We prove that each homeomorphism \( \varphi:D\to D^{\prime} \) of Euclidean domains in \( 𝕉^{n} \), \( n\geq 2 \), belonging to the Sobolev class \( W^{1}_{p,\operatorname{loc}}(D) \), where \( p\in[1,\infty) \), and having finite distortion induces a bounded composition operator from the weighted Sobolev space \( L^{1}_{p}(D^{\prime};\omega) \) into \( L^{1}_{p}(D) \) for some weight function \( \omega:D^{\prime}\to(0,\infty) \). This implies that in the cases \( p>n-1 \) and \( n\geq 3 \) as well as \( p\geq 1 \) and \( n\geq 2 \) the inverse \( \varphi^{-1}:D^{\prime}\to D \) belongs to the Sobolev class \( W^{1}_{1,\operatorname{loc}}(D^{\prime}) \), has finite distortion, and is differentiable \( {\mathcal{H}}^{n} \)-almost everywhere in \( D^{\prime} \). We apply this result to \( \mathcal{Q}_{q,p} \)-homeomorphisms; the method of proof also works for homeomorphisms of Carnot groups. Moreover, we prove that the class of \( \mathcal{Q}_{q,p} \)-homeomorphisms is completely determined by the controlled variation of the capacity of cubical condensers whose shells are concentric cubes.

我们证明\（𝕉^ {n} \），\（n \ geq 2 \）中 的欧几里得域的 每个同胚 \（\ varphi：D \ to D ^ {\ prime} \），属于Sobolev类 \ （W ^ {1} _ {p，\ operatorname {loc}}（D）\），其中 \（p \ in [1，\ infty）\）且具有有限失真，会从加权的Sobolev中引入有界合成算子将 \（L ^ {1} _ {p}（D ^ {\ prime}; \ omega）\） 换成 \（L ^ {1} _ {p}（D）\） 以获得权重函数 \（\ omega ：D ^ {\ prime} \ to（0，\ infty）\）。这意味着在 \（p> n-1 \） 和 \（n \ geq 3 \） 以及 \（p \ geq 1 \） 和 \（n \ geq 2 \）的情况下\（\ varphi ^ {-1}：D ^ {\ prime} \ to D \） 的倒数 属于Sobolev类 \（W ^ {1} _ {1，\ operatorname {loc}}（D ^ {\ prime}）\）具有有限的失真，并且是可微分的 \（{\ mathcal {H}} ^ {n} \） -几乎在\（D ^ {\ prime} \）中都是无处不在的 。我们将此结果应用于\（\ mathcal {Q} _ {q，p} \）-同胚；证明方法也适用于卡诺群的同胚。此外，我们证明 \（\ mathcal {Q} _ {q，p} \）-同胚性的类完全由壳为同心立方体的立方电容器的容量的受控变化完全确定。