A Bernstein inequality in Littlewood–Paley theory

In Appendix A of Tao's book Nonlinear Dispersive Equations, a Bernstein inequality has puzzled me a long time:

Suppose that $P_{\geq N}$ is smoothed out projection to the region $|\xi|\geq N$, where $N\in 2^{\mathbb{Z}}$ is a dyadic number, $\hat{\varphi}(\xi)$($|\hat{\varphi}(\xi)|\leq1$) is a real valued radially symmetric bump function that is equal to $1$ in ball $B(0;1)$ and supported in ball $B(0;2)$.

We define $\widehat{P_{\geq N}f}:=(1-\varphi(2\xi/N))\hat{f}(\xi)$.

Then Tao says that if $s\geq0$, we have $$\|P_{\geq N}f\|_{L^{p}(\mathbb{R}^{d})}\lesssim_{p,s,d}N^{-s}\||\nabla|^{s}P_{\geq N}f\|_{L^{p}(\mathbb{R}^{d})}.$$

How can we get this inequality?

My attempt is\begin{align}\widehat{P_{\geq N}f}=&(1-\varphi(2\xi/N))\hat{f}(\xi)\\=&|\xi|^{-s}|\xi|^{s}(1-\varphi(2\xi/N))\hat{f}(\xi)\end{align}here $|\xi|^{-s}$ is well defined because the support of $(1-\varphi(2\xi/N))$ is away from $0$. Then what should we do to deal with the above?

We know that $|\xi|^{s}(1-\varphi(2\xi/N))\hat{f}(\xi)=\widehat{|\nabla|^{s}P_{\geq N}f}$ and $|\xi|^{-s}\leq 2^{s}N^{-s}$, and then how to get Tao's result? Maybe I can just get\begin{equation}|\widehat{P_{\geq N}f}|\leq 2^{s}N^{-s}|\widehat{|\nabla|^{s}P_{\geq N}f}|.\end{equation}

If we remove the hat, we should get\begin{align}P_{\geq N}f&=\mathscr{F}^{-1}\big(|\xi|^{-s}|\xi|^{s}(1-\varphi(2\xi/N))\hat{f}(\xi)\big)\\&=\mathscr{F}^{-1}\big(|\xi|^{-s}|(1-\varphi(3\xi/N))|\xi|^{s}(1-\varphi(2\xi/N))\hat{f}(\xi)\big)\\&=\mathscr{F}^{-1}\big(|\xi|^{-s}|(1-\varphi(3\xi/N))\big)*(|\nabla|^{s}P_{\geq N}f).\end{align}

Is $\mathscr{F}^{-1}\big(|\xi|^{-s}(1-\varphi(3\xi/N))\big)\in L^{1}(\mathbb{R}^{d})$?