, Volume 17, Issue 3, pp 683-706
Date: 09 Sep 2012

Weighted Hardy operators in the local generalized vanishing Morrey spaces

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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.