= Solution
Let $\Theta(R)$ partition an $R^{-1}$-neighborhood of the parabola into $R^{-1/2}\times R^{-1}$ caps, and suppose $\widehat f_\theta$ is supported in $\theta$. The <decoupling inequality for the parabola> states that for $2\leq p<\infty$ and every $\epsilon>0$,
$$
\left\|\sum_{\theta\in\Theta(R)}f_\theta\right\|_{L^p(\mathbb R^2)}
\lesssim_{p,\epsilon}R^\epsilon
\left(1+R^{1/2-3/p}\right)
\left(\sum_{\theta\in\Theta(R)}\|f_\theta\|_{L^p(\mathbb R^2)}^2\right)^{1/2}.
$$
Solved by gpt-5.6-sol high.
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