ∘ r, for some function φ_{0} ∈ C_{c}^{∞}(M_{0}). We shall then let

(21)  π_{M}(P)φ = φ_{0}.

The fact that P is uniformly supported guarantees that φ_{0} will also have compact support in M_{0}. A more explicit description of π_{M} in the case of Lie manifolds will be obtained in the proof of Theorem 3.2, more precisely, Equation (27).

A Lie groupoid G with units M is said to integrate A if A(G) ≃ A as vector bundles over M. Recall that the groupoid G is called d–connected if G_{x} := d^{−1}(x) is a connected set, for any x ∈ M. If there exists a Lie groupoid G integrating A, then there exists also a d–connected Lie groupoid with this property. (Just take for each x the connected component of x in G_{x}.)

Our plan to show that Ψ_{1,0,V}^{∞}(M_{0}) is an algebra, is then to prove that it is the image under π_{M} of Ψ_{1,0}^{∞}(G), for a Lie groupoid G integrating A, Γ(M, A) = V. In fact, any d-connected Lie groupoid will satisfy this, by Theorem 3.2. This requires the following deep result due to Crainic and Fernandes [7] stating that the Lie algebroids associated to Lie manifolds are integrable.

Theorem 3.1. [Cranic–Fernandes] Any Lie algebroid arising from a Lie structure at infinity is actually the Lie algebroid of a Lie groupoid (i.e., it is integrable).

This Theorem should be thought of as an analog of Lie’s third theorem stating that every finite dimensional Lie algebra is the Lie algebra of a Lie group. However, the analog of Lie’s theorem for Lie algebroids does not hold: there are Lie algebroid which are not Lie algebroids to a Lie groupoid [20].

A somewhat weaker form of the above theorem, which is however enough for the proof of Melrose’s conjecture was obtained in [34].

We are now ready to state and prove the main result of this section. We refer to [17] or [35] for the concepts and results on groupoids and algebras of pseudodifferential operators on groupoids not explained below or before the statement of this theorem.

Theorem 3.2. Let M_{0} be a manifold with a Lie structure at infinity, (M, V), A = A_{V}, as above. Also, let G be a d-connected groupoid with units M and with A(G) ≃ A. Then Ψ_{1,0,V}^{m}(M_{0}) = π_{M}(Ψ_{1,0}^{m}(G)) and Ψ_{cl,V}^{m}(M_{0}) = π_{M}(Ψ_{cl}^{m}(G)).

Proof. We shall consider only the first equality. The case of classical operators can be treated in exactly the same way.

Here is first, briefly, the idea of the proof. Let P = (P_{x}) ∈ Ψ_{1,0}^{m}(G). Then the Schwartz kernels of the operators P_{x} form a smooth family of conormal distributions in I^{m}(G_{x}^{2}, G_{x}) that descends, by right invariance, to a distribution k_{P} ∈ I_{c}^{m}(G, M) (i.e., to a compactly supported distribution on G, conormal to M) called the convolution kernel of P. The map P ↦ k_{P} is an isomorphism [35] with inverse

(22)  T : I_{c}^{m}(G, M) → Ψ_{1,0}^{m}(G).

Fix a metric on A → M. The resulting exponential map (reviewed below) then gives rise, for r > 0 small enough, to an open embedding

(23)  α : (A)_{r} → G,

which is a diffeomorphism onto its image. This diffeomorphism then gives rise to an embedding

(24)  α_{∗} : I_{(r)}^{m}(A, M) := I_{c}^{m}((A)_{r}, M) → I^{m}(G, M)

such that for each χ as above

(25)  π_{M}(α_{∗}(χF_{fiber}^{−1}(a))) = a_{χ}(D) ∈ Ψ^{m}(M_{0}).

This will allow us to show that π_{M}(Ψ_{1,0}^{m}(G)) contains the linear span of all operators P of the form P = a_{χ}(D), a ∈ S_{1,0}^{m}(A^{∗}), m ∈ ℤ fixed. This reduces the problem to verifying that

(26)  π_{M}(Ψ^{−∞}(G)) = Ψ_{V}^{−∞}(M_{0}).

Using a partition of unity, this in turn will be reduced to Equation (25). Now let us provide the complete details.

Let G_{x}^{x} := d^{−1}(x) ∩ r^{−1}(x), which is a group for any x ∈ M_{0}, by the axioms of a groupoid. Then G_{x}^{x} ≃ G_{y}^{y} whenever there exists g ∈ G with d(g) = x and r(g) = y (conjugate by g). We can assume, without loss of generality, that M is connected. Let Γ := G_{x}^{x}, for some fixed x ∈ M_{0}. Our above informal description of the proof can be conveniently formalized and visualized using the following diagram whose morphisms are defined below:

[figure: S_{1,0}^{m}(A); F_{fiber}^{−1}; I^{m}(A, M); χ; I_{(r)}^{m}(A, M); α_{∗}; R; I_{(r)}^{m}(T M_{0}, M_{0}); Φ_{∗}; Ψ_{1,0}^{m}(G); ≅; e_{x}; I_{c}^{m}(G, M); l_{∗}; μ_{1}^{∗}; I^{m}(M_{0}^{2}, M_{0}); Ψ_{pr}^{m}(G_{x})^{Γ}; ≅; I_{pr}^{m}(G_{x}^{2}, G_{x})^{Γ}; ≅; r̃_{∗}; I^{m}(M_{0}^{2}, M_{0}); ≅; Ψ_{pr}^{m}(G_{x})^{Γ}; r_{∗}; Ψ_{1,0}^{m}(M_{0})]

We now define the morphisms appearing in the above diagram in such a way that it will turn out to be a commutative diagram. The bottom three rectangles will trivially turn out to be commutative. Recall that the index “pr” means “properly supported.”

First, recall that the maps F_{fiber}^{−1} (the fiberwise inverse Fourier transform), χ (the multiplication by the cut-off function χ), R (the restriction map), Φ_{∗} (induced by the inverse of the exponential map), and e_{x} (evaluation of the family (P_{y}) at y = x) have already been defined.

We let μ_{1}(g′, g) = g′g^{−1} and μ_{1}^{∗} be the map induced at the level of kernels by μ_{1} by pull-back (which is seen to be defined in this case because μ_{1} is a submersion and its range is transverse to M).

The four isomorphisms not named are the “T isomorphisms” and their inverses defined in various places earlier (identifying spaces of conormal distributions with spaces of pseudodifferential operators). More precisely, the top isomorphism is from [35] and all the other isomorphisms are the canonical identifications between pseudodifferential operators and distributions on product spaces that are conormal to the diagonal (via the Schwartz kernels). In fact, the top isomorphism T is completely determined by the requirement that the left-most square (containing e_{x}) be commutative.

