Analysis on singular spaces: Lie manifolds and operator algebras
aa r X i v : . [ m a t h . OA ] D ec ANALYSIS ON SINGULAR SPACES: LIE MANIFOLDS ANDOPERATOR ALGEBRAS
VICTOR NISTOR
Abstract.
We discuss and develop some connections between analysis onsingular spaces and operator algebras, as presented in my sequence of four lec-tures at the conference
Noncommutative geometry and applications , Frascati,Italy, June 16-21, 2014. Therefore this paper is mostly a survey paper, but thepresentation is new, and there are included some new results as well. In par-ticular, Sections 3 and 4 provide a complete short introduction to analysis onnoncompact manifolds that is geared towards a class of manifolds–called “Liemanifolds”–that often appears in practice. Our interest in Lie manifolds is dueto the fact that they provide the link between analysis on singular spaces andoperator algebras. The groupoids integrating Lie manifolds play an importantbackground role in establishing this link because they provide operator alge-bras whose structure is often well understood. The initial motivation for thework surveyed here–work that spans over close to two decades–was to developthe index theory of stratified singular spaces. Meanwhile, several other ap-plications have emerged as well, including applications to Partial DifferentialEquations and Numerical Methods. These will be mentioned only briefly, how-ever, due to the lack of space. Instead, we shall concentrate on the applicationsto Index theory.
Contents
Introduction 21. Motivation: Index Theory 41.1. An abstract index theorem 41.2. Differential and pseudodifferential operators 51.3. The Fredholm index 81.4. The Atiyah-Singer index formula 81.5. Cyclic homology and Connes’ index formula for foliations 101.6. The Atiyah-Patodi-Singer index formula 112. Motivation: Degeneration and singularity 152.1. APS-type examples: rank one 152.2. Manifolds with corners 152.3. Higher rank examples 163. Lie manifolds: definition and geometry 173.1. Lie algebroids and Lie manifolds 173.2. The metric on Lie manifolds 203.3. Anisotropic structures 24
Date : December 22, 2015.The author was partially supported by ANR-14-CE25-0012-01.Manuscripts available from http: // / ˜ nistor / AMS Subject classification (2010): 58J40 (primary) 58H05, 46L80, 46L87, 47L80.
4. Analysis on Lie manifolds 254.1. Function spaces 254.2. Pseudodifferential operators on Lie manifolds 274.3. Comparison algebras 284.4. Fredholm conditions 294.5. Pseudodifferential operators on groupoids 315. Examples and applications 325.1. Examples of Lie manifolds and Fredholm conditions 325.2. Index theory 345.3. Essential spectrum 345.4. Hadamard well posedness on polyhedral domains 35References 37
Introduction
We survey some connections between analysis on singular spaces and operatoralgebras, concentrating on applications to Index theory. The paper follows ratherclosely my sequence of four lectures at the conference
Noncommutative geometryand applications , Frascati, Italy, June 16-21, 2014. From a technical point of view,the paper mostly sets up the analysis tools needed for developing a certain approachto the Index theory of singular and non-compact spaces. These results were devel-oped by many people over several years. In addition to these older contributions,we include here also several new results tying various older results together. Inparticular, for the benefit of the reader, we have written the paper in such a waythat the third and fourth sections are to a large extend self-contained. They includemost of the needed proofs, and thus can be regarded as a very short introduction toanalysis on non-compact manifolds that focuses on applications to Lie manifolds.Lie manifolds are a class of non-compact manifolds that arise naturally in manyapplications involving non-compact and singular spaces.The main story told by this paper is, briefly, as follows. Some of the classicalAnalysis and Index theory results deal with the index of Fredholm operators. This israther well understood in the case of smooth, compact manifolds and in the case ofsmooth, bounded domains. By contrast, the non-smooth and non-compact cases aremuch less well understood. Moreover, it has become clear that the index theorems inthese frameworks require non-local invariants and (hence) cyclic homology. The fullimplementation of this program requires, however, further algebraic and analyticdevelopments. More specifically, one important auxiliary question that needs tobe answered is: “Which operators on a given non-smooth or non-compact space areFredholm.”
A convenient answer to this question involves Lie manifolds and theLie groupoids that integrate them. The techniques that were developed for thispurpose have then proved to be useful also in other mathematical areas, such asSpectral theory and the Finite Element Method.Fredholm operators play a central role in this paper for the following reasons.First of all, the (Fredholm) index is defined only for Fredholm operators, thus, inorder to state an index theorem for the Fredholm index, one needs to have examplesof Fredholm operators. In fact, the data that is needed to decide whether a givenoperator is Fredholm (principal symbol, boundary–or indicial–symbols) is also the
NALYSIS ON SINGULAR SPACES 3 data that is used for the actual computation of the index of that operator. Moreover,many interesting quantities (such as the signature of a compact manifold) are, infact, the indices of certain operators. Second, Fredholm operators have been widelyused in Partial Differential Equations (PDEs). For instance, non-linear maps whoselinearization is Fredholm play a central role in the study of non-linear PDEs. Also,Fredholm operators are useful in determining the essential spectra of Hamiltonians.Finally, the Fredholm index is the first obstruction for an operator to be invertible.As an illustration, this simple last observation is exploited in our approach tothe Neumann problem on polygonal domains (Theorem 5.14). The proof of thattheorem relies first on the calculation of the index of an auxiliary operator, thisauxiliary operator is then shown to be injective, and the final step is to augmentits domain so that it becomes an isomorphism [87].A certain point on the analysis on singular and non-compact spaces is worthinsisting upon. A typical approach to analysis on singular spaces–employed also inthis paper–is that the analysis on a singular space happens on the smooth part of thespace, with the singularities playing the important role of providing the behavior“at infinity.” Thus, from this point of view, the analysis on non-compact spaces ismore general than the analysis on singular spaces. However, for the simplicity of thepresentation, we shall usually discuss only singular spaces, with the understandingthat the results also extend to non-compact manifolds.Here are the contents of the paper. The first section is devoted to describing theIndex theory motivation for the results presented in this paper. The approach toIndex theory used in this paper is based on exact sequences of operator algebras.Thus, in the first section, we discuss the exact sequences appearing in the Atiyah-Singer index theorem, in Connes’ index theorem for foliations, and in the Atiyah-Patodi-Singer (APS) index theorem. The second section is devoted to explainingthe motivation for the results presented in this paper coming from degenerate (orsingular) elliptic partial differential equations. In that section, we just presentsome typical examples of degenerate elliptic operators that suggest how ubiquitousthey are and point out some common structures that have lead to Lie manifolds,a class of manifolds that is discussed in the third section. In this third section, weinclude the definition of Lie manifolds, a discussion of manifolds with cylindricalends (the simplest non-trivial example of a Lie manifold, the one that leads tothe APS framework), a discussion of Lie algebroids and of their relation to Liemanifolds, and a discussion of the natural metric and connection on a Lie manifold.The fourth section is a basic introduction to analysis on Lie manifolds. It beginswith discussions of the needed functions spaces, of “comparison algebras,” and ofFredholm conditions. The last section is devoted to applications, including theformulation of an index problem for Lie manifolds in periodic cyclic cohomology,an application to essential spectra, an index theorem for Callias-type operators,and the Hadamard well posedness for the Poisson problem with Dirichlet boundaryconditions on polyhedral domains.The four lectures of my presentation at the above mentioned conference weredevoted, each, to one of the following subjects: I.
Index theory , II.
Lie manifolds ,III.
Pseudodifferential operators on groupoids, and IV.
Applications , and are basedmostly on my joint works with Bernd Ammann (Regensburg), Catarina Carvalho(Lisbon), Alexandru Ionescu (Princeton), Robert Lauter (Mainz), Anna Mazzucato(Penn State) and Bertrand Monthubert (Toulouse). Nevertheless, I made an effort
V. NISTOR to put the results in context by quoting and explaining other relevant results. I havealso included significant background results and definitions to make the paper easierto read for non-specialists. I have also tried to summarize some of the more recentdevelopments. Unfortunately, the growing size of the paper has finally preventedme from including more information. Moreover, it was unpractical to provide allthe related references, and I apologize to the authors whose work has not beenmentioned enough.I would like to thank the Max Planck Institute for Mathematics in Bonn, wherepart of this work was completed. Also, I would like to thank Bernd Ammann,Ingrid and Daniel Beltit¸˘a, Karsten Bohlen, Claire Debord, Vladimir Georgescu,Marius M˘antoiu, Jean Renault, Elmar Schrohe, and Georges Skandalis for usefulcomments. 1.
Motivation: Index Theory
This paper is devoted in large part to explaining some applications of Lie man-ifolds and of their associated operator algebras to analysis on singular and non-compact spaces. The initial motivation of this author for studying analysis on sin-gular and non-compact spaces (and hence also for studying Lie manifolds) comesfrom Index theory. In this section, I will describe this initial motivation, while inthe next section, I will provide further motivation coming from degenerate partialdifferential equations. Thus, I will not attempt here to provide a comprehensiveintroduction to Index Theory, but rather to motivate the results and constructionsintroduced in this paper using it. In particular, I will stress the important role thatFredholm conditions play for index theorems. In fact, in our approach, both theindex theorem studied and the associated Fredholm conditions rely on the same ex-act sequence discussed in general in the next subsection. No results in this sectionare new.1.1.
An abstract index theorem.
An approach to Index theory is based on exact sequences of algebras of operators. We shall thus consider an abstract exactsequence(1) 0 → I → A → Symb → , in which the algebras involved will be specified in each particular application. Thesame exact sequence will be used to establish the corresponding Fredholm condi-tions. Typically, A will be a suitable algebra of operators that describes the analysison a given (class of) singular space(s). In our presentation, the algebra A will beconstructed using Lie algebroids and Lie groupoids. The choice of the ideal I alsodepends on the particular application at hand and is not necessarily determined by A . In fact, the analysis on singular spaces distinguishes itself from the analysis oncompact, smooth manifolds in that there will be several reasonable choices for theideal I .Often, in problems related to classical analysis–such as the ones that involve theFredholm index of operators–the ideal I will be contained in the ideal of compactoperators K (on some separable Hilbert space). In fact, in most applications in thispresentation, we will have I := A ∩ K . We insist, however, that this is not the onlylegitimate choice, even if it is the most frequently used one. An important otherexample is provided by taking I to be the kernel of the principal symbol map. Aswe will see below, in the case of singular and non-compact spaces, the kernel of the NALYSIS ON SINGULAR SPACES 5 principal symbol map does not consist generally of compact operators. This is thecase in the analysis on covering spaces and on foliations, which also lead naturallyto von Neumann algebras [33, 70, 76, 156].If I := A ∩ K and P ∈ A has an invertible image in A/I (that is, it is invertiblemodulo I ), then the operator P is Fredholm and a natural and far reaching questionto ask then is to compute ind( P ) := dim ker( P ) − dim coker( P ), the Fredholm indexof P , defined as the difference of the dimensions of the kernel and cokernel of P . Inany case, we see that in order to formulate an index problem, we need criteria for therelevant operators to be Fredholm, because it is the condition that P be Fredholmthat guarantees that ker( P ), the kernel of P , and coker( P ), the cokernel of P , arefinite dimensional. This is also related to the structure of the exact sequence (1).When the algebra A of the exact sequence (1) is defined using groupoids–as is thecase in this presentation–then the structure of the quotient algebra Symb := A/I is related to the representation theory of the underlying groupoid. Unfortunately,we will not have time to treat this important subject in detail, but we will provideseveral references in the appropriate places.The exact sequence (1) provides us with a boundary (or index) map(2) ∂ : K alg (Symb) → K alg ( I ) , between algebraic K -theory groups, whose calculation will be regarded as an indexformula for the reasons explained in the following subsections (see, for instance,Remark 1.4). Thus, in general, the index of an operator is an element of a K group, which explains why the usual index, which is an integer, is called the Fred-holm index in this paper. In case A and I are C ∗ -algebras, the boundary map ∂ descends to a map between the corresponding topological K -theory groups. More-over, we obtain also a map ∂ ′ : K (Symb) → K ( I ), acting also between topological K -theory groups. The maps ∂ and ∂ ′ and the maps obtained from the functorial-ity of (topological) K -groups, give rises to a six-term exact sequence of K -groups[139, 157]. Unfortunately, often the K -groups are difficult to compute, so we needto consider suitable dense subalgebras of C ∗ -algebras and their cyclic homology(see, for instance, Subsection 1.5).We begin with a quick introduction to differential and pseudodifferential opera-tors needed to fix the notation and to introduce some basic concepts. It is writtento be accessible to graduate students. We then discuss three basic index theoremsand their associated analysis (or exact sequences). These three index theorems are:the Atiyah-Singer (AS) index theorem, Connes’ index theorem for foliations, andthe Atiyah-Patodi-Singer (APS) index theorem. We will see that, at least from thepoint of view adopted in this presentation, Connes’ and APS’ frameworks extendthe Atiyah-Singer’s framework in complementary directions.1.2. Differential and pseudodifferential operators.
