Let $\psi $ be a radial real-valued $C_c ^\infty$ function supported in the unit open ball, and $\psi(x)=1$ whenever $|x| \leq \frac 1 2$. Define $\psi_N (x) = \psi (x/N) - \psi(2x/N)$ where $N = 2^m$ for some $m \in \mathbb Z$, i.e. $N$ is a dyadic number.
Let $f \in \mathcal S$ and let $P_N f $ be its Littlewood-Paley component, that is, $P_N f = \mathcal F ^{-1} (\psi_N \hat f)$. Also, let $P_{\leq N}f = \mathcal F ^{-1} (\psi _{\leq N} \hat f)$, where $\psi_{\leq N}(x)= \psi(x/N)$.
How can I prove that $P_{\leq N} f = \sum_{M \leq N} P_M f$ in $L^1$, where $M, N$ are dyadic numbers and $f \in \mathcal S$? I tried to use Young's inequality, but it does not work for $L^1$ case (although it works for $L^p$ convergence when $p>1$).
Thanks in advance!