Reiteration Formulae for the Real Interpolation Method Including limiting ${\mathcal L}$ or ${\mathcal R}$ Spaces

TL;DR

This paper characterizes limiting interpolation spaces involving ${\mathcal{L}}$ and ${\mathcal{R}}$ spaces with slowly varying functions, extending previous results to boundary cases and applying to Lorentz and Karamata spaces.

Contribution

It extends reiteration formulae to limiting cases involving ${\mathcal{L}}$ and ${\mathcal{R}}$ spaces, providing new characterizations and applications.

Findings

Characterization of limiting interpolation spaces for ${\mathcal{L}}$ and ${\mathcal{R}}$ spaces.

Extension of previous results to boundary cases $ heta_0=0$ and $ heta_1=1$.

Applications to Lorentz, small Lorentz, and Lorentz-Karamata spaces.

Abstract

We consider K-interpolation methods involving slowly varying functions. Let $\overline{A}_{\theta,*}^{\mathcal{L}}$ and $\overline{A}_{\theta,*}^{\mathcal{R}}$ $(0\leq\theta\leq1)$ be the so called ${\mathcal{L}}$ or ${\mathcal{R}}$ limiting interpolation spaces which arise naturally in reiteration formulae for the limiting cases. We characterize the interpolation spaces $\Big(\overline{A}_{\theta_0,*}^{\mathcal{L}}, *\Big)_{\eta,r,a}$, $\Big(\overline{A}_{\theta_0,*}^{\mathcal{R}}, *\Big)_{\eta,r,a}$, $\Big(*, \overline{A}_{\theta_1,*}^{\mathcal{L}}\Big)_{\eta,r,a}$, and $\Big(*, \overline{A}_{\theta_1,*}^{\mathcal{R}}\Big)_{\eta,r,a}$ $(0\leq\eta\leq1)$ for the limiting cases $\theta_0=0$ and $\theta_1=1$. This supplements the earlier papers of the authors, which only considered the case $0<\theta_0<\theta_1<1$. The proofs of most reiteration formulae are based on Holmstedt-type formulae. Applications to grand and small Lorentz spaces as well as to Lorentz-Karamata spaces are given.