We now fix some notationand recall a few basic concepts. On R n , we consider the derivations (or, which isthe same thing, vector fields ) ∂ j = ∂∂x j , j = 1 , . . . , n, and form the differentialmonomials ∂ α := ∂ α ∂ α . . . ∂ α n n , α ∈ Z n + . We let | α | := α + α + . . . + α n ∈ Z + .A differential operator of order m on R n is then an operator P : C ∞ c ( R n ) → C ∞ c ( R n )of the form(3) P u = X | α |≤ m a α ∂ α u , V. NISTOR with m minimal with this property. Sometimes P : C ∞ c ( R n ) → L ( R n ), but in thispaper we consider only operators P having smooth coefficients a α .It is easy, but important, to extend the above constructions to systems ofdifferential operators, in order to account for important operators such as: vec-tor Laplacians, elasticity, signature, Maxwell, and many others. We then take u = ( u , . . . , u k ) ∈ C ∞ c ( R n ) k = C ∞ c ( R n ; R k ) to be a smooth, compactly sup-ported section of the trivial vector bundle R k = R k × R n → R n on R n and we take a α ∈ C ∞ ( R n ; M k ( R )) to be a matrix valued function. Each coefficient a α is hence an endomorphism of the trivial vector bundle R k . Then P maps C ∞ c ( R n ; R k ) to C ∞ c ( R n ; R k ) . Let ∆ = − ∂ − . . . − ∂ n ≥ s ∈ Z + . We denote as usual H s ( R n ) := { u : R n → C , ∂ α u ∈ L ( R n ) , | α | ≤ s } = D (∆ s/ ) . As we will see below, both definitions above of Sobolev spaces extend to the caseof “Lie manifolds.” These definitions of Sobolev spaces also extend immediately tovector valued functions and, if the coefficients a α of P are bounded (together withenough derivatives, more precisely, if P ∈ W s, ∞ ( R n )), then we obtain that P maps H s ( R n ) to H s − m ( R n ).For P a differential operator of order ≤ m as in Equation (3), we let(4) σ m ( P )( x, ξ ) = X | α | = m a α ( x )( ıξ ) α ∈ C ∞ ( R n × R n ; M k ) , and call it the principal symbol of P . In particular, we have σ m +1 ( P ) = 0. Here x ∈ R n and ξ ∈ R n is the dual variable.The fact that ξ is a dual variable to x ∈ R n is confirmed by the formula fortransformations of coordinates. The principal symbol is thus seen to be a functionon T ∗ R n ≃ R n × R n . It turns out that the principal symbol σ m ( P ) of P has amuch simpler transformation formula than the (full) symbol σ ( P ) of P defined by(5) σ ( P )( x, ξ ) = X | α |≤ m a α ( x )( ıξ ) α ∈ C ∞ ( R n × R n ; M k ) . The full symbol p ( x, ξ ) := σ ( P )( x, ξ ) of P defined as above in Equation (5) is nev-ertheless important because P = p ( x, D ), where(6) p ( x, D ) u ( x ) := (2 π ) − n Z R n e ıx · ξ p ( x, ξ )ˆ u ( ξ ) dξ . There exist more general classes of functions (or symbols) p for which p ( x, D )can still be defined by the above formula (6). The resulting operator will be a pseudodifferential operator with symbol p . Let us recall the definition of the twomost basic classes of symbols for which the formula (6) defining p ( x, D ) still makessense [68, 143, 112]. For simplicity, we shall consider in the beginning only scalarsymbols, although matrix valued symbols can be handled in a completely similarway. The first class of symbols p for which the formula (6) still makes sense is theclass S m , ( R k × R N ), m ∈ R . It is defined as the space of functions a : R k + N → C that satisfy, for any i, j ∈ Z + , the estimate | ∂ ix ∂ jξ a ( x, ξ ) | ≤ C ij (1 + | ξ | ) m − j , for a constant C ij > x and ξ . Of course, in (6), we take N = n .Let us now introduce classical symbols . A function a : R k × R N → C is called eventually homogeneous of order s if there exists M > a ( x, tξ ) = t s a ( x, ξ ) NALYSIS ON SINGULAR SPACES 7 for | ξ | ≥ M and t ≥
1. A very useful class of symbols is S m cl ( R n ), defined asthe subspace of symbols a ∈ S m , ( R n ) that can be written as asymptotic series a ∼ P ∞ j =0 a m − j , meaning a − N X j =0 a m − j ∈ S m − N − , ( R n ) , with a k ∈ S k , ( R n ) eventually homogeneous of order k . If a ∈ S m cl ( R n ), the pseu-dodifferential operator a ( x, D ) is called a classical pseudodifferential operator andits principal symbol is defined by(7) σ m ( a ( x, D )) := a m and is regarded as a smooth, order m homogeneous function on T ∗ R n r “zero section” = R n r ( R n × { } ) . For Index theory, it is generally enough to consider classical pseudodifferential oper-ators. The reason is that the inverses and parametrices of classical pseudodifferen-tial operators are again classical and, if p ( x, ξ ) := P | α |≤ m p α ( ıξ ) α is a polynomial in ξ and if we let P := p ( x, D ) = P | α |≤ m p α ∂ α , then P is a classical pseudodifferentialoperator of order m .The definition of a (pseudo)differential operator P (of order ≤ m ) and of itsprincipal symbol σ m ( P ) then extend to manifolds and vector bundles by usinglocal coordinate charts. To fix notation, if E → M is a smooth vector bundle overa manifold M , we shall denote by Γ( M ; E ) the space of its smooth sections:Γ( M ; E ) := { s : M → E, s ( x ) ∈ E x } . Similarly, we shall denote by Γ c ( M ; E ) ⊂ Γ( M ; E ) the subspace of smooth, com-pactly supported sections of E over M . Sometimes, when no confusion can arise,we denote Γ( E ) = Γ( M ; E ) and, similarly, Γ c ( E ) = Γ c ( M ; E ). Getting back to ourextension of pseudodifferential operators to manifolds, we obtain this extension byreplacing as follows: R n ↔ M = a smooth manifold C ∞ c ( R n ) k ↔ sections of a vector bundle,which gives for an order m operator P acting between smooth, compactly supportedsections of E and F : P : Γ c ( M ; E ) → Γ c ( M ; F ) σ m ( P ) ∈ Γ( T ∗ M r { } ; Hom( E, F )) . Of course, σ m ( P ) is homogeneous of order m . Thus, if m = 0 and if we denote by S ∗ M := ( T ∗ M r { } ) / R ∗ + the (unit) cosphere bundle , then σ ( P ) identifies with asmooth function on S ∗ M . (The name “cosphere bundle” is due to the fact that, ifwe choose a metric on M , then the cosphere bundle identifies with the set of vectorsof length one in T ∗ M .)The main property of the principal symbol is the multiplicative property (8) σ m + m ′ ( P P ′ ) = σ m ( P ) σ m ′ ( P ′ ) , a property that is enjoyed by its extension to pseudodifferential operators (whichare allowed to have negative and non-integer orders as well). V. NISTOR
Definition 1.1.
A (classical, pseudo)differential operator P is called elliptic if itsprincipal symbol is invertible away from the zero section of T ∗ M .See [68, 128, 112, 148] for a more complete discussion of various classes of symbolsand of pseudodifferential operators. See also [8, 57, 84, 101, 120, 143].As a last ingredient before discussing the Fredholm index, we need to extend thedefinition of Sobolev spaces to manifolds. To that end, we consider also a metric g on our manifold M (or a Lipschitz equivalence class of such metrics) [12, 63].Then, for a complete manifold M , the Sobolev spaces are given by the domains ofthe powers of the (positive) Laplacian. In general, this will depend on the choiceof the metric g .1.3. The Fredholm index.
Let now M be a compact , smooth manifold, so theSobolev spaces H s ( M ) are uniquely defined. Let also P be a (classical, pseudo) dif-ferential operator of order ≤ m acting between the smooth sections of the hermitianvector bundles E and F . We denote by H s ( M ; E ) and H s ( M ; F ) the correspondingSobolev spaces of sections of these bundles.Recall that a continuous, linear operator T : X → Y acting between topologicalvector spaces is Fredholm , if and only if, the vector spaces ker( P ) := { u ∈ X, T u = 0 } and coker( P ) := Y /T X are finite dimensional. One of our model results is then thefollowing classical theorem [36, 144, 145].
Theorem 1.2.
Let P be an order m pseudodifferential operator acting between thesmooth sections of the bundles E and F on the smooth, compact manifold M and s ∈ R . Then P : H s ( M ; E ) → H s − m ( M ; F ) is Fredholm ⇔ P is elliptic. Fredholm operators appear all the time in applications (because elliptic operatorsare so fundamental). For instance, the theorem mentioned above is one of the crucialingredients in the “Hodge theory” for smooth compact manifolds, which is quiteuseful in Gauge theory.By the Open Mapping theorem, the invertibility of a continuous, linear opera-tor P : X → X acting between two Banach spaces is equivalent to the conditiondim ker( P ) = dim coker( P ) = 0. It is important then to calculate the Fredholm in-dex ind( P ) of P , defined by(9) ind( P ) := dim ker( P ) − dim coker( P ) . The reason for looking at the Fredholm index rather than looking simply at thenumbers dim ker( P ) and dim coker( P ) is that ind( P ) has better stability propertiesthan these numbers. For instance, the Fredholm index is homotopy invariant anddepends only on the principal symbol of P .1.4. The Atiyah-Singer index formula.
The index of elliptic operators on smooth,compact manifolds is computed by the Atiyah-Singer index formula [11]:
Theorem 1.3 (Atiyah-Singer) . Let M be a compact, smooth manifold and let P beelliptic, classical (pseudo)differential operator acting on sections of smooth vectorbundles on M . Then ind( P ) = h ch [ σ m ( P )] T ( M ) , [ T ∗ M ] i . NALYSIS ON SINGULAR SPACES 9
A suitable orientation has to be, of course, chosen on T ∗ M . There are manyaccounts of this theorem, and we refer the reader for instance to [57, 119, 145, 151]for more details. See [30, 76] for an approach using non-commutative geometry. Letus nevertheless mention some of the main ingredients appearing in the statementof this theorem, because they are being generalized (or need to be generalized) tothe non-smooth case. This generalization is in part achieved by non-commutativegeometry and by analysis on singular spaces. Thus, returning to Theorem 1.3, themeanings of the undefined terms in Theorem 1.3 are as follows:(i) The principal symbol σ m ( P ) of P defines a K -theory class in K ( T ∗ M ) (withcompact supports) by the ellipticity of P [11] and ch [ σ m ( P )] ∈ H evenc ( T ∗ M )is the Chern character of this class.(ii) T ( M ) ∈ H even ( M ) ≃ H even ( T ∗ M ) is the Todd class of M , so the product ch [ σ m ( P )] T ( M ) is well-defined in H evenc ( T ∗ M ).(iii) [ T ∗ M ] ∈ H evenc ( T ∗ M ) ′ is the fundamental class of T ∗ M and is chosen suchthat no sign appears in the index formula.The AS index formula was much studied and has found a number of applications.It is based on earlier work of Grothendieck and Hirzebruch and answers to a questionof Gelfand. One of the main motivations for the work presented here is the desireto extend the index formula for compact manifolds (the AS index formula) to thenoncompact and singular cases. To this end, it will be convenient to use the exactsequence formalism described in Subsection 1.1. Namely, the exact sequence (1)corresponding to the AS index formula is(10) 0 → Ψ − ( M ) → Ψ ( M ) → C ∞ ( S ∗ M ) → , where S ∗ M is the cosphere bundle of M , as before, (that is, the set of vectors oflength one of the cotangent space T ∗ M of M ). That is, in the exact sequence (1),we have I = Ψ − ( M ), A = Ψ ( M ), and Symb := A/I ≃ C ∞ ( S ∗ M ).It is interesting to point out that both the AS index formula and the Fredholmcondition of Theorem 1.2 are based on the exact sequence (10). Of course, toactually determine the index, one has to do additional work, but the informationneeded is contained in the exact sequence. This remains true for most of the otherindex theorems. We continue with some remarks. Remark . Let us see now how the exact sequence (10) and Theorem 1.3 arerelated. Recall the boundary map ∂ : K alg (Symb) → K alg ( I ) in algebraic K -theoryassociated to the exact sequence (1), see Equation (2), and let us assume that theideal I of that exact sequence consists of compact operators (i.e. I ⊂ K ). We firstconsider the natural map(11) T r ∗ : K alg ( I ) → Z , where the trace refers to the trace (or dimension) of a projection. We have, ofcourse, that T r ∗ : K alg ( K ) = K ( K ) → Z is the usual isomorphism. Then T r ∗ ◦ ∂ computes the usual (Fredholm) index , that is, we have the equality of the morphisms(12) ind = T r ∗ ◦ ∂ : K alg (Symb) ∂ −→ K alg ( I ) T r ∗ −−→ C . Indeed, if P ∈ A is invertible in A/I = Symb, then, on the one hand, P definesa class [ P ] ∈ K (Symb), and, on the other hand, P is Fredholm and its Fredholmindex is given by(13) ind( P ) = T r ∗ ◦ ∂ [ P ] . We thus see that computing the index of a Fredholm (pseudo)differential operatoron M is equivalent to computing the composite map T r ∗ ◦ ∂ : K (Symb) → C .This observation due to Connes is the starting point of the approach to indextheorems described in this paper. Remark . Let us discuss now shortly the role of the Chern character in theAS index formula. First, let us recall that the Chern character establishes anisomorphism ch : K ∗ ( M ) ⊗ C → H ∗ ( M ) ⊗ C for any compact, smooth manifold M . Moreover, in the case of the commutative algebra C ∞ ( M ), we have that K ∗ ( C ∞ ( M )) ≃ K ∗ ( M ) and hence any group morphism K ∗ ( C ∞ ( M )) → C fac-tors through the Chern character ch : K ∗ ( M ) → H ∗ ( M ) ⊗ C . Returning to theAS index formula, we have that Symb = C ∞ ( S ∗ M ) and hence the index mapind = T r ∗ ◦ ∂ : K (Symb) ≃ K ( S ∗ M ) → C can be expressed solely in termsof the Chern character. It is therefore possible to express the AS Index Formulapurely in classical terms (vector bundles and cohomology) because the quotient A/I := Symb ≃ C ∞ ( S ∗ M ) is commutative.Remark . Technically, one may have to replace the algebra A with M n ( A ) andtake P ∈ M n ( A ), but this is not an issue since the K -groups (both topologicaland algebraic) are invariant for the replacement of A with its matrix algebras.However, the approach to the index of elliptic (pseudo)differential operators usingexact sequences can be used to deal with operators P acting between sectionsof isomorphic bundles. For non-compact manifolds (and hence also for singularspaces), this is enough. For the AS index formula, however, one may have toreplace first M with M × S . For this reason, in the case of the AS index formula,Connes’ approach using the tangent groupoid may be more convenient. Remark . Elliptic operators on a smooth, compact manifolds M have certainproperties that are similar to the properties to Γ-invariant elliptic operators ona covering space Γ → ˜ M → M (so here Γ is the group of deck transformationsof ˜ M and hence ˜ M / Γ ≃ M ). The reason is that they correspond the the same Lie algebroid on M , namely T M → M . The two frameworks correspond howeverto different Lie groupoids, and their analysis is consequently also quite different.In particular, if Γ is non-trivial, one is lead to consider von Neumann algebras [33, 70, 156]. This is related to the example of foliations discussed in the nextsubsection.1.5.
