Quantization on manifolds with an embedded submanifold

TL;DR

This paper develops a calculus for elliptic problems related to smooth embeddings of manifolds with corners, using Fourier integral operators on Lie groupoids, and demonstrates its algebraic and analytical properties.

Contribution

It introduces a new calculus for relative elliptic problems on manifolds with corners, utilizing Lie groupoids, and proves its closure under composition and operator continuity.

Findings

The calculus is closed under composition.

A representation of the algebra is constructed.

Operators are continuous on Sobolev spaces.

Abstract

We investigate a quantization problem which asks for the construction of an algebra for relative elliptic problems of pseudodifferential type associated to smooth embeddings. Specifically, we study the problem for embeddings in the category of compact manifolds with corners. The construction of a calculus for elliptic problems is achieved using the theory of Fourier integral operators on Lie groupoids. We show that our calculus is closed under composition and furnishes a so-called noncommutative completion of the given embedding. A representation of the algebra is defined and the continuity of the operators in the algebra on suitable Sobolev spaces is established.