Extrapolation properties of the scale of $L_p$-spaces

A new class of extrapolation functors on the scale of $L_p$-spaces $(1<p<\infty)$ is introduced, allowing one to take for its “limiting spaces” two symmetric spaces “close” to $L_\infty$ and $L_1$. Crucial here are the extrapolation relations for the Peetre $\mathscr K$- and $\mathscr J$-functionals for the Banach couples $(L_\infty,\operatorname{Exp} L^\beta)$ and $(L_1,L(\log L)^{1/\beta})$, respectively $(\operatorname{Exp} L^\beta$ and $L(\log L)^{1/\beta}$, $\beta>0$, are Zygmund spaces).
The real method of operator interpolation is used.