It is a slightly more difficult task to define r_{∗}. We shall have to make use of a minimum of groupoid theory. Let y ∈ M be arbitrary for a moment. Since the Lie algebra of G_{y}^{y} is isomorphic to the kernel of the anchor map ϱ : A(G)_{y} → T_{y}M, we see that G_{y}^{y} is a discrete group if, and only if, y ∈ M_{0}. Then

r_{∗} : T_{y}G_{y} = A(G)_{y} → T_{y}M_{0}

is an isomorphism, if and only if, y ∈ M_{0}.

Let x ∈ M_{0} be our fixed point. Then r : G_{x} → M_{0} is a surjective local diffeomorphism. Also Γ := G_{x}^{x} acts freely on G_{x} and G_{x}/Γ = M_{0}. Hence r : G_{x} → M_{0} is a covering map with group Γ, and C^{∞}(G_{x})^{Γ} = C^{∞}(M_{0}). Let P = (P_{y}) ∈ Ψ_{1,0}^{m}(G). Since P_{x} is Γ-invariant and properly supported, the map P_{x} : C^{∞}(G_{x}) → C^{∞}(G_{x}), descends to a map C^{∞}(M_{0}) → C^{∞}(M_{0}), which is by definition r_{∗}(P). More precisely, if φ is a smooth function on M_{0}, then φ ∘ r, is a Γ-invariant function on G_{x}. Hence P (φ∘r) is defined (because P is properly supported) and is also Γ-invariant. Thus there exists a function φ_{0} ∈ C^{∞}(M_{0}) such that P (φ∘r) = φ_{0}∘r. The operator r_{∗}(P) is then given by r_{∗}(P)φ := φ_{0}. This definition of r_{∗} provides us with the following simpler definition of the vector representation π_{M}:

(27)  π_{M}(P) = r_{∗}(e_{x}(P)).

We also obtain that

(28)  π_{M} ∘ T = r_{∗} ∘ e_{x} ∘ T = r_{∗} ∘ T ∘ μ_{1}^{∗},

by the commutativity of the left-most rectangle.

The commutativity of the bottom rectangle completely determines the morphism r̃_{∗}. However, we shall also need an explicit description of this map. This can be obtained as follows. Recall that G_{x} is a covering of M_{0} with group Γ := G_{x}^{x}. Hence we can identify I_{pr}^{m}(G_{x}^{2}, G_{x})^{Γ} with I_{pr}^{m}((G_{x}^{2})^{Γ}, G_{x}^{Γ}). The map τ : (G_{x}^{2})/Γ → M_{0}^{2} is also a covering map. This allows us to identify a distribution with small support in (G_{x}^{2})/Γ with a distribution with support in a small subset of M_{0}^{2}. These identifications then extend by summation along the fibers of τ : (G_{x}^{2})/Γ → M_{0}^{2} to define a distribution τ_{∗}(u) ∈ D′(M_{0}^{2}), for any distribution u on (G_{x}^{2})/Γ whose support intersects only finitely many components of τ^{−1}(U), for any connected locally trivializing open set U ⊂ M_{0}. The morphism r̃_{∗} identifies then with τ_{∗}. Also, observe for later use that

(29) τ(g′, g) = (r(g′), r(g)) = (r(g′g^{−1}), d(g′g^{−1})) = (r(μ_{1}(g′, g)), d(μ_{1}(g′, g))).

Next, we must set l_{∗} := r̃_{∗} ∘ μ_{1}^{∗}, by the commutativity requirement. For this morphism we have a similar, but simpler, description of l_{∗}(u). Namely, l_{∗}(u) is obtained by first restricting a distribution u to d^{−1}(M_{0}) = r^{−1}(M_{0}) and then by applying to this restriction the push-forward map defined by (d, r) : d^{−1}(M_{0}) → M_{0}^{2} (that is, we sum over open sets in G_{x} covering sets in the base M_{0}^{2}). Equation (29) guarantees that this alternative description of l_{∗} satisfies l_{∗} := r̃_{∗} ∘ μ_{1}^{∗}.

To define α_{∗}, recall that we have fixed a metric on A. This metric then lifts via r : G → M to T_{vert}G ≃ r^{∗}A(G), by Equations (18) and (19). The induced metrics on the fibers of G_{y}, y ∈ M, give rise, using the (geodesic) exponential map, to maps

A_{y} ≃ A(G)_{y} = T_{y}G_{y} → G_{y}.

These maps give rise to an application (A)_{r} → G, which, by the Inverse mapping theorem, is seen to be a diffeomorphism onto its image. It, moreover, sends the zero section of A to the units of G. Then α_{∗} is the resulting map at the level of conormal distributions. (Note that G_{y} is complete.)

We have now completed the definition of all morphisms in our diagram. To prove that our diagram is commutative, it remains to prove that

l_{∗} ∘ α_{∗} = Φ_{∗} ∘ R.

This however follows from the above description of the map l_{∗}, since (d, r) is injective on α((A)_{r}) and r : G_{x} → M_{0} is an isometric covering, thus preserving the exponential maps.

The commutativity of the above diagram finally shows that

(30) a_{χ}(D) := T ∘ q_{Φ,χ}(a) = T ∘ Φ_{∗} ∘ R ∘ χ ∘ F_{fiber}^{−1}(a)

= π_{M} ∘ T ∘ α_{∗} ∘ χ ∘ F_{fiber}^{−1}(a) = π_{M}(Q),

where Q = T ∘ α_{∗} ∘ χ ∘ F_{fiber}^{−1}(a) and a ∈ S_{1,0}^{m}(A^{∗}). Thus every operator of the form a_{χ}(D) is in the range of π_{M}.

Let us notice for the rest of our argument that the definition of the vector representation π_{M} can be extended by the same formula to arbitrary right invariant families of operators P = (P_{x}), P_{x} : C^{∞}(G_{x}) → C^{∞}(G_{x}), such that the induced operator P : C_{c}^{∞}(G) → ∪C^{∞}(G_{x}) has range in C_{c}^{∞}(G). We shall use this in the following case. Let X ∈ V. Then X defines by integration a diffeomorphism of M, see Equation (15). Let X̃ be its lift to a d-vertical vector field on G (i.e., on each G_{x} we obtain a vector field, and this family of vector fields is right invariant). A result from [14, Appendix] (see also [34]) then shows that X̃ can be integrated to a global flow. Let us denote by ψ̃_{X} the family of diffeomorphisms of each G_{x} obtained in this way, as well as their action on functions. It follows then from the definition that

(31)  π_{M}(ψ̃_{X}) = ψ_{X}.

