THE INDEX OF CUSP OPERATORS ON MANIFOLDS WITH CORNERS

ROBERT LAUTER AND SERGIU MOROIANU

Abstract. We recall a Fredholm criterion for fully elliptic cusp (pseudo)differential operators on a compact manifold with corners of arbitrary codimension, acting on suitable Sobolev spaces. Then we give a formula for the index in terms of regularized “trace” functionals similar to the residue trace of Wodzicki and Guillemin.

1. Introduction

The index of elliptic pseudodifferential operators on a closed manifold was described as the integral of a characteristic form by the results of Atiyah-Singer. For manifolds with boundary, Atiyah-Patodi-Singer showed that under some boundary conditions, Dirac operators become Fredholm on suitable Sobolev spaces and computed the index. To the characteristic form from the closed case, they had to add a correction term, called the eta-invariant determined by an appropriate restriction of the operator to the boundary. The Atiyah-Patodi-Singer boundary conditions and the Sobolev spaces are completely natural in the context of “b”-pseudodifferential operators, as shown in the book [9] of Richard Melrose.

This last approach opens the gate for scores of other boundary conditions, many of them with natural extensions to manifolds with corners. Elliptic operators satisfying these conditions are Fredholm on the “correct” Sobolev spaces. The cusp structure is actually simpler to study than the classical b-structure, even though it is less natural to think of, and, in fact, the b-operators lift to cusp-operators under a transcendental blow up [13]. The index of fully elliptic cusp operators on compact manifolds with boundary was first computed in [13] as a sum of four terms. In [5], this formula was slightly modified to one similar to the Atiyah-Patodi-Singer formula. In this note, we extend the formula from [5] to an index formula for fully elliptic cusp pseudodifferential operators on manifolds with corners of arbitrary codimension (Theorem 5.2).

Let us briefly recall the idea of Melrose and Nistor [13] of writing the index of an elliptic operator in terms of functionals in Hochschild homology. Let M be a closed manifold, A an elliptic (classical) pseudo-differential operator (with scalar coefficients, for simplicity), and B a pseudodifferential operator that inverts A up to trace class remainders. Let Q be a first order, positive operator, and {Q^{τ}}_{τ∈ℂ} be the holomorphic family of complex powers of Q according to Seeley [19]. For any holomorphic family of classical pseudo-differential operators P of fixed integral order, the zeta-function Tr(P (τ)Q^{−τ}) is holomorphic for large real parts of τ, and admits a meromorphic extension to the whole plane, with at most simple poles at real, integer points. For a parametrix B of A as above we thus get:

Date: August 8, 2000.

ind (A) = Tr([A, B]) = Tr([A, B])Q^{−τ}|_{τ=0} = Tr((Q^{τ}BQ^{−τ} − B)AQ^{−τ})|_{τ=0}

(1) = Tr(τ[log Q, B]AQ^{−τ})|_{τ=0} = Res_{τ=0}Tr([log Q]AQ^{−τ}) = Tr_{R}([B, log Q]A) .

Here T r_{R} is the only trace functional on the algebra of pseudo-differential operators, called the residue trace, first discovered by Wodzicki [20] and Guillemin [3]. It has the property that it vanishes on operators of order less than − dim M. Therefore the index is expressed in terms of the full symbol of A by the above formula. Note that the term on the right hand side in (1) is a special case of the so-called Radul cocycle [18].

Similarly, the index of fully elliptic cusp operators on a manifold X with corners is given as the regularized value at 0 of a multi-dimensional version of the zeta-function, namely Tr([A, B]x^{z}Q^{−τ})_{z=0,τ=0} where x^{z} = ∏ x_{H}^{z_{H}} is the product over defining functions x_{H} of the boundary hyperfaces H of X. The index ind (A) can then be expressed as the sum of a local contribution \overline{AS}(A) determined as in (1) by finitely many terms of the complete symbol of A and a regularizer up to trace class remainders, and “boundarycontributions” that are roughly speaking determined by the restriction of A to faces of various codimension of X. We give explicit descriptions of these contributions in Theorem 5.2 in terms of appropriate functionals on the algebra of (suspended) cusp operators, and show that they are basically eta-like expressions in the sense of [11]. The main step in the proof is as usual to understand the (cusp)-trace of a commutator in terms of residue-type functionals.

Let us mention that when restricted to the diagonal z = τ = z_{H} for all H, the zeta function Tr([A, B]x^{z}Q^{−z}) has a pole of multiplicity possibly higher than 1 at z = 0. The coefficients in the Laurent expansion of such zeta functions at 0 were used in [1] for a local index formula of the associated spectral triple. However, compared with the multidimensional zeta function [5], the restricted one-dimensional version leads to a slightly more complicated index formula [13], with four terms instead of two already in the case of manifolds with boundary.

Let us stress that besides the local term \overline{AS}(A) all contributions to the index are defined in terms of the joint symbol of A, and do not depend on the choice of B. Of course, it follows that \overline{AS}(A) itself, like the index, depends only on the joint symbol of A. For surfaces, explicit expressions for the local term were recently obtained in [2].

We understand that the general form of the formula for the index of cusp pseudodifferential operators on manifolds with corners, and, in particular, the form of the boundary contributions, follows also by a homotopy argument from the corresponding formula for b-pseudodifferential operators given by Paul A. Loya [6] in his thesis; in this formula, the local contribution to the index is obtained by analyzing the heat b-trace expansion of b-pseudodifferential operators, whereas the boundary contributions come from a trace defect formula for the b-trace together with a regularizing procedure using heat kernels. Loya’s approach generalizes the methods from [9] to the case of manifolds with corners, and the formula extends earlier results of Melrose and Singer for Dirac operators on manifolds with corners to arbitrary b-pseudodifferential operators. The existence of this Atiyah-Patodi-Singer type formula for Dirac operators on manifolds with corners of arbitrary codimension was mentioned already in [12, page 95]. For the homotopy argument, note that any fully elliptic cusp pseudodifferential operator is homotopic to a fully elliptic b-pseudodifferential operators within fully elliptic operators. This was pointed out to the first author by Richard Melrose at the Oberwolfach meeting Geometric Analysis and Singular spaces (June 2000).

Let us now outline the plan of the note. In Section 2 and 3, we give a brief summary of the definitions and the Fredholm theory for cusp pseudodifferential operators on compact manifolds with corners. The analytical proofs are obvious extensions of those from the boundary (i.e. corners of codimension 1) case [8, 13], and hence are mostly omitted. Then we introduce in Section 4 certain “trace”-functionals on the (suspended) cusp calculus and discuss those properties that are essential for the index formula that is finally obtained in Section 5.

Acknowledgments: We have learnt about the cusp algebra in Richard Melrose’s seminar at MIT, and we benefited a lot from his geometric way of approaching analytic questions. We are grateful to him, Paul Loya, and Victor Nistor for providing us with preprints of their papers and for many useful remarks.

The first author would like to thank J. Cuntz, and the SFB 478, Geometrische Strukturen in der Mathematik for their warm hospitality at the University of Münster.