Cyclic homology and Connes’ index formula for foliations.
The map
T r ∗ of the basic equation ind( P ) = T r ∗ ◦ ∂ [ P ] (recall Equation (12), which is validwhen I ⊂ K ), is a particular instance of the pairing between cyclic cohomology and K -theory [30]. See also [29, 71, 90, 93, 152]. This pairing is even more importantwhen I
6⊂ K . Let us explain this. Let us denote by HP ∗ ( B ) the periodic cycliccohomology groups of an algebra B (for topological algebras, suitable topologicalversions of these groups have to be considered).Let us look again at the general exact sequence of Equation (1) and let φ be acyclic cocycle on I , that is, φ ∈ HP ( I ). A more general (higher) index theorem isthen to compute φ ∗ ◦ ∂ : K (Symb) → C . It is known that φ ∗ ◦ ∂ = ( ∂φ ) ∗ , and hence the map φ ∗ ◦ ∂ is also given by a cycliccocycle [114]. NALYSIS ON SINGULAR SPACES 11
The map φ ∗ and, in general, the approach to Index theory using cyclic homol-ogy is especially useful for foliations for the reasons that we are explaining now.We regard a foliation ( M, F ) of a smooth, compact manifold M as a sub-bundle F ⊂
T M that is integrable (that is, its space of smooth sections, denoted Γ( F ), isclosed under the Lie bracket). Connes’ construction of pseudodifferential operatorsalong the leaves of a foliation [33] then yields the exact sequence of algebras(14) 0 → Ψ − F ( M ) → Ψ F ( M ) σ −→ C ∞ ( S ∗ F ) → , where σ is again the principal symbol, defined essentially in the same manner asfor the case of smooth manifolds. In fact, for F = T M , with M a smooth, compactmanifold, this exact sequence reduces to the earlier exact sequence (10). It alsoyields a boundary (or index) map ∂ : K ( C ∞ ( S ∗ F )) = K ( S ∗ F ) → K (Ψ − F ( M )) ≃ K ( C ∞ c ( F )) , where C ∞ c ( F ) is the convolution algebra of the groupoid associated to F and wherethe topological K -groups were used. A main difficulty here is that there are fewcalculations of K (Ψ − F ( M )). These calculations are related to the Baum-Connesconjecture, which is however known not to be true for general foliations, see [67]and the references therein. See also [126, 127]. Another feature of the foliationcase is that, unlike our other examples to follow, Ψ − F ( M ) has no canonical properideals, so there are no other index maps.Unlike its K -theory, the cyclic homology of Ψ − F ( M ) is much better under-stood, in particular, it contains as a direct summand the twisted cohomology ofthe classifying space of the groupoid (graph) of the foliaton [20]. We thus havea large set of linearly independent cyclic cocycles and hence many linear maps φ ∗ : K ( C ∞ c ( F )) → C , each of which defines an index map φ ∗ ◦ ∂ : K ( C ∞ ( S ∗ F )) → C . We will not pursue further the determination of φ ∗ ◦ ∂ , but we note Connes’results in [30, 33, 76], the results of Benameur–Heitsch for Haeffliger homology [16],the results of Connes–Skandalis [32], and of myself for foliated bundles [113]. Wealso stress that in the case of foliations, it is the ideal I that causes difficulties,whereas the quotient Symb := A/I is commutative and, hence, relatively easy todeal with. The opposite will be the case in the following subsection.1.6.
The Atiyah-Patodi-Singer index formula.
A related but different type ofexample is provided by the Atiyah-Patodi-Singer (APS) index formulas [10]. Let M be an n -dimensional compact manifold with smooth boundary ∂M . By definition,this means that M is locally diffeomorphic to an open subset of [0 , × R n − . Thetransition functions for a manifold with boundary will be assumed be smooth. To M we attach the semi-infinite cylinder ∂M × ( −∞ , , yielding a manifold with cylindrical ends . The metric is taken to be a product metric g = g ∂M + dt far on the end. Kondratiev’s transform r = e t then maps the cylin-drical end to a tubular neighborhood of the boundary, such that the cylindrical endmetric becomes g = g ∂M +( r − dr ) near the boundary, since r − dr = e − t d ( e t ) = dt .Thus, on M , we consider two metrics: first, the initial, everywhere smooth metric(including up to the boundary) and, second, the modified, singular metric g thatcorresponds to the compactification of the cylindrical end manifold. We have thus obtained the simplest examples of a non-compact manifold, thatof a manifold with cylindrical ends.
We will consider on this non-compact manifoldonly differential operators with coefficients that extend to smooth functions up toinfinity (so, in particular, they have limits at infinity). Because of this, it willbe more convenient to work on M than on M ∪ ∂M × ( −∞ , r = e t . The Kondratiev transform is such that ∂ t becomes r∂ r . On M , we then take the coefficients to be smooth functions up to theboundary. Therefore, in local coordinates ( r, x ′ ) ∈ [0 , ǫ ) × ∂M on the distinguishedtubular neighborhood of ∂M , we obtain the following form for our differentialoperators (here n = dim( M )):(15) P = X | α |≤ m a α ( r, x ′ )( r∂ r ) α ∂ α x ′ . . . ∂ α n x ′ n = X | α |≤ m a α ( r∂ r ) α ∂ α ′ . Operators of this form are called totally characteristic differential operators.
Away from the boundary, the definition of the principal symbol for a totallycharacteristic differential operator is unchanged. However, in the same local coor-dinates near the boundary as in Equation (15), the principal symbol for the totallycharacteristic differential operator of Equation (15) is(16) σ m ( P ) := X | α | = m a α ξ α . Thus the principal symbols is not P | α | = m a α r α ξ α as one might first think! Otherthan the fact that this definition of the principal symbol gives the “right results,”it can be motivated by considering the original coordinates ( t, x ′ ) ∈ ( −∞ , × ∂M on the cylindrical end.The principal symbol is something that was encountered in the classical case ofthe AS-index formula as well as in the case of foliations, so it is not somethingsignificantly new in the case of manifolds with cylindrical ends–even if in that casethe definition is slightly different. However, in the case of manifolds with cylindricalends, there is another significant new ingredient, which will turn out to be bothcrucial and typical in the analysis on singular spaces. This significant new ingredientis the indicial family of a totally characteristic differential operator. To define anddiscuss the indicial family of a totally characteristic differential operator, let P beas in Equation (15) and consider the same local coordinates near the boundary asin that equation. The definition of the indicial family b P of P is then as follows (weunderline the most significant new ingredients of the definition):(17) b P ( τ ) := X | α |≤ m a α ( 0 , x ′ )( ıτ ) α ∂ α ′ . Note that b P ( τ ) is a family of differential operators on ∂M that depends on thecoefficients of P only through their restrictions to the boundary. Moreover, we seethat the indicial family b P ( τ ) of P is the Fourier transform of the operator(18) I ( P ) := X | α |≤ m a α ( 0 , x ′ ) ∂ t α ∂ α ′ , which is a translation invariant operator on ∂M × R . The operator I ( P ) is calledthe indicial operator of P [85, 110]. Note that we have denoted ∂ α = ∂ α t ∂ α ′ . NALYSIS ON SINGULAR SPACES 13
We are interested in Fredholm conditions for totally characteristic differentialoperators, so let us introduce the last ingredient for the Fredholm conditions. Letus endow M := M r ∂M with a cylindrical end metric. Since the cylindrical endmetric is complete, the Laplacian ∆ is self-adjoint, and hence we can define theSobolev space H s ( M ) as the domain of ∆ s/ , that is, H s ( M ) := D (∆ s/ ), whichturns out to be independent of the choice of the cylindrical end metric. (We con-sider all differential operators to be defined on their minimal domain, unless oth-erwise mentioned, and thus, in particular, they give rise to closed, densely definedoperators.) We then have a characterization of Fredholm totally characteristic dif-ferential operators similar to the compact case (the differences to the compact caseare underlined). Theorem 1.8.
Assume M has cylindrical ends and P is a totally characteristicdifferential operator of order m acting between the sections of the bundles E and F . Then, for any fixed s ∈ R , we have that P : H s ( M ; E ) → H s − m ( M ; F ) is Fredholm ⇔ P is elliptic and b P ( τ ) is invertible ∀ τ ∈ R . This result has a long history and related theorems are due to many people,too many to mention them all here. Nevertheless, one has to mention the pi-oneering work of Lockhart-Owen on differential operators [89] and the work ofMelrose-Mendoza for totally characteristic pseudodifferential operators [100]. Aclosely related theorem for differential operators and domains with conical pointshas appeared in a landmark paper by Kondratiev in 1967 [73]. Other important re-sults in this direction were obtained by Mazya [77] and Schrohe–Schultze [141, 142].See the books of Schulze [143], Lesch [85], and Plamenevski˘ı [128] for introductionsand more information on the topics and results of this subsection.One can easily show that I ( P ) is invertible if, and only if, ˆ P ( τ ) is invertible forall τ ∈ R . Thus the Fredholmness criterion of Theorem 1.8 can also be given thefollowing formulation that is closer to our more general result of Theorem 4.14. Theorem 1.9.
Let M and P be as in Theorem 1.8 and s ∈ R . Then P : H s ( M ; E ) → H s − m ( M ; F ) is Fredholm ⇔ P is elliptic and I ( P ) is invertible. Let us consider now a totally characteristic, twisted Dirac operator P . In case P is Fredholm, its Fredholm index is given by the Atiyah-Patodi-Singer (APS)formula [10], which expresses ind( P ) as the sum of two terms:(i) the integral over M of an explicit form, which is a local term that depends onlyon the principal symbol of the operator P , as in the case of the AS formula,and(ii) a boundary contribution that depends only on the indicial family b P ( τ ) , namelythe η -invariant, which is this time a non-local invariant. It can be expressedin terms of I ( P ) [102].We thus see that even to be able to formulate the APS-index formula, we needto know which totally characteristic operators will be Fredholm. Moreover, theingredients needed to compute the index of such an operator P (that is, its principalsymbol σ m ( P ) and the indicial operator I ( P )) are exactly the ingredients needed to decide that the given operator P is Fredholm. See [17, 80, 102, 140] for furtherresults.Let us now introduce the exact sequence of the APS index formula. First of all, A := Ψ b ( M ) is the algebra of totally characteristic pseudodifferential operators on M . One of its main properties is that the differential operators in A are exactlythe totally characteristic differential operators. See [85, 143] for a definition ofΨ ∞ b ( M ). A definition using groupoids (of a slightly different algebra) will be givenin Subsection 4.5 in a more general setting. Let r be a defining function of theboundary ∂M of M , as before. Next, the ideal is I := r Ψ − b ( M ) = Ψ b ( M ) ∩ K . Then the symbol algebra Symb :=
A/I is the fibered product(19) Symb = C ∞ ( S ∗ M ) ⊕ ∂ Ψ ( ∂M × R ) R , more precisely, Symb consists of pairs ( f, Q ) such that the principal symbol of the R invariant pseudodifferential operator Q matches the restriction of f ∈ C ∞ ( S ∗ M )to the boundary. Recalling the definition of I in Equation (18) (and extending itto totally characteristic pseudodifferential operators), we obtain the exact sequence(20) 0 → r Ψ − b ( M ) → Ψ b ( M ) σ ⊕I −−→ C ∞ ( S ∗ M ) ⊕ ∂ Ψ ( ∂M × R ) R → . (The exact sequence 0 → Ψ − b ( M ) → Ψ b ( M ) σ −→ C ∞ ( S ∗ M ) → P is Fredholm if, and only if,the pair ( σ ( P ) , I ( P )) ∈ Symb := C ∞ ( S ∗ M ) ⊕ ∂ Ψ ( ∂M × R ) R is invertible, which,in turn, is true if, and only if, P is elliptic and I ( P ) is invertible. Thus the exactsequence (20) implies Theorem 1.9.As before, composing ∂ : K (Symb) → K ( I ), I = r Ψ − ( M )), with the tracemap T r ∗ gives us the Fredholm indexind = T r ∗ ◦ ∂ : K (Symb) → C . Since
T r ∗ ◦ ∂ = ( ∂T r ) ∗ [29] (see [114] for the case when T r is replaced by a generalcyclic cocycle), we see that the APS index formula is also equivalent to the calcu-lation of the class of the cyclic cocycle ∂T r ∈ HP (Symb). This was the approachundertaken in [102, 108]. Remark . It is important to stress here first the role of cyclic homology, which isto define natural morphisms K (Symb) → C , morphisms that are otherwise difficultto come by. Also, it is important to stress that it is the noncommutativity of thealgebra of symbols Symb that explains the fact that the APS index formula isnon-local.We need to insist of the fact that in the case of the APS framework, it is thesymbol algebra Symb := A/I that causes difficulties, in large part because it is non-commutative (so the classical Chern character is not defined), whereas the ideal I ⊂ K is easy to deal with. This is an opposite situation to the one encountered forfoliations. It is for this reason that the foliation framework and the APS frameworkextend the AS framework in different directions.The approach to Index theory explained in this last subsection extends to morecomplicated singular spaces, and this has provided the author of this presentationthe motivation to study analysis on singular spaces. NALYSIS ON SINGULAR SPACES 15 Motivation: Degeneration and singularity
The totally characteristic differential operators studied in the previous subsectionappear not only in index problems, but actually arise in many practical applications.We shall now examine how the totally characteristic differential operators and otherrelated operators appear in practice. In a nut-shell, these operators can be usedto model degenerations and singularities. In this section, we introduce severalexamples. We begin with the ones related to the APS index theorem (the totallycharacteristic ones, called “rank one” by analogy with locally symmetric spaces)and then we continue with other examples. Again, no results in this section arenew.2.1.
