\displaystyle  \Big(\sum_{1 \ll j \ll N} |\widetilde{\Delta}_j f(x)|^2 \Big)^{1/2} \approx \Big(\sum_{1 \ll j \leq i} 2^{2j}N^{-1} \Big)^{1/2} \sim N^{-1/2}\,2^i. 