 Decoupling inequality for the parabola

Let $R\ge 1$ 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 $6\le p<\infty$ isFor $2\le p\le 6$, the corresponding loss is $R^{\epsilon }$. The constants are independent of $R$ and the input. This is the two-dimensional case of the Bourgain-Demeter decoupling theorem.^[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 $R^{\epsilon }$, 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.