Sharp bounds for composition with quasiconformal mappings in Sobolev spaces

Abstract

Let $\varphi$ be a quasiconformal mapping, and let $T_{\varphi }$ be the composition operator which maps $f$ to $f\circ \varphi$. Since $\varphi$ may not be bi-Lipschitz, the composition operator need not map Sobolev spaces to themselves. The study begins with the behavior of $T_{\varphi }$ on $L^p$ and $W^{1,p}$ for $1<p<\infty$. This cases are well understood but alternative proofs of some known results are provided. Using interpolation techniques it is seen that compactly supported Bessel potential functions in $H^{s,p}$ are sent to $H^{s,q}$ whenever $0<s<1$ for appropriate values of $q$. The techniques used lead to sharp results and they can be applied to Besov spaces as well.

19 pages, 5 figures