Lorentz–Shimogaki and Arazy–Cwikel theorems revisited

Léonard Cadilhac, Fedor Sukochev and Dmitriy Zanin

Vol. 328 (2024), No. 2, 227–254

DOI: 10.2140/pjm.2024.328.227

Abstract

By using the space $L_0$ of finitely supported functions as a left endpoint on the interpolation scale of $L_p$-spaces, we present a new approach to the Lorentz–Shimogaki and Arazy–Cwikel theorems which covers the whole range of $p,q\in (0,\infty ]$. In particular, we show that for $0\le p<q<r<s\le \infty$,

$$\mathrm{Int}<!--FUNCTION\; APPLICATION-->(L_q,L_r)=\mathrm{Int}<!--FUNCTION\; APPLICATION-->(L_p,L_r)\cap \mathrm{Int}<!--FUNCTION\; APPLICATION-->(L_q,L_s)$$

if the underlying space is $(0,\alpha )$, $\alpha \in (0,\infty ]$ equipped with the Lebesgue measure. As a byproduct of our result, we solve a conjecture of Levitina, Sukochev and Zanin (2020).

Keywords

interpolation, symmetric function spaces, $L_p$-spaces, majorization

Mathematical Subject Classification

Primary: 43A15, 46B70, 46M35

Milestones

Received: 15 December 2021

Revised: 6 March 2024

Accepted: 22 March 2024

Published: 30 April 2024