Singular Integrals and Commutators in Generalized Morrey Spaces

Abstract

We consider a singular an integral operator \({\fancyscript K}\) with a variable Calderon–Zygmund type kernel k(x; ξ), x ∈ ℝn, ξ ∈ ℝn\{0}, satisfying a mixed homogeneity condition of the form \( k{\left( {x;\mu ^{{\alpha _{1} }} \xi _{1} , \ldots ,\mu ^{{\alpha _{n} }} \xi _{n} } \right)} = \mu ^{{ - {\sum\nolimits_{i = 1}^n {\alpha _{i} } }}} k{\left( {x;\xi } \right)},\alpha _{i} \geqslant 1 \) and μ > 0. The continuity of this operator in Lp(ℝn) is well studied by Fabes and Riviere. Our goal is to extend their result to generalized Morrey spaces Lp,ω(ℝn), p ∈ (1,∞) with a weight ω satisfying suitable dabbling and integral conditions. A special attention is paid to the commutator \({\fancyscript C}\) [a, k] = \({\fancyscript K}\)a − a\({\fancyscript K}\) with the operator of multiplication by BMO functions.