On polyhomogeneous symbols and the Heisenberg pseudodifferential calculus

Abstract

Polyhomogeneous symbols, defined by Kohn-Nirenberg and Hörmander 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.

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