General Reiteration Theorems for $\mathcal{R}$ and $\mathcal{L}$ Clases: Mixed Interpolation of $\mathcal{R}$ and $\mathcal{L}$-spaces

Abstract

Given $E_0,E_1,F_0,F_1,E$ rearrangement invariant function spaces, $a_0$, $a_1$, $b_0$, $b_1$, $b$ slowly varying functions and $0<{\theta }_0<{\theta }_1<1$, we characterize the interpolation spaces $$({\overline{X}}_{{\theta }_0,b_0,E_0,a_0,F_0}^{\mathcal{R}},{\overline{X}}_{{\theta }_1,b_1,E_1,a_1,F_1}^{\mathcal{R}}{)}_{\theta ,b,E},({\overline{X}}_{{\theta }_0,b_0,E_0,a_0,F_0}^{\mathcal{L}},{\overline{X}}_{{\theta }_1,b_1,E_1,a_1,F_1}^{\mathcal{L}}{)}_{\theta ,b,E}$$ and $$({\overline{X}}_{{\theta }_0,b_0,E_0,a_0,F_0}^{\mathcal{R}},{\overline{X}}_{{\theta }_1,b_1,E_1,a_1,F_1}^{\mathcal{L}}{)}_{\theta ,b,E},({\overline{X}}_{{\theta }_0,b_0,E_0,a_0,F_0}^{\mathcal{L}},{\overline{X}}_{{\theta }_1,b_1,E_1,a_1,F_1}^{\mathcal{R}}{)}_{\theta ,b,E},$$ for all possible values of $\theta \in [0,1]$. Applications to interpolation identities for grand and small Lebesgue spaces, Gamma spaces and $A$ and $B$-type spaces are given.

arXiv admin note: substantial text overlap with arXiv:2010.06940