  A version of Littlewood--Paley--Rubio de Francia inequality for the
two-parameter Walsh system is proved: for any family of disjoint rectangles
$I_k = I_k^1 \times I_k^2$ in ${\mathbb{Z}_+ \times \mathbb{Z}_+}$ and a family
of functions $f_k$ with Walsh spectrum inside $I_k$ the following is true
\[\left\lVert\sum\limits_k f_k\right\rVert_{L^p} \leq C_p
\left\lVert\left(\sum\limits_k
\left|f_k\right|^2\right)^{1/2}\right\rVert_{L^p}, \qquad 1 < p \leq 2,\] where
$C_p$ does not depend on the choice of rectangles $\left\{I_k\right\}$ or
functions $\left\{f_k\right\}$. The arguments are based on the atomic theory of
two-parameter martingale Hardy spaces. In the course of the proof, a
two-parameter version of Gundy's theorem on the boundedness of operators taking
martingales to measurable functions is formulated, which might be of
independent interest.
