Fourier extension operator ($E_{\sigma }f(x)=\int e^{-ix\cdot \omega }f(\omega )d\sigma (\omega )$)

A Fourier extension operator forms an oscillatory integral from a density on a frequency surface with measure $\sigma$. The opposite sign in the phase merely reflects the output variable, so it gives identical Lp norms. In the usual dual formulation it is the adjoint of restricting the Fourier transform to that surface.

For normalized circle measure and a nonnegative cutoff near $(1,0)$, equal to one on $|\theta |\le \delta /C$ and supported on $|\theta |\le 2\delta /C$, remove the constant phase $e^{-i{\xi }_1}$. On $|{\xi }_1|\le {\delta }^{-2}$, $|{\xi }_2|\le {\delta }^{-1}$, the remaining phase has magnitude at most $2/C^2+2/C$. Taking $C$ sufficiently large keeps its real part positive and comparable to one, so the Fourier transform has magnitude at least a constant times the cap mass, which is at least $\delta /(\pi C)$. No upper bound on the cutoff is required for this lower bound.

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.