I am working on the decoupling inequality developed by Bourgain and Demeter: https://arxiv.org/abs/1604.06032.

Is there an example where we have strict inequality in Theorem 1.1, say in the case $n=2$ with $\delta^{-\alpha}$ losses in the power? Here by stricty inequality I mean $\alpha>0$ is a fixed positive constant depending on $p$ only.

This question may seem stupid, but I have not found a reference for that.

Edited: precisely, I am asking the following question.

For $g\in L^1([0,1])$ and $I\subseteq [0,1]$, define $E_I g(x,y)=\int_I g(s)e^{2\pi i (s x+s^2 y)}ds$. Is it true that for every $2\leq p\leq 6$ and every $\epsilon>0$, there is a constant $c_\epsilon>0$, depending on $p$ and $\epsilon$ only, such that for any $g\in L^1([0,1])$, any$\delta\in 2^{-2\mathbb N}$ and any ball $B$ of radius $\delta^{-1}$, we have $$ \left\| Eg\right\|_{L^p(w_B)}\geq c_\epsilon \delta^{\epsilon}\left(\sum_{j=1}^{\delta^{-1/2}}\left\|E_{[(j-1)\delta^{1/2},j\delta^{1/2}]}g\right\|^2_{L^p(w_B)}\right)^{1/2}? $$