Some sharp restriction inequalities on the sphere

Abstract: In this paper we find the sharp forms and characterize the complex-valued extremizers of the adjoint Fourier restriction inequalities on the sphere $$\big|\widehat{f \sigma}\big|{L^{p}(\mathbb{R}^{d})} \lesssim |f|{L^{q}(\mathbb{S}^{d-1},\sigma)}$$ in the cases $(d,p,q) = (d,2k, q)$ with $d,k \in \mathbb{N}$ and $q\in \mathbb{R}^+ \cup {\infty}$ satisfying: (a) $k = 2$, $q \geq 2$ and $3 \leq d\leq 7$; (b) $k = 2$, $q \geq 4$ and $d \geq 8$; (c) $k \geq 3$, $q \geq 2k$ and $d \geq 2$. We also prove a sharp multilinear weighted restriction inequality, with weight related to the $k$-fold convolution of the surface measure.