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](https://www.ems-ph.org/journals/show_abstract.php?issn=0213-2230&vol=1&iss=2&rank=1). 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](https://arxiv.org/abs/math/0306417v3) 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](https://www.ems-ph.org/journals/show_abstract.php?issn=0213-2230&vol=1&iss=2&rank=1)).** 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 kind of vector-valued Littlewood–Paley inequality for arbitrary intervals
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).