The Equations (30) and (31) then give

(32)  π_{M}(Qψ̃_{X_{1}} … ψ̃_{X_{n}}) = a_{χ}(D)ψ_{X_{1}} … ψ_{X_{n}} ∈ Ψ_{1,0,V}^{−∞}(M_{0}),

for any a ∈ S^{−∞}(A^{∗}) and Q = T ∘ α_{∗} ∘ χ ∘ F_{fiber}^{−1}(a). Also Qψ̃_{X_{1}} … ψ̃_{X_{n}} ∈ Ψ^{−∞}(G), since the product of a regularizing operator with the operator induced by a diffeomorphism is regularizing. We have thus proved that π_{M}(Ψ_{1,0}^{m}(G)) ⊃ Ψ_{1,0,V}^{m}(M_{0}). Let us now prove the opposite inclusion, that is that π_{M}(Ψ_{1,0}^{m}(G)) ⊂ Ψ_{1,0,V}^{m}(M_{0}).

Let Q ∈ Ψ_{1,0}^{m}(G) be arbitrary and let b = T^{−1}(Q). Let χ_{0} be a smooth function on G that is equal to 1 in a neighborhood of M in G and with support in α((A)_{r}) and such that χ = 1 on the support of χ_{0} ∘ α. Then b_{0} := χ_{0}b is in the range of α_{∗} ∘ χ ∘ F_{fiber}^{−1}, because any distribution u ∈ I_{(r)}^{m}(A, M) is in the range of F_{fiber}^{−1}, if r < ∞. Then the difference b − b_{0} is smooth. Because G is d-connected, we can use a similar construction as the one used to define b_{0} and a partition of unity argument to obtain that

(33)  T (b − b_{0}) = ∑_{j=1}^{l} T (b_{j})ψ̃_{X_{j1}} … ψ̃_{X_{jn}}

for some distributions b_{j} ∈ χI_{(r)}^{−∞}(A, M) and vector fields X_{jk} ∈ V. Let a_{j} be such that b_{j} = α_{∗} ∘ χ ∘ F_{fiber}^{−1}(a_{j}), for a_{0} ∈ S_{1,0}^{m}(A^{∗}) and a_{j} ∈ S_{1,0}^{−∞}(A^{∗}), if j > 0. Then Equations (30) and (32) show that

(34)  π_{M}(Q) = a_{0}(D) + ∑_{j=1}^{l} a_{j}(D)ψ_{X_{j1}} … ψ_{X_{jn}} ∈ Ψ_{1,0,V}^{m}(M_{0}).

We have thus proved that π_{M}(Ψ_{1,0}^{m}(G)) = Ψ_{1,0,V}^{m}(M_{0}), as desired. This completes our proof. □

Since the map π_{M} respects adjoints: π_{M}(P^{∗}) = π_{M}(P)^{∗}, [17], we obtain the following corollary.

Corollary 3.3. The algebras Ψ_{1,0,V}^{∞}(M_{0}) and Ψ_{cl,V}^{∞}(M_{0}) are closed under taking adjoints.

We end this section with three remarks.

Remark 3.4. Equation (33) is easily understood in the case of groups, when it amounts to the possibility of covering any given compact set by finitely many translations of a given open neighborhood of the identity. The argument in general is the same as the argument used to define the basic coordinate neighborhoods on G in [34]. The basic coordinate neighborhoods on G were used in that paper to define the smooth structure on the groupoid G.

Remark 3.5. We suspect that any proof of the fact that Ψ_{1,0,V}^{∞}(M_{0}) is closed under multiplication is equivalent to the integrability of A. In fact, Melrose has implicitly given some evidence for this in [27] for particular (M, V), by showing that the kernels of the pseudodifferential operators on M_{0} that he constructed naturally live on a modified product space M_{V}^{2}. In his case M_{V}^{2} was a blow-up of the product M × M, and hence was a larger compactification of the product M_{0} × M_{0}. The kernels of his operators naturally extended to conormal distributions on this larger product M_{V}^{2}. The product and adjoint were defined in terms of suitable maps between M_{V}^{2} and some fibered product spaces M_{V}^{3}, which are suitable blow-ups of M^{3} and hence larger compactifications of M_{0}^{3}. This in principle leads to a solution of the problem of microlocalizing V that we stated in the introduction whenever one can define the spaces M_{V}^{2} and M_{V}^{3}. Let us also mention here that Melrose’s approach usually leads to algebras that are slightly larger than ours.

Remark 3.6. Let G be a Lie groupoid such that the map π_{M} is an isomorphism and let N ⊂ M be a face of M, then we obtain a generalized indicial map

R_{N} : Ψ_{1,0,V}^{∞}(M_{0}) ≃ Ψ_{1,0}^{∞}(G) → Ψ_{1,0}^{∞}(G_{N}).

In applications, the algebras Ψ_{1,0}^{∞}(G_{N}) often turn out to be isomorphic to the algebras Ψ_{1,0,V_{1}}^{∞}(N_{0}; G) studied in the last section of this paper. In fact, this is the motivation for introducing the algebras Ψ_{1,0,V_{1}}^{∞}(N_{0}; G).

4. Properties of Ψ_{1,0,V}^{∞}(M_{0})

Theorem 3.2 has several consequences similar to the results in [21, 29, 28, 30, 38, 40].

4.1. Basic properties. We obtain that the algebras Ψ_{1,0,V}^{∞}(M_{0}) and Ψ_{cl,V}^{∞}(M_{0}) are independent of the choices made to define them and, thus, depend only on the Lie structure at infinity (M, V).

Corollary 4.1. The spaces Ψ_{1,0,V}^{m}(M_{0}) and Ψ_{cl,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}.

Proof. The space Ψ_{1,0}^{m}(G) does not depend on the metric on A or on the function χ and neither does the vector representation π_{M}. Then use Theorem 3.2. The proof is the same for classical operators. □

An important consequence is that Ψ_{1,0,V}^{∞}(M_{0}) and Ψ_{cl,V}^{∞}(M_{0}) = ∪_{m∈ℤ}Ψ_{cl,V}^{m}(M_{0}) are filtered algebras, as it is the case of the usual algebra of pseudodifferential operators on a compact manifold.

