$L^2$ boundedness of pseudodifferential operators on manifolds with ends

TL;DR

This paper establishes $L^2$ boundedness for a class of pseudodifferential operators on manifolds with ends, extending classical results to curved and non-compact geometries.

Contribution

It generalizes the Calderón-Vaillancourt theorem and constructs parametrices for elliptic operators on manifolds with ends.

Findings

Proved $L^2$ boundedness for pseudodifferential operators on manifolds with ends.

Extended classical pseudodifferential operator theory to asymptotically conical and hyperbolic geometries.

Provided a method for constructing parametrices of elliptic operators in these settings.

Abstract

We investigate properties of pseudodifferential operators on $L^2$ space on manifold with ends including asymptotically conical or hyperbolic ends. Our pseudodifferential operators are a generalization of the canonical quantization which naturally appears in the quantum mechanics on curved spaces. We prove a Calder\'on-Vaillancourt type theorem for our pseudodifferential operators and discuss a construction of parametrix of elliptic differential operators on manifolds with ends.