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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.

A Fourier transform defined only as an function in an Lp space cannot generally be evaluated on a lower-dimensional surface: changing its representative on a set of ambient Lebesgue measure zero leaves that function unchanged. Fourier restriction theory asks when the integral transform of a sufficiently regular input admits a bounded restriction to a curved surface.

For a surface $S$ with measure $\sigma$, a typical estimate is $\Vert \hat{f}{|}_S{\Vert }_{L^q(\sigma )}\le C\Vert f{\Vert }_{L^p({\mathbb{R}}^n)}$, initially for Schwartz functions. Its dual form bounds the Fourier extension operator. The surface's curvature creates oscillatory cancellation, while concentration in small caps obstructs some exponent choices.

For a circle, a cap of angular width $\delta$ gives a nearly coherent extension on a rectangle with tangential length of order ${\delta }^{-1}$ and normal length of order ${\delta }^{-2}$. Comparing these scales with input and output norms yields necessary conditions, as in the circle cap Fourier lower bound. Restriction estimates then connect oscillatory geometry with partial differential equations and decoupling inequalities.

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]

The degree-$d$ moment curve is $\gamma (t)=(t,t^2,\dots ,t^d)$ in ${\mathbb{R}}^d$. Its tangent vectors at distinct parameters are linearly independent:Factor $1,2,\dots ,d$ from the rows and apply the Vandermonde determinant formula. Parameter separation bounded below therefore gives quantitative transversality of the tangent directions.

For radial position vectors, the determinant instead equals $({\prod }_j{t}_j){\prod }_{i<j}(t_j-t_i)$. They are independent only when all parameters are nonzero as well as distinct. In particular $\gamma (0)=0$ supplies no radial direction. For $d=2$, the curve is the parabola, with tangent determinant $2(t_2-t_1)$. Its nondegeneracy is a basic source of curvature in Fourier restriction theory and decoupling inequalities.