Distributions associated to homogeneous distributions

In this paper we continue to study {it quasi associated homogeneous distributions rm{(}generalized functionsrm{)}} 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 $U_{a}f(x)=f(ax)$, $xin bR^n$, $a >0$) and its generator $sum_{j=1}^{n}x_jfrac{partial}{partial x_j}$. It is proved that $f_kin {cD}(bR^n)$ is a quasi associated homogeneous distribution of degree $lambda$ and of order $k$ if and only if $bigl(sum_{j=1}^{n}x_jfrac{partial}{partial x_j}-lambdabigr)^{k+1}f_{k}(x)=0$, or if and only if $bigl(U_a-a^lambda Ibigr)^{k+1}f_k(x)=0$, $forall , a>0$, where $I$ is a unit operator. The structure of a quasi associated homogeneous distribution is described.