EELLIPTIC OPERATORS AND K -HOMOLOGY ANNA DUWENIG
Abstract.
If a differential operator D on a smooth Hermitian vector bundle S over a compact manifold M is symmetric, it is essentially self-adjoint and soadmits the use of functional calculus. If D is also elliptic, then the Hilbertspace of square integrable sections of S with the canonical left C ( M ) -actionand the operator χ ( D ) for χ a normalizing function is a Fredholm module,and its K-homology class is independent of χ . In this expository article, weprovide a detailed proof of this fact following the outline in the book “AnalyticK-homology” by Higson and Roe. Introduction
A differential operator D acting on the sections of a smooth Hermitian vectorbundle S π → M over a compact manifold M can be regarded as an unbounded op-erator on the Hilbert space L ( M ; S ) of square integrable sections of S . If D issymmetric, then it is automatically essentially self-adjoint and hence we can usefunctional calculus. If D is also elliptic, then L ( M ; S ) with the canonical left C ( M ) -action by multiplication and the operator χ ( D ) for χ a normalizing func-tion turns out to be a Fredholm module over C ( M ) , whose K-homology class [ D ] is independent of the choice of χ . The goal of this paper is to give the details of theproof of [7, Thm. 10.6.5] in the compact case in order to make it more accessible.In particular, we compile the definitions and constructions from [7, § §
9] thatare needed to understand the theorem, we elaborate on aspects which are sparseon details (notably the proofs of Propositions 10.3.1, 10.3.5, 10.6.2 in [7]), and weprovide complete solutions to two crucial steps, namely [7, Exercise 10.9.1] and [7,Exercise 10.9.3].We start in Section 2 by defining the K-homology groups of a C ∗ -algebra A .These groups consist of equivalence classes of triples ( ν, H , F ) where ν is a repre-sentation of A on the Hilbert space H and F is a bounded operator on H withadditional properties. If A is unital, these can be stated as: F is essentially self-adjoint, is essentially unitary, and essentially commutes with the left A -action.In Section 3, we construct the Cayley Transform for densely defined self-adjointunbounded operators. We conclude that these operators allow the use of functionalcalculus.In Section 4, we first survey differential operators and prove some of their prop-erties; for example, what their commutator with a multiplication operator lookslike and that we can define their symbol independently of the choice of charts.In Subsection 4.2, we study Sobolov spaces in order to make sense of (but not Date : September 5, 2019.2010
Mathematics Subject Classification.
Primary 19K33; Secondary 58Jxx, 46Fxx.
Key words and phrases.
K-Homology, Fredholm modules, distributional Fourier Transform.The author would like to thank an anonymous referee for helpful remarks. a r X i v : . [ m a t h . K T ] S e p ANNA DUWENIG prove) G˚arding’s Inequality. In Subsection 4.3, we prove the existence of normaliz-ing functions whose distributional Fourier Transforms are supported in arbitrarilysmall intervals around 0, as is stated in [7, Exercise 10.9.3]. This is needed to showthat χ ( D ) essentially commutes with the left action.At this point, we are equipped to dive into the proof of the main theorem, cf.Theorem 5.1, which is the content of Section 5.The appendix contains a detailed proof of the existence of Friedrichs’ mollifiers,cf. [7, Exercise 10.9.1]. This tool is important to show that D is essentially self-adjoint, so that it makes sense to consider F = χ ( D ) in the main theorem.We should point out that the assumption that D be elliptic is needed solelyto invoke G˚arding’s Inequality. Therefore, we will not dwell upon ellipticity of D ,despite it being crucial for the construction of the K-homology class [ D ] and despitethe title of this paper. 2. Kasparov’s K -homology Gradings.
The material of this subsection is from [7, Appendix A].A Z / Z -grading of a vector space V is a direct sum decomposition into twosubspaces V = V + ⊕ V − , the vectorspace’s even and odd part. We will often just saythat V is graded . Equivalently, V is equipped with a vector space automorphism γ such that γ = id V , and we obtain the decomposition as V ± = { v ∈ V ∣ γ ( v ) = ± v } .An element v ∈ V is called homogeneous if it is in one of these two subspaces, andits degree is defined by ∂v = { v ∈ V + , v ∈ V − . We define V op to be V as vector space but with reversed grading, that is, ( V op ) ± ∶= V ∓ . For an endomorphism T of V , we write T op when we consider it as an endomorphismof V op . The direct sum of two graded spaces V, W is equipped with the grading ( V ⊕ W ) + ∶= V + ⊕ W + and ( V ⊕ W ) − ∶= V − ⊕ W − . A Hilbert space is graded if it is graded as a vector space, and its even and oddsubspaces are closed and mutually orthogonal. Equivalently, the grading automor-phism γ is a bounded unitary operator. Example
1: A grading of a Hilbert space H induces a grading on B(H) by deemingan operator T even (resp. odd) if T preserves (resp. reverses) the two subspaces.In terms of the grading operator γ , T is even (resp. odd) if and only if T ○ γ = γ ○ T (resp. T ○ γ = − γ ○ T ). If we think of B(H) as ( B(H + ) B(H − , H + )B(H + , H − ) B(H − ) ) , we see that T even ⇐⇒ T = ( ∗ ∗ ) , and T odd ⇐⇒ T = ( ∗∗ ) . Moreover,
B(H) ± ○ B(H) ± ⊆ B(H) + and B(H) ± ○ B(H) ∓ ⊆ B(H) − , (1) LLIPTIC OPERATORS AND K-HOMOLOGY 3 and the adjoint preserves the grading: T is even (resp. odd) if and only if T ∗ iseven (resp. odd). This makes B(H) graded as a C ∗ -algebra . Example C ∗ -algebra is the com-plex Clifford algebra : the complex unital ∗ -algebra C n is generated by n ele-ments ε , . . . , ε n which satisfy ε i ε j + ε j ε i = i ≠ j, ε ∗ i = − ε i , and ε i = − . (2)By deeming the basis { ε j ⋯ ε j k ∶ j < . . . < j k , ≤ k ≤ n } orthonormal, C n becomes aHilbert space. The left-action by multiplication is then a faithful ∗ -representationof C n on C n , which makes it a C ∗ -algebra. An element ε j ⋯ ε j k is regarded as even(resp. odd) if k is even (resp. odd).2.2. Fredholm modules.
For a separable C ∗ -algebra A , we recall the followingdefinitions from [7, § § Definition
1: A
Fredholm module over A is a triple ( ν, H , F ) consisting of(1) a representation ν ∶ A → B(H) on a separable Hilbert space H , and(2) an operator F ∈ B(H) such that ν ( a )( F ∗ − F ) , ν ( a )( F − ) , [ ν ( a ) , F ] are compact for all a ∈ A .It is sometimes helpful to be more precise and call such triples ungraded or odd Fredholm modules, in order to distinguish them from graded (sometimes also called even ) Fredholm modules : Definition
2: A graded Fredholm module is a Fredholm module ( ν, H , F ) over A such that(3) H is Z / Z -graded,(4) the operator F is odd, and all operators ν ( a ) are even. Definition
3: (1) Two (graded) Fredholm modules are called unitarily equiva-lent if there exists a (grading preserving) unitary isomorphism U betweenthe Hilbert spaces which intertwines the representations of A and the dis-tinguished bounded operators.(2) An operator homotopy between (graded) Fredholm modules ( ν, H , F ) and ( ν, H , F ) is a family {( ν, H , F t )} t ∈[ , ] of (graded) Fredholm modules suchthat [ , ] → B(H) , t ↦ F t , is norm continuous.(3) We say that a (graded) Fredholm module ( ν, H , F ′ ) is a compact perturba-tion of a (graded) Fredholm module ( ν, H , F ) if the operator ν ( a )( F − F ′ ) is compact for all a ∈ A .(4) A (graded) Fredholm module ( ν, H , F ) is called degenerate if the operators ν ( a )( F ∗ − F ) , ν ( a )( F − ) , [ ν ( a ) , F ] are zero, and not just compact, for all a ∈ A . Proposition 2.1.
Two ( graded ) Fredholm modules which are compact perturbationsof one another are operator homotopic.
ANNA DUWENIG
Proof.
The straight line from the operator F to its compact perturbation F ′ , givenby F t ∶= ( − t ) F + tF ′ for t ∈ [ , ] , can be quickly checked to give the claimedhomotopy of (graded) Fredholm modules. (cid:3) The K-Homology groups.
Since the sum of two (graded) Fredholm modules–given by the direct sum of Hilbert spaces, of representation, and of operators– isagain a (graded) Fredholm module, we arrive at the following definition for theK-homology groups:
Definition C ∗ -algebra A , let K ( A ) be theabelian group with one generator [ x ] for each unitary equivalence class of graded Fredholm modules over A , subject to the following relations:(1) If x, y are two such Fredholm modules, then [ x ⊕ y ] = [ x ] + [ y ] , and(2) two operator homotopic modules give the same class.Similarly, let K ( A ) be the abelian group with one generator for each unitary equiv-alence class of ungraded Fredholm modules over A , subject to the same relations. Remark x = ( ν, H , F ) and y = ( ν op , H op , − F op ) is homotopic to a degenerate. Sincethe degenerate modules are zero in K j ( A ) for j = ,
1, we conclude that the classes [ x ] and [ y ] in K-homology are each other’s additive inverse. Consequently, everyelement of K j ( A ) can be represented by a single (graded) Fredholm module. Remark ( ν, H , F ) for which ν ( A )H is dense in H .In view of Definition 3, we will refrain from calling such Fredholm modules “non-degenerate”. But in the case where A is unital, we can call them unital Fredholmmodules, since ν ( A )H = H is then equivalent to ν ( A ) = id H . Remark − p ( A ) of A for p >
0, built from Fredholm mod-ules with the additional datum of a p -multigrading on the Hilbert space. Thiscollection of groups satisfy Bott periodicity, that is, there exists an isomorphismK − p ( A ) → K − p − ( A ) . For our purposes, it will be sufficient to focus on K and K .For B another separable C ∗ -algebra and a unital ∗ -homomorphism α ∶ B → A ,we can turn a (graded) Fredholm module ( ν, H , F ) over A into one over B byconsidering ( ν ○ α, H , F ) . This process respects addition and unitary equivalence,and hence descends to a map on the level of K -homology,K j ( α ) = α ∗ ∶ K j ( A ) → K j ( B ) , j = , . It is easily checked that the assignment A ↦ K j ( A ) , α ↦ α ∗ , is a contravariant func-tor from the category of separable C ∗ -algebras to the category of abelian groups.3. Unbounded operators
LLIPTIC OPERATORS AND K-HOMOLOGY 5
Terminology. An unbounded operator D on a Hilbert space H is a linear mapfrom a subspace dom D ⊆ H into H . If dom D is dense, then letdom D ∗ ∶ = { η ∈ H ∣ dom D ∋ ξ ↦ ⟨ Dξ ∣ η ⟩ is bounded }= { η ∈ H ∣ ∃ χ ∈ H ∶ ∀ ξ ∈ dom D ∶ ⟨ Dξ ∣ η ⟩ = ⟨ ξ ∣ χ ⟩} . For η ∈ dom D ∗ , define D ∗ η to be the unique vector such that ⟨ Dξ ∣ η ⟩ = ⟨ ξ ∣ D ∗ η ⟩ for all ξ ∈ dom D . The operator D ∗ is linear on its domain, and is called the adjoint of D .The operator D is called ...... closed if the graph of D is a closed subset of H ⊕ H .... closable if the closure of its graph is the graph of a function. This functionis then the closure D of D .... an extension of an unbounded operator D ′ if dom D ′ ⊆ dom D and D = D ′ on dom D ′ .... symmetric if ⟨ Dξ ∣ η ⟩ = ⟨ ξ ∣ Dη ⟩ for all ξ, η ∈ dom D ; in other words, if D ∗ extends D .... self-adjoint if dom D ∗ = dom D and D ∗ ξ = Dξ for all ξ ∈ dom D .... essentially self-adjoint if D is symmetric and dom D = dom D ∗ .Note that every symmetric operator D is closable, and satisfies ⟨ Dξ ∣ ξ ⟩ ∈ R for ξ ∈ dom D . Moreover, for such D , dom D is sometimes called the minimal domain of D and dom D ∗ the maximal domain of D . Example
H = L ( R ) , the assign-ment Df = − i ∂f∂r with domain C ∞ ( R ) is an unbounded operator which is symmetricand hence closable (see [9, Chapter VIII, Section 2]). A slight variant of this ex-ample is the unbounded symmetric operator Df = − i ∂f∂θ on L ( T ) with domain C ∞ ( T ) .Many other examples of unbounded operators on Hilbert spaces can be found in[9, Chapter VIII], including some pathological ones like the last example in Section 3and Problem 4: both discuss symmetric operators, one with uncountably many andone with no self-adjoint extensions.3.2. The Cayley Transform and Borel functional calculus.Lemma 3.1 ([9, Thm. VIII.3]; [1, I.7.3.3]) . If D is a symmetric and densely de-fined unbounded operator on H , then D is self-adjoint if and only if D ± i are bothsurjective. Moreover, in that case, ( D ± i ) − is everywhere defined and a boundedoperator.Proof. Regarding the equivalence, we will actually only be interested in the forwardimplication, so let us disregard the proof of the other direction. We will follow theexplanation given in [1, I.7.3.3].As D is self-adjoint, it is closed and the domains of ( D ± i ) ∗ and D ∓ i bothcoincide with dom D . Since for all ξ, η ∈ dom D , ⟨( D ± i ) ξ ∣ η ⟩ = ⟨ Dξ ∣ η ⟩ ± ⟨ i ξ ∣ η ⟩ = ⟨ ξ ∣ Dη ⟩ ∓ ⟨ ξ ∣ i η ⟩ = ⟨ ξ ∣ ( D ∓ i ) η ⟩ , we see that D ∓ i satisfies the universal property that determines ( D ± i ) ∗ uniquely,so D ∓ i = ( D ± i ) ∗ . ANNA DUWENIG
Claim 1.
