A Littlewood-Paley question

Question

For a suitable function smooth function $\psi$ supported on $\{x\in\mathbb{R}^d: 2^{-1} \le |x| \le 2\}$, define $P_jf(x)$ as \begin{aligned} P_jf(x)=\mathcal{F}^{-1}\left(\psi(2^{-j}\xi)\mathcal{F} f(\xi)\right)(x), \end{aligned} i.e. the projection of $f$ to its frequencies localized in the range $2^{j-1}\le |\xi|\le 2^{j+1}$ by way of the forward/inverse Fourier transform. If we define $$Sf(x) = \left( \sum_{j\in \mathbb{Z}} |P_jf(x)|^2 \right)^{1/2},$$ then we know we have $$\parallel Sf\parallel_{L^p(\mathbb{R}^d)} \lesssim_{d,p} \parallel f\parallel_{L^p(\mathbb{R}^d)}.$$ However, does the following inequality, $$\sum_{j\in\mathbb{Z}} \parallel P_j f\parallel_{L^p(\mathbb{R}^d)} \lesssim_{d,p} \parallel f\parallel_{L^p(\mathbb{R}^d)},$$ hold? I'm inclined to say no.