My goal is to
show that $$\|P_j f\|_p \lesssim 2^{-js}\||\nabla |^s P_{\geq j}f\|_p$$ where $$\widehat{P_{\geq j}f}(\xi)=(1-\phi(2^{-j}\xi))\widehat{f}(\xi)$$ $$\widehat{P_jf}(\xi)=(\phi(2^{-j}\xi)-\phi(2^{-(j-1)}\xi))\widehat{f}(\xi)$$ $$\widehat{|\nabla |^sf}(\xi)=|\xi|^s\widehat{f}(\xi)$$ and where $\phi$ is smooth function with $supp(\phi)=\{\xi : \|\xi\| \leq 2 \}$ and $\phi\equiv 1$ in $\{\xi : \|\xi\|\leq 1\}$.
Honestly, I have no idea how to start. The main reason why I got stuck is that I am not able to deal with inequality of Fourier transformed function. Also, I maybe able to use some theorem about Paley-Littlewood decomposition but I am not sure. What I have done is just getting $$P_jf(x)=[[\widehat{\phi(2^{-j}\xi)-\phi(2^{-(j-1)}\xi)}]*f](x)$$ $$| \nabla |^sP_{\geq j}f(x)=\widehat{[|\xi|^s(1-\phi(2^{-j}\xi))*f]}(x)$$
I would appreciate any answer or hint for this problem.
One idea is to employ a Fourier multiplier theorem (for example the Marcinkiewicz or Mihlin Theorem). Recalling the support of $\phi$, we find \begin{align} ||P_j f||_p &= ||\mathcal{F}^{-1}\big[ \big(\phi(2^{-j}\xi)- \big(\phi(2^{-j+1}\xi) \big) \widehat{f}\big]||_p\\ &= ||\mathcal{F}^{-1}\big[ \big(\phi(2^{-j}\xi)- \big(\phi(2^{-j+1}\xi) \big) \big(1- \big(\phi(2^{-j+2}\xi) \big) \widehat{f}\big]||_p \end{align} since $$ \big(\phi(2^{-j}\xi)- \big(\phi(2^{-j+1}\xi) \big)\neq 0 \ \Rightarrow\ 2^{j-1}<|\xi|<2^{j+1} \ \Rightarrow\ \big(1- \big(\phi(2^{-j+2}\xi) \big) = 1. $$ Consequently, \begin{align*} ||P_j f||_p &= ||\mathcal{F}^{-1}\big[ \big(\phi(2^{-j}\xi)- \big(\phi(2^{-j+1}\xi) \big) \widehat{P_{\geq j-2}f}\big]||_p\\ &= ||\mathcal{F}^{-1}\Big[ \frac{\big(\phi(2^{-j}\xi)- \big(\phi(2^{-j+1}\xi) \big)}{|\xi|^s} \mathcal{F} \big[P_{\geq j-2} |\nabla|^s f \big] \Big]||_p\\ &= ||\mathcal{F}^{-1}\Big[ M^s_j(\xi)\ \mathcal{F} \big[P_{\geq j-2} |\nabla|^s f \big] \Big]||_p\\ \end{align*} with $$ M_j^s(\xi):= \frac{\big(\phi(2^{-j}\xi)- \big(\phi(2^{-j+1}\xi) \big)}{|\xi|^s}. $$ You can verify (again utilizing the support of $\phi$) that $$ \sup_{\epsilon\in\{0,1\}^n}\sup_{\xi\in\mathrm{R}^n} |{\xi_1^{\epsilon_1}\ldots\xi_n^{\epsilon_n}\partial_{\xi_1}^{\epsilon_1}\ldots\partial_{\xi_n}^{\epsilon_n}M^s_j(\xi)}|\leq C2^{-js} $$ with C independent on $j$ and $s$. The Marcinkiewicz Multiplier theorem then yields $$ ||P_j f||_p \leq C2^{-js} ||P_{\geq j-2} |\nabla|^s f||_p. $$ In this estimate a bigger part of the Fourier spectrum of $f$ is included on the right-hand side than in the OP's questions ($P_{\geq j+2}$ instead of $P_{\geq j}$). I am not sure if the estimate can be improved though.