Uniform homeomorphisms of unit spheres and Property H of Lebesgue-Bochner function spaces

Assume that the unit spheres of Banach spaces X and Y are uniformly homeomorphic. Then we prove that all unit spheres of the Lebesgue.Bochner function spaces L p (.,X) and L q (., Y) are mutually uniformly homeomorphic where 1 . p, q < .. As its application, we show that if a Banach space X has Property H introduced by Kasparov and Yu, then the space L p (.,X), 1 . p < ., also has Property H.