# Fourier multipliers in Banach function spaces with UMD concavifications

## TL;DR

This paper extends the multiplier theorem to operator-valued multipliers on Banach function spaces, introducing a new boundedness condition called ll^{r}(ll^{s})-boundedness, supported by novel Littlewood-Paley estimates.

## Contribution

It introduces ll^{r}(ll^{s})-boundedness, a new boundedness condition, and extends the multiplier theorem to Banach function spaces with UMD concavifications.

## Key findings

- Established ll^{r}(ll^{s})-boundedness implies al R-boundedness in many cases.
- Developed new Littlewood-Paley-Rubio de Francia-type estimates in Banach function spaces.
- Extended the Coifman-Rubio de Francia-Semmes multiplier theorem to operator-valued multipliers.

## 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.