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720 B. AMMANN, R. LAUTER, AND V. NISTOR
Acknowledgements. We thank Andras Vasy for several interesting discus-
sions and for several contributions to this paper. R. L. is grateful to Richard
B. Melrose for numerous stimulating conversations and explanations on pseu-
dodifferential calculi on special examples of manifolds with a Lie structure
at infinity. V. N. would like to thank the Institute Erwin Schr¨odinger in
Vienna and University Henri Poincar´e 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 Rieman-
nian 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
≥ 0onM such that
H = {x
H
=0} and dx
H
=0 on H.
For the basic facts on the analysis of manifolds with corners we refer to the
forthcoming book [25]. We shall denote by ∂M the union of all nontrivial
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,TM) 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
PSEUDODIFFERENTIAL OPERATORS
721
V⊂Γ(M, TM) 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 are 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; TM),
where r is the distance to the boundary.
(c) As another example consider a manifold with smooth boundary and con-
sider 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 nontrivial 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 defini-
tion that M = M
0
and V =Γ(M,TM), so that 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 noncompact.
Elements in the enveloping algebra Diff
∗
V
(M)ofV are called V-differential
operators on M. The order of differential operators induces a filtration
Diff
m
V
(M), m ∈ N
0
, on the algebra Diff
∗
V
(M). Since Diff
∗
V
(M)isaC
∞
(M)-
module, we can introduce V-differential operators acting between sections of
smooth vector bundles E,F → M, E,F ⊂ M × C
N
by
Diff
∗
V
(M; E, F):=e
F
M
N
(Diff
∗
V
(M))e
E
,(2)
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.
722 B. AMMANN, R. LAUTER, AND V. NISTOR
Let A → M be a vector bundle and  : A → TM a vector bundle map.
We shall also denote by  the induced map Γ(M, A) → Γ(M,TM) between
the smooth sections of these bundles. Suppose a Lie algebra structure on
Γ(M,A) is given. Then the pair (A, ) together with this Lie algebra structure
on Γ(A) is called a Lie algebroid if ([X, Y ]) = [(X),(Y )] and [X, fY ]=
f[X, Y ]+((X)f)Y for any smooth sections X and Y of A and any smooth
function f on M. The map  : A → TM is called the anchor of A. We have
also denoted by  the induced map Γ(M,A) → Γ(M,TM). We shall also write
Xf := (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

V
: A
V
−→ TM

M
(3)
such that V = 
V
(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 alge-
bras of vector fields V =Γ(A
V
) and Lie algebroids  : A → TM such that the
induced map Γ(M,A) → Γ(M, TM) is injective and has range in the Lie alge-
bra 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 TM
0
.
We now turn our attention to Riemannian structures on M
0
. Any metric
on A induces a metric on TM
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 TM
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 necessar-
ily of infinite volume and complete. Moreover, all the covariant derivatives
of the Riemannian curvature tensor are bounded. Under additional mild as-
sumptions, 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 refer-
ences therein.) A useful property is that all geometric operators on M
0
that
PSEUDODIFFERENTIAL OPERATORS
723
are associated to a metric on A are V-differential operators (i.e., in Diff
m
V
(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 boundary 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 pseudodifferen-
tial calculus on M
0
that microlocalizes the algebra of V-differential operators
Diff
∗
V
(M
0
)onM 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 : TM
0
→ M
0
the natural projection. Define
Φ:TM
0
−→ M
0
× M
0
, Φ(v):=(x, exp
x
(−v)),x= π(v).(4)
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 TM
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|
(TM
0
)
r
is a diffeomorphism onto an open
724 B. AMMANN, R. LAUTER, AND V. NISTOR
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)) ∈ (TM
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
ξ :=

1+g(ξ, ξ).(5)
We shall denote by S
m
1,0
(E) the symbols of type (1, 0) in H¨ormander’s sense [12].
Recall that they are defined, in local coordinates, by the standard estimates
|∂
α
x
∂
β
ξ
a(ξ)|≤C
K,α,β
ξ
m−|β|
,π(ξ) ∈ K,
where K is a compact subset of M trivializing E (i.e., π
−1
(K)  K × R
n
) and
α and β are multi-indices. If a ∈ S
m
1,0
(E), then its image in S
m
1,0
(E)/S
m−1
1,0
(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| are large. A symbol a ∈ S
m
1,0
(E) will be called classical if there
exist symbols a
k
∈ S
m−k
1,0
(E), homogeneous of degree m − k, such that a −

N−1
j=0
a
k
∈ S
m−N
1,0
(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
π : A
∗
→ M be the canonical projections. Then the inverse of
the Fourier transform F
−1
fiber
, along the fibers of A
∗
gives a map
F
−1
fiber
: S
m
1,0
(A
∗
) −→ C
−∞
(A):=C
∞
c
(A)

, F
−1
fiber
a, ϕ := a, F
−1
fiber
ϕ,(6)
where a ∈ S
m
1,0
(A
∗
), ϕ is a smooth, compactly supported function, and
F
−1
fiber
(ϕ)(ξ):=(2π)
−n

π(ζ)=π(ξ)
e
iξ,ζ
ϕ(ζ) dζ.(7)
Then I
m
(A, M ) is defined as the image of S
m
1,0
(A
∗
) through the above map. We
shall call this space the space of distributions on A conormal to M. The spaces
I
m
(TM
0
,M
0
) and I
m
(M
2
0
, Δ
M
0
)=I
m
(M
2
0
,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,
PSEUDODIFFERENTIAL OPERATORS
725
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 pseu-
dodifferential operators. For example, since we have
I
m
(M
2
0
,M
0
) ⊆C
−∞
(M
2
0
):=C
∞
c
(M
2
0
)

