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

TL;DR

This paper characterizes the interpolation spaces for certain rearrangement invariant spaces when specific parameters are equal, extending previous work and applying results to grand and small Lebesgue spaces.

Contribution

It completes the study of interpolation spaces for ${ m R}$ and ${ m L}$-spaces by addressing the case where parameters are equal, which was previously unresolved.

Findings

Provides a full characterization of the interpolation spaces when parameters are equal.

Recovers and generalizes known identities for grand and small Lebesgue spaces.

Extends the theoretical framework of rearrangement invariant space interpolation.

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.