Proposition 4.2. Using the above notation, we have 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_{cl,V}^{m}(M_{0})Ψ_{cl,V}^{m′}(M_{0}) ⊆ Ψ_{cl,V}^{m+m′}(M_{0}) ,

for all m, m′ ∈ ℂ ∪ {−∞}.

Proof. Use Theorem 3.2 and the fact that π_{M} preserves the product. □

Part (i) of the following result is an analog of a standard result about the bcalculus [28], whereas the second formula is the independence of diffeomorphisms of the algebras Ψ_{cl,V}^{∞}(M_{0}), in the framework of manifolds with a Lie structure at infinity. Recall that if X ∈ Γ(A), we have denoted by ψ_{X} := Ψ_{X}(1, ·) : M → M the diffeomorphism defined by integrating X (and specializing at t = 1).

Proposition 4.3. (i) Let x be a defining function of some hyperface of M. Then

x^{s}Ψ_{1,0,V}^{m}(M_{0})x^{−s} = Ψ_{1,0,V}^{m}(M_{0}) and x^{s}Ψ_{cl,V}^{m}(M_{0})x^{−s} = Ψ_{cl,V}^{m}(M_{0})

for any s ∈ ℂ.

(ii) Similarly,

ψ_{X}Ψ_{1,0,V}^{m}(M_{0})ψ_{X}^{−1} = Ψ_{1,0,V}^{m}(M_{0}) and ψ_{X}Ψ_{cl,V}^{m}(M_{0})ψ_{X}^{−1} = Ψ_{cl,V}^{m}(M_{0}),

for any X ∈ Γ(A).

Proof. We have that x^{s}Ψ_{cl}^{m}(G)x^{−s} = Ψ_{cl}^{m}(G), for any s ∈ ℂ, by [17]. A similar result for type (1, 0) operators is proved in the same way as in [17]. This proves (a) because π_{M}(x^{s}P x^{−s}) = x^{s}π_{M}(P)x^{−s}.

Similarly, using the notations of Theorem 3.2, we have ψ̃_{X}Ψ_{cl}^{m}(G)ψ̃_{X}^{−1} = Ψ_{cl}^{m}(G), for any X ∈ Γ(A) = V. By the diffeomorphism invariance of the space of pseudodifferential operators, ψ̃_{X}P ψ̃_{X}^{−1} defines a right invariant family of pseudodifferential operators on G for any such right invariant family P = (P_{x}), as in the proof of Theorem 3.2. To check that the family P_{1} := ψ_{X}P ψ_{X}^{−1} has a compactly supported convolution kernel, denote by (G)_{a} = {g, dist(g, d(g)) ≤ a}. Then observe that supp(ψ̃_{X}P ψ̃_{X}^{−1}) ⊂ G_{d+2∥X∥} whenever supp(P) ⊂ (G)_{d}. Then use Equation (31) to conclude the result.

The proof for type (1, 0) operators is the same. □

Let us notice that the same proof gives (ii) above for any diffeomorphism of M_{0} that extends to an automorphism of (M, A). Recall that an automorphism of the Lie algebroid π : A → M is a morphism of vector bundles (φ, ψ), φ : M → M, ψ : A → A, such that φ and ψ are diffeomorphisms, π ∘ ψ = φ ∘ π, and we have the following compatibility with the anchor map ϱ:

ϱ ∘ ψ = φ_{∗} ∘ ϱ.

4.2. Mapping properties. Let H^{s}(M_{0}) be the domain of (1 + Δ)^{s/2}, where Δ is the (positive) Laplace operator on M_{0} defined by the metric, if s ≥ 0. The space H^{−s}(M_{0}), s ≥ 0, is defined by duality, the duality form being the pairing of distributions with test functions.

Corollary 4.4. Each operator P ∈ Ψ_{1,0,V}^{m}(M_{0}), P : C_{c}^{∞}(M_{0}) → C^{∞}(M_{0}), extends to a continuous linear operators P : C^{∞}(M) → C^{∞}(M) and P : H^{s}(M_{0}) → H^{s−m}(M_{0}). The space H^{m}(M_{0}), m ≥ 0, identifies with the domain of P with the graph topology and H^{−m}(M_{0}) = P L^{2}(M_{0})+L^{2}(M_{0}), for any elliptic P ∈ Ψ_{1,0}^{m}(M_{0}).

Proof. The first part is a direct consequence of the definitio of ∈ Ψ_{1,0,V}^{m}(M_{0}) is properly supported. The last part follows from the results of [1] and [3].

We now sketch the proof for the benefit of the reader. It follows from the explicit form of the kernels of operators T ∈ Ψ_{1,0,V}^{−n−1}(M_{0}), n = dim(M_{0}), that such a T is bounded on L^{2}(M_{0}). Using the symbolic properties of the algebra Ψ_{1,0,V}^{∞}(M_{0}), namely Proposition 2.6 and Proposition 4.2, it then follows that any T ∈ Ψ_{1,0,V}^{0}(M_{0}) is bounded on L^{2}(M_{0}) (the details are the same as in [17] or [3]). Using again the symbolic properties of Ψ_{1,0,V}^{∞}(M_{0}), it is proved as in [3] that the domain of the closure of P and P L^{2}(M_{0}) + L^{2}(M_{0}) are independent of P elliptic of order m. Let us denote by H_{m} the domain of the closure of P and H_{−m} = P L^{2}(M_{0}) + L^{2}(M_{0}). Then it is proved in [3] that T : H_{r} → H_{r−m} is bounded, for any T of order m. In [1] it is proved using partitions of unity that T : H^{r}(M_{0}) → H^{r−m}(M_{0}) is bounded for any T of order m. This shows that H_{r} = H^{r}(M_{0}) for any r ∈ ℝ. □

4.3. Quantization. We have the following quantization properties of the algebra Ψ_{1,0,V}^{∞}(M_{0}).

For any X ∈ Γ(A), denote by a_{X} : A^{∗} → ℂ the function defined by a_{X}(ξ) = ξ(X). Then there exists a unique Poisson structure on A^{∗} such that {a_{X}, a_{Y} } = a_{[X,Y]}. It is related to the Poisson structure { · , · }^{T^{∗}M} on T^{∗}M via the formula

{f_{1} ∘ ϱ^{∗}, f_{2} ∘ ϱ^{∗}}^{T^{∗}M} = {f_{1}, f_{2}} ∘ ϱ^{∗},

where ϱ^{∗} : T^{∗}M → A^{∗} denotes the dual to the anchor map ϱ. In particular, in { · , · } and { · , · }^{T^{∗}M} coincide on M_{0}.

