# 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 $$1 we have
$$\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!

• What do you mean by $2I_{k,j} \subset I_k$? (What is $2I_{k,j}$? do you mean fixing the center and doubling the width?) Jun 6, 2022 at 15:05
• @WillieWong Yes! I forgot to point that out. Thanks!
– Feng
Jun 6, 2022 at 15:06
• @WillieWong I'm not sure about your meaning. I tried to follow the proof of classcial Littlewood-Paley theorem: first use smoother kernels and then come back to non-smooth kernels. But it failed when I tried to come back to non-smooth kernels. Emm... Would you mind elaborating more? I'm not familiar with this topic. Thanks in advance!
– Feng
Jun 6, 2022 at 15:23
• @WillieWong I’m asking the details of the formula $\|S^\Omega f\|_p \sim \|S^{Well(\Omega)}f\|_p$ in page 8 of Lacey’s paper. I can understand everything in that sketch of proof before this formula. He said in the paper that “by a vector-valued Littlewood-Paley inequality”, and I want to know which one is this inequality, how is it proved and how is it used here.
– Feng
Jun 6, 2022 at 15:46

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: $$$$\|S^\Omega f\|_{L^p} = \Big\| \Big( \sum_{\omega\in\Omega} |S_\omega f|^2 \Big)^{1/2} \Big\|_{L^p}\asymp \Big\| \Big( \sum_{\omega\in\Omega} |S^{\mathrm{Well}(\omega)}S_\omega f|^2 \Big)^{1/2} \Big\|_{L^p} = \Big\| S^{\mathrm{Well}(\Omega)} f \Big\|_{L^p} \ \text{(1)}.$$$$ The inequality we used here is $$\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).

• It is not clear to me how (2.11) is applied though. Can you clarify? To go from $[-1/2,1/2]$ to arbitrary $\omega$ in $\Omega$, some scaling is involved in the affine transformation. Furthermore, I am not sure about your comment about lacunary intervals: the way $\mathrm{Well}([-1/2,1/2])$ is defined it does not meet your description of something lacunary. Jun 6, 2022 at 17:22
• And this inequality in turn is not obvious but I believe that this is what Lacey meant by "vector-valued Littlewood--Paley inequality". Jun 6, 2022 at 18:33
• Thanks for the new edit. Jun 6, 2022 at 18:55
• Thanks for the answer! I did realize that I need to use the inequality $$\Big\| \Big( \sum_{\omega\in\Omega} |S^{\mathrm{Well(\omega)}}f_\omega|^2 \Big)^{1/2} \Big\|_{L^p} \lesssim \Big\| \Big( \sum_{\omega\in\Omega} |f_\omega|^2 \Big)^{1/2} \Big\|_{L^p}$$ and I tried to mimic the proof of the vector-valued LP inequality that I mentioned in OP, which doesn’t involve $A_p$-weights, only involves vector-valued singular integral theory, to prove this inequality. But I failed. After reading your answer, now I’m pretty sure that I need to learn something new!
– Feng
Jun 6, 2022 at 21:29
• For the passage from weighted estimates to vector valued ones, another (easier on the eyes) reference is RdF's 1980 paper "VECTOR VALUED INEQUALITIES FOR OPERATORS IN $L^p$ SPACES" (Bull. LMS, 1980). It is also very short. Jun 7, 2022 at 15:40