  We prove the $\ell^s$-boundedness of a family of integral operators with an
operator-valued kernel on UMD Banach function spaces. This generalizes and
simplifies earlier work by Gallarati, Veraar and the author, where the
$\ell^s$-boundedness of this family of integral operators was shown on Lebesgue
spaces. The proof is based on a characterization of $\ell^s$-boundedness as
weighted boundedness by Rubio de Francia.