APS-type examples: rank one.
Let us denote by ρ the distance to theorigin in R d . Here is a list of examples of totally characteristic operators. Example . In our three examples below, the first one is a true totally character-istic operator, whereas the other two require us to remove the factor ρ − first.(1) The elliptic generator L of the Black-Scholes PDE ∂ t − L [146] Lu := σ x ∂ x u + rx∂ x u − ru . (2) The Laplacian in polar coordinates ( ρ, θ )∆ u = ρ − (cid:0) ρ ∂ ρ u + ρ∂ ρ u + ∂ θ u (cid:1) . (3) The Schr¨odinger operator in spherical coordinates ( ρ, x ′ ), x ′ ∈ S , − (∆ + Zρ ) u = − ρ − (cid:0) ρ ∂ ρ u + 2 ρ∂ ρ u + ∆ S u + Zρu (cid:1) . A similar expansion is valid for elliptic operators in generalized spherical co-ordinates in arbitrary dimensions and was used by Kondratiev in [73] to studydomains with conical points. Kondratiev’s paper is widely used since it providesthe needed analysis facts to deal with polygonal domains, the main testing groundfor numerical methods.2.2.
Manifolds with corners.
For more complicated examples we will need man-ifolds with corners. Recall that M is a manifold with corners if, and only if, M is locally diffeomorphic to an open subset of [0 , n . The transition functions of M are supposed to be smooth, as in the case of manifolds with smooth boundary.A manifold with boundary is a particular case of a manifold with corners, but weagree in this paper that a smooth manifold does not have boundary (or corners),since we regard the corners (or boundary) as some sort of singularity.A point p ∈ M is called of depth k if it has a neighborhood V p diffeomorphic to[0 , k × ( − , n − k by a diffeomorphism φ p : V p → [0 , k × ( − , n − k mapping p to the origin: φ p ( p ) = 0. A connected component F of the set of points of depth k will be called an open face (of codimension k ) of M . The set of points of depth 0of M is called the interior of M , is denoted M , and its connected components arealso considered to be an open faces of M . The closure in M of an open face F of M will be called a closed face of M . A closed face of M may not be a manifoldwith corners in its own. The union of the proper faces of M is denoted by ∂M andis called the boundary of M . Thus M := M r ∂M . The following set of vector fields will be useful when defining Lie manifolds:(21) V b := { X ∈ Γ( M ; T M ) , X is tangent to all boundary faces of M } . Let us notice that in the case of manifolds with boundary, the totally characteristicdifferential operators on M , see Equation (15), are generated by C ∞ ( M ) and thevector fields X ∈ V b .2.3. Higher rank examples.
We now continue with more complicated examples,which we call “higher rank” examples, again by analogy with locally symmetricspaces. In general, the natural domains for these higher rank examples will bemanifolds with corners.
Example . There are no “higher rank” example in dimension one, so we beginwith an example in dimension two.(1) The simplest non-trivial example is the Laplacian∆ H = y ( ∂ x + ∂ y )on the hyperbolic plane H = R × [0 , ∞ ), whose metric is y − ( dx + dy ).(2) The Laplacian on the hyperbolic plane is closely related to the SABR Par-tial Differential Equation (PDE) due to Lesniewsky and collaborators [62].The SABR PDE is also a parabolic PDE ∂ t − L associated to a stochasticdifferential equation, with2 L := y (cid:0) x ∂ x + 2 ρνx∂ x ∂ y + ν ∂ y (cid:1) , with ρ and ν real parameters. Stochastic differential equations providemany interesting and non-trivial examples of degenerate parabolic PDEsthat can be treated using Lie manifolds.(3) A related example is that of the Laplacian in cylindrical coordinates ( ρ, θ, z )in three dimensions:(22) ∆ u = ρ − (cid:0) ( ρ∂ ρ ) u + ∂ θ u + ( ρ∂ z ) (cid:1) . Ignoring the factor ρ − , which amounts to a conformal change of metric, wesee that our differential operator (that is, ρ ∆) is generated by the vectorfields ρ∂ ρ , ∂ θ , and ρ∂ z , and that the linear span of these vector fields is a Lie algebra . The resultingpartial differential operators are usually called edge differential operators .This example can be used to treat the behavior near edges of polyhedraldomains of elliptic PDEs. This behavior is more difficult to treat than thebehavior near vertices. For a boundary value problems in a three dimen-sions wedge of dihedral angle α , the natural domain is [0 , α ] × [0 , ∞ ) × R ,a manifold with corners of codimension two.We thus again see that Lie algebras of vector fields are one of the main ingredientsin the definition of the differential operators that we are interested in. More relatedexamples will be provided below as examples of Lie manifolds.Degenerate elliptic equations have many applications in Numerical Analysis, see[13, 38, 41, 88, 87], for example. NALYSIS ON SINGULAR SPACES 17 Lie manifolds: definition and geometry
Motivated by the previous two sections, we now give the definition of a Liemanifold largely following [7]. We also introduce a slightly more general class ofmanifolds than in [7] by allowing the manifold with corners appearing in the defi-nition to be noncompact. We also slightly simplify the definition of a Lie manifoldbased on a comment of Skandalis. We thus define our Lie manifolds using Lie al-gebroids and then we recover the usual definition in terms of Lie algebras of vectorfields. I have tried to make this section as self-contained as possible, thus includingmost of the proofs, some of which are new.3.1.
Lie algebroids and Lie manifolds.
We have found it convenient to intro-duce Lie manifolds and “open manifolds with a Lie structure at infinity” in termsof Lie algebroids, which we will recall shortly. First, recall that we use the followingnotation, if E → X is a smooth vector bundle, we denote by Γ( X ; E ) (respectively,by Γ c ( X ; E )) the space of smooth (respectively, smooth, compactly supported) sec-tions of E . Sometimes, when no confusion can arise, we simply write Γ( E ), or,respectively, Γ c ( E ). We now introduce Lie algebroids. Lie algebroids were intro-duced by Pradines [129]. See also [65, 66] for some basic results on Lie algebroidsand Lie groupoids. We refer to [92, 103] for further material and references to Liealgebroids and groupoids. Definition 3.1. A Lie algebroid A → M is a real vector bundle over a manifoldwith corners M together with a Lie algebra structure on Γ( M ; A ) (with bracket[ , ]) and a vector bundle map ̺ : A → T M , called anchor , such that the inducedmap ̺ ∗ : Γ( M ; A ) → Γ( M ; T M ) satisfies the following two conditions:(i) ̺ ∗ ([ X, Y ]) = [ ̺ ∗ ( X ) , ̺ ∗ ( Y )] and(ii) [ X, f Y ] = f [ X, Y ] + ( ̺ ∗ ( X ) f ) Y , for all X, Y ∈ Γ( M ; A ) and f ∈ C ∞ ( M ).For further reference, let us recall here the isotropy of a Lie algebroid. Definition 3.2.
Let ̺ : A → T M be a Lie algebroid on M with anchor ̺ . Thenthe kernel ker( ̺ x : A x → T x M ) of the anchor is the isotropy of A at x ∈ M .The isotropy at any point can be shown to be a Lie algebra. See [9] for general-izations. Recall that we denote by ∂M the boundary M , that is, the union of itsproper faces, and by M := M r ∂M its interior. Definition 3.3.
A pair (
M , A ) consisting of a manifold with corners M and a Liealgebroid A → M is called an open manifold with a Lie structure at infinity if itsanchor ̺ : A → T M satisfies the following properties:(i) ̺ : A x → T x M is an isomorphism for all x ∈ M := M r ∂M and(ii) V := ̺ ∗ (Γ( M ; A )) ⊂ V b .If M is compact, then the pair ( M , A ) will be called a
Lie manifold .Condition (ii) means that the Lie algebra of vector fields V := ̺ ∗ (Γ( A )) consistsof vector fields tangent to all faces of M . One of the main reason for introducingopen manifolds with a Lie structure at infinity is in order to be able to localize Liemanifolds. Thus, if ( M , A ) is a Lie manifold and V ⊂ M is an open subset, then( V, A | V ) will not be a Lie manifold, in general, but will be an open manifold witha Lie structure at infinity. Thus the Lie manifolds are exactly the compact openmanifold with a Lie structure at infinity. Lie manifolds were introduced in [7]. By extension, M and M := M r ∂M in Definition 3.3 will also be called openmanifolds with a Lie structure at infinity. We shall write Γ( A ) instead of Γ( M ; A )when no confusion can arise, also, we shall usually write Γ( A ) instead of ̺ ∗ (Γ( A )).We have the following “trivial” example. Example . The “example zero” of a Lie manifold is that of a smooth, compactmanifold M = M (no boundary or corners) and is obtained by taking A = T M ,thus V = Γ( T M ) = Γ( M ; T M ) = V b . Then ( M, A ) is a (trivial) example of a Liemanifold. This example of a Lie manifold provides the framework for the AS IndexTheorem. Similarly, every smooth manifold M is an open manifold with a Liestructure at infinity by taking M = M and A = T M . Example . Let M be a manifold with corners such that its interior M := M r ∂M identifies with the quotient of a Lie group G by a discrete subgroup Γ and the actionof G on G/ Γ by left multiplication extends to an action of G on M . Let g be the Liealgebra of G . Then A := M × g with anchor given by the infinitesimal action of G is naturally a Lie algebroid. Note that the action of the Lie algebra g preserves thestructure of faces of M and hence ̺ ∗ (Γ( A )) ⊂ V b . We call the corresponding man-ifold with a Lie structure at infinity a group enlargement. The simplest example isthat of G = M = R ∗ + acting on [0 , ∞ ]. Many interesting Lie manifolds arising inpractice are, locally, group enlargements, see for instance [54, 55] for some examplescoming from quantum mechanics.Let ( M , A ) be an open manifold with a Lie structure at infinity. In applications,it is easier to work with the vector fields V := ̺ ∗ (Γ( A )), associated to a Lie manifold( M, A ), than with the Lie algebroid A → M . We shall then use the followingalternative definition of Lie manifolds. Proposition 3.6.
Let us consider a pair ( M , V ) consisting of a compact manifoldwith corners M and a subspace V ⊂ Γ( M ; T M ) of vector fields on M that satisfy:(i) V is closed under the Lie bracket [ , ] ;(ii) Γ c ( M ; T M ) ⊂ V ⊂ V b ;(iii) C ∞ ( M ) V = V and V is a finitely-generated C ∞ ( M ) –module;(iv) V is projective (as a C ∞ ( M ) –module).Then there exists a Lie manifold ( M , A ) with anchor ̺ such that ̺ ∗ (Γ( M ; A )) = V .Conversely, if ( M , A ) is a Lie manifold, then V := ̺ ∗ (Γ( M ; A )) satisfies theconditions (i)–(iv) above.Proof. Let (
M , V ) be as in the statement. Since V is a finitely generated, projective C ∞ ( M )–module, the Serre–Swan Theorem implies then that there exists a finitedimensional vector bundle A V → M , uniquely defined up to isomorphism, suchthat(23) V ≃ Γ( M ; A V ) , as C ∞ ( M )–modules. Let I x := { φ ∈ C ∞ ( M ) , φ ( x ) = 0 } be the maximal ideal cor-responding to x ∈ M . The fibers ( A V ) x , x ∈ M , of the vector bundle A V → M are given by ( A V ) x = V /I x V . Since Γ( M ; A V ) ≃ V ⊂ Γ( M ; T M ), we automaticallyobtain for each x ∈ M a map( A V ) x := V /I x V → Γ( M ; T M ) /I x Γ( M ; T M ) = T x M .
