Singular Integrals and Commutators in Generalized Morrey Spaces

Abstract

We consider a singular an integral operator ${\fancyscript K}$ with a variable Calderón–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 L ^p(ℝⁿ) is well studied by Fabes and Rivière. Our goal is to extend their result to generalized Morrey spaces L ^p,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.