Singular Integrals and Commutators in Generalized Morrey Spaces

We consider a singular an integral operator K with a variable Calderón.Zygmund type kernel k(x; .), x . .n, . . .n\{0}, satisfying a mixed homogeneity condition of the form k(x;..1.1,.,..n.n)=...ni=1.ik(x;.),.i.1 and . > 0. The continuity of this operator in L p(.n) is well studied by Fabes and Rivière. Our goal is to extend their result to generalized Morrey spaces L p,.(.n), p . (1,.) with a weight . satisfying suitable dabbling and integral conditions. A special attention is paid to the commutator C [a, k] = K a . a K with the operator of multiplication by BMO functions.