Fibered Cusp b-Pseudodifferential Operators and its Applications

TL;DR

This paper introduces a new class of pseudodifferential operators on manifolds with corners, generalizing existing calculi, and proves their Fredholm properties, index theorems, and applications to non-closed manifolds.

Contribution

It defines fibered cusp b-pseudodifferential operators via blow-up techniques, extending the $ ext{Mazzeo-Melrose}$ calculus, and establishes their Fredholm criteria and index theorems.

Findings

Defined a new pseudodifferential calculus $ ext{Psi}^*_{ ext{Phi},b}(X)$

Proved Fredholm conditions for the new operators

Established an index theorem for non-closed $ ext{Z}/k$-manifolds

Abstract

Let $X$ be a smooth compact manifold with corners which has two embedded boundary hypersurfaces $\partial_0 X , \partial_1 X$, and a fiber bundle $\phi:\partial_0 X \to Y$ is given. By using the method of blowing up, we define a pseudodifferential culculus $\Psi ^* _{\Phi,b} (X)$ generalizing the $\Phi$-calculus of Mazzeo and Melrose and the (small) $b$-calculus of Melrose. We discuss the Fredholm condition of such operators and prove the relative index theorem. And as its application, the index theorem of "non-closed" $\mathbb{Z}/k$ - manifolds is proved.

Equations

$$P:L_b^2(X)\to L_b^2(X)$$

$$0\to {\Psi }_b^{-1}(X)\to {\Psi }_b^0(X)\stackrel{\sigma }{\to }{S}^0({}^b{T}^{\ast }X)\to 0$$

$$0\to x{\Psi }_b^{-1}(X)\to {\Psi }_b^0(X)\stackrel{N}{\to }{\Psi }_{b,\mathrm{inv}}^0(\widetilde{\partial X})\to 0$$

$$\mathrm{ind}(x^{{\beta }_1}P{x}^{-{\beta }_1})-\mathrm{ind}(x^{{\beta }_2}P{x}^{-{\beta }_2})=\frac{1}{2\pi i}\mathrm{tr}\oint \hat{N}(P{)}^{-1}(\lambda )\frac{\partial \hat{N}(P)}{\partial \lambda }(\lambda )d\lambda ,$$

$$P:L_{\Phi }^2(X)\to L_{\Phi }^2(X)$$

$$0\to {\Psi }_{\Phi }^{-1}(X)\to {\Psi }_{\Phi }^0(X)\stackrel{\sigma }{\to }{S}^0({}^{\Phi }{T}^{\ast }X)\to 0$$

$$0\to x{\Psi }_{\Phi }^{-1}(X)\to {\Psi }_{\Phi }^0(X)\stackrel{N}{\to }{\Psi }_{\mathrm{sus}{(}^{\Phi }NY)}^0(\partial X)\to 0$$

$$\{P\in {\Psi }_{sc}^0(X)\mid \text{P is fully elliptic}\}/\text{homotopy}\to K(D(TX),\partial D(TX)),$$

$$P:L_{\Phi ,b}^2(X)\to L_{\Phi ,b}^2(X)$$

$$0\to {\Psi }_{\Phi ,b}^{-1}(X)\to {\Psi }_{\Phi ,b}^0(X)\stackrel{\sigma }{\to }{S}^0({}^{\Phi ,b}{T}^{\ast }X)\to 0$$

$$0\to x_0{\Psi }_{\Phi ,b}^0(X)\to {\Psi }_{\Phi ,b}^0(X)\stackrel{N_0}{\to }{\Psi }_{\mathrm{sus}({}^{\Phi ,b}NY)}^0({\partial }_{0}X)\to 0$$

$$0\to x_1{\Psi }_{\Phi ,b}^0(X)\to {\Psi }_b^0(X)\stackrel{N_1}{\to }{\Psi }_{\Phi ,\mathrm{inv}}^0(\widetilde{{\partial }_{1}X})\to 0$$

$$\mathrm{ind}(x_1^{{\beta }_1}P{x}_1^{-{\beta }_1})-\mathrm{ind}(x_1^{{\beta }_2}P{x}_1^{-{\beta }_2})=\frac{1}{2\pi i}\mathrm{tr}\oint \widehat{N_1}(P{)}^{-1}(\lambda )\frac{\partial \widehat{N_1}(P)}{\partial \lambda }(\lambda )d\lambda ,$$

$${\Psi }_{sc,b,\mathbb{Z}/k}^0(X)=\{P\in {\Psi }_{\Phi ,b}^0(X)\mid \text{ N1(P) can be written as a direct sum of k copies of some operator on Z }\}.$$

$$\{P\in {\Psi }_{sc,b,\mathbb{Z}/k}^{\ast }(X)\mid \text{ σ(P) and N0(P) are invertible}\}/\text{homotopy}\to K(D(\overline{TX}),{\partial }_{0}D(\overline{TX})),$$

$$\partial X={\partial }_{0}X\cup {\partial }_{1}X,\angle X={\partial }_{0}X\cap {\partial }_{1}X$$

$$\angle X\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$M:=\{(x_0,x_1,y,z)\in {\mathbb{R}}_{\ge 0}\times {\mathbb{R}}_{\ge 0}\times {\mathbb{R}}^{k-1}\times {\mathbb{R}}^l\},{\partial }_{0}M:=\{x_0=0\},{\partial }_{1}M:=\{x_1=0\},$$

$$N:=\{(x_1,y)\in {\mathbb{R}}_{\ge 0}\times {\mathbb{R}}^{k-1}\},\varphi :{\partial }_{1}M\to N$$

$${\mathcal{V}}_{\Phi ,b}(X)=\{V\in \mathcal{V}(X)\mid V{x}_0\in x_0^2{C}^{\infty }(X),V{|}_{{\partial }_{0}X}\text{ is tangent to the fibers of ϕ, V∣∂1X is tangent to ∂1X }\},$$

$$B:=\{x_0\mid \text{x0 is a boundary defining function of ∂0X}\},$$

$$G:=\{\alpha \in C^{\infty }(X)\mid \text{α>0 and α∣∂0X is constant on each fiber of ϕ }\}$$

$${}^{\Phi ,b}NY:=\mathrm{span}\{x_0^2\frac{\partial }{\partial x_0},x_0{x}_1\frac{\partial }{\partial x_1},x_0\frac{\partial }{\partial y_i}\}.$$

$${\varphi }^{\ast }({}^{\Phi ,b}NY)=\ker ({}^{\Phi ,b}TX{|}_{{\partial }_{0}X}\to TX{|}_{{\partial }_{0}X}).$$

$$x_0^2\frac{\partial }{\partial x_0}=\frac{1}{\alpha }\cdot {\tilde{x}}_0^2\frac{\partial }{\partial {\tilde{x}}_0}$$

$$x_0{x}_1\frac{\partial }{\partial x_1}=\frac{1}{{\alpha }^2}\frac{\partial \alpha }{\partial x_1}\cdot {\tilde{x}}_0^2{x}_1\frac{\partial }{\partial {\tilde{x}}_0}+\frac{1}{\alpha }\cdot {\tilde{x}}_0{x}_1\frac{\partial }{\partial x_1}$$

$$x_0\frac{\partial }{\partial y_i}=\frac{1}{{\alpha }^2}\frac{\partial \alpha }{\partial y_i}\cdot {\tilde{x}}_0^2\frac{\partial }{\partial {\tilde{x}}_0}+\frac{1}{\alpha }\cdot {\tilde{x}}_0\frac{\partial }{\partial y_i}$$

$$(\alpha ,\tau ,\eta )\in G\times (\underline{\mathbb{R}}\oplus {}^{b}TY)\mapsto (\tau /\gamma +d\gamma \cdot \eta /{\gamma }^2,\eta /\gamma )\in \underline{\mathbb{R}}\oplus {}^{b}TY,$$

$$(\tau ,{\sigma }_1{x}_1\frac{\partial }{\partial x_1},{\eta }_i\frac{\partial }{\partial y_i},x_0)\in (\underline{\mathbb{R}}\oplus {}^{b}TY)\underset{G}{\times }B\mapsto (\tau x_0^2\frac{\partial }{\partial x_0},{\sigma }_1{x}_0{x}_1\frac{\partial }{\partial x_1},x_0\frac{\partial }{\partial y_i})\in {}^{\Phi ,b}NY,$$

$$X_b^2=[X^2;({\partial }_{0}X{)}^2,({\partial }_{1}X{)}^2],{\beta }_b:X_b^2\to X^2$$

Full text

1. 1 Introduction
2. 2 The definition of fibered cusp b-pseudodifferential operators
3. 3 Symbols and normal operators
4. 4 the Fredholm condition and the relative index theorem
5. 5 Application to $\mathbb{Z}$/k-manifolds

1 Introduction

To investigate the index problems on a singular space, it is important to define a suitable pseudodifferential calculus adapted to singularities. Let $X$ be a smooth compact manifold with corners which has two embedded boundary hypersurfaces $\partial_{0}X,\partial_{1}X$, and a fiber bundle $\phi:\partial_{0}X\to Y$ is given. For such $X$, we define a pseudodifferential culculus $\Psi^{*}_{\Phi,b}(X)$ generalizing the $\Phi$-calculus of Mazzeo and Melrose [13] or the $b$-calculus of Melrose [14], when $\partial_{1}X$ or $\partial_{0}X$ is empty (respectively). We call an element of $\Psi^{*}_{\Phi,b}(X)$ a fibered cusp $b$-pseudodifferential operator. The purpose of this paper is to give a relative index formula for fibered cusp $b$-pseudodifferential operators, and as its application, to prove the index theorem of “non-closed” $\mathbb{Z}/k$ - manifolds. For simplicity, we use the 0-th order operators $\Psi^{0}_{\Phi,b}(X)$ for the most argument. Before considering our general pseudodifferential culculus $\Psi^{0}_{\Phi,b}(X)$, let us review the $b$-calculus and the $\Phi$-calculus.

