On unconditional bases in certain Banach function spaces (Q558302)

Banach function spaces (BFS) are considered, i.e., Banach spaces \(X\) of Lebesgue measurable real-valued functions on \([0,1]\) such that, roughly speaking, the norm on \(X\) is monotone, depends only on the absolute value of the function, and \(X\) is continuously embedded between \(L^\infty [0,1]\) and \(L^1 [0,1]\). If \(X\) is separable, the dual (in the sense of Köthe) \(X'\) coincides with the usual dual space \(X^\ast\). The author introduces the following notions: A BFS \(X\) has the singularity property (a) at a point \(t_0 \in [0,1]\) if there is a constant \(c>0\) such that \(\| \chi _{( t_0-\varepsilon , t_0+ \varepsilon )\cap [0,1]}\| _X\geq c\) for every \(\varepsilon >0\), (b) in a subset \(E \subseteq [0,1]\) of Lebesgue measure zero if there is a constant \(c>0\) such that \(\| \chi _A\| _X \geq c\) for every open set \(A\supseteq E\). The main result of this paper is the following Theorem. Let \(F=\{f_n\}_{n=1}^\infty \) and \(G=\{g_n\}_{n=1}^\infty \) be systems in \(X\) and \(X^\ast\), respectively, with \(\langle f_n,g_m \rangle =\delta_{nm}\), where \(X\) is a separable BFS and \(X^\ast\) has the singularity property at a point \(t_0\). If \(C([t_0-\varepsilon , t_0+ \varepsilon ]\cap [0,1])\) is in the linear span of \(G\) for some \(\varepsilon >0\), then \(F\) cannot be an unconditional basis for the space \(X\). Moreover, the author investigates some examples like Lorentz and Marcinkiewicz spaces and presents a BFS \(X\) which has the singularity property in a zero set \(E\), but does not have the singularity property at any point of \([0,1]\).