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

TL;DR

This paper characterizes complex interpolation spaces involving R and L classes within rearrangement invariant spaces, with applications to grand, small Lebesgue, Gamma, and $A$ and $B$-type spaces.

Contribution

It provides new interpolation formulas for R and L classes in rearrangement invariant spaces, extending previous results and applying to various specialized function spaces.

Findings

Derived explicit interpolation formulas for R and L classes.

Applied results to grand, small Lebesgue, Gamma, and $A$, $B$-spaces.

Enhanced understanding of the structure of interpolation spaces.

Abstract

Given $E_0, E_1, E, F$ rearrangement invariant spaces, $a, b, b_0, b_1$ slowly varying functions and $0\leq \theta_0<\theta_1\leq 1$, we characterize the interpolation space $$(\overline{X}_{\theta_0,b_0,E_0}, \overline{X}^{\mathcal R}_{\theta_1, b_1,E_1,a,F})_{\theta,b,E} \quad \text{and} \quad (\overline{X}^{\mathcal L}_{\theta_0, b_0,E_0,a,F}, \overline{X}_{\theta_1,b_1,E_1})_{\theta,b,E},$$ 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.