Reference request: Commutator estimate

Question

I am wondering if the following commutator estimate is true, and in such case, where can I find a proof for it. Let $N\in 2^{\mathbb{Z}}$ dyadic, and let's denote by $P_N$ the standard Littlewood-Paley projectors, that is, $P_N$ projects into frequencies $\{\vert\xi\vert\sim N\}$. Finally, consider two functions in the Schwartz class $f,g\in \mathcal{S}(\mathbb{R})$. Then, there exists $C>0$ such that $$ \big\Vert [P_N\partial_x,g]f\big\Vert_{L^2(\mathbb{R})}\leq C\Vert g_x\Vert_{L^\infty(\mathbb{R})}\Vert f\Vert_{L^2(\mathbb{R})}, $$ where $[\cdot,\cdot]$ stands for the commutator operator $[A,B]=AB-BA$. Does anyone knows if such inequality holds and where can I find it?