2. Cusp structures and cusp differential operators

Let us first define cusp differential operators. Let X be a manifold with corners in the sense of Melrose [16]. We assume that each boundary hyperface H of X is given as the zero set of a defining function x_{H} ∈ C^{∞}(X, \overline{ℝ}_{+}). In other words, H = \{x_{H} = 0\}, and dx_{H} ≠ 0 on H. We write M_{k}(X) for the set of all boundary faces of codimension k of X. The cusp structure associated to a fixed family of defining functions is the set V_{c} of all vector fields V on X satisfying V ∈ V_{T} + ∏_{H∈M_{1}(X)} x_{H}^{2}V(X), where V_{T} (x_{H}) = 0 for H ∈ M_{1}(X). It is easy to see that V_{c} is a Lie algebra, which depends on the functions x_{H} only up to terms vanishing quadratically at H. Moreover, V_{c} is the set of all smooth sections of a vector bundle, called ^{c}T X, and the inclusion maps ^{c}T X canonically (but not isomorphically) to T X. We agree to write ^{c}Ω^{α}(X) for Ω^{α}(^{c}T X) where Ω^{α} is the functor of α-densities.

Following [13], we define the algebra of cusp differential operators Diff_{c}(X) as the enveloping algebra of V_{c}. Since V_{c} is a C^{∞}(X)–module, we can analogously define cusp differential operators acting between sections of vector bundles E, F over X.

In the next Section, we are going to describe the construction of an algebra of cusp pseudodifferential operators that contains Diff_{c}(X); moreover, we explain a notion of (full) ellipticity that characterizes the Fredholm operators within the cusp algebra. This algebra has been constructed in [13] for manifolds with boundary. We give below an overview of the general construction, which is based on Melrose’s theory of blow-up and conormal distributions on manifolds with corners.

3. The algebra Ψ_{c}(X)

We recall that classical pseudo-differential operators on a closed manifold M can be defined as distributional sections over M × M of the density bundle 1 ⊗ ΩM, classical conormal to the diagonal. We are going to construct the cusp version X_{c}^{2} of X × X using iterated blow-ups. Then cusp pseudo-differential operators are just extendible distributions on this space, classical conormal to the pre-image of the diagonal, with values in a suitable density bundle. Recall that the (real spherical) blow-up of an interior submanifold Y of a manifold with corners X is defined as a set by [X; Y] = (X \ Y) ∪ SY, i.e. by replacing Y with the sphere bundle of the normal bundle of Y in X. The two pieces are glued using the geodesic rays of any Riemannian metric. There is a canonical smooth blow-down map β : [X; Y] → X which maps X \Y identically onto itself, and projects SY onto Y via the bundle map. For blowing up submanifolds that also meet the boundary, one has roughly speaking to replace the normal bundle by the inward pointing normal bundle. For a quick review on blow-up techniques we refer to [7], [9], or [10, Section 18]; more details can be found in the forthcoming book of Melrose [16].

Let us start with X × X. The family

B := ⋃_{H∈M_{1}(X)} H × H ⊆ (∂X)^{2}

of products of hypersurfaces intersect normally, in the sense of Melrose [16]. This allows us to define the iterated blow-up of all these submanifolds. The result is called X_{b}^{2}, the b–double space. It comes naturally equipped with a blow-down map β_{b}^{2} : X_{b}^{2} ⟶ X^{2}. Note that (up to a canonical diffeomorphism) the space X_{b}^{2} is independent of the order of the elements in M_{1}(X). To each hyperface H ∈ M_{1}(X) there corresponds a new boundary hyperface ff^{b}(H) of X_{b}^{2}. By [9, Lemma 4.1], the choice of the cusp structure yields a natural identification ff^{b}(H) = [−1, 1] × H_{b}^{2}; thus, the fiber diagonal submanifolds Φ^{2}(H) := {0}×H_{b}^{2} ⊆ ff^{b}(H) are well-defined. Since B_{1} := ⋃_{H∈M_{1}(X)} Φ^{2}(H) ⊆ X_{b}^{2} is again a normal family with Φ^{2}(H) and Φ^{2}(H′) intersecting transversally for H, H′ ∈ M_{1}(X), H ≠ H′, the cusp-double space X_{c}^{2} defined by

β_{c}^{2} : X_{c}^{2} := [X_{b}^{2}; B_{1}] ⟶ X_{b}^{2} ⟶^{β_{b}^{2}} X^{2}

does not depend on the order of the elements in M_{1}(X). To each H ∈ M_{1}(X) there corresponds a boundary hyperface ff^{c}(H) of X_{c}^{2}. The union ff^{c} := ⋃_{H∈M_{1}(X)} ff^{c}(H) is

called the cusp-front face. The lifted diagonal Δ_{c} := \overline{(β_{c}^{2})^{−1}(Δ \ ∂X^{2})}^{X_{c}^{2}} is a p-submanifold of X_{c}^{2} that intersects the boundary ∂X_{c}^{2} only in ff^{c}, and does it transversally. Restricting the blow-down map β_{c}^{2} to the lifted diagonal Δ_{c} induces a canonical diffeomorphism Δ_{c} ⟶^{≅} X. Moreover, lifting cusp vector fields from X to X_{c}^{2} through (β_{c}^{2})^{∗}p_{1}^{∗} induces an isomorphism between the normal bundle to the lifted diagonal Δ_{c} and ^{c}T X. In this sense, the cusp blow-up resolves the degeneracy of cusp vector fields.

For the sake of notational simplicity, we first define cusp operators acting on sections of the cusp half density bundle ^{c}Ω^{1/2}. The results of this section hold for operators between sections of arbitrary bundles as well. The changes are mainly notational.

Let ^{c}KD^{1/2} := x_{ff^{c}}^{−1}Ω^{1/2} be the cusp kernel half-density bundle. Here, we have x_{ff}^{c} := ∏_{H∈M_{1}(X)} x_{ff^{c}(H)}, where x_{ff^{c}(H)} is a defining function for the face ff^{c}(H) of X_{c}^{2}.

Definition 3.1. For m ∈ ℂ, let Ψ_{c}^{m}(X) be the space of distributional sections of the cusp kernel half-density bundle ^{c}KD^{1/2} on X_{c}^{2}, classical conormal of order m to Δ_{c}, vanishing rapidly to all boundary faces that do not intersect Δ_{c}, and extendible across all boundary faces.

One defines similarly the p-suspended cusp algebra on X. Namely, consider distributions on (ℝ^{p})^{2} × X_{c}^{2}, classical conormal of order m to Δ_{ℝ^{p}} × Δ_{c}, translation–invariant in the ℝ^{p} directions, and rapidly decreasing along lines not parallel to Δ_{ℝ^{p}}. In fact, later on we will mainly use the isomorphic space Ψ_{c}^{m}(X; ℝ^{p}) of parameter-dependent cusp operators that is obtained by applying the ℝ^{p}–Fourier transform to the convolution kernel of a p-suspended cusp operator.

The cusp differential operators of order m ∈ ℕ_{0} are exactly the differential operators in Ψ_{c}^{m}(X), i.e. those distributions supported on Δ_{c}. Hence, Ψ_{c}^{m}(X) is a sensible pseudo-differential extension of Diff_{c}(X).