Proposition 4.5. We have that

σ^{(m+m′−1)}([P, Q]) = {σ^{(m)}(P), σ^{(m′)}(Q)}

for any P ∈ Ψ_{1,0}^{m}(M_{0}) and any Q ∈ Ψ_{1,0}^{m′}(M_{0}), where { · , · } is the usual Poisson bracket on A^{∗}.

Proof. The Poisson structure on T^{∗}M_{0} is induced from the Poisson structure on A^{∗}. In turn, the Poisson structure on T^{∗}M_{0} determines the Poisson structure on A^{∗}, because T^{∗}M_{0} is dense in A^{∗}. The desired result then follows from the similar result that is known for pseudodifferential operators on M_{0} and the Poisson bracket on T^{∗}M_{0}. □

We conclude with the following result, which is independent of the previous considerations, but sheds some light on them. The invariant differential operators on G are generated by d–vertical invariant vector fields on G, that is by Γ(A(G)). We have by definition that π_{M} = ϱ : Γ(M; A(G)) → Γ(M; T M), and hence π_{M} maps the algebra of invariant differential operators onto G to Diff_{V}^{∗}(M_{0}). In particular, the proof of Theorem 3.2 (more precisely Equation (30)) can be used to prove the following result, which we will however prove also without making appeal to Theorem 3.2.

Proposition 4.6. Let X ∈ Γ(A) and denote by a_{X}(ξ) = ξ(X) the associated linear function on A^{∗}. Then a_{X,χ} ∈ S^{1}(A^{∗}) and a_{X}(D) = −iX. Moreover,

{a_{χ}(D), a = polynomial in each fiber } = Diff_{V}^{∗}(M_{0}).

Proof. We continue to use a fixed metric on A to trivialize any density bundle. Let u = F_{fiber}^{−1}(a), where a ∈ S_{cl}^{m}(A^{∗}) polynomial in each fiber. By the Fourier inversion formula (and integration by parts), u is supported on M, which is the same thing as saying that u is a distribution the form ⟨u, f⟩ = ∫_{M} P_{0}f(x)dvol(x), with P_{0} a differential operator acting along the fibers of A → M and f ∈ C_{c}^{∞}(A). It then follows from the definition of a_{χ}(D), from the formula above for u = F_{fiber}^{−1}(a), and from the fact that χ = 1 in a neighborhood of the support of u that

(35)  a_{χ}(D)f(x) = [P_{0}f(exp_{x}(−v))]|_{v=0}, v ∈ T_{x}M_{0}.

Let X_{1}, X_{2}, …, X_{m} ∈ Γ(A) and

(36)  a = a_{X_{1}}a_{X_{2}} … a_{X_{m}} ∈ S^{m}(A^{∗}).

Then the differential operator P_{0} above is given by the formula

P_{0}f(x) = ∫_{A_{x}^{∗}} a(ξ)F^{−1}f(ξ),

with the inverse Fourier transform F^{−1} being defined along the fiber A_{x}. Hence

P_{0} = i^{m}X_{1}X_{2} … X_{m},

with each X_{j} being identified with the family of constant coefficients differential operators along the fibers of A → M that acts along A_{x} as the derivation in the direction of X_{j}(x).

For any X ∈ A, we shall denote by ψ_{tX} the one parameter subgroup of diffeomorphisms of M generated by X. (Note that ψ_{tX} is defined for any t because M is compact and X is tangent to all faces of M.) We thus obtain an action of ψ_{tX} on functions by [ψ_{tX}(f)](x) = f(exp(tX)x). Then the differential operator P_{0} is associated to a as in Equation (36) is given by

(37)  P_{0}(f ∘ exp)|_{M} = i^{m}[∂_{1}∂_{2} … ∂_{m}ψ_{t_{1}X_{1}+t_{2}X_{2}+...+t_{m}X_{m}}f]|_{t_{1}=...=t_{m}=0}.

Then Equations (35) and (37) give

(38)  a_{χ}(D)f = i^{m}[∂_{1} … ∂_{m} exp(−t_{1}X_{1} − … − t_{m}X_{m})f]|_{t_{1}=...=t_{m}=0},

In particular, a_{X}(D) = −iX, for any X ∈ Γ(A).

This proves that

(39)  a_{χ}(D) ∈ Diff_{V}^{∗}(M_{0}),

by the Campbell-Hausdorff formula [10, 36], which states that a_{χ}(D) is generated by X_{1}, X_{2}, …, X_{n} (and their Lie brackets), and hence that it is generated by V, which was assumed to be a Lie algebra.

Let us prove now that any differential operator P ∈ Diff_{V}^{∗}(M_{0}) is of the form a_{χ}(D), for some polynomial symbol a on A^{∗}. This is true if P has degree zero. Indeed, assume P is the multiplication by f ∈ C^{∞}(M). Lift f to an order zero symbol on A^{∗}, by letting this extension to be constant in each fiber. Then P = f(D). We shall prove our statement by induction on the degree m of P. By linearity, we can reduce to the case P = i^{−m}X_{1} … X_{m}, where X_{1}, …, X_{m} ∈ Γ(A). Let a = a_{X_{1}} … a_{X_{m}}. Then

σ_{m}(a_{χ}(D))(ξ) = a(ξ) = X_{1}(ξ) … X_{m}(ξ) = σ_{m}(P),

and hence Q := a_{χ}(D) − i^{−m}X_{1} … X_{m} ∈ Diff_{V}^{m−1}(M_{0}). By the induction hypothesis, Q = b_{χ}(D) for some polynomial symbol of order at most m − 1 on A^{∗}. This completes the proof. □

From this we obtain the following corollary.

Corollary 4.7. Let Diff(M_{0}) be the algebra of all differential operators on M_{0}. Then

Ψ_{1,0,V}^{∞}(M_{0}) ∩ Diff(M_{0}) = Diff_{V}^{∗}(M_{0}).

Proof. We know from the above proposition that

Ψ_{1,0,V}^{∞}(M_{0}) ∩ Diff(M_{0}) ⊃ Diff_{V}^{∗}(M_{0}).

Conversely, assume P ∈ Ψ_{1,0,V}^{m}(M_{0}) ∩ Diff(M_{0}). We shall prove by induction on m that P ∈ Diff_{V}^{m}(M_{0}). If m = 0 then P is the multiplication with a smooth function f on M_{0}. But then f = σ^{(0)}(P) ∈ S^{0}(A^{∗}) is constant along the fibers of A^{∗} → M, and hence f ∈ C^{∞}(M) ⊂ Diff_{V}^{∗}(M_{0}).