First we review the $b$-calculus. ([14],[17]) Let $X$ be a compact manifold with boundary, and $x$ be its boundary defining function. Then we can define a small calculus of $b$-pseudodifferential operators. Each element $P\in\Psi^{0}_{b}(X)$ defines a bounded operator.

[TABLE]

There are two important homomorphisms:the symbol map $\sigma$ and the normal map $N$ (or the indicial map). These maps are $*$-homomorphisms of filtered algebras which make the following sequences exact.

[TABLE]

[TABLE]

Where ${}^{b}T^{*}X$ is a vector bundle over $X$ which is non-canonically isomorphic to $T^{*}X$, and $S^{0}(^{b}T^{*}X)$ is a space of symbols of order [math] over ${}^{b}T^{*}X$. $\widetilde{\partial X}$ is a compactification of the positive normal bundle of $\partial X\hookrightarrow X$ which is non-canonically diffeomorphic to $\partial X\times[0,1]$, and $\Psi^{m}_{b,\mathrm{inv}}(\tilde{X})$ is an algebra of $b$-pseudodifferential operators on $\tilde{X}$ which are invariant under the action of $(0,\infty)$ on $\tilde{X}$.

We say $P\in\Psi^{0}_{b}(X)$ is elliptic if $\sigma(P)$ is invertible, and fully-elliptic if in addition $\hat{N}(P)(\lambda)$ is invertible for all $\lambda\in\mathbb{R}$. Where $\hat{N}(P)$ is a Mellin transform of $N(P)$, which is a $\Psi^{0}(\partial X)$-valued entire holomorphic function. It is known that $P:L^{2}_{b}(X)\to L^{2}_{b}(X)$ is Fredholm if and only if $P$ is fully-elliptic.

The relative index theorem in [14] combined with the operator-valued logarithmic residue theorem [8] gives the following result for a elliptic $P$.

[TABLE]

where $\mathrm{tr}$ is the trace, $\beta_{1},\beta_{2}\notin-\mathrm{ImSpec}(\hat{N}(P))\ (i=1,2)$ , $\beta_{2}>\beta_{1}$, and the path of integral is chosen so that its interior contains all poles of $\hat{N}(P)^{-1}(\lambda)$ such that $\beta_{1}<-\mathrm{Im}(\lambda)<\beta_{2}$. And \mathrm{Spec}(\hat{N}(P)):=\{\lambda\in\mathbb{C}\mid\text{ \hat{N}(P)(\lambda) is not invertible }\} is a discrete set.

Recall that a “closed” $\mathbb{Z}/k$-manifold $X$ is a manifold with boundary such that $\partial X$ is a disjoint union of $k$ copies of a closed manifold $Y$ , $\partial X=kY$. Freed and Melrose [7] introduced a subalgebra of $\Psi^{0}_{b}(X)$, sconsists of $P\in\Psi^{0}_{b}(X)$ for which $N(P)$ can be written as a direct sum of $k$ copies of some operator. For such operator $P$, $\mathrm{ind}(x^{\beta}Px^{-\beta})\mod k\in\mathbb{Z}/k$ is independent of $\beta$ because right hand side of the formula (1) is always a multiple of $k$. And they proved the index theorem which asserts that this $\mathbb{Z}/k$-valued index can be written in terms of topological K theory.

Secondly we review the $\Phi$-calculus [13]. Let $X$ be a compact manifold with boundary, and a fiber bundle $\phi:\partial X\to Y$ is given. Such $X$ is called a manifold with fibered boundary. Fix a boundary defining function $x$ of $\partial X$. Then we can define a calculus of $\Phi$-pseudodifferential operators (or fibered cusp pseudodifferential operatos) $\Psi^{*}_{\Phi}(X)$, which is a filtered $*$-algebra. Each element $P\in\Psi^{0}_{\phi}(X)$ defines a bounded operator.

[TABLE]

There exists two homomorphisms, a symbol map $\sigma$ and a normal map $N$ which make the following sequences exact.

[TABLE]

[TABLE]

Where ${}^{\Phi}T^{*}X$ is a vector bundle over $X$ which is non-canonically isomorphic to $T^{*}X$, ${}^{\Phi}NY$ is a vector bundle over $Y$ which is non-canonically isomorphic to $\mathbb{R}\oplus T^{*}Y$ and $\Psi^{0}_{\mathrm{sus}(^{\Phi}NY)}(\partial X)$ is a space of ${}^{\Phi}NY$-suspended pseudodifferential operators on $\partial X$ of order 0.

We say $P\in\Psi^{0}_{\phi}(X)$ is elliptic if $\sigma(P)$ is invertible, and fully elliptic in addition $N(P)$ is invertible. It is shown that $P:L^{2}_{\Phi}(X)\to L^{2}_{\Phi}(X)$ is Fredholm if and only if $P$ is fully-elliptic.

In an extreme case when $\phi$ is an identity map $\phi:\partial X\to\partial X$, the calculus $\Psi^{0}_{sc}(X):=\Psi^{0}_{\Phi}(X)$ is called scattering calculus. In this case, as outlined in [16], the index problem of fully elliptic operator is reduced to the Atiyah-Singer index theorem. Let us explain it briefly. We can define a map

[TABLE]

where $\partial D(TX)=D(TX|_{\partial X})\cup S(TX)$ and $D$ or $S$ means a disk or a sphere bundle of the vector space. The composition of this map and the topological index map [3] $\mathrm{t\mathchar 45\relax ind}:K(D(TX),\partial D(TX))\to\mathbb{Z}$ gives the index of fully elliptic operator.

Finally let us move on to our general calculus. Let $X$ be a smooth compact manifold with corners which has two embedded boundary hypersurfaces $\partial_{0}X,\partial_{1}X$ , and a fiber bundle $\phi:\partial_{0}X\to Y$ is given. Suppose that the fiber $Z$ of $\phi$ is a closed manifold. Fix a boundary defining function $x_{0}$ of $\partial_{0}X$ and $x_{1}$ of $\partial_{1}X$. Extending the notion in [7] we call such $X$ also a manifold with fibered boundary.

We define a pseudodifferential calculus $\Psi^{*}_{\Phi,b}(X)$ of fibered cusp $b$-pseudodifferential operators. Each element $P\in\Psi^{0}_{\Phi,b}(X)$ defines a bounded operator.

[TABLE]

We define a symbol map $\sigma$ and two normal maps $N_{0},N_{1}$ with respect to two boundaries $\partial_{0}X,\partial_{1}X$, which make the following sequences are exact.

[TABLE]

[TABLE]

[TABLE]

Where ${}^{\Phi,b}T^{*}X$ is a vector bundle over $X$ which is non-canonically isomorphic to $T^{*}X$ , ${}^{\Phi,b}NY$ is a vector bundle over $Y$ which is non-canonically isomorphic to $\mathbb{R}\oplus^{b}T^{*}Y$. $\widetilde{\partial_{1}X}$ is a compactification of the normal bundle of the embedding $\partial_{1}X\hookrightarrow X$ which is non-canonically diffeomorphic to $\partial_{1}X\times[0,1]$. Because $\widetilde{\partial_{1}X}$ is also a manifold with fibered boundary so we can define $\Psi^{0}_{\Phi}(\widetilde{\partial_{1}X})$, and “$\mathrm{inv}$” in $\Psi^{0}_{\Phi,\mathrm{inv}}(\widetilde{\partial_{1}X})$ means the invariance under the action of $(0,\infty)$.

We say $P\in\Psi^{m}_{\Phi,b}(X)$ is elliptic when $\sigma(P)$ is invertible, and fully-elliptic when in addition $N_{0}(P)$ and $\hat{N_{1}}(P)(\lambda)\ (\lambda\in\mathbb{R})$ are invertible. Where $\hat{N_{1}}(P)$ is a Mellin transform of $N_{1}(P)$, which is a $\Psi^{0}_{\Phi}(\partial_{1}X)$-valued entire holomorphic function.

And we prove the Fredholm condition.

Theorem 1**.**

For $P:L^{2}_{\Phi,b}(X)\to L^{2}_{\Phi,b}(X)$ is Fredholm if and only if $P$ is fully elliptic.

We also prove the relative index theorem.

Theorem 2**.**

Let $P\in\Psi^{0}_{\Phi,b}(X)$ and suppose $\sigma(P)$ and $N_{0}(P)$ are invertible. Take any $\beta_{i}\notin-\mathrm{ImSpec}(\hat{N_{1}}(P))\ (i=1,2)$, $\beta_{2}>\beta_{1}$. Then,

[TABLE]

where the path of integral is chosen so that its interior contains all poles of $\hat{N_{1}}(P)^{-1}(\lambda)$ such that $\beta_{1}<-\mathrm{Im}(\lambda)<\beta_{2}$.