Let us briefly describe the Fredholm theory for cusp pseudodifferential operators on compact manifolds with corners. Besides some technical points specific to the cusp algebra, most of the arguments are formal manipulations of symbols; they were described in detail for the double-edge calculus in [4]. For a comprehensive treatment of the cusp calculus on manifolds with boundary we refer to [13, Appendix]. Let us first observe that the distributional kernel κ_{A} ∈ Ψ_{c}^{m}(X) corresponds to a continuous linear operator A : Ċ^{∞}(X,^{c}Ω^{1/2}) ⟶ Ċ^{∞}(X,^{c}Ω^{1/2}) by a variant of the Schwartz kernel theorem. In particular, composition of cusp operators is well-defined. For a map F : M_{1}(X) → ℂ (called a weight system), consider x^{F} := ∏_{H∈M_{1}(X)} x_{H}^{F(H)} as an operator on Ċ^{∞}(X,^{c}Ω^{1/2}). Since we get the kernel of x^{−F}Ax^{F} by multiplying the kernel of A with a function that is 1 at Δ_{c}, smooth up to all faces of X_{c}^{2} whose intersection with Δ_{c} is non-empty, and polynomially bounded at all other boundary faces of X_{c}^{2}, we have x^{−F}Ψ_{c}^{m}(X)x^{F} = Ψ_{c}^{m}(X). This allows us to define cusp operators with a more general behavior at the boundary, namely

Ψ_{c}^{m,F}(X) := x^{−F}Ψ_{c}^{m}(X) = Ψ_{c}^{m}(X)x^{−F} .

Accordingly, the spaces Ψ_{c}^{m,F}(X; ℝ^{p}) are defined.

Using a triple version of the spaces X_{c}^{2} we prove that the cusp-calculus is closed under composition of operators on C^{∞}(X,^{c}Ω^{1/2}); more precisely, we have

Ψ_{c}^{m_{1},F_{1}}(X) × Ψ_{c}^{m_{2},F_{2}}(X) ⟶ Ψ_{c}^{m_{1}+m_{2},F_{1}+F_{2}}(X) : (A, B) ⟼ A ∘ B ,

By duality, cusp operators act also on C^{−∞}(X,^{c}Ω^{1/2}) with C^{∞}(X,^{c}Ω^{1/2}) as an invariant subspace. For more refined mapping properties, we need weighted cusp-Sobolev spaces. Let us denote by L^{2}(X,^{c}Ω^{1/2}) the space of square-integrable c-half densities, and define

H_{c}^{m,F}(X,^{c}Ω^{1/2}) := {f ∈ C^{−∞}(X,^{c}Ω^{1/2}) : Ψ_{c}^{m,F}(X)f ⊆ L^{2}(X,^{c}Ω^{1/2})} .

Let us note that the spaces H_{c}^{m,F}(X,^{c}Ω^{1/2}) are Hilbert(able) spaces, in which the space Ċ^{∞}(X,^{c}Ω^{1/2}) is dense. The main result on boundedness is summarized in the next proposition.

Proposition 3.2. Let s, m ∈ ℂ and F, G : M_{1}(X) ⟶ ℂ be arbitrary. Then each A ∈ Ψ_{c}^{m,F}(X) induces a bounded linear operator

A : H^{s,G}(X,^{c}Ω^{1/2}) ⟶ H^{s−m,G−F}(X,^{c}Ω^{1/2}) .

To understand which cusp operators act as Fredholm operators, we need to understand the behavior of the kernels up to the first order at the diagonal and the faces of X_{c}^{2}. As in the classical case, Fourier transform along the fibers of the normal bundle NΔ_{c} to the cusp diagonal (which is canonically isomorphic to the cusp tangent bundle ^{c}T X) leads to the principal symbol map

cσ : Ψ_{c}^{m,F}(X) ⟶ x^{−F}S^{[m]}(^{c}T^{∗}X) ≅ C^{∞}(^{c}S^{∗}X) ,

where S^{[m]}(^{c}T^{∗}X) stands for the space of smooth functions on ^{c}T^{∗}X \ {0} homogeneous of degree m in the fibers, and the last isomorphism is given by the choice of the defining functions x_{H}, H ∈ M_{1}(X), and a defining function for the cusp cosphere bundle ^{c}S^{∗}X in the radial compactification of ^{c}T^{∗}X. This leads to the following short exact sequence

0 ⟶ Ψ_{c}^{m−1,F}(X) ⟶ Ψ_{c}^{m,F}(X) ⟶^{^{c}σ} C^{∞}(^{c}S^{∗}X) ⟶ 0 . (2)

Similarly, we obtain the corresponding sequence for parameter-dependent cusp-operators

0 ⟶ Ψ_{c}^{m−1,F}(X; ℝ^{p}) ⟶ Ψ_{c}^{m,F}(X) ⟶^{^{c}σ̃} C^{∞}(^{c}S^{∗}X_{p}) ⟶ 0 , (3)

where ^{c}S^{∗}X_{p} := ((^{c}T^{∗}X × ℝ^{p}) \ {0}) /ℝ_{+}.

In order to analyze the behavior at the boundary faces, let us introduce the following notation. For E ⊆ M_{1}(X), let δ_{E} : M_{1}(X) → {0, 1} be the characteristic function of E. For G ∈ M_{k}(X) and for F : M_{1}(X) ⟶ ℂ, we write E(G) := {H ∈ M_{1}(X) : G ⊆ H} and F_{G} : M_{1}(G) → ℂ : H ↦ F(H′), where H′ ∈ M_{1}(X) is the unique boundary hyperface of X with H = H′ ∩ G. Note furthermore, that the part ff^{c}(G) of the front face ff^{c} that is lifted from G × G is canonically identified with the lift to G_{c}^{2} of the normal bundle N_{X}(G) of G in X. The choice of a cusp structure fixes an isomorphism of this bundle to ℝ^{E(G)}.

Taking the leading part (in a Laurent expansion) of a kernel at ff^{c}(G) and integrating it along the fibers of N_{X}(G) yields a morphism (·)_{G} : Ψ_{c}^{m,F}(X) → Ψ_{c}^{m,F_{G}}(G). For H ∈ M_{1}(X) and ξ ∈ ℝ, the function exp(iξ/x_{H}) acts as a multiplier on Ċ^{∞}(X,^{c}Ω^{1/2}), and we have exp(iξ/x_{H})Ψ_{c}^{m,F}(X) exp(−iξ/x_{H}) ⊆ Ψ_{c}^{m,F}(X). The leading part of exp(iξ/x_{H})A exp(−iξ/x_{H}) at ff^{c}(G) is given by e^{−iS_{H}ξ}κ_{A}|_{ff^{c}(G)} where κ_{A}|_{ff^{c}(G)} is the leading part of the kernel of A, and (S_{H})_{H∈E(G)} are the canonical coordinates in ℝ^{E(G)}. Therefore, the following map is well-defined:

I_{G,X}^{F} : Ψ_{c}^{m,F}(X) ⟶ Ψ_{c}^{m,F_{G}}(G; ℝ^{E(G)}) : A ⟼ [ℝ^{E(G)} ∋ ξ ↦ (e^{iξ/x}Ae^{−iξ/x})_{G}] ,

where e^{iξ/x} := ∏_{H∈E(G)} e^{iξ_{H}/x_{H}}. We call I_{G,X}^{F} the indicial family at G; it induces the following short exact sequence.

0 ⟶ ∑_{H∈E(G)} Ψ_{c}^{m,F−δ_{{H}}}(X) ⟶ Ψ_{c}^{m,F}(X) ⟶^{I_{G,X}^{F}} Ψ_{c}^{m,F_{G}}(G; ℝ^{E(G)}) ⟶ 0 . (4)

In the sequel we will write I_{G} for I_{G,X}^{F} when the ambient space X and the weight system are clear from the context. The principal symbol and the indicial families are related by the following set of compatibility conditions

(5)  ^{c}σ̃(I_{G}(A)) = ^{c}σ(A)|_{^{c}S^{∗}X|_{G}} ,
(6)  I_{F,X}^{F}(A)(ξ_{E(F)}) = I_{F,G}^{F_{G}} (I_{G,X}^{F}(A)(ξ_{E(G)})) (ξ_{E(F)\E(G)}) ,

whenever F, G are boundary faces of X with F ⊆ G, and ξ_{E(F)} ∈ ℝ^{E(F)}, ξ_{E(G)} ∈ ℝ^{E(G)}, and ξ_{E(F)\E(G)} ∈ ℝ^{E(F)\E(G)}.

