Singular Integrals and Commutators in Generalized Morrey Spaces

The purpose of this paper is to study singular integrals whose kernels k(x; ξ) are variable, i.e. they depend on some parameter x ∈ R and in ξ ∈ R \ {0} satisfy mixed homogeneity condition of the form k(x;μξ1, . . . , μ ξn) = μ − ∑ n i=1 ik(x; ξ) with positive real numbers αi ≥ 1 and μ > 0. The continuity of these operators in L(R) is well studied by Fabes and Rivière. Our goal is to extend their results in generalized Morrey spaces with a weight satisfying suitable dabbling and integral conditions. A special attention is paid also of the commutators of the kernel with functions of bounded and vanishing mean oscillation.