Quasiconformal mappings and Royden algebras in space

Abstract:

On every open connected set $G$ in Euclidean $n$-space ${R^n}$ and for every index $p > 1$, we define the Royden $p$-algebra ${M_p}(G)$. We use results by F. W. Gehring and W. P. Ziemer to prove that two such sets $G$ and $G’$ are quasiconformally equivalent if and only if their Royden $n$-algebras are isomorphic as Banach algebras. Moreover, every such algebra isomorphism is given by composition with a quasiconformal homeomorphism between $G$ and $G’$. This generalizes a theorem by M. Nakai concerning Riemann surfaces. In case $p \ne n$, the only homeomorphisms which induce an isomorphism of the $p$-algebras are the locally bi-Lipschitz mappings, and for $1 < p < n$, every such isomorphism arises this way. Under certain restrictions on the domains, these results extend to the Sobolev space $H_p^1(G)$ and characterize those homeomorphisms which preserve the $H_p^1$ classes.