Pseudodifferential operators on manifolds with fibred corners
aa r X i v : . [ m a t h . DG ] D ec PSEUDODIFFERENTIAL OPERATORS ON MANIFOLDS WITHFIBRED CORNERS
CLAIRE DEBORD, JEAN-MARIE LESCURE, AND FR´ED´ERIC ROCHON
Abstract.
One way to geometrically encode the singularities of a stratifiedpseudomanifold is to endow its interior with an iterated fibred cusp metric. Forsuch a metric, we develop and study a pseudodifferential calculus generalizingthe Φ-calculus of Mazzeo and Melrose. Our starting point is the observa-tion, going back to Melrose, that a stratified pseudomanifold can be ‘resolved’into a manifold with fibred corners. This allows us to define pseudodifferen-tial operators as conormal distributions on a suitably blown-up double space.Various symbol maps are introduced, leading to the notion of full ellipticity.This is used to construct refined parametrices and to provide criteria for themapping properties of operators such as Fredholmness or compactness. Wealso introduce a semiclassical version of the calculus and use it to establisha Poincar´e duality between the K -homology of the stratified pseudomanifoldand the K -group of fully elliptic operators. Contents
Introduction 11. Manifolds with fibered corners and stratified pseudomanifolds 42. Vector fields on manifolds with fibred corners 103. The definition of S -pseudodifferential operators 134. Groupoids 175. Action of S -pseudodifferential operators 216. Suspended S -operators 247. Symbol Maps 288. Composition 319. Mapping properties 3610. The semiclassical S -calculus 4611. Poincar´e duality 53References 62 Introduction
To study linear elliptic equations on singular spaces, it is very helpful to havea pseudodifferential calculus adapted to the geometry of the singularities. Indeed,such a tool allows one to construct refined parametrices to geometric operators likethe Laplacian, leading to a precise description of the space of solutions and typi-cally having important consequences and applications in spectral theory, scatteringtheory, index theory and regularity theory. This has also applications to studycertain non-linear elliptic equations, see for instance [26],[15], [45] for recent works in that direction. Over the years, various types of pseudodifferential calculi havebeen introduced on non-compact and singular spaces, see for instance [24], [8], [29],[48], [22], [23] [20], [18], [13] and [44]. Such a diversity of calculi comes from thefact that different types of singularities usually require quite different treatments.Still, many of the examples above are concerned with a particular class of singu-lar spaces: stratified pseudomanifolds. The notion of stratified pseudomanifold isrelatively easy to describe and has the advantage of including many important ex-amples of singular spaces, going from manifolds with corners to algebraic varieties.One could therefore hope for a relatively uniform treatment of pseudodifferentialoperators in this context. However, it is necessary to first choose a Riemannianmetric geometrically encoding the singularities. There are two natural choices. Topresent these two choices, let us first consider a stratified pseudomanifold of depthone, that is, with only one singular stratum, see Figure 1 and Figure 2. ❈❈❈❈❈❈❈❈❈❈❈❈ E ✘✘✘✘✘✘❳❳❳❳❳❳✘✘✘✘✘✘❳❳❳❳❳❳✘✘✘✘✘✘❳❳❳❳❳❳ ✲ rL Figure 1. g ed ❈❈❈❈❈❈❈❈❈❈❈❈ E L ✲ r Figure 2. g fc The first choice, going back to Cheeger [7], is to consider an incomplete edgemetric, a prototypical example being a metric which in a neighborhood of thesingular stratum takes the form(1) g ed = dr + g E + r g L , where r is the distance to the singular stratum, g E is a Riemannian metric onthe singular stratum (the edge) and g L is a choice of metric on the link. In thissetting, a pseudodifferential calculus was developed independently by Mazzeo [22]and Schulze [48]. In [22], the metric which is really used as a starting point is infact the conformally related metric(2) e g ed = g ed r = dr r + g E r + g L , a complete edge metric, which has the virtue of defining a Lie algebra of vectorfields ‘generating’ the pseudodifferential calculus.Alternatively, one can consider a fibred cusp metric to encode the singularity,which is a certain type of complete Riemannian metric of finite volume on the SEUDODIFFERENTIAL OPERATORS ON MANIFOLDS WITH FIBRED CORNERS 3 regular stratum. A prototypical example of such metric is one which near thesingular stratum takes the form(3) g fc = dr r + g E + r g L . For such a metric, a pseudodifferential calculus was introduced by Mazzeo andMelrose [23] starting with a Lie algebra of smooth vector fields associated to theconformally related metric(4) g fb = g fc r = dr r + g E r + g L . Both (1) and (3) have analogs on general stratified pseudomanifolds by iteratingthe definition. The generalization of (1) is called an iterated edge metric [1] . Animportant source of examples of iterated edge metrics is given by certain constantcurvature metrics [26] and by K¨ahler-Einstein metrics singular along a divisor [15].On the other hand, we call the analog of (3) for a general stratified pseudomanifoldan iterated fibred cusp metric , see Definition 2.3 below. For stratified pseudoman-ifolds of depth one, natural examples of such metrics are given by certain fibredcusp K¨ahler-Einstein metrics, see [45].For iterated edge metrics, an associated pseudodifferential calculus has beenintroduced in [37] and [38] for operators of order zero and was used in [36]. Thereis also a recent survey [47] by Schulze giving a nice description of how his methodscan be adapted to stratified pseudomanifolds of higher depth. Adopting a Liegroupoid point of view, one can obtain a pseudodifferential calculus by applyingthe general method of [39] and [2], which works for both iterated edge metrics anditerated fibred cusp metrics. This latter approach is suitable for certain applicationsin index theory, but the properness condition on the support of the operators makesit less appealing for the construction of refined parametrices. Still, in certain cases,this can be avoided by introducing a length function, see [43].In this paper, we propose to systematically develop and study a calculus of pseu-dodifferential operators on stratified pseudomanifolds equipped with an iteratedfibred cusp metric. We call it the S -calculus. Our approach takes its inspirationfrom [23], which deals with the case of a stratified pseudomanifold of depth 1. Inparticular, we start with a Lie algebra of smooth vector fields associated to iter-ated fibred corner metrics , a type of metrics conformally related to iteratedfibred cusp metrics. To be able to consider stratified pseudomanifolds of arbitrarydepth, our starting point is the idea, going back to Melrose (see [1]), that a strati-fied pseudomanifold can be resolved by a manifold with fibred corners. This allowsus to use blow-up techniques in a systematic way to construct the double space onwhich the Schwartz kernels of the operators can naturally be defined.To prove that this pseudodifferential calculus is closed under composition, wediverge from [23] and follow an approach closer in spirit to [18]. Beside the ‘usual’principal symbol, we introduce a ‘noncommutative’ symbol for each singular stra-tum of the stratified pseudomanifold by restricting on a corresponding front facein the double space. This lead to a simple Fredholm criterion for polyhomoge-neous pseudodifferential operators: an operator is Fredholm when acting on suit-able Sobolev spaces if and only if it is elliptic and its ‘noncommutative’ symbolsare invertible for each stratum. We say such operators are fully elliptic. For fullyelliptic operators, we are able to construct a refined parametrix giving rise to a CLAIRE DEBORD, JEAN-MARIE LESCURE, AND FR´ED´ERIC ROCHON corresponding regularity result. This refined parametrix can also be used to showthat our calculus is spectrally invariant, namely, that an invertible operator (whenacting on suitable Sobolev spaces) has its inverse also contained in the calculus.Along the way, we have a parallel discussion that keeps track of the underlyingLie groupoid and relates our approach with the one of [39] and [2]. This becomesparticularly useful at the end of the paper, where we establish a Poincar´e dualitybetween the fully elliptic S -operators and the K-homology of the stratified pseudo-manifold. In [36], such a Poincar´e duality was obtained using the pseudodifferentialoperators of [37] and [38]. Using instead groupoids, the first two authors introducedin [10] the noncommutative tangent space of a stratified pseudomanifold and showedits K-theory is Poincar´e dual to the K-homology of the stratified pseudomanifold.In fact, they showed more generally that the C ∗ -algebra of the noncommutativetangent space is dual in the sense of KK-theory to the C ∗ -algebra of continuousfunctions of the underlying stratified pseudomanifold.A key feature of our approach is the introduction of the semiclassical S -doublespace and its associated semiclassical S -calculus. This allows us to define a contin-uous family groupoid T FC X playing the role in our context of the noncommutativetangent space of [10]. By looking at the associated algebra of pseudodifferentialoperators, we can then provide a simple way of relating classes of fully elliptic S -operators with elements of the K -theory of T FC X (see Theorem 10.6 below). Thisallows us to use a hybrid combination of the operator theoretic methods of [36](see also [33]) and the groupoid approach of [10] to obtain our Poincar´e dualityresult in KK -theory, Theorem 11.1. In Theorem 11.4, we also provide an interpre-tation of this result in terms of a quantization map for full symbols of fully elliptic S -operators, relating in this way the points of view of [10] and [36].The paper is organized as follows. In Section 1, we introduce the definition ofmanifolds with fibered corners and recall from [1] how they can be used to ‘resolve’stratified pseudomanifolds. In Section 2, we introduce a natural class of vectorfields defined on a manifold with fibered corners X . This leads to the notion of S -pseudodifferential operators in Section 3. In Section 4, we review the definitionof groupoid and explain its relevance to the present context. In Section 5, wedescribe how S -pseudodifferential operators act on smooth functions. Section 6is about suspended operators, which are used in Section 7 to introduce varioussymbol maps for S -operators. In section 8, we prove that the composition of two S -operators is again a S -operator. In Section 9, we introduce natural Sobolev spaceson which S -operators act and provide criteria to determine when a S -operator isbounded, compact or Fredholm. In Section 10, we introduce the semiclassical S -double space and the associated semiclassical S -calculus, as well as the Lie groupoid T FC X . This is used to obtain a relationship between classes of fully elliptic S -operators and elements of the K-theory of T FC X . Finally, in Section 11, we establisha Poincar´e duality in KK -theory between T FC X and the stratified pseudomanifold S X associated to X and interpret it in terms of a quantization map. Aknowledgement.
The authors are very grateful to Thomas Krainer for manyhelpful conversations. Manifolds with fibered corners and stratified pseudomanifolds
Let X be a manifold with corners as defined in [27]. In particular, we areassuming that each boundary hypersurface H ⊂ X is embedded in X . This means SEUDODIFFERENTIAL OPERATORS ON MANIFOLDS WITH FIBRED CORNERS 5 that there exists a boundary defining function x H ∈ C ∞ ( X ) such that x − H (0) = H , x H is positive on X \ H and the differential dx H is nowhere zero on H . In sucha situation, one can choose a smooth retraction r H : N H → H , where N H isa (tubular) neighborhood of H in X such that ( r H , x H ) : N H → H × [0 , ∞ ) isa diffeomorphism on its image. We call ( N H , r H , x H ) a tube system for H . Asmooth map φ : X → Y between manifolds with corners is said to be a fibration if it is a locally trivial surjective submersion. Definition 1.1.
Let X be a compact manifold with corners and H , . . . , H k an ex-haustive list of its set of boundary hypersurfaces M X . Suppose that each bound-ary hypersurface H i is the total space of a smooth fibration π i : H i → S i wherethe base S i is also a compact manifold with corners. The collection of fibrations π = ( π , . . . , π k ) is said to be an iterated fibration structure if there is a partialorder on the set of hypersurfaces such that(i) for any subset I ⊂ { , . . . , k } with ∩ i ∈ I H i = ∅ , the set { H i | i ∈ I } is totallyordered.(ii) If H i < H j , then H i ∩ H j = ∅ , π i ( H i ∩ H j ) = S i with π i : H i ∩ H j → S i a surjective submersion and S ji := π j ( H i ∩ H j ) ⊂ S j is one of theboundary hypersurfaces of the manifold with corners S j . Moreover, thereis a surjective submersion π ji : S ji → S i such that on H i ∩ H j we have π ji ◦ π j = π i .(iii) The boundary hypersurfaces of S j are exactly the S ji with H i < H j . Inparticular if H i is minimal, then S i is a closed manifold.A manifold with fibred corners is a manifold with corners X together with aniterated fibration structure π . A smooth map ψ : X → X ′ between two manifoldswith fibred corners ( X, π ) and ( X ′ , π ′ ) is said to be a diffeomorphism of mani-folds with fibred corners if it is a diffeomorphism of manifolds with corners andif for each H i ∈ M X , there is H ′ µ ( i ) ∈ M X ′ and a diffeomorphism ψ i : S i → S ′ µ ( i ) inducing a commutative diagram H i ψ / / π i (cid:15) (cid:15) H ′ µ ( i ) π ′ µ ( i ) (cid:15) (cid:15) S i ψ i / / S ′ µ ( i ) . Remark 1.2.
With the previous notation, for any j , S j is naturally a manifoldwith fibered corners. The hypersurfaces are the S ji with fibration π ji : S ji → S i for any i such that H i < H j . The same goes for the fibres of π i . Precisely, if x belongs to S i let L xi := π − i ( x ). Then L xi is a manifold with fibered corners, wherethe boundary hypersurfaces are the L xi ∩ H j with H i < H j and the correspondingfibration comes from the restriction of π j . Notice that in the special case where H i is maximal, the fibre L xi is a closed manifold. Definition 1.3.
A family of tube system ( N i , r i , x i ) for H i , i = 1 , . . . , k is an iterated fibred tube system of the manifold with fibred corners X if the followingcondition holds for H i < H j , r j ( N i ∩ N j ) ⊂ N i , x i ◦ r j = x i , π i ◦ r i ◦ r j = π i ◦ r i on N i ∩ N j , A more standard terminology would be smooth fibre bundle.
CLAIRE DEBORD, JEAN-MARIE LESCURE, AND FR´ED´ERIC ROCHON and the restriction to H j of the function x i is constant on the fibres of π j .If X is equipped with an iterated fibred tube system, then for each H i ∈ M X ,we have an induced iterated fibred tube system on each fibre of π i : H i → S i .Similarly, there is an induced iterated fibred tube system on the base S i .To see that manifolds with fibred corners always admit iterated fibred tubesystems, it is useful to describe tube systems in terms of vector fields. Given a tubesystem ( N H , r H , x H ) for the boundary hypersurface H , one can naturally associateto it a vector field ξ H ∈ C ∞ ( N H ; T X ) such that( r H , x H ) ∗ ξ H = ∂∂x H . Clearly, the tube system can be recovered from this vector field by considering itsflow. More generally, if η H ∈ C ∞ ( X ; T X ) is a vector field which is inner pointingan nowhere vanishing on H , but tangent to all other boundary hypersurfaces, wecan construct a tube system ( N ′ H , r ′ H , x ′ H ) such that( r ′ H , x ′ H ) ∗ η H = ∂∂x ′ H by considering the flow of η H for some short period of time E H . Thus, to obtainan iterated fibred tube system, it suffices to associate a vector field ξ H i to eachboundary hypersurface H i in such a way that,(i) The restriction ξ H i | H i is inner pointing and nowhere vanishing on H i ;(ii) If H i < H j , then ξ H i is tangent to H j and there is a vector field ξ ji ∈C ∞ ( S j , T S j ) such that ( π j ) ∗ ( ξ H i | H j ) = ξ ji , while ξ H j is tangent to thefibres of the fibration π i : H i → S i on H i . Moreover, in a neighborhood of H i ∩ H j , we have that [ ξ H i , ξ H j ] = 0.Indeed, the flows of these vector fields generates tube system for each boundaryhypersurface. The condition that [ ξ H i , ξ H j ] insures that the flows of ξ H i and ξ H j commute, so that in particular x i ◦ r j = x i near H i ∩ H j , while requiring ξ H j to betangent to the fibres of π i : H i → S i insures that π i ◦ r i ◦ r j = π i ◦ r j near H i ∩ H j .On the other hand, the condition ( π j ) ∗ ( ξ H i | H j ) = ξ ji insures that x i | H j is constanton the fibres of π j : H j → S j . Thus, by shrinking the tube systems if necessary, wecan insure they form an iterated fibred tube system. Lemma 1.4.
A manifold with fibred corners always admit an iterated fibred tubesystem.Proof.
By the discussion above, it suffices to find vector fields ξ H i ∈ C ∞ ( X ; T X )for each boundary hypersurface H i ∈ M X in such a way that conditions (i) and(ii) above are satisfied. This requires to construct the vectors fields ξ ij on S j aswell.Recall that the depth of X is the highest codimension of a boundary face of X .If X has depth zero, that is, if X is a smooth manifold, the lemma is trivial. Wecan thus proceed by induction on the depth of X to prove the lemma. In particular,for each i , the base S i of the fibration π i is a manifold with fibred corners of depthsmaller than the one of X and we can assume we have vector fields satisfying ( i )and ( ii ) on S i . We can denote the vector field associated to the boundary face S ji of S j by ξ ji . Proceeding by recurrence on the partial order of the S i to constructthese vector fields, starting with S i minimal, we can assume furthermore that for H i < H j < H k , we have SEUDODIFFERENTIAL OPERATORS ON MANIFOLDS WITH FIBRED CORNERS 7 (iii) ( π kj ) ∗ ξ ki | S kj = ξ ji .To construct the vector fields ξ H i on X , we can proceed by recurrence on M X starting with maximal elements. For a maximal element H i , we just choose ξ H i ∈C ∞ ( X ; T X ) such that ξ H i | H i is inner pointing and nowhere vanishing on H i andis tangent to the fibres of π j : H j → S j for H j = H i .Suppose now that H i is a hypersurface such that for all H j ∈ M X with H j >H i , the corresponding vector field ξ H j has been constructed in such a way thatconditions ( i ) and ( ii ) hold. To construct ξ H i , we first define it on the maximalelements in { H ∈ M X ; H > H i } . Let H j be such a maximal element. We choose ξ H i | H j ∈ C ∞ ( H j ; T H j ) in such a way that on H j , we have that ( π j ) ∗ ( ξ H i | H j ) = ξ ji .This can be done by also requiring at the same time that [ ξ H i | H j , ξ H k | H j ] = 0near H i ∩ H j ∩ H k for H k such that H j > H k > H i . Indeed, we can do so byfirst constructing ξ H i | H j recursively on the boundary faces of H j , starting with theboundary faces of smallest dimension, and extending the definition at the next levelusing the flow of vector fields ξ H k for H j > H k > H i whenever possible. Thanksto the fact condition (ii) is satisfied by the vector fields ξ H k , this can be achievedconsistently. In this process, we also require that ξ H i | H j be tangent to the fibres of π l : H l → S l for H l < H i . Using the flow of ξ H j , we can then extend the definitionof ξ H i to a neighborhood of H i ∩ H j .Doing this for all maximal elements in { H ∈ M X ; H > H i } , we then proceedrecursively to extend the definition of ξ H i in a neighborhood of the other hypersur-faces of { H ∈ M X ; H > H i } . Thus, let H k > H i be given and suppose that ξ H i has already been defined in a neighborhood of H j for H j > H i such that H j > H k .Then ξ H i | H k is already defined in a neighborhood of H j ∩ H k . As before, proceedingby recurrence on the boundary faces of H k , we can extend the definition of ξ H i | H k to all of H k in such a way that ( π k ) ∗ ( ξ H i | H k ) = ξ ki and [ ξ H i | H k , ξ H l | H j ] = 0 near H k ∩ H l for H k > H l > H i . We can also require ξ H i | H k to be tangent to thefibres of π l : H l → S l for H l < H i . Using the flow of ξ H k , we can then extend thedefinition of ξ H i to a neighborhood of H k ∩ H i . Because the already defined vectorfields satisfy condition (ii), this extension is consistent with the previous ones.Thus, proceeding recursively, we can extend the definition of ξ H i to a neighbor-hood of [ { H ∈ M X ; H>H i } H i ∪ H. We can then extend this definition further in such a way that ξ H i is tangent to thefibres of π k : H k → S k for H k < H i . This give us a vector field ξ H i which togetherwith the already existing vector fields satisfies conditions (i) and (ii), completingthe inductive step and the proof. (cid:3) As observed by Melrose and subsequently described in [1], there is a correspon-dence between manifolds with fibered corners and stratified pseudomanifolds. Forthe convenience of the reader and in order to set up the notation, we will reviewthe main features of this correspondence and refer to [1] for further details.Let us first recall what are pseudomanifolds. We will use the notations andequivalent descriptions given by A. Verona in [53] and used by the first two authorsin [10].
