Strong Factorizations between Couples of Operators on Banach Function Spaces

Let $T\colon X_1\to Y_1$ and $S\colon X_2\to Y_2$ be two continuous linear operators between Banach function spaces related to a finite measure space. Under some lattice requirements on the spaces involved, we give characterizations by means of inequalities of when $T$ can be strongly factorized through $S$, that is, $T=M_g\circ S\circ M_f$ with $M_f\colon X_1\to X_2$ and $M_g\colon Y_2\to Y_1$ being multiplication operators defined by some measurable functions $f$ and $g$. In particular, we study the cases when $S$ is a composition operator or a kernel operator.