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.
Domaines
Analyse classique [math.CA]
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bilinear rubio de francia with smooth AND non-smooth rectangles.pdf (391.52 Ko)
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