On sharp Fourier extension from spheres in arbitrary dimensions (Harmonic Analysis and Nonlinear Partial Differential Equations)

We prove a new family of sharp L2 (Sd-1)T L4 (Rd) Fourier extension inequalities from the unit sphere Sd-1 G Rd, valid in arbitrary dimensions d > 3. § 1. IntroductionThe purpose of this short note is to establish the following result.Theorem 1.1. Iげ怯(Sd-1) は怯(Sd-1)Equality holds in (1.1) if and only if f is a constant function.In (1.1),fb denotes the Fourier extension operator from the unit sphere Sd-1 G Rd, (1.2) fb(x) = / f(心'心 db(3).