For ξ ∈ dom D , we have ∥( D ± i ) ξ ∥ ≥ ∥ ξ ∥ , so that D ± i is bounded belowby . In particular, D ± i is injective.Proof of claim. For ξ ∈ dom ( D ± i ) = dom D , we have because of D = D ∗ ∥( D ± i ) ξ ∥ = ∥ Dξ ∥ ± ⟨ i ξ ∣ Dξ ⟩ ± ⟨ Dξ ∣ i ξ ⟩ + ∥ i ξ ∥ = ∥ Dξ ∥ + ∥ ξ ∥ . (3)Injectivity is now clear. (cid:3) Claim 2.
Since D ± i is bounded below and D is closed, the range of D ± i is closed.Proof of claim. A straightforward computation shows that D ± i is closed be-cause D is. If T ξ n → η for T ∶= D ± i and some ξ n ∈ dom T = dom D , then ( T ξ n ) n is a Cauchy sequence. The previous claim shows ∥ T ( ξ n − ξ m )∥ ≥ ∥ ξ n − ξ m ∥ , so we see that ( ξ n ) n is also Cauchy and hence converges to some ξ . As T is closedand ( ξ n , T ξ n ) n is a sequence in its graph that converges, we must have ξ ∈ dom T and T ξ n → T ξ . (cid:3) Claim 3. D ± i has dense range.Proof of claim. If ξ ∈ range ( D ± i ) ⊥ , then ⟨( D ± i ) ν ∣ ξ ⟩ = = ⟨ ν ∣ ⟩ for all ν ∈ dom D .In particular, ξ is in dom ( D ± i ) ∗ with 0 = ( D ± i ) ∗ ξ = ( D ∓ i ) ξ . Thus,range ( D ± i ) = range ( D ± i ) ⊥⊥ ⊃ ker ( D ∓ i ) ⊥ . Since ker ( D ∓ i ) = { } by Claim 1, D ± i thus indeed has dense range. (cid:3) All in all, we have shown that D ± i is both injective on dom D and surjec-tive. Therefore, there exists a linear map ( D ± i ) − ∶ H → dom D ⊆ H which is inverseto D ± i . Lastly, since ∥( D ± i ) ξ ∥ ≥ ∥ ξ ∥ , we conclude ∥( D ± i ) − ∥ ≤ (cid:3) Definition D a densely defined unbounded operator on H , the spectrum σ ( D ) of D is defined as σ ( D ) ∶= C ∖ { z ∈ C ∣ D − z is injective on dom ( D − z ) = dom D with dense range, and ( D − z ) − is bounded } . Remark
4: Note that the proof of Lemma 3.1 also works for any other z ∈ C ∖ R inplace of i . Thus we have shown that, if D is self-adjoint, σ ( D ) ⊆ R . Also, it followsfrom Claim 2 that, if D is closed and D − z is bounded below, then D − z has closedrange. So if z ∉ σ ( D ) , then the range of D − z is all of H . Definition c ∶ R → S ∖ { } , c ( t ) = t + i t − i , with inverse c − ( z ) = i z + i z − i . If D is a densely defined self-adjoint operator on H , then Lemma 3.1 shows that itmakes sense to define c ( D ) ∶= ( D + i )( D − i ) − ∶ H → H , and that this map is an isomorphism of H . It is called the Cayley Transform of D .From Equation (3), we see that ∥( D + i ) ξ ∥ = ∥( D − i ) ξ ∥ , so c ( D ) is even a unitary.Moreover, it does not have 1 in its spectrum: if c ( D ) ξ = ξ , then for ξ ′ = ( D − i ) − ξ LLIPTIC OPERATORS AND K-HOMOLOGY 7 we have ( D + i ) ξ ′ = ( D − i ) ξ ′ , that is i ξ ′ = − i ξ ′ . Thus, ξ ′ = ξ =
0, so thatwe have shown that c ( D ) − η ∈ H is arbitrary,let ξ ∶= − i ( D − i ) η and compute ( c ( D ) − ) ξ = ( D + i )( D − i ) − ξ − ξ = − i ( D + i ) η + i ( D − i ) η = η, so we have shown that c ( D ) − U is a unitary which does not have 1 as eigenvalue, then U − ξ ∈ range ( U − ) ⊥ , then ⟨( U − ) η ∣ ξ ⟩ = η ∈ H , so ( U ∗ − ) ξ =
0. Injectivity of U − ξ =
0. Therefore, the so-called inverse Cayley Transform of U defined by c − ( U ) ∶= i ( U + )( U − ) − ∶ range ( U − ) → H , is densely defined. Lemma 3.2 ([4, 3.5. Corollary]) . The inverse Cayley Transform of a unitary whichdoes not have as eigenvalue is a self-adjoint operator.Proof. A quick computation shows that c − ( U ) is symmetric, so we only need tocheck that the domain of its adjoint is contained in range ( U − ) . If ξ ∈ dom ( c − ( U )) ∗ ,then there exists η ∈ H such that for all ν ′ ∈ range ( U − ) , we have ⟨ c − ( U ) ν ′ ∣ ξ ⟩ = ⟨ ν ′ ∣ η ⟩ . In other words, for every ν ′ = ( U − ) ν , ⟨ i ( U + ) ν ∣ ξ ⟩ = ⟨( U − ) ν ∣ η ⟩ . Since this holds for every ν ∈ H , it follows that − i ( U ∗ + ) ξ = ( U ∗ − ) η . By applying i U to both sides, we get ξ + U ξ = ( + U ) ξ = ( − U ) i η = i η − U i η. Rearranging and adding ξ to both sides yields2 ξ = ( i η − U i η − U ξ ) + ξ = ( − U )( i η + ξ ) , so ξ ∈ range ( U − ) as claimed. (cid:3) If 1 ∉ σ ( U ) , then it follows from our comment in Remark 4 that c − ( U ) isactually everywhere defined and bounded. One can check that c − ( c ( D )) = D and c ( c − ( U )) = U, so we have found: Proposition 3.3 ([4, 3.5. Corollary; 3.1. Theorem]) . The Cayley Transform is a bi-jective map from the densely defined, self-adjoint operators to the unitary operatorswhich do not have as eigenvalue. The Cayley Transform makes it possible to extend the Borel functional calculusfor normal operators to densely defined, self-adjoint operators. It has the followingproperties:
ANNA DUWENIG
Proposition 3.4 (Functional Calculus; [1, I.7.4.5. Thm, I.7.4.7. Def.]) . For D adensely defined, self-adjoint operator on H , there exists a linear map { h ∶ R → C Borel measurable } —→ { densely defined unbounded operators on H} h z→ h ( D ) with the following properties: (1) id R ( D ) = D . (2) If h ≥ , then h ( D ) is positive. (3) If ∣ h ∣ = , then h ( D ) is unitary. (4) h ( D ) ∗ = h ( D ) ; in particular, if h is real-valued, then h ( D ) is self-adjoint. (5) If h is bounded and continuous, then ∥ h ( D )∥ = ∥ h ∥ ∞ . (6) If h n is a uniformly bounded sequence of functions which converges point-wise to h , then h n ( D ) → h ( D ) strongly. Lemma 3.5 (special case of [7, Lemma 10.6.2]) . Suppose D is an unbounded, es-sentially self-adjoint operator on H , and T ∈ B(H) preserves dom D and satis-fies T D = − DT . If f ∈ C b ( R ) is odd, then T f ( D ) = − f ( D ) T , and if f is even,then T f ( D ) = f ( D ) T .Proof. Let us first set some notation: the decomposition of a function f into itseven and odd part is given by f e ( x ) = f ( x ) + f (− x ) f o ( x ) = f ( x ) − f (− x ) , so that f = f e + f o . Let us denote by ˜ f ( x ) ∶= f e − f o = f (− x ) . The claim can now be rephrased to
T f ( D ) = ˜ f ( D ) T . In other words, T gradedcommutes with f ( D ) when C b ( R ) has the Z / Z -grading into even and odd func-tions.Claim 1. It suffices to show the claim for elements of C ( R ) .Proof of claim. For f ∈ C b ( R ) , take functions f n ∈ C ( R ) converging pointwise to f . By Property (6) of Functional Calculus, we have strong convergence f n ( D ) → f ( D ) and also ˜ f n ( D ) → ˜ f ( D ) , so for every h ∈ H , we get T f ( D ) h = T ( lim n →∞ f n ( D ) h ) = lim n →∞ T f n ( D ) h = lim n →∞ ˜ f n ( D ) T h = ˜ f ( D ) T h, where we used the assumption that
T f n ( D ) = ˜ f n ( D ) T (cid:3) By the Stone-Weierstrass Theorem [2], either of the functions ψ ± ( x ) ∶= i ± x = ψ ∓ ( x ) generate C ( R ) as a C ∗ -algebra.Claim 2. It suffices to show that T graded commutes with ψ ± ( D ) . LLIPTIC OPERATORS AND K-HOMOLOGY 9
Proof of claim.
For a fixed f ∈ C ( R ) , assume ψ = ∑ n,k ∈ N × a n,k ψ n + ( ψ + ) k = ∑ n,k ∈ N × a n,k ψ n + ψ k − is such that ∥ f − ψ ∥ ∞ < (cid:15). The properties of continuous functional calculus shows that, if T graded com-mutes with g ( D ) for g some continuous function, then it also graded commuteswith g n ( D ) for positive powers of g . Thus, we have T ψ n ± ( D ) = ( ˜ ψ ± ) n ( D ) T byassumption, which implies ∥ T f ( D ) − ˜ f ( D ) T ∥ ≤ ∥ T f ( D ) − T ψ ( D )∥ + ∥ T ψ ( D ) − ˜ f ( D ) T ∥= ∥ T ( f − ψ )( D )∥ + ∥ ˜ ψ ( D ) T − ˜ f ( D ) T ∥≤ ∥ T ∥ ⋅ ∥ f − ψ ∥ ∞ + ∥ ˜ ψ − ˜ f ∥ ∞ ⋅ ∥ T ∥ < (cid:15) ∥ T ∥ . Since this is possible for any (cid:15) , this implies
T f ( D ) = ˜ f ( D ) T as wanted. (cid:3) As ( i ± D ) T = T ( i ∓ D ) by assumption, we get T ψ ∓ ( D ) = ψ ± ( D ) T. (4)Since ψ ± (− x ) = ψ ∓ ( x ) , we can see that ψ e + = ψ e − and ψ o + = − ψ o − . As a consequence,2 ( ψ + + ψ − ) = ( ψ e + + ψ o + ) + ( ψ e − + ψ o − ) = ψ e + , so ψ + + ψ − = ψ e + , and 2 ( ψ + − ψ − ) = ( ψ e + + ψ o + ) − ( ψ e − + ψ o − ) = ψ o + , so ψ + − ψ − = ψ o + . From Equation (4), it thus follows that
T ψ e + ( D ) = T ( ψ + + ψ − )( D ) = ( ψ − + ψ + )( D ) T = ψ e + ( D ) T and T ψ o + ( D ) = T ( ψ + − ψ − )( D ) = ( ψ − − ψ + )( D ) T = − ψ o + ( D ) T. In other words, T graded commutes with ψ + ( D ) . (cid:3) Elliptic operators
Notation : We will write λ for Lebesgue measure on R n , ∥ ⋅ ∥ C k for the Euclideannorm on C k , and ∥ ⋅ ∥ for L -norms. Definition
7: A vector bundle S π → M over a smooth manifold M is called smooth if S is also a manifold and π is a smooth map. We write Γ ( M ; S ) for the sectionsof this bundle, i.e. Γ ( M ; S ) ∶= { v ∶ M → S ∣ v p ∈ S p for all p ∈ M } , and we write Γ ∞ ( M ; S ) resp. Γ c ( M ; S ) for the smooth resp. compactly supportedsections.A smooth vector bundle S π → M is called Hermitian if, for each p ∈ M , thereis an inner product (⋅ ∣ ⋅) S p on the fibre S p ∶= π − ( p ) , and these inner products varysmoothly : for every u, v ∈ Γ ∞ ( M ; S ) , the map M ∋ p ↦ ( u ( p ) ∣ v ( p )) S p ∈ C is smooth. In the following, we will fix a smooth Hermitian complex vector bundle S π → M of rank k over a smooth manifold M of dimension n . Let us denote the norminduced by the inner product (⋅ ∣ ⋅) S p on S p by ∥ ⋅ ∥ S p . An example to keep in mindis the case where M is spin c and S is its spinor bundle.We further assume that we are given a nowhere-vanishing smooth measure µ on M , that is, µ is a Borel measure such that for every chart ( U, ϕ ) of M , there existsa smooth function f ∶ ϕ ( U ) → ( , ∞) such that d ( ϕ ∗ µ U ) = f d λ ϕ ( U ) . This meansfor a ( ϕ ∗ µ U ) -integrable function h ∶ ϕ ( U ) → C that ∫ U h ○ ϕ d µ = ∫ ϕ ( U ) h ⋅ f d λ. Moreover, since f does not vanish, we can also consider g = f and get for λ ϕ ( U ) -integrable h ∫ U ( h ⋅ g ) ○ ϕ d µ = ∫ ϕ ( U ) h d λ. (5) Remark
5: For technical reason, there will be the standing assumption that thereexists a number L so that we have for all of the above mentioned Radon-Nikodymderivatives the inequality ∥ f ∥ ∞ , ∥ g ∥ ∞ ≤ L .We construct the Hilbert space L ( M ; S ) as the completion of Γ ∞ c ( M ; S ) withrespect to the norm coming from the inner product ⟨ u ∣ v ⟩ ∶= ∫ M ( u ( p ) ∣ v ( p )) S p d µ ( p ) . For a subset U ⊆ M , we will write L ( U ; S ) for the completion of the smoothsections whose compact support is contained in U . Lastly, let M ∶ C ( M ) → B( L ( M ; S )) , g ↦ M g , be the representation of C ( M ) which, on the dense subspace Γ ∞ c ( M ; S ) , is givenby pointwise multiplication.4.1. Differential operators.
