Sequence spaces \(l_{p,q}\) in probabilistic characterizations of weak type operators (Q1781351)

Let \(T: X\to L_0([0,1],{\mathcal M},{\mathfrak m})\) be an operator (not necessarily linear) defined on a quasi-Banach space \(X\) and taking values in the space of Lebesgue-measurable real functions. In this paper, the author studies factorization theorems for linear and superlinear (= quasilinear) operators with values in \(L_0\). He proves this type of factorization with the help of the Lorentz sequence spaces \(l_{p,q}\). He obtains characteristic properties of sequences of functions belonging to bounded sets in the spaces \(L_{p,\infty}\) for \(0< p<\infty\) and \(0< p< q\). Further, he uses sequences of independent random variables to distinguish weak type operators from operators factorizable through \(L_{p,\infty}\).