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Lorentz-Shimogaki-Arazy-Cwikel Theorem Revisited

Léonard Cadilhac, Fedor Sukochev, Dmitriy Zanin

Abstract

We present a new approach to Lorentz-Shimogaki and Arazy-Cwikel Theorems which covers all range of $p,q\in (0,\infty]$ for function spaces and sequence spaces. As a byproduct, we solve a conjecture of Levitina and the last two authors.

Lorentz-Shimogaki-Arazy-Cwikel Theorem Revisited

Abstract

We present a new approach to Lorentz-Shimogaki and Arazy-Cwikel Theorems which covers all range of for function spaces and sequence spaces. As a byproduct, we solve a conjecture of Levitina and the last two authors.

Paper Structure

This paper contains 9 sections, 30 theorems, 186 equations.

Table of Contents

  1. Introduction
  2. Interpolation for the couple $(L_0,L_q)$
  3. Construction of contractions on ($L_0$,$L_q$) and ($L_p$,$L_\infty$)
  4. Partition lemmas
  5. Construction of operators
  6. Interpolation spaces for the couple $(L_p,L_q)$
  7. Interrpolation spaces for couples of $\ell^p$-spaces
  8. An interpolation theorem for the couple $(\ell^p,\ell^q)$
  9. Upper Boyd index

Key Result

Theorem 1.1

Let $E\subset\mathcal{X}$ be a quasi-Banach function space (a priori, not necessarily symmetric). Let $p,q\in (0,\infty)$ such that $p<q$. Then:

Theorems & Definitions (72)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5: Bounded operator on a couple of quasi-Banach function spaces
  • Definition 1.6: Interpolation space between quasi-Banach function spaces
  • Theorem 1.7
  • proof
  • Definition 1.8: Bounded operator on a couple $(L_0(\Omega),Y)$ for a quasi-Banach function space $Y$
  • Definition 1.9: Interpolation space for a couple $(L_0(\Omega),Y)$ for a quasi-Banach function space $Y$
  • ...and 62 more