Duals of limiting interpolation spaces

TL;DR

This paper determines the dual spaces of certain limiting real interpolation spaces involving Banach space couples, extending classical results by Lions and Peetre to more general limiting cases.

Contribution

It establishes the duality of limiting real interpolation $K$- and $J$-spaces with slowly varying functions, generalizing classical duality results.

Findings

Duals of limiting interpolation spaces are characterized.

Extends classical duality results to limiting cases.

Provides a framework for duality in more general interpolation settings.

Abstract

The aim of the paper is to establish duals of the limiting real interpolation $K$- and $J$-spaces $(X_0,X_1)_{0,q,v;K}$ and $(X_0,X_1)_{0,q,v;J}$, where $(X_0,X_1)$ is a compatible couple of Banach spaces, $1\le q<\infty$, $v$ is a slowly varying function on the interval $(0,\infty)$, and the symbols $K$ and $J$ stand for the Peetre $K$- and $J$-functionals. In the case of the classical real interpolation method $(X_0,X_1)_{\theta,q}$, where $\theta \in (0,1)$ and $1\le q < \infty$, this problem was solved by Lions and Peetre.