Extensions of homogeneous distributions on deformations to the normal cone
Moudrik Chamoux
TL;DR
The paper addresses extending $a$-homogeneous distributions from $\operatorname{DNC}(M,V)\setminus V\times\mathbb{R}$ to the full deformation space $\operatorname{DNC}(M,V)$. It uses Meyer's weak-homogeneity extension framework together with a polar-coordinate analysis along the normal bundle to reduce the extension problem to canonical one-dimensional and fiberwise harmonic components. The main contribution is a constructive proof that every $a$-homogeneous distribution admits an $a$-homogeneous extension, with a detailed description of possible extensions and an analysis of when the extension can be canonical, including connections to the tangent/adiabatic groupoid formalism of Van Erp and Yuncken. The results illuminate how homogeneity interacts with the Deformation to the Normal Cone structure, providing tools for pseudodifferential calculus on filtered manifolds and related groupoids.
Abstract
On a deformation to the normal cone $\operatorname{DNC}(M,V)$ we show that given a distribution $u\in\mathcal{D}'(\operatorname{DNC}(M,V)\setminus V\times\mathbb{R})$ if $u$ is homogeneous of order $a$ for the zoom action, then it admits an $a$-homogeneous extension $\widetilde{u}\in\mathcal{D}'(\operatorname{DNC}(M,V))$. We describe all such extensions and discuss briefly about how it translates to the work of Van Erp and Yuncken in arXiv:2303.15787 . The technique used come from the results on the extension of weakly homogeneous distributions provided by Yves Meyer in the 90s.