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

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Published in:Transactions of the American Mathematical Society 2019-04, Vol.371 (7), p.4837-4868
Main Authors: Amenta, Alex, Lorist, Emiel, Veraar, Mark
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Language:English
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description 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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title Fourier multipliers in Banach function spaces with UMD concavifications
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