On the ${\mathrm{\ell }}^s$-boundedness of a family of integral operators

In this paper we prove an ${\mathrm{\ell }}^s$-boundedness result for integral operators with operator-valued kernels. The proofs are based on extrapolation techniques with weights due to Rubio de Francia. The results will be applied by the first and third author in a subsequent paper where a new approach to maximal $L^p$-regularity for parabolic problems with time-dependent generator is developed.