On nonatomic Banach lattices and Hardy spaces

Abstract

We are interested in the question when a Banach space $X$ with an unconditional basis is isomorphic (as a Banach space) to an order-continuous nonatomic Banach lattice. We show that this is the case if and only if $X$ is isomorphic as a Banach space with $X({\ell }_2)$. This and results of J. Bourgain are used to show that spaces $H_1({\mathbf{T}}^n)$ are not isomorphic to nonatomic Banach lattices. We also show that tent spaces introduced in \cite{4} are isomorphic to Rad $H_1$.