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!