Let $X$ be a $\mathbb{Z}/k$-manifold, i.e. $X$ is a smooth compact manifold with corners which has two embedded boundary hypersurfaces $\partial_{0}X,\partial_{1}X$, and $\partial_{1}X=kY$ for some manifold with boundary $Y$. Then $X$ is a manifold with Baas-Sullivan singularity [4], and $X$ is called closed as a manifold with Baas-Sullivan singularity if $\partial_{0}X$ is empty. We regard $X$ as a manifold with fibered boundary by setting $\phi=\mathrm{Id}_{\partial_{0}X}:\partial_{0}X\to\partial_{0}X$. And define $\Phi^{0}_{sc,b}:=\Phi^{0}_{\Phi,b}$. We define a subalgebra $\Psi^{0}_{sc,b,\mathbb{Z}/k}(X)$ of $\Psi^{0}_{sc,b}(X)$ which is compatible to the structure as a $\mathbb{Z}/k$-manifold, by setting

[TABLE]

For $P\in\Psi^{0}_{sc,b,\mathbb{Z}/k}(X)$, $\mathrm{ind}_{\beta}(P)\mod k\in\mathbb{Z}/k$ is independent of $\beta$ because the right hand side of (2) is always a multiple of $k$.

We can define a map

[TABLE]

where the overlines mean the identification of $k$ copies, and $\overline{TX}\to\overline{X}$ is a vector bundle. $\partial D(TX)=S(TX)\cup D(TX|_{\partial_{0}X})\cup D(TX|_{\partial_{1}X})$, and $\partial_{0}D(\overline{TX}):=S(\overline{TX})\cup D(\overline{TX|_{\partial_{0}X}})$. As in the case of Atiyah-Singer [3] or Freed-Melrose [7], there exists a topological index map $\mathrm{t\mathchar 45\relax ind}:K(D(\overline{TX}),\partial_{0}D(\overline{TX}))\to\mathbb{Z}/k$ [20]. And we prove the composition of these two maps gives the $\mathbb{Z}/k$-valued index of the operator $P$.

2 The definition of fibered cusp b-pseudodifferential operators

In this section, based on the discussion in [13], we define the fibered cusp $b$-pseudodifferential operators.

Let $X$ be a smooth compact manifold with corners which has two embedded boundary hypersurfaces. Thus, following relations hold where $\partial_{0}X$ and $\partial_{1}X$ are the boundary hypersurfaces.

[TABLE]

Suppose a fiber bundle $\phi:\partial_{0}X\to Y$ is given, where $Y$ is a compact manifold with boundary and each fiber $Z$ is a compact manifold without boundary. Suppose further $\phi$ maps the boundary to the boundary, thus restricts to a fiber bundle $\phi|_{\angle X}:\angle X\to\partial Y$, and the following diagram commutes.

[TABLE]

We say $X$ is a manifold with fibered boundary in this case.

The whole following discussion can be applied in the case when each fiber $\phi^{-1}(y)$ varies on connected components of $Y$, but for simplicity, we assume they are all diffeomorphic to a single closed manifold $Z$.

Take any boundary defining functions $x_{0},x_{1}\in C^{\infty}(X)$ for $\partial_{0}X,\partial_{1}X$ respectively. We assume that $x_{1}|_{\partial_{0}X}$ is constant on each fiber of $\phi$, which is always possible by taking boundary defining function of $\partial Y\subset Y$ and pulling it back to $\partial_{0}X$ and extend it to $X$.

To describe local properties of $X$, it is convenient to introduce “model space” $M$ as follows.

[TABLE]

[TABLE]

where $\phi$ is the projection. Then $M$ is a manifold with fibered boundary (without a compactness assumption).

For any manifold with fibered boundary $X$ and $p\in X$, their exists an open neighbourhood $p\in U$ and diffeomorphism onto open subset of $M$ which preserves the structure of manifold with fibered boundary. Where “preserving the structure” means it preserves $\partial_{0}$, $\partial_{1}$ and $\phi$, and when $p\in\partial_{0}X$ or $p\in\partial_{1}X$, it preserves the function $x_{0}$ or $x_{1}$.

We define fibered cusp $b$-vector fields on $X$ as follows:

[TABLE]

where $\mathcal{V}(X)$ is the space of smooth vector fields on $X$.

When $X=M$, it is straightforward to check $\mathcal{V}_{\Phi,b}(M)$ is freely generated by $x_{0}^{2}\cfrac{\partial}{\partial x_{0}},x_{0}x_{1}\cfrac{\partial}{\partial x_{1}},x_{0}\cfrac{\partial}{\partial y_{i}},\cfrac{\partial}{\partial z_{j}}$ over $C^{\infty}(M)$. Thus their exist a smooth vector bundle ${}^{\Phi,b}TX$ over $X$ and the isomorphism $\Gamma(X,^{\Phi,b}TX)\simeq\mathcal{V}_{\Phi,b}(X)$.

The map ${}^{\Phi,b}TX\to TX$, induced by $\mathcal{V}{}^{\Phi,b}(X)\hookrightarrow\mathcal{V}(X)$ defines the Lie algebroid structure on $X$. Such a structure is called a Lie structure at infinity in [1] and [2].

Although the space $\mathcal{V}_{\Phi,b}(X)$ depends on the choice of boundary defining function $x_{0}$ of $\partial_{0}X$, the full information of $x_{0}$ is not needed to determine $\mathcal{V}_{\Phi,b}(X)$, and we can prove the following lemma by direct calculation.

Lemma 1**.**

Two choices of boundary defining function of $\partial_{0}X$, $x_{0}$ and $\tilde{x}_{0}$ defines a same space $\mathcal{V}_{\Phi,b}(X)$ if and only if $\tilde{x}_{0}/x_{0}=\alpha\in C^{\infty}(X)$ satisfies $\alpha|_{\partial_{0}X}=\phi^{*}\gamma$ for some $\gamma\in C^{\infty}(Y)$.

On the set

[TABLE]

the group

[TABLE]

acts by multiplication. The above lemma implies that fixing $\mathcal{V}_{\Phi,b}(X)$ is equivalent to fixing the $G$ orbit in $B$.

From now on, we fix the Lie algebroid $\mathcal{V}_{\Phi,b}(X)$, or equivalently, the $G$ orbit in $B$.

Next we define the vector bundle over $Y$ by using the local coordinate:

[TABLE]

By definition,

[TABLE]

We want to describe this vector bundle without using coordinate. Note that if we fix $x_{0}$, ${}^{\Phi,b}NY$ is clearly isomorphic to $\underline{\mathbb{R}}\oplus{}^{b}TY$. If $\tilde{x}_{0}=\alpha x_{0}$ is another choice of boundary defining function where $\alpha\in G$. The coordinate exchange is given as following.

[TABLE]

[TABLE]

[TABLE]

The group $G$ acts on $\underline{\mathbb{R}}\oplus{}^{b}TY$ by

[TABLE]

where $\gamma\in C^{\infty}(Y)$ is defined by $\phi^{*}\gamma=\alpha^{-1}|_{\partial_{0}X}$ and $d\gamma\cdot\eta$ means a paring of $T^{*}Y$ and ${}^{b}TY$. By the above formula of coordinate exchange, the map

[TABLE]

is a well-defined isomorphism. This gives a coordinate-free definition of ${}^{\Phi,b}NY$.

Let $X^{2}_{b}$ be a smooth manifold obtained by blowing up $\partial_{0}X\times\partial_{0}X$ and $\partial_{1}X\times\partial_{1}X$ in $X\times X$.

[TABLE]

The order of two blow-ups does not matter because $\partial_{0}X\times\partial_{0}X$ and $\partial_{1}X\times\partial_{1}X$ intersects transversely (see [15]). $X^{2}_{b}$ has 6 boundary hypersurfaces $L_{0},F_{0},R_{0},L_{1},F_{1}$ and $R_{1}$, which corresponds to $\partial_{0}X\times X,\partial_{0}X\times\partial_{0}X,X\times\partial_{0}X,\partial_{1}X\times X,\partial_{1}X\times\partial_{1}X$ , and $X\times\partial_{1}X$ respectively, where $L$, $F$ or $R$ stands for left, front or right.

Define $\Phi:=\partial_{0}X\underset{Y}{\times}\partial_{0}X=\{(w,w^{\prime})\in\partial_{0}X\times\partial_{0}X\mid\phi(w)=\phi(w^{\prime})\}\subset(\partial_{0}X)^{2}$. Then $\Phi$ can be lifted to $\Phi_{b}\subset(\partial_{0}X)^{2}_{b}$. The smooth function $x_{0}^{\prime}/x_{0}:X^{2}_{b}\to[0,\infty]$ is independent of the choice of $x_{0}$ when restricted to $F_{0}$ and there is a diffeomorphism $(\partial_{0}X)^{2}_{b}\simeq\{x_{0}^{\prime}/x_{0}=1\}\cap F_{0}$.

