Extension of isometries on the unit sphere of L p spaces

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.