Extension of isometries on the unit sphere of L p space

In this paper we study the isometric extension problem and show that every surjective isometry between the unit spheres of L p(µ) (1 < p < ., p . 2) and a Banach space E can be extended to a linear isometry from L p(µ) onto E, which means that if the unit sphere of E is (metrically) isometric to the unit sphere of L p(µ), then E is linearly isometric to L p(µ). We also prove that every surjective 1-Lipschitz or anti-1-Lipschitz map between the unit spheres of L p(µ1, H 1) and L p(µ2, H 2) must be an isometry and can be extended to a linear isometry from L p(µ1, H 1) to L p(µ2, H 2), where H 1 and H 2 are Hilbert spaces