# Distributions associated to homogeneous distributions

Research paper by **A. V. Kosyak, V. I. Polischook, V. M. Shelkovich**

Indexed on: **03 May '12**Published on: **03 May '12**Published in: **Mathematics - Classical Analysis and ODEs**

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#### 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.