General reiteration theorems for $${{\mathcal {R}}}$$ and $${{\mathcal {L}}}$$ classes: mixed interpolation of $${{\mathcal {R}}}$$ and $${{\mathcal {L}}}$$-spaces

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