Assume now that the statement is proved for P of order < m. We shall prove it then for P of order m. Then a := σ^{(m)}(P) is a polynomial symbol in S^{m}(A^{∗}). Thus a_{χ}(D) ∈ Diff_{V}^{m}(M_{0}), by Proposition 4.6. But then σ^{(m)}(P − a_{χ}(D)) = 0, by Lemma 2.2, and hence P − a_{χ}(D) ∈ Ψ_{1,0,V}^{m−1}(M_{0}) ∩ Diff(M_{0}). By the induction hypothesis P − a_{χ}(D) ∈ Diff_{V}^{m−1}(M_{0}). This completes the proof. □

5. Group actions and semi-classical limits

One of the most convenient features of manifolds with a Lie structure at infinity is that questions on the analysis on these manifolds often reduce to questions on the analysis on simpler manifolds. These simpler manifolds are manifolds of the same dimension but endowed with certain non-trivial group actions. Harmonic analysis techniques then allow us to ultimately reduce our questions to analysis on lower dimensional manifolds with a Lie structure at infinity. In this section, we discuss the algebras Ψ_{1,0,V}^{∞}(M_{0}, G) that generalize the algebras Ψ_{1,0,V}^{∞}(M_{0}) when group actions are considered. These algebras are necessary for the reductions mentioned above and are typically the range of (generalized) indicial maps. Then we discuss a semi-classical version of the algebra Ψ_{1,0,V}^{∞}(M_{0}).

5.1. Group actions. We shall consider the following setting. Let M_{0} be a manifold with a Lie structure at infinity (M, A), and V = Γ(A), as above. Also, let G be a Lie group with Lie algebra 𝔤 := Lie(G). We shall denote by 𝔤_{M} the bundle M × 𝔤 → M. Then

(40)  V_{G} := V ⊕ C^{∞}(M, 𝔤) ≃ Γ(A ⊕ 𝔤_{M})

has the structure of a Lie algebra with respect to the bracket [ · , ·] which is defined such that on C^{∞}(M, 𝔤) it coincides with the pointwise bracket, on V it coincides with the original bracket, and, for any X ∈ V, f ∈ C^{∞}(M), and Y ∈ 𝔤, we have

[X, f ⊗ Y] := X(f) ⊗ Y .

(Here f ⊗ Y denotes the function ξ : M → 𝔤 defined by ξ(m) = f(m)Y ∈ 𝔤.)

The main goal of this subsection is to indicate how the results of the Section (2) extend to V_{G}, after we replace A with A ⊕ 𝔤_{M}, M_{0} with M_{0} × G, and M with M × G. The resulting constructions and definitions will yield objects on M × G that are invariant with respect to the action of G on itself by right translations.

We now proceed by analogy with the construction of the operators a_{χ}(D) in Subsection 2.3. First, we identify a section of V_{G} := V ⊕ C^{∞}(M, 𝔤) ≃ Γ(A ⊕ 𝔤_{M}) with a right G-invariant vector field on M_{0} × G. At the level of vector bundles, this corresponds to the map

(41)  p : T (M_{0} × G) = T M_{0} × T G → T M_{0} × 𝔤,

where the map T G → 𝔤 is defined using the trivialization of T G by right invariant vector fields. Let p_{1} : M × G → M be the projection onto the first component and p_{1}^{∗}A be the lift of A to M × G via p_{1}.

The map p defined in the Equation (41) can then be used to define the lift

(42)  p^{∗}(u) ∈ I^{m}(p_{1}^{∗}A ⊕ T G, M × G),

for any distribution u ∈ I^{m}(A ⊕ 𝔤_{M}, M). In particular, p^{∗}(u) will be a right Ginvariant distribution. Then we define R to be the restriction of distributions from p_{1}^{∗}A ⊕ T G to distributions on T M_{0} × T G = T (M_{0} × G).

We endow M_{0} × G with the metric obtained from a metric on A and a right invariant metric on G. This allows us to define the exponential map, thus obtaining, as in Section 2, a differentiable map

(43)  Φ : (T M_{0} × T G)_{r} = (T (M_{0} × G))_{r} → (M_{0} × G)^{2}

that is a diffeomorphism onto an open neighborhood of the diagonal, provided that r < r_{0}, where r_{0} is the injectivity radius of M_{0} × G. We shall denote as before by

Φ_{∗} : I_{c}^{m}((T M_{0} × T G)_{r}, M_{0} × G) → I_{c}^{m}((M_{0} × G)^{2}, M_{0} × G)

the induced map on conormal distributions.

The inverse Fourier transform will give a map

(44)  F_{fiber}^{−1} : S_{1,0}^{m}(A^{∗} ⊕ 𝔤_{M}^{∗}) ⟶ I^{m}(A ⊕ 𝔤_{M}, M),

defined by the same formula as before (Equation (6)). Finally, we shall also need a smooth function χ on A ⊕ 𝔤_{M} that is equal to 1 in a neighborhood of the zero section and has support inside (A ⊕ 𝔤_{M})_{r}.

We can then define the quantization map in the G-equivariant case by

(45) q_{Φ,χ,G} := Φ_{∗} ∘ R ∘ p^{∗} ∘ χ ∘ F_{fiber}^{−1} : S_{1,0}^{m}(A^{∗} ⊕ 𝔤_{M}^{∗}) ⟶ I^{m}((M_{0} × G)^{2}, M_{0} × G).

The main difference with the definition in Equation (11) is that we included the map p^{∗}, which is the lift of distributions in I^{m}(A ⊕ 𝔤_{M}, M) to G-invariant distributions in I^{m}(p_{1}^{∗}A ⊕ T G, M_{0} ⊗ G), see Equation (42). Then

(46)  a_{χ}(D) = T ∘ q_{Φ,χ,G},

as before.

With this definition of the quantization map, all the results of the previous sections remain valid, with the appropriate modifications. In particular, we obtain the following definition of the algebra of G-equivariant pseudodifferential operators associated to (A, M, G).

Definition 5.1. For m ∈ ℝ, the space Ψ_{1,0,V}^{m}(M_{0}, G) of G-equivariant pseudodifferential operators generated by the Lie structure at infinity (M, A) is the linear space of operators C_{c}^{∞}(M_{0} × G) → C_{c}^{∞}(M_{0} × G) generated by a_{χ}(D), a ∈ S_{1,0}^{m}(A^{∗} ⊕ 𝔤_{M}^{∗}), and b_{χ}(D)ψ_{X_{1}} … ψ_{X_{k}}, b ∈ S^{−∞}(A^{∗} ⊕ 𝔤_{M}^{∗}) and X_{j} ∈ Γ(A ⊕ 𝔤_{M}).

