# 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.

## Key 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.