On extension of isometries on the unit spheres of Lp-spaces for 0 < p ≤ 1

摘要

This paper gives an affirmative answer to Tingley's problem in the spaces ~(Lp)(μ) where 0<1 by showing that every isometry from the unit sphere of ~(Lp)(μ) onto the unit sphere of X which is a subspace of an ~(Lp)(ν)-space can be extended to a linear isometry on the whole space ~(Lp)(μ). Especially, by employing the technique underlying the proof of the case 0<1, we can prove that every surjective isometry between the unit spheres of ~(L1)(μ) and a Banach space F can also be linearly and isometrically extended to the whole space.