Reiteration Theorem for ${\mathcal R}$ and ${\mathcal L}$-spaces with the same parameter

Abstract: Let $E, F, E_0, E_1$ be rearrangement invariant spaces; let $a, \mathrm{b}, \mathrm{b}0, \mathrm{b}_1$ be slowly varying functions and $0< \theta_0,\theta_1<1$. We characterize the interpolation spaces $$\Big(\overline{X}^{\mathcal R}{\theta_0,\mathrm{b}0,E_0,\mathrm{a},F}, \overline{X}^{\mathcal L}{\theta_1,\mathrm{b}1,E_1,\mathrm{a},F}\Big){\eta,\mathrm{b},E}:, \quad 0\leq\eta\leq1,$$ when the parameters $\theta_0$ and $\theta_1$ are equal (under appropriate conditions on $\mathrm{b}_i(t)$, $i=0,1$). This completes the study started in \cite{Do2020,FMS-RL3}, which only considered the case $\theta_0<\theta_1$. As an application we recover and generalize interpolation identities for grand and small Lebesgue spaces.