Fourier multipliers in Banach function spaces with UMD concavifications

We prove various extensions of the Coifman–Rubio de Francia–Semmes multiplier theorem to operator-valued multipliers on Banach function spaces. Our results involve a new boundedness condition on sets of operators which we call ℓ r ( ℓ s ) {\ell ^{r}(\ell ^{s})} -boundedness, which implies R \mathcal {R} -boundedness in many cases. The proofs are based on new Littlewood–Paley–Rubio de Francia-type estimates in Banach function spaces which were recently obtained by the authors.