Abstract

We define the scale \({{\mathcal{Q}}_{p}}\), \(n - 1 < p < \infty \), of homeomorphisms of spatial domains in \({{\mathbb{R}}^{n}}\), a geometric description of which is due to the control of the behavior of the p-capacity of condensers in the image through the weighted p-capacity of the condensers in the preimage. For p = n the class \({{\mathcal{Q}}_{n}}\) of mappings contains the class of so-called Q-homeomorphisms, which have been actively studied over the past 25 years. An equivalent functional and analytic description of these classes \({{\mathcal{Q}}_{p}}\) is obtained. It is based on the problem of the properties of the composition operator of a weighted Sobolev space into a nonweighted one induced by a map inverse to some of the class \({{\mathcal{Q}}_{p}}\).

中文翻译：

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加权Sobolev空间上的合成算子和$$ {{\ mathcal {Q}} _ {p}} $$-同胚

摘要

我们定义规模\（{\ mathcal Q}} _ {P} {{} \），\（N - 1 <P <\ infty \），在空间域中的同胚\（{{\ mathbb {R} } ^ {n}} \），其几何描述是由于通过原像中电容器的加权p容量控制了图像中电容器p容量的行为所致。对于p = n，映射的类\（{{\ mathcal {Q}} _ {n}} \）包含所谓的Q-同胚性，在过去25年中已经对其进行了积极的研究。这些类的等效功能和分析描述\（{{\ mathcal {Q}} _ {p}} \）获得。它基于一个问题，该问题是将加权Sobolev空间的合成算符的性质转化为由与某些类\（{{\ mathcal {Q}} _ {p}} \）逆的映射得出的非加权空间。