Stein–Tomas restriction theorem

Remarks

By duality, the estimate is equivalent to $\norm{\widehat{g \, d\sigma}}_{L^{p'}(\R^n)} \le C \norm{g}_{L^2(S)}$, with $1/p + 1/p' = 1$. A $TT^*$ argument reduces this to the bound $\norm{f * \hat\sigma}_{L^{p'}} \le C^2 \norm{f}_{L^p}$. One splits $\hat\sigma$ into dyadic pieces in $\abs{x}$. Each piece is bounded from $L^1$ to $L^\infty$ by the decay of the Fourier transform of surface measure, and on $L^2$ by the size of its Fourier transform. Interpolating these two bounds, the pieces sum when $p < 2(n+1)/(n+3)$. Peter Tomas proved the range $p < 2(n + 1)/(n + 3)$, and Elias Stein obtained the endpoint.

Example: for the circle in $\R^2$ the range is $1 \le p \le 6/5$, and for the sphere in $\R^3$ it is $1 \le p \le 4/3$.

The exponent $2(n + 1)/(n + 3)$ is sharp for the $L^2(S)$ norm on the left. The Knapp example takes $\hat f$ to be a smooth bump adapted to a cap of $S$ of radius $\delta$. Such a cap lies in a slab of thickness about $\delta^2$. Comparing both sides as $\delta \to 0$ forces $p \le 2(n+1)/(n+3)$. Without curvature, for a flat piece of a hyperplane, no estimate with $p > 1$ holds.

Going deeper

For a student

This is the main known case of the Fourier restriction problem with $L^2$ norm on the surface, and it is sharp in that setting. It is used to prove Strichartz estimates for the Schrödinger and wave equations, which are restriction estimates for paraboloids and cones. The restriction conjecture asks for more general $L^q(S)$ norms and a larger range of $p$.