Consider the classical dyadic decomposition of $\mathbb{R}$ $$\{\Phi_j\}_{j\in\mathbb{Z}}\in \mathcal{S}(\mathbb{R}) \quad\text{ where }\quad \operatorname{supp} \widehat{\Phi}_j\subset A_j:=\{\xi\in\mathbb{R}:2^{j-1}<|\xi|<2^{j+1}\}.$$ Define $\Delta_jf=\Phi_j\ast f$.
Fix $j\in\mathbb{Z}$. Let $k,l\in \mathbb{Z}$ with $|k-l|\leq1$. We know that $$ [\Delta_j(\Delta_kf\cdot\Delta_l g) ]\hat{}(\xi)=\widehat{\Phi}_l(\xi)\cdot[(\widehat{\Phi}_k(\xi)\cdot\widehat{f}(\xi))\ast(\widehat{\Phi}_l(\xi)\cdot\widehat{g}(\xi))]$$ and $\operatorname{supp}(\widehat{\Phi}_k(\xi)\cdot\widehat{f}(\xi))\ast(\widehat{\Phi}_l(\xi)\cdot\widehat{g}(\xi))\subset A_k+A_l$.
My question: Can I affirm that there exists $N_0\in\mathbb{N}$ such that $\Delta_j(\Delta_kf\cdot\Delta_l g)=0$ if $|k-j|>N_0$?