Research Article Received 21 July 2011 Published online 19 May 2012 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/mma.2532 MOS subject classification: 46E30; 42B35; 42B25; 47B38; 45E05; 76G25 Hardy and singular operators in weighted generalized Morrey spaces with applications to singular integral equations D. Lukkassen a , A. Meidell a , L.-E. Persson b and N. Samko c * † Communicated by T. Qian We study the weighted boundedness of the multi-dimensional Hardy-type and singular operators in the generalized Morrey spaces L p,' .R n , w/, defined by an almost increasing function '.r/ and radial type weights. We obtain sufficient conditions, in terms of numerical characteristics, that is, index numbers of the weight functions and the function '. In rela- tion with the wide usage of singular integral equations in applications, we show how the solvability of such equations in the generalized Morrey spaces depends on the main characteristics of the space, which allows to better control both the singularities and regularity of solutions. Copyright © 2012 John Wiley & Sons, Ltd. Keywords: weighted generalized Morrey space; weighted Hardy operators; Matuszewska–Orlicz indices; singular integral operators; normal solvability; Fredholm index; closed form solution 1. Introduction In the last two decades, there was an increased interest on Morrey spaces L p, ./ :D 8 ˆ < ˆ : f : sup x2,r>0 1 r  Z eB.x,r/ jf .y/j p dy < 1 9 > = > ; , (1.1) where e B.x, r/ D B.x, r/ \ , B.x, r/ Dfy 2 R n : jy  xj < rg, introduced in [1] in 1938, as well as to their generalizations and study of operators in such spaces. It was always of great interest to study generalizations of the classical Lebesgue spaces. The splash of interest in particular to Morrey- type spaces during the last decade has, as a background, advances in harmonic analysis, which allow to consider operators in such spaces in more generality and also to fill in some gaps in the theory of Morrey spaces. Another general reason of this interest (historically the first one) is the usage of Morrey-type spaces in the estimation of regularity of solutions of PDE. We also hope to later develop applications of such spaces in the homogeneity theory; we refer for instance to [2] and [3] with respect to that theory. Our main interest in this paper is to study weighted Morrey-type spaces. In this paper, we deal with the generalized Morrey spaces L p,' … ./ over an open set   R n defined by the norm kf k p,' :D sup x2…,r>0 0 B @ 1 '.r/ Z eB.x,r/ jf .y/j p dy 1 C A 1 p , (1.2) a Narvik University College , PO Box 385, N-8505, Narvik, Norway b Luleå University of Technology, SE 971 87, Luleå, Sweden c Departamento de Matemtica, Centro CEAF, Instituto Superior Técnico, IST ID Av. Rovisco Pais, 1, 1049-003, Lisbon, Portugal *Correspondence to: N. Samko, Departamento de Matemtica, Centro CEAF, Instituto Superior Técnico, IST ID Av. Rovisco Pais, 1, 1049-003 Lisbon, Portugal. † E-mail: nsamko@gmail.com 1300 Copyright © 2012 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2012, 35 1300–1311