On extension of isometries on the unit spheres of [formula omitted]-spaces for [formula omitted

Abstract

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