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

Abstract

AbstractGiven $$E_0, E_1, F_0, F_1, E$$ E 0 , E 1 , F 0 , F 1 , E rearrangement invariant function spaces, $$a_0$$ a 0 , $$a_1$$ a 1 , $$\mathrm {b}_0$$ b 0 , $$\mathrm {b}_1$$ b 1 , $$\mathrm {b}$$ b slowly varying functions and $$0< \theta _0<\theta _1<1$$ 0 < θ 0 < θ 1 < 1 , we characterize the interpolation spaces $$\begin{aligned} \big ({\overline{X}}^{{\mathcal {R}}}_{\theta _0,\mathrm {b}_0,E_0,a_0,F_0}, {\overline{X}}^{{\mathcal {R}}}_{\theta _1, \mathrm {b}_1,E_1,a_1,F_1}\big )_{\theta ,\mathrm {b},E},\quad \big ({\overline{X}}^{{\mathcal {L}}}_{\theta _0, \mathrm {b}_0,E_0,a_0,F_0}, {\overline{X}}^{\mathcal L}_{\theta _1,\mathrm {b}_1,E_1,a_1,F_1}\big )_{\theta ,\mathrm {b},E} \end{aligned}$$ ( X ¯ θ 0 , b 0 , E 0 , a 0 , F 0 R , X ¯ θ 1 , b 1 , E 1 , a 1 , F 1 R ) θ , b , E , ( X ¯ θ 0 , b 0 , E 0 , a 0 , F 0 L , X ¯ θ 1 , b 1 , E 1 , a 1 , F 1 L ) θ , b , E and $$\begin{aligned} \big ({\overline{X}}^{{\mathcal {R}}}_{\theta _0,\mathrm {b}_0,E_0,a_0,F_0}, {\overline{X}}^{{\mathcal {L}}}_{\theta _1, \mathrm {b}_1,E_1,a_1,F_1}\big )_{\theta ,\mathrm {b},E},\quad \big ({\overline{X}}^{{\mathcal {L}}}_{\theta _0, \mathrm {b}_0,E_0,a_0,F_0}, {\overline{X}}^{\mathcal R}_{\theta _1,\mathrm {b}_1,E_1,a_1,F_1}\big )_{\theta ,\mathrm {b},E}, \end{aligned}$$ ( X ¯ θ 0 , b 0 , E 0 , a 0 , F 0 R , X ¯ θ 1 , b 1 , E 1 , a 1 , F 1 L ) θ , b , E , ( X ¯ θ 0 , b 0 , E 0 , a 0 , F 0 L , X ¯ θ 1 , b 1 , E 1 , a 1 , F 1 R ) 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.