NALYSIS ON SINGULAR SPACES 19
These maps piece together to yield a bundle map (anchor) ̺ : A V → T M that makes A V → M a Lie algebroid. The anchor map ̺ is an isomorphism over the interior M of M since Γ c ( M ; T M ) ⊂ V , which is part of Assumption (ii). Since V ⊂ V b , againby Assumption (ii), we obtain that ( M , A V ) is indeed a Lie manifold.Conversely, let ( M , A ) be a Lie manifold with anchor ̺ : A → T M . We needto check that V := ̺ ∗ (Γ( M ; A )) satisfies conditions (i)–(iv) of the statement. In-deed, V := ̺ ∗ (Γ( M ; A )) is a Lie algebra because Γ( M ; A ) is a Lie algebra and ̺ ∗ : Γ( M ; A ) → Γ( M ; T M ) is an injective Lie algebra morphism. So Condition (i)is satisfied. To check the second conditions, we notice that Definition 3.3(i) (isomor-phism over the interior) gives that Γ c ( M ; T M ) ⊂ V . Since we have by assumption V ⊂ V b , we see that Condition (ii) is also satisfied. Finally, Conditions (iii) and (iv)are satisfied since the space of smooth sections of a finite dimensional vector bundledefines a projective module over the algebra of smooth functions on the base, againby the Serre-Swan theorem. (cid:3) Let (
M , V ) as in the statement of the above proposition, Proposition 3.6. We call V its structural Lie algebra of vector fields and we call the Lie algebroid A V → M introduced in Equation (23) the the Lie algebroid associated to ( M , V ). The alter-native characterization of Lie manifolds in Proposition 3.6 is the one that will beused in our examples. Remark . It is worthwhile pointing out that the condition that V be a finitelygenerated, projective C ∞ ( M )–module in Proposition 3.6 together with the fact thatthe anchor ̺ is an isomorphism over the interior of M are equivalent to the followingcondition, where n = dim( M ):For every point p ∈ M , there exist a neighborhood V p of p in M and n -vector fields X , X , . . . , X n ∈ V such that, for any vectorfield Y ∈ V , there exist smooth functions φ , φ , . . . , φ n ∈ C ∞ ( M )such that(24) Y = φ X + φ X + . . . + φ n X n on V p , with φ i | V p uniquely determined.The vector fields X , X , . . . , X n are then called a local basis of V on V p . (Thisis the analog in our case of the well known fact from commutative algebra that amodule is projective if, and only if, it is locally free.)In the next example, we shall need the defining functions of a “hyperface.” A hy-perface is a proper face H ⊂ M of maximal dimension (dimension dim( H ) = dim( M ) − defining function of a hyperface H of M is a function x such that H = { x = 0 } and dx = 0 on H . The hyperface H ⊂ M is called embedded if it has adefining function. The existence of a defining function is a global property, becauselocally one can always find defining functions, a fact that will be needed in theexample below.The simplest example of a non-compact Lie manifold is that of a manifold withcylindrical ends. The following example generalizes this example to the higher rankcase. It is a basic example to which we will come back later. Example . Let M a compact manifold with corners and V = V b . Let us checkthat ( M , V b ) is a Lie manifold. We shall use Proposition 3.6. Condition (i) is easilyverified since the Lie bracket of two vector fields tangent to a submanifold is againtangent to that submanifold. Condition (ii) in the Proposition 3.6 is even easier since, by definition, vector fields that are zero near the boundary ∂M are containedin V b . Clearly, V is a C ∞ ( M ) module. The only non-trivial fact to check is that V is finitely generated and projective as an C ∞ ( M ) module. This is actually theonly fact that we still need to check. To verify it, let us fix a corner point p ofcodimension k (that is, p belongs to an open face F of codimension k ). Then, in aneighborhood of p , we can find k defining functions r , r , . . . , r k of the hyperfacescontaining p such that a local basis of V around p (see Remark 3.7) is given by(25) r ∂ r , r ∂ r , . . . , r k ∂ r k , ∂ y k +1 , . . . , ∂ y n , where y k +1 , . . . , y n are local coordinates on the open face F of dimension k con-taining p , so that ( r , r , . . . , r k , y k +1 , . . . , y n ) provide a local coordinate system ina neighborhood of p in M . If M has a smooth boundary, then V b generates the to-tally characteristic differential operators, which were introduced in Equation (15),and hence this example corresponds to a manifold with cylindrical ends. In fact,we will see that the natural Riemannian metric of a manifold with (asymptotically)cylindrical ends. This example was studied also by Debord and Lescure [42, 45],Melrose and Piazza [99], Monthubert [105], Schulze [143], and many others.By [7], every vector field X ∈ V that has compact support in M gives rise to aone parameter group of diffeomorphisms exp( tX ) : M → M , t ∈ R . We denote byexp( V ) the subgroup of diffeomorphisms generated by all exp( X ) with X ∈ V andcompact support in M . The results in [9] show that exp( V ) acts by Lie automor-phisms of V (the condition (iv) of Proposition 3.6 that V be a projective module isnot necessary). Also, it would be interesting to see how the groups exp( V ) fit intothe general theory of infinite dimensional Lie groups [111].3.2. The metric on Lie manifolds.
As seen from the example of manifolds withcylindrical ends, Lie manifolds have an intrinsic geometry. We now discuss someresults in this direction following [7] and we extend them to open manifolds with aLie structure at infinity (this extension is straightforward, but needed). Thus, fromnow on, (
M , A ) will be an open manifold with a Lie structure at infinity. (Thus wewill not assume (
M , A ) to be a Lie manifold, unless explicitly stated.)
Definition 3.9.
Let (
M , A ) be an open manifold with a Lie structure at infinity.A metric on
T M is called compatible (with the structure at infinity) if it extendsto a metric on A → M .We shall need the following lemma. Lemma 3.10.
Let ( M , A ) be an open manifold with a Lie structure at infinity withcompatible metric g . Assume M to be paracompact. Then there exists a smoothmetric h on T M such that h ≤ g .Proof. Let us choose an arbitrary metric h on M (or, more precisely, on T M ).For each p ∈ M , let U p ⊂ V p be open neighborhoods of p in M such that V p hascompact closure and contains the closure of U p . Since V p has compact closureand the anchor map ̺ is continuous, we obtain that there exists M p > h ( ξ ) ≤ M p g ( ξ ) for every ξ ∈ A | V p . Let us choose I ⊂ ∂M such that { U p , p ∈ I } isa locally finite covering of the boundary ∂M . Let V be the complement of ∪ p ∈ I U p and let ( φ p ) p ∈ I ∪{ } be a smooth, locally finite partition of unity on M subordinated NALYSIS ON SINGULAR SPACES 21 to the covering ( V p ) p ∈ I ∪{ } . Then, if we define h = X p ∈ I φ p M − p h + φ g , the metric h will satisfy h ≤ g everywhere, as desired. (cid:3) Let us fix from now on a metric g on A , which restricts to a compatible Riemann-ian metric on M . The inner product of two vectors (or vector fields) X, Y ∈ Γ( M ; T M )in this metric will be denoted (
X, Y ) ∈ C ∞ ( M ) and the associated volume form d vol g . Of course, if X, Y ∈ V := ̺ ∗ (Γ( M ; A )), then ( X, Y ) ∈ C ∞ ( M ). We nowwant to investigate some properties of the metric g . For simplicity, we writeΓ( T M ) = Γ( M ; T M ) and ̺ ∗ (Γ( A )) = Γ( M ; A ). Let us consider the Levi-Civitaconnection(26) ∇ g : Γ( T M ) → Γ( T M ⊗ T ∗ M )associated to the metric g . Recall that an A –connection on a vector bundle E → M (see [7] and the references therein) is given by a differential operator ∇ such that(27) ∇ fX ( f ξ ) = f (cid:0) f ∇ X ( ξ ) + X ( f ) ξ (cid:1) for all f, f ∈ C ∞ ( M ) and ξ ∈ Γ( M ; E ). The following proposition from [7] givesthat the Levi-Civita connection extends to an “ A -connection.” Proposition 3.11.
Let ( M , A ) be an open manifold with a Lie structure at infinityand g be a compatible metric M . The Levi-Civita connection associated to the com-patible metric g extends to a linear differential operator ∇ = ∇ g : ̺ ∗ (Γ( A )) → Γ( A ⊗ A ∗ ) , satisfying(i) ∇ X ( f Y ) = X ( f ) Y + f ∇ X ( Y ) ,(ii) X ( Y, Z ) = ( ∇ X Y, Z ) + ( Y, ∇ X Z ) , and(iii) ∇ X Y − ∇ Y X = [ X, Y ] ,for all X, Y, Z ∈ V = ̺ ∗ (Γ( A )) .Proof. We recall the proof for the benefit of the reader. Since the metric g actuallycomes from an metric on A by restriction to T M ⊂ A , we see that(28) φ ( Z ) := ([ X, Y ] , Z ) − ([ Y, Z ] , X )+([ Z, X ] , Y )+ X ( Y, Z )+ Y ( Z, X ) − Z ( X, Y ) , defines a smooth function on M for any Z ∈ V and that this smooth functiondepends linearly on Z . Hence there exists a smooth section V ∈ V such that φ ( Z ) = ( V, Z ) for all Z ∈ V . We then define ∇ X Y := V . By the definition of ∇ and by the classical definition of the Levi-Civita connection, ∇ extends the Levi-Civita connection. Since the Levi-Civita connection satisfies the properties that weneed to prove (on M ), by the density of M in M , we obtain that ∇ satisfies thesame properties. (cid:3) We continue with some remarks
Remark . An important consequence of the above proposition is that each ofthe covariant derivatives ∇ k R of the curvature R extends to a tensor defined onthe whole of M . If M is compact (that is, if ( M , A ) is a Lie manifold), it followsthat the curvature and all its covariant derivatives are bounded . It turns out alsothat the radius of injectivity of M is positive [6, 43], and hence M has boundedgeometry . We next discuss the divergence of a vector field, which is needed to define ad-joints.
Remark . Another important consequence of the existence of an extension ofthe Levi-Civita connection to M is the definition of the divergence of a vectorfield. Indeed, let us fix a point p ∈ M and a local orthonormal basis X , . . . , X n of A on some neighborhood of p in M ( n = dim( M )). We then write ∇ X i X = P nj =1 c ij ( X ) X j and define(29) div( X ) := − n X j =1 c jj ( X ) , which is a smooth function on the given neighborhood of p that does not dependon the choice of the local orthonormal basis ( X i ) used to define it. Consequently,this formula defines a global function div( X ) ∈ C ∞ ( M ).We now introduce differential operators on open manifolds with a Lie structureat infinity. The desire to study these operators is the main reason why we areinterested in Lie manifolds. Definition 3.14.
Let (
M , A ) be an open manifold with a Lie structure at infinityand V := Γ( M ; A ). The algebra Diff( V ) is the algebra of differential operators on M generated by the operators of multiplication with functions in C ∞ ( M ) and bythe directional derivatives with respect to vector fields X ∈ V .In [9], the definition of Diff( V ) was extended to the case of not necessarily pro-jective C ∞ ( M )-modules V .Clearly, in our first example, Example 3.8, the resulting algebra of differentialoperators Diff( V ) = Diff( V b ) for M a manifold with boundary is the algebra oftotally characteristic differential operators. We shall see several other examples inthis paper. The differential operators in Diff( V ) can be regarded as acting eitheron functions on M or on functions on M := M r ∂M . When it comes to classes ofmeasurable functions–say Sobolev spaces–this makes no difference. However, thefact that Diff( V ) maps C ∞ ( M ) to C ∞ c ( M ) is a non-trivial property that does notfollow from the L -mapping properties of Diff( V ) on M . We have the followingsimple remark on the local structure of operators in Diff( V ). Remark . Every P ∈ Diff( V ) of order at most m can be written as a sum ofdifferential monomials of the form X α X α . . . X α k k , where X i ∈ V , k ≤ m , and α is a multi-index. If Y , Y , . . . , Y n are vector fields in V forming a local basis around p ∈ M (so dim( M ) = n ), then every P ∈ Diff( V ) of order at most m can be writtenin a neighborhood of p in M uniquelly as P = X | α |≤ m a α Y α Y α . . . Y α n n , a α ∈ C ∞ c ( M ) . This follows from the Poincar´e-Birkhoff-Witt theorem of [118].The next remark states that the algebra Diff( V ) is closed under adjoints. Remark . We shall denote the inner product on L ( M ; vol g ) by ( , ) L . Let P ∈ Diff( V ). The formal adjoint P ♯ of P is then defined by(30) ( P f , f ) L = ( f , P ♯ f ) L , f , f ∈ C ∞ c ( M ) . NALYSIS ON SINGULAR SPACES 23
Let X ∈ V := ̺ ∗ (Γ( A )). Since div( X ) ∈ C ∞ ( M ) of Equation (29) extends theclassical definition on M , we have that Z M X ( f ) d vol g = Z M f div( X ) d vol g . In particular, the formal adjoint of X is(31) X ♯ = − X + div( X ) ∈ Diff( V ) , and hence Diff( V ) is closed under formal adjoints.We can consider matrices of operators in Diff( V ) and operators acting on bundles. Remark . We can extend the definition of Diff( V ) by considering the spaceDiff( V ; E, F ) of operators acting between smooth sections of the vector bundles
E, F → M . This can be done either by embedding the vector bundles E and F intotrivial bundles or by looking at a local basis. The formal adjoint of P ∈ Diff( V ; E, F )is then an operator P ♯ ∈ Diff( V ; F ∗ , E ∗ ). We shall write Diff( V ; E ) := Diff( V ; E, E ).Typically E and F will have hermitian metrics and then we identify E ∗ with E and F ∗ with F . In particular, if E is a Hermitian bundle, then Diff( V ; E ) is an algebraclosed under formal adjoints.We are ready now to prove that all geometric operators on M that are asso-ciated to a compatible metric g are generated by V := ̺ ∗ (Γ( A )) [7]. (Recall thata compatible metric on M is a metric coming from a metric on the Lie algebroid A of our Lie manifold ( M , A ) by restriction to
T M .) In particular, we have thefollowing result [7].
Proposition 3.18.
We have that the de Rham differential d on M extends toa differential operator d ∈ Diff( V ; Λ q A ∗ , Λ q +1 A ∗ ) . Similarly, the extension ∇ ofthe Levi-Civita connection to an A -valued connection defines a differential operator ∇ ∈ Diff( V ; A, A ⊗ A ∗ ) . Proof.