The space Ψ_{cl,V}^{m}(M_{0}, G) of classical G-equivariant pseudodifferential operators generated by the Lie structure at infinity (M, A) is defined similarly, but using classical symbols a.

With this definition, all the results on the algebras Ψ_{1,0,V}^{m}(M_{0}) and Ψ_{cl,V}^{m}(M_{0}) extend right away to the spaces Ψ_{1,0,V}^{m}(M_{0}, G) and Ψ_{cl,V}^{m}(M_{0}, G). In particular, these spaces are algebras, are independent of the choice of the metric on A used to define them, and have the usual symbolic properties of the algebras of pseudodifferential operators.

The only thing that maybe needs more explanations is with what we replace π_{M} in the G-equivariant case, because in the G-equivariant case we no longer use the vector representation. Let G be a groupoid integrating A, Γ(A) = V. Then G × G integrates A ⊕ 𝔤_{M}. If P = (P_{x}) ∈ Ψ_{1,0}^{m}(G × G), then we consider π_{0}(P) to be the operator induced by P_{x} on (G_{x}/G_{x}^{x}) × G, x ∈ M_{0}, the later space being a quotient of (G × G)_{x}. We shall use then π_{0} instead of π_{M} in the G-equivariant case. (By the proof of Theorem 3.2, π_{0} = π_{M}, if G is reduced to a point.)

5.2. Indicial maps. The main reason for considering the algebras Ψ_{1,0,V}^{m}(M_{0}, G) and their classical counterparts is the following. Let (M, V), V = Γ(M, A), be a manifold with a Lie structure at infinity. Let N_{0} ⊂ M be a submanifold such that T_{x}N_{0} = ϱ(A_{x}) for any x ∈ N_{0}. Moreover, assume that N_{0} is completely contained in an open face F ⊂ M such that N := \overline{N_{0}} is a submanifold with corners of F and N_{0} = N ∖ ∂N. Then the restriction A|_{N_{0}} is such that the Lie bracket on V = Γ(A) descends to a Lie bracket on Γ(A|_{N_{0}}). (This is due to the fact that the space I of functions vanishing on N is invariant for derivations in V. Then IV is an ideal of V, and hence V/IV ≃ Γ(A|_{N}) is naturally a Lie algebra.)

Assume now that there exists a Lie group G and a vector bundle A_{1} → N such that A|_{N} ≃ A_{1} ⊕ 𝔤_{N} and V_{1} := V|_{N} ≃ Γ(A_{1}). Then V_{1} is a Lie algebra and (N_{0}, N, A_{1}) is also a manifold with a Lie structure at infinity. In many cases (certainly for many of the most interesting examples) one obtains for any Lie group H a natural morphism

(47)  R_{N} : Ψ_{1,0,V}^{m}(M_{0}; H) → Ψ_{1,0,V_{1}}^{m}(N_{0}; G × H).

For example, the generalizations of the morphisms considered in [17] are of the form (47). However, we do not know exactly what are the conditions under which the morphism R_{N} above is defined.

Let 𝔥 = Lie H and 𝔥_{N} = M × Lie H. Then, at the level of kernels the morphism defined by Equation (47) corresponds to the restriction maps

r_{N} : I^{m}(A^{∗} ⊕ 𝔥_{N}^{∗}, M) → I^{∗}(A^{∗}|_{N} ⊕ 𝔥_{N}^{∗}, N) ≃ I^{∗}(A_{1}^{∗} ⊕ 𝔤_{N} ⊕ 𝔥_{N}^{∗}, N)

in the sense that R_{N}(a_{χ}(D)) = (r_{N}(a))_{χ}(D).

5.3. Semi-classical limits. We now define the algebra Ψ_{1,0,V}^{m}(M_{0}[[h]]), an element of which will be, roughly speaking, a semi-classical family of operators (T_{t}), T_{t} ∈ Ψ_{1,0,V}^{m}(M_{0}) t ∈ (0, 1]. See [46] for some applications of semi-classical analysis.

Definition 5.2. For m ∈ ℝ, the space Ψ_{1,0,V}^{m}(M_{0}[[h]]) of pseudodifferential operators generated by the Lie structure at infinity (M, A) is the linear space of families of operators T_{t} : C_{c}^{∞}(M_{0} × G) → C_{c}^{∞}(M_{0} × G), t ∈ (0, 1], generated by

a_{χ}(t, tD) , a ∈ S_{1,0}^{m}([0, 1] × A^{∗} ⊕ 𝔤_{M}^{∗}),

and

b_{χ}(t, tD)ψ_{tX_{1}(t)} … ψ_{tX_{k}(t)}, b ∈ S^{−∞}([0, 1] × A^{∗} ⊕ 𝔤_{M}^{∗}), X_{j} ∈ Γ([0, 1] × A ⊕ 𝔤_{M}).

The space Ψ_{cl,V}^{m}(M_{0}[[h]]) of semi-classical families of pseudodifferential operators generated by the Lie structure at infinity (M, A) is defined similarly, but using classical symbols a.

Thus we consider families of operators (T_{t}), T_{t} ∈ Ψ_{1,0,V}^{m}(M_{0}), defined in terms of data a, b, X_{k}, that extends smoothly to t = 0, with the interesting additional feature that the cotangent variable is rescaled as t → 0.

Again, all the results on the algebras Ψ_{1,0,V}^{m}(M_{0}) and Ψ_{cl,V}^{m}(M_{0}) extend right away to the spaces Ψ_{1,0,V}^{m}(M_{0}[[h]]) and Ψ_{cl,V}^{m}(M_{0}[[h]]), except maybe Proposition 4.6 and its corollary, Corollary 4.7, that need to be properly reformulated.

Another variant of the above constructions is to consider families of manifolds with a Lie structure at infinity. The necessary changes are obvious though, and we will not discuss them here.

References

[1] B. Ammann, A. Ionescu, and V. Nistor, Sobolev spaces and regularity for polyhedral domains, Preprint 2004/2005, ArXiv:math.AP/0402321.

[2] B. Ammann, R. Lauter, and V. Nistor, On the geometry of Riemannian manifolds with a Lie structure at infinity, Int. J. Math. Math. Sci. (2004), no. 1-4, 161–193.

[3] B. Ammann, R. Lauter, V. Nistor, and A. Vasy, Complex powers and non-compact manifolds, Comm. Part. Diff. Eq. 29 (2004), no. 5-6, 671–705.