Let O : M_{1}(X) → ℂ : H ↦ 0 be the zero-weight. Then it is immediate that for A = x^{−F}A_{0} = A_{1}x^{−F} ∈ Ψ_{c}^{m,F}(X) we have

I_{G,X}^{F}(A) = I_{G,X}^{O}(A_{0}) = I_{G,X}^{O}(A_{1}). (7)

To simplify matters, we combine the principal symbol and the indicial families into the joint symbol

τ := ^{c}σ ⊕ ⨁_{G∈M(X)} I_{G,X}^{F} : Ψ_{c}^{m,F}(X) ⟶ Q ⊆ C^{∞}(^{c}S^{∗}X) ⊕ ⨁_{G∈M(X)} Ψ_{c}^{m,F_{G}}(G; ℝ^{E(G)}) ,

where the space Q consists simply of all tuples satisfying the compatibility conditions (5) and (6) for all F, G. A combination of (2) and (4) yields the following exact sequence for the joint symbol

0 ⟶ Ψ_{c}^{m−1,F−δ_{M_{1}(X)}}(X) ⟶ Ψ_{c}^{m,F}(X) ⟶^{τ} Q ⟶ 0 . (8)

Thus, a formal symbolic argument shows that an operator A ∈ Ψ_{c}^{m,F}(X) whose joint symbol is invertible, is invertible up to elements in the totally residual space

⋂_{p,F} Ψ_{c}^{p,F}(X) = Ψ_{c}^{−∞,−∞}(X) ≅ Ċ^{∞}(X^{2},^{c}Ω^{1/2})

and has a pseudo-inverse B ∈ Ψ_{c}^{−m,−F}(X). The invertibility of τ(A) is equivalent to the invertibility of^{c}σ(A)(ζ) for all ζ ∈ ^{c}S^{∗}X, and I_{G,X}^{F}(A)(ξ) : H_{c}^{m,F_{G}}(G,^{c}Ω^{1/2}) → L^{2}(G,^{c}Ω^{1/2}) for all ξ ∈ ℝ^{E(G)} and all G ∈ M_{k}(X). Since inverses of invertible cusp operators are again cusp operators, one can show inductively that in the last condition it is enough to have invertibility only for those G ∈ M_{1}(X). Cusp operators with invertible joint symbols are called fully elliptic.

Theorem 3.3. Let A ∈ Ψ_{c}^{m,F}(X) be arbitrary. Then we have

(a) A : H^{s,G}(X,^{c}Ω^{1/2}) ⟶ H^{s−m,G−F}(X,^{c}Ω^{1/2}) is compact if and only if τ(A) = 0.

(b) A : H^{s,G}(X,^{c}Ω^{1/2}) ⟶ H^{s−m,G−F}(X,^{c}Ω^{1/2}) is a Fredholm operator if and only if it is fully elliptic. In that case, we find a Fredholm inverse up to Ψ_{c}^{−∞,−∞}(X) in Ψ_{c}^{−m,−F}(X).

(c) If τ(A) is invertible, and Au ∈ H^{s,G}(X,^{c}Ω^{1/2}) for some u ∈ H^{∞,∞}(X,^{c}Ω^{1/2}), then we have u ∈ H^{s+m,F+G}(X,^{c}Ω^{1/2}); in particular, we have N(A) ⊆ Ċ^{∞}(X,^{c}Ω^{1/2}) and N(A^{∗)} ⊆ Ċ^{∞}(X,^{c}Ω^{1/2}), and the index of the Fredholm operator A

ind A = dim N(A) − dim N(A^{∗})

does neither depend on the choice of the Sobolev order s nor on the weight G.

Note the similarities but also the significant differences to the corresponding results for the b-calculus on manifolds with corners [6, 14, 15].

Let us close this section with another property of the cusp calculus that will be used in the computation of the index in Section 5. For each hyperface H ∈ M_{1}(X), the function log x_{H} acts as an multiplier on Ċ^{∞}(X,^{c}Ω^{1/2}), hence the commutator with log x_{H} is well-defined.

Lemma 3.4. For A ∈ Ψ_{c}^{m,F}(X) and H ∈ M_{1}(X), we have

\frac{d}{dz} (x_{H}^{z}Ax_{H}^{−z}) |_{z=0} = [log x_{H}, A] ∈ Ψ_{c}^{m−1,F−δ_{\{H\}}}(X) .

Moreover, we have

I_{H,X}^{F−δ_{{H}}}([log x_{H}, A])(ξ_{H}) = i∂_{ξ_{H}}I_{H,X}^{F}(A)(ξ_{H}) . (9)

Proof: From Equation (8), the first claim follows if we prove that the principal symbol and the indicial families at hyperfaces of the left-hand side vanish. This is clear from (7) and from the invariance to conjugation of the principal symbol. Now the second claim is easy to check for cusp differential operators. The general proof is straightforward in local coordinates, so we will omit it.

In the sequel we will write D_{H}(A) := [log x_{H}, A]. For S = {H_{1}, …, H_{k}} ⊆ M_{1}(X), we abbreviate D_{S} := D_{H_{1}} ∘ ⋯ ∘ D_{H_{k}}.

Remark 3.5. Let S = {H_{1}, …, H_{k}} ⊆ M_{1}(X) be arbitrary. Then an iterated application of Lemma 3.4 yields

D_{H_{1}} ∘ ⋯ ∘ D_{H_{k}} : Ψ_{c}^{m,F}(X) ⟶ Ψ_{c}^{m−k,F−δ_{S}}(X)

and for each connected component G of ⋂_{j=1}^{k} H_{j} and any A ∈ Ψ_{c}^{m,F}(X), we have

I_{G,X}^{F−δ_{S}}(D_{H_{1}} ∘ ⋯ ∘ D_{H_{k}}(A))((ξ_{H})_{H∈E(G)}) = i^{k}∂_{ξ_{H_{1}}} ⋯ ∂_{ξ_{H_{k}}} I_{G,X}^{F}(A)((ξ_{H})_{H∈E(G)}) .

4. Trace-like functionals on the cusp calculus

In this section, we will use several times the existence of a meromorphic extension of a given holomorphic function defined as the trace of a holomorphic family of cusp operators. The proofs are all very similar, and consist of a Taylor–type expansion combined with the following trivial lemma.

