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. 



On the unconditional basis on the Hardy space $H^{1}$ and the Lorentz function space $L_{w,1}$
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.


