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We prove the boundedness of a smooth bilinear Rubio de Francia operatorassociated with an arbitrary collection of squares (with sides parallel to theaxes) 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, weshow 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 fornegative values of $s'$, which correspond to quasi-Banach spaces $L^s$.

A bilinear Rubio de Francia inequality for arbitrary squares