Stein’s restriction conjecture (L^∞→L^q) for curved hypersurfaces

Determine the optimal range of q for which the extension operator E_S associated to a curved hypersurface S (e.g., unit sphere or compact paraboloid patch) satisfies ∥E_S g∥_{L^q(R^d)} ≲ ∥g∥_{L^∞(S)}, and prove that this holds if and only if q > 2d/(d−1).

Background

The conjecture reflects the improved behavior of extension from curved surfaces, contrasted with the L^2→L^q Tomas–Stein theorem; it is open in dimensions d ≥ 3 and central to harmonic analysis.

It aligns with wave-packet phenomena and underpins many advances (e.g., decoupling), with consequences for large-value sets of oscillatory solutions to dispersive PDEs.