Singular integral operators on non-compact manifolds and analysis on polyhedral domains
Victor Nistor
TL;DR
This work develops a Lie-manifold framework to extend elliptic analysis to non-compact and polyhedral spaces via a Melrose-style pseudodifferential calculus. It constructs the algebra $\Psi_{\mathcal{V}}^{\infty}(M)$ by quantizing the Lie algebroid $A$ through a groupoid $\mathcal{G}$, proving closure under composition and a Fredholm criterion that reduces invertibility to ellipticity and model operators on strata $M_\alpha\times G_\alpha$. The key contributions include the definition of type I Lie manifolds with pragmatic Fredholm conditions, a suite of concrete examples linking to the $b$-, edge-, and scattering-calculi, and explicit spectral results for Laplacians and Dirac operators on manifolds with multi-cylindrical ends. These results enable boundary-value problems and nonlinear PDE analysis on non-smooth, non-compact domains while providing detailed spectral data aligned with established theories. The framework integrates geometric, algebro-groupoid, and analytic methods to generalize Melrose's program to polyhedral settings and beyond.
Abstract
We review the definition of a Lie manifold $(M, \VV)$ and the construction of the algebra $Ψ\sp{\infty}\sb{\VV}(M)$ of pseudodifferential operators on a Lie manifold $(M, \VV)$. We give some concrete Fredholmness conditions for pseudodifferential operators in $Ψ\sp{\infty}\sb{\VV}(M)$ for a large class of Lie manifolds $(M, \VV)$. These Fredholmness conditions have applications to boundary value problems on polyhedral domains and to non-linear PDEs on non-compact manifolds. As an application, we determine the spectrum of the Dirac operator on a manifold with multi-cylindrical ends.