Definition
8: A ( first order linear ) differential operator acting on the sections of S is a C -linear map D ∶ Γ ∞ ( M ; S ) → Γ ∞ ( M ; S ) such that a): if u, v ∈ Γ ∞ ( M ; S ) agree on an open set U , then Du, Dv also agree on U ,and b): for a coordinate chart of M that also trivializes S , say S ∣ U U × C k C k M ⊇ U V ⊆ R nψπ ↺ ≈ Ψpr pr ≈ ϕ (6) LLIPTIC OPERATORS AND K-HOMOLOGY 11 there exist functions A , . . . , A n , B ∈ C ∞ ( U, M k ( C )) such that for all p ∈ U and all u ∈ Γ ∞ ( M ; S ) , we have ( Du )( p ) = n ∑ j = Ψ − ( p, A j ( p ) ⋅ ∂ j ( ψ ○ u ○ ϕ − ) ∣ ϕ ( p ) )+ Ψ − ( p, B ( p ) ⋅ ( ψ ○ u )( p )) . (7)We will from now on regard such a differential operator as an unbounded oper-ator on L ( M ; S ) with dense domain Γ ∞ c ( M ; S ) . By abuse of terminology, we willsay “differential operator on M ”, tacitly assuming a fixed Hermitian bundle S . Example § S over a manifold M , consisting of Clifford T M -modules andequipped with a connection, ∇ . Then one can locally define a Dirac operator / ∂ M by ( / ∂ M u )( p ) = n ∑ i = c p ( ∂∂ϕ i ∣ p ) ⋅ ∇ ∂∂ϕ i ( u ) p , (8)where u is a smooth compactly supported section of S , ϕ is a chart of M around p ,and c denotes the Clifford action of the tangent vector ∂∂ϕ i on S . We immediatelysee that / ∂ M is a first order differential operator. One can further show (see [10,Prop. 3.11]) that / ∂ M is symmetric.A bundle S with such structure would be the spinor bundle of a spin c manifold.In the example M = T with its canonical spin c structure, the spinor bundle is thetrivial line bundle, T × C , so the domain, Γ ∞ c ( M ; S ) , of / ∂ T = − i ∂∂θ is then just C ∞ ( T ) , smooth functions on the circle. Lemma 4.1.
Let D be a symmetric differential operator on M and let u ∈ dom D ∗ have compact support K . Then the support of D ∗ u is contained in K .Proof. Let w k be a sequence in dom D = Γ ∞ c ( M ; S ) which converges to u in L -norm.If we take K = ⋂ ∞ k = V k for open nested sets V k + ⊆ V k ⊆ M (see Lemma 5.11 for aconstruction), Urysohn gives us smooth [ , ] -valued functions ρ k with supp ( ρ k ) ⊆ V k which are 1 on K . Note that u k ∶= ρ k ⋅ w k is also in dom D , and since u issupported in K , we see ∥ u − u k ∥ = ∫ K ∥ u ( p ) − u k ( p )∥ S p d µ + ∫ M ∖ K ∥ u k ( p )∥ S p d µ ≤ ∫ K ∥ u ( p ) − w k ( p )∥ S p d µ + ∫ M ∖ K ∥ w k ( p )∥ S p d µ = ∥ u − w k ∥ , so u k also converges to u . As u k is supported in V k , we get from Property a) ofdifferential operators that Du k is supported in V k , too. We know that Du k = D ∗ u k converges to D ∗ u in L -norm, so by choosing an appropriate subsequence, we can assume (∗) in the following computation:1 k (∗) > ∥ D ∗ u − Du k ∥ = ∫ V k ∥ D ∗ u ( p ) − Du k ( p )∥ S p d µ + ∫ M ∖ V k ∥ D ∗ u ( p )∥ S p d µ ≥ ∫ M ∖ V k ∥ D ∗ u ( p )∥ S p d µ. Now, note that V k + m ⊆ V k for any m , and hence ∫ M ∖ V k ∥ D ∗ u ( p )∥ S p d µ ≤ ∫ M ∖ V k + m ∥ D ∗ u ( p )∥ S p d µ < k + m . It follows that ∫ M ∖ V k ∥ D ∗ u ( p )∥ S p d µ = k , and as M ∖ K = ⋃ k M ∖ V k , ∫ M ∖ K ∥ D ∗ u ( p )∥ S p d µ ≤ ∑ k ∫ M ∖ V k ∥ D ∗ u ( p )∥ S p d µ = . We conclude that D ∗ u is also supported in K . (cid:3) Lemma 4.2. If D is a differential operator on M which is locally given by Equa-tion (7) , and if g ∈ C ∞ ( M ) , then [ D, M g ] can locally be written as [ D, M g ] u ( p ) = n ∑ j = ∂ j ( g ○ ϕ − ) ∣ ϕ ( p ) ⋅ Ψ − ( p, A j ( p ) ⋅ ( ψ ○ u ( p ))) . (9) In particular, if K ⊆ M is compact, then [ D, M g ] extends to a bounded operator on L ( K ; S ) .Proof. It suffices to consider those D that locally look like only one of the summandsin Equation (7). Given a chart ( U, ϕ ) and a trivialization Ψ of S , if ( Du )( p ) = Ψ − ( p, B ( p ) ⋅ ( ψ ○ u )( p )) , B ∈ C ∞ ( U, M k ( C )) , then D is itself only a multiplication operator (albeit by a matrix), and so it in factcommutes with M g . So consider the case in which ( Du )( p ) = Ψ − ( p, A ( p ) ⋅ ∂ j ( ψ ○ u ○ ϕ − ) ∣ ϕ ( p ) ) , A ∈ C ∞ ( U, M k ( C )) , for some 1 ≤ j ≤ n . We compute for u ∈ Γ ∞ ( M ; S ) and p ∈ U : [ D, M g ] u ( p ) = D ( gu )( p ) − g ( p )( Du )( p )= Ψ − ( p, A ( p ) ⋅ ∂ j ( ψ ○ ( gu ) ○ ϕ − ) ∣ ϕ ( p ) )− g ( p ) Ψ − ( p, A ( p ) ⋅ ∂ j ( ψ ○ u ○ ϕ − ) ∣ ϕ ( p ) ) . As g ( p ) is just a scalar and Ψ − ( p, ⋅ ) and ψ are linear, we get [ D, M g ] u ( p ) = Ψ − ( p, A ( p ) ⋅ ∂ j (( g ○ ϕ − ) ⋅ ( ψ ○ u ○ ϕ − )) ∣ ϕ ( p ) )− Ψ − ( p, g ( p ) ⋅ A ( p ) ⋅ ∂ j ( ψ ○ u ○ ϕ − ) ∣ ϕ ( p ) ) . By the product rule, ∂ j (( g ○ ϕ − ) ⋅ ( ψ ○ u ○ ϕ − )) ∣ ϕ ( p ) = ∂ j ( g ○ ϕ − ) ∣ ϕ ( p ) ( ψ ○ u ( p )) + g ( p ) ∂ j ( ψ ○ u ○ ϕ − ) ∣ ϕ ( p ) , LLIPTIC OPERATORS AND K-HOMOLOGY 13 so we arrive at [ D, M g ] u ( p ) = Ψ − ( p, ∂ j ( g ○ ϕ − ) ∣ ϕ ( p ) ⋅ A ( p ) ⋅ ( ψ ○ u ( p ))) . (10) (cid:3) Definition
9: The symbol σ D of a differential operator D is the R -vector bundlemorphism σ D ∶ T ∗ M → End ( S ) defined as follows: given a cotangent vector ξ ∈ T ∗ p M at p , take a chart ( U, ϕ ) around p ∈ M and a trivialization of S U as in Diagram (6). Suppose D locally looksas in Equation (7), and write ξ = ∑ nj = ξ j d ϕ jp , where { d ϕ jp } j denotes the basis of T ∗ p M that is dual to the basis { ∂∂ϕ j ∣ p } j of T p M . Then we define for η ∈ S p , σ D ( p, ξ ) η ∶= Ψ − ⎛⎝ p, n ∑ j = ξ j A j ( p ) ψ ( η )⎞⎠ . Remark
6: In Lemma 4.2, we have actually shown that [ D, M g ] u ( p ) = σ D ( p, d g ∣ p )( u ( p )) . Lemma 4.3.
The definition of σ D does not depend on the choice of Ψ or ϕ .Proof. First, assume that Ω is another trivialization of S U , and let ω ∶= pr ○ Ω.Since the fibre maps of both Ψ and Ω are linear isomorphisms, there exists a smoothmap H ∶ U → GL k ( C ) given by C k S p C kH ( p ) Ω ( p, ⋅ )≅ Ψ ( p, ⋅ )≅ . Moreover, we can write D also in the form ( Du )( p ) = n ∑ j = Ω − ( p, E j ( p ) ⋅ ∂ j ( ω ○ u ○ ϕ − ) ∣ ϕ ( p ) )+ Ω − ( p, E ( p ) ⋅ ( ω ○ u )( p )) , for all u ∈ Γ ∞ ( M ; S ) . By clever choices of u and some use of the product rule, onecan conclude that A j ( p ) = H ( p ) E j ( p ) H − ( p ) for each 1 ≤ j ≤ n . Therefore, for any η ∈ S p , A j ( p ) ψ ( η ) = H ( p ) E j ( p ) ω ( η ) and so Ψ − ( p, A j ( p ) ψ ( η )) = Ψ − ( p, H ( p ) E j ( p ) ω ( η )) = Ω − ( p, E j ( p ) ω ( η )) . We see from this that σ D ( p, ξ ) does not depend on the choice of Ψ.Next, let γ be another chart around p . Again, we can write D in the form ( Du )( p ) = n ∑ l = Ψ − ( p, F l ( p ) ⋅ ∂ l ( ψ ○ u ○ γ − ) ∣ γ ( p ) )+ Ψ − ( p, F ( p ) ⋅ ( ψ ○ u )( p )) . We get that n ∑ j = A j ( p ) ⋅ ∂ j ( ψ ○ u ○ ϕ − ) ∣ ϕ ( p ) = n ∑ l = F l ( p ) ⋅ ∂ l ( ψ ○ u ○ γ − ) ∣ γ ( p ) = n ∑ l = F l ( p ) ⋅ n ∑ j = ∂ j ( ψ ○ u ○ ϕ − ) ∣ ϕ ( p ) ∂ l ( ϕ ○ γ − ) j ∣ γ ( p ) , and so another clever choice of u yields A j ( p ) = n ∑ l = ∂ l ( ϕ ○ γ − ) j ∣ γ ( p ) F l ( p ) . Moreover, if ξ = ∑ l ν l d γ lp , then ν l = ξ ( ∂∂γ l ∣ p ) = n ∑ j = ξ j ∂ l ( ϕ ○ γ − ) j ∣ γ ( p ) . Combined, we have for any v ∈ C kn ∑ j = ξ j A j ( p ) v = n ∑ j = ξ j ( n ∑ l = ∂ l ( ϕ ○ γ − ) j ∣ γ ( p ) F l ( p )) v = n ∑ l = ν l F l ( p ) v, and so we conclude that σ D ( p, ξ ) also does not depend on the choice of ϕ . (cid:3) Definition
10: We say that a differential operator is elliptic if its symbol σ D mapseach ( p, ξ ) in T ∗ M with ξ ≠ S p . Example
5: The Dirac operator we mentioned in Example 4 is elliptic: using Equa-tion (8), one can show that its symbol is given by σ / ∂ M ( p, ξ ) = − ∥ ξ ∥ , see [7, 11.1.1 Def.] or [8, Lemma 5.1].4.2. Sobolev Spaces.
