Extensions of homogeneous distributions on deformations to the normal cone

On a deformation to the normal cone $\mathrm{DNC}(M,V)$ we show that given a distribution $u\in {\mathcal{D}}^{\mathrm{\prime }}(\mathrm{DNC}(M,V)\setminus V\times \mathbb{R})$ if $u$ is homogeneous of order $a$ for the zoom action, then it admits an $a$-homogeneous extension $\tilde{u}\in {\mathcal{D}}^{\mathrm{\prime }}(\mathrm{DNC}(M,V))$. 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.