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Duals of limiting interpolation spaces

Manvi Grover, Bohumír Opic

TL;DR

The paper extends Lions–Peetre duality to the limiting real interpolation spaces with $K$- and $J$-functionals and slowly varying weights, describing explicit dual representations at $\theta=0$ (and related variants) for compatible Banach couples. It develops a framework of slowly varying transforms and auxiliary tools to relate $(X_0,X_1)'_{0,q,\cdot;K}$ and $(X_0',X_1')_{0,q',\cdot;J}$ (and their $K$-space counterparts), including discrete characterizations via $\lambda_{\theta,q,w}$ and density considerations. The main contributions are the several duality theorems and their corollaries across $K$- and $J$-spaces, along with variants and density/dedicated domain results that unify and extend classical real interpolation dualities to the limiting setting. These results deepen the understanding of interpolation with function parameters and have potential applications in analysis where limiting interpolation spaces arise, such as endpoint estimates and density theorems. Throughout, the authors provide detailed proofs (sections 5–10) and discuss an alternative CP discrete method in the concluding remarks.

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 $(X_0,X_1)_{θ,q}$, where $θ\in (0,1)$ and $1\le q < \infty$, this problem was solved by Lions and Peetre.

Duals of limiting interpolation spaces

TL;DR

The paper extends Lions–Peetre duality to the limiting real interpolation spaces with - and -functionals and slowly varying weights, describing explicit dual representations at (and related variants) for compatible Banach couples. It develops a framework of slowly varying transforms and auxiliary tools to relate and (and their -space counterparts), including discrete characterizations via and density considerations. The main contributions are the several duality theorems and their corollaries across - and -spaces, along with variants and density/dedicated domain results that unify and extend classical real interpolation dualities to the limiting setting. These results deepen the understanding of interpolation with function parameters and have potential applications in analysis where limiting interpolation spaces arise, such as endpoint estimates and density theorems. Throughout, the authors provide detailed proofs (sections 5–10) and discuss an alternative CP discrete method in the concluding remarks.

Abstract

The aim of the paper is to establish duals of the limiting real interpolation - and -spaces and , where is a compatible couple of Banach spaces, , is a slowly varying function on the interval , and the symbols and stand for the Peetre - and -functionals. In the case of the classical real interpolation method , where and , this problem was solved by Lions and Peetre.

Paper Structure

This paper contains 11 sections, 38 theorems, 295 equations.

Table of Contents

  1. Introduction
  2. Notation, definitions and preliminaries
  3. Main Results
  4. Auxiliary assertions
  5. Proof of Theorem \ref{['DT0S']}
  6. Proof of Theorem \ref{['DTJ1']}
  7. Proof of Theorem \ref{['DTJ11']}
  8. Proofs of Theorem \ref{['DT0S1']} and Remark \ref{['remark100*']}
  9. Proofs of Theorem \ref{['DTJ111']} and Remark \ref{['remark100J*']}
  10. Proofs of Theorem \ref{['DTJ111.1']} and and Remark \ref{['remark100J*4']}
  11. Concluding remarks

Key Result

Lemma 2.1

(OG) Let $b,b_1,b_2 \in SV(0,\infty)$ and let $d(t):=b(1/t)$ for all $t>0 .$(i) Then $\, b_1 b_2, d \in SV(0,\infty)$ and $b^r \in SV(0,\infty)$ for each $r \in {\mathbb R}.$(ii) If $\,\varepsilon$ and $\kappa$ are positive numbers, then there are positive constants $c_\varepsilon$ and $C_\varepsilo (iii) If $\,\alpha>0$ and $q \in (0,\infty]$, then, for all $t>0,$ (iv) If $\,q \in (0,\infty]$, a

Theorems & Definitions (64)

  • Lemma 2.1
  • Remark 2.2
  • Definition 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Remark 2.6
  • Definition 2.7: the space $\lambda_{\theta ,q,w}$
  • Remark 2.8
  • Theorem 3.1: 1. duality theorem for K-spaces and $\theta=0$
  • Theorem 3.2: 1. duality theorem for J-spaces and $\theta=0$
  • ...and 54 more