  Let $H_\omega f$ be the Fourier restriction of $f\in L^2(\mathbb{R})$ to an
interval $\omega\subset \mathbb{R}$. If $\Omega$ is an arbitrary collection of
pairwise disjoint intervals, the square function of $\{H_\omega f: \omega \in
\Omega\}$ is termed the Rubio de Francia square function $T^\Omega$. This
article proves a pointwise bound for $T^\Omega f$ by a sparse operator
involving local $L^2$-averages. A pointwise bound for the smooth version of
$T^\Omega$ by a sparse square function is also proved. These pointwise
localization principles lead to quantified $L^p(w)$, $p>2$ and weak $L^p(w)$,
$p\geq 2$ norm inequalities for $T^\Omega$. In particular, the obtained weak
$L^p(w)$ norm bounds are new for $p\geq 2$ and sharp for $p>2$. The proofs rely
on sparse bounds for abstract balayages of Carleson sequences, local
orthogonality and very elementary time-frequency analysis techniques. The paper
also contains two results related to the outstanding conjecture that $T^\Omega$
is bounded on $L^2(w)$ if and only if $w\in A_1$. The conjecture is verified
for radially decreasing even $A_1$ weights, and in full generality for the
Walsh group analogue of $T^\Omega$.
