I am studying Paley-Littlewood theorem in Harmonic analysis, and I met an exercise. I would like to construct a function $f$ as a counterexample to show that the inequality
\begin{equation}
\| f \|^2\_p \leq C \sum\_{j \in \mathbb{Z}} \| P\_j f\|^2\_p,
\end{equation}
is not true when $p<2,$ and here $C$ is a constant number only related to $f.$ Here,$P\_j$ is the Paley-Littlewood operator. More explicitly, for $\beta(x) \in C\_0^{\infty}(\mathbf{R}^n)$ and ${\rm supp}\,{
\beta} \subset \{1/2<|x|<2 \},$ and satisfying $1=\sum\_{j \in \mathbb{Z}} \beta(2^{-j}x), x \neq 0.$ We define the Fourier transform of $P\_j f$ is $\beta(2^{-j}\xi)\hat{f}(\xi).$


My attempt is, the function $f$ has the following form: $f(x)=e^{2 \pi imx }\phi(x),$ here $m$ is a constant number to be determined, and $\phi(x)$ is a Schwartz function. If $m$ is properly selected, then for some $P\_j$, we have $P\_j f=0.$ But I am stuck here.


Can anybody help me? Some suggestions or reference books are welcomed!



How to give a counterexample of this estimate related to Paley-Littlewood theorem?
This question is really about non-coincidence of different function spaces: the right hand side in your inequality is equal to the the square of the norm in (homogeneous) Besov space $\dot{B}\_{p}^{0,2}$. And by Littlewood--Paley theorem, the left-hand side is equivalent to the (square of) norm in Tribel--Lizorkin space $\dot{F}\_{p}^{0,2}$ which is defined as
\begin{equation}
\Big(\int \Big( \sum\_j |P\_j f|^2 \Big)^{p/2}\Big)^{2/p},
\end{equation}
which is bigger than
$$
\sum\_j \|P\_j f\|^2\_p
$$
if $p<2$ by Minkowski's integral inequality. So, if such inequality was true then we would have $\|f\|\_{\dot{B}\_{p}^{0,2}}\asymp \|f\|\_{\dot{F}\_p^{0,2}}$.


The fact that different function spaces do not coincide is standard, different counterexamples can be found probably in many different books about function spaces, for example in the book by Sawano "Theory of Besov Spaces" (see Theorem 2.16 there).


