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!