We want to construct the Sobolev space associated to ourvector bundle. Recall first that for f ∈ C ∞ c ( R n , C ) , the Sobolev norm is defined by ∥ f ∥ , R n ∶= ∥ f ∥ + n ∑ i = ∥ ∂f∂x i ∥ . Take an atlas of M whose charts are small enough to also allow smooth, fibrewiseisometric trivializations as in Diagram (6). For a compact subset K of M , let {( U i , ϕ i )} li = be a subcover of charts, and denote the corresponding trivialisationsof S by Ψ i = π × ψ i . These induce mapsΨ ∗ i ∶ Γ ∞ ( U i ; S ∣ U i ) → ( C ∞ ( V i )) k which send a section v ∶ U i → S ∣ U i to the map R n ⊇ V i ∋ x ↦ ψ i ( v ( ϕ − i ( x ))) . As explained in Lemma 5.12, we can pick smooth compactly supported functions ρ , . . . , ρ l ∶ M → [ , ] such that supp ( ρ i ) ⊆ U i and l ∑ i = ρ i ( p ) = p ∈ K. LLIPTIC OPERATORS AND K-HOMOLOGY 15
We define for u ∈ Γ ∞ ( K ; S ) (that is, sections of the bundle supported in K ): ∥ u ∥ ∶= l ∑ i = ∥ Ψ ∗ i ( ρ i ⋅ u )∥ , R n . Though this norm relies heavily on the choices involved, its equivalence class doesnot. We define L ( K ; S ) to be the completion of Γ ∞ ( K ; S ) with respect to thisnorm. Let us gather some facts about Sobolev spaces that we will need later: Lemma 4.4.
For K ⊆ M compact, there exists a number c > such that for all u ∈ L ( K ; S ) , ∥ u ∥ ≤ c ∥ u ∥ . Proof.
Since ∥ f ∥ , R n ≥ ∥ f ∥ , we get ∥ u ∥ ≥ l ∑ i = ∥ Ψ ∗ i ( ρ i ⋅ u )∥ . Let f i , g i = f i be as in Equation (5) for ( U i , ϕ i ) . Recall that we assumed inRemark 5 that ∥ f i ∥ ∞ ≤ L for some number L and all i . For v ∈ Γ ∞ c ( U i ; S ∣ U i ) , wehave ∥ Ψ ∗ i ( v )∥ = ∫ R n ∥ ψ i ( v ( ϕ − i ( x )))∥ C k d λ = ∫ U i ∥ ψ i ( v ( p ))∥ C k ( g i ○ ϕ )( p ) d µ, and since ψ i is isometric, we get ∥ Ψ ∗ i ( v )∥ ≥ L ∫ U i ∥ v ( p )∥ S p d µ = L ∥ v ∥ . Thus, ∥ u ∥ ≥ √ L l ∑ i = ∥ ρ i ⋅ u ∥ . Furthermore, ( l ∑ i = ∥ ρ i ⋅ u ∥ ) ≥ l ∑ i = ∥ ρ i ⋅ u ∥ = l ∑ i = ⎛⎜⎝∫ K ρ i ( p ) ∥ u ( p )∥ S p d µ ⎞⎟⎠= ∫ K ( l ∑ i = ρ i ( p ) ) ∥ u ( p )∥ S p d µ ≥ ∫ K l ( l ∑ i = ρ i ( p )) ∥ u ( p )∥ S p d µ = l ∫ K ∥ u ( p )∥ S p d µ = l ∥ u ∥ , so that all in all ∥ u ∥ ≥ √ L ⋅ l ∥ u ∥ . (cid:3) Proposition 4.5 ([12, IV.2.2] - without proof) . Every differential operator D on M has a continuous extension to an operator L ( K ; S ) → L ( K ; S ) where K ⊆ M is any compact subset. Corollary 4.6.
For D a symmetric differential operator on M and K ⊆ M compact, L ( K ; S ) is contained in the domain of D ∗ .Proof. For u ∈ L ( K ; S ) , we need to show that there exists C > ∣⟨ u ∣ Dv ⟩∣ ≤ C ⋅ ∥ v ∥ for all v ∈ Γ ∞ c ( M ; S ) = dom D . Let ( u n ) n be a sequence in Γ ∞ ( K ; S ) converging to u in ∥ ⋅ ∥ . By Proposition 4.5, ( Du n ) n converges in L ( M ; S ) , so the ∥ ⋅ ∥ -norm ofthe sequence is bounded by some number N . For 0 ≠ v , take some big enough n such that ∥ u − u n ∥ ≤ ∥ v ∥ c (∥ Dv ∥ + ) where c > ∣⟨ u ∣ Dv ⟩∣ ≤ ∣⟨ u − u n ∣ Dv ⟩∣ + ∣⟨ u n ∣ Dv ⟩∣ = ∣⟨ u − u n ∣ Dv ⟩∣ + ∣⟨ Du n ∣ v ⟩∣≤ ∥ u − u n ∥ ∥ Dv ∥ + ∥ Du n ∥ ∥ v ∥ < ( + N ) ∥ v ∥ . (cid:3) We will need the following propositions later, but we will not prove them here.
Proposition 4.7 (Rellich Lemma; [7, 10.4.3], [12, IV.1.2] - without proof) . For K ⊆ M compact, the inclusion L ( K ; S ) ↪ L ( K ; S ) is a compact operator. Proposition 4.8 (G˚arding’s Inequality; [7, 10.4.4] - without proof) . Suppose M is compact. If D is an elliptic differential operator on M , then there is a constant c > such that, for all u ∈ L ( M ; S ) , c ⋅ ∥ u ∥ ≤ ∥ u ∥ + ∥ Du ∥ . As mentioned in the introduction, to be able to invoke G˚arding’s Inequality isthe reason why we need to assume ellipticity of D in our main theorem.4.3. Fourier Transforms and Normalizing functions.
Most of the statementsbelow can be found in [5, Chapters 8 and 9].
Notation : For f ∶ R n → C and x, y ∈ R n , let ( τ x f )( y ) ∶= f ( y − x ) and ˜ f ( y ) ∶= f (− y ) . For f ∈ L ( R n ) , its Fourier and inverse Fourier Transform are given byˆ f ( x ) = ∫ R n e − π i x ⋅ y f ( y ) d y and ˇ f ( x ) = ∫ R n e π i x ⋅ y f ( y ) d y. (11)If f, g ∈ L , then ∫ R n ˆ f ( x ) g ( x ) d x = ∫ R n f ( x ) ˆ g ( x ) d x. (12)As a consequence, one can show that if f, ˆ f are both L , then the inversion formula holds: for almost every x ∈ R n , we have f ( x ) = ˇˆ f ( x ) = ∫ R n e π i x ⋅ y ˆ f ( y ) d y. LLIPTIC OPERATORS AND K-HOMOLOGY 17
Definition
11: The
Schwartz space S consists of those smooth functions on R n which have rapidly decaying derivatives. To be more precise, define for N ∈ N anda multi-index α , ∥ φ ∥ N,α ∶= sup x ∈ R n ( + ∥ x ∥) N ∣ ∂ α φ ( x )∣ . (13)Then S ∶= { φ ∈ C ∞ ( R n ) ∣ for any N ∈ N , α multi-index ∶ ∥ φ ∥ N,α < ∞} . When equipped with the seminorms given in Definition 13, S becomes a Fr´echetspace, cf. [5, 8.2. Proposition]. The Fourier Transform then maps S continuouslyinto itself and, because of the inversion formula, is hence an isomorphism of S (cf. [5, 8.28 Cor.]). Definition
12: A distribution F is a functional on C ∞ c ( R n ) . We will denote by ⟪ F , φ ⟫ the value of F at the point φ ∈ C ∞ c ( R n ) , and let D ′ be the space of distri-butions. The support of F is the complement of the maximal open subset U ⊆ R n for which ⟪ F , φ ⟫ = φ such that supp ( φ ) ⊆ U . A distribution F is tempered if it extends con-tinuously to all of S . As C ∞ c ( R n ) is dense in S (cf. [5, 9.9 Prop.]), the space oftempered distributions is the dual space S ′ of S . Example
6: If f ∶ R n → C is locally integrable (that is, integrable on compact sets),then it defines a distribution by ⟪ f , φ ⟫ ∶= ∫ R f ( x ) φ ( x ) d x for φ ∈ C ∞ c ( R n ) .If ψ ∈ C ∞ c ( R n ) , then ∫ R ( f ∗ ψ )( x ) φ ( x ) d x = ∫ R ∫ R f ( x ) ψ ( y − x ) φ ( x ) d y d x = ∫ R f ( x )( φ ∗ ˜ ψ )( x ) d x, and so the above Example justifies the following definition: Definition
13 ([5, p. 285]): If F ∈ D ′ and ψ ∈ C ∞ c ( R n ) , we define for φ ∈ C ∞ c ( R n ) , ⟪ F ∗ ψ , φ ⟫ ∶= ⟪ F , φ ∗ ˜ ψ ⟫ . One can show (see [5, 9.3 Prop.]) that this distribution is actually given by inte-gration against the function F ∗ ψ defined by F ∗ ψ ( x ) ∶= ⟪ F , τ x ˜ ψ ⟫ . Lemma 4.9. If F ∈ D ′ has compact support and if ψ ∈ C ∞ c ( R n ) , the function F ∗ ψ is a smooth compactly supported function.Proof. Regarding smoothness, see [5, 9.3a) Prop.]. If we let A be the closure of supp ( F ) + supp ( ψ ) , then A is compact by assumption. For x ∉ A , the function τ x ˜ ψ ∶ y ↦ ψ ( x − y ) is supported outside of supp ( F ) , so that F ∗ ψ ( x ) = ⟪ F , τ x ˜ ψ ⟫ = . (cid:3) Example f ∶ R → C is measurable and bounded,then it defines a tempered distribution by ⟪ f , φ ⟫ ∶= ∫ R f ( x ) φ ( x ) d x for φ ∈ S . Indeed, since sup x ∈ R ( + ∥ x ∥) ∣ φ ( x )∣ = ∥ φ ∥ , < ∞ , we have ∫ R ∣ f ( x ) φ ( x )∣ d x ≤ ∫ R ∥ f ∥ ∞ ∥ φ ∥ , ( + ∣ x ∣) d x ≤ ∫ R ∥ f ∥ ∞ ∥ φ ∥ , + ∣ x ∣ d x = ∥ f ∥ ∞ ∥ φ ∥ , π < ∞ , so the measurable function f φ is integrable, and ⟪ f , φ ⟫ is well-defined and contin-uous.The advantage of tempered distributions over other distributions is the followingdefinition: Definition
14: If F is a tempered distribution, we define its Fourier and inverseFourier Transform by ⟪ ˆ F , φ ⟫ ∶= ⟪
F , ˆ φ ⟫ , and ⟪ ˇ F , φ ⟫ ∶= ⟪
F , ˇ φ ⟫ for φ ∈ S . Because of Equation (12), we see that, if F is integration against an L -function, then both ˆ F and ˇ F agree with the definition given in Definition 11. More-over, we again have the inversion formula ˆˇ F = ˇˆ F = F . Lemma 4.10.
Suppose we are given an even, integrable function h ∶ R → C . Thenthe assignment pv (∫ h ( t ) t ) ∶ C ∞ c ( R ) —→ C ,ϕ z→ lim (cid:15) → + ⎛⎝ − (cid:15) ∫ −∞ h ( t ) t ϕ ( t ) d t + ∞ ∫ (cid:15) h ( t ) t ϕ ( t ) d t ⎞⎠ , extends continuously to S . Furthermore, the Fourier Transform of this tempereddistribution is given by integration against the ( well-defined ) function ζ ( x ) ∶= ∞ ∫ −∞ sin ( tx ) i t h ( t π ) d t. Proof.