Lemma 4.1. The integrals ∫_{1}^{∞} t^{−z−1}dt and ∫_{0}^{1} t^{z−1}dt are absolutely convergent and define holomorphic functions in the domain {z ∈ ℂ : Re z > 0}; they extend meromorphically to the whole complex plane with a simple pole at z = 0

Let F : M_{1}(X) → ℕ_{0} be a weight system. We are going to extend the usual integral

∫_{X} : Ċ^{∞}(X,^{c}Ω^{1}) ⟶ ℂ (10)

to the space x^{F}C^{∞}(X,^{c}Ω^{1}). For f ∈ C^{∞}(X,^{c}Ω^{1}), the function G(f) : z ↦ ∫_{X} x^{z}f is holomorphic in the domain {z ∈ ℂ^{M_{1}(X)} : Re z_{H} + F(H) > 1 for all H ∈ M_{1}(X)}. Moreover, G(f) extends meromorphically to ℂ^{M_{1}(X)} with at most simple poles at the points z_{H} = 1 − F(H) − j, j ∈ ℕ_{0}. Let ^{c-}∫_{X}^{x} f be the regularized value of G(f) at z_{H} = 0, H ∈ M_{1}(X). Note that ^{c-}∫_{X}^{x} extends (10) as desired but depends on the choice of the defining functions x_{H} of H ∈ M_{1}(X) provided F(H) ≤ 1. However, the dependence is “controlled” in the following sense: if codim X = 1, i.e. X is a manifold with boundary, and x′ = x + bx^{2} with 0 < b ∈ C^{∞}(X) and 1 + bx > 0 are two cusp-equivalent boundary defining functions for X, then we have

c-∫ ^{x} f −^{c-}∫_{X}^{x′} f = Res_{z=0}G(ln(1 + bx)f) = \{ \begin{matrix}∫_{∂X} b|_{∂X}f|_{∂X} & , F(∂X) = 0 \\ 0 & , F(∂X) = 1 \end{matrix} . (11)

The dependence on the choice of the defining functions x_{H} for higher codimensions can be expressed inductively; we skip the details and, for the sake of simplicity, we fix for the rest of the paper a system of defining functions x_{H}, H ∈ M_{1}(X).

Similarly, we can also extend the integral to x^{F}C^{∞}(X,^{c}Ω^{1}) for a general weight system F : M_{1}(X) → ℂ, but the dependence on the defining functions becomes more complicated involving also derivatives of b_{H} and f. Therefore, we restrict ourselves to a weight system F : M_{1}(X) → ℕ_{0} as above.

We are now going to use the cusp integral to define an extension of the trace for trace class operators to arbitrary elements in Ψ_{c}^{m,−F}(X) of sufficiently negative order. To this end, first observe that when restricted to Δ_{c} ≅ X, the cusp kernel half-density bundle on X_{c}^{2} is naturally isomorphic to the cusp density bundle ^{c}Ω^{1}(X). Thus, for any m < − dim X, we can define

cTr : Ψ_{c}^{m,−F}(X) ⟶ ℂ : A ⟼ ^{c-}∫_{X}^{x} κ_{A}|_{Δ_{c}} . (12)

Note that even though the kernel of A is only continuous because of m < − dim X, it restricts to an element in x^{F}C^{∞}(X,^{c}Ω^{1}) by the properties of conormal distributions. We call ^{c}Tr the cusp trace. Since it is defined in terms of the cusp integral, the cusp trace also depends on the choice of the defining functions x_{H}, H ∈ M_{1}(X). Finally, let us note that for A ∈ Ψ_{c}^{m,−F}(X), m < − dim X, the identity f(0) = ∫_{ℝ^{n}} F(f)(ζ)dζ where F is the Fourier transform, gives

cTr(A) = ^{c-}∫_{^{c}T^{∗}X}^{x} aω_{c}^{dim(X)} (13)

for any symbol a ∈ S^{m}(^{c}T^{∗}X) satisfying F^{−1}a = κ_{A} near Δ_{c}, where ω_{c} is the (degenerate) canonical symplectic form on ^{c}T^{∗}X induced from T^{∗}X through the natural map of vector bundles. The cusp integral for the non-compact manifold ^{c}T^{∗}X is either defined as for X taking additional growth conditions at infinity into account, or by considering the radial compactification of ^{c}T^{∗}X which is a compact manifold with corners.

Recall that for A ∈ Ψ_{c}^{m,−F}(X; ℝ^{p}), we have A(ξ) ∈ Ψ_{c}^{m,−F}(X). Using the inequality

sup_{ζ∈ℝ^{p}}⟨ζ⟩^{−ℓ}⟨ζ, ξ⟩^{m} ≤ C_{ℓ}⟨ξ⟩^{m−ℓ} for all ξ ∈ ℝ^{p} and ℓ < 0 ,

we obtain for m < − dim X, any α > 0 with m + α < − dim X, and A ∈ Ψ_{c}^{m,−F}(X; ℝ^{p})

|^{c}Tr(A(ξ))| ≤ C(α, A)⟨ξ⟩^{−α} for all ξ ∈ ℝ^{p} .

Therefore, for any m < − dim X −p, we obtain a well-defined, continuous linear functional

c\overline{Tr} : Ψ_{c}^{m,−F}(X; ℝ^{p}) ⟶ ℂ : A ⟼ ∫_{ℝ^{p}} ^{c}Tr(A(ξ))dξ (14)

which is not a trace for positive codimension of X because of the trace defect of the cusp trace ^{c}Tr.

Let us now extend this functional to parameter-dependent cusp operators of arbitrary order m. To this end, fix a positive operator Q ∈ Ψ_{c}^{1}(X; ℝ^{p}). As in [3, Theorem 5.5], we construct a holomorphic family ℂ ∋ τ ↦ Q^{τ} ∈ Ψ_{c}^{τ}(X; ℝ^{p}) such that Q^{τ_{1}}Q^{τ_{2}} = Q^{τ_{1}+τ_{2}} and Q^{0} = E : ξ ↦ id_{X}. Then, for A ∈ Ψ_{c}^{m,−F}(X; ℝ^{p}), the function H_{Q}(A) : τ ↦ ^{c}\overline{Tr}(AQ^{−τ}) is holomorphic in the domain \{τ ∈ ℂ : Re τ > m + p + dim X\}, and admits a meromorphic extension to the whole plane with at most simple poles at τ = m + dim X + p − j, j ∈ ℕ_{0}. Let us denote by ^{c}\overline{Tr}_{R}(A) resp. ^{c}\overline{Tr}_{Q}(A) the residue resp. the regularized value of H_{Q}(A) at τ = 0.

By the very definition, ^{c}\overline{Tr}_{Q} is an extension of ^{c}\overline{Tr} that depends on the choice of Q. The residue, however, is independent on Q and local in the sense that it vanishes on operators of sufficiently negative order; in fact, we have the following explicit formula.

Proposition 4.2. Let n := dim X and A ∈ Ψ_{c}^{m,−F}(X; ℝ^{p}) be arbitrary. Then modulo Ψ_{c}^{−∞,−F}(X; ℝ^{p}), A has an expansion A ∼ ∑_{j≤p} a_{j} with a_{j} ∈ x^{F}S^{[j]}(^{c}T^{∗}X × ℝ^{p}), and we have

c\overline{Tr}_{R}(A) = ^{c-}∫_{^{c}S^{∗}X_{p}}^{x} a_{−p−n}ι_{R}(ω_{c}^{n} ∧ dξ) , (15)

where R is the radial vector field in the fibers of ^{c}T^{∗}X × ℝ^{p}.

We will omit the proof, since it does not differ significantly from the residue–trace formula of Wodzicki and Guillemin.

Let us mention that for a closed manifold X the functional ^{c}\overline{Tr} in (14) has been considered in [11]. However, the extension to arbitrary orders given in [11] is slightly different from ^{c}\overline{Tr}_{Q} introduced above. See also Nistor’s preprint [17] for a discussion of this discrepancy.

Similarly, let now Q ∈ Ψ_{c}^{1}(X) be positive, and ℂ ∋ τ ↦ Q^{τ} ∈ Ψ_{c}^{τ}(X) be the corresponding holomorphic family of complex powers. For z ∈ ℂ^{M_{1}(X)}, let us denote by x^{z} ∈ Ψ_{c}^{0,z}(X) the multiplier ∏_{H∈M_{1}(X)} x_{H}^{z_{H}}. Note that x^{z} does not depend on an order on M_{1}(X).

