Hardy and singular operators in weighted generalized Morrey spaces with applications to singular integral equations

We study the weighted boundedness of the multi-dimensional Hardy-type and singular operators in the generalized Morrey spaces Lp,f(Rn,w), defined by an almost increasing function f(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 f. In relation 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 (c) 2012 John Wiley & Sons, Ltd.