The proof of this theorem is to see that the classical formulas for thesegeometric operators extend to M , provided that T M is replaced by A . For instance,for the de Rham differential, let ω ∈ Γ( M ; Λ k A ∗ ) and X , . . . , X k ∈ V , and use theformula( dω )( X , . . . , X k ) = q X j =0 ( − j X j ( ω ( X , . . . , ˆ X j , . . . , X k )) + X ≤ i 04 V. NISTOR and z smooth functions on the given neighborhood V . Since the only derivativein this formula is X and X ∈ V , this proves the desired statement for ∇ . (cid:3) Let us consider a vector bundle E → M . If E has a metric, then an A –connection ∇ ∈ Diff( V ; E, E ⊗ A ∗ ) is said to preserve the metric if(32) ( ∇ X ( ξ ) , ζ ) E + ( ξ, ∇ X ( ζ )) E = X ( ξ, ζ ) E for all X ∈ V and ξ, ζ ∈ Γ( M ; E ). In particular, it follows from Proposition 3.11that the extension of the Levi-Civita connection to an A –connection on A preservesthe metric used to define it. We then have the following theorem Theorem 3.19. We continue to consider the fixed metric on A and its associ-ated compatible metric g on M . Let E → M be a hermitian bundle with a metricpreserving A -connection. Then ∆ E := ∇ ∗ ∇ ∈ Diff( V ; E ) . Similarly, ∆ g := d ∗ d ∈ Diff( V ) . Proof. This follows from the fact that Diff( V ) and its vector bundle analogues areclosed under formal adjoints and from Proposition 3.18. (cid:3) Thus, in order to study geometric operators on a Lie manifold, it is enough tostudy the properties of differential operators generated by V . It should be noted,however, that a Riemannian manifold may have different compactifications to a Liemanifold. An example is R n , which can either be compactified to ([ − , n , V b ) (aproduct of manifolds with cylindrical ends) or it can be radially compactified toyield an asymptotically euclidean manifold. (See Example 5.3.)Theorem 3.19 also gives the following. Remark . Similarly, since d ∈ Diff( V ; E, F ), we get that the Hodge operator d + d ∗ ∈ Diff( V ; Λ ∗ A ∗ ), the same is true for the signature operator. More generally,let W → M be a Clifford bundle with admissible connection in Diff( V ; W, W ⊗ A ∗ ).Then the associated Dirac operators are also generated by V . If W is the Cliffordbundle associated to a Spin c -structure on A , then the Levi-Civita connection on W is in Diff( V ; W, W ⊗ A ∗ ). All these statements seem to be more difficult to provedirectly in local coordinates. See [7, 82] for proofs.3.3. Anisotropic structures. It is very important in applications to extend theprevious frameworks to include anisotropic structures [13]. We introduce them nowfor the purpose of later use. Definition 3.21. An anisotropic structure on an open manifold ( M , A ) with a Liestructure at infinity is an open manifold manifold with a Lie structure at infinity( M , B ) (same M ) together with a vector bundle map A → B that is the identity over M and makes Γ( A ) = Γ( M ; A ) an ideal (in Lie algebra sense) of Γ( B ) = Γ( M ; B ).We shall denote W := ̺ ∗ (Γ( B )), so V := ̺ ∗ (Γ( A )) satisfies [ X, Y ] ⊂ V for all X ∈ W and Y ∈ V . Recall the groups exp( V ) and exp( W ) introduced at the end ofSubsection 3.1 (and recall that they are generated by compactly supported vectorfields), then we have the following. Remark . The group exp( W ) acts on M by Lipschitz diffeomorphisms, it acts on A by Lie algebroid morphism, it acts on V := ̺ ∗ (Γ( A )) by Lie algebra morphisms,and it acts on on Diff( V ) by algebra morphisms. Moreover,exp( V ) ⊂ exp( W ) NALYSIS ON SINGULAR SPACES 25 is a normal subgroup. 4. Analysis on Lie manifolds Our main interest is in the analytic properties of the differential operators inDiff( V ). In this section, we introduce our function spaces following [6] and discussFredholm conditions. Throughout this section, ( M , A ) will denote an open manifoldwith a Lie structure at infinity and Lie algebroid A with anchor ̺ : A → T M . Also,by V := ̺ ∗ (Γ( A )) we shall denote the structural Lie algebra of vector fields on M .4.1. Function spaces. We review in this subsection the needed definitions of func-tion spaces. Let ( M , A ) be our given open manifold with a Lie structure at infinityand let g be a compatible metric on the interior M of M (that is, coming froma metric on A denoted with the same letter, see Definition 3.9). Let ∇ be theLevi-Civita connection acting on the tensor powers of the bundles A and A ∗ . Wethen define, for m ∈ Z + , the Sobolev spaces as in [4, 6, 59, 63]:(33) H m ( M ) = { u : M → C , ∇ k u ∈ L ( M ; A ∗⊗ k ) , ≤ k ≤ m } . Remark . In general, the Sobolev spaces H m ( M ) will depend on the choice ofthe metric g , but if M is compact (that is, if ( M , A ) is a Lie manifold), then theyare independent of the choice of the metric, as we shall see below. It is interestingto notice that if denote by d vol g the volume form (1-density) associated to g . If h is another such compatible metric, then d vol h /d vol g and d vol g /d vol h extendto smooth, bounded functions on M . Hence the space L ( M ) := L ( M ; vol g ) isindependent of the choice of the compatible metric g .The spaces H m ( M ) behave well with respect to anisotropic structures. Proposition 4.2. Let ( M , A ) be an open manifold with a Lie structure at infinityand with an anisotropic structure ( M , B ) , such that W := Γ( B ) ⊃ Γ( A ) . Then exp( W ) acts by bounded operators on H m ( M ) .Proof. This follows from the fact that exp( W ) is generated by vector fields withcompact support in M . (cid:3) We now consider some alternative definitions of these Sobolev spaces in particularcases. We first consider the case of complete manifolds [4, 6, 59, 63, 64]. Remark . Let us assume that ( M , A ) and the compatible metric g are such that M is complete and let ∆ g be (positive) Laplacian associated to the metric g . Then H s ( M ) coincides with the domain of (1 + ∆ g ) s/ (we use the geometer’s Laplacian,which is positive).In the bounded geometry case we can consider partitions of unity. Remark . Let us assume that ( M , A ) and the compatible metric g are such that M := M r ∂M is of bounded geometry. Then the definition of the Sobolev spaceson M can be given using a choice of partition of unity with bounded derivatives asin [6], for example, to patch the locally defined classical Sobolev spaces. See also[4, 6, 49, 59, 74, 75, 147].Finally, if M is an open subset of a Lie manifold, we have yet the followingdefinition. Remark . Let U ⊂ M be an open subset of a Lie manifold ( M , A ) (so M iscompact) and let, as usual, V denote the structural Lie algebra of vector fieldsΓ( M , A ), then [7](34) H m ( U ) := { u : U → C , X X . . . X k u ∈ L ( U ) , k ≤ m, X j ∈ V} , so the Sobolev spaces H m ( U ) will be independent of the chosen compatible metricon the Lie manifold.Each of theses definitions of the Sobolev spaces has its own advantages anddisadvantages. For instance, the definition (34) has the advantage that it im-mediately gives the boundedness of operators P ∈ Diff( V ), see Lemma 4.6. Let H − s ( M ) := ( H s ( M )) ∗ and extend the definition of Sobolev space to s non-integerby interpolation. Lemma 4.6. Let us assume that A is endowed with a metric such that the re-sulting metric g on T M ⊂ A is of bounded geometry. Let P ∈ Diff( V ) of order ord( P ) ≤ m and with coefficients that are compactly supported in M . Then themap P : H s ( M ) → H s − m ( M ) is bounded for all s ∈ R . Note that if M is compact (i.e. ( M , A ) is a Lie manifold), then all P ∈ Diff( V )have compactly supported coefficients. Let us denote by ( E ) r the set of vectors oflength < r , where E is a real or complex vector bundle endowed with a metric. Proof. Let K ⊂ M be a compact subset such that the coefficients of P are zerooutside K . Let us choose a compact neighborhood L of K in M and let r be thedistance from K to the complement of L in a metric h on M such that h ≤ g , whichexists by Lemma 3.10. Then r > 0, because K is compact. Moreover, the distancefrom K to the complement of L in the metric g is ≥ r since h ≤ g . Let us fix r lessthan the injectivity radius of M and with r > r > 0. For every p ∈ M , we thenconsider the exponential map exp : T p M → M , which is a diffeomorphism from (theopen ball of radius r ) ( T p M ) r onto its image. Thus P gives rise to a differentialoperator P p on each of the open balls ( T p M ) r . Using the results of [147], it sufficesto show that the coefficients of P in any of these balls of radius r are uniformlybounded. Indeed, this is a consequence of the following lemma, where the supportof the resulting map is contained in L . (cid:3) The following lemma (see [8]) underscores the additional properties that the Liemanifolds enjoy among the class of all manifolds with bounded geometry. Lemma 4.7. Let us use the notation of the proof of the previous lemma and denotefor any p ∈ M by P p the differential operator on ( T p M ) r induced by the exponentialmap. Then the map M ∋ p → P p extends to a compactly supported smooth functiondefined on M such that P p is a differential operator on ( A p ) r . We define the anisotropic Sobolev spaces in a similar way. Remark . Let ( M , A ) be a Lie manifold with an anisotropic structure ( M , B ),and let W := Γ( B ) ⊃ Γ( A ). Then we define(35) H m + q W ( M ) := { u : M → C , X X . . . X k Y , Y . . . Y l u ∈ L ( M ) , for all X j ∈ W , ≤ j ≤ k ≤ m, and for all Y i ∈ V , ≤ i ≤ l ≤ q } . The spaces H m + q W ( M ) are again independent of the chosen compatible metric onthe Lie manifold. NALYSIS ON SINGULAR SPACES 27 Pseudodifferential operators on Lie manifolds. Let us begin by recallingthe definition of a tame submanifold with corners from [7] (we changed slightly theterminology). Definition 4.9. Let M be a manifold with corners and L ⊂ M be a submanifold.We shall say that L is a tame submanifold with corners of M if L is a manifold withcorners (in its own) that intersects transversely all faces of M and such that eachopen face F of L is the open component of a set of the form F ∩ L , where F is anopen face of M (of the same codimension as F ).The closed faces of a manifold with corners M are thus not tame submanifoldswith corners of M even if they happen to be manifolds with corners. Also, thediagonal of the n -dimensional cube [ − , n is not a tame submanifold with corners.However, { } × [ − , n − is a tame submanifold with corners of [ − , n . In fact,all tame submanifolds with corners L ⊂ M have a tubular neighborhood [6, 8].This tubular neighborhood allows us then to define the space I m ( M , L ) of classicalconormal distributions as in [68] or as in [118] for manifolds with corners as follows.Let V → X be a real vector bundle. A distribution on V is (classically) conormal to X if its fiberwise Fourier transform is a classical symbol on V ∗ → M . Weshall denote the set of these distributions corresponding to a symbol of order ≤ m by I m ( V, M ). We shall denote by I mc ( V, M ) ⊂ I m ( V, M ) the subset of conormaldistributions with compact support. We extend the definition of I mc ( V, M ) to thecase of a tame submanifold M ⊂ V by localization.Let us fix a compatible metric on M , the interior of our open manifold witha Lie structure at infinity ( M , A ). Also, let us fix r > M . As in the proof of Lemma 4.6, the exponential map then defines adiffeomorphism from the set ( T M ) r of vectors of length < r to ( M × M ) r , whichis an open neighborhood of the diagonal in M × M . This allows us to define anatural bijection(36) Φ : I mc (( T M ) r , M ) → I mc (( M × M ) r , M ) , Similarly, we obtain by restriction an inclusion(37) I mc (( A ) r , M ) → I m (( T M ) r , M ) . Recall the group of diffeomorphisms exp( V ) defined at the end of Subsection 3.1.Then we define as in [8](38) Ψ m V ( M ) := Φ( I mc (( A ) r , M )) + Φ( I −∞ c (( A ) r , M )) exp( V ) . Then Ψ m V ( M ) is independent of the parameter r > g used to define it and we have the following result [8]. Theorem 4.10. Let ( M , A ) be a Lie manifold, M := M r ∂M and V := ̺ ∗ (Γ( A )) .We have Ψ m V ( M )Ψ m ′ V ( M ) ⊂ Ψ m + m ′ V ( M ) . The subspace Ψ m V ( M ) is closed underadjoints and the principal symbol σ m : Ψ m V ( M ) → S mcl ( A ∗ ) /S m − cl ( A ∗ ) is surjectivewith kernel Ψ m − V ( M ) . Moreover, any P ∈ Ψ m V ( M ) defines a bounded operator H s ( M ) → H s − m ( M ) . If an anisotropic structure with structural vector fields W isgiven, then the group exp( W ) acts by degree preserving automorphisms on Ψ m V ( M ) . The group exp( W ) is seen to act on Ψ m V ( M ) since it acts by Lipschitz diffeomor-phisms of M . The proof of the above theorem is too long to include here. See [8]for details. Let us just say that it is obtained by realizing Ψ ∗V ( M ) as the image of a groupoid pseudodifferential operator algebra [8, 105, 107, 118] for any Lie groupoidintegrating the Lie algebroid A defining the Lie manifold ( M , A ) [43, 44, 117].The algebra Ψ ∗V ( M ) has the property that its subset of differential operatorscoincides with Diff( V ). It also has the good symbolic properties that answer to aquestion of Melrose [8, 101].4.3. Comparison algebras. We continue to denote by ( M , A ) an open manifoldwith a Lie structure at infinity and by V := ̺ ∗ (Γ( A )). For simplicity, we shallassume that M is connected. We now recall from [106] the comparison C ∗ -algebra A ( U, V ) associated to an open subset U ⊂ M . Its definition extends to openmanifolds with a Lie structure at infinity by Lemma 4.6 that justifies the followingdefinition. Definition 4.11. Let us assume that A has a metric g such that the induced metricon M is of bounded geometry and let U ⊂ M be an open subset. Then A ( U ; V ) isthe norm closed subalgebra of the algebra B ( L ( M ; vol g )) of bounded operators on L ( M ; vol g ) generated by all the operators