Let ϕ ∈ C ∞ c ( R ) . For any (cid:15) >
0, the following two integrals exist since h isintegrable, and are equal because h is even: − (cid:15) ∫ −∞ h ( t ) t ϕ ( ) d t = − ∞ ∫ (cid:15) h ( t ) t ϕ ( ) d t. LLIPTIC OPERATORS AND K-HOMOLOGY 19
Therefore, we may rewrite ⟪ pv (∫ h ( t ) t ) , ϕ ⟫ = lim (cid:15) → + ⎛⎝ − (cid:15) ∫ −∞ ϕ ( t ) − ϕ ( ) t h ( t ) d t + ∞ ∫ (cid:15) ϕ ( t ) − ϕ ( ) t h ( t ) d t ⎞⎠= ∞ ∫ −∞ ϕ ( t ) − ϕ ( ) t h ( t ) d t, where the last line holds because t ↦ ϕ ( t )− ϕ ( ) t can be smoothly extended at 0 bythe value ϕ ′ ( ) by L’Hˆopital. We therefore get ∣⟪ pv (∫ h ( t ) t ) , ϕ ⟫∣ ≤ ∞ ∫ −∞ ∣ ϕ ( t ) − ϕ ( ) t h ( t )∣ d t ≤ ∥ h ∥ L ⋅ sup t ∈ R ∣ ϕ ( t ) − ϕ ( ) t ∣≤ ∥ h ∥ L ⋅ sup t ∈ R ∣ ϕ ′ ( t )∣ by the Mean Value Theorem. In particular, the value is finite for ϕ ∈ C ∞ c ( R ) andin fact also for ϕ ∈ S . Moreover, given ϕ k ∈ S converging to 0, the above line meansthat ∣⟪ pv (∫ h ( t ) t ) , ϕ k ⟫∣ ≤ ∥ h ∥ L ⋅ ∥ ϕ k ∥ , k →∞ —→ , so we have shown that our functional extends continuously to S .Regarding ζ , first note that the (scaled) sinc function R × ∋ t ↦ sin ( tx ) t can becontinuously extended at 0 by assigning it the value x , and that it is bounded by ∣ x ∣ .Hence, since h is integrable, we see that ζ ( x ) is actually a finite number, so ζ iswell-defined. To check that ζ is the Fourier transform, we can equivalently showthat ˇ ζ = pv (∫ h ( t ) t ) , so consider ⟪ ˇ ζ , ϕ ⟫ = ⟪ ζ , ˇ ϕ ⟫ = ∞ ∫ −∞ ζ ( x ) ˇ ϕ ( x ) d x = ∞ ∫ −∞ ⎛⎝ ∞ ∫ −∞ sin ( tx ) i t h ( t π ) d t ⎞⎠ ˇ ϕ ( x ) d x. As mentioned above, ∣ sin ( tx ) i t ∣ ≤ ∣ x ∣ , so since h and x ˇ ϕ are integrable (the latterbecause ϕ ∈ S ), we can use the Dominated Convergence Theorem to get ⟪ ˇ ζ , ϕ ⟫ = lim (cid:15) → + ∞ ∫ −∞ ⎛⎝ − (cid:15) ∫ −∞ sin ( tx ) i t h ( t π ) d t + ∞ ∫ (cid:15) sin ( tx ) i t h ( t π ) d t ⎞⎠ ˇ ϕ ( x ) d x. Now again, for any (cid:15) >
0, the following two integrals exist since h is integrable, andare equal because h is even: − (cid:15) ∫ −∞ cos ( xt ) t h ( t π ) d t = − ∞ ∫ (cid:15) cos ( xt ) t h ( t π ) d t. Therefore, with the previous computation, ⟪ ˇ ζ , ϕ ⟫ = lim (cid:15) → + ∞ ∫ −∞ ⎛⎝ − (cid:15) ∫ −∞ − e i tx t h ( t π ) d t + ∞ ∫ (cid:15) − e i tx t h ( t π ) d t ⎞⎠ ˇ ϕ ( x ) d x = lim (cid:15) → + ∞ ∫ −∞ ⎛⎝ ∞ ∫ (cid:15) e − π i tx h ( t ) t d t + − (cid:15) ∫ −∞ e − π i tx h ( t ) t d t ⎞⎠ ˇ ϕ ( x ) d x. A standard use of Tonelli’s and Fubini’s Theorem shows that we can interchangethe order of integration, so that ⟪ ˇ ζ , ϕ ⟫ = lim (cid:15) → + ⎡⎢⎢⎢⎢⎣ − (cid:15) ∫ −∞ ⎛⎝ ∞ ∫ −∞ e − π i tx ˇ ϕ ( x ) d x ⎞⎠ h ( t ) t d t + ∞ ∫ (cid:15) ⎛⎝ ∞ ∫ −∞ e − π i tx ˇ ϕ ( x ) d x ⎞⎠ h ( t ) t d t ⎤⎥⎥⎥⎥⎦ . Since ϕ ∈ S , we know that the inversion formula holds: for almost every t , we have ϕ ( t ) = ˆˇ ϕ ( t ) = ∞ ∫ −∞ e − π i xt ˇ ϕ ( x ) d x. Therefore, ⟪ ˇ ζ , ϕ ⟫ = lim (cid:15) → + ⎡⎢⎢⎢⎢⎣ − (cid:15) ∫ −∞ ϕ ( t ) h ( t ) t d t + ∞ ∫ (cid:15) ϕ ( t ) h ( t ) t d t ⎤⎥⎥⎥⎥⎦ = ⟪ pv (∫ h ( t ) t ) , ϕ ⟫ . (cid:3) Definition
15: A smooth function χ ∶ R → [− , ] is a normalizing function if(1) χ is odd,(2) for x > χ ( x ) >
0, and(3) for x → ±∞ , we have χ ( x ) → ± Lemma 4.11.
For every (cid:15) > , there exists a normalizing function χ whose ( dis-tributional ) Fourier transform is supported in (− (cid:15), (cid:15) ) .Proof. We will follow the instructions in [7, Exercise 10.9.3].Fix an even function g ∈ C ∞ c ( R , R ) such that g ∗ g ( ) = π . One could, forexample, take a rescaled version of the function t ↦ { exp (− − t ) if ∣ t ∣ < , f ∶= g ∗ g , and define χ ( x ) ∶= ∞ ∫ −∞ sin ( xt ) t f ( t ) d t, which is well-defined (see proof of Lemma 4.10), odd, and smooth.Now, recall that for any a >
0, the sinc function is the Fourier transform of ascaled characteristic function, namelysin ( πat ) πt = ˆ [− a,a ] ( t ) . Using [5, Lemma 8.25], we can thus rewrite χ for positive x as follows: χ ( x ) = π ∞ ∫ −∞ ˆ [− x π , x π ] ( t ) f ( t ) d t = π ∞ ∫ −∞ [− x π , x π ] ( t ) ˆ f ( t ) d t = π x π ∫ − x π ˆ f ( t ) d t = π x π ∫ − x π ˆ g ( t ) d t, from which we see that χ ( x ) ≥
0. Moreover, since g is non-zero and smooth withcompact support, ˆ g does not vanish on any interval (cf. [5, p. 293]). The above LLIPTIC OPERATORS AND K-HOMOLOGY 21 equality hence gives χ ( x ) > x >
0, so χ satisfies Property 2 of normalizingfunctions. Furthermore, π x π ∫ − x π ˆ g ( t ) d t ≤ π ∞ ∫ −∞ ˆ g ( t ) d t = π ∥ ˆ g ∥ (∗) = π ∥ g ∥ = πf ( ) = , where (∗) holds because of the Plancherel Theorem (see [5, 8.29]), so we have shownthat χ is indeed [− , ] -valued. Next, the Dominated Convergence Theorem allowsus to computelim x →∞ χ ( x ) = lim x →∞ ∫ sin ( t ) t f ( tx ) d t DCT = ∫ sin ( t ) t f ( ) d t = , so we have shown Property 3 of normalizing functions.From Lemma 4.10, we see thatˇ χ = pv (∫ f ( πt ) i t ) , so ⟪ ˆ χ , ϕ ⟫ = ⟪ ˇ χ , ˜ ϕ ⟫ = lim (cid:15) → + ⎛⎝ − (cid:15) ∫ −∞ f ( πt ) i t ϕ (− t ) d t + ∞ ∫ (cid:15) f ( πt ) i t ϕ (− t ) d t ⎞⎠ . Thus, if ϕ has support disjoint from the support of t ↦ f (− πt ) = f ( πt ) , then ⟪ ˆ χ , ϕ ⟫ =
0. In other words, the support of ˆ χ is contained in π supp ( f ) , which iscompact.Lastly, out of χ with Fourier Transform supported in, say, (− b, b ) , we wantto construct another normalizing function whose Fourier transform is supportedin (− (cid:15), (cid:15) ) . Let T ( x ) ∶= (cid:15) xb and χ ∶= χ ○ T . As (cid:15), a are positive, this is again anormalizing function, and we compute for ϕ ∈ S , ⟪ ˆ χ , ϕ ⟫ = ⟪ χ , ˆ ϕ ⟫ = ∞ ∫ −∞ χ ( x ) ˆ ϕ ( x ) d x = ∞ ∫ −∞ ( χ ○ T )( x ) ˆ ϕ ( x ) d x = ∞ ∫ −∞ χ ( x )( ˆ ϕ ○ T − )( x )( T − ) ′ ( x ) d x. From [5, Thm. 8.2b)] we know that ( ˆ ϕ ○ T − ) ⋅ ( T − ) ′ = ( ϕ ○ T ) ˆ . If ϕ is now supported outside of (− (cid:15), (cid:15) ) , so that ϕ ○ T is supported outside of (− b, b ) ,then the above computations yield ⟪ ˆ χ , ϕ ⟫ = ∞ ∫ −∞ χ ( x )( ϕ ○ T ) ˆ ( x ) d x = ⟪ ˆ χ , ϕ ○ T ⟫ = . This proves that ˆ χ is supported in (− (cid:15), (cid:15) ) . (cid:3) Lemma 4.12 ([7, Prop. 10.3.5]) . If D is an essentially self-adjoint differentialoperator on M and ψ a bounded Borel function on R whose Fourier transform hascompact support, then for all u, v ∈ Γ ∞ c ( M ; S ) , we have ⟨ ψ ( D ) u ∣ v ⟩ = ⟪ ˆ ψ , s ↦ ⟨ e πisD u ∣ v ⟩⟫ . Proof.
We follow the idea given in [7, Prop. 10.3.5]. If we first take ψ ∈ S , then ψ = ˇˆ ψ , so that ⟨ ψ ( D ) u ∣ v ⟩ = ⟨(∫ e π i sD ˆ ψ ( s ) d s ) u ∣ v ⟩ = ∫ ⟨ e π i sD u ∣ v ⟩ ˆ ψ ( s ) d s. Since for functions in L ( R ) , the classical Fourier transform coincides with thedistributional Fourier transform , the above equation can be rewritten as ⟨ ψ ( D ) u ∣ v ⟩ = ⟪ ˆ ψ , g ⟫ , where g ( s ) ∶= ⟨ e π i sD u ∣ v ⟩ , which was to be shown. Using the inversion formulafor ψ = ˆ ψ ∈ S once more, we could also write this as ⟨ ˇ ψ ( D ) u ∣ v ⟩ = ⟪ ψ , g ⟫ (14)for ψ ∈ S arbitrary Now let us take a general ψ as specified in the lemma. Asexplained in Example 7, ψ gives rise to a tempered distribution, denoted by F for now. In particular, it makes sense to speak of its Fourier transform. Fixsome φ ∈ C ∞ c ( R , R ) with ∫ φ ( x ) d x =
1, and define φ t ( x ) ∶= t φ ( xt ) . Since we haveassumed ˆ F to have compact support, ˆ F ∗ φ t ∈ C ∞ c ( R ) by Lemma 4.9, so thatEquation (14) implies ⟨( ˆ F ∗ φ t ) ˇ ( D ) u ∣ v ⟩ = ⟪ ˆ F ∗ φ t , g ⟫ . (15)If we can now show that(1) ( ˆ F ∗ φ t ) ˇ = ψ ⋅ ˇ φ t ,(2) lim t → ⟨( ψ ⋅ ˇ φ t )( D ) u ∣ v ⟩ = ⟨ ψ ( D ) u ∣ v ⟩ , and(3) lim t → ⟪ ˆ F ∗ φ t , g ⟫ = ⟪ ˆ F , g ⟫ ,then ⟨ ψ ( D ) u ∣ v ⟩ = lim t → ⟨( ˆ F ∗ φ t ) ˇ ( D ) u ∣ v ⟩ (15) = lim t → ⟪ ˆ F ∗ φ t , g ⟫ = ⟪ ˆ F , g ⟫ , so we would be done. ad (1): By virtue of [5, p. 283], it suffices to check that the functions induce thesame distribution: we recall that ˜ φ ( x ) = φ (− x ) , and compute for f ∈ C ∞ c , ⟪( ˆ F ∗ φ t ) ˇ , f ⟫ = ⟪ ˆ F ∗ φ t , ˇ f ⟫ = ⟪ ˆ F , ˇ f ∗ ˜ φ t ⟫ = ⟪ F , ( ˇ f ∗ ˜ φ t ) ˆ ⟫= ⟪ F , ˆˇ f ⋅ ˆ˜ φ t ⟫ = ∞ ∫ −∞ ψ ( x ) f ( x ) ˆ˜ φ t ( x ) d x = ⟪ ψ ⋅ ˇ φ t , f ⟫ . ad (2): Using Property (6) of Functional Calculus, it is sufficient to show that {∥ ψ ⋅ ˇ φ t ∥ ∞ } t is bounded and that ψ ⋅ ˇ φ t converges pointwise to ψ : first ofall, ˇ φ t ( x ) = ˆ φ (− tx ) implies ∥ ψ ⋅ ˇ φ t ∥ ∞ ≤ ∥ ψ ∥ ∞ ⋅ ∥ ˆ φ ∥ ∞ < ∞ . Secondly, suppose supp ( φ ) ⊆ [− a, a ] , and take h ∈ C ∞ c such that h ∣[− a,a ] ≡ φ t = φ t ⋅ h for t ≤
1. Since φ t —→ δ in D ′ for t —→ y ∈ R , f y ( x ) ∶= e π i xy h ( x ) ∈ C ∞ c and getˇ φ t ( y ) = ∞ ∫ −∞ e π i xy φ t ( x ) h ( x ) d x = ⟪ φ t , f y ⟫ t → —→ ⟪ δ , f y ⟫ = f y ( ) = . LLIPTIC OPERATORS AND K-HOMOLOGY 23 ad (3):
Recall that we defined g ( s ) ∶= ⟨ e π i sD u ∣ v ⟩ for fixed compactly supportedsections u, v . Since ∂ m g is bounded, it follows from [5, Thm. 8.14(c)] that ∂ m ( g ∗ ˜ φ t ) = ( ∂ m g ) ∗ ˜ φ t t → —→ ∂ m g uniformly on compact sets. Let us take h ∈ C ∞ c such that 0 ≤ h ≤ h ∣ supp ( ˆ F ) ≡
1, where we use that ˆ F is compactly supported. As each ∂ i h has compact support, we get ∥( ∂ i h ) ⋅ ( ∂ m ( g ∗ ˜ φ t ) − ∂ m g )∥ ∞ t → —→ . It follows by the product rule that, for any k , ∥ ∂ k ( h ⋅ [ g ∗ ˜ φ t ]) − ∂ k ( h ⋅ g )∥ ∞ t → —→ , that is, h ⋅ [ g ∗ ˜ φ t ] t → —→ h ⋅ g in C ∞ ( R ) . As ˆ F has compact support, it is inthe dual space of C ∞ ( R ) . Since h ∣ supp ( ˆ F ) ≡
1, we therefore get ⟪ ˆ F ∗ φ t , g ⟫ = ⟪ ˆ F , g ∗ ˜ φ t ⟫ = ⟪ ˆ F , h ⋅ [ g ∗ ˜ φ t ]⟫ t → —→ ⟪ ˆ F , h ⋅ g ⟫ = ⟪ ˆ F , g ⟫ , which finishes our proof. (cid:3) The Main Theorem
Theorem 5.1 (special case of [7, Thm. 10.6.5]) . Let D be a symmetric ellipticdifferential operator on a smooth and compact manifold M . Let H ∶= L ( M ; S ) andlet M be the representation of C ( M ) on H by multiplication. For χ a normalizingfunction and F ∶= χ ( D ) , the triple ( M , H , F ) is a Fredholm module. Moreover, itsclass in K ( C ( M )) does not depend on the choice of χ and can hence be denotedby [ D ] . This is the theorem we are set out to prove. As a first step, let us see that wehave functional calculus at our disposal, so that χ ( D ) makes sense. Proposition 5.2 ([7, Lemma 10.2.5]) . Let D be a symmetric differential opera-tor on a smooth manifold M and let u ∈ L ( M ; S ) be compactly supported. Then u ∈ dom D if and only if u ∈ dom D ∗ . In particular, if M is compact, then D isessentially self-adjoint. To prove Proposition 5.2, we need the following two lemmas.
