A bilinear Rubio de Francia inequality for arbitrary squares

We prove the boundedness of a smooth bilinear Rubio de Francia operator associated with an arbitrary collection of squares (with sides parallel to the axes) in the frequency plane$$\left(f, g \right)\mapsto \left( \sum\_{\omega \in \Omega}\left| \int\_{\mathbb{R}^2} \hat{f}(\xi) \hat{g}(\eta) \Phi\_{\omega}(\xi, \eta) e^{2 \pi i x\left(\xi+\eta \right)} d \xi d \eta\right|^r \right)^{1/r},$$ provided $r\textgreater{}2$. More exactly, we show that the above operator maps $L^p \times L^q \to L^s$ whenever $p, q, s'$ are in the "local $L^{r'}$" range, i.e. $\displaystyle \frac{1}{p}+\frac{1}{q}+\frac{1}{s'}=1$, $\displaystyle0 \leq \frac{1}{p}, \frac{1}{q} \textless{}\frac{1}{r'}$, and $\displaystyle\frac{1}{s'}\textless{}\frac{1}{r'}$. Note that we allow for negative values of $s'$, which correspond to quasi-Banach spaces $L^s$.