Proposition 4.3. For each holomorphic function A : ℂ^{M_{1}(X)} × ℂ → Ψ_{c}^{m,−F}(X), the associated function F_{Q}(A) : (z, τ) ↦ T r(A(z, τ)x^{z}Q^{−τ}) is holomorphic in the domain {(z, τ) ∈ ℂ^{M_{1}(X)}×ℂ : Re z_{H} > 1−F(H), Re τ > m+n}, and it extends to a meromorphic function with at most simple poles in each variable at the points z_{H} = 1 − F(H) − j_{H}, τ = m + n − ℓ, for j_{H}, ℓ ∈ ℕ_{0}.

This justifies the definition of the following functionals as the coefficients of the expansion of F_{Q}(A) near the origin:

Definition 4.4. Let A ∈ Ψ_{c}^{m}(X). For ε ∈ {0, 1}, and S ⊆ M_{1}(X), we define T̂r_{S,ε}(A) by

(16) F_{Q}(A)(z, τ) = \frac{1}{τ}T̂r_{∅,1}(A) + ∑_{∅≠S⊆M_{1}(X)} ∑_{ε=0,1} \frac{1}{z^{δ_{S}}τ^{ε}} T̂r_{S,ε}(A) + Regular(0) .

Here Regular(0) means a remainder which is regular at the origin in each variable.

Let us now consider the functionals T̂r_{S,ε} : Ψ_{c}^{m}(X) ⟶ ℂ in more detail. Note that we have Ψ_{c}^{m,−F}(X) ⊆ Ψ_{c}^{m}(X) for any weight system F : M_{1}(X) → ℕ_{0}.

We first observe that the functionals T̂r_{S,ε} with ε = 1 are local, i.e. they vanish on cusp operators of order −∞, whereas the functionals with ε = 0 are not local in this sense. For ∅ ≠ S ⊆ M_{1}(X), let us denote by G_{S} ⊆ M_{|S|}(X) the set of connected components of ⋂_{H∈S} H.

Proposition 4.5. Let ∅ ≠ S ⊆ M_{1}(X) and A ∈ Ψ_{c}^{m,−δ_{S}}(X) be arbitrary. Then we have

T̂r_{S,0}(A) = ∑_{G∈G_{S}} ^{c}\overline{Tr}_{Q_{G}} (I_{G}(A)) , T̂r_{S,1}(A) = ∑_{G∈G_{S}} ^{c}\overline{Tr}_{R} (I_{G}(A)) ,

where Q_{G} := I_{G}(Q) ∈ Ψ_{c}^{1}(G; ℝ^{E(G)}).

Proof: Let \overline{S} := M_{1}(X) \ S be the complement of S in M_{1}(X), and assume first that Re τ and, for H ∈ \overline{S}, Re z_{H} are large.

Because A ∈ Ψ_{c}^{m,−δ_{S}}(X), we have Ax^{zδ_{M_{1}(X)}}Q^{−τ} = Bx^{(z+1)δ_{S}} for some cusp operator B ∈ Ψ_{c}^{m−τ,−zδ\overline{S}}(X). As in (13), let b ∈ S^{m−τ}(^{c}T^{∗}X) be such that F^{−1}b = κ_{B} near Δ_{c}. Then b is non-singular at the faces H ∈ S, and moreover we can assume that the restriction b_{G} of b at G ∈ G_{S} is the full symbol of I_{G}(B). Since the degenerate symplectic volume form is a multiple of x^{−2δ_{S}}, from (13) and Lemma 4.1 it follows that the coefficient of z^{−δ_{S}} in the Laurent expansion of F_{Q}(A) is given by

∑_{G∈G_{S}} ∫_{ℝ^{E(G)}} Tr (I_{G}(B)(ξ)) dξ .

This depends holomorphically on the parameters τ and z_{H} for large Re τ and Re z_{H}, H ∈ \overline{S}. Thus, the coefficient of z^{−δ_{S}} is given (as a function of τ) by the regularized value at z_{H} = 0 of the meromorphic extension with respect to the variables (z_{H})_{H∈\overline{S}} of this sum. On the other hand, we know

I_{G}(B) = I_{G}^{−δ_{S}}(A)x^{zδ\overline{S}}Q_{G}^{−τ} ,

since the indicial family operator is multiplicative, and I_{G}(x_{H}) equals 1 if H ∈ S, and x_{H} otherwise. Thus, the regularized value at z_{H} = 0 is given by replacing Tr with the cusp trace ^{c}Tr:

τ ⟼ ∑_{G∈G_{S}} ^{c}Tr (I_{G}(A)(ξ)Q_{G}(ξ)^{−τ}) dξ = ∑_{G∈G_{S}} ∫_{ℝ^{E(G)}} ^{c}\overline{Tr}(I_{G}(A)Q_{G}^{−τ}) (17)

provided the real part of τ is large enough. Now T̂r_{S,ε}(A) is either the residue (ε = 1) or the regularized value (ε = 0) at τ = 0 of the terms on the right-hand side in (17). Therefore, a look at the definitions of ^{c}\overline{Tr}_{R} and ^{c}\overline{Tr}_{Q_{G}} completes the proof.

Let us point out that the expressions for T̂r_{S,ε}(A) for arbitrary A ∈ Ψ_{c}^{m}(X) are much more complicated. Fortunately, Proposition 4.5 is all what we need in order to understand the various contributions to the index of fully elliptic operators. It remains to consider the case S = ∅ and ε = 1. Here, in fact, we recover the formula for the Wodzicki residue.

Proposition 4.6. Let A ∈ Ψ_{c}^{m}(X) be arbitrary. Then the image of A under the quotient map Ψ_{c}^{m}(X) → Ψ_{c}^{m}(X)/Ψ_{c}^{−∞}(X) has an expansion of the form ∑_{j≤m} a_{j} with a_{j} ∈ S^{[j]}(^{c}T^{∗}X), and we have

T̂r_{∅,1}(A) = ^{c-}∫_{^{c}S^{∗}X}^{x} a_{−n}ι_{R}ω_{c}^{n} , (18)

where R is the radial vector field in the fibers of ^{c}T^{∗}X.

Proof: We fix z ∈ ℂ^{M_{1}(X)} with Re z_{H} large, and apply either the well-known formula for closed manifolds to a closed extension X̃ of X and an extension of the operator Ax^{z}, or follow step by step the proof of this formula. Note that in the proof we need only a finite number of derivatives of the local symbols up to a certain order. This smoothness for Ax^{z} is obtained by taking Re z_{H} large enough. To obtain the coefficient of τ^{−1} it remains to take the regularized value at z = 0 which leads to the cusp integral in the final formula (18).

We are now going to describe on which subspaces of Ψ_{c}^{ℤ}(X) := ⋃_{m∈ℤ} Ψ_{c}^{m}(X) the functionals T̂r_{S,ε} act as traces, in the sense that they vanish on commutators. For S ⊆ M_{1}(X), let us denote by \overline{S} := M_{1}(X) \ S its complement.

Proposition 4.7. Let A_{j} ∈ Ψ_{c}^{m_{j},−F_{j}}(X), j = 1, 2, be arbitrary with F_{j} : M_{1}(X) ⟶ ℕ_{0}.

