On a problem of Lions concerning real interpolation spaces. The quasi-Banach case

Abstract

We prove that, under a mild condition on a couple (A0,A1) of quasi-Banach spaces, all real interpolation spaces (A0,A1)θ,p with 0<θ<1 and 0<p≤∞ are different from each other. In the Banach case and for 1≤p≤∞ this was shown by Janson, Nilsson, Peetre and Zafran, thus solving an old problem posed by J.-L. Lions. Moreover, we give an application to certain spaces which are important objects in Operator Theory and which consist of bounded linear operators whose approximation numbers belong to Lorentz sequence spaces.