Quasiconformal mappings and Royden algebras in space

Abstract

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