On Bilipschitz Extensions in Real Banach Spaces

Suppose that E and E' denote real Banach spaces with dimension at least 2, that D≠E and D'≠E' are bounded domains with connected boundaries, that f:D→D' is an M-QH homeomorphism, and that D' is uniform. The main aim of this paper is to prove that f extends to a homeomorphism f¯:D¯→D¯' and f-∣∂D is bilipschitz if and only if f is bilipschitz in D¯. The answer to some open problems of Väisälä is affirmative under a natural additional condition