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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.

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.

The angular-frequency Fourier transform of a finite measure, or a complex measure of finite total variation, is defined by the displayed integral. It is bounded by total variation and continuous by dominated convergence. An integrable density recovers the ordinary Fourier transform of a function. A measure supported on a curved surface instead leads to a Fourier extension operator.