Lemma 5.3 ([7, Lemma 1.8.1] - without proof) . If D is a closable unboundedoperator, then u ∈ dom D if and only if there exists a sequence { u j } j in dom D suchthat u j → u and {∥ Du j ∥} j is bounded. Lemma 5.4 ([7, Exercise 10.9.1]) . For K ⊆ M compact, there exist for sufficientlysmall (cid:15) > t > , operators F t ∶ L ( K ; S ) → L ( M ; S ) which satisfy (1) ∥ F t ∥ ≤ C for some constant C and all t , (2) ∀ u ∈ L ( K ; S ) ∶ lim t → F t u = u in L ( M ; S ) , (3) ∀ u ∈ L ( K ; S ) ∶ F t u is smooth with compact support, and (4) for any differential operator D on M , [ D, F t ] extends to a bounded operator L ( K ; S ) → L ( M ; S ) , and its norm is bounded independent of t . We remark that the constant in Property (1) is usually supposed to be 1, but C is good enough for us. For a proof of the existence of these so-called Friedrichs’mollifiers , see the appendix on p. 31.
Proof of Proposition . Since the minimal domain of D is always contained inthe maximal domain, let us take u ∈ dom D ∗ with compact support (pick anyrepresentative). According to Lemma 5.3, we need to find a sequence of v n in dom D which converges to u in the Hilbert space and such that {∥ Dv n ∥} n is bounded. Letus take F t as in Lemma 5.4 for K ∶= supp ( u ) , let t n be a sequence converging to 0,and let v n ∶= F t n u . Since v n ∈ Γ ∞ c ( M ; S ) by Property (3) of the mollifiers, it is inthe domain of D , and by Property (2), v n → u in L ( M ; S ) . It remains to see whythe sequence D ( v n ) is bounded:By Lemma 4.1, D ∗ u is in L ( K ; S ) , so F t ( D ∗ u ) makes sense. Moreover, byProperty (4) of the mollifiers, we also have that [ D, F t ] u has a well-defined meaning.All in all, we can therefore write D ( v n ) = D ∗ ( F t n u ) = F t n ( D ∗ u ) + [ D, F t n ] u. Because of Property (1) and Property (4) of { F t } t , there exists C > t , we have ∥ F t ∥ , ∥[ D, F t ]∥ < C . Hence ∥ D ( v n )∥ ≤ ∥ F t n ( D ∗ u )∥ + ∥[ D, F t n ] u ∥ ≤ C ⋅ (∥ D ∗ u ∥ + ∥ u ∥) , so the sequence is indeed bounded. (cid:3) The next proposition will show that the class [ D ] does not depend on the choiceof normalizing function χ , and that χ ( D ) − Proposition 5.5 ([7, Prop. 10.4.5, Lemma 10.6.3]) . If D is a symmetric elliptic dif-ferential operator on a compact manifold M , and ϕ ∈ C ( R ) , then ϕ ( D )∶ L ( M ; S ) → L ( M ; S ) is a compact operator. In particular, if χ , χ are nor-malizing functions, then the operators χ ( D ) and χ ( D ) differ only by a compactoperator.Proof. We first want to show that dom D = L ( M ; S ) by showing the followingcontainments: dom D ⊆ L ( M ; S ) ⊆ dom D ∗ = dom D. By Proposition 5.2, our symmetric operator is essentially self-adjoint (which ex-plains the equality on the right), and Corollary 4.6 gives us L ( M ; S ) ⊆ dom D ∗ .Now suppose u ∈ dom D , that is, ( u, Du ) ∈ Γ ( D ) = Γ ( D ) . This means there is asequence ( u j ) j ∈ dom D such that u j —→ u and Du j —→ Du in L ( M ; S ) . In par-ticular, ( u j ) j is Cauchy in L ( M ; S ) , so G˚arding’s inequality implies that ( u j ) j is also Cauchy with respect to ∥ ⋅ ∥ (remember that M is assumed compact). AsL ( M ; S ) is (by definition) complete with respect to this norm, ( u j ) j thus has a ∥ ⋅ ∥ -limit in L ( M ; S ) . The Rellich lemma, for example, shows that this limit mustcoincide with u , so we have shown u ∈ L ( M ; S ) . All in all, dom D = L ( M ; S ) .Now let us focus on the function ψ ( x ) = ( i + x ) − . Since the domain of D isdense and D is self-adjoint, Lemma 3.1 implies that i + D has full range. Thus,for every u ∈ L ( M ; S ) , there exists v ∈ dom D = L ( M ; S ) such that ( i + D ) v = u .Since D is self-adjoint, we know that ∥( i + D ) v ∥ = ∥ v ∥ + ∥ Dv ∥ , LLIPTIC OPERATORS AND K-HOMOLOGY 25 see Equation (3). Hence it follows from G˚arding’s inequality and the properties ofFunctional Calculus that, for some c > c ⋅ ∥ ψ ( D ) u ∥ = c ⋅ ∥ v ∥ ≤ ∥ v ∥ + ∥ Dv ∥ ≤ √ ∥( i + D ) v ∥ = √ ∥ u ∥ . In other words, ψ ( D ) is a bounded operator L ( M ; S ) → L ( M ; S ) , and thus bythe Rellich lemma, it is a compact operator L ( M ; S ) → L ( M ; S ) . Lastly, if wetake an arbitrary ϕ ∈ C ( R ) , then for any (cid:15) >
0, there are finitely many a i,j ∈ C suchthat XXXXXXXXXXX ϕ − m ∑ i,j = a i,j ψ i ψ j XXXXXXXXXXX ∞ < (cid:15), because ψ generates C ( R ) as a C ∗ -algebra. By Property (5) of Functional Calculus,we get for f ∶= ϕ − ∑ mi,j = a i,j ψ i ψ j that ∥ f ( D )∥ = ∥ f ∥ ∞ < (cid:15). This means that the operator ϕ ( D ) is approximated by compact operators and ishence itself compact. (cid:3) The remaining work before the proof of Theorem 5.1 on page 28 will culminatein Proposition 5.9, which says that [ χ ( D ) , M f ] is compact for f ∈ C ( M ) . Proposition 5.6 ([7, Prop. 10.3.1]) . If D is an essentially self-adjoint differentialoperator on M , and if W is an open neighborhood of a compact set K ⊆ M , thenthere exists (cid:15) > such that ∀ ∣ s ∣ < (cid:15), ∀ u ∈ L ( K ; S ) ∶ supp ( e i sD u ) ⊆ W. Proof.
We will follow the proof given in [7]. Let g ∈ C ∞ c ( M, [ , ]) be such that g ∣ K ≡ g ∣ M ∖ W ≡ . Pick f ∈ C ∞ ( R , [ , ]) non-decreasing such thatfor t < ∶ f ( t ) < , and for t ≥ ∶ f ( t ) = . We have shown in Lemma 4.2 that [ D, M g ] is bounded (even on all of L ( M ; S ) since g is compactly supported), so let c > ∥[ D, M g ]∥ . We use this to define for s ∈ R + and p ∈ M : h s ( p ) ∶= f ( g ( p ) + cs ) , L s ∶= { p ∈ M ∣ h s ( p ) = } . We will deal with positive s only; for negative s , do the same construction for − D .Claim 1. If t ≤ s , then L t ⊆ L s .Proof of claim. An element p is in L t exactly if f ( g ( p ) + ct ) =
1. By choice of f ,this means g ( p ) + ct ≥
1. As s ≥ t and c is positive, this implies g ( p ) + cs ≥ f ( g ( p ) + cs ) =
1. Therefore, p ∈ L s . (cid:3) Claim 2.
For ≤ s < c , we have K ⊆ L ⊆ L s ⊆ W .Proof of claim. For the first inclusion, use g ∣ K ≡ f ( g ( p )) = p ∈ K by choice of f . The second inclusion follows from the above computation. Forthe last inclusion, recall that, if p ∉ W , then g ( p ) = g . Since cs < s , we therefore have h s ( p ) = f ( cs ) < f . (cid:3) Let us write ˙ h s to denote˙ h s ( p ) ∶= ∂ s ( s ↦ h s ( p )) ∣ s = cf ′ ( g ( p ) + cs ) for p ∈ M . Since c is positive and f is non-decreasing, we have ˙ h s ( p ) ≥ s and p .Claim 3. [ D, M h s ] = c M ˙ h s [ D, M g ] . Proof of claim.