By regarding $\Phi_{b}$ as a submanifold of $\{x_{0}^{\prime}/x_{0}=1\}\cap F_{0}$, we define

[TABLE]

$X^{2}_{\Phi,b}$ has 7 boundary hypersurfaces $L_{0},F_{0},R_{0},L_{1},F_{1},R_{1}$ and $FF_{0}$ where the new hypersurface $FF_{0}$ corresponds to $\Phi_{b}$.

For a model space $M$, we want to describe these blowing-ups explicitly using coordinates.

[TABLE]

First the coordinate on $M^{2}_{b}\setminus L_{0}\setminus L_{1}$ is given as following.

[TABLE]

[TABLE]

In this coordinate, $\Phi_{b}=\{x_{0}=0,s_{0}=1,s_{1}=1,y=y^{\prime}\}$. Thus, we can give an explicit coordinate on $M^{2}_{\Phi,b}\setminus L_{0}\setminus R_{0}$ around $FF_{0}$ as following.

[TABLE]

[TABLE]

Proposition 1**.**

$\mathcal{G}:=X^{2}_{\Phi,b}\setminus L_{0}\setminus R_{0}\setminus L_{1}\setminus R_{1}$* has a structure of a Lie groupoid by extending the Lie groupoid structure on $\mathrm{int}\,X^{2}$. The set of units of $\mathcal{G}$ is the lifted diagonal $\Delta_{\Phi,b}\subset X^{2}_{\Phi,b}$, and the associated Lie algebroid $A(\mathcal{G})$ is ${}^{\Phi,b}TX$.*

Proof.

Recall that the Lie algebroid structure of $\mathrm{int}\,X^{2}$ is given as follows:

[TABLE]

[TABLE]

[TABLE]

where $d,r,\mu,u$ and $\iota$ are domain, range, multiplication, unit, and inversion map respectively.

By using the coordinate on $X^{2}_{\Phi,b}$ described above, we can compute:

[TABLE]

[TABLE]

where

[TABLE]

Because $x_{0}^{\prime}/x_{0}=1-x_{0}u_{0}$ and $x_{1}^{\prime}/x_{1}=1-x_{0}u_{1}$ are smooth on $\mathcal{G}$, $d,r,\mu,u$ and $\iota$ can be extended to $\mathcal{G}$. These maps satisfy the axiom of the Lie groupoid as it satisfy on the dense subset $\mathrm{int}\,X^{2}$.

Clearly, the set of units of $\mathcal{G}$ is the lifted diagonal

[TABLE]

By definition, $A(\mathcal{G})$ is spanned by the restrictions of $\partial/\partial u_{0},\partial/\partial u_{1},\partial/\partial v$ and $\partial/\partial w$ to $\Delta_{\Phi,b}$.

[TABLE]

Thus, $A(\mathcal{G})={}^{\Phi,b}TX$. ∎

The Lie groupoid structure of $\mathcal{G}$ can be described simpler if we use the notion of blow-up of a Lie groupoid([6]). Let $\mathcal{G}_{0}\rightrightarrows X$ be the groupoid obtained by blowing up $X^{2}\rightrightarrows X$ at $Hol(\partial_{0}X,\mathcal{F})\rightrightarrows\partial_{0}X$, where $\mathcal{F}$ is the foliation defined by the fibration $\phi:\partial_{0}X\to Y$ and $Hol$ is the holonomy groupoid. This groupoid $\mathcal{G}_{0}$ is described more precisely in [5]. Let $\mathcal{G}_{1}\rightrightarrows X$ be the groupoid obtained by blowing up $X^{2}\rightrightarrows X$ at $(\partial_{1}X)^{2}\rightrightarrows\partial_{1}X$, which is called a puff groupoid in [18]. Then $\mathcal{G}$ is the fiber product of $\mathcal{G}_{0}$ and $\mathcal{G}_{1}$.

Definition 1**.**

Let $E$ and $F$ are vector bundles over $X$, and $m\in\mathbb{R}$ be an arbitrary real number. Then, the space of fibered cusp b-pseudodifferential operator of order m from $E$ to $F$ is defined as follows.

[TABLE]

Where $\pi_{L},\pi_{R}:X^{2}_{\Phi,b}\to X$ are left and right projection, $E^{\prime}$ is a dual of $E$, ${}^{\Phi,b}\Omega:=|\Lambda^{\dim X}|({}^{\Phi,b}T^{*}X)$ , $I^{m}$ is a space of conormal distribution, and $P\equiv 0$ means a vanishing of infinite order.

In this paper we only consider classical or one-step polyhomogeneous conormal distribution.

The space of uniformly supported pseudodifferential operator $\Psi^{m}_{c}(\mathcal{G};E,F)$ defined in [19] and [2] is, by definition,

[TABLE]

By definition, $\Psi^{m}_{c}(\mathcal{G};E,F)\subset\Psi^{m}_{\Phi,b}(X;E,F)$.

Let $\dot{C}^{\infty}(X;E):=x_{0}^{\infty}x_{1}^{\infty}C^{\infty}(X;E)$ be a space of smooth section of $E$ which vanishes in infinite order on $\partial_{0}X$ and $\partial_{1}X$. By general theory of conormal distributions [15], for $u\in\dot{C}^{\infty}(X;E)$ we can see that $Pu:=(\pi_{L})_{*}P\pi^{*}_{R}u$ defines continuous linear operators $\dot{C}^{\infty}(X;E)\to\dot{C}^{\infty}(X;F)$ where $(\pi_{L})_{*}$ is a fiber integral.

To give an explicit description of $P$, we assume $P$ is supported in the coordinate patch $\{(x_{0},x_{1},y,z,u_{0},u_{1},v,w)\}$.

By the condition $P\equiv 0$ on $L_{0}\cup R_{0}\cup L_{1}\cup R_{1}$ , $P$ decreases rapidly as $u_{0}^{2}+u_{1}^{2}+|v|^{2}\to\infty$. Thus we can take a fourier transform with respect to $u_{0},u_{1},v,w$ , and $P$ can be written as following.

[TABLE]

Note that $\pi_{R}^{*}({}^{\Phi,b}\Omega)$ is generated by $x_{0}^{\prime-k-2}x_{1}^{\prime-1}|dx_{0}^{\prime}dx_{1}^{\prime}dy^{\prime}dz^{\prime}|$. Its restriction to the fiber of $\pi_{L}$ is $\frac{x_{0}^{k+2}x_{1}}{x_{0}^{\prime k+2}x_{1}^{\prime}}du_{0}du_{1}dvdw$, and the coefficient $\frac{x_{0}^{k+2}x_{1}}{x_{0}^{\prime k+2}x_{1}^{\prime}}$ is a non-zero smooth function, thus we can absorb this coefficient in the symbol term.

For a function $u(x_{0},x_{1},y,z)$, the action of $P$ is given by

[TABLE]

For any complex numbers $\alpha$ and $\beta$, $x_{0}^{\alpha}x_{1}^{\beta}Px_{0}^{-\alpha}x_{1}^{-\beta}\in\Psi^{m}_{\Phi,b}(X;E,F)$, because $x_{0}^{\prime}/x_{0}=1-x_{0}u_{0}$ , $x_{1}^{\prime}/x_{1}=1-x_{0}u_{1}$ and these derivatives are smooth up to $FF_{0}$ and $F_{1}$ and at most polynomial order up to other boundary hypersurfaces. Thus $P$ also defines an operator $x_{0}^{\alpha}x_{1}^{\beta}{C}^{\infty}(X;E)\to x_{0}^{\alpha}x_{1}^{\beta}{C}^{\infty}(X;F)$. In particular, we can obtain three operators,

[TABLE]

[TABLE]

and

[TABLE]

such that $(Pu)|_{\partial_{0}X}=P|_{\partial_{0}X}u|_{\partial_{0}X}$, $(Pu)|_{\partial_{1}X}=P|_{\partial_{1}X}u|_{\partial_{1}X}$ and $(Pu)|_{\angle X}=P|_{\angle X}u|_{\angle X}$ for $u\in C^{\infty}(X;E)$.

Lemma 2**.**

$P|_{\partial_{0}X}\in\Psi^{m}_{\mathrm{fiber}}(\partial_{0}X;E,F)$* , $P|_{\partial_{1}X}\in\Psi^{m}_{\Phi}(\partial_{1}X;E,F)$ , $P|_{\angle X}\in\Psi^{m}_{\mathrm{fiber}}(\angle X;E,F)$ where $\Psi^{m}_{\mathrm{fiber}}$ is the space of a family of m-th pseudodifferential operators on each fiber of $\phi$ which depends smoothly on the base points, and $\Psi^{m}_{\Phi}$ is the space of m-th fibered cusp pseudodifferential operator.*

Proof.

By using partition of unity, we can assume $P$ is supported in the coordinate patch. Let $p$ is a symbol of $P$ as in (5), the symbols of $P|_{\partial_{0}X}$, $P|_{\partial_{1}X}$ and $P|_{\partial_{0}X}$ are $p(0,x_{1},y,z,0,0,0,\zeta)$ , $p(x_{0},0,y,z,\sigma_{0},0,\eta,\zeta)$ and $p(0,0,y,z,0,0,0,\zeta)$ respectively. ∎

As in [13], we can construct a blow-up $X^{3}_{\Phi,b}$ of $X^{3}$

[TABLE]

By using this manifold, we can prove the following proposition exactly parallel as in [13] or [5].