CLAIRE DEBORD, JEAN-MARIE LESCURE, AND FR´ED´ERIC ROCHON
Let S X be a locally compact separable metrizable space. Recall that a C ∞ -stratification of S X is a pair ( S , N ) such that,(1) S = { s i } is a locally finite partition of S X into locally closed subsets of S X ,called the strata, which are smooth manifolds such that s ∩ ¯ s = ∅ if and only if s ⊂ ¯ s . In that case we will write s ≤ s and s < s if moreover s = s .(2) N = { S N s , π s , ρ s } s ∈ S is the set of control data, namely S N s is an openneighborhood of s in S X , π s : S N s → s is a continuous retraction and ρ s : S N s → [0 , + ∞ [ is a continuous map such that s = ρ − s (0). The map ρ s is either surjective or the constant zero function.Moreover if S N s ∩ s = ∅ , then the map( π s , ρ s ) : S N s ∩ s → s × ]0 , + ∞ [is a smooth proper submersion.(3) For any strata s, t such that s < t , the set π t ( S N s ∩ S N t ) is included in S N s and the equalities π s ◦ π t = π s and ρ s ◦ π t = ρ s hold on S N s ∩ S N t .(4) For any two strata s and s the following two statements hold, s ∩ ¯ s = ∅ if and only if S N s ∩ s = ∅ ; S N s ∩ S N s = ∅ if and only if s ⊂ ¯ s , s = s or s ⊂ ¯ s . A stratification gives rise to a filtration. Indeed, if S X j is the union of strata ofdimension ≤ j , then, ∅ ⊂ S X ⊂ · · · ⊂ S X n = S X. We call n the dimension of S X and X • := S X \ S X n − the regular part of S X . Thestrata included in X • are called regular while strata included in S X \ X • are called singular . The set of singular (resp. regular) strata is denoted S sing (resp. S reg ). Definition 1.5. A stratified pseudomanifold is a triple ( S X, S , N ) where S X isa locally compact separable metrizable space, ( S , N ) is a C ∞ -stratification on S X and the regular part X • is a dense open subset of S X .Given a stratified pseudomanifold, notice that the closure of each of its stratais also naturally a stratified pseudomanifold. Given a manifold with fibred corners X , there is a simple way of extracting a stratified pseudomanifold. Let H , . . . , H k an exhaustive list of its boundary hypersurfaces and let π = ( π , . . . , π k ) be theiterated fibration structure on X . On X , consider the equivalence relation x ∼ y ⇐⇒ x = y or x, y ∈ H i with π i ( x ) = π i ( y ) for some H i . We denote by S X the quotient space of X by the previous equivalence relation and q : X → S X the quotient map. By construction, the restriction of q to X \ ∂X is a homeomorphism. We claim that S X is naturally a stratified pseudomanifold.Indeed, for any i ∈ I let σ i := π i ( H i \ S H k Let ( S X, S , N ) be a stratified pseudomanifold. Then the depth d ( s )of a stratum s is the biggest k such that one can find k different strata s , · · · , s k − such that s < s < · · · < s k − < s k := s. The depth of the stratification ( S , N ) of X is d ( X ) := sup { d ( s ) , s ∈ S } . A stratum whose depth is 0 will be called minimal. Remark 1.7. This definition is consistent with the notion of depth for mani-folds with corners, which constitute a particular type of stratified pseudomanifolds.Moreover, if X is a manifold with fibred corners of depth k , then its associatedstratified pseudomanifold S X also has depth k . Notice that different conventionsfor the depth are also common, see for instance [1].Let ( S X, S , N ) be a stratified pseudomanifold. For any singular stratum s , set L s := ρ − s (1). From [53], we know there is an isomorphism between S N s and L s × [0 , + ∞ [ / ∼ s where ∼ s is the equivalence relation induced by ( x, ∼ s ( y, 0) when π s ( x ) = π s ( y ). This local triviality around strata enables to make the unfoldingprocess of [6] (see also [10] for a complete description). If s is minimal, one canconstruct a pseudomanifold( S X \ s ) ∪ L s × [ − , ∪ ( S X \ s )using the gluing coming from the trivialization of the neighborhood S N s of s . If M isthe set of minimal strata of S X and m = S s ∈ M s is the union of the minimal strataof S X , then one can more generally construct the double stratified pseudomanifold2 X = ( S X \ m ) ∪ G s ∈ M L s × [ − , ! ∪ ( S X \ m )by gluing L s × [ − , 1] for s ∈ M via the trivialization of the neighborhood S N s of s . Since all the minimal strata of S X are involved, notice that the depth of2 X is one less that the one of S X . The stratified pseudomanifold 2 X also comeswith an involution τ interchanging the two copies of S X \ m with fixed point setidentified with L m = F s ∈ M L s × { } . This fixed point set is naturally a stratifiedpseudomanifold and come with a surjective map L m → m induced by the retractionsin each neighborhood S N s for s ∈ M .If S X has depth k , we can repeat this process k times to obtain a stratified pseu-domanifold 2 k X of depth 0, in other words, a smooth manifold. The manifold 2 k X comes with a continuous surjective map p : 2 k X → S X . At the j th step of thisunfolding process, we get a stratified pseudomanifold 2 j X with an involution τ j and a fixed point set equipped with a surjective map as before. This lift to 2 k X to give k involutions τ , . . . , τ k with k fixed point sets given by smooth hypersurfaces R , . . . , R k equipped with smooth fibrations on each of their connected components.The various bases of these fibrations are simply the smooth manifolds correspondingto the unfolded strata of S X . The complement 2 k X \ ( S kj =1 R j ) consists of 2 k copiesof X • . The closure of any one of these copies is naturally a manifold with corners FC X with boundary hypersurfaces given by parts of the hypersurfaces R , . . . , R k and have corresponding induced fibrations with bases given by manifolds with cor-ners. These fibrations give FC X a structure of manifold with fibred corners. We call FC X the manifold with fibered corners associated to the pseudomanifold S X .Up to the identifications described below, the two previous operations are mu-tually inverse. Precisely, starting with a stratified pseudomanifold S X and letting FC X be the associated manifold with fibered corners, we have for any x, y in FC X that x ∼ y if and only if p ( x ) = p ( y ). In other words, the map p factors througha homeomorphism FC X/ ∼ −→ S X which is a diffeomorphism in restriction to eachstrata and with respect to the control data. Conversely, starting with a manifoldwith fibered corners X and letting S X be its associated stratified pseudomanifold,it can be seen that FC X is isomorphic to the original manifold with fibred corners X by noticing that the unfolding process described above has an analog for manifoldswith fibred corners obtained by gluing along boundary hypersurfaces and with thesame resulting smooth manifold 2 k X .2. Vector fields on manifolds with fibred corners Let X be a manifold with corners with H , . . . , H k an exhaustive list of theboundary hypersurfaces of X . For each i ∈ { , . . . , k } , let x i ∈ C ∞ ( X ) be aboundary defining function for H i . Recall that V b ( X ) := { ξ ∈ Γ( T X ) ; ξx i ∈ x i C ∞ ( X ) ∀ i } is the Lie algebra of b -vector fields. Notice in particular that a b -vector field ξ ∈V b ( X ) is necessarily tangent to all the boundary hypersurfaces of X . Suppose that π = ( π , . . . , π k ) is an iterated fibration structure on X . Definition 2.1. The space V S ( X ) of S -vector fields on the manifold with fibredcorners ( X, π ) is(2.1) V S ( X ) := { ξ ∈ V b ( X ); ξ | H i is tangent to the fibres of π i : H i → S i and ξx i ∈ x i C ∞ ( X ) ∀ i } . Remark 2.2. This definition depends on the choice of boundary defining functions.To lighten the discussion, we might sometime not mention explicitly the choice ofboundary defining functions, but the use of S -vector fields and related conceptsalways presuppose such a choice has been made to start with. Moreover, we willusually assume the boundary defining functions are induced by an iterated fibredtube system.As can be seen directly from the definition, V S ( X ) is a C ∞ ( X ) module and isclosed under the Lie bracket. It is therefore a Lie subalgebra of Γ( T X ). Moregenerally, the space Diff k S ( X ) of S -differential operators of order k is the spaceof operators on C ∞ ( X ) generated by C ∞ ( X ) and product of up to m elements of V S ( X ). SEUDODIFFERENTIAL OPERATORS ON MANIFOLDS WITH FIBRED CORNERS 11 Away from the boundary, a S -vector field is just like a usual vector field in C ∞ ( X ; T X ). On the other hand, near a point p ∈ ∂X , it is useful to introducea system of coordinates in which S -vector fields admits a simple description. Tothis end, let H i , . . . , H i ℓ be the boundary hypersurfaces of X containing the point p ∈ ∂X . After relabelling if necessary, we can assume H , . . . , H ℓ are the boundaryhypersurfaces containing p and that(2.2) H < H < · · · < H ℓ . Let x i ∈ C ∞ ( X ) be the chosen boundary defining function for H i . Consider a smallneighborhood of p where for each i ∈ { , . . . , ℓ } , the fibration π i : H i → S i restrictsto be trivial. Consider then tuples of functions y i = ( y i , . . . , y k i i ) for 1 ≤ i ≤ ℓ and z = ( z , . . . , x q ) such that(2.3) ( x , y , . . . , x ℓ , y ℓ , z )form coordinates near p with the property that on H i , ( x , . . . , x i − , y , . . . , y i )induce coordinates on the base S i of the fibration π i : H i → S i such that π i corresponds to the map(2.4) ( x , . . . , b x i , . . . x ℓ , y, z ) → ( x , . . . , x i − , y , . . . , y i ) , where the b notation above the variable x i means it is omitted. Thus, the coordi-nates ( x i +1 , . . . , x ℓ , y i +1 , . . . , y ℓ , z ) restrict to give coordinates on the fibres of thefibration π i . With such coordinates, the Lie algebra V S ( X ) is locally spanned over C ∞ ( X ) by the vector fields(2.5) ∂∂z j , w i x i ∂∂x i , w i ∂∂y n i i , for j ∈ { , . . . , q } , i ∈ { , . . . , ℓ } , n i ∈ { , . . . , k i } , where w i = Q ℓm = i x m . Thus, inthese coordinates, a S -vector field ξ ∈ V S ( X ) is of the form(2.6) ξ = ℓ X i =1 a i x i w i ∂∂x i + ℓ X i =1 k i X j =1 b ij w i ∂∂y ji + q X m =1 c m ∂∂z m , with a i , b ij , c m ∈ C ∞ ( X ).Since V S ( X ) is a C ∞ ( M )-module, there exists a smooth vector bundle π T X → X and a natural map ι π : π T X → T X which restricts to an isomorphism on X \ ∂X such that(2.7) V S ( X ) = ι π C ∞ ( X ; π T X ) . At a point p ∈ X , the fibre of π T X above p can be defined by(2.8) π T p X = V S / I p · V S , where I p ⊂ C ∞ ( X ) is the ideal of smooth functions vanishing at p . Although themap ι π : π T X → T X fails to be an isomorphism of vector bundles, notice that π T X is nevertheless isomorphic to T X , but not in a natural way.Unless the hypersurface H i has no boundary, notice that the kernel of the naturalmap π T X | H i → T X | H i does not form a vector bundle over H i . To obtain a vectorbundle on H i , we need to introduce an intermediate vector bundle in between π T X and T X . Let X i = X ∪ H i X be the manifold with corners obtained by gluing two copies of X along the boundaryhypersurface H i . The manifold X i naturally has an iterated fibration structureinduced from the one of X . Hoping this will lead to no confusion, we will alsodenote this iterated fibration by π . We then have a corresponding Lie algebra V S ( X i ) of S -vector fields as well as an associated S -tangent vector bundle π T X i .Consider then the restriction of this vector bundle to one of the two copies of X inside X i , π \ π i T X = π T X i | X . Away from H i , the vector bundle π \ π i T X is canonically isomorphic to π T X . How-ever, seen as a subspace C ∞ ( X ; T X ), the space of sections C ∞ ( X ; π \ π i T X ) isslightly bigger than V S ( X ). We have in fact the following natural sequence ofmaps π T X / / π \ π i T X a π \ πi / / T X. When restricted to the boundary hypersurface H i , the first map π T X → π \ π i T X issuch that its kernel π N H i is naturally a vector bundle over H i . This vector bundleis the pullback of a vector bundle on S i . To see this, set π T H i := { ξ ∈ π \ π i T X (cid:12)(cid:12)(cid:12) H i ; a π \ π i ( ξ ) ∈ T H i } . The fibration π i : H i → S i induces the short exact sequence0 / / π T ( H i \ S i ) / / π T H i ( π i ) ∗ / / π ∗ i π T S i / / , where π T S i is the S -tangent bundle of S i and π T ( H i \ S i ) is such that its restrictionto each fibre F i of the fibration π i is the S -tangent bundle π T F i of that fibre.In particular, this induces a canonical identification π ∗ i π T S i = π T H i / π T ( H i /S i ).Now, using the vector field x i ∂∂x i induced by a tube system for H i , we have anatural decomposition(2.9) π N H i ∼ = ( π T H i / π T ( H i /S i )) × R . This means a tube system for H i induces an isomorphism of vector bundles(2.10) π N H i ∼ = π ∗ i π N S i where π N S i = π T S i × R , and thus, a corresponding fibration(2.11) F i π N H i ( π i ) ∗ (cid:15) (cid:15) π N S i . An iterated fibred corner metric (or S -metric ) is a choice of metric g π forthe vector bundle π T X . Via the map ι π : π T X → T X , it restricts to give acomplete Riemannian metric on X \ ∂X . In the local coordinates (2.3), a specialexample of such a metric would be(2.12) g π = ℓ X i =1 dx i ( x i w i ) + ℓ X i =1 k i X j =1 ( dy ji ) w i + q X m =1 dz m . The Laplacian associated to a S -metric is an example of S -differential operator oforder 2. Similarly, if π T X → X has a spin structure, then the corresponding Diracoperator associated to a S -metric is a S -differential operator of order 1. SEUDODIFFERENTIAL OPERATORS ON MANIFOLDS WITH FIBRED CORNERS 13 The S -density bundle associated to a manifold with fibred corners is thesmooth real line bundle π Ω with fibre above p ∈ X given by(2.13) π Ω p = { u : Λ dim X ( π T p X ) → R ; u ( tω ) = | t | u ( ω ) , ∀ ω ∈ Λ dim X ( π T p X ) , t = 0 } . A S -density is an element of C ∞ ( X ; π Ω). In particular, the volume form of a S -metric is naturally a S -density. Via the map ι π : π T X → T X , a S -density restrictsto give a density on the interior of X . Let ν ∈ C ∞ ( X ; Ω) be a non-vanishing densityon X , where Ω = Ω( T X ) is the density bundle associated to T X . On X \ ∂X , a S -density ν π can be written in terms of ν as(2.14) ν π = k Y i =1 x S i i ! − hν, for some h ∈ C ∞ ( X ) . As indicated in the introduction, S -metrics are conformally related to anothertype of metrics geometrically encoding the singularities of the stratified pseudo-manifold. Definition 2.3. On a manifold with fibred corners ( X, π ) with a boundary definingfunction x H specified for each boundary hypersurface H ∈ M X , an iteratedfibred cusp metric g ifc is a metric of the form g ifc = x g π , x = Y H ∈ M X x H , where g π is a S -metric.3. The definition of S -pseudodifferential operators Let X be a manifold with fibred corners. Let H , . . . , H k be an exhaustive listof its boundary hypersurfaces with x , . . . , x k ∈ C ∞ ( X ) a choice of correspondingboundary defining functions and π i : H i → S i the corresponding fibrations. Con-sider the Cartesian product X = X × X with the projections pr R : X × X → X and pr L : X × X → X on the right and left factors respectively. Then x ′ i := pr ∗ R x i and x i := pr ∗ L x i are boundary defining functions for X × H i and H i × X respec-tively. The b -double space X b is the space obtained from X by blowing up thep-submanifolds H × H , . . . , H k × H k ,(3.1) X b := [ X ; H × H ; . . . ; H k × H k ] , with blow-down map β b : X b → X . Near H i × H i , this amounts to the introductionof polar coordinates r i := q x i + ( x ′ i ) , ω i = x i r i , ω ′ i = x ′ i r i , where r i is a boundary defining function for the ‘new’ hypersurface B i := β − b ( H i × H i ) ⊂ X b introduced by the blow-up, while near B i , the functions ω i and ω ′ i are boundarydefining functions for the lifts of the ‘old’ boundary hypersurfaces. Notice thatsince H i × H i and H j × H j are transversal as p -submanifolds for i = j , the diffeo-morphism class of X b stays the same if we change the order in which we blow up(cf. Proposition 5.8.2 in [27] or [25, p.21]). ff π ✻ x ✲ x ′ (cid:0)(cid:0) (cid:0)(cid:0) (cid:0)(cid:0) (cid:0)(cid:0) ∆ π Figure 3. The π -double spaceTo define the π -double space, consider the fibre diagonal on the p -submanifold H i × H i ,(3.2) D π i = { ( h, h ′ ) ∈ H i × H i ; π i ( h ) = π i ( h ′ ) } . To lift this p -submanifold to the front face B i , notice that B i = SN + ( H i × H i )is by definition a quarter of circle bundle on H i × H i giving a canonical decompo-sition(3.3) B i = ( H i × H i ) × [ − , s i , with s i := ω i − ω ′ i . Thus, we can define a lift of D π i to B i by(3.4) ∆ i := { ( h, h ′ , ∈ H i × H i × [ − , s i ; π i ( h ) = π i ( h ′ ) } . The space ∆ i is a p -submanifold of B i and X b . To obtain the π -double space,it suffices to blow up ∆ i in X b for i ∈ { , . . . , k } . As opposed to the definition of X b , the order in which the blow-ups are performed is important, different ordersleading to different diffeomorphism classes of manifolds with corners. Fortunately,our assumptions on the partial order of hypersurfaces of X give us a systematicway to proceed.More precisely, assume that the hypersurfaces of X are labeled in such a waythat(3.5) i < j = ⇒ H i < H j or H i ∩ H j = ∅ . With this convention, we define the π -double space by(3.6) X π := [ X b ; ∆ ; . . . ; ∆ k ] . See Figure 3 for a picture of X π when X is a manifold with boundary. Noticethat the order in which we blow up is not completely determined by (3.5), but adifferent choice of ordering satisfying (3.5) would amount in commuting the blow-ups of p -submanifolds which do not intersect, an operation which does not affectthe diffeomorphism class of X π .We have corresponding blow-down maps(3.7) β π − b : X π → X b , β π := β b ◦ β π − b : X π → X . We denote the ‘new’ hypersurface introduced by blowing up ∆ i by(3.8) ff π i := ( β π − b ) − (∆ i ) ⊂ X π . SEUDODIFFERENTIAL OPERATORS ON MANIFOLDS WITH FIBRED CORNERS 15 The p -submanifold ff π i is called the front face associated to the boundary hyper-surface H i . Let also ∆ π := β − π ( ◦ ∆ X )denote the lift of the diagonal ∆ X ⊂ X × X to X π , where ◦ ∆ X is the interior of∆ X . On X π , we are particularly interested in the lift of the Lie algebra V S ( X ) withrespect to the natural maps(3.9) π L = pr L ◦ β π : X π → X, π R = pr R ◦ β π : X π → X. Lemma 3.1. The lifted diagonal ∆ π is a p -submanifold of X π . Furthermore, thelifts to X π of the Lie algebra V S ( X ) via the maps π L and π R are transversal to thelifted diagonal ∆ π .Proof. The result is trivial in the interior of X π . Thus, let p ∈ ∆ π ∩ ∂X π be given.We need to show that the lemma holds in a neighborhood of p in X π . Moreover,by symmetry, we only have to prove the result for the lift of V S ( X ) with respect tothe map π L . Let H i , . . . , H i ℓ be the boundary hypersurfaces of X