Fourier restriction theory

Fourier restriction theory studies estimates for Fourier transforms restricted to curved submanifolds and for the corresponding extension operators.

A decoupling inequality controls an $L^p$ norm of a function with Fourier support near a curved set by an ${\ell }^2$ sum of norms of pieces supported near smaller caps.

For Fourier supports in $R^{-1/2}\times R^{-1}$ caps along the parabola,

Functions with pairwise disjoint, or uniformly finitely overlapping, Fourier supports are almost orthogonal in $L^2$ by the Plancherel theorem.

A function whose Fourier support has widths $N$ in spatial-frequency directions and $N^2$ in a time-frequency direction varies only on the dual scales $N^{-1}$ and $N^{-2}$. Convolution with an adapted Schwartz kernel makes this quantitative.

Constructive interference occurs when many oscillatory phases nearly agree, making their exponential terms add with comparable arguments and producing a large value of the sum.

Kakeya inequalities bound overlaps of long thin tubes with controlled directions.

For three transverse families of $R^{1/2}\times R^{1/2}\times R$ tubes in ${\mathbb{R}}^3$,

The degree-$d$ moment curve is $\gamma (t)=(t,t^2,\dots ,t^d)$. Tangent or radial directions from separated parameter intervals are quantitatively transverse by the Vandermonde determinant.

The cubic moment curve is ${\mathcal{M}}^3=\{(t,t^2,t^3):0\le t\le 1\}\subset {\mathbb{R}}^3$.