Proposition 2**.**

Let $E,F,G$ are vector bundles over X , $m,m^{\prime}\in\mathbb{R}$ , $P\in\Psi^{m}_{\Phi,b}(X;E,F)$ and $Q\in\Psi^{m^{\prime}}_{\Phi,b}(X,F,G)$ , then $Q\circ P\in\Psi^{m+m^{\prime}}_{\Phi,b}(X;E,G)$.

3 Symbols and normal operators

In this section, we define the symbol $\sigma$ and normal operators $N_{0}$ and $N_{1}$. Essentially, $N_{0}(P)$ and $N_{1}(P)$ are restriction of the kernel of $P$ to $FF_{0}$ and $F_{1}$. A similar notion is called a normal operator in [13], an indicial operator in [14], and just a symbol in [5].

As described in [15],[13],[19], we can obtain a symbol homomorphism

[TABLE]

Where $S^{m}({}^{\Phi,b}T^{*}X;\mathrm{Hom}\,(E,F))$ is a space of bundle homomorphisms ${}^{\Phi,b}T^{*}X\setminus 0\to\mathrm{Hom}\,(E,F)$ which are homogeneous of degree $m$. The sequence

[TABLE]

is exact.

Next, we consider a normal operator at $\partial_{0}X$.

$p\in Y$ , $(\tau,\tilde{\eta})\in(\underline{\mathbb{R}}\oplus T^{*}Y)_{y}$. Fix any real valued $f\in C^{\infty}(Y)$ such that $f(p)=\tau$ , $df(p)=\tilde{\eta}$ , and real valued $\tilde{f}\in C^{\infty}(X)$ such that $\phi^{*}f=\tilde{f}|_{\partial_{0}X}$ . Define

[TABLE]

.

[TABLE]

Thus, for $\tau=f(x_{1},y)$,$\tilde{\sigma}=\frac{\partial}{\partial x_{1}}f(x_{1},y)$ , $\xi_{i}=\frac{\partial}{\partial y_{i}}f(x_{1},y)$, the symbol of $\tilde{N}(\tau,\sigma,\xi)$ is given by $p(0,x_{1},y,z,-\tau,x_{1}\tilde{\sigma},\xi,\eta)$, in particular $\tilde{N_{0}}$ is well-defined and does not depend on the choice of $f$ or $\tilde{f}$ and depends smoothly in $(\tau,\tilde{\eta})$.

Note that if $(\tilde{\sigma},\xi_{i})=(\tilde{\sigma}dx_{1},\sum\xi_{i}dy_{i})$ is a coordinate for $T^{*}Y$, then $(x_{1}\tilde{\sigma}dx_{1}/x_{1},\sum\xi_{i}dy_{i})$ is a coordinate for ${}^{b}{T^{*}Y}$. So if $\beta_{b}:TY\to{}^{b}TY$ is a blow-down map, the above symbol expression implies that there is a unique map $\hat{N_{0}}:(\underline{\mathbb{R}}\oplus T^{*}Y)_{y}\to\Psi^{m}(\phi^{-1}(p);E,F)$ such that $\tilde{N_{0}}\circ\beta_{b}=\hat{N_{0}}$, i.e. $\hat{N_{0}}(\tau,\sigma,\xi)=\tilde{N}(\tau,\sigma/x_{1},\xi)$, and its symbol is given by $p(0,x_{1},y,z,-\tau,\sigma,\xi,\eta)$.

By the fourier transform, it turns out that $\hat{N_{0}}$ defines a suspended pseudodifferential operator (see [13]). $N_{0}=N_{0}(P)\in\Psi^{m}_{\mathrm{sus}({}^{\Phi,b}NY)}(\partial X)$ , ${}^{\Phi}NY\simeq\underline{\mathbb{R}}\oplus{}^{b}T^{*}Y$. And we say $N_{0}(P)$ is a normal operator of $P$ on $\partial_{0}X$.

We obtain the exact sequence

[TABLE]

We consider a normal operator of $P$ at $\partial_{1}X$. For $P\in\Psi^{m}_{\Phi,b}(X;E,F)$ , $\lambda\in\mathbb{C}$ define

[TABLE]

Obviously, $\hat{N_{1}}:\mathbb{C}\to\Psi^{m}_{\Phi}(\partial_{1}X;E,F)$ is an entire holomorphic function.

Let $\widetilde{\partial_{1}X}\simeq\partial_{1}X\times[0,\infty]$ is the compactification of the positive normal bundle of $\partial_{1}X\subset X$, then $\widetilde{\partial_{1}X}$ obviously has a structure of a manifold with fibered boundary. Define

[TABLE]

Note that the first front faces $F_{1}\subset X^{2}_{\Phi,b}$ and $F_{1}\subset\widetilde{\partial_{1}X}^{2}_{\Phi,b}$ are canonically diffeomorphic, so by Mellin transformation, it turns out that $\hat{N_{1}}(P)$ defines $N_{1}(P)\in\Psi^{m}_{\Phi,b,\mathrm{inv}}(\widetilde{\partial_{1}X})$ and the following sequence is exact (see [14] for more detail).

[TABLE]

In a local coordinate, define $t:=\log(1-x_{0}X_{1})/x_{0}$, then t is smooth up to $x_{0}=0$, and $X_{1}=(1-e^{tx_{0}})/x_{0}$ is also smooth up to $x_{0}=0$. And $\frac{x_{1}^{\prime i\lambda}}{x_{1}^{i\lambda}}=e^{ix_{0}\lambda t}$

Thus by changing coordinate form $X_{1}$ to $t$ , $(x_{0},x_{1},X_{0},t,y,Y,z,Z)$ also gives a coordinate. Define the symbol $\tilde{p}$ of $P$ with respect to this coordinate by following.

[TABLE]

Then the symbol of $\hat{N_{1}}(P)(\lambda)$ is $\tilde{p}(x_{0},0,y,z,\sigma_{0},-x_{0}\lambda,\eta,\zeta)$.

The normal operators $N_{0},N_{1}$ can be thought as the restriction to $\partial_{1}X,\partial_{0}X$, and the symbol $\sigma$ can be thought as the restriction to the boundary $S({}^{\Phi,b}T^{*}X)$ of ${}^{\Phi,b}T^{*}X$ at infinity. We want to consider further restriction to the intersection of these two.

As in the case of $\sigma$, we can define symbol maps

[TABLE]

and

[TABLE]

To consider the restriction to $\angle X$, as $\widetilde{\partial_{1}X}$ is also a manifold with fibered boundary, we can define a normal operator on $\partial_{0}(\widetilde{\partial_{1}X})=\widetilde{\angle X}$, $N_{0,1}:\Psi^{m}_{\Phi,b,\mathrm{inv}}(\widetilde{\partial_{1}X};E,F)\to\Psi^{m}_{\mathrm{sus}({}^{\Phi,b}N\widetilde{\partial Y})}(\widetilde{\angle X};E,F)$.

Where $\widetilde{\partial Y}\simeq\partial Y\times[0,\infty]$ ,$\widetilde{\angle X}\simeq\angle X\times[0,\infty]$ Note that for $Q\in\Psi^{m}_{\Phi,b,\mathrm{inv}}(\widetilde{\partial_{1}X};E,F)$, $N_{0,1}(Q)$ is also equivariant to the action of $(0,\infty)$, and $(0,\infty)$ acts on $\widetilde{\angle X}$ or $\widetilde{\partial Y}$ by multiplication. So the restriction of $N_{0,1}$ to the any fiber of $\phi$ gives the same value in $\Psi^{m}_{\mathrm{sus}({}^{\Phi,b}N{\partial Y})}(\angle X;E,F)$, where ${}^{\Phi,b}N{\partial Y}:={}^{\Phi,b}NY|_{\partial Y}\simeq\underline{\mathbb{R}}\oplus\underline{\mathbb{R}}\oplus T^{*}\partial Y$.

Thus we can define

[TABLE]

On the other hand we can define

[TABLE]

by restriction.

We can also define the symbol map.

[TABLE]

Define following maps by restrictions of symbols.

[TABLE]

[TABLE]

[TABLE]

[TABLE]

In summary, we defined 12 maps $\sigma,\sigma_{0},\sigma_{1},\sigma_{0,1},N_{0},N_{1},N_{0,1}N_{1,0}$ and four restriction maps. It is obvious from the definition that any pair of compositions which is defined on same spaces coincide, e.g. $N_{0,1}N_{0}=N_{1,0}N_{1}$ or $\sigma|_{\partial_{1}X}=\sigma_{0}{N_{0}}$. And exact sequences exist for all of these 12 maps as shown in the case $\sigma,N_{0}$ and $N_{1}$, but we will omit here.

Finally we can consider the joint symbol $J^{m}$ which is defined as follows.

[TABLE]

Then the following sequence is exact by the diagram chasing.

[TABLE]

4 the Fredholm condition and the relative index theorem

In this section we prove the Fredholm condition and the relative index theorem for fibered cusp $b$-pseudodifferential operators.

