 Solution

Let $\Theta (R)$ partition an $R^{-1}$-neighborhood of the parabola into $R^{-1/2}\times R^{-1}$ caps, and suppose ${\hat{f}}_{\theta }$ is supported in $\theta$. The decoupling inequality for the parabola states that for $2\le p<\infty$ and every $ϵ>0$,