Composition of $q$-quasiconformal mappings and functions in Orlicz–Sobolev spaces

Abstract

Let $\Omega\subset\mathbb{R}^{n}$, $q\geq n$ and $\alpha\geq0$ or $1<q\leq n$ and $\alpha\leq0$. We prove that the composition of $q$-quasiconfomal mapping $f$ and function $u\in WL^{q}\log^{\alpha}L_{\operatorname{loc}}(f(\Omega))$ satisfies $u\circ f\in WL^{q}\log^{\alpha}L_{\operatorname{loc}}(\Omega)$. Moreover, each homeomorphism $f$ which introduces continuous composition operator from $WL^{q}\log^{\alpha}L$ to $WL^{q}\log^{\alpha}L$ is necessarily a $q$-quasiconformal mapping. As a new tool, we prove a Lebesgue density type theorem for Orlicz spaces.