The ${\mathrm{\ell }}^s$-boundedness of a family of integral operators on UMD Banach function spaces

We prove the ${\mathrm{\ell }}^s$-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 ${\mathrm{\ell }}^s$-boundedness of this family of integral operators was shown on Lebesgue spaces. The proof is based on a characterization of ${\mathrm{\ell }}^s$-boundedness as weighted boundedness by Rubio de Francia.