(a) If F_{1}(H)+F_{2}(H) ≥ 2 for all H ∈ \overline{S}, then we have T̂r_{S,1}([A_{1}, A_{2}]) = 0. In particular, the restriction Tr_{S,1} of T̂r_{S,1} to Ψ_{c}^{ℤ,−δ\overline{S}}(X) is a trace.

(b) If S ≠ ∅, m_{1} + m_{2} < − dim X, and F_{1}(H) + F_{2}(H) ≥ 2 for all H ∈ \overline{S}, then we have T̂r_{S,0}([A_{1}, A_{2}]) = 0. In particular, for 2m > dim X, the restriction Tr_{S,0} of T̂r_{S,0} to Ψ_{c}^{−m,−δ\overline{S}}(X) has the trace property.

Proof: Fix an order S = \{1, …, k\} on S. Because of [A_{1}, A_{2}] ∈ Ψ_{c}^{m_{1}+m_{2}−1,−2δ\overline{S}}(X) the function F_{Q}([A_{1}, A_{2}]) is analytic near z_{H} = 0 for H ∈ \overline{S}. Therefore, Tr_{S,1}([A_{1}, A_{2}]) is the coefficient of z^{−δ_{S}}τ^{−1} in the Laurent expansion of the meromorphic extension of

Tr[A_{1}, A_{2}]x^{zδ_{S}}Q^{−τ} = TrA_{1}(A_{2} − x_{1}^{z_{1}}A_{2}x_{1}^{−z_{1}})x_{1}^{z_{1}} ⋯ x_{k}^{z_{k}}Q^{−τ} + … + TrA_{1}x_{1}^{z_{1}} ⋯ x_{k}^{z_{k}}(A_{2} − Q^{−τ}A_{2}Q^{τ})Q^{−τ} .

On the other hand, each term (A_{2} − x_{j}^{z_{j}}A_{2}x_{j}^{−z_{j}}), j = 1, …, k, resp. (A_{2} − Q^{−τ}A_{2}Q^{τ}) is an analytic family of cusp operators of fixed order that vanishes at z_{j} = 0 or τ = 0, therefore, none of the terms contributes non-trivially to the coefficient of z^{−δ_{S}}τ^{−1}, hence Tr_{S,1}([A_{1}, A_{2}]) = 0, which completes the proof of (a). The proof of (b) is completely similar, hence we omit it.

As a consequence, we see that only Tr_{M_{1}(X),1} is a trace on Ψ_{c}^{ℤ}(X). When X is a manifold without boundary, we get just one functional, the residue trace T r_{R} of Wodzicki and Guillemin. For a manifold with boundary, we get the trace functionals Tr_{∂,σ} = Tr_{M_{1}(X),1}, T̂r_{σ} = T̂r_{∅,1}, and T̂r_{∂} = T̂r_{M_{1}(X),0} of Melrose and Nistor [13].

5. The index formula

We are now going to express the index of a cusp Fredholm operator on X in terms of the functionals introduced in the previous section. We start with some preliminary results. Similar to the commutators D_{H} with log x_{H}, H ∈ M_{1}(X), we introduce for B ∈ Ψ_{c}^{m,F}(X) the commutator with “log Q” where Q ∈ Ψ_{c}^{1}(X) is as in the previous section.

D_{Q}(B) := [log Q, B] := \frac{d}{dτ} (Q^{τ}AQ^{−τ})|_{τ=0} .

This is well-defined because τ ↦ Q^{−τ}BQ^{τ} ∈ Ψ_{c}^{m,F}(X) is an analytic family of fixed order. Since the principal symbol of Q^{−τ}BQ^{τ} vanishes, we even have

(19) D_{Q}(B) = [log Q, B] ∈ Ψ_{c}^{m−1,F}(X) for B ∈ Ψ_{c}^{m,F}(X) .

The indicial operators are continuous, thus, we obtain for any G ∈ M_{k}(X)

(20)  I_{G}(D_{Q}(B)) = D_{Q_{G}}(I_{G}(B))
:= [log Q_{G}, I_{G}(B)]
:= \frac{d}{dτ} (Q_{G}^{τ}I_{G}(B)Q_{G}^{−τ})|_{τ=0} ∈ Ψ_{c}^{m−1,F_{G}}(G; ℝ^{E(G)}) ,

where we have set again Q_{G} := I_{G,X}^{O}(Q) ∈ Ψ_{c}^{1}(G; ℝ^{E(G)}).

Let A ∈ Ψ_{c}^{m,F}(X) and B ∈ Ψ_{c}^{m′,F′}(X) be cusp operators such that m + m′ ∈ ℤ and F + F′ : M_{1}(X) → ℤ. Assume that [A, B] is of trace-class. Our basic example is a fully elliptic operator A together with a parametrix B up to trace class remainders.

For technical reasons let us fix temporarily an order on M_{1}(X), i.e. a bijection {1, …, ℓ} → M_{1}(X), and, for each subset S = {j_{1}, …, j_{k}} ⊆ {1, …, ℓ} = M_{1}(X) with j_{1} < ⋯ < j_{k}, define D_{S} := D_{j_{1}} ∘ ⋯ ∘ D_{j_{k}} for S ≠ ∅, and D_{∅} = id.

The next proposition is the main step in the computation of the index. For similar results in the b-setting we refer to [6, Theorem 6.6] and [9, Proposition 5.9].

Proposition 5.1.

Tr[A, B] = T̂r_{∅,1}(D_{Q}(B)A) − ∑_{∅≠S⊆M_{1}(X)} T̂r_{S,0}(AD_{S}(B)) .

Proof: We start by noting that T r([A, B]) = F_{Q}([A, B])|_{z=0,τ=0}. The following identity holds by Proposition 4.6 in an open subset of ℂ^{ℓ+1}, hence everywhere by analytic continuation:

F_{Q}([A, B])(z, τ) = Tr (A(B − x_{1}^{z_{1}}Bx_{1}^{−z_{1}})x_{1}^{z_{1}} ⋯ x_{ℓ}^{z_{ℓ}}Q^{−τ}) + ⋯
+ Tr (Ax_{1}^{z_{1}} ⋯ x_{ℓ−1}^{z_{ℓ−1}}(B − x^{z_{ℓ}}Bx^{−ℓ})x_{ℓ}^{z_{ℓ}}Q^{−τ})
+ Tr ((Q^{τ}BQ^{−τ} − B)Ax_{1}^{z_{1}} ⋯ x_{ℓ}^{z_{ℓ}}Q^{−τ}) .

We are interested in evaluating the right-hand side at z_{1} = … = z_{ℓ} = τ = 0. Each term is not going to be regular, but of course the evaluation of the sum is the sum of the regularized values of each term. By Proposition 4.3, we can neglect in these regularized values the trace of any holomorphic multiple of z_{j}^{2}, for some j, or of τ^{2}. For instance, (B−x_{1}^{z_{1}}Bx_{1}^{−z_{1}})+z_{1}D_{1}(B) is a holomorphic multiple of z_{1}^{2}. This means that the regularized values of T r(A(B − x_{1}^{z_{1}}Bx_{1}^{−z_{1}})x_{1}^{z_{1}}x_{2}^{z_{2}} ⋯ x_{ℓ}^{z_{ℓ}}Q^{−τ}) and of −z_{1}T r(AD_{1}(B)x_{1}^{z_{1}} ⋯ x_{ℓ}^{z_{ℓ}}Q^{−τ}) at z_{1} = … = z_{ℓ} = τ = 0 are the same. A corresponding equality holds for the last term with x_{1} replaced by Q.

