The general Fourier restriction problem remains open

Recorded orientation, not proved here. For $f\in L^{\mathrm{\infty }}(S^{n-1})$ the restriction conjecture asks for $\mathrm{\parallel }\widehat{fd\sigma }{\mathrm{\parallel }}_{L^q({\mathbb{R}}^n)}{\lesssim }_q\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}$ for all $q>2n\mathrm{/}(n-1)$; the constant density shows this range is best possible. The conjecture is known for $n=2$ (Fefferman and Zygmund) and remains open for $n\ge 3$. The Stein-Tomas theorem of this page settles only the $L^2$-density line $q\ge 2(n+1)\mathrm{/}(n-1)$, which lies strictly above the conjectured $L^{\mathrm{\infty }}$ range; no claim here settles the general restriction problem, and the Knapp examples of the companion page constrain every such estimate.

The operators in the display are those of Fourier restriction and adjoint extension operators; the necessity of the $L^2$-density threshold recorded here is the theorem Knapp necessary condition for spherical L2 restriction, and the companion page's counterexample exhibits the same cap family in the limit $\delta \downarrow 0$.