Abstract
Let X be a Banach function space, L∞ [0, 1] ⊂ X ⊂ L1[0, 1]. It is proved that if dual space of X has singularity property in closed set E ⊂ [0, 1] then: 1) there exists no orthonormal basis in C[0, 1], which forms an unconditional basis in X in metric of L1[0, 1] space, 2) for the Hardy-Littlewood maximal operator M we have 
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Kopaliani, T. The Singularity Property of Banach Function Spaces and Unconditional Convergence in L1[0, 1]. Positivity 10, 467–474 (2006). https://doi.org/10.1007/s11117-005-0001-6
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DOI: https://doi.org/10.1007/s11117-005-0001-6