Distributions associated to 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 $({\sum }_{j=1}^n{x}_j\frac{\partial }{\partial x_j}-\lambda {)}^{k+1}{f}_k(x)=0$, or if and only if $(U_a-a^{\lambda }I{)}^{k+1}{f}_k(x)=0$, $\forall a>0$, where $I$ is a unit operator. The structure of a quasi associated homogeneous distribution is described.

13 pages