Sequence spaces $l_{p,q}$ in parabolistic characterizations of the weak type operators

Not necessarily linear operators $T\colon X\mapsto L_\circ([0,1],\mathscr M,\mathbf m)$ defined on the quasi-Banach space $X$ and taking values in the space of real-valued Lebesgue measurable functions are considered in this paper. Factorization theorems for linear and superlinear operators with values in the space $L_\circ$ are proved with the help of Lorentz sequence spaces $l_{p,q}$. In this way sequences of functions belonging to a fixed bounded set in the spaces $L_{p,\infty}$ are characterized for $0<p<\infty, 0<q\le p$. The possibility to distinguish weak type operators (bounded in the space $L_{p,\infty}$) from the operators factorizable through $L_{p,\infty}$ is obtained in terms of secuences of independent random variables. A criterion is established for an operator to be symmetrically order bounded in $L_{p,r}, 0<r\le\infty$. Some refinements for the translation invariant sets and operators are obtained.