A bilinear Rubio de Francia inequality for arbitrary rectangles

Abstract

Let $\mathcal{R}$ be a collection of disjoint dyadic rectangles $R$ with sides parallel to the axes, let ${\pi }_R$ denote the non-smooth bilinear projection onto $R$ \[ π_R (f,g)(x):=\iint \mathbf{1}_{R}(ξ,η) \widehat{f}(ξ) \widehat{g}(η) e^{2πi (ξ+ η) x} dξdη\] and let $r>2$. We show that the bilinear Rubio de Francia operator associated to $\mathcal{R}$ given by \[ f,g \mapsto \Big(\sum_{R\in\mathscr{R}} |π_{R} (f,g)|^r \Big)^{1/r} \] is $L^p\times L^q\to L^s$ bounded whenever $1/p+1/q=1/s$, $r^{\prime }<p,q<r$. This extends from squares to rectangles a previous result by the same authors, and as a corollary extends in the same way a previous result from Benea and the first author for smooth projections, albeit in a reduced range.

29 pages, 2 figures