Distributions associated to homogeneous distributions

TL;DR

This paper characterizes quasi associated homogeneous distributions in multiple dimensions using dilatation operators and their generators, extending previous one-dimensional results to higher dimensions.

Contribution

It provides a multidimensional characterization of quasi associated homogeneous distributions via dilatation operators and their generators, building on earlier one-dimensional work.

Findings

Characterization of these distributions using the dilatation operator $U_a$

Equivalence of conditions involving the degree $$ and the order $k$

Structural description of quasi associated homogeneous distributions

Abstract

In this paper we continue to study {\it quasi associated homogeneous distributions \rm{(}generalized functions\rm{)}} which were introduced in the paper by V.M. Shelkovich, Associated and quasi associated homogeneous distributions (generalized functions), J. Math. An. Appl., {\bf 338}, (2008), 48-70. [arXiv:math/0608669]. For the multidimensional case we give the characterization of these distributions in the terms of the dilatation operator $U_{a}$ (defined as $U_{a}f(x)=f(ax)$, $x\in \bR^n$, $a >0$) and its generator $\sum_{j=1}^{n}x_j\frac{\partial}{\partial x_j}$. It is proved that $f_k\in {\cD}'(\bR^n)$ is a quasi associated homogeneous distribution of degree $\lambda$ and of order $k$ if and only if $\bigl(\sum_{j=1}^{n}x_j\frac{\partial}{\partial x_j}-\lambda\bigr)^{k+1}f_{k}(x)=0$, or if and only if $\bigl(U_a-a^\lambda I\bigr)^{k+1}f_k(x)=0$, $\forall \, a>0$, where $I$ is a unit operator. The structure of a quasi associated homogeneous distribution is described.