of the form φ P (1 + ∆ g ) − k φ , where φ i ∈ C ∞ c ( U ), P ∈ Diff( V ) is a differential operator of order ≤ k , and ∆ g is theLaplacian on M (not on U !) associated to the metric g .In case an anisotropic structure is given, the group exp( W ) acts by automor-phisms on the comparison algebra A ( M ; V ). We shall need the following lemmathat follows right away from the results in [7] and [82]. By a pseudodifferential op-erator on a manifold, we shall mean one that is obtained by the usual quantizationformula in any coordinate system. Lemma 4.12. Let us use the notation and the assumptions of Definition 4.11 andlet T := φ P (1 + ∆ g ) − k φ . Then T is contained in the norm closure of Ψ V ( M ) and is a pseudodifferential operator of order ≤ with principal symbol σ ( T ) = φ σ k ( P )(1 + | ξ | ) − k φ . Moreover, the principal symbol depends continuously on T , and hence extends to acontinuous, surjective morphism σ : A ( U ; V ) → C ( S ∗ A | U ) . As in [106], we obtain the following result. Theorem 4.13. Let ( M , A ) be a connected open manifold with a Lie structure atinfinity with a compatible metric of bounded geometry. Then A ( M ; V ) contains thealgebra K ( L ( M )) of all compact operators on L ( M ) and is contained in the normclosure of Ψ V ( M ) .Proof. We recall the proof for the benefit of the reader. The inclusion of A ( M ; V )in the norm closure of Ψ V ( M ) follows from Lemma 4.12.Let φ , φ ∈ C ∞ c ( M ) and P ∈ Diff( V ) be a differential operator of order at most2 k − 1, then the composition operator φ P (1 + ∆) − k φ is a non-zero compact op-erator and belongs to A ( M ; V ), by the definition. The role of the cut-off func-tions is to decrease the support of the distribution kernel of P (1 + ∆) − k . Since φ P (1 + ∆) − k φ is compact, we thus obtain that A ( M ; V ) contains non-zero com-pact operators. Let ξ , ξ ∈ L ( M ) be nonzero. Then we can find φ , φ , and P asabove such that T := φ P (1 + ∆) − k φ satisfies ( T ξ , ξ ) = 0. Hence A ( M ; V ) hasno non-trivial invariant subspace. Hence A ( M ; V ) contains all compact operatorsbecause any proper subalgebra of the algebra of compact operators has an invariantsubspace. (cid:3) NALYSIS ON SINGULAR SPACES 29 Fredholm conditions. Theorem 4.13 allows us, in principle, to study theFredholm property of operators in A ( M ; V ). Let us denote by K = K ( L ( M ) theideal of compact operators in A ( M ; V ). Recall then Atkinson’s classical result[50] that states that T ∈ A ( M ; V ) is Fredholm if, and only if, its image T + K in A ( M ; V ) / K is invertible.Usually it is difficult to check directly that T + K is invertible in A ( M ; V ) / K ,and, instead, one checks the invertibility of operators of the form π ( T ), where π ranges through a suitable family of representations of A ( M ; V ) / K . Exactly what arethe needed properties of the family of representations of A ( M ; V ) / K was studied in[116, 135]. Let us recall the main conclusions of that paper. Let us consider a familyof representations F and π ∈ F . It is not enough for the family F to be faithful sothat the invertibility of all π ( T ) implies the Fredholmness of T . For this implicationto be true, the necessary condition is that the family F be invertibility sufficient [116, 135]. An equivalent condition (in the separable case) is that the family F be exhausting , in the sense that every irreducible representation of A ( M ; V ) / K isweakly contained in one of the representations π ∈ F .This approach was used (more or less explicitly) in [40, 46, 53, 54, 55, 81, 82,94, 104, 130, 135, 136, 141], and in many other papers. However, in order for thisapproach to be effective, we need to have a good understanding of the representationtheory of the quotient A ( M ; V ) / K . This seems to be difficult in general, at leastwithout using groupoids. Thus we shall replace the comparison algebra A ( M ; V )with the norm closure of the algebra Ψ V ( M ). The algebra Ψ V ( M ) is defined in thenext section.There are many general results that yield Fredholm conditions for operators. Weformulate now one such result that is sufficient in most applications. We shall makesome the following assumptions on the Lie manifold ( M , A )(a) We assume that there exists a filtration(39) ∅ =: U − ⊂ U ⊂ . . . ⊂ U k ⊂ U k +1 ⊂ . . . ⊂ U N := M of M with open sets such that each S k := U k r U k − is a manifold (possibly withcorners) and that there exists submersions p k : S k → B k of manifolds (possiblywith corners) whose fibers are orbits of exp( V ).(b) We assume that, for each k = 0 , . . . , N , there exists a Lie algebroid A k → B k with zero anchor map such that A | S k ≃ p ↓↓ k ( A k ), the pull-back of A k by thesubmersion p k [66]. In particular, we have the isomorphism of vector bundles A | S k ≃ ker( p k ) ∗ ⊕ p ∗ k ( A k ) . (c) Let us denote by ( Z α ) α ∈ J the family of orbits Z α = exp( V ) p of V and by G α thesimply-connected Lie group that integrates the Lie algebra ( A k ) p k ( p ) ≃ ker( ̺ p ),for any p ∈ Z α ⊂ S k . Also, let us denote by G the disjoint union(40) G := ∪ α ∈ J Z α × Z α × G α with the induced groupoid structure. We assume that the groupoid exponentialmap makes G a Hausdorff Lie groupoid [117]. In particular, G is a manifold(possibly with corners).Under the above assumptions, the results in [72, 81, 82, 116, 132, 134] give thefollowing theorem. Theorem 4.14. Let I be an index parametrizing the set of obits of V on ∂M .(So J = I ∪ { } .) We can associate to each P ∈ Diff( V ; E, F ) a family ( P α ) , of G α -invariant operators P α on Z α × G α . If all the groups G α , α ∈ I are amenable,then the following Fredholm condition holds. P : H s ( M ) → H s − m ( M ) is Fredholm ⇔ P is ellipticand all P α : H s ( Z α × G α ) → H s − m ( Z α × G α ) , α ∈ I, are invertible . Proof. (Sketch) The exact sequence of (full) C ∗ -algebras associated to an opensubset of the set of units of a groupoid [132] tells us that Prim( C ∗ ( G )) is the disjointunion of the sets Prim( C ∗ ( G S k )). We have that C ∗ ( G ) ≃ C ∗ r ( G ), because C ∗ ( G S k ) ≃ C ∗ r ( G S k ) for each k , by the amenability of the groups G α . This shows that the set { Ind( λ α ) } of representations of C ∗ ( G ) induced from the regular representations λ α of G α is an exhausting set of representations of C ∗ ( G ) ≃ C ∗ r ( G ). Each π α := Ind( λ α )is the regular representation of C ∗ r ( G ) associated to (any unit in) the orbit Z α , with π being the vector representation on L ( M ).The assumptions imply that S = U = M and that G = { e } . Hence G S isthe pair groupoid S × S and C ∗ ( G M ) ≃ K , the algebra of compact operatorson L ( M ). We obtain that C ∗ ( G ) / K ≃ C ∗ ( G ∂M ) ≃ C ∗ r ( G ∂M ) and the regularrepresentations of C ∗ ( G ∂M ) form an invertibility sufficient set of representations of C ∗ ( G ∂M ). This gives that a ∈ C ∗ ( G ) ≃ C ∗ r ( G ) is Fredholm if, and only if, π α ( a ) isinvertible for all α corresponding to orbits in ∂M .We shall apply these observations to the algebra Ψ ∞ ( G ) of pseudodifferentialoperators on groupoids, which is recalled in the next subsection. The fact that G isHausdorff implies that the vector representation of C ∗ ( G ) (associated to the orbit M ⊂ M ) is injective, by [72]. We shall use then the vector representation to identifyΨ ∞ ( G ) and C ∗ r ( G ) with their images under the vector representation. In, particular, P is given by a family of operators ( P x ), x ∈ M , with operators corresponding tounits in the same orbit unitarily equivalent. We then let P α := P x , for some x inthe orbit corresponding to α .Let a := (1 + ∆) ( s − m ) / P (1 + ∆) − s/ ∈ Ψ ( G ) [81, 82]. We then have that P isFredholm if, and only if, a is Fredholm on L ( M ), which, in turn, is true, if, andonly if, π α ( a ) is invertible for all Z α ⊂ ∂M . Since π x ( a ) := (1 + π x (∆)) ( s − m ) / P x (1 + π x (∆)) − s/ , acting on Z α × G α , the result follows from the fact that the set of arrows of G withdomain x ∈ Z α is Z α × G α . (cid:3) This theorem is closely related to the representations of Lie groupoids, see [21,27, 28, 51, 52, 69, 133, 154]. For our result, we need Hausdorff groupoids, seehowever also [72, 153] for some results on non-Hausdorff groupoids. More generalFredholm conditions can be obtained along the same lines, but the result mentionedhere, although having a rather long list of assumptions, is easy to prove and to use.More references to earlier results will be given in the next section when discussingexamples. Remarks . We continue with a few remarks.(1) If M is compact and smooth (so without corners), then I = ∅ , and werecover Theorem 1.2. As we will explain below, we also recover Theorem1.9. Each operator P α is “of the same kind” as P (Laplace, Dirac, ... ) NALYSIS ON SINGULAR SPACES 31 and can be recovered by “freezing the coefficients” at the orbit Z α . Thetheorem allows us to reduce some questions on M to questions on P α and G α . Because of the G α -invariance of our operators, we can use resultson harmonic analysis on G α to obtain an inductive procedure to studygeometric operators on M [91, 121].(2) Each open face F of M is invariant for exp( V ), and hence, if an orbit Z α intersects F , then it is completely contained in F . In particular, the set oforbits I identifies with the disjoint union of the sets B k for k = 1 , , . . . , N .(3) The sections of ker( p k ) ∗ act by derivation on the sections of p ∗ k ( A , ). Also, itfollows that for any p ∈ F , the isotropy Lie algebra ker( ̺ p ) is canonicallyisomorphic to the Lie algebra ( A k ) p k ( p ) , see Definition 3.2.(4) We note that our assumptions on ( M , A ) imply that the groupoid G con-sidered in our assumptions must coincide with the one introduced by ClaireDebord [43, 44].4.5. Pseudodifferential operators on groupoids. Let us briefly recall, for thebenefit of the reader, the definition of pseudodifferential operators on a Lie groupoid G . Let d : G → M be the domain map and G x = d − ( x ). Then Ψ m ( G ), m ∈ R , consists of smooth families ( P x ) x ∈ M of classical, order m pseudodifferentialoperators ( P x ∈ Ψ m ( G x )) that are right invariant with respect to multiplication byelements of G and are “uniformly supported.” To define what uniformly supportedmeans, let us observe that the right invariance of the operators P x implies thattheir distribution kernels K P x descend to a distribution k P ∈ I m ( G , M ) [104, 118].The family P = ( P x ) is called uniformly supported if, by definition, k P has compactsupport in G .Groupoids simplify the study of pseudodifferential operators on singular andnon-compact spaces. For instance, one obtains a straightforward definition of the“generalized indicial operators” as restrictions to invariant subsets [81]. More pre-cisely, let N ⊂ M be an invariant subset for G , that is, d − ( N ) = r − ( N ), and let G N := d − ( N ). Let us now assume that P ∈ Ψ m ( G ) is given by the family ( P x ) x ∈ M ,then the N –indicial family I N ( P ) := ( P x ) x ∈ N is defined simply as the restrictionof P to N and is in Ψ m ( G N ). See [47] for an extension of these results in relation tothe adiabatic groupoid. See also [1, 19, 18, 98] for results on the Boutet-de-Montvelcalculus in the framework of groupoids.In order to be able to use the machinery of Lie groupoids in analysis, one hasto sometimes first integrate the Lie algebroid that naturally arises in the analysisproblem at hand. That is, given a Lie algebroid A → M , one wants to find a Liegroupoid G such that A ( G ) ≃ A . Such a groupoid A does not always exist, andwhen it exists, it is not unique. Moreover, the choice of the groupoid G dependson the analysis problem one is interested to solve. For example, the Lie algebroid T M → M for a smooth, compact manifold M has the pair groupoid M × M as a minimal integrating groupoid and has P ( M ), the path groupoid of M asthe maximal integrating groupoid [43, 44, 117]. The first groupoid leads to theusual analysis on compact, smooth manifolds (the AS-framework), whereas thesecond one leads to the analysis of invariant operators on f M , the universal coveringspace of M (with group of deck transformation π ( M )). Both these examples areexamples of d -connected integrating groupoids (i.e. the fibers of the domain map d are connected). There are analysis problems, however, when one is lead to non- d -connected groupoids [25]. There are many works dealing with pseudodifferential operators on groupoids,on singular spaces, or with the related C ∗ -algebras, see for example [3, 15, 26, 37,46, 58, 86, 95, 96, 109, 122, 131, 135, 136, 149, 150, 155].5. Examples and applications We now discuss some applications. They are included just to give an idea of themany possible applications of Lie manifolds, so we will be short, but we refer to theexisting literature for more details. We begin with some examples.5.1. Examples of Lie manifolds and Fredholm conditions. We now includeexamples of Lie manifolds and show how to use Theorem 4.14. The followingexamples cover many of the examples appearing in practice. Example . We now review our first, basic example, Example 3.8, in view ofthe new results. Recall that V = V b := the space of vector fields on M that aretangent to ∂M . Near the boundary, a local basis is given by Equation (25) ofExample 3.8, and hence Diff( V b ) is the algebra of totally characteristic differentialoperators. If M has a smooth boundary and we denote by r the distance to theboundary (in some everywhere smooth metric–including the boundary), then atypical compatible metric on M is given near the boundary by ( r − dr ) + h , where h is a metric smooth up to the boundary. Hence the geometry is that of a manifoldwith cylindrical ends.We have that