Sharp bounds for composition with quasiconformal mappings in Sobolev spaces

TL;DR

This paper investigates the behavior of composition operators induced by quasiconformal mappings on Sobolev and related function spaces, providing sharp bounds and extending results to Besov spaces.

Contribution

It offers new sharp bounds for composition operators with quasiconformal mappings on Sobolev and Besov spaces, including alternative proofs and interpolation techniques.

Findings

Composition operators map $H^{s,p}$ to $H^{s,q}$ under certain conditions.

Results extend to Besov spaces with sharp bounds.

Provides alternative proofs for known results in $L^p$ and $W^{1,p}$ spaces.

Abstract

Let $\phi$ be a quasiconformal mapping, and let $T_\phi$ be the composition operator which maps $f$ to $f\circ\phi$. Since $\phi$ may not be bi-Lipschitz, the composition operator need not map Sobolev spaces to themselves. The study begins with the behavior of $T_\phi$ 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.