Extrapolatory description for the Lorentz and Marcinkiewicz spaces “close” to $L_\infty$

We introduce the notion of a rearrangement invariant extrapolation space on $[0,1]$. We obtain some sufficient conditions (also necessary in some cases) for the Marcinkiewicz and Lorentz spaces to be extrapolation spaces. We generalize and improve the previous results, which enables us to determine the possible limits of such description of spaces and thereby to establish assertions of the Yano-type theorem. In particular, we present some examples of the spaces “close” to $L_\infty$ in a sense but lacking this description.