Let $X$ be a manifold with fibered boundary. In this section fix a Riemannian metric $g$ on ${}^{\Phi,b}TX$, then $g$ can be considered as a singular metric on $TX$ and $\mathrm{int}\,X$ is a Riemannian manifold with respect to that metric. And we also assume that every complex vector bundle $E$ on $X$ has a hermitian metric $h$. Then, we can define the $L^{2}$ space.

[TABLE]

For $u,v\in L^{2}_{\Phi,b}(X;E)$, we can define the inner product by $(u,v):=\int h(u,v)dg$. and $L^{2}_{\Phi,b}(X;E)$ is a Hilbert space with respect to this inner product.

Let $P\in\Psi^{m}_{\Phi,b}(X;E,F)$ then its formal adjoint $P^{*}\in\Psi^{m}_{\Phi,b}(X;F,E)$ can be defined.

To prove $P$ is bounded on $L^{2}$, we need some technical preparations. Consider the normal operator on $\partial_{1}X$, $N_{1}:\Psi^{m}_{\Phi,b}(X;E,F)\to\Psi^{m}_{\Phi,b,\mathrm{inv}}(\widetilde{\partial_{1}X};E,F)$, then as illustrated in [17], we can define a section of $N_{1}$ as following. Fix a diffeomorphism $\widetilde{\partial_{1}X}\simeq\partial_{1}X\times[0,\infty]$ and a collar neighbourhood $\partial_{1}X\times[0,\infty]\hookrightarrow X$ and take a smooth function $\psi$ on $X$ which is supported in a small neighbourhood of $\partial_{1}X$ and identically equal to 1 around the neighbourhood of $\partial_{1}X$. Then the multiplication by $\psi$ can be regarded as a operator $M_{\psi}:C^{\infty}(\widetilde{\partial_{1}X})\to C^{\infty}(X)$. Define $S:\Psi^{m}_{\Phi,b,\mathrm{inv}}(\widetilde{\partial_{1}X};E,F)\to\Psi^{m}_{\Phi,b}(X;E,F)$ by $S(B)=M_{\psi}BM_{\psi}^{*}$. By definition $S$ is smooth with respect to the Frécht space topology.

As in the case of $b$-calculus, for $B\in\Psi^{0}_{\Phi,b,\mathrm{inv}}(\widetilde{\partial_{1}X};E,F)$ and $u\in\dot{C}^{\infty}(\widetilde{\partial_{1}X};E)$, the action of $B$ is characterized as follows.

[TABLE]

Where $\hat{u}$ and $\widehat{Bu}$ are Mellin transform of $u$ or $\widehat{Bu}$, and $\hat{B}(\lambda)=\hat{N_{1}}(B)(\lambda)$.

By the coordinate representation given in section 3, for $\lambda\in\mathbb{R}$ $\hat{B}(\lambda)$ is bounded with respect to the Frćhet space topology on $\Psi^{0}_{\Phi}(\partial_{1}X;E,F)$. And by [13], the embedding $\Psi^{0}_{\Phi}(\partial_{1}X;E,F)\to\mathcal{L}(L^{2}_{\Phi}(\partial_{1}X;E),L^{2}_{\Phi}(\partial_{1}X;F))$ is bounded, so $||\hat{B}(\lambda)||$ is bounded.

Thus $B$ is also bounded because Mellin transform is an isomorphism on a $L^{2}$ space.

Proposition 3**.**

For $P\in\Psi^{0}_{\Phi,b}(X;E,F)$, $P$ is bonded as an operator $L^{2}_{\Phi,b}(E)\to L^{2}_{\Phi,b}(F)$. And the inclusion $\Psi^{0}_{\Phi,b}(X;E,F)\to\mathcal{L}(L^{2}_{\Phi,b}(E),L^{2}_{\Phi,b}(F))$ is bounded.

Proof.

First, we show the proposition holds for $P\in x_{0}^{N}x_{1}^{N}\Psi^{-N}_{\Phi,b}(X;E,F)$ and $N>0$ is sufficiently large. In this case the kernel of $P$ blows down and can be written as a continuous kernel on $X^{2}$ and the boundedness is obvious.

Secondly, we show the proposition holds for $P\in x_{0}^{\epsilon}x_{1}^{\epsilon}\Psi^{-\epsilon}_{\Phi,b}(X;E,F)$ for any $\epsilon>0$. Because $||P||^{2}=||P^{*}P||$ and $P^{*}P\in x_{0}^{2\epsilon}x_{1}^{2\epsilon}\Psi^{-2\epsilon}$ , using this discussion recursively, the boundedness follows by first step.

Lastly, we consider the general case $P\in\Psi^{0}_{\Phi,b}(X;E,F)$. As noted above $N_{1}(P)$ is bounded with respect to $L^{2}$ norms, so $S(N_{1}(P))=M_{\psi}S(N_{1}(P))M_{\psi}^{*}$ is also bounded with respect to $L^{2}$ norms, because $M_{\psi}$ is obviously bounded with respect to $L^{2}$ norms. By replacing $P$ by $P-S(N_{1}(P))$, we can assume that $N_{1}(P)=0$ i.e. $P\in x_{1}\Psi^{0}_{\Phi,b}(X;E,F)$

Take sufficiently large $C>0$ so that $C-N_{0}(P^{*}P)$ and $C-\sigma(P^{*}P)$ are positive. Then, we can find a formally self adjoint operator $A\in\Psi^{0}_{\Phi,b}(X;E,E)$ such that $N_{0}(A)=\sqrt{C-N_{1}(P^{*}P)}$ ,$N_{1}(A)=\sqrt{C}$ and $\sigma(A)=\sqrt{C-\sigma(P^{*}P)}$. Note that $\sqrt{C-N_{1}(P^{*}P)}$ can be defined because the calculus of the suspended pseudodifferential operator is closed under holomorphic functional calculus. Set $B:=C-P^{*}P-A^{2}$ , then $N_{0}(B)=N_{1}(B)=\sigma(B)=0$ so $B\in x_{0}x_{1}\Psi^{-1}_{\Phi,b}(X;E,E)$. Thus $B$ is $L^{2}$ bounded by second step. $||Pu||^{2}=(P^{*}Pu,u)=-(Bu,u)+C||u||^{2}-||Au||^{2}\leq(||B||+||C||)\cdot||u||^{2}$

∎

As $\Psi^{0}_{c}(\mathcal{G};E,F)\subset\Psi^{0}_{\Phi,b}(X;E,F)$ is obviously dense with respect to the $L^{2}$ norm, following propositions can be reduced to the general theory of pseudodifferential operator of a groupoid [19, 10].

By similar arguments we can also prove the following proposition.

Proposition 4**.**

Any one of the 12 maps in the last part of the section 3, including $\sigma$, $N_{0}$ and $N_{1}$, is bounded with respect to the $L^{2}$ norm.

Theorem 3**.**

$P\in\Psi^{0}_{\Phi,b}(X;E,F)$* is Fredholm if and only if $\sigma(P)$ and $N_{0}(P)$ are invertible and $\hat{N_{1}}(P)(t)$ is invertible for all $t\in\mathbb{R}$.*

Proof.

As described in [10], by diagram chasing, the $L^{2}$ completion of the exact sequence (6) is also exact,

[TABLE]

where $\mathcal{K}$ is the space of compact operators.

Thus $P$ is Fredholm if and only if $j(P)$ is invertible if and only if $\sigma(P),N_{0}(P),N_{1}(P)$ are invertible in its $L^{2}$ closure.

Because $S^{0}({}^{\Phi,b}T^{*}X;E,F)$ and $\Psi^{0}_{\mathrm{sus}({}^{\Phi,b}NY)}(\partial_{0}X;E,F)$ are closed under holomorphic functional calculus, $\sigma(P)$ and $N_{0}(P)$ are invertible if and only if they are invertible in its completion.

For $N_{1}(P)\in\Psi^{0}_{\Phi,b,\mathrm{inv}}(\widetilde{\partial_{1}X};E,F)$ , there is an injective $*$-homomorphism defined by the Mellin transform.

[TABLE]

where $C_{b}(\mathbb{R},\Psi^{0}_{\Phi}(\partial_{1}X;E,F))$ is a space of bounded continuous function from $\mathbb{R}$ to $\Psi^{m}_{\Phi}(\partial_{1}X;E,F)$. Obviously, the completion of $C_{b}(\mathbb{R},\Psi^{0}_{\Phi}(\partial_{1}X;E,F))$ is $C_{b}(\mathbb{R},\overline{\Psi}^{0}_{\Phi}(\partial_{1}X;E,F))$ and the above map extends to an injective *-homomorphism.

[TABLE]

Thus $N_{1}(P)$ is invertible in its completion if and only if its image in $C_{b}(\mathbb{R},\overline{\Psi}^{0}_{\Phi}(\partial_{1}X;E,F))$ is invertible, and the claim follows. ∎

Finally, we move on to the proof of the relative index theorem.

Lemma 3**.**

Let $P\in\Psi^{0}_{\Phi,b}(X;E,F)$ and suppose that $\sigma(P)$ and $N_{0}(P)$ are invertible. Then there is a parametrix $Q\in\Psi^{-m}_{\Phi,b}(X;F,E)$ such that $PQ-\mathrm{Id}\in x_{0}^{\infty}\Psi^{-\infty}_{\Phi,b}(X;F,F)$ , $QP-\mathrm{Id}\in x_{0}^{\infty}\Psi^{-\infty}_{\Phi,b}(X;E,E)$.