,
we can associate to a distribution in K ∈ I
m
(M
2
0
,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 H¨ormander [11], [12] states that T
K
is a pseudod-
ifferential 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
T : I
m
(M
2
0
,M
0
) → Hom(C
∞
c
(M
0
), C
−∞
(M
0
)).(8)
Recall now that (A)
r
denotes the set of vectors of norm <rof the vector
bundle A. We agree to write I
m
(r)
(A, M ) for all k ∈ I
m
(A, M ) with supp k ⊆
(A)
r
. The space I
m
(r)
(TM
0
,M
0
) is defined in an analogous way. Then restriction
defines a map
R : I
m
(r)
(A, M ) −→ I
m
(r)
(TM
0
,M
0
).(9)
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
Φ
∗
: I
m
(r)
(TM
0
,M
0
) → I
m
(M
2
0
,M
0
).(10)
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
m
c
(Y,X) denotes the space of compactly supported conormal
distributions, so that I
m
(r)
(A, M )=I
m
c
((A)
r
,M). Occasionally, we shall use
the double subscripts “cl,pr” and “cl,c.” Note that “c” implies “pr”.
726 B. AMMANN, R. LAUTER, AND V. NISTOR
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
m
1,0
(A
∗
) −→ I
m
(M
2
0
,M
0
),
q
Φ,χ
(a):=Φ
∗

R

χF
−1
fiber
(a)

.(11)
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
),
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 .(12)
Specializing to the case of Euclidean space M
0
= R
n
with the standard metric
we have τ(x, y)=x − y, and hence
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 ,(13)
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
.Ifχ
1
and χ
2
are smooth functions with
support (A)
r
and χ
j
=1in a neighborhood of M ⊆ A, then (χ
1
− χ
2
)F
−1
fiber
(a)
is a smooth function, and hence a
χ
1
(D) − a
χ
2
(D) has a smooth Schwartz
kernel. Moreover, the map S
m
1,0
(A
∗
) →C
∞
(A) that maps a ∈ S
m
1,0
(A
∗
) to
(χ
1
− χ
2
)F
−1
fiber
(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
−1
fiber
(a) and χ
2
F
−1
fiber
(a) are
contained in the diagonal Δ
M
0
and χ
1
− χ
2
vanishes there, we have that
(χ
1
− χ
2
)F
−1
fiber
(a) is a smooth function.
To prove the continuity of the map S
m
1,0
(A
∗
)  a → (χ
1
− χ
2
)F
−1
fiber
(a) ∈
C
∞
(A), it is enough, using a partition of unity, to assume that A → M is a triv-
ial 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).
PSEUDODIFFERENTIAL OPERATORS
727
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
m
1,0
(A
∗
) and each
χ ∈C
∞
c
((A)
r
) with χ =1close to M ⊆ A, the distribution q
Φ,χ
(a) is the
Schwartz-kernel of a pseudodifferential operator a
χ
(D) on M
0
, which is prop-
erly supported if r<∞ and has principal symbol σ
(μ)
(a) ∈ S
m
1,0
(E)/S
m−1
1,0
(E).
If a ∈ S
μ
cl
(A
∗
), then a
χ
(D) is a classical pseudodifferential operator.
Proof. Denote also by χ : I
m
(TM
0
,M
0
) → I
m
(r)
(TM
0
,M
0
) the “multipli-
cation by χ” map. Then
a
χ
(D)=T ◦ Φ
∗
◦R◦χ ◦F
−1
fiber
(a):=T
Φ
∗
(R(χF
−1
fiber
(a)))
= T ◦ q
Φ,χ
(a)(14)
where T is defined as in Equation (8). Hence a
χ
(D) is a pseudodifferential
operator by H¨ormander’s result mentioned above [11], [12] (stating that the
distribution conormal to the diagonal is exactly the Schwartz kernel of pseu-
dodifferential 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 → TM to the interior A|
M
0
is the identity. (This also follows
from 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
m
1,0
(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 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.
728 B. AMMANN, R. LAUTER, AND V. NISTOR
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
m
1,0
(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
ψ
X
: C
∞
(M) →C
∞
(M).(15)
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.Form ∈ R, the space Ψ
m
1,0,V
(M
0
)ofpseudodiffer-
ential 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
m
1,0
(A
∗
), and
b
χ
(D)ψ
X
1
ψ
X
k
, b ∈ S
−∞
(A
∗
) and X
j
∈ Γ(A), ∀j.
Similarly, the space Ψ
m
cl,V
(M
0
)ofclassical pseudodifferential operators gen-
erated 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
m
1,0,V
(M
0
) and Ψ
∞
cl,V
(M
0
):=∪
m∈
Z
Ψ
m
cl,V
(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
σ
(m)
:Ψ
m
1,0,V
(M
0
)/Ψ
m−1
1,0,V
(M
0
) → S
m
1,0
(A
∗
)/S
m−1
1,0
(A
∗
)(16)
and
σ
(m)
:Ψ
m
cl,V
(M
0
)/Ψ
m−1
cl,V
(M
0
) → S
m
cl
(A
∗
)/S
m−1
cl
(A
∗
).(17)
Proof. This follows from the classical case of the spaces Ψ
m
(M
0
)by
Lemma 2.2.
PSEUDODIFFERENTIAL OPERATORS
729
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 ∈Gwill 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 →Gsatisfying 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., mul-
tiplication, 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,
T
vert
G := ∪
x∈M
T G
x
=kerd
∗
,(18)
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,