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 $\overline{A}_{θ,*}^{\mathcal{L}}$ and $\overline{A}_{θ,*}^{\mathcal{R}}$ $(0\leqθ\leq1)$ 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 $\Big(\overline{A}_{θ_0,*}^{\mathcal{L}}, *\Big)_{η,r,a}$, $\Big(\overline{A}_{θ_0,*}^{\mathcal{R}}, *\Big)_{η,r,a}$, $\Big(*, \overline{A}_{θ_1,*}^{\mathcal{L}}\Big)_{η,r,a}$, and $\Big(*, \overline{A}_{θ_1,*}^{\mathcal{R}}\Big)_{η,r,a}$ $(0\leqη\leq1)$ for the limiting cases $θ_0=0$ and $θ_1=1$. This supplements the earlier papers of the authors, which only considered the case $0<θ_0<θ_1<1$. 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.