In this paper we prove a Sobolev–Spanne type ∁M{x0}p(·),ω(Ω)→∁M{x0}q(·),ω(Ω)\documentclass[12pt]{minimal}
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\begin{document}$${\,^{^{\complement }}\!\mathcal M}_{\{x_0\}}^{p(\cdot ),\omega } (\varOmega )\rightarrow {\,^{^{\complement }}\!\mathcal M}_{\{x_0\}}^{q(\cdot ),\omega } (\varOmega )$$\end{document}-theorem for the potential operators Iα\documentclass[12pt]{minimal}
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\begin{document}$$I^{\alpha }$$\end{document}, where ∁M{x0}p(·),ω(Ω)\documentclass[12pt]{minimal}
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\begin{document}$${\,^{^{\complement }}\!\mathcal M}_{\{x_0\}}^{p(\cdot ),\omega }(\varOmega )$$\end{document} is local “complementary” generalized Morrey spaces with variable exponent p(x), ω(r)\documentclass[12pt]{minimal}
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\begin{document}$$\omega (r)$$\end{document} is a general function defining the Morrey-type norm and Ω\documentclass[12pt]{minimal}
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\begin{document}$$\varOmega $$\end{document} is an open unbounded subset of Rn\documentclass[12pt]{minimal}
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\begin{document}$${{\mathbb {R}}^n}$$\end{document}. In addition, we prove the boundedness of the commutator of potential operators [b,Iα]\documentclass[12pt]{minimal}
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\begin{document}$$[b,I^{\alpha }]$$\end{document} in these spaces. In all cases the conditions for the boundedness are given in terms of Zygmund-type integral inequalities on ω(x,r)\documentclass[12pt]{minimal}
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\begin{document}$$\omega (x,r)$$\end{document}, which do not assume any assumption on monotonicity of ω(x,r)\documentclass[12pt]{minimal}
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\begin{document}$$\omega (x,r)$$\end{document} in r.
