The singularity property of Banach function spaces and unconditional convergence in \(L^1[0,1]\). (Q850592)

A Banach function space \(X\) on \([0,1]\) is said to have the singularity property in a set \(E\) of measure zero if there is a constant \(c > 0\) such that \(\| \chi_U\| _X \geq c\) for every open set \(U \supset E\). The main result of the paper states that if \(X'\) -- the associate space to \(X\) -- has the singularity property in some closed set and \((f_n)_1^\infty\) is an orthonormal basis of continuous functions, then \((f_n)_1^\infty\) does not form an unconditional basis of \(X\) even in the sense of \(L_1\)-norm. Applying this result to the Franklin system, the author deduces that, under the conditions of the main theorem, \(\sup\{\| Mf\| _{L_1[0,1]}: \| f\| _X \leq 1\} = \infty\), where \(M\) stands for the Hardy--Littlewood maximal operator.