A kind of vector-valued Littlewood–Paley inequality for arbitrary intervals The title may be inappropriate, and I apologize for that.
I'm writing a reading report on my harmonic analysis course. My topic is the Littlewood-Paley inequality for arbitrary intervals, which was proved in Rubio de Francia, 1985. I found it hard to understand the proof of Lemma 2.3 in that paper, beacuse I've learnt nothing about $A_p$-weights. However, I found that page 8 of Lacey’s paper proves the same lemma using a vector-valued Littlewood–Paley inequality, in a very short space (only two lines are related). So I'm asking about the details of the proof using vector-valued Littlewood–Paley inequality.
Now I restate the main problem here.
For any set $I\in\mathbb R$, we define the operator $\widehat{S_If}=\chi_I\hat f$. For every interval $I$ and $c>0$, we denote by $cI$ the interval with the same center as $I$ and length: $|cI|=c|I|$.
Given disjoint intervals $\{I_k=(a_k,b_k)\}_{k\in\mathbb Z}$, we define that the Whitney decomposition of each $I_k$ consists of $\{I_{k,j}\}_{j\in\mathbb Z}$, where
$$I_{k,j}=\begin{cases}
    \left[a_k+(b_k-a_k)\frac{2^{-j}}3, a_k+(b_k-a_k)\frac{2^{-j+1}}3\right], & j\geq 0,\\
    \left[b_k-(b_k-a_k)\frac{2^{j+1}}3, b_k-(b_k-a_k)\frac{2^{j}}3\right], & j\leq -1.
\end{cases}$$
Note that $2I_{k,j}\subset I_k$ and $\sum_{k,j}\chi_{2I_{k,j}}(x)\leq 5$ for all $x\in\mathbb R$. We are going to prove that

Theorem (Lemma 2.3 in Rubio de Francia, 1985). Given disjoint intervals $\{I_k\}_{k\in\mathbb Z}$ and its Whitney decomposition as above, then for all $1<p<\infty$ we have
$$\left\|\left(\sum_{k,j\in\mathbb Z}\left|S_{I_{k,j}}f\right|^2\right)^{\frac12}\right\|_p\sim_p\left\|\left(\sum_{k\in\mathbb Z}|S_{I_k}f|^2\right)^{\frac12}\right\|_p,\qquad f\in L^p(\mathbb R).$$

Page 8 of Lacey’s paper sketches the proof of this Theorem, as he said, “by a vector-valued Littlewood-Paley inequality”. But he didn’t say which one the vector-valued LP inequality is.
I have known a kind of vector-valued Littlewood–Paley inequality, which states that
$$\left\|\left(\sum_{k,j\in\mathbb Z}\left|S_{\Delta_j}f_k\right|^2\right)^{\frac12}\right\|_p\lesssim_p\left\|\left(\sum_{k\in\mathbb Z}|f_k|^2\right)^{\frac12}\right\|_p,\qquad f\in L^p(\mathbb R),$$
where $\Delta_j=(-2^{j+1},-2^j]\cup[2^j,2^{j+1})$ is the Littlewood–Paley dyadic intervals.
But I don't know how to use this vector-valued Littlewood–Paley inequality to prove the Theorem. I tried to mimic the proof of the above inequality, but I failed since it became very messy.
Maybe this is not the one used in Lacey’s proof. I want to know which one does Lacey use, and how it is used to prove the Theorem.
Any help would be appreciated!
 A: Here you need to apply slightly non-standard Littlewood--Paley inequality. It is well known (however, an exact reference does not come to my mind immediately but I believe any proof of standard L.--P. inequality works equally well in this case) that the Littlewood--Paley inequality holds not only for the intervals $[2^j, 2^{j+1})$ but also for arbitrary lacunary intervals in $\mathbb{R}$, that is, you may take the intervals $[\lambda_k,\lambda_{k+1})$ as long as $\lambda_{k+1}/\lambda_k\ge c > 1$ (or $\lambda_{k+1}/\lambda_k\le c < 1$; and you can also shift and dilate such collection). And the collection $\mathrm{Well}(\Omega)$ is obviously lacunary in such sence (or at least a union of two lacunary collections).
Now, we apply this observation to get:
\begin{equation}
\|S^\Omega f\|_{L^p} = \Big\| \Big( \sum_{\omega\in\Omega} |S_\omega f|^2 \Big)^{1/2} \Big\|_{L^p}\asymp \Big\| \Big( \sum_{\omega\in\Omega} |S^{\mathrm{Well}(\omega)}S_\omega f|^2 \Big)^{1/2} \Big\|_{L^p} = \Big\| S^{\mathrm{Well}(\Omega)} f \Big\|_{L^p} \ \text{(1)}.
\end{equation}
The inequality we used here is
$$
\Big\| \Big( \sum_{\omega\in\Omega} |S^{\mathrm{Well(\omega)}}f_\omega|^2 \Big)^{1/2} \Big\|_{L^p} \lesssim \Big\| \Big( \sum_{\omega\in\Omega} |f_\omega|^2 \Big)^{1/2} \Big\|_{L^p}
$$
for arbitrary functions $f_\omega$ --- this is the kind of vector-valued Littlewood--Paley inequality Lacey (and Rubio de Francia) used. I am not sure that it can be proved easier than by following the path Rubio de Francia mentioned, that is, using the uniform boundedness of operators $S^{\mathrm{Well}(\omega)}$ in $L^2(w)$ for $w\in A_2$. The key observation here is that the collections $\mathrm{Well}(\omega)$ are obtained from one another by some affine transformation so this uniform boundedness is not surprising once you know the boundedness of just one operator, say $S^{\mathrm{Well}([-1/2. 1/2])}$ on $L^2(w)$. The reverse inequality in the middle of formula (1) follows by duality.
Rubio de Francia mentioned the weighted Littlewood--Paley inequality because in general weighted norm inequalities imply vector-valued ones. One of possible references for these things is the book by Rubio de Francia and Garcia-Cuerva "Weighted norm inequalities and related topics" (here you may apply Theorem 6.4 from page 519).
