Are there any (at least mildly) explicit counterexamples to the statement
$$
\sum_{m \in \mathbb{Z}} \|P_m f\|_p \lesssim \|f\|_p?
$$
(Or some good reason to expect this to be false?).

Here $P_m$ is the $m$-th Littlewood-Paley projection, that is
$$
\widehat{P_m f} = \psi_m \widehat{f}
$$
with $\psi_m(\xi) = \psi(\xi/2^m)$, $\psi(\xi) = \phi(\xi)-\phi(2\xi)$ and $\phi$ is
a real radial Schwartz function supported on the closed centered ball of radius $2$ and which equals $1$ on the closed centered ball of radius $1$. $P_m$ being (initially at least) defined for Schwartz functions.

*Edit: Of course, I am interested in the non-trivial case $p \neq 2$.*