General Reiteration Theorems for $${{\mathcal {R}}}$$ and $${{\mathcal {L}}}$$ Classes: Case of Right $${{\mathcal {R}}}$$-Spaces and Left $${{\mathcal {L}}}$$-Spaces

Abstract

AbstractGiven $$E_0, E_1, E, F$$ E 0 , E 1 , E , F rearrangement invariant spaces, $$a, \mathrm {b}, \mathrm {b}_0, \mathrm {b}_1$$ a , b , b 0 , b 1 slowly varying functions and $$0\le \theta _0<\theta _1\le 1$$ 0 ≤ θ 0 < θ 1 ≤ 1 , we characterize the interpolation spaces $$\begin{aligned} ({\overline{X}}_{\theta _0,\mathrm {b}_0,E_0}, {\overline{X}}^{{\mathcal {R}}}_{\theta _1, \mathrm {b}_1,E_1,a,F})_{\theta ,\mathrm {b},E}\quad \text {and}\quad ({\overline{X}}^{{\mathcal {L}}}_{\theta _0, \mathrm {b}_0,E_0,a,F}, {\overline{X}}_{\theta _1,\mathrm {b}_1,E_1})_{\theta ,\mathrm {b},E}, \end{aligned}$$ ( X ¯ θ 0 , b 0 , E 0 , X ¯ θ 1 , b 1 , E 1 , a , F R ) θ , b , E and ( X ¯ θ 0 , b 0 , E 0 , a , F L , X ¯ θ 1 , b 1 , E 1 ) θ , b , E , for all possible values of $$\theta \in [0,1]$$ θ ∈ [ 0 , 1 ] . Applications to interpolation identities for grand and small Lebesgue spaces, Gamma spaces and A and B-type spaces are given.