# Weighted Hardy operators in the local generalized vanishing Morrey spaces

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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Weighted Hardy operators in the local generalized vanishing Morrey spaces

, Volume 17 (3) – Sep 9, 2012
24 pages

/lp/springer_journal/weighted-hardy-operators-in-the-local-generalized-vanishing-morrey-kgCEIlPICi
Publisher
Springer Journals
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-012-0199-z
Publisher site
See Article on Publisher Site

### 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.

### Journal

PositivitySpringer Journals

Published: Sep 9, 2012

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