$L^p$-norm estimate of Littlewood-Paley multiplier operator 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.
 A: 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.
