The Regularity of Inverses to Sobolev Mappings and the Theory of \( \mathcal{Q}_{q,p} \) -Homeomorphisms

Сибирский математический журнал

article

S. K. Vodopyanov¹ 

[1] Sobolev Institute of Mathematics

关键词: quasiconformal analysis;    Sobolev space;    composition operator;    capacity estimate;   

DOI  :  10.1134/S0037446620060051

学科分类：数学（综合）

来源: Izdatel stvo Instituta Matematiki Rossiiskoi Akademii Nauk

【 摘 要 】

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.

【 授权许可】

CC BY