the orbits Z α are the open faces of M , except M itself. Thegroups are G α ≃ R k , where k is the codimension of the corresponding face (so allare commutative Lie groups). In the case of a smooth boundary, the Z α ’s are theconnected components of the boundary, G α = R , and P α is the restriction of I ( P )to a translation invariant operator on Z α × R . See also [85, 99, 100, 105, 107, 142]for just a sample of the many papers on this particular class of manifolds.In the following examples, M will be a compact manifold with smooth boundary ∂M . The following example is that of an asymptotically hyperbolic space and hasthe feature that it leads to non-commutative groups G α . Example . Let M be a compact manifold with smooth boundary ∂M and defin-ing function r . We proceed as in Example (3.8). The structural Lie algebra ofvector fields is V = r Γ( T M ) = the space of vector fields on M that vanish on theboundary. Using the same notation as in the Example (3.8), near a point of theboundary ∂M = { r = 0 } , a local basis is given by(41) r∂ r , r∂ y , . . . , r∂ y n , so V is a finitely generated, projective C ∞ ( M )–module. Since V is also closed underthe Lie bracket and Γ c ( M ; T M ) ⊂ V ⊂ V b , we have that ( M , V ) defines indeed aLie manifold.The orbits Z α ⊂ ∂M are reduced to points, so we can take I := ∂M , and G α = T α ∂M ⋊ R is the semi-direct product with R acting by dilations on the vector space T α ∂M , α ∈ I . The pseudodifferential calculus Ψ ∗V ( M )for this example was defined also by[78], Lauter-Moroianu [79], Mazzeo [97], and Schulze [143]. The metric is asymp-totically hyperbolic . See also [2, 56, 61].The following example covers, in particular, R n with the usual Euclidean metricand with the radial compactification. NALYSIS ON SINGULAR SPACES 33 Example . As in the previous example, M is a compact manifold with smoothboundary ∂M = { r = 0 } . We shall take now V = r V b = the space of vector fields on M that vanish on the boundary ∂M and whose normal covariant derivative to theboundary also vanishes. Using the same notation as in the previous two examples,at the boundary ∂M , a local basis is given by(42) r ∂ r , r∂ y , . . . , r∂ y n . Again the orbits Z α are reduced to points, so α ∈ I := ∂M , but this time G α = T α M = T α ∂M × R is commutative. See also [22, 101, 120, 141]. If ∂M = S n − ,the resulting geometry is that of an asymptotically Euclidean manifold. In partic-ular, R n with the radial compactification fits into the framework of this example. Example . As in the previous two examples, M is a compact manifold withsmooth boundary ∂M = { r = 0 } . To construct our Lie algebra of vector fields V = V e , we assume that we are given a smooth fibration π : ∂M → B , and we let V e to be the space of vector fields on M that are tangent to the fibers of π : ∂M → B .By choosing a product coordinate system on a small open subset of the boundary,a local basis is then given by(43) r∂ r , r∂ y , . . . , r∂ y k , ∂ y k +1 , . . . , ∂ y n . Here k is such that the fibers of π : ∂M → B have dimension n − k . Thus, when k = 1 (so the fibration is over a point, that is, π : ∂M → pt ), we recover ourfirst example, Example 5.1. On the other hand, when k = n (so the fibrationis π : ∂M → ∂M ), we recover our second example, Example 5.2. For n = 3 and k = 2, we recover the edge differential operators of Example 2.2 (3) (see Equation(22)). We note that V := V e ⊂ V b =: W yields a typical example of an anisotropicstucture.In general, in this example, the set of orbits is I = { α } = B , Z α = π − ( α ), and G α = T α B ⋊ R is a solvable Lie group with R acting by dilations. The geometryis related to that of locally symmetric spaces. Differential operators of this kindappear in the study of behavior at the edge of boundary value problems. Thisexample generalizes the second example (Example 5.2) and the same references arevalid for this example as well. See however also [60] for possibly the first paper onthis type of examples.We conclude with some less standard examples. Example . Let us assume that we are in the same framework as in the previousexample, Example, 5.4, but we replace the fibration of ∂M with a foliation. Thenthe resulting Lie manifold may fail to satisfy Theorem 4.14. See however [137]. Itis interesting to notice that in this case, the resulting class of Riemann manifoldsleads naturally to the study of foliation algebras.Our last example in this subsection is on a manifold with corners. Example . Let A → M be a Lie algebroid (we do not assume Γ( M ; A ) ⊂ V b )and let φ : M → [0 , ∞ ) be a smooth function such that { φ = 0 } = ∂M . We define V := φ Γ( M ; A ). Then ( M , V ) defines a Lie manifold.A related example deals with the N -body problem in Quantum Mechanics [48]and can be used to give a new proof of the classical HWZ-theorem on the essentialsspectrum of these operators. This is too long and technical to include here, however. Index theory. Let now ( M , A ) be a Lie manifold and let f be the productof the defining functions of all its faces. We consider then the exact sequence(44) 0 → f Ψ − ( G ) → Ψ ( G ) → Symb → , which gives rise as before to the map ∂ : K (Symb) → K ( I ) . The Fredholmindex problem is in this case to compute T r ∗ ◦ ∂ : K (Symb) → Z . Since φ ∗ ◦ ∂ = ψ ∗ , where ψ = ∂φ ∈ HP (Symb), by Connes’ results, the Fredholmindex problem is equivalent to computing the class of ψ in periodic cyclic homology.This is a difficult problem that is still largely unsolved. Undoubtedly, excision incyclic theory will play an important role [39]. See also [29, 30, 31, 113, 114, 123, 124,125, 138, 152]. Instead of this general problem, we shall look now at a particular,but relevant case [23, 24]. Definition 5.7. We say that a be a Lie manifold ( M , W ) is asymptotically com-mutative if all vectors in W vanish on ∂M and all isotropy Lie algebras ker( q x ) arecommutative.Let x , x , . . . , x k be the defining functions of all the hyperfaces of M and f = x a x a . . . x a k k for some positive integers a j . Then, for any Lie manifold, theproduct ( M , V ) W := f V defines an asymptotically commutative Lie manifold ( M , W ).If ( M , W ) is asymptotically commutative, then the algebra Symb is commuta-tive. Its completion will be of the form C (Ω), as in the work of Cordes [34, 35].Since the algebra Symb is commutative in this case, it is possible then to computethe index of Fredholm operators using classical invariants [24]. As an application,one obtains also the index of Dirac operators coupled with potentials of the form f − V , where V is invertible at infinity on any Lie manifold (not just asymptoticallycommutative) [24].5.3. Essential spectrum. We now present some applications to essential spectra.We use the notation introduced in Subsection 4.3. The applications to essentialspectra of operators are based on the fact that for a self-adjoint operator D wehave that(45) λ ∈ σ ess ( D ) ⇔ D − λ is not Fredholm.We shall consider the case of self-adjoint operators D affiliated to A ( M, V ) (that is,satisfying ( D + ı ) − ∈ A ( M, V ) [40, 53]). Then we can use Theorem 4.14 to studywhen D − λ is (or is not) Fredholm.We shall use these ideas for the open manifolds modeled by the Lie algebra ofvector fields ( M , V b ) of Example (5.1) and the associated positive Laplacian ∆ M [83]. Theorem 5.8. Let M be a manifold with corners and let ( M , V b ) be the Lie man-ifold of Example (5.1) . We endow M , the interior of M , with the induced metric.Let ∆ M be the associated positive Laplacian on M . Assuming that M = M , wehave σ (∆ M ) = [0 , ∞ )A complete characterization of the spectrum (multiplicity of the spectral mea-sure, discreteness of the point spectrum, absence of continuous singular spectrum)is wide open, in spite of its importance. NALYSIS ON SINGULAR SPACES 35 Similarly, let \ D be the Dirac operator associated to a Clif f ( A )-bundle over M .Then [115] Theorem 5.9. The Dirac operator \ D on M = M r ∂M is invertible if, and onlyif, for any open face F (including the interior face M ), the associated Dirac operator \ D F , has no harmonic spinors (that is, it has zero kernel). The proof uses Theorem 4.14 and the fact that the resulting operators P α arealso Dirac operators.Many similar results were obtained in Quantum Mechanics by Georgescu andhis collaborators [40, 53, 55]. In fact, certain problems related to the N –bodyproblem can be formulated in terms of a suitable compactifications of X := R n toa manifold with corners M on which X still acts and such that the Lie algebra ofvector fields V is obtained from the action of X [54]. See also [48].5.4. Hadamard well posedness on polyhedral domains. This type of appli-cation [14] is of a different nature and does not use pseudodifferential operatorsor other operator algebras. It uses only Lie manifolds and their geometry. Letthen Ω ⊂ R n be an open, bounded subset of with boundary ∂ Ω. We shall considerthe “simplest” boundary value problem on Ω, the Poisson problem with Dirichlet boundary conditions:(46) ( − ∆ u = fu | ∂ Ω = 0 . We refer to [14] for further references and details not included here. Recall then thefollowing classical result, which we shall refer to as the basic well-posedness theorem(for ∆ on smooth domains) Theorem 5.10. Let us assume that ∂ Ω is smooth. Then the Laplacian ∆ definesan isomorphism ∆ : H s +1 (Ω) ∩ { u | ∂ Ω = 0 } → H s − (Ω) , s ≥ . A useful consequence (easy to contradict for non-smooth domains) is: Corollary 5.11. If f and ∂ Ω are smooth, then the solution u of the Poissonproblem with Dirichlet boundary conditions is also smooth. It has been known for a very long time that the basic well posedness theorem does not extend to the case when ∂ Ω is non-smooth . This can be immediately seenfrom the following example. Example . Let us assume that Ω is the unit square, that is Ω = (0 , . If u is smooth, then ∂ x u (0 , 0) = 0 = ∂ y u (0 , f (0 , 0) = ∆ u (0 , 0) = 0. Bychoosing f (0 , = 0, we will thus obtain a solution u that is not smooth.In view of the many practical applications of the basic well-posedness theorem,we want to extend it in some form to non-smooth domains. Assume now Ω ⊂ R n isa polyhedral domain. Exactly what a polyhedral domain means in three dimensionsis subject to debate. In this presentation, we shall use the definition in [14] interms of stratified spaces (we refer to that paper–a version of which paper was firstcirculated in 2004 as an IMA preprint–for the exact definition). The key technicalpoint in that paper is to replace the classical Sobolev spaces H m (Ω), introduced in Equation (33) with weighted versions as in Kondratiev’s paper [73]. Let us thendenote by ρ the distance function to the singular part of the boundary and define K ma (Ω) := { u, ρ | α |− a ∂ α u ∈ L (Ω) , | α | ≤ m } . (Notice the appearance of the factor ρ !) Thus, in two dimensions, ρ ( x ) is thedistance from x ∈ Ω to the vertices of Ω, whereas in three dimensions, ρ ( x ) is thedistance from x ∈ Ω to the set of edges of Ω. Theorem 5.13. Let Ω ⊂ R n be a bounded polyhedral domain and m ∈ Z + . Thenthere exists η > such that ∆ : K m +1 a +1 (Ω) ∩ { u | ∂ Ω = 0 } → K m − a − (Ω) , is an isomorphism for all | a | < η . In two dimensions, this result is due to Kondratiev [73].The proof of Theorem 5.13 is based on a study of the properties of a Lie manifoldwith boundary Σ(Ω) canonically associated to Ω by a blow-up procedure. Theweighted Sobolev spaces K ma (Ω) can be shown to coincide with the usual Sobolevspaces associated to Σ(Ω). See [6] for the definition of Lie manifolds with boundary.General blow-up procedures for Lie manifolds were studied in [5]. It can be shownthat the class of Lie manifolds satisfying Theorem 4.14 is closed under blow-ups withrespect to tame Lie submanifolds. Since most practical applications deal with Liemanifolds that are obtained by such a blow-up procedure from a smooth manifold,that establishes Theorem 4.14 in most cases of interest.The blow-up procedure is an inductive procedure that consists in successively re-placing cones of the form CL := [0 , ǫ ) × L/ ( { } × L ) with their associated cylinders[0 , ǫ ) × L. No well posedness result similar to Theorem 5.13 holds for the Neumann problem (normal derivative at the boundary is zero):(47) ( − ∆ u = f∂ ν u = 0 , where ν is a continuous unit normal vector field at the boundary. In fact, in threedimensions, the above problem is never Fredholm.Here is however a variant of Theorem 5.13 that has been proved proved useful inpractice. Let us consider a polygonal domain Ω and, for each vertex P of Ω, let usconsider a function χ P ∈ C ∞ ( R ) that is equal to 1 around the vertex P , dependsonly on the distance to P , and has small support. Let W s be the linear span of thefunctions χ P , where P ranges through the set of vertices of Ω. Let { } ⊥ be thespace of functions with integral zero. Then we have the following result [87]. Theorem 5.14. Let Ω ⊂ R be a connected, bounded polygonal domain and m ∈ Z + .Then there exists η > such that ∆ : (cid:16) K m +1 a +1 (Ω) ∩ { ∂ ν u = 0 } + W s (cid:17) ∩ { } ⊥ → K m − a − (Ω) ∩ { } ⊥ , is an isomorphism for all < a < η . The proof of this theorem is based on an index theorem on polygonal domains,more precisely, a relative index theorem as follows. NALYSIS ON SINGULAR SPACES 37 Proof. (Sketch) Let us denote by ∆ a the operator for a fixed value of the weight a .Then one knows by [73] (or an analysis similar to the one needed for the APS indexformula), that ∆ a is Fredholm if, and only if, a = kπ/α , where k ∈ Z and α rangesthrough the values of the angles of our domain Ω. (For the Dirichlet problem onehas a similar condition for a , except that k = 0.) 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