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Article Dans Une Revue International Mathematics Research Notices Année : 2020

A bilinear Rubio de Francia inequality for arbitrary rectangles

Résumé

Let $\mathscr{R}$ be a collection of disjoint dyadic rectangles $R$, let $\pi_R$ denote the non-smooth bilinear projection onto $R$ \[ \pi_R (f,g)(x):=\iint 1_{R}(\xi,\eta) \widehat{f}(\xi) \widehat{g}(\eta) e^{2\pi i (\xi + \eta) x} \, d \xi d\eta \] and let $r>2$. We show that the bilinear Rubio de Francia operator associated to $\mathscr{R}$ given by \[ f,g \mapsto \Big(\sum_{R\in\mathscr{R}} |\pi_{R} (f,g)|^r \Big)^{1/r} \] is $L^p \times L^q \rightarrow L^s$ bounded whenever $1/p + 1/q = 1/s$, $ r' < p,q < r $. This extends from squares to rectangles a previous result by the same authors in [7], and as a corollary extends in the same way a previous result from [2] for smooth projections, albeit in a reduced range.
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Dates et versions

hal-01858507 , version 1 (20-08-2018)

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  • HAL Id : hal-01858507 , version 1

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Frederic Bernicot, Marco Vitturi. A bilinear Rubio de Francia inequality for arbitrary rectangles. International Mathematics Research Notices, 2020. ⟨hal-01858507⟩
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