Proof.

We construct the right parametrix $Q$ inductively. Take $Q_{0}\in\Psi^{0}_{\Phi,b}(X;F,E)$ so that $\sigma_{0}(Q_{0})=\sigma_{0}(P)^{-1}$, $N_{0}(Q_{0})=\sigma(P)^{-1}$. Then $PQ_{0}-\mathrm{Id}\in x_{0}\Psi^{-1}_{\Phi,b}(X;F,F)$.

Set $R_{0}:=(PQ_{0}-\mathrm{Id})/x_{0}\in\Psi^{-1}_{\Phi,b}(X;E,E)$ . And take $Q_{1}\in\Psi^{-1}_{\Phi,b}(X;F,E)$ such that $\sigma_{-1}(Q_{1})=-\sigma_{0}(P)^{-1}\sigma_{-1}(R_{0})$, $N_{0}(Q_{1})=-N_{0}(P)^{-1}N_{0}(R_{0})$. By definition of $Q_{1}$, $P(Q_{0}+x_{0}Q_{1})-\mathrm{Id}=x_{0}(R_{0}+PQ_{1})\in x_{0}^{2}\Psi^{-2}_{\Phi,b}(X;F,F)$.

Suppose we constructed $Q_{1},\dots Q_{n}$ such that $Q_{m}\in\Psi^{-m}_{\Phi,b}(X;F,E)$ and $P(\sum_{0}^{n}x_{0}^{m}Q_{m})-\mathrm{Id}\in x_{0}^{n+1}\Psi^{-n-1}_{\Phi,b}(X;F,F)$. Set $R_{n}:=(P(\sum_{0}^{n}x_{0}^{m}Q_{m})-\mathrm{Id})/x_{0}^{n+1}$. And take $Q_{n+1}\in\Psi^{-n-1}_{\Phi,b}(X;F,E)$ such that $\sigma_{-n-1}(Q_{1})=-\sigma_{0}(P)^{-1}\sigma_{-n-1}(R_{n})$, $N_{0}(Q_{1})=-N_{0}(P)^{-1}N_{0}(R_{n})$. Then as above, $P(\sum_{0}^{n+1}x_{0}^{m}Q_{m})-\mathrm{Id}\in x_{0}^{n+2}\Psi^{-n-1}_{\Phi,b}(X;F,F)$.

Finally define $Q:=\sum_{0}^{\infty}Q_{m}$ by an asymptotic sum. Then $Q\in\Psi^{0}_{\Phi,b}(X;F,E)$ and $PQ-\mathrm{Id}\in\underset{m}{\cap}x_{0}^{m}\Psi^{-m}_{\Phi,b}(X;E,E)=x_{0}^{\infty}\Psi^{-\infty}(X;F,F)$. As we can construct left parametrix similarly, $Q$ is actually right and left parametrix.

∎

For the above parametrix $Q$, define $S=\mathrm{Id}-PQ\in x_{0}^{\infty}\Psi^{-\infty}_{\Phi,b}(X;F,F)$. Note that $x_{0}^{\infty}\Psi^{-\infty}_{\Phi,b}(X;F,F)=x_{0}^{\infty}\Psi^{-\infty}_{b}(X;F,F)$, because the kernel of any element of $x_{0}^{\infty}\Psi^{-\infty}_{\Phi,b}(X;F,F)$ vanishes on $FF_{0}$ in infinite order and blows down to the kernel on $X^{2}_{b}$.

We can see $\hat{N_{1}}(S)(\lambda)=\mathrm{Id}-\hat{N_{1}}(P)(\lambda)\hat{N_{1}}(Q)(\lambda)$ rapidly decreases as $|\mathrm{Re}{\lambda}|\to\infty$ by the theory of $b$ - calculus. Thus as in [14], we can prove the following lemma.

Lemma 4**.**

Let $P\in\Psi^{0}_{\Phi,b}(X;E,F)$ and suppose that $\sigma(P)$ and $N_{0}(P)$ are invertible. $\hat{N_{1}}(P)(\lambda)^{-1}$ is a meromorphic map from $\mathbb{C}$ to $\Psi^{0}_{\Phi,b}(X;F,E)$ such that for any $N>0$, there exists $C>0$ such that $\hat{N_{1}}(P)^{-1}(\lambda)$ exists and bounded on \{\lambda\in\mathbb{C}\mid\text{|\mathrm{Re}\lambda|>C$and$|\mathrm{Im}\lambda|<N }\}.

*In particular,the number of poles in the strip $\{\lambda\in C\mid|\mathrm{Im}\lambda|<N\}$ is finite. *

Let $P\in\Psi^{0}_{\Phi,b}(X;E,F)$, and suppose that $\sigma(P)$ and $N_{0}(P)$ are invertible. Obviously, $\sigma(x_{1}^{\alpha}Px_{1}^{-\alpha})=\sigma(P)$. And because $x_{1}$ is constant on each fiber of $\phi$ , $N_{0}(P)$ commutes with $x_{1}^{\alpha}$ and $N_{0}(x_{1}^{\alpha}Px_{1}^{-\alpha})=N_{0}(P)$.

For $\beta\in\mathbb{R}$, $\hat{N_{1}}(x_{1}^{\beta}Px_{1}^{-\beta})(\lambda)=\hat{N_{1}}(\lambda+i\beta)$. By theorem 3 $x_{1}^{\beta}Px_{1}^{-\beta}$ is Fredholm if and only if $\beta\notin-\mathrm{Im}\mathrm{Spec}(\hat{N_{1}}(P))$. Where \mathrm{Spec}(\hat{N_{1}}(P)):=\{\lambda\in\mathbb{C}\mid\text{ \hat{N_{1}}(P)(\lambda) is not invertible}\} which is discrete by lemma 4.

Thus, exactly as in [14], we can prove the relative index theorem.

Theorem 4**.**

Let $P\in\Psi^{0}_{\Phi,b}(X;E,F)$ and suppose that $\sigma(P)$ and $N_{0}(P)$ are invertible, $\beta_{i}\notin-\mathrm{Im}\mathrm{Spec}_{b}(P)\ (i=1,2)$ $\beta_{2}>\beta_{1}$ .Then,

[TABLE]

where $\mathrm{ind}$ is the index of Fredholm operator, $\mathrm{tr}$ is the trace, and the integral path is chosen so that its interior contains all poles of $\hat{N}(P)^{-1}(\lambda)$ such that $\beta_{1}<-\mathrm{Im}(\lambda)<\beta_{2}$.

For $P\in\Psi^{0}_{\Phi,b}(X;E,F)$ and $I:=[\delta,\gamma]\subset\mathbb{R}$ , $\delta<\gamma$ be a closed interval. Define a norm by $||P||_{I}:=\sup_{\alpha\in I}||x_{1}^{\alpha}Px_{1}^{-\alpha}||$. And $\overline{\Psi^{0}_{\Phi,b}}^{I}(X;E,F)$ be a completion with respect to that norm. Then $\hat{N_{1}}$ extends.

[TABLE]

where $\mathbb{R}\times iI=\{\lambda\in C\mid\delta\leq\mathrm{Im}(\lambda)\leq\gamma\}$ and $\mathrm{Hol}_{b}$ is a space of bounded continuous function which is holomorphic in the interior.

To see $\hat{N_{1}}$ extends to the completion, let $P\in\overline{\Psi^{0}_{\Phi,b}}^{I}(X;E,F)$ and $P_{n}\in\Psi^{0}_{\Phi,b}(X;E,F)$ such that $||P-P_{n}||_{I}\to 0$, then $\hat{N_{1}}(P_{n})|_{\mathbb{R}\times iI}$ is a Cauchy sequence by the definition of the norm, and uniformly converges to some $\hat{N_{1}}(P)$. Because uniform limit of holomorphic function is holomorphic, $\hat{N_{1}}(P)$ is holomorphic in the interior.

Note that $\sigma$ and $N_{0}$ also extends because $\sigma(x_{1}^{\alpha}Px_{1}^{-\alpha})=\sigma(P)$ and $N_{0}(x_{1}^{\alpha}Px_{1}^{-\alpha})=N_{0}(P)$.

And theorem 4 can be extended to the completion $P\in\overline{\Psi^{0}_{\Phi,b}}^{I}(X;E,F)$, because both hand side of the identity are continuous with respect to $||\cdot||_{I}$ and is an integer.

5 Application to $\mathbb{Z}$/k-manifolds

In this section we fix an isomorphism $TX={}^{\Phi,b}T^{*}X$ for simplicity. Suppose $X$ is a $\mathbb{Z}/k$ manifold with boundary , i.e. $X$ is a manifold with corner and $\partial X=\partial_{0}X\cup\partial_{1}X\ ,\ \angle X=\partial_{0}X\cap\partial_{1}X$ and the diffeomorhpism $\partial_{1}X\simeq kZ$ is given, where $Z$ is a manifold with boundary and $kZ$ is a disjoint union of $k$ copies of $Z$. For $\phi=\mathrm{Id}:\partial_{0}X\to\partial_{0}X$ , we regard $X$ as a manifold with fibered boundary. And we write $\Psi^{0}_{sc,b}(X;E,F)=\Psi^{0}_{\Phi,b}(X;E,F)$ in this case.