containing thepoint π L ( p ) ∈ ∂X . After relabelling if necessary, we can assume H , . . . , H ℓ are thehypersurfaces containing π L ( p ) and that(3.10) H < H < · · · < H ℓ . Near π L ( p ) ∈ ∂X , let ( x i , y i , z ) be coordinates as in (2.3). Recall that in suchcoordinates, the Lie algebra V S ( X ) is locally spanned over C ∞ ( X ) by the vectorfields(3.11) ∂∂z j , w i x i ∂∂x i , w i ∂∂y n i i , for j ∈ { , . . . , q } , i ∈ { , . . . , ℓ } , n i ∈ { , . . . , k i } , where w i = Q ℓm = i x m .On X , we can then consider the coordinates(3.12) x i , y i , z, x ′ i , y ′ i , z ′ , where ( x i , y i , z ) is seen as the pullback of our coordinates from the left factor of X , while ( x ′ i , y ′ i , z ′ ) is the pullback of our coordinates from the right factor of X .On the b -double space X b , we can then consider the local coordinates(3.13) r i = x ′ i , s i = x i − x ′ i x ′ i , y i , y ′ i , z, z ′ , In these coordinates, we have that x i = r i ( s i + 1) , w i = w ′ i σ i , where w ′ i = ℓ Y j = i r j , σ i = ℓ Y j = i ( s j + 1) . Thus, under the map pr L ◦ β b , the vector fields of (3.11) lift to(3.14) ∂∂z j , w ′ i σ i ( s i + 1) ∂∂s i , w ′ i σ i ∂∂y n i i . Finally, on X π , we can consider the local coordinates near p given by(3.15) r i = x ′ i , S i = s i w ′ i , Y i = y i − y ′ i w ′ i , y ′ i , z, z ′ . In these coordinates, the lifted diagonal ∆ π is given by the subset { S i = 0 , Y i =0 , z = z ′ } . In particular, this shows it is a p -submanifold. From (3.14), we also seethat under the map π L , the vector fields of (3.11) lift to(3.16) ∂∂z j , σ i ( S i w ′ i + 1) ∂∂S i , σ i ∂∂Y n i i , where σ i = ℓ Y j = i ( S j w ′ j + 1) . These vector fields are clearly transversal to the lifted diagonal ∆ π = { S i = 0 , Y i =0 , z = z ′ } , which completes the proof. (cid:3) Corollary 3.2. The natural diffeomorphism ∆ π ∼ = X induced by the map π L (oralternatively by the map π R ) is covered by natural identifications N ∆ π ∼ = π T X, N ∗ ∆ π ∼ = π T ∗ X, where N ∆ π is the normal bundle of ∆ π in X π . Remark 3.3. It can also be proved that the Lie algebra V S ( X ) lifts via π L or π R to give a Lie subalgebra of V b ( X π ). Near ∆ π , this follows from the local description(3.16). Since we do not need this fact elsewhere on X π , we omit the proof.On the π -double space, the Schwartz kernels of S -differential operators admit asimple description. Let us first describe the Schwartz kernel of the identity operator.In the local coordinates (3.12) near ∆ X ∩ ( ∂X × ∂X ) , the Schwartz kernel of theidentity operator can be written as(3.17) K Id = ℓ Y i =1 δ ( x i − x ′ i ) δ ( y i − y ′ i ) dx i dy ′ i ! δ ( z − z ′ ) dz ′ = δ ( x − x ′ ) δ ( y − y ′ ) δ ( z − z ′ ) dx ′ dy ′ dz ′ , where in the second line we suppressed the subscripts to lighten the notation. Usingthe coordinates (3.13) on X b , this Schwartz kernel becomes(3.18) K Id = δ ( s ) δ ( y − y ′ ) δ ( z − z ′ ) dx ′ dy ′ z ′ w ′ , w ′ = ℓ Y j =1 x ′ j . Finally, using the local coordinates (3.15) on the π -double space X π , this becomes(3.19) K Id = δ ( S ) δ ( Y ) δ ( z − z ′ ) dx ′ dy ′ dz ′ Q ℓi =1 ( x ′ i )( w ′ i ) k i +1 , w ′ i = ℓ Y j = i x ′ j , = δ ( S ) δ ( Y ) δ ( z − z ′ ) dx ′ dy ′ dz ′ Q ℓi =1 ( x ′ i ) S i , = δ ( S ) δ ( Y ) δ ( z − z ′ ) π ∗ R ( ν π ) , where ν π ∈ C ∞ ( X ; π Ω) is a non-vanishing S -density on X . Let D (∆ π ) = C ∞ ( X π ) · µ be the space of smooth δ -functions on the p -submanifold ∆ π ⊂ X π , where µ is anon-vanishing delta function with smooth coefficient as in (3.19). From the localcomputation (3.19), we see that(3.20) K Id ∈ D (∆ π ) · π ∗ R ( ν π ) . SEUDODIFFERENTIAL OPERATORS ON MANIFOLDS WITH FIBRED CORNERS 17 Thus, if P ∈ Diff k S ( X ) is a S -differential operator of order k , we see from Lemma 3.1that(3.21) K P = π ∗ L P · K Id ∈ D k (∆ π ) · π ∗ R ( ν π ) , where D k (∆ π ) is the space of delta functions of order at most k , namely(3.22) D k (∆ π ) = Diff k ( X S ) · D (∆ π ) . In fact, since π ∗ L ( V S ( X )) is transversal to ∆ π by Lemma 3.1, the space of Schwartzkernels of S -differential operators of order k is precisely given by(3.23) D k (∆ π ) · π ∗ R ( ν π ) . If E and F are smooth complex vector bundles on X and Diff k S ( X ; E, F ) is thespace of S -differential operators of order k acting from C ∞ ( X ; E ) to C ∞ ( X ; F ), thenworking in local trivializations, we can in a similar way identify the correspondingspace of Schwartz kernels with(3.24) D k (∆ π ) · C ∞ ( X π ; β ∗ π Hom( E, F ) ⊗ π ∗ R ( π Ω)) , where Hom( E, F ) = pr ∗ L ( F ) ⊗ pr ∗ R ( E ∗ ).Since delta functions are a special type of conormal distributions, this suggests todefine S -pseudodifferential operators of order k acting from sections of E to sectionsof F by(3.25) Ψ m S ( X ; E, F ) := { K ∈ I m ( X π ; ∆ π ; β ∗ π (Hom( E ; F )) ⊗ π ∗ Rπ Ω); K ≡ ∂X π \ ff π } . Here, ff π := ∪ ki =1 ff π i and I m ( X π ; ∆ π ; β ∗ π (Hom( E, F )) ⊗ π ∗ Rπ Ω) is the space ofconormal distributions of order m at ∆ π . The notation K ≡ ∂X π \ ff( X π )means that its Taylor series identically vanishes at ∂X π \ ff( X π ). We can similarlydefine the space of polyhomogeneous (or classical) S -pseudodifferential operators oforder m by(3.26) Ψ m S − ph ( X ; E, F ) := { K ∈ I m ph ( X π ; ∆ π ; β ∗ π (Hom( E ; F )) ⊗ π ∗ R ( π Ω)); K ≡ ∂X π \ ff π } , where I m ph ( X π ; ∆ π ; β ∗ π (Hom( E, F )) ⊗ π ∗ Rπ Ω) is the space of polyhomogeneous conor-mal distributions of order m at ∆ π .With these definitions, notice that there are natural inclusions(3.27) Diff k S ( X ; E, F ) ⊂ Ψ k S − ph ( X ; E, F ) ⊂ Ψ k S ( X ; E, F ) . Groupoids We refer to [42, 21] for the classical definitions and constructions related togroupoids and their Lie algebroids. We recall here the basic definitions needed forthis paper.A groupoid is a small category in which every morphism is an isomorphism.Let us make this notion more explicit. A groupoid G is a pair ( G (0) , G (1) ) of setstogether with structural morphisms: the unit u : G (0) → G (1) , the source and range s, r : G (1) → G (0) , the inverse ι : G (1) → G (1) , and the multiplication µ which isdefined on the set G (2) of pairs ( α, β ) ∈ G (1) × G (1) such that s ( α ) = r ( β ). Here,the set G (0) denotes the set of objects (or units) of the groupoid, whereas the set G (1) denotes the set of morphisms of G . The identity morphism of any object of G enables one to identify that object with a morphism of G . This leads to theinjective map u : G (0) → G . Each morphism g ∈ G has a “source” and a “range.”The inverse of a morphism α is denoted by α − = ι ( α ). The structural maps satisfythe following properties,(i) r ( αβ ) = r ( α ) and s ( αβ ) = s ( β ), for any pair ( α, β ) in G (2) ,(ii) s ( u ( x )) = r ( u ( x )) = x , u ( r ( α )) α = α , αu ( s ( α )) = α ,(iii) r ( α − ) = s ( α ), s ( α − ) = r ( α ), αα − = u ( r ( α )), and α − α = u ( s ( α )),(iv) the partially defined multiplication µ is associative.We shall need groupoids with additional structures. Definition 4.1. A Lie groupoid (resp. locally compact groupoid ) is a groupoid G = ( G (0) , G (1) , s, r, µ, u, ι )such that G (0) and G (1) are manifolds with corners (resp. locally compact spaces),the structural maps s, r, µ, u, and ι are differentiable (resp. continuous), the sourcemap s is a submersion (resp. surjective and open) and G x := s − ( x ), x ∈ M , areall Hausdorff manifolds without corners (resp. locally compact Hausdorff spaces).We will also encounter the notion of continuous family groupoid ([41]). Definition 4.2. A locally compact groupoid G is a continuous family groupoid when it is covered by open sets U with homeomorphisms Φ f = ( f, φ f ) : U → f ( U ) × U f where f ∈ { r, s } and U f ⊂ R n such that the following holds:(1) for all ( U, Φ f ) and ( V, Ψ f ) as above such that W = U ∩ V = ∅ , the mapΨ f ◦ Φ − f : Φ f ( W ) −→ Ψ f ( W ) is of class C , ∞ , which means that x ψ f ◦ φ − f ( x, · ) is continuous from f ( W ) to C ∞ ( φ f ( W ) , ψ f ( W )) (which hasthe topology of uniform convergence on compacta of all derivatives);(2) The inversion and product maps are locally C , ∞ in the above sense.We say that ( U, Φ f ) is a C , ∞ local chart for ( G , f ).A simple example of Lie groupoid is the pair groupoid associated to a smoothmanifold M . It is obtained by taking G (0) = M , G (1) = M × M , s ( x, y ) = y , r ( x, y ) = x , ( x, y )( y, z ) = ( x, z ), u ( x ) = ( x, x ) and with inverse ι ( x, y ) = ( y, x ).Like vector bundles, groupoids can be pulled back. More precisely, let G ⇒ M be a locally compact Hausdorff groupoid with source s and range r . If f : N → M is a surjective map, the pullback groupoid ∗ f ∗ ( G ) ⇒ N of G by f is by definitionthe set(4.1) ∗ f ∗ ( G ) := { ( x, γ, y ) ∈ N × G × N | r ( γ ) = f ( x ) , s ( γ ) = f ( y ) } with the structural morphisms given by(1) the unit map x ( x, f ( x ) , x ),(2) the source map ( x, γ, y ) y and range map ( x, γ, y ) x ,(3) the product ( x, γ, y )( y, η, z ) = ( x, γη, z ) and inverse ( x, γ, y ) − = ( y, γ − , x ).The results of [35] apply to show that the groupoids G and ∗ f ∗ ( G ) are Moritaequivalent when f is surjective and open.The infinitesimal object associated to a Lie groupoid is its Lie algebroid, whichwe define next. SEUDODIFFERENTIAL OPERATORS ON MANIFOLDS WITH FIBRED CORNERS 19 Definition 4.3. A Lie algebroid A over a manifold M is a vector bundle A → M ,together with a Lie algebra structure on the space Γ( A ) of smooth sections of A anda bundle map ̺ : A → T M whose extension to sections of these bundles satisfies(i) ̺ ([ X, Y ]) = [ ̺ ( X ) , ̺ ( Y )], and(ii) [ X, f Y ] = f [ X, Y ] + ( ̺ ( X ) f ) Y ,for any smooth sections X and Y of A and any smooth function f on M .The map ̺ is called the anchor map of A . Note that we allow the base M inthe definition above to be a manifold with corners.Now, let G = G (1) s ⇒ r G (0) be a Lie groupoid. We denote by T s G the subbundleof T G (1) of s -vertical tangent vectors. In other words, T s G is the kernel of thedifferential T s of s .For any α in G (1) , let R α : G r ( α ) → G s ( α ) be the right multiplication by α . Atangent vector field Z on G (1) is right invariant if it satisfies,– Z is s -vertical, namely T s ( Z ) = 0.– For all ( α, β ) in G (2) , Z ( αβ ) = T R β ( Z ( α )).The Lie algebroid AG of a Lie groupoid G is defined as follows [21],– The fibre bundle AG → G (0) is the restriction of T s G to G (0) . In otherwords, AG = ∪ x ∈G (0) T x G x is the union of the tangent spaces to the s -fibreat the corresponding unit.– The anchor ρ : AG → T G (0) is the restriction of the differential T r of r to AG .– If Y : U → AG is a local section of AG , where U is an open subset of G (0) ,we define the local right invariant vector field Z Y associated with Y by Z Y ( α ) = T R α ( Y ( r ( α ))) for all α ∈ G U := r − ( U ) . The Lie bracket is then defined by[ , ] : Γ( AG ) × Γ( AG ) −→ Γ( A G )( Y , Y ) [ Z Y , Z Y ] G (0) where [ Z Y , Z Y ] denotes the s -vertical vector field obtained with the usualbracket and [ Z Y , Z Y ] G (0) is the restriction of [ Z Y , Z Y ] to G (0) . Remark 4.4. When G is a continuous family groupoid, the vector bundle AG →G (0) as defined above still exists. Indeed, the fibres G x , x ∈ G (0) are smoothmanifolds and we still can set(4.2) AG = G x ∈G (0) T x G x . This vector bundle is smooth in the sense of [41] and it is called the Lie algebroidof G again.In this paper, a central example of Lie algebroid is given by π T X with anchormap given by the natural map ι π : π T X → T X . In fact, the space ◦ X π ∪ ◦ ff π has anatural structure of Lie groupoid with Lie algebroid naturally identified with π T X under the identification X ∼ = ∆ π . More precisely, we set(4.3) G (0) π = ∆ π , G (1) π = ◦ X π ∪ ◦ ff π . For α ∈ G (1) π with β π ( α ) = ( x , x ), we define the source and range of α by(4.4) s ( α ) = x , r ( α ) = x . The map(4.5) ι : ◦ X × ◦ X → ◦ X × ◦ X ( x, x ′ ) ( x ′ , x )extends in a unique way to a smooth map ι : G (1) π → G (1) π defining on G (1) π theinverse map. Similarly, the natural multiplication map on the groupoid ◦ X × ◦ X extends to give a composition map(4.6) µ : G (2) π → G (1) π where(4.7) G (2) π = { ( α, β ) ∈ G (1) π × G (1) π ; r ( β ) = s ( α ) } . To see that the Lie algebroid of G π is precisely π T X , it suffices to use Corollary 3.2and to notice that AG π is isomorphic to N ∆ π , a fact that follows from the obser-vation that the source map of G π is a surjective submersion equal to the identitymap when restricted to units. The Lie groupoid G π admits a decomposition intosimpler groupoids. Indeed, for each boundary hypersurface H i of X , notice thatthe subgroupoid (ff π i ∩ ◦ ff π ) \ [ H i If a measured groupoid G is the finite disjoint union of measure-wise amenable (see [3, Definition 3.3.1] ) groupoids G i , that is, G = ⊔ i ∈ I G i and G (0) = ⊔ i ∈ I G (0) i , where everything is assumed to be borelian, then G is measurewiseamenable. In particular, C ∗ ( G ) is nuclear and equal to C ∗ r ( G ) .Proof. The fact G is measurewise amenable follows from [3, Proposition 5.3.4] ap-plied to the Borel map q : G (0) → I defined by q ( x ) = i if x ∈ G (0) i . By [3, 6.2.14],we then have that C ∗ ( G ) is nuclear and equal to C ∗ r ( G ). (cid:3) For instance, this criterion can be applied to the groupoid G π . Lemma 4.6. The groupoid G π is measurewise amenable. In particular, C ∗ ( G π ) isnuclear and equal to C ∗ r ( G π ) .Proof. By Lemma 4.5, it suffices to observe that G π can be written as a disjointunion of topologically amenable (and thus measurewise amenable, by [3]) groupoids,(4.8) G π = ( ◦ X × ◦ X ) G k G i =1 ( H i × π i π T S i × π i H i ) | G i × R ! , where G i = H i \ ( ∪ j>i H j ). The topological amenability of the various subgroupoidson the right-hand side can be justified as follows,(i) A vector bundle is topologically amenable as a bundle of abelian groups;(ii) Topological amenability is preserved under equivalence of groupoids ([3]).For instance, given a vector bundle E → S and a locally trivial fibre bundle p : H → S , the groupoid ( H × p E × p H ) ⇒ H is equivalent as a groupoid tothe vector bundle E , and thus is topologically amenable;(iii) The cartesian product of amenable groupoids is amenable. (cid:3) Action of S -pseudodifferential operators Let us first consider the space Ψ m S ( X ; E, F ) in (3.25) in the simpler situationwhere E = F = C . Notice that (3.25) can alternatively be rewritten as(5.1) Ψ m S ( X ) = I m ( X π ; ∆ π ) · C ∞ ff π ( X π ; π ∗ R ( π Ω)) , where C ∞ ff π ( X π ; π ∗ R ( π Ω)) is the space of smooth sections vanishing with all theirderivatives at all boundary faces except those contained in ff π . To describe theaction of S -operators on functions, we will need the following result about thepushforward of conormal distributions. Lemma 5.1. The map π L = pr L ◦ β π : X π → X induces a continuous linear map ( π L ) ∗ : I m ( X π ; ∆ π ) · C ∞ ff π ( X π ; Ω) → C ∞ ( X ; Ω) . Proof. If K ∈ I m ( X π ; ∆ π ) · C ∞ ff π ( X π ; Ω) is supported near the lifted diagonal, thenthe result follows from general properties of conormal distributions together withthe fact the map π L is transversal to ∆ π . Thus, using a cut-off function, we canassume K ∈ C ∞ ff π ( X π ; Ω). To proceed further, notice that π L is a b -fibration (werefer to [27] for a definition). Indeed, as a blow-down map, β π is a surjective b -submersion. Since the projection pr L : X → X is also clearly a surjective b -submersion, so is the composite π L = pr L ◦ β π . Thus, according to Proposition 2.4.2in [27], π L is a b -fibration provided no boundary hypersurface of X π is mappedto a boundary face of X of codimension greater than one. This is clear for the‘old’ hypersurfaces in X π , while the ‘new’ hypersurfaces are mapped under β π to boundary faces of X of codimension 2 which are then mapped under pr L toboundary faces of codimension 1 under the projection pr L .The lemma can then be seen as a special case of the Push-forward Theorem of[28] for b -fibrations. Precisely, the lemma is a consequence of this theorem combinedwith the fact π − L ( H i ) ∩ ff π = ff π i for all boundary hypersurfaces H i ⊂ X . (cid:3) Since the previous lemma is dealing with smooth densities, it cannot be applieddirectly to the space of conormal distributions Ψ m S ( X ). Lemma 5.2. The tensor product identification pr ∗ L Ω ⊗ pr ∗ R π Ω ≡ Ω on the interiorof X extends to give an isomorphism of spaces of sections C ∞ ff π ( X π ; β ∗ π (pr ∗ L Ω ⊗ pr ∗ R π Ω)) = C ∞ ff π ( X π ; Ω) Proof. It suffices to notice that the singular factors of sections of β ∗ π (pr ∗ L Ω ⊗ pr ∗ R π Ω)all arise at faces not contained in ff π , and so are absorbed by the infinite ordervanishing at these faces. This can be seen using the local coordinates. Indeed, inthe coordinates (3.12), an element of C ∞ ( X π ; pr ∗ L Ω ⊗ pr ∗ R ( π Ω) is of the form hdxdydzdx ′ dy ′ dz ′ Q ℓi =1 ( x ′ i ) S i = hdxdydzdx ′ dy ′ dz ′ Q ℓi =1 x ′ i ( w ′ i ) k i +1 , for some h ∈ C ∞ ( X π ) . Thus, in the coordinates of (3.15), it takes the form e hdSdY dzdx ′ dy ′ dz ′ for some e h ∈ C ∞ ( X π ) , and the only possible singular terms occur when S i → ∞ or Y i → ∞ , that is, atfaces not contained in ff π . (cid:3) SEUDODIFFERENTIAL OPERATORS ON MANIFOLDS WITH FIBRED CORNERS 23 We can then define a push-forward map(5.2) ( π L ) ∗ : I m ( X π ; ∆ π ) · C ∞ ff π ( X π ; π ∗ Rπ Ω) → C ∞ ( X )by requiring that for K ∈ I m ( X π ; ∆ π ) · C ∞ ff π ( X π ; π ∗ Rπ Ω) and any non-vanishingsection v ∈ C ∞ ( X ; Ω),(5.3) v · ( π L ) ∗ K = ( π L ) ∗ ( π ∗ L v · K ) , where the right hand side of (5.3) is in C ∞ ( X ; Ω) thanks to Lemma 5.1 andLemma 5.2. This push-forward map provides a way to make S -pseudodifferentialoperators act on functions. To state the main result of this section, we still needto introduce some notation. If M X is the set of boundary hypersurfaces and A ⊂ M X is a subset, then set x A = Y H ∈ A x H where x H ∈ C ∞ ( X ) is a choice of boundary defining function for H . For any A ⊂ M X , consider the space˙ C ∞ A ( X ; E ) = \ k ∈ N x kA C ∞ ( X ; E )of smooth sections on X vanishing with all their derivatives on each boundaryhypersurface H ∈ A . When A = M X , this gives the space˙ C ∞ ( X ; E ) = ˙ C ∞ M X ( X ; E )of smooth sections vanishing with all their derivatives on ∂X . It is also usefulto use the notation C ∞ A ( X ; E ) = ˙ C ∞ M X \ A ( X ; E ). Thus, for A = M X , we have C ∞ M X ( X ; E ) = C ∞ ( X ; E ).Each space ˙ C ∞ A ( X ; E ) comes with a natural structure of Fr´echet space inducedfrom the one of C ∞ ( X ; E ). The corresponding space of distributions ˙ C −∞ A ( X ; E ) isdefined to be the dual of C ∞ A ( X ; E ∗ ⊗ Ω). Similarly, we use the notation C −∞ A ( X ; E )to denote the dual of ˙ C ∞ A ( X ; E ∗ ⊗ Ω). Proposition 5.3. Via the push-forward map (5.2) , an element P ∈ Ψ m S ( X ; E, F ) defines a continuous linear map P : C ∞ ( X ; E ) → C ∞ ( X ; F ) . For each subset A ⊂ M X , this map restricts to give a continuous linear map P : ˙ C ∞ A ( X ; E ) → ˙ C ∞ A ( X ; F ) . These maps extend by continuity in the distributional topology to linear maps P : ˙ C −∞ A ( X ; E ) → ˙ C −∞ A ( X ; F ) for all subsets A ⊂ M X .Proof. The first assertion is a consequence of Lemma 5.1 and Lemma 5.2. Usinga partition of unity subordinate to a covering by open sets over which E and F restrict to be trivial, we can reduce to the case E = F = C to prove the secondassertion. Let A ⊂ M X be given. Since the function ( x A x ′ A ) ∈ C ∞ ( X \ ∂X ) pulls back to X π to give a function which is smooth on ff π and has only finite order singularitiesat hypersurfaces not in ff π , we see that P ∈ Ψ m S ( X ; E, F ) = ⇒ e P k = x kA ◦ P ◦ x − kA ∈ Ψ m S ( X ; E, F )for all k ∈ N . On the other hand, given u ∈ ˙ C ∞ A ( X ; E ), we can write it as u = x kA e u k for some e u k ∈ C ∞ ( X ; E ), so that x − kA P u = e P k e u k ∈ C ∞ ( X ; F ) = ⇒ P u ∈ x kA C ∞ ( X ; F ) . Since k ∈ N is arbitrary, this means P u ∈ ˙ C ∞ A ( X ; F ).For the proof of the last assertion, choose a non-vanishing density in C ∞ ( X ; π Ω)as well as Hermitian metrics for E and F . These then define a L -inner product forsections of E and F . To see the action of P ∈ Ψ m ( X ; E, F ) extends to distributions,it suffices to notice that from (3.25), the formal adjoint of P ∈ Ψ m ( X ; E, F ) withrespect to this L -inner product is an element of Ψ m S ( X ; F, E ), so that the actionof P on distributions can be defined by duality. (cid:3) The following proposition can be interpreted as a dual statement to the Schwartzkernel theorem. Proposition 5.4. A continuous linear operator A : ˙ C ∞ ( X ) → C −∞ ( X ) induces acontinuous linear map A : C −∞ ( X ) → ˙ C ∞ ( X ) if and only if its Schwartz kernel K A is an element of ˙ C ∞ ( X × X ; pr ∗ R Ω X ) where Ω X is the density bundle on X and pr R : X × X → X is the projection on the right factor.Proof. One proceeds as in the proof of Proposizione 1.2 in [40]. Namely, it sufficesto notice that if L ( C −∞ ( X ) , ˙ C ∞ ( X )) denotes the space of continuous linear maps(with C −∞ ( X ) equipped with the strong dual topology), then (see [52]) L ( C −∞ ( X ) , ˙ C ∞ ( X )) ∼ = ˙ C ∞ ( X ; Ω X ) b ⊗ ˙ C ∞ ( X ) ∼ = ˙ C ∞ ( X × X ; pr ∗ R Ω X ) . (cid:3) Let us denote by ˙Ψ −∞ S ( X ) the space of operators with Schwartz kernel in ˙ C ∞ ( X × X ; pr ∗ R Ω X ). From the definition of S -operators, it is clear that we have the iden-tification x ∞ Ψ −∞ S ( X ) = ˙Ψ −∞ S ( X ) where x = Q ki =1 x i . From Proposition 5.4, weimmediately obtain the following. Corollary 5.5. For A ∈ ˙Ψ −∞ S ( X ) and B ∈ Ψ m S ( X ) , we have AB ∈ ˙Ψ −∞ S ( X ) , BA ∈ ˙Ψ −∞ S ( X ) . Suspended S -operators Before describing the symbol maps associated to S -operators, we first need todiscuss how to suspend them in the sense of [30]. To this end, let ( X, π ) be amanifold with fibred corners and let H , . . . , H k be its boundary hypersurfaceswith corresponding boundary defining functions x , . . . , x k . Let V be a Euclidean SEUDODIFFERENTIAL OPERATORS ON MANIFOLDS WITH FIBRED CORNERS 25 vector space, that is, a finite dimensional real vector space with inner product h· , ·i V . Consider on V the function ρ V ( v ) = (1 + h v, v i V ) − , v ∈ V. Let V be the radial compactification of V as defined in [31], so that ρ V extendsto be a boundary defining function for ∂V ⊂ V . We can regard V as a manifoldwith fibred corners, the fibration on the boundary being given by the identity mapId : ∂V → ∂V . We can get a new manifold with fibred corners ( V × X, ̟ ) bytaking the Cartesian product of V and X . The iterated fibration structure ̟ of V × X is naturally induced from those of V and X as follows. The fibration ̟ onthe boundary hypersurface Z = ∂V × X is given by the projection on ∂V , whilethe fibration of the boundary hypersurface Z i = V × H i is given by ̟ i = Id × π i .The partial order on the boundary hypersurfaces of V × X is specified by requiringthat for all i, j ∈ { , . . . , k } ,(6.1) Z < Z i , Z i < Z j ⇐⇒ H i < H j . Finally, the boundary defining function of Z i = V × H i is taken to be the pullbackof x i to V × X , while we choose the boundary defining function x of Z to be thepullback of ρ V to V × X .Let E and F be smooth complex vector bundles on V × X obtained by pullingback complex vector bundles on X to V × X . Consider then the space Ψ m S ( V × X ; E, F ) of S -operators of order m acting from sections of E to sections of F . Fromthe previous section, we know that an operator P ∈ Ψ m S ( V × X ; E, F ) induces acontinuous linear map(6.2) P : S ( V × X ; E ) → S ( V × X ; F )where S ( V × X ; E ) = ˙ C ∞ Z ( V × X ; E ) is the space of smooth sections of E vanishingwith all their derivatives at the boundary hypersurface Z = ∂V × X , and similarly S ( V × X ; F ) = ˙ C ∞ Z ( V × X ; F ). Given v ∈ V , consider the diffeomorphism(6.3) T v : V × X → V × X ( w, p ) ( w + v, p )obtained by translating by v . Since E is the pullback of a vector bundle defined on X , we have a corresponding action(6.4) T ∗ v : S ( V × X ; E ) → S ( V × X ; E ) ψ ψ ◦ T v For the same reason, we have an action T ∗ v : S ( V × X ; F ) → S ( V × X ; F ). Definition 6.1. The space Ψ m S − sus( V ) ( X ; E, F ) of V -suspended S -operatorsof order m on X acting from sections of E to sections of F is the subspace ofoperators P in Ψ m S ( V × X ; E, F ) such that for all v ∈ V , T ∗− v ◦ P ◦ T ∗ v = P. When V = R p , we use the notation Ψ m S − sus( p ) ( X ; E, F ) = Ψ m S − sus( R p ) ( X ; E, F ) andsay the corresponding operators are p -suspended.In terms of the Schwartz kernel K P seen as a distribution on V × X , thetranslation invariance in this definition means that for all v ∈ V ,(6.5) T ∗ ( v,v ) K P = K P where T ( v,v ) is the diffeomorphism(6.6) T v : V × X → V × X ( w, w ′ , p, p ′ ) ( w + v, w ′ + v, p, p ′ ) . If(6.7) a : V → V ( v, v ′ ) ( v − v ′ )denotes the projection onto the anti-diagonal of V , this means that K P is thepullback via the map a × Id : V × X → V × X of a distribution on V × X .To accurately describe this distribution, notice first that parallel transport withrespect to the Euclidean metric on V gives a canonical identification of vectorbundles T V = V × T V = V × V extending naturally to a trivialization(6.8) Id T V ∼ = V × V. Using this identification and Corollary 3.2, one can see that the linear isomorphism(6.9) L : V × V → V × V ( v ′ , w ) ( v ′ + w, v ′ )naturally extends to give an identification Id T V ∼ = G (1)Id ( V ) of non-compact mani-folds with boundary, where G (1)Id ( V ) = ( V \ ∂V ) ∪ (ff Id \ ∂ ff Id )is the Lie groupoid associated to V . Since a ◦ L ( v ′ , w ) = w , this means the map a can be extended to a map(6.10) a : G (1)Id ( V ) → V by composing the identification G (1)Id ( V ) ∼ = Id T V ∼ = V × V with the projectionpr : V × V → V on the second factor.On the other hand, the ̟ -double space is naturally given by(6.11) ( V × X ) ̟ = V × X π , where V is the Id-double space of the manifold with fibred boundary V . Considerthen the map(6.12) α = a × Id : G (1)Id ( V ) × X π → V × X π . In terms of this map, the translation invariance condition in Definition 6.1 meansthat as a distribution on G (1)Id ( V ) × X π , the Schwartz kernel of a V -suspended S -operator is the pullback of a distribution on V × X π . More precisely, we haveobtained the following. Lemma 6.2. The space of Schwartz kernels of V -suspended S -operators of order m acting from sections of E to sections of F is given by Ψ m S − sus( V ) ( X ; E, F ) = { α ∗ K ; K ∈ I m ( V × X π ; { } × ∆ π ; V ) ,K ≡ at ( V × ∂X π \ ff π ) ∪ ( ∂V × X π ) } , SEUDODIFFERENTIAL OPERATORS ON MANIFOLDS WITH FIBRED CORNERS 27 where V = pr ∗ ( β ∗ π Hom( E, F ) ⊗ π ∗ Rπ Ω) ⊗ pr ∗ Ω with pr : V × X π → V and pr : V × X π → X π the natural projections. From that perspective, the action of anoperator P ∈ Ψ m S − sus( V ) ( X ; E, F ) on a section u ∈ S ( V × X ; E ) is given by P u = ( ̟ L ) ∗ ( α ∗ K P · ̟ ∗ R u ) , where ̟ L and ̟ R are the analog of the maps (3.9) for the manifold with fibredcorners V × X . Seen as a distribution on V × X π , it is possible to take the Fourier transform inthe V -factor of the Schwartz kernel K P of a V -suspended S -operator P ,(6.13) K b P (Υ) = Z V e − i Υ · v K P ( v ) , Υ ∈ V ∗ . We will call Υ ∈ V ∗ the suspension parameter . This gives for each Υ ∈ V ∗ the Schwartz kernel K b P (Υ) of a S -operator b P (Υ) ∈ Ψ m S ( X ; E, F ). Similarly, if ν denotes the translation invariant density on V associated to our choice of innerproduct h· , ·i V , then we can define the Fourier transform(6.14) F E : S ( V × X ; E ) → S ( V ∗ × X ; E )by(6.15) b u (Υ) = F E ( u )(Υ) = Z V e − i Υ · v u ( v ) ν, Υ ∈ V ∗ , with inverse Fourier transform given by(6.16) u ( v ) = F − E ( b u )( v ) = 1(2 π ) dim V Z V ∗ e i Υ · v b u (Υ) ν ∗ , where ν ∗ is the density on V ∗ dual to ν . With these definitions, we have as expectedthat the action of P on S ( V × X ; E ) can be described by(6.17) c P u (Υ) = b P (Υ) b u (Υ) , ∀ Υ ∈ V ∗ . In other words, the Fourier transform of P is given by(6.18) b P = F F ◦ P ◦ F − E . If Q ∈ Ψ m S − sus( V ) ( X ; G, E ) is another V -suspended operator, where G is a complexvector bundle on V × X given by the pullback of a complex vector bundle on X ,then we have in particular that(6.19) \ P ◦ Q (Υ) = b P (Υ) ◦ b Q (Υ) . That is, under the Fourier transform, the convolution product in the V -factor be-comes pointwise composition. Since an operator P can be recovered from b P bytaking the inverse Fourier transform, we see that b P completely describes the oper-ator P . It is important however to notice that the Fourier transform of an operator P ∈ Ψ m S − sus( V ) ( X ; E, F ) is not an arbitrary smooth family of S -operators. For in-stance, as can be readily seen by taking the Fourier transform of K P in directionsconormal to ∆ π ⊂ X π , me must have that(6.20) ( D α Υ b P )(Υ) ∈ Ψ m −| α | S ( X ; E, F ) , ∀ α ∈ N dim V , ∀ Υ ∈ V ∗ . For operators of order −∞ , we can completely characterize the image of the Fouriertransform. It is given by smooth families of S -operators V ∗ ∋ Υ b P (Υ) ∈ Ψ −∞ S ( X ; E, F ) such that for any Fr´echet semi-norm k · k of the space Ψ −∞ S ( X ; E, F ), we have(6.21) sup Υ k Υ α D β Υ b P k < ∞ , ∀ α, β ∈ N dim V . For operators of order m ∈ R , one has more generally that for any Fr´echet semi-norm k · k of Ψ m S ( X ; E, F ), the Fourier transform b P of a suspended operator P ∈ Ψ m S − sus( V ) ( X ; E, F ) must satisfy(6.22) sup Υ k (1 + | Υ | ) | α |− m D α Υ b P k < ∞ ∀ α ∈ N dim V . In this latter case however, these conditions are not sufficient to fully characterizethe image of the Fourier transform.The discussion above has a straightforward generalizations to families. Namely,consider a fibration(6.23) F H φ (cid:15) (cid:15) S where S is a manifold with corners and where the fibres are manifolds with fibredcorners. We suppose that the fibration is locally trivial in the sense that for each s ∈ S , there is a neighborhood U of s , a manifold with fibred corners F and adiffeomorphism ψ : φ − ( U ) → U × F inducing a commutative diagram(6.24) φ − ( U ) ψ / / φ " " ❋❋❋❋❋❋❋❋❋ U × F pr U | | ②②②②②②②②② U such that for all u ∈ U , the restriction ψ : φ − ( u ) → { u } × F is a diffeomorphism of manifold with fibred corners. For such a fibration, we canconsider the space of fibrewise S -operators of order m Ψ m S ( H/S ; E, F ) where E and F are smooth complex vector bundles on H . If moreover V → S is a smoothEuclidean vector bundle, that is, a smooth real vector bundle equipped with afibrewise inner product, we can then consider the space of fibrewise V -suspended S -operators Ψ m S − sus( V ) ( H/S ; E, F ). Thus, an operator P ∈ Ψ m S − sus( V ) ( H/S ; E, F ) isa smooth family S ∋ s P s ∈ Ψ m S − sus( V s ) ( φ − ( s ); E, F )of fibrewise V -suspended S -operators, where V s is the fibre of V above s ∈ S .7. Symbol Maps As for other calculi of pseudodifferential operators on singular spaces, varioussymbol maps can be defined. The ordinary symbol map can be defined in termsof the principal symbol map for conormal distributions introduced by H¨ormander(see Theorem 18.2.11 in [14]),(7.1) I m ( Y, Z ; Ω Y ) σ m / / S [ M ] ( N ∗ Z ; Ω ( N ∗ Z )) SEUDODIFFERENTIAL OPERATORS ON MANIFOLDS WITH FIBRED CORNERS 29 with M = m − dim Y + dim Z where φ : N ∗ Z → Z is the natural projectionand(7.2) S [ M ] ( N ∗ Z ) = S M ( N ∗ Z ) /S M − ( N ∗ Z ) , where S M ( N ∗ Z ) is the usual space of functions ψ ∈ C ∞ ( N ∗ Z ) such that in a localtrivialization N ∗ Z | U ∼ = U × R nξ with local variable u in U ,(7.3) sup u,ξ | D αu D βξ ψ | (1 + | ξ | ) M −| β | < ∞ ∀ α, β ∈ N n . In our case, Y = X π and Z = ∆ π . By Corollary 3.2, N ∗ ∆ π ∼ = π T ∗ X . Since π Ω R isnaturally isomorphic to π Ω L ⊗ π Ω R when restricted to the diagonal and since thesingular symplectic form of π T ∗ X provides a natural trivialization of Ω( π T ∗ X ), weget a map(7.4) Ψ m S ( X ; E, F ) σ m / / S [ m ] ( π T ∗ X ; φ ∗ Hom( E, F ))inducing a short exact sequence(7.5)0 / / Ψ m − S ( X ; E, F ) / / Ψ m S ( X ; E, F ) σ m / / S [ m ] ( π T ∗ X ; φ ∗ Hom( E, F )) / / . Here, φ : π T ∗ X → X is the bundle projection. When we consider instead poly-homogeneous pseudodifferential operators of degree m , the principal symbol is ahomogeneous section of degree m on π T ∗ X \ { } , so it defines a map(7.6) Ψ m S − ph ( X ; E, F ) σ m / / C ∞ ( π S ∗ X ; Λ m ⊗ φ ∗ Hom( E, F ))where Λ is the dual of the tautological real line bundle of π S ∗ X . Definition 7.1. An operator P ∈ Ψ m S ( X ; E, F ) is elliptic if its principal symbol σ m ( P ) is invertible.To study the asymptotic behavior of S -operators at each boundary hypersur-face, it is also useful to introduce other symbols, that is, normal operators in theterminology of [23]. Those additional symbol maps are defined by restricting theSchwartz kernel of the operator to the various front faces,(7.7) σ ∂ i : Ψ m S ( X ; E, F ) → Ψ m ff πi ( H i ; E, F )with(7.8) Ψ m ff πi ( H i ; E, F ) = n K ∈ I m (ff π i , ∆ ff πi ; β ∗ π (Hom( E, F )) ⊗ π ∗ Rπ Ω | ff πi ); K ≡ ∂ ff π i ∩ ∂ ff π ) } , where ∆ ff πi = ff π i ∩ ∆ π . The symbol map σ ∂ i clearly induces a short exact sequence(7.9) 0 / / x i Ψ m S ( X ; E, F ) / / Ψ m S ( X ; E, F ) σ ∂i / / Ψ m ff πi ( H i ; E, F ) / / , where x i is the boundary defining function of H i . Remark 7.2. Since β ∗ π ( x i x ′ i ) is equal to 1 on ff π i , notice that for z ∈ C , P ∈ Ψ m S ( X ; E, F ) = ⇒ P i,z := x zi P x − zi ∈ Ψ m S ( X ; E, F ) with σ ∂ i ( P i,z ) = σ ∂ i ( P ) . Clearly, the space G (1)ff πi = ff π i \ ( ∂ ff π i ∩ ∂ ff π ) has a natural Lie groupoid structureinduced from the one of G (1) π with units given by G (0)ff πi = ∆ ff πi . The conormaldistributions in Ψ m ff π ( X ) which have compact support on G (1)ff πi can be understoodas elements of the algebra Ψ ∗ ( G (1)ff πi ) of pseudodifferential operators associated tothe Lie groupoid G (1)ff πi .The space Ψ m ff πi ( H i ; E, F ) can also be interpreted as a space of suspended S -operators. To see this, notice that since the fibres of the fibration π i : H i → S i arenaturally manifolds with fibre corners with typical fibre F i having iterated fibrationstructure π F i , we can form the fibrewise π F i -double space(7.10) ( F i ) π Fi ( H i × π i H i ) π Fi (cid:15) (cid:15) S i . If π N S i denotes the radial compactification of the vector bundle π N S i → S i definedin (2.10), then notice that the front face ff π i is naturally identified with the totalspace of the fibration obtained from the fibration (7.10) by pulling it back to π N S i .This means we have a natural fibration(7.11) ( F i ) π Fi ff π i (cid:15) (cid:15) π N S i . With this identification, the Schwartz kernels in (7.8) corresponds to the Schwartzkernels of π N S i -suspended S -operators associated to the fibration π i : H i → S i ,that is,(7.12) Ψ m ff πi ( H i ; E, F ) = Ψ m S − sus( π NS i ) ( H i /S i ; E, F ) . Recalling the identification (2.10), we see that, as a suspended operator, the symbol σ ∂ i ( P ) has a natural action on Schwartz sections,(7.13) σ ∂ i ( P ) : S ( π N H i ; E ) → S ( π N H i ; F ) . Definition 7.3. An operator P ∈ Ψ m S ( X ; E, F ) is said to be fully elliptic if it iselliptic and if for all i ∈ { , . . . , k } , σ ∂ i ( P ) is invertible as a map σ ∂ i ( P ) : S ( π N H i ; E ) → S ( π N H i ; F ) . A V -suspended S -operator P ∈ Ψ m S − sus( V ) ( X ; E, F ) is said to be fully elliptic if, asa S -operator in Ψ m S ( V × X ; E, F ), it is elliptic and if for all boundary hypersurfacesof the form Z i = V × H i , the corresponding symbol σ ∂ i ( P ) is invertible as a map σ ∂ i ( P ) : S ( ̟ N F i ; E ) → S ( ̟ N F i ; F ) . If H i and H j are two hypersurfaces such that H i < H j , then the associated sym-bols σ ∂ i and σ ∂ j satisfy a certain compatibility condition, namely, their respectiverestrictions to ff π i ∩ ff π j agree. From the point of view of suspended operators, this SEUDODIFFERENTIAL OPERATORS ON MANIFOLDS WITH FIBRED CORNERS 31 means that the restriction of σ ∂ j ( P ) ∈ Ψ m ff πj ( X ) to π N ∗ S j | S ji is the symbol of thesuspended family σ ∂ i ( P ) associated to the face H j ∩ H i .8. Composition To show that S -operators compose nicely, various strategy could be used. Oneapproach consists in defining pseudodifferential operators using Lie groupoids as in[39], in which case the fact the composition of operators in the calculus remainsin the calculus follows directly from the definition. As indicated earlier, the dis-advantage with such an approach is that the inverse of an invertible operator isnot typically within the algebra. Another approach, developed by Melrose andcollaborators (see for instance [23]), is to consider a triple space suitably blownup where composition can be represented by a pushforward map coming from a b -fibration. The result then follows from the description in [27] of general mappingproperties that such pushforward maps satisfy. Such an approach is likely to workin our context, but might involves a rather complicated triple space. Instead, wewill proceed by less geometric means and follow the approach of [18] by workinglocally and using a proof by induction on the dimension of the manifold with fibredcorners. Theorem 8.1. Let E, F, G be smooth complex vector bundles on a manifold withfibred corners X . Then for A ∈ Ψ m S ( X ; F, G ) and B ∈ Ψ n S ( X ; E, F ) , we have that A ◦ B ∈ Ψ m + n S ( X ; E, G ) , with σ ∂ i ( A ◦ B ) = σ ∂ i ( A ) ◦ σ ∂ i ( B ) for each hypersurface H i ⊂ X of X . Moreover, the