A bilinear Rubio de Francia inequality for arbitrary rectangles

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 $R$ $${\pi }_R(f,g)(x):=\iint {\mathbf{1}}_R(\xi ,\eta )\widehat{f}(\xi )\widehat{g}(\eta )e^{2\pi i(\xi +\eta )x}d\xi d\eta$$ and let $r>2$. We show that the bilinear Rubio de Francia operator associated to $\mathcal{R}$ given by $$f,g\mapsto (\sum _{R\in \mathcal{R}}\mathrm{\mid }{\pi }_R(f,g){\mathrm{\mid }}^r{)}^{1\mathrm{/}r}$$ is $L^p\times L^q\to L^s$ bounded whenever $1\mathrm{/}p+1\mathrm{/}q=1\mathrm{/}s$, $r^{\mathrm{\prime }}<p,q<r$. 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.