Distributions associated to homogeneous distributions

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In this paper we continue to study quasi associated homogeneous distributions (generalized functions) which were introduced in the paper by V.M. Shelkovich, Associated and quasi associated homogeneous distributions (generalized functions), J. Math. An. Appl., 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 Uₐ (defined as Uₐf(x)=f(ax), x∈ ⁿ, a >0) and its generator ∑ⱼ₌₁ⁿxⱼ∂/∂ xⱼ. It is proved that fₖ∈ '(ⁿ) is a quasi associated homogeneous distribution of degree λ and of order k if and only if (∑ⱼ₌₁ⁿxⱼ∂/∂ xⱼ-λ)ᵏ⁺¹fₖ(x)=0, or if and only if (Uₐ-aλI)ᵏ⁺¹fₖ(x)=0, ∀ a>0, where I is a unit operator. The structure of a quasi associated homogeneous distribution is described.