Singular Integrals and Commutators in Generalized Morrey Spaces

We consider a singular an integral operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\fancyscript K}$$\end{document} with a variable Calderón–Zygmund type kernel k(x; ξ), x ∈ ℝn, ξ ∈ ℝn\{0}, satisfying a mixed homogeneity condition of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ k{\left( {x;\mu ^{{\alpha _{1} }} \xi _{1} , \ldots ,\mu ^{{\alpha _{n} }} \xi _{n} } \right)} = \mu ^{{ - {\sum\nolimits_{i = 1}^n {\alpha _{i} } }}} k{\left( {x;\xi } \right)},\alpha _{i} \geqslant 1 $$\end{document} and μ > 0. The continuity of this operator in Lp(ℝn) is well studied by Fabes and Rivière. 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\fancyscript C}$$\end{document} [a, k] = \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\fancyscript K}$$\end{document}a − a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\fancyscript K}$$\end{document} with the operator of multiplication by BMO functions.