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 $$({\overline{X}}_{{\theta }_0,{\mathrm{b}}_0,E_0,\mathrm{a},F}^{\mathcal{R}},{\overline{X}}_{{\theta }_1,{\mathrm{b}}_1,E_1,\mathrm{a},F}^{\mathcal{L}}{)}_{\eta ,\mathrm{b},E},0\le \eta \le 1,$$ 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.