Extensions of homogeneous distributions on deformations to the normal cone

TL;DR

This paper extends the theory of homogeneous distributions on deformations to the normal cone, providing a method to extend distributions of a given homogeneity order and relating it to recent work by Van Erp and Yuncken.

Contribution

It introduces a new approach to extend homogeneous distributions on deformations to the normal cone, building on Meyer's techniques and connecting to recent research by Van Erp and Yuncken.

Findings

All homogeneous extensions of distributions are characterized.

The method applies to distributions homogeneous under the zoom action.

Connections to recent work by Van Erp and Yuncken are discussed.

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.