Homeomorphisms of Finite Inner Distortion: Composition Operators on Zygmund-Sobolev and Lorentz-Sobolev Spaces

Abstract

Let $p > n-1$ and $\alpha\in\mathsf{R}$ and suppose that $f:\Omega\stackrel{\rm onto\,\,}\longrightarrow\Omega^\prime$ is a homeomorphism in the Zygmund-Sobolev space ${\it WL}^{p}\log^{\alpha} L_{\mathop{\rm loc}\nolimits} (\Omega{,}\mathsf{R}^n)$. Define $r{=}\frac p {p-n+1}$. Assume that $u{\in}{\it WL}^r\log^{-\alpha(r-1)} L_{\mathop{\rm loc}\nolimits}(\Omega)$. Then $u\circ\smash{f^{-1}}\in {\rm BV}_{\mathop{\rm loc}\nolimits}(\Omega^\prime)$. We obtain a similar result whenever $f$ is a homeomorphism in the Lorentz-Sobolev space ${\it WL}^{p,q}_{\mathop{\rm loc}\nolimits} (\Omega,\mathsf{R}^n)$ with $p,q>n-1$ and $u\in {\it WL}^{r,s}_{\mathop{\rm loc}\nolimits}(\Omega)$ with $r$ as before and $s=\frac q {q-n+1}$. Moreover, if we further assume that $f$ has finite inner distortion we obtain in both cases $u\circ \smash{f^{-1}}\in W^{1,1}_{\mathop{\rm loc}\nolimits}(\Omega^\prime)$