On nonatomic Banach lattices and Hardy spaces

Kalton, N. J.; Wojtaszczyk, P.

doi:10.1090/S0002-9939-1994-1181168-6

On nonatomic Banach lattices and Hardy spaces

Authors: N. J. Kalton and P. Wojtaszczyk
Journal: Proc. Amer. Math. Soc. 120 (1994), 731-741
MSC: Primary 46B42; Secondary 42B30, 46E15
DOI: https://doi.org/10.1090/S0002-9939-1994-1181168-6
MathSciNet review: 1181168
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Abstract: We are interested in the question when a Banach space 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 is isomorphic as a Banach space with $X({\ell _2})$ . This and results of 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 by Coifman, Meyer, and Stein are isomorphic to $\operatorname{Rad} \;{H_1}$ .

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