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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.

Ciprian Demeter coauthored the decoupling inequality with Jean Bourgain. It bounds global Fourier contributions by a quadratic sum of contributions from small caps near a curved hypersurface. The theorem turns geometric curvature into an analytic inequality and yields consequences for Strichartz estimate for the free Schrödinger equation and discrete restriction problems.^[ref]

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.

Jean Bourgain proved the decoupling inequality with Ciprian Demeter. Their theorem controls a function with Fourier support near a curved hypersurface through contributions from smaller frequency caps. The result connects curvature with improved summation estimates, with applications to restriction and Strichartz estimate for the free Schrödinger equation.^[ref]