= Decoupling inequality for the parabola
Let $R\geq1$ and decompose the Fourier support near a fixed compact arc of $(\xi,\xi^2)$ into caps with tangential width $R^{-1/2}$ and normal width $R^{-1}$. For the associated pieces $f_\theta$, the <decoupling inequality> for $6\leq p<\infty$ is
$$
\left\|\sum_\theta f_\theta\right\|_p\leq C_{p,\varepsilon}R^{\varepsilon+1/4-3/(2p)}
\left(\sum_\theta\|f_\theta\|_p^2\right)^{1/2}.
$$
For $2\leq p\leq6$, the corresponding loss is $R^\varepsilon$. The constants are independent of $R$ and the input. This is the two-dimensional case of the <Bourgain-Demeter decoupling theorem>.<The proof of the l2 Decoupling Conjecture by Jean Bourgain and Ciprian Demeter>[Theorem1.1, printed p352, n=2 and delta=1/R]{ref}
At $p=2$, <Fourier-support almost orthogonality> already supplies a bounded constant. At the critical exponent $p=6$, the theorem keeps only the arbitrarily small power loss $R^\varepsilon$, despite allowing many interacting caps. This is much stronger than applying the <triangle inequality> and then <Cauchy-Schwarz> to about $R^{1/2}$ pieces. The cap dimensions and the convention $\delta=R^{-1}$ determine the power of $R$; confusing $R$ with its square changes that exponent.
Back to article page
