Isomorphic classification of atomic weak L^p spaces

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 $f^{\ast }$ 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.

References in corpus (1)

• Isomorphisms of certain weak L^p spaces