Quantization on filtered manifolds

TL;DR

This paper develops a pseudodifferential calculus on filtered manifolds, defining symbol classes and proving properties like composition and continuity, extending previous groupoid-based approaches.

Contribution

It introduces a new pseudodifferential calculus on filtered manifolds with symbol classes and local quantization, aligning with existing groupoid-based methods.

Findings

Established a calculus with composition and adjoint properties.

Proved continuity on adapted Sobolev spaces.

Showed equivalence with van Erp and Yuncken's groupoid calculus.

Abstract

In this article, we develop a pseudodifferential calculus on a general filtered manifold M . The symbols are fields of operators $\sigma$(x, $\pi$) parametrised by x $\in$ M and the unitary dual G x M of the osculating Lie group G x M . We define classes of symbols and a local quantization formula associated to a local frame adapted to the filtration. We prove that the collection of operators on M coinciding locally with the quantization of symbols enjoys the essential properties of a pseudodifferential calculus: composition, adjoint, parametrices, continuity on adapted Sobolev spaces. Moreover, we show that the polyhomogeneous subcalculus coincides with the calculus constructed by van Erp and Yuncken via groupoids.