Publication at Faculty of Mathematics and Physics |
2012
Abstract
Let $\Omega\subset\rn$, $q\geq n$ and $\alpha\geq 0$ or $1<q\leq n$ and $\alpha\leq 0$. We prove that the composition of $q$-quasiconfomal mapping $f$ and function $u\in WL^q\log^{\alpha}L_{\loc}(f(\Omega))$ satisfies $u\circ f\in WL^q\log^{\alpha}L_{\loc}(\Omega)$.
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.