Positivity

, Volume 17, Issue 3, pp 683–706 | Cite as

Weighted Hardy operators in the local generalized vanishing Morrey spaces

Article

Abstract

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.

Keywords

Generalized weighted Morrey space Local Morrey spaces  Weighted Hardy inequalities Weighted Hardy operators Bary-Stechkin classes Matuszewska-Orlicz indices 

Mathematics Subject Classification (2000)

46E30 42B35 42B25 47B38 

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Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Departamento de Matematica, Centro CEAFInstituto Superior TécnicoLisboaPortugal
  2. 2.Luleå University of TechnologyLuleåSweden

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