Isomorphic classification of $L_{p,q}$-spaces, II

TL;DR

This paper completes the isomorphic classification of $L_{p,q}$-spaces on general measure spaces by identifying new subspaces and extending previous results on resonant measure spaces.

Contribution

It provides a full isomorphic classification of $L_{p,q}$-spaces on arbitrary $\sigma$-finite measure spaces, building on prior work on resonant spaces.

Findings

Identification of new subspaces of $L_{p,q}$-spaces

Complete classification on general measure spaces

Extension of previous classifications on resonant spaces

Abstract

This is a continuation of the papers [Kuryakov-Sukochev, JFA, 2015] and [Sadovskaya-Sukochev, PAMS, 2018], in which the isomorphic classification of $L_{p,q}$, for $1< p<\infty$, $1\le q<\infty$, $p\ne q $, on resonant measure spaces, has been obtained. The aim of this paper is to give a complete isomorphic classification of $L_{p,q}$-spaces on general $\sigma$-finite measure spaces. Towards this end, several new subspaces of $L_{p,q}(0,1)$ and $L_{p,q}(0,\infty)$ are identified and studied.