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We prove various extensions of the Coifman-Rubio de Francia-Semmes multipliertheorem to operator-valued multipliers on Banach function spaces. Our resultsinvolve a new boundedness condition on sets of operators which we call$\ell^{r}(\ell^{s})$-boundedness, which implies $\mathcal{R}$-boundedness inmany cases. The proofs are based on new Littlewood-Paley-Rubio de Francia-typeestimates in Banach function spaces which were recently obtained by theauthors.

Fourier multipliers in Banach function spaces with UMD concavifications