In general, we want to describe the regularized values of

−z_{s}T r (Ax_{1}^{z_{1}} ⋯ x_{s−1}^{z_{s−1}}D_{s}(B)x_{s}^{z_{s}} ⋯ x_{ℓ}^{z_{ℓ}}Q^{−τ}) resp. (21)

τTr (D_{Q}(B)Ax_{1}^{z_{1}} ⋯ x_{ℓ}^{z_{ℓ}}Q^{−τ}) . (22)

Definition 4.4 immediately applies to (22), and shows that regularized value is given by

T̂r_{∅,1}(D_{Q}(B)A) .

To use Definition 4.4 in (21) as well, we would like to bring all x_{j}^{z_{j}} to the right. Each such operation gives rise to a derivation:

x_{j}^{z_{j}}C = Cx_{j}^{z_{j}} + (x_{j}^{z_{j}}Cx_{j}^{−z_{j}} − C)x_{j}^{z_{j}} .

We neglect again holomorphic multiples of z_{j}^{2}, and replace (x_{j}^{z_{j}}Cx_{j}^{−z_{j}} −C) by z_{j}D_{j}(C). In other words, we can replace in the above evaluations the term x_{j}^{z_{j}}C by (1 + z_{j}D_{j})(C)x_{j}^{z_{j}}, for j = 1, …, ℓ. By iterating this scheme, the regularized value of (21) becomes the regularized value of

T r (−z_{s}A(1 + z_{1}D_{1}) ∘ ⋯ ∘ (1 + z_{s−1}D_{s−1}) ∘ D_{s}(B)x_{1}^{z_{1}} ⋯ x_{ℓ}^{z_{ℓ}}Q^{−τ}) . (23)

By definition, the regularized value of (23) is given by

−∑^{s} T̂r_{S,0} (AD_{S}(B))

where the sum ∑^{s} is taken over all subsets S = {j_{1} < ⋯ < j_{q}} ⊆ {1, …, s} with j_{q} = s. If we let s vary, we get all nonempty subsets S exactly once, which proves the result.

Let now A ∈ Ψ_{c}^{m,F}(X) be a fully elliptic cusp operator, and B ∈ Ψ_{c}^{−m,−F}(X) be any operator which inverts A up to trace-class remainders. Then the trace of [A, B] does not depend on the choice of B, and equals in fact the index of A. Moreover, we have σ(B) = σ(A)^{−1} and I_{G}(B) = I_{G}(A)^{−1} for all G ∈ M_{k}(X).

We are now going to analyze more closely the various contributions to the index given in Proposition 5.1. As in [13], let us denote by

\overline{AS}(A) := T̂r_{∅,1}(D_{Q}(B)A)

the term that resembles the Atiyah-Singer contribution of the closed case discussed (1) in the introduction. It can be computed in terms of finitely many local expressions of the full symbols of A and of the parametrix B. A priori we need several terms from the full symbol expansion of A to compute it, even though the result clearly depends only on the joint symbol of A.

For ∅ ≠ S ⊆ M_{1}(X) with |S| = k, we have AD_{S}(B) ∈ Ψ_{c}^{−k,−δ_{S}}(X) by Remark 3.5. Thus, Proposition 4.5 yields

(24)  T̂r_{S,0}(AD_{S}(B)) = ∑_{G∈G_{S}} i^{k} ^{c}\overline{Tr}_{Q_{G}} (I_{G}(A)∂_{ξ}^{δ_{S}}I_{G}(B)) .

Summarizing we end up with the following formula for the index of a fully elliptic cusp operator.

Theorem 5.2. Let A ∈ Ψ_{c}^{m,F}(X) be fully elliptic, and B ∈ Ψ_{c}^{−m,−F}(X) be a parametrix inverting A up to trace class remainders. Then we have

(25)  ind A = \overline{AS}(A) − ∑_{∅≠S⊆M_{1}(X)} ∑_{G∈G_{S}} i^{|S|}^{c}\overline{Tr}_{Q_{G}} (I_{G}(A)∂_{ξ}^{δ_{S}}I_{G}(A)^{−1}) .

It is worth stressing that besides the local term \overline{AS}(A) all contributions to the index in (25) are defined in terms of the joint symbol τ(A) and its inverse only.

When X is a closed manifold, the right-hand side of (25) has just one term, namely \overline{AS}(A) = T r_{R}([log Q, B]A), where T r_{R} is the residue trace of Wodzicki and Guillemin. We can write it as the integral on the cosphere bundle of the part of homogeneity −n in the symbol of A[log Q, B] – see also Proposition 4.6. This result appeared in [13].

For manifolds with boundary, we get the following formula, similar to the Atiyah-Patodi-Singer formula:

(26)  ind (A) = T̂ r_{∅,1}([log Q, B]A) − T̂ r_{{1},0}(A[log x, B])
(27)  = \overline{AS}(A) − i ∑_{H∈M_{1}(X)} \overline{Tr}_{Q_{H}} ((I_{H}(A)∂_{ξ}I_{H}(A)^{−1}) .

In that case, the index of A was first computed in [13] as a sum of four terms using the one-dimensional zeta-function ζ(A)(z, τ) = Tr(Ax^{z}Q^{−τ}). In [5], it has been shown that this sum simplifies slightly to (26) when using the two-variable zeta function. Moreover, the cusp integral used to compute \overline{AS}(A) converges without regularization, i.e. the meromorphic extension G of the function z ⟼ ∫_{X} x^{z}(D_{Q}(B)A)_{−n}ι_{R}ω_{c}^{n} is actually analytic for \{Re z > −1\}, i.e. \overline{AS}(A) = G(0) [5, Proposition 3].

Let us finally describe a trivial but interesting modification of (27)^{1}. Replace \overline{Tr}_{Q_{H}} by the Hadamard type regularization \overline{Tr}_{M} of \overline{Tr} : Ψ^{−∞}(H; ℝ) → ℂ described in [11], and let T_{H} := \overline{Tr}_{Q_{H}} − \overline{Tr}_{M} for H ∈ M_{1}(X). It is known that \overline{Tr}_{M} has the trace property. Then the index formula (27) can be rewritten in the form

ind (A) = \overline{AS}′(A) − i ∑_{H∈M_{1}(X)} η_{H}(A) ,

where now

\overline{AS}′(A) := \overline{AS}(A) − i ∑_{H∈M_{1}(X)} T_{H}(I_{H}(A)∂_{ξ}I_{H}(A)^{−1} and
η_{H}(A) := \overline{Tr}_{M}(I_{H}(A)∂_{ξ}I_{H}(A)^{−1}) for H ∈ M_{1}(X)

have the index property

\overline{AS}′(A_{1}A_{2}) = \overline{AS}′(A_{1}) + \overline{AS}′(A_{2}) and η_{H}(A_{1}A_{2}) = η_{H}(A_{1}) + η_{H}(A_{2})

for all H ∈ M_{1}(X), and any two fully elliptic operators A_{1} and A_{2}. Moreover, by [17, Proposition 2] the term \overline{AS}′(A) is still local in the sense that it is invariant when adding terms of sufficiently negative order.

1The authors are indebted to Richard Melrose for this observation.

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Fachbereich 17 - Mathematik, Universität Mainz, D-55099 Mainz, Germany. currently: Universität Münster, SFB 478, Hittorfstraße 27, D-48149 Münster, Germany. e-mail: lauter@mathematik.uni-mainz.de Institutul de Matematică al Academiei Române, PO. Box 1-764, RO-70700, Bucharest, Romania e-mail: Sergiu.Moroianu@imar.ro