Strong extrapolation spaces and interpolation

We introduce a new class of rearrangement invariant spaces on the segment $[0,1]$ which contains the most common extrapolation spaces with respect to the $L_p$-scale as $p\to\infty$. We characterize the class and demonstrate under certain conditions that the Peetre $\mathscr K$-functional has extrapolatory description in the couple $(E,L_\infty)$ if and only if $E$ belongs to the class. By way of application, we establish a new extrapolation theorem for the bounded operators in $L_p$.