= Bourgain-Demeter decoupling theorem
{c}
{title2=$p_c=2(n+1)/(n-1)$}
The Bourgain-Demeter decoupling theorem controls a <function> whose <Fourier support> lies near a positively curved hypersurface by the contributions from small frequency caps. For a compact $C^2$ hypersurface in $\mathbb R^n$, $n\geq2$, with positive-definite <second fundamental form>, decompose its $\delta$-neighborhood into caps of tangential width $\delta^{1/2}$ and normal thickness $\delta$. If $f_\theta$ denotes the corresponding Fourier pieces, the critical estimate is
$$
\|f\|_{L^{p_c}}\leq C_\varepsilon\delta^{-\varepsilon}\left(\sum_\theta\|f_\theta\|_{L^{p_c}}^2\right)^{1/2},\qquad p_c=\frac{2(n+1)}{n-1},\quad\varepsilon>0.
$$
The constant is independent of $\delta\in(0,1)$ and $f$, for the fixed surface and chosen cap decomposition. The same arbitrarily small loss holds for $2\leq p\leq p_c$; supercritical exponents require an additional scale power.<The proof of the l2 Decoupling Conjecture by Jean Bourgain and Ciprian Demeter>[Theorem1.1 and equation(3), printed pp351–352]{ref}
For a parabola in $\mathbb R^2$, $p_c=6$. A naive <triangle inequality> sums all cap norms, whereas decoupling sums their squares before taking a square root. Curvature makes this much stronger control possible. The theorem links <Fourier restriction theory> with estimates for oscillatory sums and dispersive equations.
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