Isomorphic classification of atomic weak L^p spaces

TL;DR

This paper classifies the isomorphic types of weak L^p spaces over purely atomic measure spaces, showing they are either ℓ^∞ or ℓ^1 when the measure space is countably generated and σ-finite.

Contribution

It provides a complete isomorphic classification of weak L^p spaces in the atomic case, extending understanding of their structure in relation to measure space properties.

Findings

Weak L^p spaces over purely atomic measure spaces are classified up to isomorphism.

Infinite-dimensional weak L^p spaces over countably generated σ-finite measure spaces are isomorphic to ℓ^∞ or ℓ^1.

The classification results clarify the structure of weak L^p spaces in atomic settings.

Abstract

Let $\msp$ be a measure space and let $1 < p < \infty$. The {\em weak $L^p$}\/ space $\wlp$ consists of all measurable functions $f$ such that \[ \|f\| = \sup_{t>0}t^{\frac{1}{p}}f^*(t) < \infty,\] where $f^*$ is the decreasing rearrangement of $|f|$. It is a Banach space under a norm which is equivalent to the expression above. In this paper, we pursue the problem of classifying weak $L^p$ spaces isomorphically when $\msp$ is purely atomic. It is also shown that if $\msp$ is a countably generated $\sigma$-finite measure space, then $\wlp$ (if infinite dimensional) must be isomorphic to either $\ell^\infty$ or $\seq$. The results of this article were presented at the conference in Columbia, Missouri in May, 1994.