On polyhomogeneous symbols and the Heisenberg pseudodifferential calculus

TL;DR

This paper characterizes polyhomogeneous symbols without asymptotic expansions and applies this to show the equivalence of two Heisenberg pseudodifferential calculi on contact manifolds.

Contribution

It provides a new simple characterization of polyhomogeneous functions using homogeneity and extends the Heisenberg calculus equivalence to filtered manifolds.

Findings

Polyhomogeneous symbols are restrictions of homogeneous functions modulo Schwartz functions.

The characterization applies to arbitrary graded dilations.

Heisenberg calculus on contact manifolds coincides with groupoid calculus.

Abstract

Polyhomogeneous symbols, defined by Kohn-Nirenberg and H\"ormander in the 60's, play a central role in the symbolic calculus of most pseudodifferential calculi. We prove a simple characterisation of polyhomogeneous functions which avoids the use of asymptotic expansions. Specifically, if $U$ is open subset of $\mathbb{R}^d$, then a polyhomogeneous symbol on $U \times \mathbb{R}^d$ is precisely the restriction to $t=1$ of a function on $U \times \mathbb{R}^{d+1}$ which is homogeneous for the dilations of $\mathbb{R}^{d+1}$ modulo Schwartz class functions. This result holds for arbitrary graded dilations on the vector space $\mathbb{R}^d$. As an application, using the generalisation of A.~Connes' tangent groupoid for a filtered manifold, we show that the Heisenberg calculus of Beals and Greiner on a contact manifold or a codimension 1 foliation coincides with the groupoid calculus of Van Erp and the second author.