Weighted Hardy operators in the local generalized vanishing Morrey spaces

In this paper we study $p\rightarrow q$ -boundedness of the multi-dimensional Hardy type operators in the vanishing local generalized Morrey spaces $V\mathcal L ^{p,\varphi }_\mathrm{{loc}}(\mathbb R ^n,w)$ defined by an almost increasing function $\varphi (r)$ and radial type weight $w(|x|)$ . We obtain sufficient conditions, in terms of some integral inequalities imposed on $\varphi $ and $w$ , for such a boundedness. In the case where the function $\varphi (r)$ and the weight are power functions, these conditions are also necessary.