A phase transition phenomenon between the isometric and isomorphic extension problems for hölder functions between l_p spaces

It is shown that it is possible to extend α Hölder maps from subsets of L_p to L_q (1 <p, q≤2) isometrically if and only if α≤p/q*, and isomorphically if and only if α≤p/2. It is also proved that the set of as which allow an isomorphic extension for α Hölder maps from subsets of X to Y is monotone when Y is a dual Banach space. Finally, the isometric and isomorphic extension problems for Hölder functions between L_p and L_q is studied for general p, q≥ 1, and a question posed by K. Ball is solved by showing that it is not true that all Lipschitz maps from subsets of Hilbert space into normed spaces extend to the whole of Hilbert space.