Fourier extension estimate ($\Vert E_{\sigma }f{\Vert }_{L^q({\mathbb{R}}^n)}\lesssim \Vert f{\Vert }_{L^p(\sigma )}$)

An extension estimate is a norm bound for the Fourier extension operator, uniform over its input. The exponents encode the curvature and scale of the frequency surface. For normalized circle measure, a cap of angular width $\delta$ produces size comparable to $\delta$ on a rectangle of dimensions ${\delta }^{-2}\times {\delta }^{-1}$. This single-cap example forces $p\ge 4$ for a finite diagonal estimate with the same exponent $p$ on both sides.

On the unit circle, the arclength-measure transform satisfies $|\hat{\sigma }(x)|\lesssim (1+|x|{)}^{-1/2}$. This follows by removing angular neighborhoods of width $|x{|}^{-1/2}$ around the two stationary points and integrating by parts on the remainder. Gaussian positivity for even extension moments bounds the weighted fourth moment for any bounded $f$ by that for $f=1$. Radial integration of $(1+r{)}^{-2}{e}^{-\pi r^2/R^2}$ with area element $rdr$ costs only $\log R$, proving the displayed local estimate. Modulated disjoint caps and random signs convert it into the planar Kakeya neighborhood lower bound.