have $$\begin{aligned}
\frac 1{|I|}\int_I {|\psi_0^k-\langle\psi_0^k\rangle_{I} |}^q\frac 1{|I|}\int_I {|\phi -\langle\phi \rangle_{I} |}^q\le 4^{q+1} \langle |\psi_0^k|^q\rangle_I \langle |\phi |^q\rangle_I.\end{aligned}$$ We first consider the case when $\ell(I)\le 1$ and we may further assume that $I\subset (0,1)$. Since $q\eta_k<1$ we know that $|x|^{-q\eta_k}\in A_2$ and hence, by $I\subset(0,1),$ $$4^{q+1} \langle |\psi_0^k|^q\rangle_I \langle |\phi |^q\rangle_I\le 4^{7/2} c_k^{5/2}[|x|^{-q\eta_k}]_{A_2}.$$It remains to consider the case when $\ell(I)>1$. Since certainly $(0,1)\cap I\neq \emptyset$ (as otherwise there is nothing to prove) we know that $(0,1)\subset 3I$. Then due to that $\phi$ is a periodic function we have $$4^{q+1} \langle |\psi_0^k|^q\rangle_I \langle |\phi |^q\rangle_I\le  4^{q+1} \langle |\psi_0^k|^q\rangle_{(0,1)} \langle |\phi |^q\rangle_{(0,1)}\le  4^{7/2} c_k^{5/2}[|x|^{-q\eta_k}]_{A_2}.$$Therefore, we conclude that $$S_q\le  4^{7/2} c_k^{5/2}[|x|^{-q\eta_k}]_{A_2}.$$ On the other hand, since $\psi_0^k \phi_0\le c_k$, then by similar arguments as in Proposition \[prop1\] we have $$T_q\le 4^{7/2} c_k^{5/2}[|x|^{-q\eta_k}]_{A_2}.$$ Hence by Theorem \[upperbound\] (see below) we know that the commutator $[\phi,[\psi_k,H]]$ is bounded on $L^2$ with norm $\sim c_k^{5/2} [|x|^{-q\eta_k}]_{A_2}\|M_{q'}\|_{L^2\rightarrow L^2}^2$. Thus, we may further demand the constant $c_{k}$ to be so small that ${\big\|[\phi,[\psi_k,H]]\big\|_{L^2\to L^2}} \leq 2^{-k}$. Then $[\phi,[\psi, H]]$ also is bounded on $L^2$: $${\big\| [\phi,[\psi,H]]\big\|_{L^2\to L^2}} = {\big\|[\phi,[\sum_{k\in 4\mathbb N+2}\psi_k,H]]\big\|_{L^2\to L^2}} \leq \sum_{k\in4\mathbb N+2}{\big\|[\phi,[\psi_k,H]]\big\|_{L^2\to L^2}} \leq \sum_{k=1}^{\infty}2^{-k}=1.$$

It remains to check that the pair $(\psi,\phi)$ is precisely what 