[4] C. Bacuta, V. Nistor, and L. Zikatanov, Improving the rate of convergence of ‘high order finite elements’ on polygons, Numerische Mathematik 100 (2005), no. 2, 165–184.

[5] C. Bär, The Dirac operator on hyperbolic manifolds of finite volume, J. Diff. Geom. 54 (2000), no. 3, 439–488.

[6] M. Costabel, Boundary integral operators on curved polygons, Ann. Mat. Pura Appl. (4) 133 (1983), 305–326.

[7] M. Crainic and R. Fernandes, Integrability of Lie brackets, Ann. of Math. (2) 157 (2003), no. 2, 575–620.

[8] M. Dauge, Elliptic boundary value problems on corner domains, Lecture Notes in Mathematics, vol. 1341, Springer-Verlag, Berlin, 1988, Smoothness and asymptotics of solutions.

[9] C. Epstein, R.B. Melrose, and G. Mendoza, Resolvent of the Laplacian on strictly pseudoconvex domains, Acta Math. 167 (1991), 1–106.

[10] G. Hochschild, The structure of Lie groups. (Holden-Day Series in Mathematics), San Francisco-London-Amsterdam: Holde-Day, Inc. IX, 230 p. , 1965.

[11] L. Hörmander, Fourier integral operators I, Acta Math. 127 (1971), 79–183.

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

[13] M. Karoubi, Homologie cyclique et K-theorie, Astérisque 149 (1987), 1–147.

[14] A. Kumpera and D. Spencer, Lie Equations, vol. 1, Princeton Univ. Press., 1972.

[15] R. Lauter, Pseudodifferential analysis on conformally compact spaces, Mem. Amer. Math. Soc., vol. 163, 2003.

[16] R. Lauter and S. Moroianu, Fredholm theory for degenerate pseudodifferential operators on manifolds with fibered boundaries, Comm. Part. Diff. Eq. 26 (2001), 233–283.

[17] R. Lauter and V. Nistor, Analysis of geometric operators on open manifolds: a groupoid approach, Quantization of Singular Symplectic Quotients (N.P. Landsman, M. Pflaum, and M. Schlichenmaier, eds.), Progress in Mathematics, vol. 198, Birkhäuser, Basel - Boston - Berlin, 2001, pp. 181–229.

[18] J. Lewis, Layer potentials for elastostatics and hydrostatics in curvilinear polygonal domains, Trans. Amer. Math. Soc. 320 (1990), no. 1, 53–76.

[19] J. Lewis and C. Parenti, Pseudodifferential operators of Mellin type, Comm. Part. Diff. Eq. 8 (1983), no. 5, 477–544.

[20] K. Mackenzie, Lie groupoids and Lie algebroids in differential geometry, Lecture Notes Series, vol. 124, London Mathematical Society, 1987.

[21] R. Mazzeo, Elliptic theory of differential edge operators. I., Comm. Part. Diff. Eq. 16 (1991), no. 10, 1615–1664.

[22] R. Mazzeo and R.B. Melrose, Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature, J. Funct. Anal. 75 (1987), 260–310.

[23] R. Mazzeo and R.B. Melrose, Pseudodifferential operators on manifolds with fibered boundaries, Asian J. Math. 2 (1998), 833–866.

[24] R. Mazzeo and A. Vasy, Analytic continuation of the resolvent of the Laplacian on SL(3)/SO(3), Amer. J. Math. 126 (2004), 821–844.

[25] R.B. Melrose, Analysis on manifolds with corners, book: in preparation.

[26] R.B. Melrose, Transformation of boundary value problems, Acta Math. 147 (1981), 149–236.

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

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

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

[30] R.B. Melrose and G. Mendoza, Elliptic operators of totally characteristic type, MSRI Preprint.

[31] G. Mendoza, Strictly pseudoconvex b-CR manifolds, Comm. Part. Diff. Eq. 29 (2004), no. 9-10, 1437–1503.

[32] M. Mitrea and V. Nistor, Boundary layer potentials on manifolds with cylindrical ends, ESI Preprint no. 1244, 2002.

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

[34] V. Nistor, Groupoids and the integration of Lie algebroids, J. Math. Soc. Japan 52 (2000), 847–868.

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

[36] A. Onishchik, Lie groups and Lie algebras I. Foundations of Lie theory. Lie transformation groups. Transl. from the Russian by A. Kozlowski., Encyclopaedia of Mathematical Sciences.

20. Berlin: Springer-Verlag., 1993.

[37] C. Parenti, Operatori pseudodifferentiali in ℝ^{n} e applicazioni, Annali Mat. Pura ed App. 93 (1972), 391–406.

[38] E. Schrohe, Spectral invariance, ellipticity, and the Fredholm property for pseudodifferential operators on weighted Sobolev spaces., Ann. Global Anal. Geom. 10 (1992), no. 3, 237–254.

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

[40] B.W. Schulze, Boundary value problems and singular pseudo-differential operators., Wiley-Interscience Series in Pure and Applied Mathematics. Chichester: John Wiley & Sons., 1998.

[41] M. Shubin, Pseudodifferential operators and spectral theory, Springer Verlag, Princeton, N.J., 1987.

[42] S. Simanca, Pseudo-differential operators, Pitman research notes in mathematics, vol. 171, Longman Scientific & Technical, Harlow, Essex, 1990.

[43] M. Taylor, Pseudodifferential operators, Princeton Mathematical Series, vol. 34, Princeton University Press, Princeton, N.J., 1981.

[44] M. Taylor, Partial differential equations, Applied Mathematical Sciences, vol. I–III, Springer-Verlag, New York, 1995-1997.

[45] A. Vasy, Propagation of singularities in many-body scattering, Ann. Sci. École Norm. Sup. (4) 34 (2001), no. 3, 313–402.

[46] A. Vasy and M. Zworski, Semiclassical estimates in asymptotically Euclidean scattering, Comm. Math. Phys. 212 (2000), no. 1, 205–217.

[47] J. Wunsch, Propagation of singularities and growth for Schrödinger operators, Duke Math. J. 98 (1999), 137–186.

Bernd Ammann, L’Institut Élie Cartan (IÉCN), Université Henri Poincaré, B.P. 239, F-54506 Vandoeuvre-Les-Nancy, Cedex, France

E-mail address: ammann@iecn.u-nancy.fr

Robert Lauter, Universität Mainz, Fachbereich 17 – Mathematik, D - 55099 Mainz, Germany

E-mail address: lauter@mathematik.uni-mainz.de

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

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