On the subinvariance of uniform domains in Banach spaces

Suppose that E and E ′ denote real Banach spaces with dimension at least 2 , that D ⊂ E and D ′ ⊂ E ′ are domains, and that f : D → D ′ is a homeomorphism. In this paper, we prove the following subinvariance property for the class of uniform domains: suppose that f is a freely quasiconformal mapping and that D ′ is uniform. Then the image f ( D 1 ) of every uniform subdomain D 1 in D under f is still uniform. This result answers an open problem of Väisälä in the affirmative.