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

Abstract

Given E_0, E_1, E, F rearrangement invariant spaces, a, mathrm {b}, mathrm {b}_0, mathrm {b}_1 slowly varying functions and 0le theta _0<theta _1le 1, we characterize the interpolation spaces (X¯θ0,b0,E0,X¯θ1,b1,E1,a,FR)θ,b,Eand(X¯θ0,b0,E0,a,FL,X¯θ1,b1,E1)θ,b,E,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} ({\\overline{X}}_{\ heta _0,\\mathrm {b}_0,E_0}, {\\overline{X}}^{{\\mathcal {R}}}_{\ heta _1, \\mathrm {b}_1,E_1,a,F})_{\ heta ,\\mathrm {b},E}\\quad \ ext {and}\\quad ({\\overline{X}}^{{\\mathcal {L}}}_{\ heta _0, \\mathrm {b}_0,E_0,a,F}, {\\overline{X}}_{\ heta _1,\\mathrm {b}_1,E_1})_{\ heta ,\\mathrm {b},E}, \\end{aligned}$$\\end{document}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.