induced map Ψ m S ( X ; F, G ) × Ψ n S ( X ; E, F ) → Ψ m + n S ( X ; E, G ) is continuous with respect to the natural Fr´echet topology on each space. A similarresult holds for polyhomogeneous S -operators. To describe the inductive step in the proof of this theorem, consider, for p ∈ N ,the new manifold with corners R p × X where R p is the radial compactification of R p as described in [31]. A natural boundary defining function for the boundary ∂ R p ∼ = S p − is given by (1 + r ) − where r is the Euclidean distance from theorigin.Notice that R p × X has a natural structure of manifold with fibred cornersinduced from the one of X . Indeed, the fibration on the boundary hypersurface Z = ∂ R p × X is given by the projection on ∂ R p , while on the boundary hypersurface Z i = R p × H i , where H i ⊂ X is a hypersurface of X with fibration π i : H i → S i ,the fibration is given by Id × π i : R p × H i → R p × S i . Lemma 8.2. Suppose that the conclusion of Theorem 8.1 holds for the manifoldwith fibred corners X . Then for A ∈ Ψ m S ( R p × X ) and B ∈ Ψ n S ( R p × X ) , we have A ◦ B ∈ Ψ m + n S ( R p × X ) , with σ ∂ j ( A ◦ B ) = σ ∂ j ( A ) ◦ σ ∂ j ( B ) for all boundary hypersurfaces Z j ⊂ R p × X . Proof. Using the Fourier transform on R p , we can describe the action of operators A ∈ Ψ m S ( R p × X ) and B ∈ Ψ n S ( R p × X ) on u ∈ ˙ C ∞ ( R p × X ) by(8.1) Au ( t ) = 1(2 π ) p Z e i ( t − t ′ ) · τ a ( t ; τ ) u ( t ′ ) dt ′ dτ,Bu ( t ) = 1(2 π ) p Z e i ( t − t ′ ) · τ b ( t ; τ ) u ( t ′ ) dt ′ dτ. Here, a and b are operator-valued symbols,(8.2) a ∈ C ∞ ( R p ; Ψ m S − sus( p ) ( X )) , b ∈ C ∞ ( R p ; Ψ n S − sus( p ) ( X )) , where Ψ ℓ S − sus( p ) ( X ) is the space of R p -suspended S -operators of order ℓ on X andthe variable τ ∈ R p in (8.1) is seen as the suspension parameter.If we forget that a and b are operator-valued, then there symbol class is the oneintroduced in [40] and [49] (see also [31]). In this setting, there are standard methodsto study the composition of operators, see for instance the proof of Proposizione 1.4in [40], or in the context of the Weyl calculus, the proof of Theorem 29.1 in [50].Since the operator-valued symbols are such that(8.3) a ∈ C ∞ ( R pt ; Ψ m S − sus( p ) ( X )) = ⇒ D αt D βτ a ∈ (1 + t ) − | α | C ∞ ( R pt ; Ψ m −| β | S − sus( p ) ( X )) , where τ is the suspension parameter, these methods have a straightforward gener-alization.Indeed, let c ( t, τ ) be the operator-valued symbol such that(8.4) ABu ( t ) = 1(2 π ) p Z e i ( t − t ′ ) · τ c ( t, τ ) u ( t ′ ) dt ′ dτ. As in [40], for each N ∈ N , we have(8.5) c ( t, τ ) = X | α | Suppose that the conclusions of Theorem 8.1 hold for all manifoldswith fibred corners Y of dimension less than the one of X . Suppose that A ∈ Ψ m S ( X ) and B ∈ Ψ n S ( X ) are such that their Schwartz kernels are supported inside the set β − π (cid:0) ν i ( π − i ( V i ) × [0 , ǫ i )) (cid:1) ⊂ X π where V i ⊂ S i \ ∂S i is some open set in the interior of the base S i of the fibration π i : H i → S i . Then A ◦ B ∈ Ψ m + n S ( X ) with σ ∂ j ( A ◦ B ) = σ ∂ j ( A ) ◦ σ ∂ j ( B ) for all hypersurfaces H j ⊂ X . Proof. Let F i be the typical fibre of the fibration(8.10) F i H iπ i (cid:15) (cid:15) S i . Then as described in § 1, the fibre F i is naturally a manifold with fibred corners.Since dim F i < dim X , it is part of our assumptions that Ψ m S ( F i ) ◦ Ψ n S ( F i ) ⊂ Ψ m + n S ( F i ). The strategy of the proof is to reduce composition of the operators A and B to Lemma 8.2.Let { W q } q ∈Q be a finite covering of the closure of V i in S i \ ∂S i by open sets in S i \ ∂S i diffeomorphic to open balls and such that the fibration (8.10) restricts to atrivial fibration over each S i . Let ϕ q ∈ C ∞ c ( W q ) be functions which restricts to givea partition of unity on V i and let ˜ ϕ q ∈ C ∞ c ( W q ) be functions such that ϕ q ˜ ϕ q = ϕ q .Let ψ q = ( ν i ) ∗ pr ∗ π ∗ i ϕ q , ˜ ψ q = ( ν i ) ∗ pr ∗ π ∗ i ˜ ϕ q , be the corresponding pulled back functions on U i = ν i ( H i × [0 , ǫ i )) where pr : H i × [0 , ǫ i ) → H i is the projection on the left factor. Then we can write theoperator A as(8.11) A = X q Aψ q = X q (cid:16) ˜ ψ q Aψ q + (1 − ˜ ψ q ) Aψ q (cid:17) . Since ˜ ψ q ψ q = ψ q , the Schwartz kernel of the second term is supported away fromthe diagonal in X , which means it is an element of ˙Ψ −∞ S ( X ). Thus, we have that(8.12) A ≡ X q ˜ ψ q Aψ q mod ˙Ψ −∞ S ( X ) . Similarly, we have that(8.13) ψ q B = ψ q B ˜ ψ q + ψ q B (1 − ˜ ψ q ) ≡ ψ q B ˜ ψ q mod ˙Ψ −∞ S ( X ) . Thus, using Corollary 5.5, we see that(8.14) AB ≡ X q ˜ ψ q Aψ q B mod ˙Ψ −∞ S ( X )= X q ˜ ψ q Aψ q ˜ ψ q B = X q ( ˜ ψ q A ˜ ψ q ) ψ q B ≡ X q ( ˜ ψ q A ˜ ψ q )( ψ q B ˜ ψ q ) mod ˙Ψ −∞ S ( X ) . This means we can assume both K A and K B are supported in the subset β − π (cid:0) ν i ( π − i ( W q ) × [0 , ǫ )) (cid:1) ⊂ X π . Since we are assuming W q is diffeomorphic to an open ball, this means there existsan embedding(8.15) ι q : W q ֒ → S p i − , where p i − S i = dim W q . Since the fibration π i is trivial when restricted to W q , we can assume π − i ( W q ) = F i × W q SEUDODIFFERENTIAL OPERATORS ON MANIFOLDS WITH FIBRED CORNERS 35 with π i given by projecting on the right factor. The embedding (8.15) can beextended to an embedding(8.16) W q × [0 , ǫ i ) ι q × Id / / S p i − × [0 , ǫ i ) / / R p i where the second map is the standard collar neighborhood of S p i − = ∂ R p i − in theradial compactification R p i − of R p i using the boundary defining function √ r +1 where r is the distance from the origin.Via these identifications, this means we can regard A and B as operators actingon functions of R p i × F i , more precisely: A ∈ Ψ m S ( R p i × F i ), B ∈ Ψ n S ( R p i × F i ).The result then follows by applying Lemma 8.2. (cid:3) We have now all the ingredients to prove the composition theorem. Proof of Theorem 8.1. By using a partition of unity, we can work locally in opensets where the vector bundles E , F and G are trivial. Thus, without loss of gener-ality, we can assume that E = F = G = C and A ∈ Ψ m S ( X ), B ∈ Ψ n S ( X ).Since the case where dim X = 0 is trivial, we can assume by induction on thedimension that the theorem is true for manifolds with fibred corners of dimensionless than the one of X . For each boundary hypersurface H i of X , consider thetubular neighborhood ν i : H i × [0 , ǫ i ) x i → U i ⊂ X of (8.9). Let also χ i , e χ i , b χ i ∈C ∞ c ( U i ) ⊂ C ∞ ( X ) be non-negative cut-off functions such that b χ i ≡ H i , χ i b χ i = b χ i and e χ i χ i = χ i . Using the cut-off functions χ i , e χ i and b χ i , we can rewritethe composition of A and B as(8.17) AB = Aχ i B + A (1 − χ i ) B = e χ i Aχ i B + (1 − e χ i ) Aχ i B + b χ i A (1 − χ i ) B + (1 − b χ i ) A (1 − χ i ) B. Since χ i b χ i = b χ i and e χ i χ i = χ i , the Schwartz kernels of (1 − e χ i ) Aχ i and b χ i A (1 − χ i )are both supported away from the diagonal in X × X , which means the operators(1 − e χ i ) Aχ i and b χ i A (1 − χ i ) are both in ˙Ψ −∞ S ( X ). Thus, using Corollary 5.5, wesee that modulo operators in ˙Ψ −∞ S ( X ), we have(8.18) AB ≡ e χ i Aχ i B + (1 − b χ i ) A (1 − χ i ) B mod ˙Ψ −∞ S ( X ) . Similarly, if χ ′ i ∈ C ∞ c ( U i ) is such that b χ i χ ′ i = χ ′ i and χ ′ i ≡ H i , then we canwrite the operator B as(8.19) B = b χ i Bχ i + b χ i B (1 − χ i ) + (1 − b χ i ) Bχ ′ i + (1 − b χ i ) B (1 − χ ′ i ) ≡ b χ i Bχ i + (1 − b χ i ) B (1 − χ ′ i ) mod ˙Ψ −∞ S ( X ) . If ˇ χ i ∈ C ∞ c ( U i ) is another cut-off function such that ˇ χ i e χ i = e χ i , then we can alsowrite B as(8.20) B = e χ i B ˇ χ i + e χ i B (1 − ˇ χ i ) + (1 − e χ i ) Bχ i + (1 − e χ i ) B (1 − χ i ) ≡ e χ i B ˇ χ i + (1 − e χ i ) B (1 − χ i ) mod ˙Ψ −∞ S ( X ) . Substituting (8.19) and (8.20) in (8.18), we see by Corollary 5.5 that(8.21) AB ≡ ( e χ i Aχ i )( e χ i B ˇ χ i ) + e χ i Aχ i (1 − e χ i ) B (1 − χ i ) + (1 − b χ i ) A (1 − χ i ) b χ i Bχ i + (1 − b χ i ) A (1 − χ i )(1 − b χ i ) B (1 − χ ′ i ) mod ˙Ψ −∞ S ( X ) ≡ ( e χ i Aχ i )( e χ i B ˇ χ i ) + (1 − b χ i ) A (1 − χ i )(1 − b χ i ) B (1 − χ ′ i ) mod ˙Ψ −∞ S ( X ) . Thus, from (8.21), we can reduce the problem of composition to two situations, • K A and K B are supported near ff π i ; • K A and K B are supported away from ff π i .In particular, if H i is a minimal hypersurface with respect to the partial order ofhypersurfaces of X , then the first term on the right hand side of (8.21) can betaken care of by Lemma 8.3. In fact, starting with the minimal hypersurfaces H i and proceeding recursively on the partial order of boundary hypersurfaces of X using (8.21) and Lemma 8.3 at each step, we can reduce to the case where K A and K B are supported away from ff π i for all i . Adding operators in ˙Ψ −∞ S ( X ) ifnecessary, we can even reduce to the case the Schwartz kernels of A and B havecompact support in ( X \ ∂X ) . By doubling X to get a smooth closed manifold, thisreduces to the standard result about composition of pseudodifferential operators onclosed manifolds. It is straightforward to check that polyhomogeneity is preservedunder composition. (cid:3) Mapping properties Let ( X, π ) be a manifold with fibred corners. Let H , . . . , H k be its boundaryhypersurfaces with choice of boundary defining functions x , . . . , x k . As for theΦ-calculus of [23], an important ingredient in the study of mapping properties of S -operators is the construction of a parametrix for fully elliptic operators. We willalso need such a result for S -suspended operators, in which case the notation˙Ψ −∞ S − sus( V ) ( X ; E , E ) = { A ∈ Ψ −∞ S − sus( V ) ( X ; E , E ); b A (Υ) ∈ ˙Ψ −∞ S ( X ; E , E ) ∀ Υ ∈ V ∗ } , for E and E complex vector bundles over X , is useful to describe the error term. Proposition 9.1 (Parametrix) . If P ∈ Ψ m S ( X ; E, F ) is fully elliptic, then thereexists Q ∈ Ψ − m S ( X ; F, E ) such that Id − QP ∈ ˙Ψ −∞ S ( X ; E ) , Id − P Q ∈ ˙Ψ −∞ S ( X ; F ) . Moreover, ker P ⊂ ˙ C ∞ ( X ; E ) and ker P ∗ ⊂ ˙ C ∞ ( X ; F ) . Similarly, if V is an Eu-clidean vector space and P ∈ Ψ m S − sus( V ) ( X ; E, F ) is fully elliptic, then there exists Q ∈ Ψ − m S − sus( V ) ( X ; F, E ) such that Id − QP ∈ ˙Ψ −∞ S − sus( V ) ( X ; E ) , Id − P Q ∈ ˙Ψ −∞ S − sus( V ) ( X ; F ) . Proof. Using this proposition and Corollary 9.2 below and proceeding by inductionon the dimension of X , we can assume that σ ∂ i ( P ) − ∈ Ψ − m ff πi ( H i ; F, E ). Thismeans we can choose Q ∈ Ψ − m S ( X ; F, E ) such that σ − m ( Q ) = σ m ( P ) − and σ ∂ j ( Q ) = σ ∂ j ( P ) − . Then we have(9.1) Id − Q P ∈ x Ψ − S ( X ; E ) , Id − P Q ∈ x Ψ − S ( X ; F ) . Suppose for a proof by induction that we have defined Q ℓ ∈ x ℓ Ψ − m − ℓ S ( X ; F, E ) for ℓ ≤ n such that ˜ Q n = Q + · · · + Q n satisfies(9.2) Id − ˜ Q n P ∈ x n +1 Ψ − n − S ( X ; E ) , Id − P ˜ Q n ∈ x n +1 Ψ − n − S ( X ; F ) . SEUDODIFFERENTIAL OPERATORS ON MANIFOLDS WITH FIBRED CORNERS 37 Then, setting ˜ Q n +1 = ˜ Q n + Q n +1 , we would like to find Q n +1 ∈ x n +1 Ψ − m − n − S ( X ; F, E )such that(9.3) Id − ˜ Q n +1 P = Id − ˜ Q n P − Q n +1 P ∈ x n +2 Ψ − n − S ( X ; E ) , that is, such that(9.4) Q n +1 P = Id − ˜ Q n P modulo x n +2 Ψ − n − S ( X ; E ) . Thus, taking Q n +1 = (Id − ˜ Q n P ) ˜ Q n will give(9.5) Id − ˜ Q n +1 P ∈ x n +2 Ψ − n − S ( X ; E )with ˜ Q n +1 = ˜ Q n + Q n +1 . As one can check, we will also have that(9.6) Id − P Q n +1 ∈ x n +2 Ψ − n − S ( X ; F ) . We can then define Q to be the asymptotic sum of the Q ℓ giving the desiredparametrix. If f ∈ ker P , then(9.7) P f = 0 ⇒ QP f = 0 ⇒ f = (Id − QP ) f ∈ ˙ C ∞ ( X ; E )since Id − QP ∈ ˙Ψ −∞ S ( X ; E ). There is a similar argument for the kernel of P ∗ .For fully elliptic V -suspended S -operators, the proof is similar and is left to thereader. (cid:3) Corollary 9.2. If V is an Euclidean vector space and P ∈ Ψ m S − sus( V ) ( X ; E, F ) is afully elliptic V -suspended operators which is invertible as a map P : S ( V × X ; E ) →S ( V × X ; F ) , then it has an inverse in Ψ − m S − sus( V ) ( X ; F, E ) .Proof. Let P ∈ Ψ m S − sus( V ) ( X ; E, F ) be as in the statement of the corollary and let Q ∈ Ψ − m S − sus( V ) ( X ; F, E ) be the parametrix of Proposition 9.1. in particular, wehave that b P (Υ) b Q (Υ) = Id + b R (Υ) , ∀ Υ ∈ V ∗ , where R ∈ ˙Ψ −∞ S − sus( V ) ( X ; F ) . By (6.21) we see that b R (Υ) is small for | Υ | large, so that there exists K > b R (Υ) is invertible for | Υ | > K with inverse of the form Id + b S (Υ),where b S (Υ) = ∞ X k =1 ( − k b R (Υ) k ∈ ˙Ψ −∞ S ( X ; F )satisfies (6.21). Thus, for | Υ | > K , we have that(9.8) b P (Υ) − = b Q (Υ)(Id + b S (Υ)) . Now, the invertibility of P clearly implies the invertibility of b P (Υ) for all Υ ∈ V ∗ .Using the parametrix Q , we have(9.9) b P (Υ) − = b P (Υ) − ( b P (Υ) b Q (Υ) − b R (Υ)) = b Q (Υ) − b P (Υ) − b R (Υ) . By Proposition 5.4, we must have b P (Υ) − b R (Υ) ∈ ˙Ψ −∞ S ( X ; F, E ) for all Υ ∈ V ∗ .Thus, from (9.8) and (9.9), we see that b P (Υ) − = b Q (Υ) + c W (Υ) , where W ∈ ˙Ψ −∞ S − sus( V ) ( X ; F, E ) is such that c W (Υ) = b Q (Υ) b S (Υ) for | Υ | > K .Taking the inverse Fourier transform, we finally obtain that P − = Q + W ∈ Ψ − m S − sus( V ) ( X ; F, E ) . (cid:3) As we will see, this last corollary will be useful to study the action of S -operatorson square integrable functions. Precisely, let g π be a choice of S -metric and let dg π ∈ C ∞ ( X ; π Ω) be its volume form. Let L g π ( X ) be the corresponding space offunctions on X \ ∂X that are square integrable with respect to the density dg π .To establish the L -boundedness of S -pseudodifferential operators of order zero, wewill, as in [23], follow the standard trick of H¨ormander relying on the constructionof an approximate square root. Proposition 9.3. If B ∈ Ψ S ( X ) is formally self-adjoint with respect to a positive S -density ν on X , then there exists C > sufficiently large so that C + B = A ∗ A + R for some A ∈ Ψ S ( X ) and R ∈ ˙Ψ −∞ S ( X ) .Proof. The proof is by induction on the depth of X . The case where X is a closedmanifold is well-known and the case where X is a manifold with boundary is provenby Mazzeo and Melrose in [23].For i ∈ { , . . . , k } , let ν S i be a positive section of Ω( π N S i ) and write ν | H i = ν F i ⊗ ν S i where ν F i is a positive density in the fibres of the fibration π i : H i → S i . Thenthe suspended family of S -pseudodifferential operators ˆ σ ∂ i ( B )( η ) with η ∈ π N ∗ S i is formally self-adjoint with respect to the density ν F i . By our induction hypothesis(see Corollary 9.6), for C > σ ∂ i ( B )( η ) + C ) ∈ Ψ S ( F i ) has a uniquepositive square root so that ( C + σ ∂ i ( B )) also has a unique positive square root inΨ πi ( X ). Similarly, ( C + σ ( B )) has unique positive square root provided C > A ∈ Ψ S ( X ) such that(9.10) σ ( A ) = ( C + σ ( B )) , σ ∂ i ( A ) = ( C + σ ∂ i ( B )) , i ∈ { , . . . , k } . Replacing A by ( A + A ∗ ) if necessary, we can assume that A is formally self-adjoint with(9.11) C + B − A ∈ x Ψ − S ( X ) . To get an error term in ˙Ψ −∞ S ( X ), we can proceed by induction. Thus, assume thatwe have found a formally self-adjoint operator A ℓ ∈ Ψ S ( X ) such that(9.12) C + B − A ℓ = R ℓ +1 ∈ x ℓ +1 Ψ − ℓ − S ( X ) . Writing A ℓ +1 = A ℓ + Q ℓ where the formally self-adjoint operator Q ℓ ∈ x ℓ +1 Ψ − ℓ − S ( X )is to be found, we have(9.13) C + B − A ℓ +1 = R ℓ +1 − Q ℓ A ℓ − A ℓ Q ℓ − Q ℓ = R ℓ +1 − Q ℓ A ℓ − A ℓ Q ℓ modulo x ℓ +2 Ψ − ℓ − S ( X ). First, this means we need to solve(9.14) σ − ℓ − ( R ℓ +1 ) = 2 σ ( A ℓ ) σ − ℓ − ( Q ℓ ) , SEUDODIFFERENTIAL OPERATORS ON MANIFOLDS WITH FIBRED CORNERS 39 which clearly has a formally self-adjoint solution Q ℓ, ∈ x ℓ +1 Ψ − ℓ − S ( X ). Thus,replacing A ℓ by A ℓ, = A ℓ + Q ℓ, in (9.13), this means we have to solve (9.13) with R ℓ +1 replaced by R ℓ +1 , ∈ x ℓ +1 Ψ − ℓ − S ( X ).Proceeding by induction on i ∈ { , . . . , k } and with the convention that x = 1and w i = Q ij =1 x j , assume more generally we have found formally self-adjointoperators Q ℓ,j ∈ w j − x ℓ +1 Ψ − ℓ − S ( X ) for j ≤ i − A ℓ,i − = A ℓ + P i − j =0 Q ℓ,j satisfies(9.15) C + B − A ℓ,i − = R ℓ +1 ,i − ∈ w i − x ℓ +1 Ψ − ℓ − S ( X ) . To find Q ℓ,i , write Q ℓ,i = x ℓ +1 i ˜ Q ℓ,i where ˜ Q ℓ,i ∈ x − ℓ − i w i − x ℓ +1 Ψ − ℓ − S ( X ). UsingRemark 7.2, this means we need to solve(9.16) ˆ σ ∂ i ( x − ℓ − i R ℓ +1 ,i − ) = ˆ σ ∂ i ( ˜ Q ℓ,i )ˆ σ ∂ i ( A ) + ˆ σ ∂ i ( A )ˆ σ ∂ i ( ˜ Q ℓ,i ) . As pointed out in [23], this is solvable with e Q ℓ,i formally self-adjoint as ˆ σ ∂ i ( A ) ispositive and (9.16) is the linearization of the square root equation(9.17) (ˆ σ ∂ i ( A ) + ˆ σ ∂ i ( ˜ Q ℓ,i )) = ˆ σ ∂ i ( x − ℓ − i R ℓ +1 ,i − ) + ˆ σ ∂ i ( A ) . Thus, we can find Q ℓ,i such that (9.16) satisfied. Replacing Q ℓ,i by Q ∗ ℓ,i + Q ℓ,i ifnecessary, we can assume furthermore that Q ℓ,i is formally self-adjoint. Thus,taking A ℓ +1 = A ℓ + P ki =0 Q ℓ,i insures that A ℓ +1 = A ∗ ℓ +1 and(9.18) C + B − A ℓ +1 ∈ x ℓ +2 Ψ − ℓ − S ( X ) . We can then define A as an asymptotic sum specified by the A ℓ . (cid:3) Theorem 9.4. Any element P ∈ Ψ S ( X ; E , E ) defines a bounded linear operatorfrom H = L ( X ; E ) to H = L ( X ; E ) with L -norms defined by a positive S -density on X and Hermitian metrics on E and E . Furthermore, the map Ψ S ( X ; E , E ) → L ( H , H ) is continuous.Proof. Considering a local trivialization if necessary, we can assume that E = E = C and H = H = L ( X ). Then B = − P ∗ P ∈ Ψ S ( X ) is formally self-adjoint. Bythe previous proposition, there exists C > A ∈ Ψ S ( X ) formally self-adjointsuch that(9.19) C − P ∗ P = A ∗ A + R for some R ∈ x ∞ Ψ −∞ S ( X ). Thus, given u ∈ ˙ C ∞ ( X ), we have(9.20) k P u k = C k u k − k Au k − h u, Ru i≤ C k u k + | h u, Ru i| ≤ C ′ k u k , where the fact elements of ˙Ψ −∞ S ( X ) are in L ( H ) has been used. Thus, there is awell-defined linear map(9.21) Ψ S ( X ) → L ( H ) . Since the map Ψ S ( X ) ∋ A 7→ h u, Av i H = K A ( π ∗ L ( uν π ) ⊗ π ∗ R ( v ))is continuous for all u, v ∈ ˙ C ∞ ( X ), where ν π is the S -density used to define the L -norm, we see that the graph of the linear map (9.21) is closed with the respectto the topology induced by the norms A 7→ |h u, Av i| . Since this topology is weaker than the norm topology, this means the graph of this map is also closed when weuse the norm topology on L ( H ). The map (9.21) is therefore continuous by theclosed graph theorem. (cid:3) There is a similar result for suspended S -operators. Let V be a Euclidean vectorspace and let g V be the corresponding Euclidean metric. On the manifold withfibred corners V × X , consider the ̟ -metric g ̟ = pr ∗ g V + pr ∗ g π where pr : V × X → V and pr : V × X → X are the projections on the first andsecond factors respectively. Corollary 9.5. Any element P ∈ Ψ S − sus( V ) ( X ; E , E ) defines a bounded linearoperator from H = L g ̟ ( V × X ; E ) to H = L g ̟ ( V × X ; E ) with L -norm definedby a volume form dg ̟ and Hermitian metrics on E and E . Furthermore, the map Ψ S − sus( V ) ( X ; E , E ) → L ( H , H ) is continuous.Proof. Since our proof of Theorem 9.4 is by induction on the depth of X and sincethe inductive step is not yet completed, we cannot at this stage simply use thestatement of Theorem 9.4 for the manifold with fibred corners V × X to obtain theresult. Instead, consider the Fourier transform of P ,Υ b P (Υ) ∈ Ψ S ( X ; E , E ) , Υ ∈ V ∗ . By Theorem 9.4, we know that for each Υ ∈ V ∗ , the operator b P (Υ) induces acontinuous linear map b P (Υ) : L g π ( X ; E ) → L g π ( X ; E ) . Let g V ∗ be the metric on V ∗ which is dual to g V and let g ̟ ∗ = pr ∗ g V ∗ + pr ∗ g π be the corresponding metric on V ∗ × X . Since the Fourier transform induces anisomorphism of Hilbert spaces F i : L g ̟ ( V × X ; E i ) → L g ̟ ∗ ( V ∗ × X ; E i ) , we conclude from (6.22) and Theorem 9.4 that P ∈ L ( H , H ). The continuityof the map Ψ S − sus( V ) ( X ; E , E ) → L ( H , H ) can be proved in the same way asbefore. (cid:3) As a family of suspended operators, the symbol σ ∂ i ( P ) of an operator P ∈ Ψ S ( X ; E, F ) will act on the Banach space L g π ( π N H i /S i ; E ) obtained by takingthe closure of the space of Schwartz sections S ( π N H i ; E ) with respect to the norm(9.22) k f k L gπ ( π NH i /S i ; E ) = sup s ∈ S i k f | φ − i ( s ) k L gπ ( φ − i ( s ); E ) , f ∈ S ( π N H i ; E ) , where φ i = π i ◦ ν i : π N H i → S i and ν i : π N H i → H i is the vector bundleprojection. On each fibre of φ i , the L -norm of a section of E is specified by achoice of Hermitian metric on E and the natural density induced by g π . Thus, SEUDODIFFERENTIAL OPERATORS ON MANIFOLDS WITH FIBRED CORNERS 41 from Corollary 9.5, we see that the symbol σ ∂ i ( P ) of an operator P ∈ Ψ S ( X ; E, F )naturally induce a continuous linear map(9.23) σ ∂ i ( P ) : L g π ( π N H i /S i ; E ) → L g π ( π N H i /S i ; F ) . Notice that the Banach space L g π ( π N H i /S i ; E ) also has a natural structure of C ( S i )-Hilbert module.To complete the inductive step necessary to the proof of Proposition 9.3, we cannow use this fact with Corollary 9.2 to construct the unique positive square root ofthe operator C + B in Proposition 9.3 and its suspended versions. Corollary 9.6. Given a formally self-adjoint operator B ∈ Ψ S − sus( V ) ( X ) , thereexists a positive constant C such that C + B is invertible and has a well-definedformally self-adjoint positive definite square root in Ψ S − sus( V ) ( X ) .Proof. From Corollary 9.5, we know that B gives a bounded operator(9.24) B : L g ̟ ( V × X ) → L g ̟ ( V × X ) . Thus, taking C big enough, we can define the square root of C + B as a boundedoperator by(9.25) C (1 + BC ) = C ∞ X j =0 f ( j ) (0) j ! (cid:18) BC (cid:19) j using the power series of f ( x ) = (1 + x ) at x = 0. To see it is an element ✻ ✲ ✫✪✬✩ ✻ Γ r C xy Figure 4. of Ψ S − sus( V ) ( X ), we can use the alternative representation in terms of a contourintegral(9.26) ( C + B ) = 12 πi Z Γ λ ( λ − ( C + B )) − dλ where Γ is an anti-clockwise circle centered at C and radius r such that k B k L ( H ) For δ > , an operator A ∈ Ψ − δ S ( X ; E ) is compact when acting on H = L g π ( X ; E ) if and only if σ ∂ j ( A ) = 0 for all j ∈ { , . . . , k } . In particular, apolyhomogeneous S -operator A ∈ Ψ S − ph ( X ; E ) of order zero is compact when actingon L g π ( X ; E ) if and only if A ∈ x Ψ − S − ph ( X ; E ) .Proof. Without loss of generality, we can assume E = C is the trivial vector bundle.By definition, the space of compact operators K ( H ) is the closure in L ( H ) of oper-ators of finite ranks. Clearly, since ˙ C ∞ ( X ) is dense in L g π ( X ), we can as well define K ( H ) as the closure of finite rank operators represented by an element of ˙Ψ −∞ S ( X ).These operators of finite rank are certainly dense in ˙Ψ −∞ S ( X ). Thus, K ( H ) is givenby the closure of ˙Ψ −∞ S ( X ) in L ( H ). Since the map Ψ S ( X ) → L ( H ) is continuous,we conclude that the closure of ˙Ψ −∞ S ( X ) in Ψ − δ S ( X ), namely, x Ψ − δ S ( X ), is includedin K ( H ).Conversely, let A ∈ Ψ − δ S ( X ) be a compact operator. Suppose for a contradictionthat σ ∂ i ( A ) = 0 for some i ∈ { , . . . , k } . This means that we can find y i ∈ S i anda function f ∈ C ∞ c ( π N y i H i ) such that(9.27) σ ∂ i ( A ) | y i f = 0 . Without loss of generality, we can assume in fact that y i ∈ S i \ ∂S i . Let V bea small neighborhood of y i ∈ S i such that the fibration π i : H i → S i is trivial over V , namely, there is a diffeomorphism ψ : π − i ( V ) → F i × V inducing a commutativediagram(9.28) π − i ( V ) ψ / / π i " " ❋❋❋❋❋❋❋❋❋ V × F i pr L | | ②②②②②②②②② V where pr L : V × F i → V is the projection on the left factor. Let ι i : H i × [0 , ǫ ) x i → X be a tubular neighborhood of H i in X compatible with the boundary definingfunction x i . Using the diffeomorphism ψ , we can identify the open set ι i ( π − i ( V ) × [0 , ǫ ) x i ) ⊂ X with the open set(9.29) V × F i × [0 , ǫ ) x i . Choosing V to be smaller if needed, we can assume it is diffeomorphic to an openball in the Euclidean space. Let y be a choice of coordinates on V such that thepoint y i ∈ V corresponds to y = 0. On the open set V × (0 , ǫ ) x i , consider thecoordinates(9.30) u = 1 x i , v = yx i . Considering alternatively v and u as linear coordinates on the vector space π i N y i S i = T y i S i × R u , we regard V × (0 , ǫ ) x i as an open subset in π i N y i S i , and consequentlywe can regard U = V × F i × (0 , ǫ ) x i as a subset of π i N y i S i × F i = π i N y i H i . For k ∈ N , consider the new function(9.31) f k ( u, v, z ) = f ( u − k, v, z ) , z ∈ F i , obtained by translating f in the u variable. Since we assume that the support of f is compact, by taking k sufficiently large, we can insure that the support of f is contained in the open set U . In fact, since the operator σ ∂ i ( A ) is translation SEUDODIFFERENTIAL OPERATORS ON MANIFOLDS WITH FIBRED CORNERS 43 invariant, we will still have that (9.27) holds after translating f in the u variable,so without loss of generality, we can assume that the support of f k is contained in U for all k ∈ N . Again, by translation invariance of σ ∂ i ( A ), we will have that σ ∂ i ( A ) | y i f k = 0for all k ∈ N . Since the function f k is supported in U , we can also regard it asa function on X . Let χ ∈ C ∞ c ( V × F i × [0 , ǫ ) x i ) be a cut-off function such that χ ≡ { y i } × F i × { } . Thus, if we consider the operator P = χA ∈ Ψ − δ S ( X ), P will also obviously be compact, and we will have that σ ∂ i ( P ) | y i = σ ∂ i ( A ) | y i . Now, thanks to the cut-off function χ , the action of P on f ∈ C ∞ c ( U ) ⊂ C ∞ ( X ) isgiven by: P f k ( u, v, z ) = Z U K P ( u, v, u ′ , v ′ , z, z ′ ) f k ( u ′ , v ′ , z ′ ) du ′ dv ′ dz ′ , where the integral is in the sense of distributions. Similarly, the action of σ ∂ i ( P ) | y i can be described by σ ∂ i ( P ) | y i f k ( u, v, z ) = Z U K σ ∂i ( P ) | yi ( u, v, u ′ , v ′ , z, z ′ ) f k ( u ′ , v ′ , z ′ ) du ′ dv ′ dz ′ Since as a function on U ⊂ X , the support of the function f k is uniformly approach-ing the fibre π − i ( y i ) ⊂ H i as k → + ∞ , we see from the definition of the normaloperator that we must have that as k tends to infinity,(9.32) P f k − σ ∂ i ( P ) f k → L -norm defined by the S -metric g + du + dv + g F i , where g F i is a choice of S -metric on F i . By translation invariance of this metricand of σ ∂ i ( P ) | y i , we have that, on π i N y i H i , k σ ∂ i ( P ) | y i f k k L = k σ ∂ i ( P ) | y i f k L = 0If we restrict σ ∂ i ( P ) | y i f k to U , we still clearly have thatlim k →∞ k σ ∂ i ( P ) | y i f k k L ( U ) = k σ ∂ i ( P ) | y i f k L ( πi N yi S i ) = 0 . On the other hand, σ ∂ i ( P ) | y i f k being moved to infinity as k → ∞ , we see thatit converges pointwise to zero everywhere on U , so that the sequence σ ∂ i ( P ) f k cannot converge in L . We conclude from (9.32) that the sequence P f k also failsto converge in L . Since by translation invariance of the metric, the sequence f k is bounded in L , this contradicts the fact P is a compact operator. To avoida contradiction, we must conclude that σ ∂ i ( A ) = 0 for all i ∈ { , . . . , k } , whichcompletes the proof. (cid:3) More generally, there are natural Sobolev spaces associated to S -operators. Asbefore, let g π be a S -metric on X and let E → X be a complex vector bundle witha Hermitian metric, so that we have a corresponding space L g π ( X ; E ) of squareintegrable sections. For m > 0, we define the associated S -Sobolev space by(9.33) H m S ( X ; E ) = { f ∈ ˙ C −∞ ( X ; E ) ; P f ∈ L g π ( X ; E ) ∀ P ∈ Ψ m S ( X ; E ) } , while for m < 0, we define it by(9.34) H m S ( X ; E ) = { f ∈ ˙ C −∞ ( X ; E ); f = N X i =1 P i g i , g i ∈ L g π ( X ; E ) , P i ∈ Ψ − m ( X ; E ) } . If V is a Euclidean vector space, we define the corresponding V -suspended S -Sobolevspace by(9.35) H m S − sus( V ) ( X ; E ) = H m S ( V × X ; E ) . These spaces can be given the structure of a Hilbert space using fully elliptic op-erators. More precisely, for m > 0, let A m ∈ Ψ m S ( X ; E ) be a choice of elliptic S -operator and consider the formally self-adjoint operator D m ∈ Ψ m S ( X ; E ) definedby(9.36) D m = A ∗ m A m + Id E . Lemma 9.8. For m > , the operator D m is fully elliptic and invertible. Inparticular, its inverse D − m := ( D m ) − is an element of Ψ − m S ( X ; E ) .Proof. Let H , . . . , H k be the boundary hypersurfaces of X and suppose that theyare labelled in such a way that H i < H j = ⇒ i < j. We will first prove by induction on i ∈ { , . . . , k } starting with i = k that σ ∂ i ( D m )is fully elliptic and invertible. For i = k , the fibres of the fibration π k : H k → S k are closed manifolds, so that in this case, σ ∂ k ( D m ) is automatically fully ellipticsince it is elliptic. Thus, for i ∈ { , . . . , k } , the inductive step we need to show isthat if σ ∂ i ( D m ) is fully elliptic, then it is invertible. To see this, fix s ∈ S i andconsider the π N s S i -suspended operator σ ∂ i ( A m ) s above s . For a fixed Υ ∈ π N ∗ s S i ,consider the operator Q = \ σ ∂ i ( D m )(Υ) ∈ Ψ m S ( π − i ( s ); E ) . Thus, if B = \ σ ∂ i ( A m )(Υ), we have that Q = B ∗ B + Id E . By Proposition 9.1, if Qu = 0, then u ∈ ˙ C ∞ ( π − i ( s ); E ). Thus, we have in particular(9.37) Qu = 0 = ⇒ h u, B ∗ Bu + u i L , = ⇒ k Bu k L + k u k L = 0 , = ⇒ u ≡ . Thus, since Q is formally self-adjoint, we have that ker Q = ker Q ∗ = { } , so that Q is invertible. Since Υ ∈ N ∗ s S i was arbitrary, this means that σ ∂ i ( D m ) s is invertible.Thus, since s ∈ S i was arbitrary, this means that σ ∂ i ( D m ) is invertible, whichcompletes the inductive step.With this argument, we see D m is fully elliptic. In particular, by Proposition 9.1,if D m u = 0, then u ∈ ˙ C ∞ ( X ; E ). We can then show D m is invertible using a similarargument as in (9.37), which completes the proof. (cid:3) SEUDODIFFERENTIAL OPERATORS ON MANIFOLDS WITH FIBRED CORNERS 45 Using the operator D m with D m = ( D − m ) − for m < D = Id E , we canthen define an inner product on H m S ( X ; E ) by(9.38) h u, v i H m S ( X ; E ) = h D m u, D m v i L gπ ( X ; E ) , with corresponding norm(9.39) k u k H m S ( X ; E ) = k D m u k L gπ ( X ; E ) . Using Theorem 9.4, it is straightforward to check H m S ( X ; E ) is precisely the closureof ˙ C ∞ ( X ; E ) with respect to this norm. Proposition 9.9. Any S -pseudodifferential operator P ∈ Ψ m S ( X ; E, F ) induces abounded linear map P : x ℓ H p S ( X ; E ) → x ℓ H p − m S ( X ; F ) for p, ℓ ∈ R .Proof. Thinking of E and F as subbundles of a bigger bundle H , we reduce to thecase where E = F . The result then follows from Theorem 9.4 by noticing P = x ℓ D m − p e P D p x − ℓ with e P = D p − m x − ℓ P x ℓ D − p ∈ Ψ S ( X ; E ) . (cid:3) In particular, we conclude from Proposition 9.9 that for all ℓ ∈ R and p ∈ R , theoperator D m induces an isomorphism(9.40) D m : x ℓ H p S ( X ; E ) → x ℓ H p − m S ( X ; F ) Proposition 9.10. We have a continuous inclusion x ℓ H m S ( X ; E ) ⊂ x ℓ ′ H m ′ S ( X ; E ) if and only if ℓ ≥ ℓ ′ and m ≥ m ′ . The inclusion is compact if and only if ℓ > ℓ ′ and m > m ′ .Proof. The fact that these are continuous inclusions follows from the isomorphism(9.40) and Proposition 9.9. The statement about compactness follows by using theisomorphism (9.40) and the fact that for ǫ > 0, the operator x ǫ D − ǫ ∈ x ǫ Ψ − ǫ S ( X ; E )is a compact operator from L g π ( X ; E ) to itself. (cid:3) By the parametrix construction of Proposition 9.1 as well as Proposition 9.9 andProposition 9.10, an operator P ∈ Ψ m S ( X ; E, F ) is Fredholm as an operator(9.41) P : x ℓ H p + m S ( X ; E ) → x ℓ H p S ( X ; F )whenever it is fully elliptic. When P is polyhomogeneous, it is also possible toestablish the converse. Theorem 9.11. An operator P ∈ Ψ m S − ph ( X ; E, F ) induces a Fredholm operator P : x ℓ H p + m S ( X ; E ) → x ℓ H p S ( X ; F ) if and only if it is fully elliptic.Proof. We will follow the approach of [19, Theorem 4]. First, by considering insteadthe operator ˜ P = x − ℓ D p P D − p − m x ℓ , we can assume that P is of order 0 and is seenas a bounded operator P : L g π ( X ; E ) → L g π ( X ; F ) . Furthermore, by considering instead the operator (cid:18) P ∗ P (cid:19) : L g π ( X ; E ⊕ F ) → L g π ( X ; E ⊕ F ) , we can reduce to the case E = F with P self-adjoint. By Theorem 9.4, we have acontinuous linear map ι : Ψ S − ph ( X ; E ) → L ( H , H ) , where H = L g π ( X ; E ). Let P S − ph ( X ; E ) be the image of this map and P S − ph ( X ; E )its closure in L ( H , H ). Now, the principal symbol induces a continuous linear map σ : Ψ S − ph ( X ; E ) → C ∞ ( π S ∗ X ; hom( E )) . Using instead the C -topology on C ∞ ( π S ∗ X ; hom( E )), this extends to a homomor-phism of C ∗ -algebras σ : P S − ph ( X ; E ) → C ( π S ∗ X ; hom( E )) . Similarly, the symbol map σ ∂ i induces a continuous linear map σ ∂ i : P S − ph ( X ; E ) → P πi − ph ( H i ; E ) , where P πi − ph ( H i ; E ) is the closure of Ψ πi − ph ( H i ; E ) in L ( H i , H i ) with H i theBanach space L g π ( π N H i /S i ; E ) introduced in (9.23). By Theorem 9.7, this inducesan injective map(9.42)( σ , k M i =1 σ ∂ i ) : P S − ph ( X ; E ) / K ֒ → C ( π S ∗ X ; hom( E )) ⊕ ( k M i =1 P πi − ph ( X ; E ))where K ⊂ L ( H , H ) is the subspace of compact operators. Since this is an injectivemap of C ∗ -algebras mapping the identity to the identity, it is a standard fact(see for instance Proposition 1.3.10 in [12]) that an element of P S − ph ( X ; E ) / K isinvertible if and only if its image under the map (9.42) is invertible. Since a boundedoperator in L ( H , H ) is Fredholm if and only if it is invertible in L ( H , H ) / K , theresult follows. (cid:3) The semiclassical S -calculus Consider the manifold with corner X π × [0 , ǫ where ǫ should be consideredas a semiclassical parameter. The semiclassical π -double space is obtained byblowing up the p -submanifold ∆ π × { } ,(10.1) X π − sl = [ X π × [0 , ǫ ; ∆ π × { } ]with blow-down map(10.2) β sl : X π − sl → X π × [0 , ǫ . See Figure 5 for a picture of the semiclassical π -double space when X is a manifoldwith boundary. We denote the ‘new’ boundary face obtained via this blow-up by(10.3) ff = β − (∆ π × { } ) ⊂ X π − sl . We also denote by(10.4) T ǫ =0 = β − ( X π × { } \ (∆ π × { } )the lift of the ‘old’ face X π × { } to X π − sl . SEUDODIFFERENTIAL OPERATORS ON MANIFOLDS WITH FIBRED CORNERS 47 ✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑ ff π − sl ✻ x ✲ x ′ ✑✑✑✰ ǫ (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) ff T ǫ =0 T ǫ =0 Figure 5. The semiclassical π -double spaceNotice that ff \ (ff ∩ T ǫ =0 ) is naturally diffeomorphic to N ∆ π ∼ = π T X and thatff is diffeomorphic to the radial compactification of π T X → X . We will also denotethe lift of ff π i × [0 , 1] to X π − sl by(10.5) ff π i − sl = β − (ff π i × [0 , ∼ = [ff π i × [0 , π ∩ ff π i ) × { } ] . It will be useful to consider the spaces(10.6) ff π − sl = k [ i =1 ff π i − sl , ff sl = ff π − sl ∪ ff as well as the lift of ∆ π × [0 , 1] to X π − sl ,(10.7) ∆ sl = β − (∆ π × (0 , . Let also ff ∆ sl = ff sl ∪ ( X π × { } ) be the union of all the hypersurfaces of ∂X π − sl having a non-empty intersection with ∆ sl .We can now define the space of semiclassical S -pseudodifferential operators of order m by(10.8) Ψ m S − sl ( X ; E, F ) = (cid:8) K ∈ I m ( X π − sl , ∆ sl ; β ∗ sl p ∗ β ∗ π (Hom( E, F ) ⊗ π ∗ R ( π Ω))) K ≡ ∂X π − sl \ ff ∆ sl (cid:9) , where p : X π × [0 , → X π is the projection on the first factor. Polyhomogeneoussemiclassical S -operators can be defined in a similar way.As for S -pseudodifferential operators, there is a corresponding semiclassical Liegroupoid(10.9) G (0) π − sl = ∆ sl , G (1) π − sl = ◦ X π − sl ∪ ◦ ff ∆ sl , where ◦ ff ∆ sl = ff ∆ sl \ ∂ ff ∆ sl is the interior of ff ∆ sl as a subset of ∂X π − sl . Clearly, ∆ sl is naturally identified with X × [0 , and range of α ∈ G (1) π − sl with p ◦ β sl ( α ) = ǫ and β π ◦ p ◦ β sl ( α ) = ( x , x ) ∈ X by(10.10) d ( α ) = ( x , ǫ ) , r ( α ) = ( x , ǫ ) , where p : X π × [0 , → [0 , 1] is the projection on the second factor. Since G π − sl isa Lie groupoid, any choice of a metric on π T X × [0 , 1] provides a (smooth) Haarsystem on G π − sl [41], giving to it the structure of a measured groupoid. As inthe proof of Lemma 4.6, observe that G π − sl can be written as a disjoint union ofmeasurewise amenable groupoids,(10.11) G π − sl = ( π T X ) G ( ◦ X × ◦ X ) × (0 , ǫ k G i =1 ( H i × π i π T S i × π i H i ) | G i × (0 , ǫ × R , where G i = H i \ ( ∪ j>i H j ). Thus, by Lemma 4.5, we conclude that G π − sl is mea-surewise amenable with C ∗ ( G π − sl ) nuclear and equal to C ∗ r ( G π − sl ).In the terminology of [9], G π − sl is the tangent groupoid of G π . From [2] and[39], there is a calculus of pseudodifferential operators associated to this groupoid.It corresponds to operators in Ψ ∗ π − sl ( X ; E, F ) with Schwartz kernel compactlysupported in G (1) π − sl . As for G (1) π , the inverse map ι and the composition mapscomes from the natural smooth extensions of the corresponding maps on the Liegroupoid ◦ X × ◦ X × [0 , 1] with domain and range given by d ( x , x , ǫ ) = ( x , ǫ ) and r ( x , x , ǫ ) = ( x , ǫ ).There are many symbol maps associated to Ψ m S − sl ( X ; E, F ). There is the obviousone associated to conormal distributions. With the natural identification of N ∗ ∆ sl with π T X × [0 , m S − sl ( X ; E, F ) σ m / / S [ m ] ( N ∗ ∆ sl ; φ ∗ Hom( E, F ))where φ is the composition of the natural maps N ∗ ∆ sl → ∆ sl and ∆ sl = X × [0 , → X . This gives a short