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

中文翻译：

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无界集合上局部“互补”广义变量指数莫雷空间中潜在算子及其交换子的有界性

在本文中，我们证明了 Sobolev–Spanne 类型 $${\,^{^{\complement }}\!\mathcal M}_{\{x_0\}}^{p(\cdot ),\omega } (\ varOmega )\rightarrow {\,^{^{\complement }}\!\mathcal M}_{\{x_0\}}^{q(\cdot ),\omega } (\varOmega )$$-theorem for the潜在运算符 $$I^{\alpha }$$，其中 $${\,^{^{\complement }}\!\mathcal M}_{\{x_0\}}^{p(\cdot ),\ omega }(\varOmega )$$ 是具有可变指数 p(x) 的局部“互补”广义 Morrey 空间，$$\omega (r)$$ 是定义 Morrey 型范数的一般函数，$$\varOmega $$是 $${{\mathbb {R}}^n}$$ 的开无界子集。此外，我们证明了这些空间中潜在算子 $$[b,I^{\alpha }]$$ 的交换子的有界性。在所有情况下，有界条件都是根据 $$\omega (x,r)$$ 上的 Zygmund 型积分不等式给出的，