A vector bundle $E$ over $X$ is called $\mathbb{Z}/k$-vector bundle if $E|_{\partial_{1}X}=kE_{Z}$ for some vector bundle $E_{Z}\to Z$. Fix a $\mathbb{Z}/k$-vector bundle structure on $TX$. Define $\overline{X}$ be a quotient of $X$ obtained by identifying $k$ copies of $Z$ in $X$. Then $TX\to X$ descends to a vector bundle $\overline{TX}\to\overline{X}$.

Let $E,F$ are $\mathbb{Z}/k$- vector bundle over $X$, and $P\in\Psi^{0}_{\Phi,b}(X;E,F)$. Define

[TABLE]

Where $kQ=Q\oplus Q\dots\oplus Q\in\Psi^{0}_{sc,b}(\widetilde{\partial_{1}X};E,F)$ is defined by using isomorphism $\partial_{1}X\simeq kZ$.

$\hat{N_{0}}(P)$ is a bundle homomorphism over $\partial_{0}X$ , $\hat{N_{0}}(P):\underline{\mathbb{R}}\oplus T(\partial_{1}X)\to\underset{y\in\partial_{0}X}{\cup}\Psi^{0}(\phi^{-1}(y);E,F)$. And note that $\phi^{-1}(y)$ is one point set in this case. So $\Psi^{0}(\phi^{-1}(y);E,F)\simeq\mathrm{Hom}\,(E,F)$. Under this identification, by the compatibility of $N_{0}$ and $\sigma$, $\sigma(P)(\xi)=\lim_{t\to\infty}\hat{N_{0}}(t\xi)$ for $\xi\in S(TX|_{\partial_{0}X})$.

By above observations, there is a map

[TABLE]

Where we regard $\sigma(P)\cup N_{0}(P)$ as a bundle isomorphism between $E$ and $F$ over $S(\overline{TX})\cup D(\overline{TX})|_{\overline{\partial_{0}X}}$.

Let $P\in\sc$ and suppose that $\sigma(P)$ and $N_{0}(P)$ are invertible. Then the right hand side of the identity in theorem 4 is always a multiple of $k$, so $\mathrm{ind}(x_{1}^{\beta}Px_{1}^{-\beta})\mod k\in\mathbb{Z}/k$ does not depends on $\beta\notin-\mathrm{Im}\mathrm{Spec}(\hat{N_{1}}(P)(\lambda))$. And more strongly following lemma holds.

Lemma 5**.**

Let $P,Q\in\sc$ and suppose that $\sigma(P),\sigma(Q)$ and $N_{0}(P),N_{1}(P)$ are invertible. If $(\sigma(P),N_{0}(P))$ and $(\sigma(Q),N_{0}(Q))$ are homotopic in the space of invertible joint symbols, then $\mathrm{ind}(x_{1}^{\beta}Px_{1}^{-\beta})\equiv\mathrm{ind}(x_{1}^{\beta}Qx_{1}^{-\beta})\mod k$ for $\beta\notin-\mathrm{Im}\mathrm{Spec}(\hat{N_{1}}(P)(\lambda))\cup-\mathrm{Im}\mathrm{Spec}(\hat{N_{1}}(Q)(\lambda))$

Proof.

Let $(s_{t},n_{t})$, $0\leq t\leq 1$ be a homotopy such that $(s_{0},n_{0})=(\sigma(P),N_{0}(P))$ , $(s_{1},n_{1})=(\sigma(Q),N_{0}(Q))$. Take any lift $R_{t}$ of $(s_{t},n_{t})$ , i.e. $(\sigma(R_{t}),N_{0}(R_{t}))=(s_{t},n_{t})$. Combining the homotopies $(1-t)P+R_{0}$ and $tQ+(1-t)R_{1}$, we can get a homotopy $S_{t}$ such that $S_{0}=P$, $S_{1}=Q$ and $(\sigma(S_{t}),N_{0}(S_{t}))$ is invertible for all $0\leq t\leq 1$. Partition the interval sufficiently small $[t_{0},t_{1}],\dots,[t_{m-1},t_{m}]$ ,$0=t_{0}<t_{1}<\dots<t_{m-1}<t_{m}=1$ so that we can choose $\beta_{0},\dots,\beta_{m-1}$ such that $x^{\beta_{i}}S_{t}x^{-\beta_{i}}$ is Fredholm on $[t_{i},t_{i+1}]$.

Then $\mathrm{ind}(x^{\beta_{i}}S_{t}x^{-\beta_{i}})$ constant on $[t_{i},t_{i+1}]$ and $\mathrm{ind}(x^{\beta_{i}}S_{t_{i}}x^{-\beta_{i}})\equiv\mathrm{ind}(x^{\beta_{i-1}}S_{t_{i}}x^{-\beta_{i-1}})\mod k$. So $\mathrm{ind}(x^{\beta_{0}}Px^{-\beta_{0}})\equiv\mathrm{ind}(x^{\beta_{m-1}}Qx^{-\beta_{m-1}})\mod k$ and the claim is proved. ∎

As in [7] or [20], we can define a topological index map.

[TABLE]

And the index theorem can be proved.

Theorem 5**.**

Let $P\in\sc$ and suppose that $\sigma(P)$ and $N_{0}(P)$ are invertible, then $\mathrm{ind}(x_{1}^{\beta}Px_{1}^{-\beta})\mod k=\mathrm{t\mathchar 45\relax ind}(s(P))\in\mathbb{Z}/k$ , $\beta\notin-\mathrm{Im}\mathrm{Spec}(\hat{N_{1}}(\lambda))$.

Proof.

We will demonstrate two different ways to prove the theorem.

The first method is to reduce the case when $\partial_{0}X$ is empty as in [16]. Embed $X$ into $Y$ ,where $Y$ is $\mathbb{Z}/k$- manifold such that $\partial_{0}Y=\phi$, e.g. we can take $Y$ as a double of $X$. Choose $G$ so that $F\oplus G\simeq\underline{\mathbb{C}}^{n}$ , by replacing $P$ by $P\oplus\mathrm{Id}_{G}$, we can assume $F=\underline{\mathbb{C}}^{n}$ is a trivial bundle. Because $D(TX|_{\partial_{0}X})$ is homotopy equivalent to $\partial_{0}X$, by replacing $P$ to homotpic element, we can assume that $\hat{N_{1}}(P):TX|_{\partial_{0}X}\to\mathrm{Hom}\,(E,F)$ is constant on each fiber and given by some bundle isomorphism $\theta:E\simeq F={\mathbb{C}}^{n}$.

Using $\theta$, we can extend $E$ onto $Y$ in obvious way. Take a cut-off function $\phi$ such that $\phi\equiv 1$ near $\partial_{0}X$. If we choose $\phi$ so that its support is sufficiently small, $Q:=\theta\phi+P(1-\phi)$ is homotopic to $P$. $Q$ can be extended to a b-pseudodifferential operator $\tilde{Q}$ on $Y$ And by construction, under the excision map $K(D(\overline{TX}),S(\overline{TX})\cup D(\overline{TX})|_{\overline{\partial_{0}X}})\to K(D(\overline{TY}),S(\overline{TY}))$, $\sigma(Q)$ is mapped to $\sigma(\tilde{Q})$ thus, $\mathrm{t\mathchar 45\relax ind}(s(Q))=\mathrm{t\mathchar 45\relax ind}(s(\tilde{Q}))$. Obviously, $\mathrm{ind}(x_{1}^{\beta}Qx_{1}^{-\beta})=\mathrm{ind}(x_{1}^{\beta}\tilde{Q}x_{1}^{-\beta})$ for all $\beta\notin-\mathrm{Im}\mathrm{Spec}(\hat{Q}(\lambda))$.

Because $\partial_{0}Y$ is empty, by [7], $\mathrm{t\mathchar 45\relax ind}(s(\tilde{Q}))=\mathrm{ind}(x_{1}^{\beta}\tilde{Q}x_{1}^{-\beta})\mod k$ and the theorem is proved.

For the second method, we only give a outline.

We define the analytic index $\mathrm{a\mathchar 45\relax ind}:K(D(\overline{TX}),S(\overline{TX})\cup D(\overline{TX})|_{\overline{\partial_{0}X}})\to\mathbb{Z}/k$ by $\mathrm{a\mathchar 45\relax ind}(s(P))=\mathrm{ind}(x^{\beta}Px^{-\beta})\mod k$ , $\beta\notin-\mathrm{Im}\mathrm{Spec}(\hat{P}(\lambda))$. Then it is well-defined. And we can prove that $\mathrm{a\mathchar 45\relax ind}$ satisfies the axioms as in [3], [7] or [20].

For the part of the axiom about multiplication, we need to pay some attention. Let $W$ be a closed manifold, and $P\in\Psi^{m}_{\Phi,b}(X;E,F)$ , $m>0$ . Then in general, as in [3], $P\boxtimes\mathrm{Id}_{W}\notin\Psi^{m}_{\Phi,b}(X\times W;E,F)$ but it is contained in the completion $\overline{\Psi^{m}_{\Phi,b}}^{I}(X\times Z;E,F)$ defined in section 4 .

∎