For D locally as in Equation (7), we have shown in Equation (9)that ([ D, M h s ] u )( p ) = n ∑ j = ∂ j ( h s ○ ϕ − ) ∣ ϕ ( p ) ⋅ Ψ − ( p, A j ( p ) ⋅ ( ψ ○ u ( p ))) , and similarly, ( c M ˙ h s [ D, M g ] u ) ( p ) = c ˙ h s ( p ) n ∑ j = ∂ j ( g ○ ϕ − ) ∣ ϕ ( p ) ⋅ Ψ − ( p, A j ( p )⋅( ψ ○ u ( p ))) . If we write h s = f ○ k s where k s ( p ) ∶= g ( p ) + cs , then the chain rule gives ∂ j ( h s ○ ϕ − ) ∣ ϕ ( p ) = f ′ ( k s ( p )) ∂ j ( k s ○ ϕ − ) ∣ ϕ ( p ) = c ˙ h s ( p ) ∂ j ( g ○ ϕ − ) ∣ ϕ ( p ) for each 1 ≤ j ≤ n , which implies the claim. (cid:3) Because of Claim 3, we have M ˙ h s − i [ D, M h s ] = c M ˙ h s ( c − i [ D, M g ]) . By choice of c , we see that c ⋅ ≥ ∥ i [ D, M g ]∥ ⋅ ≥ i [ D, M g ] , so c − i [ D, M g ] ≥ . By Lemma 4.2, [ D, M g ] is a multiplication operator, so itcommutes with M ˙ h s . As ˙ h s is non-negative, we have therefore shown that M ˙ h s − i [ D, M h s ] ≥ . (16)Since it suffices to prove the proposition for u ∈ Γ ∞ ( K ; S ) , fix such u and de-fine u s ∶= e i sD u . Since ( ∂ s u s ) ∣ s = i Du s , we have ∂ s ⟨ h s ⋅ u s ∣ u s ⟩ ∣ s = ⟨ ∂ s ( h s ⋅ u s ) ∣ s ∣ u s ⟩ + ⟨ h s ⋅ u s ∣ ( ∂ s u s ) ∣ s ⟩= ⟨ ˙ h s ⋅ u s + h s ⋅ i Du s ∣ u s ⟩ + ⟨ h s ⋅ u s ∣ i Du s ⟩= ⟨ ˙ h s ⋅ u s + i h s ⋅ Du s ∣ u s ⟩ − ⟨ i D ( h s ⋅ u s ) ∣ u s ⟩ as D ⊆ D ∗ = ⟨( M ˙ h s − i [ D, M h s ]) u s ∣ u s ⟩ ≥ ⟨ h s ⋅ u s ∣ u s ⟩ is an increasing function, and in particular for s ≥ ⟨ h s ⋅ u s ∣ u s ⟩ ≥ ⟨ h ⋅ u ∣ u ⟩ = ⟨ h ⋅ u ∣ u ⟩ (∗) = ⟨ u ∣ u ⟩ = ⟨ u s ∣ u s ⟩ , where (∗) holds because h = f ○ g is 1 on K ⊇ supp ( u ) , and the last equality comesfrom e i sD being a unitary. Since 1 ≥ h s ≥
0, this means ∥ u s ∥ ≥ ∥√ h s u s ∥ = ⟨ h s ⋅ u s ∣ u s ⟩ ≥ ⟨ u s ∣ u s ⟩ = ∥ u s ∥ . LLIPTIC OPERATORS AND K-HOMOLOGY 27
Therefore, ∫ M ∥ u s ( p )∥ S p d µ = ∫ M ∥√ h s ( p ) u s ( p )∥ S p d µ. Again, since 1 ≥ h s ≥
0, we have ∥ u s ( p )∥ S p ≥ ∥√ h s ( p ) u s ( p )∥ S p , and hence theequality of integrals implies ( u s ( p ) ∣ u s ( p )) S p = (√ h s ( p ) ⋅ u s ( p ) ∣ √ h s ( p ) ⋅ u s ( p )) S p , so ∥√ − h s ( p ) u s ( p )∥ S p = . (This equality is actually true not only almost everywhere but for all p ∈ M sincewe are dealing with smooth functions.) This implies that h s u s = u s . In particu-lar, supp ( u s ) has to be contained in the set on which h s is 1, that is, supp ( e i sD u ) = supp ( u s ) ⊆ L s ⊆ W for s < c by Claim 2. This finishes the proof of Proposition 5.6. (cid:3) Corollary 5.7 ([7, Cor. 10.3.3]) . Let D be an essentially self-adjoint differentialoperator on a manifold M . Let f , f be bounded functions on M with disjointsupports, and suppose supp ( f ) is compact. Then there exists (cid:15) > such that ∀ ∣ s ∣ < (cid:15) ∶ M f ○ e i sD ○ M f = . Proof.
By assumption, K ∶= supp ( f ) is compact. Since the support of f is disjointfrom K , the set W ∶= M ∖ supp ( f ) is an open neighborhood of K . By Proposi-tion 5.6, there exists an (cid:15) > ∀ ∣ s ∣ < (cid:15), ∀ v ∈ L ( K ; S ) , supp ( e i sD v ) ⊆ W. For any u ∈ L ( M ; S ) , we know that M f u is supported in K , so e i sD M f u issupported in W . As W = M ∖ supp ( f ) , we hence get M f e i sD M f u = u ∈ Γ ∞ ( M ; S ) . (cid:3) Lemma 5.8 (Kasparov’s lemma; [7, 5.4.7] - without proof) . Suppose X is compactHausdorff, ν ∶ C ( X ) → B(H) a non-degenerate representation, and T ∈ B(H) . If ν ( f ) T ν ( f ) is compact for every f , f ∈ C ( X ) with disjoint support, then [ T, ν ( f )] is compact for every f ∈ C ( X ) . Proposition 5.9 (special case of [7, Lemma 10.6.4]) . If D a symmetric ellipticdifferential operator on a compact manifold M , χ a normalizing function, and f ∈ C ( M ) , then [ χ ( D ) , M f ] is compact.Proof of Proposition . Since M is a non-degenerate representation of C ( M ) onL ( M ; S ) , Kasparov’s lemma says that it suffices to show that, for all f , f ∈ C ( M ) with disjoint supports, M f χ ( D ) M f is compact. Moreover, because of Proposi-tion 5.5, we can actually show this for any normalizing function, and do not needto use the given χ .So let us fix such f , f . By Corollary 5.7, there exists (cid:15) > ∀ ∣ s ∣ < (cid:15) ∶ M f e π i sD M f = . By Lemma 4.11, we can take a normalizing function χ with supp ( ˆ χ ) ⊆ (− (cid:15), (cid:15) ) .We then get by Lemma 4.12 that, for all ˜ u, ˜ v ∈ Γ ∞ ( M ; S ) and g ( s ) ∶= ⟨ e π i sD ˜ u ∣ ˜ v ⟩ , ⟨ χ ( D ) ˜ u ∣ ˜ v ⟩ = ⟪ ˆ χ , g ⟫ . (17)If we choose ˜ u ∶= f ⋅ u and ˜ v ∶= f ⋅ v for u, v ∈ Γ ∞ ( M ; S ) , then g ( s ) = ⟨ e π i sD ( f ⋅ u ) ∣ f ⋅ v ⟩ = ⟨ M f ○ e π i sD ○ M f ( u ) ∣ v ⟩ , so that g ( s ) = ∣ s ∣ < (cid:15) by choice of (cid:15) , and hence ⟪ ˆ χ , g ⟫ = supp ( ˆ χ ) ⊆ (− (cid:15), (cid:15) ) . Thus, Equation (17) gives ⟨ M f χ ( D ) M f u ∣ v ⟩ =
0. We conclude that the sameeven holds for u, v ∈ L ( M ; S ) , so that we have proved M f χ ( D ) M f = (cid:3) Finally, we can prove Theorem 5.1.
Proof of Theorem . F is self-adjoint by Property (4) of Functional Calculus be-cause χ is real-valued. Since χ is a normalizing function, χ − ∈ C ( R ) , so Propo-sition 5.5 implies that ( χ − )( D ) = F − [ χ ( D ) , M f ] = [ F, M f ] is compact for any f ∈ C ( M ) , so we have shown that theproperties of a Fredholm module are satisfied. Lastly, if χ is another normalizingfunction, then by Proposition 5.5 again, χ ( D ) differs from χ ( D ) only by a compactoperator. This means that ( M , H , χ ( D )) is a compact perturbation of ( M , H , F ) .Therefore, they determine the same K-homology class by Proposition 2.1. (cid:3) Remark
7: There is an obvious extension of Theorem 5.1 to even K-homology: if S is equipped with a smooth idempotent vector bundle automorphism γ S (that is, S is Z / Z -graded), then the map γ ∶ Γ ∞ c ( M ; S ) → Γ ∞ c ( M ; S ) , γu ( p ) ∶= γ S ( u ( p )) , extends to a grading operator of H = L ( M ; S ) with respect to which the left C ( M ) -action is even. If we further assume that D is odd, then Lemma 3.5 impliesthat F is odd as well, so that the Fredholm module ( M , H , F ) is actually graded.Again, the corresponding class in K ( C ( M )) only depends on D . Example
8: As discussed in Example 4 and Example 5, the Dirac operator / ∂ M of a spin c manifold M is an unbounded, symmetric elliptic differential operator.Moreover, in case the dimension of the manifold is even, the spinor bundle is actuallygraded and / ∂ M is an odd operator. Consequentially, if the manifold is compact, / ∂ M determines a class [ / ∂ M ] in the even or odd K-homology of C ( M ) , dependingon whether dim ( M ) is even or odd.An interesting consequence of Theorem 5.1 is that it gives rise to maps onK-theory: if ⟨ ⋅ , ⋅ ⟩∶ K j ( A ) × K j ( A ) → Z , ( j = , D on a smooth and compact manifold M gives rise toa map K ( C ( M )) → Z , x ↦ ⟨ x, [ D ]⟩ , LLIPTIC OPERATORS AND K-HOMOLOGY 29 by pairing a K-theory class with the K-homology class [ D ] constructed above. Ifthe vector bundle S over M which is underlying D is graded and if D is odd, thenwe get a map K ( C ( M )) → Z , x ↦ ⟨ x, [ D ]⟩ . Appendix
In order to prove Lemma 5.4, we first need the following version for R n : Lemma 5.10.
There exist operators ˜ F t ∶ L ( R n ) → L ( R n ) such that (a) ∥ ˜ F t ∥ ≤ , (b) ∀ u ∈ L ( R n ) ∶ lim t → ˜ F t u = u in L ( R n ) , (c) ∀ u ∈ L ( R n ) ∶ ˜ F t u is smooth, (d) if u has compact support, then so does ˜ F t u , and (e) for all ≤ k ≤ n and f ∈ C ∞ ( R n ) with bounded partial derivatives, theoperator [ f ⋅ ∂∂x k , ˜ F t ] extends to a bounded operator whose norm is boundedindependent of t .Proof. Pick a smooth function φ ∶ R n → R + with compact support and ∫ R n φ d λ = φ t ( x ) ∶= t − n φ ( xt ) , which has the same properties as φ . Set ˜ F t u = φ t ∗ u for u ∈ L ( R n ) , that is: ˜ F t u ( x ) = t − n ∫ R n φ ( x − yt ) u ( y ) d λ ( y ) . By [13, IV 9.4], we have ∥ ˜ F t ∥ ≤ sup {∥ φ t ∥ L ⋅ ∥ u ∥ ∶ ∥ u ∥ ≤ } = ∥ φ t ∥ L = , so Property (a) holds. Moreover, Property (b) and (c) follow from [13, Satz IV 9.5]and [13, Korollar IV 9.7] respectively. It is well known that supp ( ˜ F t u ) ⊆ supp ( φ t ) + supp ( u ) , so that Property (d) follows from φ t having compact support. It remains to checkProperty (e):Using integration by parts and the fact that φ is compactly supported, we cancompute for u ∈ L ( R n )[ f ⋅ ∂∂x k , ˜ F t ] u ( x ) = ∫ y ∈ R n [ t n + ∂φ∂x k ∣ x − yt ( f ( x ) − f ( y )) + t n φ ( x − yt ) ∂f∂x k ∣ y ] u ( y ) d y. In other words, [ f ⋅ ∂∂x k , ˜ F t ] is an integral transform with kernel k t ( x, y ) = t n + ∂φ∂x k ∣ x − yt ( f ( x ) − f ( y )) + t n φ ( x − yt ) ∂f∂x k ∣ y . As stated in [6, Thm. 5.2], the so-called
Schur’s test says that, ifsup x ∈ R n ∥ k t ( x, ⋅)∥ L ≤ α and sup y ∈ R n ∥ k t (⋅ , y )∥ L ≤ β, then the integral transform extends to a bounded operator whose norm is boundedby √ αβ . We claim that, if for all 1 ≤ j ≤ n and supp ( φ ) ⊆ [− a, a ] we have ∥ ∂f∂x j ∥ ∞ < C, then α = β = C ( na ∥ ∂φ∂x k ∥ L + ∥ φ ∥ L ) do the trick.For x, y ∈ R n such that ∥ x − y ∥ ≤ at , repeated application of the Mean ValueTheorem (see the proof of[11, Thm. 5.3.10], for example) gives ∣ f ( x ) − f ( y )∣ ≤ n ∑ j = at ∥ ∂f∂x j ∥ ∞ ≤ natC. For these x, y , we compute ∣ k t ( x, y )∣ ≤ t n + ∣ ∂φ∂x k ∣ x − yt ( f ( x ) − f ( y ))∣ + t n ∣ φ ( x − yt ) ∂f∂x k ∣ y ∣≤ t n + ∣ ∂φ∂x k ∣ x − yt ∣ ⋅ natC + t n ∣ φ ( x − yt )∣ ⋅ C = t n C [∣ ∂φ∂x k ∣ x − yt ∣ na + ∣ φ ( x − yt )∣] For all other x, y , we have k t ( x, y ) = φ is supported within [− a, a ] .This means that the above calculation and a substitution shows that ∥ k t ( x, ⋅)∥ L , ∥ k t (⋅ , y )∥ L ≤ C ( na ∥ ∂φ∂x k ∥ L + ∥ φ ∥ L ) , and we are done. (cid:3) Lemma 5.11.
