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Journal of Function Spaces and Applications
Volume 2012 (2012), Article ID 283285, 19 pages
http://dx.doi.org/10.1155/2012/283285
Research Article

Weighted Hardy Operators in Complementary Morrey Spaces

1Department of Technology, Narvik University College, P.O. Box 385, 8505 Narvik, Norway
2Department of Mathematics, Luleå University of Technology, SE 921 87 Luleå, Sweden
3Departamento de Matematica, Universidade do Algarve, 6005-139 Faro, Portugal

Received 23 July 2012; Accepted 20 September 2012

Academic Editor: Alois Kufner

Copyright © 2012 Dag Lukkassen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the weighted -boundedness of the multidimensional weighted Hardy-type operators and with radial type weight , in the generalized complementary Morrey spaces defined by an almost increasing function . We prove a theorem which provides conditions, in terms of some integral inequalities imposed on and , for such a boundedness. These conditions are sufficient in the general case, but we prove that they are also necessary when the function and the weight are power functions. We also prove that the spaces over bounded domains Ω are embedded between weighted Lebesgue space with the weight and such a space with the weight , perturbed by a logarithmic factor. Both the embeddings are sharp.