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Journal of Function Spaces and Applications
Volume 2013, Article ID 809704, 12 pages
http://dx.doi.org/10.1155/2013/809704
Research Article

Weak Type Inequalities for Some Integral Operators on Generalized Nonhomogeneous Morrey Spaces

1Department of Mathematics, Bandung Institute of Technology, Bandung 41032, Indonesia
2Department of Mathematics and Information Sciences, Tokyo Metropolitan University, Minami-Ohsawa 1-1, Hachioji, Tokyo 192-0397, Japan
3Department of Mathematics, Jenderal Soedirman University, Purwokerto 53122, Indonesia

Received 9 September 2013; Accepted 20 November 2013

Academic Editor: Vagif Guliyev

Copyright © 2013 Hendra Gunawan 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 prove weak type inequalities for some integral operators, especially generalized fractional integral operators, on generalized Morrey spaces of nonhomogeneous type. The inequality for generalized fractional integral operators is proved by using two different techniques: one uses the Chebyshev inequality and some inequalities involving the modified Hardy-Littlewood maximal operator and the other uses a Hedberg type inequality and weak type inequalities for the modified Hardy-Littlewood maximal operator. Our results generalize the weak type inequalities for fractional integral operators on generalized non-homogeneous Morrey spaces and extend to some singular integral operators. In addition, we also prove the boundedness of generalized fractional integral operators on generalized non-homogeneous Orlicz-Morrey spaces.