It is a classical result in harmonic analysis that


$$ \|\|P\_kf\|\_{\ell^2\_k}\|\_{L^p\_x}\approx\|f\|\_{L^p} $$


for $p\in(1,\infty)$, where $P\_k$ is the Littlewood-Paley decomposition onto frquency $\approx 2^k$.


What if I replace the $\ell^2$ norm in $k$ by the $\ell^q$ norm? Is it possible that in addition to the trivial estimate derived from the embedding of $\ell^q$ in $\ell^2$ (or vice versa), there are some nontrivial value of $(p,q)$ for this to hold?



$\ell^q$ analog of square function
The following counterexamples are taken from examples 6.1.10 and 6.1.11 in the textbook L. Grafakos, *Classical Fourier Analysis (Third Edition)*.



> 
> **Claim 1.** Fix $1<p<\infty$ and $q<2$. Then the inequality
>  $$\| (\sum\_{j\in\mathbb{Z}} |P\_{j}(f)|^{q})^{1/q}\|\_{L^{p}} \lesssim\_{p,q} \|f\|\_{L^{p}}$$
>  cannot hold.
> 
> 
> 


*Proof.* Let $\psi$ be a $C^{\infty}$ function on $\mathbb{R}$ such $\hat{\psi}\geq 0$,


$$\hat{\psi}(\xi) = \begin{cases} 1 & {\xi \in [9/8,15/8]} \\ 0 & {\xi\notin [7/8,17/8]} \end{cases},$$
and
$$\sum\_{j\in\mathbb{Z}} \hat{\psi}(2^{-j}\xi)^{2}=1, \qquad \xi>0.$$
Extend $\hat{\psi}$ to be an even function on $\mathbb{R}$. Let $P\_{j}:=\psi(2^{-j}D)$.


Now let $\varphi$ be a Schwartz function such that $\hat{\varphi}\geq 0$ and $\mathrm{supp}(\hat{\varphi}) \subset [11/8,13/8]$. Fix $N\gg 1$, and define


$$f\_{j}(x) := e^{2\pi\frac{12}{8}2^{j}x}\varphi(x), \qquad j=1,\ldots,N.$$


Since $\widehat{f}\_{j}(\xi) = \hat{\varphi}(\xi-\frac{12}{8}2^{j})$ is supported on the set
$$[\frac{11}{8} + \frac{12}{8}2^{j}, \frac{13}{8}+\frac{12}{8}2^{j}]\subset [\frac{9}{8}2^{j}, \frac{15}{8}2^{j}], j\gg 1,$$
$\hat{\psi}(2^{-j}\xi)=1$ on the support of $\hat{f}\_{j}$.


Now define
$$f := \sum\_{j=3}^{N}f\_{j}.$$
We have the lower bound
$$\|(\sum\_{j}|P\_{j}(f)|^{q})^{1/q}\|\_{L^{p}}\geq \|(\sum\_{j=3}^{N} |f\_{j}|^{q})^{1/q}\|\_{L^{p}} = (N-3)^{1/q}\|\varphi\|\_{L^{p}}.$$
By the Littlewood-Paley inequality, we have the upper bound
$$\|f\|\_{L^{p}} \lesssim\_{p} \|(\sum\_{j}|P\_{j}f|^{2})^{1/2}\|\_{L^{p}} = (N-3)^{1/2}\|\varphi\|\_{L^{p}}.$$


For $N\gg 1$, we obtain a contradiction. $\Box$



> 
> **Claim 2.** Fix $1<p<\infty$ and $2<q<\infty$. Then the inequality
>  $$\|g\|\_{L^{p}} \lesssim\_{p,q} \|(\sum\_{j} |P\_{j}g|^{q})^{1/q}\|\_{L^{p}}$$
>  cannot hold.
> 
> 
> 


*Proof.* This follows from duality and Claim 1. $\Box$.


