High-Low Paraproduct Convergence in $L^p$

Question

For $f, g \in \mathcal{S}$, define $f_k = P_k f$, $g^k = \sum_{j=-\infty}^{k-3}P_j g$, where $P_j$ is the Littlewood-Paley projection. Is it true that

$$ P(f, g) = \sum_{j=-\infty}^\infty f_j g^j $$

converges in $L^p$ and satisfies the Holder type inequality

$$ \|P(f, g)\|_{L^p} \le C \|f\|_{L^q}\|g\|_{L^r} $$

for $\dfrac{1}{p} = \dfrac{1}{q} + \dfrac{1}{r}$ ? Why?