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Question 1. Does the Hardy space $H^{1}$ have an unconditional basis? This problem appeared in S.Kwapien and A.Pelczynski's paper: Some linear topological properties of the hardy spaces $H^{p}$, Compositio Mathematica.33(1976),261-288. I do not know whether this problem was solved.

Question 2. Does the Lorentz function space $L_{w,1}(0,1)$ or $L_{w,1}(0,+\infty)$ have an unconditional basis?

Thank you.

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Maurey proved the existenc of an unconditional basis as in question 1 (Acta Math. 1980) and then Wojtaszczyk (Ark. f. Math. 1982) gave an explicit example---the Franklin system.

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  • $\begingroup$ A fine overview of the whole subject which was opened up by Maurey's key paper can also be found in the monograph by P. Müller on the isomorphism theory of Hardy spaces. $\endgroup$
    – dalry
    Commented Nov 5, 2015 at 15:15

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