exact sequence(10.13)0 / / Ψ m − S − sl ( X ; E, F ) / / Ψ m S − sl ( X ; E, F ) σ m / / S [ m ] ( N ∗ ∆ sl ; φ ∗ Hom( E, F )) / / . We say an operator P ∈ Ψ m S − sl ( X ; E, F ) is elliptic if its principal symbol σ m ( P ) isinvertible.Restriction to boundary hypersurfaces of X π − sl leads to other types of symbols.Restricting to the hypersurface ff , we get the short exact sequence(10.14)0 / / ǫ Ψ m S − sl ( X ; E, F ) / / Ψ m S − sl ( X ; E, F ) σ ǫ =0 / / Ψ m ff ( X ; E, F ) / / . On the other hand, restricting to the face ff π i − sl for i ∈ { , . . . , k } , we get the shortexact sequence(10.15)0 / / x i Ψ m S − sl ( X ; E, F ) / / Ψ m S − sl ( X ; E, F ) σ ff πi − sl / / Ψ m ff πi − sl ( X ; E, F ) / / . Combining the symbol maps σ ff πi and σ ǫ =0 , that is, restricting to the hypersurfaceff sl , we also get the short exact sequence(10.16)0 / / ǫx Ψ m S − sl ( X ; E, F ) / / Ψ m S − sl ( X ; E, F ) σ ffsl / / Ψ m ff sl ( X ; E, F ) / / . SEUDODIFFERENTIAL OPERATORS ON MANIFOLDS WITH FIBRED CORNERS 49 Finally, a symbol of particular importance is obtained by restricting at the face X π × { } , giving the short exact sequence(10.17)0 / / (1 − ǫ )Ψ m S − sl ( X ; E, F ) / / Ψ m S − sl ( X ; E, F ) σ ǫ =1 / / Ψ m S ( X ; E, F ) / / . In fact, more generally, for ǫ ∈ (0 , A ∈ Ψ m S − sl ( X ; E, F ) to thehypersurface X π × { ǫ } to get an operator A ǫ ∈ Ψ m S ( X ; E, F ). This gives us a wayof composing semiclassical S -operators,( A ◦ B ) ǫ := A ǫ ◦ B ǫ . Proposition 10.1. If E, F and G are smooth vector bundles on X , then Ψ m S − sl ( X ; F, G ) ◦ Ψ n S − sl ( X ; E, F ) ⊂ Ψ m + n S − sl ( X ; E, G ) and the induced map is continuous with respect to the natural Fr´echet topology.Furthermore, the various symbol maps induce composition laws in such a way thatthey become algebra homomorphisms. A similar result holds for polyhomogeneoussemiclassical S -operators.Proof. We can employ the same strategy as in the proof of Theorem 8.1 and proceedby induction on the dimension of the manifold with fibred corners. Notice that thesecond part of the proof of Theorem 8.1 (starting with Lemma 8.3) mostly involvepartitions of unity and has a direct generalization to semiclassical S -operators.This means the proposition follows from Lemma 10.2 below, which is an analog ofLemma 8.2 for semiclassical S -operators. (cid:3) Lemma 10.2. Suppose that the conclusion of Proposition 10.1 holds for the man-ifold with fibred corners X . Then it also holds for the manifold with fibred corner R p × X defined just before Lemma 8.2.Proof. The proof is similar to the one of Lemma 8.2. To avoid repetition, we willfocus on the parts that require changes. First, without loss of generality, we canassume E = F = G = C . Using the Fourier transform on R p , we can describe theaction of operators A ∈ Ψ m S − sl ( R p × X ) and B ∈ Ψ n S − sl ( R p × X ) by(10.18) A ǫ u ( t ) = 1(2 πǫ ) p Z e i ( t − t ′ ) · τǫ a ( t ; τ ) u ( t ′ ) dt ′ dτ,B ǫ u ( t ) = 1(2 πǫ ) p Z e i ( t − t ′ ) · τǫ b ( t ; τ ) u ( t ′ ) dt ′ dτ. Here, a and b are operator-valued symbols,(10.19) a ∈ C ∞ ( R p ; Ψ m S − sl − sus( p ) ( X )) , b ∈ C ∞ ( R p ; Ψ n S − sl − sus( p ) ( X )) , where Ψ ℓ S − sl − sus( p ) ( X ) is the space of R p -suspended semiclassical S -operators oforder ℓ on X , and the variable τ ∈ R p in (8.1) is seen as the suspension parameter.Precisely, as for suspended S -operators, the space Ψ ℓ S − sl − sus( p ) ( X ) can be definedas the subspace of Ψ ℓ S − sl ( R p × X ) consisting of operators that are unchanged bytranslations in R p . These operator-valued symbols are such that(10.20) a ∈ C ∞ ( R pt ; Ψ m S − sl − sus( p ) ( X )) = ⇒ D αt D βτ a ∈ (1+ t ) − | α | C ∞ ( R pt ; Ψ m −| β | S − sl − sus( p ) ( X )) , so that the techniques of [40] can be applied. More precisely, using the change ofvariable ξ = τǫ , we deduce from (8.4), (8.5) and (8.6) that(10.21) A ǫ ◦ B ǫ u ( t ) = 1(2 πǫ ) p Z e i ( t − t ′ ) · τǫ c ( t, τ ) u ( t ′ ) dt ′ dτ, where c ( t, τ ) is an operator-valued symbol which for N ∈ N can be written in theform(10.22) c ( t, τ ) = X | α | An operator P ∈ Ψ m S − sl ( X ; E, F ) is said to be elliptic if σ m ( P )is invertible. It is said to be fully elliptic if it is elliptic and σ ǫ =1 ( P ) is a fullyelliptic S -operator.A natural sub-groupoid of G (1) π − sl is obtained by considering the interior of ff sl (asa subset of ∂X π − sl ),(10.25) T FC X = ◦ ff sl = ff sl \ ∂ ff sl , where ∂ ff sl := ff sl ∩ ( ∂X π − sl ) \ ff sl . The groupoid T FC X also contains ff \ (ff ∩ T ǫ =0 ) as a subgroupoid. It inheritsfrom the Lie structure of G π − sl the structure of a continuous family groupoid ([41]).There is also an induced continuous Haar system once a Haar system is fixed on G π − sl .As for G π − sl , the groupoid T FC X can be written as a disjoint union of topologi-cally amenable groupoids,(10.26) T FC X = π T X ⊔ ki =1 ( H i × π i π T S i × π i H i ) | G i × (0 , ǫ × R , SEUDODIFFERENTIAL OPERATORS ON MANIFOLDS WITH FIBRED CORNERS 51 where G i = H i \ ( ∪ j>i H j ). Thus, we conclude from Lemma 4.5 that T FC X ismeasurewise amenable with C ∗ ( T FC X ) nuclear and equal to C ∗ r ( T FC X ).As we will now describe, the K -theory of T FC X corresponds to the stable ho-motopy classes of fully elliptic polyhomogeneous S -operators. For this purpose, wewill restrict our attention to fully elliptic polyhomogeneous S -operators of orderzero. This is not a serious restriction. if P ∈ Ψ m S − ph ( X ; E, F ) is fully elliptic, wecan replace it by the fully elliptic operator P (∆ E + 1) − m ∈ Ψ S − ph ( X ; E, F ), where∆ E ∈ Ψ S − ph ( X ; E ) is some (positive) Laplacian associated to a choice of S -metricon X and a choice of Hermitian metric on E . Definition 10.4. Two fully elliptic operators P ∈ Ψ S − ph ( X ; E , E ) and P ∈ Ψ S − ph ( X ; E , F ) are homotopic if they can be connected by a continuous familyof fully elliptic polyhomogeneous S -operators P t ∈ Ψ S − ph ( X ; E t , F t ) , t ∈ [0 , . We say instead that P and P are stably homotopic if they become homotopicafter the addition to each of them of the identity operator Id H acting on the sectionsof some complex vector bundle H → X .Stable homotopies induce an equivalence relation and we denote by FE S ( X )the set of fully elliptic operators modulo stable homotopies. This set is in fact anabelian group with addition given by direct sum and inverse given by the parametrixconstruction of Proposition 9.1. It can be identified with the K-theory of a mappingcone. To see this, let us use the notation of the proof of Theorem 9.11 and denoteby A = P S − ph ( X ) the closure of Ψ S − ph ( X ) in L ( H , H ), where H = L g π ( X ). Thealgebra A contains the subalgebra K ⊂ L ( H , H ) of compact operators so that wecan consider the quotient map(10.27) q : A → A / K . The algebra A = C ( X ) of continuous functions on X is another subalgebra of A .Denote also by q : A → A / K the restriction of the quotient map to A . Let(10.28) C q = { ( a , a ) ∈ A ⊕ C ([0 , A / K ); q ( a ) = a (0) } be the mapping cone of the map q : A → A / K . Consider also the mapping cylinder(10.29) C + q = { ( a , a ) ∈ A ⊕ C ([0 , A / K ); q ( a ) = a (0) } . By Theorem 9.11, a fully elliptic operator P ∈ Ψ S − ph ( X ; E, F ), defines a K -classin K (C + q , C q ) ∼ = K (C q ) . This K -class only depends on the stable homotopy class of P so that there is awell-defined group homomorphism(10.30) σ C q : FE S ( X ) → K (C q ) . Proposition 10.5. The map σ C q is a group isomorphism.Proof. This can be seen as a particular case of a result of Savin [46, Theorem 4].Alternatively, since Theorem 9.11 identifies FE S ( X ) with the relative K -group K ( q )associated to the homomorphism q : A → A / K (see for instance [4] or [16] for adefinition of K ( q )), we can follow instead the approach in [16, Theorem 3.29]. This consists in noticing that the map σ C q naturally fits into a commutative diagram ofexact sequences,(10.31) K ( A ) / / s (cid:15) (cid:15) K ( A / K ) / / s (cid:15) (cid:15) FE S ( X ) σ C q (cid:15) (cid:15) / / K ( A ) / / Id (cid:15) (cid:15) K ( A / K ) Id (cid:15) (cid:15) K ( SA ) / / K ( S ( A / A )) / / K (C q ) / / K ( A ) / / K ( A \ K ) , where the bottom row is the Puppe sequence associated to q : A → A / K and s denotes the suspension isomorphism. The result then follows by applying thefive-lemma to this diagram. (cid:3) The group FE S ( X ) can also be related with the K -theory of the groupoid T FC X . Indeed, given a fully elliptic S -operator P ∈ Ψ S − ph ( X ; E, F ), let P ∈ Ψ S − sl − ph ( X ; E, F ) be a corresponding fully elliptic semiclassical S -operator suchthat σ ǫ =1 ( P ) = P . The full ellipticity insures that σ ff sl ( P ) ∈ Ψ sl − ph ( X ; E, F )defines a K -class in K ( P sl − ph ( X ) , C ∗ ( T FC X )) ∼ = K ( C ∗ ( T FC X )) , where C ∗ ( T FC X ) = C ∗ r ( T FC X ) is the reduced C ∗ -algebra of the groupoid T FC X and P sl − ph ( X ) is the C ∗ -algebra obtained by taking the closure Ψ sl − ph ( X ) withrespect to the reduced norm of the groupoid T FC X , see for instance [19, p.641].This K -class only depends on the stable homotopy class of P , so that there isin fact a well-defined group homomorphism(10.32) σ nc : FE S ( X ) → K ( C ∗ ( T FC X )) . Theorem 10.6. The map σ nc in (10.32) is an isomorphism of abelian groups.Proof. By Proposition 10.5, it suffices to construct a natural identification between K ( C ∗ ( T FC X )) and K (C q ) inducing a commutative diagram(10.33) K ( C ∗ ( T FC X )) ∼ = (cid:15) (cid:15) FE S ( X ) σ nc ♦♦♦♦♦♦♦♦♦♦♦ σ C q ' ' ❖❖❖❖❖❖❖❖❖❖❖ K (C q ) . To construct this natural identification, consider the algebra(10.34) ( σ ⊕ σ ff sl )(Ψ S − sl − ph ( X )) ⊂ C ∞ ( S ( N ∗ ∆ sl )) ⊕ Ψ sl − ph ( X )and let B be its C ∗ -closure in C ( S ( N ∗ ∆ sl )) ⊕P sl − ph ( X ). The symbol σ ǫ =1 restrictsto give a map(10.35) σ ǫ =1 : B → C ( S ( π T ∗ X )) ⊕ k M i =1 P πi − ph ( H i ) ! , where H , . . . , H k is an exhaustive list of the boundary hypersurfaces of X and P πi − ph ( H i ) is the C ∗ -closure of Ψ πi − ph ( H i ) with respect to the reduced norm SEUDODIFFERENTIAL OPERATORS ON MANIFOLDS WITH FIBRED CORNERS 53 (see [19, p.641]) for the groupoid ff π i ∩ ◦ ff π . There is a natural inclusion ι : C ( X ) ֒ →B . Let B be the kernel of the map (10.35) and consider the subalgebraˆ B = { b ∈ B | σ ff ( b ) ∈ C ( X ) } . Clearly, there is a natural identification K ( ˆ B ) ∼ = K (C ι ), where C ι is the map-ping cone of the natural inclusion ι : C ( X ) → σ ǫ =1 ( B ). On the other hand, thecommutative diagram of short exact sequences(10.36) 0 / / S ( σ ǫ =1 ( B )) / / Id (cid:15) (cid:15) ˆ B / / (cid:127) _ (cid:15) (cid:15) C ( X ) / / (cid:127) _ (cid:15) (cid:15) / / S ( σ ǫ =1 ( B )) / / B / / P − ph ( X ) / / K -theory. Since the inclusion C ( X ) ⊂ P − ph ( X ) induces isomorphisms in K -theory,we conclude by the five-lemma that the inclusion ˆ B ⊂ B also induces isomorphismsin K -theory. This means there are natural identifications(10.37) K ( B ) ∼ = K ( ˆ B ) ∼ = K (C ι ) ∼ = K (C q ) , where we have used Theorem 9.11 in the last step. On the other hand, the principalsymbol induces a short exact sequence(10.38) 0 / / C ∗ ( T FC X ) / / B σ / / C ( S ( π T ∗ X ) × [0 , / / . Since the quotient is contractible, this induces a natural identification K ( C ∗ ( T FC X )) ∼ = K ( B ) , so that we obtain the desired identification by combining this with (10.37). Thanksto the naturality of our construction, one can readily check it induces a commutativediagram as in (10.33). (cid:3) Poincar´e duality This last section will involve some Kasparov bivariant K -theory. The unfamiliarreader may for instance have a look at [51, 5, 11]. We are using the notations of [5]and [11].Let P ∈ Ψ S − ph ( X ; E, F ) be a fully elliptic operator and let Q be a parametrixfor P as constructed in Proposition 9.1. Set H = L g π ( X, E ) ⊕ L g π ( X, F ) and P = (cid:18) QP (cid:19) . By Theorem 9.11, the operator P is bounded and Fredholm on H . Let C ∞ π ( X ) ⊂ C ∞ ( X ) be the subalgebra of smooth functions on X which areconstants along the fibres of the fibration π i for each boundary hypersurface H i of X . Clearly, we have a dense inclusion C ∞ π ( X ) ⊂ C ( S X ). Denote by m : C ( S X ) →L ( H ) the representation given by multiplication.For f ∈ C ∞ π ( X ), m ( f ) is naturally a S -operator of order 0. The commutator[ m ( f ) , P ] is a S -operator of order − σ ∂ i ([ m ( f ) , P ]) = 0 for all i . Hence, by Theorem 9.7, the commutator [ m ( f ) , P ] is a compact operator. By the densityof C ∞ π ( X ) in C ( S X ), we conclude more generally that [ m ( f ) , P ] ∈ K ( H ) for all f ∈ C ( S X ). Since P − Id ∈ K ( H ), this means ( H , m , P ) is a Kasparov ( C ( S X ) , C )-module. We denote by(11.1) [ P ] = [( H , m , P )] ∈ KK ( C ( S X ) , C ) = K ( S X ) . the corresponding Kasparov ( C ( S X ) , C )-cycle.It is straightforward to check that this Kasparov cycle only depends on the stablehomotopy class of P . This means this procedure defines a homomorphism of abeliangroups(11.2) quan : FE S ( X ) → K ( S X ) . Using the identification of Theorem 10.6, this can be seen as defining a homomor-phism of abelian groups(11.3) PD := quan ◦ σ − : K ( C ∗ ( T FC X )) → K ( S X ) . This map establishes a Poincar´e duality between T FC X and S X . This can be de-scribed in a systematic way using Kasparov bivariant K-theory.We first recall that two separable C ∗ -algebras A and B are Poincar´e dual in K -theory if there exist α ∈ KK ( A ⊗ B, C ) and β ∈ KK ( C , A ⊗ B ) (minimal tensorproducts are understood) such that β ⊗ A α = 1 B and β ⊗ B α = 1 A . Once such an α is given, the element β completing the Poincar´e duality is unique. The element α (resp. β ) is called the Dirac (resp. dual-Dirac) element of the Poincar´e duality.For any C ∗ -algebras C, D , they provide isomorphisms · ⊗ A α : KK ( C, A ⊗ D ) −→ KK ( B ⊗ C, D ) , with inverses given by β ⊗ B · : KK ( B ⊗ C, D ) −→ KK ( C, A ⊗ D ) . We are interested in the special case where A = C ∗ ( T FC X ) and B = C ( S X ). Toconstruct a Dirac element, consider the groupoid G ′ π − sl = G π − sl \ ff π ×{ ǫ = 1 } . Itenters in the short exact sequence(11.4) 0 −→ C ∗ ( ◦ X × ◦ X × (0 , −→ C ∗ ( G ′ π − sl ) ev FC −→ C ∗ ( T FC X ) −→ , where ev FC is the obvious evaluation map induced by the inclusion T FC X ⊂ G ′ π − sl .The ideal is contractible so by classical arguments [ev FC ] is invertible in KK -theoryand we set(11.5) ∂ FC X = [ev FC ] − ⊗ [ev ǫ =1 ] ⊗ [ ◦ µ ] − ∈ KK ( C ∗ ( T FC X ) , C ) . Here, ev ǫ =1 : C ∗ ( G ′ π − sl ) → C ∗ ( ◦ X × ◦ X ) is the obvious evaluation map at ǫ = 1and the homomorphism ◦ µ is defined by λ λq where q is a rank one self-adjointprojection and [ ◦ µ ] − is thus the Morita equivalence C ∗ ( ◦ X × ◦ X ) ∼ C .The natural inclusion C ∞ π ( X ) ⊂ Ψ S − sl − ph ( X ) extends to an inclusion ι : C ( S X ) ֒ →P S − sl − ph ( X ), where P S − sl − ph ( X ) is the C ∗ -closure of Ψ S − sl − ph ( X ) with respect SEUDODIFFERENTIAL OPERATORS ON MANIFOLDS WITH FIBRED CORNERS 55 to the reduced norm for the groupoid G π − sl . This can be used to define a ‘zerosections’ homomorphism(11.6) Ψ FC π − sl : C ( S X ) ⊗ C ∗ ( G π − sl ) −→ C ∗ ( G π − sl ) g ⊗ a ι ( g ) a. By restriction to T FC X , we also get a map(11.7) Ψ FC X : C ( S X ) ⊗ C ∗ ( T FC X ) −→ C ∗ ( T FC X ) . Consider then the following Kasparov cycle,(11.8) D FC X = [Ψ FC X ] ⊗ ∂ FC X ∈ KK ( C ( S X ) ⊗ C ∗ ( T FC X ) , C ) . Theorem 11.1. The Kasparov cycle D FC X is the Dirac element of a Poincar´e dualitybetween C ∗ ( T FC X ) and C ( S X ) .Proof. The groupoid T FC X is slightly different, but nevertheless intimately relatedto the noncommutative tangent space of [10] (see Corollary 11.3 below). At the costof clarifying this relationship, it is therefore possible to transfer the Poincar´e dualityresult of [10] to our context. To have instead a more self-contained approach, wewill adapt the proof of [10] to our context. Really, this should be thought as ahybrid of the groupoid approach of [10] and the operator theoretic approach of [36](see also [33]).Let H , . . . , H k be an exhaustive list of the boundary hypersurfaces of X suchthat i < j, H i ∩ H j = 0 = ⇒ H i < H j . Set X = X and consider the non-compact manifolds with fibred corners(11.9) X j := X \ j [ i =1 H i ! , for j ∈ { , . . . , k } . Let C ( S X j ) = { f ∈ C ( S X ); f | q ( H i ) = 0 for i ≤ j } be the corresponding space of continuous functions on the associated stratifiedpseudomanifold, where q : X → S X is the natural quotient map. Finally, set T FC X = T FC X and consider the subgroupoid T FC X j := T FC X \ ◦ j [ i =1 ff π i − sl ! , where the interior is taken as a subset of ∂X π − sl . Clearly, the morphism Ψ FC X restricts to give a morphismΨ FC X j : C ( S X j ) ⊗ C ∗ ( T FC X j ) → C ∗ ( T FC X ) , allowing us to define the following Kasparov cycle, D FC X j = [Ψ FC X j ] ⊗ ∂ FC X ∈ KK ( C ( S X j ) ⊗ C ∗ ( T FC X j ) , C ) . Now, for j ∈ { , . . . , k } , we have two natural short exact sequences of C ∗ -algebras,0 / / C ( S X j ) / / C ( S X j − ) α / / C ( S j \ ∂S j ) / / , (11.10) 0 / / C ∗ ( H j ) / / C ∗ ( T FC X j − ) β / / C ∗ ( T FC X j ) / / , (11.11) where H j ⊂ ff π j − sl is the subgroupoid given by H j = T FC X j − \ T FC X j . It is naturally Morita equivalent to the groupoid π T S i . For this latter groupoid,we have a natural Kasparov cycle given by D MC S j = [Ψ MC S j ] ⊗ ∂ FC S j ∈ KK ( C ( S j \ ∂S j ) ⊗ C ∗ ( π T S j ) , C ) , where Ψ MC S j : C ( S j \ ∂S j ) ⊗ C ∗ ( π T S j ) → C ∗ ( T FC S j )is the morphism obtained by restriction of Ψ FC S j . Using the Morita equivalencebetween H j and π T S j , this gives a corresponding Kasparov cycle D MC H j ∈ KK ( C ( S j \ ∂S j ) ⊗ C ∗ ( H j ) , C ). This cycle can be defined alternatively by D MC H j = [Ψ MC H j ] ⊗ ∂ FC X ,where Ψ MC H j : C ( S j \ ∂S j ) ⊗ C ∗ ( H j ) → C ∗ ( T FC X )is the morphism obtained by restriction of Ψ FC X .Now, the cycle D FC j − , D FC j and D MC H j can be used to obtain a diagram intertwiningthe six-term exact sequences in KK-theory associated to the short exact sequences(11.10) and (11.11),(11.12) ... (cid:15) (cid:15) ... (cid:15) (cid:15) KK q ( A, B ⊗ C ( S X j )) ⊗ C ( S Xj ) D FC Xj / / (cid:15) (cid:15) KK q ( A ⊗ C ∗ ( T FC X j ) , B ) (cid:15) (cid:15) KK q ( A, B ⊗ C ( S X j − )) (cid:15) (cid:15) ⊗ C ( S Xj − D FC Xj − / / KK q ( A ⊗ C ∗ ( T FC X j − ) , B ) (cid:15) (cid:15) KK q ( A, B ⊗ C ( S j \ ∂S j )) ∂ α (cid:15) (cid:15) ⊗ C ( Sj \ ∂Sj ) D MC H j / / KK q ( A ⊗ C ∗ ( H j ) , B ) ∂ β (cid:15) (cid:15) ... ... , where A and B are C ∗ -algebras.The result then follows from the following two claims. Claim 1. The diagram (11.12) is commutative up to sign. Claim 2. The Kasparov cycles D FC X k = D MC X and