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.