 Bourgain-Demeter decoupling theorem

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 ${\mathbb{R}}^n$, $n\ge 2$, 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 isThe 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 $2\le p\le p_c$; supercritical exponents require an additional scale power.^[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.