Fourier multipliers in Banach function spaces with UMD concavifications

Amenta, Alex; Lorist, Emiel; Veraar, Mark

doi:10.1090/tran/7520

Fourier multipliers in Banach function spaces with UMD concavifications
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by Alex Amenta, Emiel Lorist and Mark Veraar PDF

Trans. Amer. Math. Soc. 371 (2019), 4837-4868 Request permission

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.

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