Fourier multipliers in Banach function spaces with UMD concavifications

Abstract

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 ${\ell }^r({\ell }^s)$-boundedness, which implies $\mathcal{R}$-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.

References in corpus (1)

• On the vector-valued Littlewood-Paley-Rubio de Francia inequality

Cited by in corpus (1)

• Vector-valued extensions of operators through multilinear limited range extrapolation