For K a compact subset of a manifold M , we can write K = ⋂ ∞ k = V k for some open sets V k + ⊆ V k ⊆ M .Proof. For K contained in some chart ( U, ϕ ) , we have ϕ ( K ) = ∞ ⋂ k = ˜ V k , where ˜ V k ∶= ⋃ x ∈ ϕ ( K ) B k ( x ) . If we then let N be smaller than the distance of the compact set ϕ ( K ) to the closedset R n ∖ ϕ ( U ) , then for k ≥ N we have ˜ V k ⊆ ϕ ( U ) , and hence K = ∞ ⋂ k = N V k , where V k ∶= ϕ − ( ˜ V k ) . Now, for arbitrary K , take finitely many open sets U , . . . , U l which cover K suchthat K ∩ U i is contained in a chart. From the above, we get (after re-indexing) K = K ∩ U ∪ . . . ∪ K ∩ U l = ( ∞ ⋂ k = V k ) ∪ . . . ∪ ( ∞ ⋂ k = V lk ) ⊂ ∞ ⋂ k = ( V k ∪ . . . ∪ V lk ) . Since each family { V in } n is nested, we also have ( ∞ ⋂ k = V k ) ∪ . . . ∪ ( ∞ ⋂ k = V lk ) ⊃ ∞ ⋂ k = ( V k ∪ . . . ∪ V lk ) , and hence K = ⋂ ∞ k = ( V k ∪ . . . ∪ V lk ) . (cid:3) Lemma 5.12.
For K a compact subset of a manifold M and { U i } li = an open coverof K in M , there exist smooth compactly supported functions ρ , . . . , ρ l ∶ M → [ , ] such that supp ( ρ i ) ⊆ U i and ∑ li = ρ i ( p ) = for all p ∈ K . LLIPTIC OPERATORS AND K-HOMOLOGY 31
Proof.
With U ∶= M ∖ K , take a partition of unity { ρ i } li = of M subordinate tothe cover { U i } li = . Since M = U ∪ U ∪ . . . ∪ U l , we get from [3, Lemma 1.4.8] thatthere exists an open cover { V i } li = of M with V i ⊆ U i for all 0 ≤ i ≤ l . Now since K ∩ V i ⊆ V i ⊆ U i for i ≠
0, and K ∩ V i ⊆ K is compact, we know by [5, Prop. 4.31]that there exists a precompact open set W i such that K ∩ V i ⊆ W i ⊆ W i ⊆ U i . Note that the collection of W i ’s covers all of K , so that we can take a smoothpartition of unity { ρ i } li = of M which is subordinate to { M ∖ K } ∪ { W i } li = . Inparticular, since W i is precompact and supp ( ρ i ) ⊆ W i for i >
0, we know that those ρ ’s have compact support. Moreover, it follows from supp ( ρ ) ⊆ M ∖ K that for p ∈ K = l ∑ i = ρ i ( p ) = l ∑ i = ρ i ( p ) . (cid:3) Lemma (Lemma 5.4) . For M and S as specified at the beginning of Section , andany K ⊆ M compact, there exist operators F t ∶ L ( K ; S ) → L ( M ; S ) for sufficientlysmall (cid:15) > t > which satisfy (1) ∥ F t ∥ ≤ C for some constant C and all t , (2) ∀ u ∈ L ( K ; S ) ∶ lim t → F t u = u in L ( M ; S ) , (3) ∀ u ∈ L ( K ; S ) ∶ F t u is smooth with compact support, and (4) for any differential operator D on M , [ D, F t ] extends to a bounded operator L ( K ; S ) → L ( M ; S ) , and its norm is bounded independent of t .Proof of Lemma . Take an atlas A of M whose charts are small enough to also allow smoothtrivializations of S which are isometries on the fibres, S ∣ U U × C k M ⊇ U R nπ ≈ pr ≈ Let {( U i , ϕ i )} li = be finitely many of those charts which cover the compact set K , and let { ρ i } li = be as in Lemma 5.12. For our trivializations, we will writeΨ i ∶ S ∣ U i ≈ —→ U i × C k , ψ i ∶= pr ○ Ψ i . Moreover, let f i , g i = f i ∶ R n → ( , ∞) be such that for all h ∈ C ∞ c ( R n ) and E ⊆ U i Borel, we have ∫ E h ○ ϕ i d µ = ∫ ϕ i ( E ) h ⋅ f i d λ and ∫ E ( h ⋅ g i ) ○ ϕ i d µ = ∫ ϕ i ( E ) h d λ. We have assumed in Remark 5 that ∥ f i ∥ ∞ , ∥ g i ∥ ∞ ≤ L for some number L . Inparticular, we have for v ∈ ⊕ k L c ( R n ) and any 1 ≤ i ≤ l , ∫ U i ∥ v ○ ϕ i ( p )∥ C k d µ = ∫ R n ∥ v ( x )∥ C k ⋅ f i ( x ) d λ ≤ ∥ v ∥ ⋅ L. (18) For u ∈ L ( U i ; S ) , since ψ i is isometric we get ∫ R n ∥( ψ i ○ u ○ ϕ − i )( x )∥ C k d λ = ∫ R n ∥( u ○ ϕ − i )( x )∥ S ϕ − i ( x ) d λ = ∫ U i ∥ u ( p )∥ S p g i ( ϕ i ( p )) d µ ≤ ∥ u ∥ ⋅ L. (19)For 1 ≤ i ≤ l , we define ⊕ k L ( R n ) L ( U i ; S ) F it ∶ L c ( U i ; S ) ⊕ k L c ( R n ) ⊕ k C ∞ c ( R n ) Γ ∞ c ( U i ; S ) u ψ i ○ u ○ ϕ − i , v Ψ − i ( ⋅ , v ○ ϕ i )⊕ kj = w j ⊕ kj = ˜ F t w j ⊆ ⊆ Notice that, indeed, F it takes values in Γ ∞ c ( U i ; S ) : since ϕ i is a diffeomorphism, ψ i ○ u ○ ϕ − i is compactly supported when u is, and in particular, all of its componentfunctions are compactly supported. By Property (c) of ˜ F t , ˜ F t w j is smooth, and byProperty (d), it is compactly supported when w j is. Since ϕ i and Ψ i are smooth, sois Ψ − i ( ⋅ , v ○ ϕ i ) for smooth v , and again, since ϕ i is a diffeomorphism, we concludethat Ψ − i ( ⋅ , v ○ ϕ i ) has compact support for compactly supported v .Now let F t ∶ L ( K ; S ) → L ( M ; S ) , F t u ∶= l ∑ i = F it ( ρ i ⋅ u ) . By the above explanation, F t actually takes values in Γ ∞ c ( M ; S ) because the ρ i are compactly supported. Hence, F t satisfies Property (3), and we need to checkthe other properties. By abuse of notation, we will write ˜ F t for the operator ⊕ k ˜ F t . ad Property (1): For u ∈ L ( U i ; S ) , we compute ∥ F it u ∥ = ∫ U i ∥ Ψ − i ( p, ˜ F t ( ψ i ○ u ○ ϕ − i ) ○ ϕ i ( p ))∥ S p d µ = ∫ U i ∥ ˜ F t ( ψ i ○ u ○ ϕ − i ) ○ ϕ i ( p )∥ C k d µ as Ψ i ( p, ⋅ ) is an isometry ≤ ∥ ˜ F t ( ψ i ○ u ○ ϕ − i )∥ ⋅ L by Eq. (18) ≤ ∥ ψ i ○ u ○ ϕ − i ∥ ⋅ L ≤ ∥ u ∥ ⋅ L since ∥ ˜ F t ∥ ≤ ∥ F it ∥ ≤ L , so that ∥ F t ∥ = sup ∥ u ∥ ≤ ∥ l ∑ i = F it ( ρ i u )∥ ≤ l ∑ i = sup ∥ u ∥ ≤ ∥ F it ( ρ i u )∥ ≤ l ∑ i = ∥ F it ∥ ≤ l ⋅ L =∶ C. ad Property (2): As Ψ i ( p, ⋅ ) is an isometry, we have for u in L ( U i ; S ) , p ∈ U i : ∥( F it u − u )( p )∥ S p = ∥ Ψ − i ( p, ˜ F t ( ψ i ○ u ○ ϕ − i ) ○ ϕ i ( p )) − u ( p )∥ S p = ∥ ˜ F t ( ψ i ○ u ○ ϕ − i ) ○ ϕ i ( p ) − ( ψ i ○ u ○ ϕ − i )( ϕ i ( p ))∥ C k . LLIPTIC OPERATORS AND K-HOMOLOGY 33
By Equation (18), it therefore follows that ∥ F it u − u ∥ ≤ ∥ ˜ F t ( ψ i ○ u ○ ϕ − i ) − ( ψ i ○ u ○ ϕ − i )∥ ⋅ L, so that Property (b) of ˜ F t implies that lim t → F it u = u in L -norm. Therefore, forarbitrary u ∈ L ( K ; S ) , ∥ F t u − u ∥ = ∥ l ∑ i = ( F it ( ρ i u ) − ρ i u )∥ ≤ l ∑ i = ∥ F it ( ρ i u ) − ρ i u ∥ —→ . ad Property (4): Suppose D is a differential operator acting on the sections of S .Since ρ i u is supported in U i for u ∈ Γ ∞ ( M ; S ) , we know that D ( ρ i u ) ∈ Γ ∞ ( U i ; S ) by Property a) of differential operators. Therefore, F it ( D ( ρ i u )) also makes sense,and we can write [ F t , D ] u = l ∑ i = F it ( ρ i Du ) − D ( F it ( ρ i u ))= l ∑ i = F it ( ρ i Du ) − F it ( D ( ρ i u )) + F it ( D ( ρ i u )) − D ( F it ( ρ i u ))= l ∑ i = F it [ M ρ i , D ] u + [ F it , D ] ( ρ i u ) , that is, [ F t , D ] = l ∑ i = F it [ M ρ i , D ] + [ F it , D ] M ρ i . (20)In order to check that [ F t , D ] extends to an operator that is bounded indepen-dent of t , we will show that F it [ M ρ i , D ] and [ F it , D ] M ρ i do. As was shown inLemma 4.2, [ M ρ i , D ] is a bounded operator on L ( K ; S ) , and since F it is boundedindependent of t (namely by L , as was shown above), so is F it [ M ρ i , D ] . It remainsto show that u ↦ [ F it , D ] ( ρ i u ) for a fixed but arbitrary 1 ≤ i ≤ l is bounded inde-pendent of t . It suffices to consider those D that (locally) look like only one ofthe summands in Equation (7). First, recall that for a C k -vector valued function w on R n , we have ψ i ○ Ψ − i ( ⋅ , w ( ⋅ )) = w ( ⋅ ) , so for u i ∶= ρ i u and p ∈ U i , we compute ( F it Du i ) ( p ) = Ψ − i ( p, ˜ F t ( ψ i ○ ( Du i ) ○ ϕ − i ) ○ ϕ i ( p ))= Ψ − i ( p, ˜ F t (( A ○ ϕ − i ) ⋅ ∂ j ( ψ i ○ u i ○ ϕ − i )) ○ ϕ i ( p )) and ( DF it u i ) ( p ) = Ψ − i ( p, A ( p ) ∂ j ( ψ i ○ ( F it u i ) ○ ϕ − i ) ∣ ϕ i ( p ) )= Ψ − i ( p, A ( p ) ∂ j ( ˜ F t ( ψ i ○ u i ○ ϕ − i )) ∣ ϕ i ( p ) . ) If we write ˜ A ∶= A ○ ϕ − i and v i ∶= ψ i ○ u i ○ ϕ − i , then this means ∥[ F it , D ] u i ( p )∥ S p = ∥( ˜ F t ( ˜ A ⋅ ∂ j v i ) − ˜ A ⋅ ∂ j ( ˜ F t v i )) ○ ϕ i ( p )∥ C k . Hence by Equation (18), ∥[ F it , D ] u i ∥ ≤ ∥[ ˜ F t , ˜ A ⋅ ∂ j ] v i ∥ ⋅ L. Note that v i is supported in the compact set κ ∶= ϕ i ( supp ( ρ i )) ⊆ R n . Therefore,Property (e) of ˜ F t implies that [ ˜ F t , ˜ A ⋅ ∂ j ] extends to an operator on ⊕ k L ( κ ) which is bounded by, say, c independent of t . Since moreover ∥ v i ∥ = ∥ ψ i ○ u i ○ ϕ − i ∥
22 (19) ≤ ∥ u i ∥ ⋅ L, we conclude ∥[ F it , D ] u i ∥ ≤ c ⋅ ∥ v i ∥ ⋅ L ≤ c ⋅ ∥ u i ∥ ⋅ L ≤ c ⋅ ∥ u ∥ ⋅ L . As neither L nor c depend on t , and this holds true for every 1 ≤ i ≤ l , we aredone. (cid:3) Remark
8: Note that we do not mind our construction in the proof of Lemma 5.4to be highly dependent on our choice of atlas and partition of unity.
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Anna Duwenig, Department of Mathematics and Statistics, University of Victoria,Victoria, BC, Canada V8P 5C2
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