Singular Integrals and Commutators in Generalized Morrey Spaces

We consider a singular an integral operator K with a variable Calderon-Zygmund type kernel k(x;ξ),x ∈ R~n,ξ ∈ R~n\{0},satisfying a mixed homogeneity condition of the form k(x;μ~(α_1)ξ1,…, μ~(α_n)ξn)=μ~-Σ_i~n=1~(α_i)k(x;ξ),α_i≥1 and μ0.The continuity of this operator in L~p(R~n)is well studied by Fabes and Riviere.Our goal is to extend their result to generalized Morrey spaces L~(p,w)(R~n), p ∈(1,∞)with a weight ω satisfying suitable dabbling and integral conditions.A special attention is paid to the commutator l[a,k]=Ka-aK with the operator of multiplication by BMO functions.