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

Abstract

We consider K-interpolation methods involving slowly varying functions. Let ${\overline{A}}_{\theta ,\ast }^{\mathcal{L}}$ and ${\overline{A}}_{\theta ,\ast }^{\mathcal{R}}$ $(0\le \theta \le 1)$ 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 $({\overline{A}}_{{\theta }_0,\ast }^{\mathcal{L}},\ast {)}_{\eta ,r,a}$, $({\overline{A}}_{{\theta }_0,\ast }^{\mathcal{R}},\ast {)}_{\eta ,r,a}$, $(\ast ,{\overline{A}}_{{\theta }_1,\ast }^{\mathcal{L}}{)}_{\eta ,r,a}$, and $(\ast ,{\overline{A}}_{{\theta }_1,\ast }^{\mathcal{R}}{)}_{\eta ,r,a}$ $(0\le \eta \le 1)$ 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.