D MCS j for j ∈ { , . . . , k } are Diracelements. Indeed, using the Morita equivalence between π T S j and H j , we see that D MC H j is also a Dirac element. Thus, starting with j = k and applying the five-lemma to(11.12), we find that the map KK q ( A, B ⊗ C ( S X k − )) ⊗ C ( S Xk − D FC Xk − / / KK q ( A ⊗ C ∗ ( T FC X k − ) , B ) SEUDODIFFERENTIAL OPERATORS ON MANIFOLDS WITH FIBRED CORNERS 57 is an isomorphism. By [10, Lemma 2], this implies D FC X k − is a Dirac element.Repeating this argument for j = k − , k − , . . . , 1, we find more generally that D FC X j is a Dirac element for all j ∈ { , , . . . , k } . In particular, D FC X = D FC X is aDirac element.Thus, it remains to prove the two claims, which we do below. (cid:3) Proof of Claim 1. The proof of the commutativity of the squares not involvingboundary homomorphisms is straightforward and left to the reader. To obtain thecommutativity of the remaining squares, we need to show that(11.13) ∂ α ⊗ C ( S X j ) D FC X j = ∂ β ⊗ C ∗ ( H j ) D MC H j , where ∂ α ∈ KK ( C ( S j \ ∂S j ) , C ( S X j )) and ∂ β ∈ KK ( C ∗ ( T FC X j ) , C ∗ ( H j )) arethe boundary homomorphisms associated to the short exact sequences (11.10) and(11.11). From the definition of D FC j and D MC H j , this means we need to show that(11.14) ∂ α ⊗ C ( S X j ) [Ψ FC X j ] = ∂ β ⊗ C ∗ ( H j ) [Ψ MC H j ]in KK ( C ( S j \ ∂S j ) ⊗ C ∗ ( T FC X j ) , C ∗ ( T FC X )). To see this, consider the subgroupoid L j := T FC X j ∩ ff ∩ ff π j − sl ⊂ π T X | H j . Thus, there is a natural restriction homo-morphism C ∗ ( T FC X j ) → C ∗ ( L j ). There is also an obvious multiplication homomor-phism C ( S j \ ∂S j ) ⊗ C ( L j ) → C ( π T X | ◦ H j ) . Let also N H j be a tubular neighborhood of H j coming from an iterated fibred tubesystem and set W = ◦ N H j . The tube system of H j induces an identification(11.15) C ( R ) ⊗ C ∗ ( π T X | ◦ H j ) → C ∗ ( T W ) . On the other hand, the short exact sequence0 / / C ∗ ( ◦ ff π j − sl ) / / C ∗ ( ◦ ff π j − sl ∪ π T X | ◦ H j ) / / C ∗ ( π T X | ◦ H j ) / / , induces a boundary homomorphism in KK ( C ∗ ( π T X | ◦ H j ) , C ∗ ( ◦ ff π j − sl )). By com-posing with the inclusion C ∗ ( ◦ ff π i − sl ) ⊂ C ∗ ( H j ), this induces a morphism ∂ ∈ KK ( C ( R ) ⊗ C ∗ ( π T X | ◦ H j ) , C ∗ ( H j )). Using the identification (11.15), this gives acorresponding element in ∂ ′ ∈ KK ( C ∗ ( T W ) , C ∗ ( H j )) inducing a commutative di-agram of Kasparov cycles C ( R ) ⊗ C ( π T X | ◦ H j ) / / ∂ ' ' PPPPPPPPPP C ∗ ( T W ) ∂ ′ (cid:15) (cid:15) C ∗ ( H j ) . The result then follows by noticing this fits into a bigger diagram commutative upto sign involving the Kasparov cycles of (11.14),(11.16) C ( R ) ⊗ C ( S j \ ∂S j ) ⊗ C ( T FC X j ) (cid:15) (cid:15) ∂ α ⊗ C ( S Xj ) [Ψ FC Xj ] , , ∂ β ⊗ C∗ ( H j ) [Ψ MC H j ] r r C ( R ) ⊗ C ∗ ( π T X | ◦ H j ) u u ❦❦❦❦❦❦❦❦❦❦❦❦❦ ∂ ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙ C ∗ ( T W ) ∂ ′ / / ) ) ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ C ∗ ( H j ) u u ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ C ∗ ( T FC X ) . (cid:3) For Claim 2, this is the Poincar´e duality for manifolds with corners obtained in[32]. The result of [32] is not formulated in terms of Dirac elements, but this can beremedied easily by using the semiclassical b -double space (or the semiclassical cuspdouble space). For the convenience of the reader, we will provide a brief outline.First, the semiclassical b -double space is defined by X b − sl = [ X b × [0 , ǫ ; ∆ b × { } ] , where ∆ b ⊂ X b is the lifted diagonal. Denote the new face obtained by this blow-up by ff ,b . Notice that the b -tangent bundle is naturally included in ff ,b . If ff b − sl is the union of all the boundary hypersurfaces intersecting the lift of ∆ b × [0 , 1] in X b − sl , we get a corresponding groupoid T b X := ◦ ff b − sl \ ( ◦ ff b − sl ∩ X b × { } ) . Using evaluation maps as in the fibred corners case, one can define a naturalKasparov cycle ∂ bX ∈ KK ( C ∗ ( T b X ) , C ). There is also a ‘zero sections’ mor-phism Ψ bX : C ( X ) ⊗ C ∗ ( T b X ) → C ∗ ( T b X ), and so a corresponding Kasparov cycle D bX = [Ψ bX ] ⊗ ∂ bX in KK ( C ( X ) ⊗ C ∗ ( T b X ) , C ).Let H , . . . , H k be an exhaustive list of boundary hypersurfaces of X and set X j = X \ j [ i =1 H i , X ′ j = X \ k [ i = j +1 H i , with the convention that X = X = X ′ k . Then, by restriction of D bX , we obtaincorresponding cycles D bX j ∈ KK ( C ( X j ) ⊗ C ∗ ( b T X ′ j ) , C ) , where b T X ′ j = b T X (cid:12)(cid:12) X ′ j . Since D MC X = D bX k , Claim 2 is a consequence of the following proposition. Proposition 11.2. If X is a compact manifold with corners and H , . . . , H k isan exhaustive list of its boundary hypersurfaces, then the Kasparov cycle D bX j is aDirac element for all j ∈ { , , . . . , k } . SEUDODIFFERENTIAL OPERATORS ON MANIFOLDS WITH FIBRED CORNERS 59 Proof. From [17], we know that D bX is a Dirac element. This suggests to proceed byinduction on the depth of X . Thus, assume the proposition is true for all manifoldswith corners of depth less than the one of X . If N H j = H j × [0 , 1] is a collarneighborhood of H j in X , then, after making obvious identifications, the inclusion N H j ⊂ X induces two natural short exact sequences of C ∗ -algebras,0 / / C ( X j ) / / C ( X j − ) / / C ( ˆ H j × [0 , / / , (11.17) 0 / / C ∗ ( b T ˇ H j × T (0 , / / C ∗ ( b T X ′ j − ) / / C ∗ ( b T X ′ j ) / / , (11.18)where ˆ H j = H j \ j − [ i =1 ( H i ∩ H j ) ! , ˇ H j = H j \ k [ i = j +1 ( H i ∩ H j ) . By our inductive assumption, the cycle D b ˆ H j ∈ KK ( C ( ˆ H j ) ⊗ C ∗ ( b T ˇ H j ) , C ) is aDirac element. On the other hand, D b [0 , ∈ KK ( C ([0 , ⊗ C ∗ ( T (0 , , C ) is aDirac element by the result of [17]. This means the corresponding cycle D b ˆ H j × [0 , = D b ˆ H j ⊗ D b [0 , ∈ KK ( C ( ˆ H j × [0 , ⊗ C ∗ ( b T ˇ H j × T (0 , , C )is a Dirac element. Now, this Dirac element combines with D bX j and D bX j − to givea diagram intertwining the six-term exact sequences in KK -theory associated to(11.17) and (11.18),(11.19) ... (cid:15) (cid:15) ... (cid:15) (cid:15) KK q ( A, B ⊗ C ( X j )) ⊗ C ( Xj ) D bXj / / (cid:15) (cid:15) KK q ( A ⊗ C ∗ ( b T X ′ j ) , B ) (cid:15) (cid:15) KK q ( A, B ⊗ C ( X j − )) (cid:15) (cid:15) ⊗ C ( Xj − D bXj − / / KK q ( A ⊗ C ( b T X ′ j − ) , B ) (cid:15) (cid:15) KK q ( A, B ⊗ C ( ˆ H j × [0 , (cid:15) (cid:15) ⊗ C ( ˆ Hj × [0 , D b ˆ Hj × [0 , / / KK q ( A ⊗ C ∗ ( b T ˇ H j × T (0 , , B ) (cid:15) (cid:15) ... ... , where A and B are C ∗ -algebras. Using a similar method as for (11.12), it can beshown that this diagram is commutative up to sign. Thus, starting with j = 0and applying the five-lemma recursively to (11.19) as well as [10, Lemma 2], weconclude that D bX j is a Dirac element for all j ∈ { , , . . . , k } . (cid:3) Since the noncommutative tangent space T S X of [10] is also Poincar´e dual tothe stratified pseudomanifold S X , Theorem 11.1 has the following consequence. Corollary 11.3. The C ∗ -algebras C ∗ ( T S X ) and C ∗ ( T FC X ) are KK -equivalent.Proof. Let D S X ∈ KK ( C ∗ ( T S X ) ⊗ C ( S X ) , C ) be the Dirac element of [10] thatprovides the Poincar´e duality between C ∗ ( T S X ) and C ( S X ). Denote by( D FC X ) − ∈ KK ( C , C ∗ ( T FC X ) ⊗ C ( S X )) , ( D S X ) − ∈ KK ( C , C ∗ ( T S X ) ⊗ C ( S X )) , the dual-Dirac elements of D FC X and D S X respectively. Then the element α = ( D S X ) − ⊗ C ( S X ) D FC X ∈ KK ( C ∗ ( T FC X ) , C ∗ ( T S X ))is a KK -equivalence between C ∗ ( T FC X ) and C ∗ ( T S X ) with inverse α − = ( D FC X ) − ⊗ C ( S X ) D S X ∈ KK ( C ∗ ( T S X ) , C ∗ ( T FC X )) . (cid:3) The map PD in (11.3) can be described in terms of the Dirac element D FC X . Theorem 11.4. If P ∈ Ψ S − ph ( X ; E, F ) is a fully elliptic operator, then (11.20) σ nc ( P ) ⊗ C ∗ ( T FC X ) D FC X = [ P ] . In particular, the map PD in (11.3) is an isomorphism of abelian groups.Proof. Let P ∈ Ψ S − ph ( Y ; E, F ) be a fully elliptic operator and let Q be a parametrixfor P as constructed in Proposition 9.1. Let P ∈ Ψ S − sl − ph ( X ; E, F ) and Q ∈ Ψ S − sl − ph ( X ; F, E ) be fully elliptic semiclassical S -operators such that σ ǫ =1 ( P ) = P and σ ǫ =1 ( Q ) = Q . Without loss of generality, we can choose Q such that(11.21) PQ − ∈ Ψ −∞ S − sl ( X ; F ) , QP − ∈ Ψ −∞ S − sl ( X ; E ) . By construction, a := P| T FC X is a pseudodifferential operator on the groupoid T FC X of order 0, so it gives a (bounded) morphism between the C ∗ ( T FC X )-Hilbert modules C ∗ ( T FC X, E ) and C ∗ ( T FC X, F ). Reverting E and F , the same is true for b := Q| T FC X so we get a bounded morphism a = (cid:18) ba (cid:19) ∈ L ( C ∗ ( T FC X, E ⊕ F )) . Since σ ǫ =1 ( P ) = P and σ ǫ =1 ( Q ) = Q , we have that a | ǫ =1 is invertible with inverse b | ǫ =1 so that a − Id ∈ K ( C ∗ ( T FC X, E ⊕ F )). This means(11.22) (cid:0) C ∗ ( T FC X, E ⊕ F ) , a (cid:1) is a Kasparov ( C , C ∗ ( T FC X ))-cycle. Its class in K ( C ∗ ( T FC X )) is the element σ nc ( P )defined in (10.32).Similarly, we get a K -theory class associated with P . As before, T := (cid:18) QP (cid:19) ∈ L ( C ∗ ( G ′ π − sl , E ⊕ F ))and T − ∈ K ( C ∗ ( G ′ π − sl , E ⊕ F )), so that[ T ] = (cid:0) C ∗ ( G ′ π − sl , E ⊕ F ) , T (cid:1) ∈ K ( C ∗ ( G ′ π − sl )) . The cycle [ T ] is such that(11.23) [ T ] ⊗ [ev FC ] = (ev FC ) ∗ [ T ] = σ nc ( P ) . SEUDODIFFERENTIAL OPERATORS ON MANIFOLDS WITH FIBRED CORNERS 61 In order to achieve the computation proving (11.20), we observe that the homo-morphism (11.6) naturally induces a map(11.24) Ψ FC π − sl ′ : C ( S X ) ⊗ C ∗ ( G ′ π − sl ) −→ C ∗ ( G ′ π − sl )leading to the equality of homomorphisms(11.25) ev FC ◦ Ψ FC π − sl ′ = Ψ FC X ◦ (Id C ( S X ) ⊗ ev FC ) . Now, using the basic properties of the Kasparov product, we have, σ nc ( P ) ⊗ C ∗ ( T FC X ) D FC X = ([ T ] ⊗ [ev FC ]) ⊗ C ∗ ( T FC X ) D FC X , by (11.23),= τ C ( S X ) ([ T ] ⊗ [ev FC ]) ⊗ D FC X = τ C ( S X ) ([ T ]) ⊗ [Id C ( S X ) ⊗ ev FC ] ⊗ D FC X = τ C ( S X ) ([ T ]) ⊗ [Ψ FC π − sl ′ ] ⊗ [ev ǫ =1 ] ⊗ [ ◦ µ ] − , by (11.8), (11.25) . The next step requires some details. We have τ C ( S X ) ( T ) = (cid:18) C ( S X ) ⊗ C ∗ ( G ′ π − sl , E ⊕ F ) , l, Id ⊗ (cid:18) QP (cid:19)(cid:19) where C ( S X ) ⊗ C ∗ ( G ′ π − sl , E ⊕ F ) has the obvious right C ( S X ) ⊗ C ∗ ( G ′ π − sl )-modulestructure and the representation l is defined by: l ( f )( g ⊗ ξ ) = ( f g ) ⊗ ξ . We thenhave, τ C ( S X ) ( T ) ⊗ Ψ FC π − sl ′ = [ C ( S X ) ⊗ C ∗ ( G ′ π − sl , E ⊕ F )] ⊗ Ψ FC π − sl ′ C ∗ ( G ′ π − sl ) , l ⊗ Id , (Id ⊗ (cid:18) QP (cid:19) ) ⊗ Id ! . (11.26)By construction, C ∗ ( G ′ π − sl , E ⊕ F ) is a finitely generated projective Hilbert C ∗ ( G ′ π − sl )-module, so we can choose a self-adjoint idempotent e ∈ M r ( C ∗ ( G ′ π − sl )) such that C ∗ ( G ′ π − sl , E ⊕ F ) = e C ∗ ( G ′ π − sl ) r . This choice provides a Hilbert C ∗ ( G ′ π − sl )-moduleisomorphism[ C ( S X ) ⊗ C ∗ ( G ′ π − sl , E ⊕ F )] ⊗ Ψ FC Y − sl ′ C ∗ ( G ′ π − sl ) ≃ C ∗ ( G ′ π − sl , E ⊕ F )under which the representation l ⊗ Id corresponds to ν : C ( S X ) → L ( C ∗ ( G ′ π − sl , E ⊕ F )) defined by ν ( f )( e ( b , . . . , b r )) := e (Ψ FC π − sl ′ ( f, b ) , . . . , Ψ FC π − sl ′ ( f, b r )) , and the operator (Id ⊗ (cid:18) QP (cid:19) ) ⊗ Id simply corresponds to (cid:18) QP (cid:19) . In otherwords, we have the equality(11.27) τ C ( S X ) ([ T ]) ⊗ [Ψ FC Y − sl ′ ] = (cid:20)(cid:18) C ∗ ( G ′ π − sl , E ⊕ F ) , ν, (cid:18) QP (cid:19)(cid:19)(cid:21) in KK ( C ( S X ) , C ∗ ( G ′ π − sl )). It follows that,(11.28) σ nc ( P ) ⊗ C ∗ ( T FC X ) D FC X = (cid:20)(cid:18) C ∗ ( G ′ π − sl , E ⊕ F ) , ν, (cid:18) QP (cid:19)(cid:19)(cid:21) ⊗ [ev ǫ =1 ] ⊗ [ ◦ µ ] − = (cid:20)(cid:18) C ∗ ( ◦ X × ◦ X, E ⊕ F ) , ν ǫ =1 , (cid:18) QP (cid:19)(cid:19)(cid:21) ⊗ [ ◦ µ ] − = (cid:20)(cid:18) L π ( X, E ⊕ F ) , m , (cid:18) QP (cid:19)(cid:19)(cid:21) = [ P ] . (cid:3) References [1] P. Albin, E. Leichtnam, R. Mazzeo, and P. Piazza, The signature package on Witt spaces ,preprint, arXiv:1112.0989.[2] B. Ammann, R. Lauter, and V. Nistor, Pseudodifferential operators on manifolds with Liestructure at infinity , Annals of Mathematics (2007), 717–747.[3] C. Anantharaman-Delaroche and J. Renault, Amenable groupoids , L’EnseignementMath´ematique, 2000, Volume 36 of Monographies de L’Enseignement Math´ematique.[4] I. Androulidakis and G. Skandalis, The analytic index of elliptic pseudodifferential operatorson a singular foliation , preprint, arXiv:1004.3797.[5] Bruce Blackadar, K -theory for operator algebras , Cambridge University Press, 1998.[6] J.-P. Brasselet, G. Hector, and M. Saralegi, Th´eor`eme de de Rham pour les vari´et´es strat-ifi´ees , Ann. Global Anal. Geom. (1991), no. 3, 211–243. MR 1143404 (93g:55009)[7] J. Cheeger, Spectral geometry of singular riemannian spaces , J. Differential Geom. (1983),no. 4, 575–657.[8] G.A. Mendoza C.L. Epstein and R.B. Melrose, Resolvent of the Laplacian on strictly pseu-doconvex domains. , Acta Math. (1991), 1–106.[9] A. Connes, Noncommutative Geometry , Academic Press, San Diego, CA, 1994.[10] Claire Debord and Jean-Marie Lescure, K -duality for stratified pseudomanifolds , Geom.Topol. (2009), no. 1, 49–86. MR 2469513 (2009h:19005)[11] , Index theory and groupoids , Geometric and topological methods for quantum fieldtheory, Cambridge Univ. Press, 2010, pp. 86–158.[12] J. Dixmier, Les C ∗ -alg`ebres et leurs repr´esentations , Gauthier-Villars, Paris, 1964.[13] D. Grieser and E. Hunsicker, Pseudodifferential calculus for generalized Q -rank 1 locallysymmetric spaces I , J. Funct. Anal. (2009), no. 12, 3748–3801.[14] L. H¨ormander, The analysis of linear partial differential operators. vol. 3 , Springer-Verlag,Berlin, 1985.[15] T. Jeffres, R. Mazzeo, and Y. Rubinstein, K¨ahler-Einstein metrics with edge singularities ,preprint, arXiv:1105.5216.[16] M. Karoubi, K-Theory , Springer-Verlag, 2008 (reprint of the 1978 edition).[17] G.G. Kasparov, Equivariant KK-theory and the Novikov conjecture , Invent. Math. (1988),147–201.[18] T. Krainer, Elliptic boundary problems on manifolds with polycylindrical ends , J. Funct.Anal. (2007), 351–386.[19] R. Lauter, B. Monthubert, and V. Nistor, Pseudodifferential analysis on continuous familygroupoids , Documenta Math. (2000), 625–655.[20] R. Lauter and S. Moroianu, Fredholm theory for degenerate pseudodifferential operators onmanifold with fibred boundaries , Comm. Partial Differential Equations (2001), 233–283.[21] Kirill C. H. Mackenzie, General theory of Lie groupoids and Lie algebroids , London Mathe-matical Society Lecture Note Series, vol. 213, Cambridge University Press, Cambridge, 2005.MR 2157566 (2006k:58035)[22] R. Mazzeo, Elliptic theory of differential edge operators. I. , Comm. Partial Differential Equa-tions (1991), no. 10, 1615–1664. SEUDODIFFERENTIAL OPERATORS ON MANIFOLDS WITH FIBRED CORNERS 63 [23] R. Mazzeo and R. B. Melrose, Pseudodifferential operators on manifolds with fibred bound-aries , Asian J. Math. (1999), no. 4, 833–866.[24] R. Mazzeo and R.B. Melrose, Meromorphic extension of the resolvent on complete spaceswith asymptotically constant negative curvature , J. Funct. Anal. (1987), no. 2, 260–310.[25] , Analytic surgery and the eta invariant , Geom. Funct. Anal. (1995), no. 1, 14–75.[26] R. Mazzeo and G. Montcouquiol, Infinitesimal rigidity of cone-manifolds and the Stokerproblem for hyperbolic and Euclidean polyhedra , J. Differential Geom. (2011), no. 3, 525–576.[27] R.B. Melrose, Differential analysis on manifolds with corners Calculus of conormal distributions on manifolds with corners , Int. Math. Res. Notes (1992), 51–61.[29] , The Atiyah-Patodi-Singer index theorem , A. K. Peters, Wellesley, Massachusetts,1993.[30] , The eta invariant and families of pseudodifferential operators , Math. Res. Lett. (1995), no. 5, 541–561. MR 96h:58169[31] , Geometric scattering theory , Cambridge University Press, Cambridge, 1995.[32] R.B. Melrose and P. Piazza, Analytic K-theory for manifolds with corners , Adv. in Math (1992), 1–27.[33] R.B. Melrose and F. Rochon, Index in K-theory for families of fibred cusp operators , K-theory (2006), 25–104.[34] Bertrand Monthubert and Fran¸cois Pierrot, Indice analytique et groupo¨ıdes de Lie , C. R.Acad. Sci. Paris S´er. I Math. (1997), 193–198.[35] Paul S. Muhly, Jean N. Renault, and Dana P. Williams, Equivalence and isomorphism forgroupoid C ∗ -algebras , J. Operator Theory (1987), 3–22.[36] V. Nazaikinskii, A. Savin, and B. Sternin, Homotopy classification of elliptic operators onstratified manifolds , Izv. Math. (2007), no. 6, 1167–1192.[37] , Pseudodifferential operators on stratified manifolds , Differ. Uravn. (2007), no. 4,519–532.[38] , Pseudodifferential operators on stratified manifolds II , Differ. Uravn. (2007),no. 5, 685–696.[39] V. Nistor, A. Weinstein, and P. Xu, Pseudodifferential operators on groupoids , Pacific J.Math. (1999), 117–152.[40] C. Parenti, Operatori pseudodifferenziali in R n e applicazioni , Annali Mat. Pura et Appl. (1972), 359–389.[41] Alan L. T Paterson, Continuous family groupoids , Homology, Homotopy and Applications (2000), 89–104.[42] Jean Renault, A groupoid approach to C ∗ -algebras , Lecture Notes in Mathematics, vol. 793,Springer, Berlin, 1980. MR 584266 (82h:46075)[43] Victor Nistor Robert Lauter, Bertrand Monthubert, Spectral invariance for certain algebrasof pseudodifferential operators , Journal of the Institut of Math. Jussieu (2005), 405–442.[44] F. Rochon, Pseudodifferential operators on manifolds with foliated boundaries , preprint,arXiv:1009.4272.[45] F. Rochon and Z. Zhang, Asymptotics of complete K¨ahler metrics of finite volume onquasiprojective manifolds , preprint, arXiv:1106.0873.[46] Anton Savin, Elliptic operators on manifolds with singularities and K-homology , K-theory (2005), 71–98.[47] B.-W. Schulze, The iterative structure of corner operators , preprint, arXiv:09.05.0977.[48] , Pseudo-differential operators on manifolds with singularities , North-Holland, Ams-terdam, 1991.[49] M.A. Shubin, Pseudodifferential operators on R n , Sov. Math. Dokl. (1971), 147–151.[50] , Pseudodifferential operators and spectral theory , Springer, 2001.[51] Georges Skandalis, Kasparov’s bivariant K -theory and applications , Exposition. Math. (1991), 193–250.[52] F. Treves, Topological Vector Spaces, Distributions and Kernels , Academic Press, New York,1967.[53] Andrei Verona, Stratified mappings—structure and triangulability , Lecture Notes in Mathe-matics, vol. 1102, Springer-Verlag, Berlin, 1984. MR 771120 (86k:58010) Laboratoire de Math´ematiques, Universit´e Blaise Pascal E-mail address : [email protected] Laboratoire de Math´ematiques, Universit´e Blaise Pascal E-mail address : [email protected] Department of mathematics, Australian National University E-mail address ::