In this paper we find the sharp forms and characterize the complex-valued extremizers of the adjoint Fourier restriction inequalities on the spherein the cases (d, p, q) = (d, 2k, q) with d, k ∈ N and q ∈ R + ∪ {∞} satisfying: (a) k = 2, q ≥ 2 and 3 ≤ d ≤ 7;(b) k = 2, q ≥ 4 and d ≥ 8; (c) k ≥ 3, q ≥ 2k and d ≥ 2. We also prove a sharp multilinear weighted restriction inequality, with weight related to the k-fold convolution of the surface measure.Building up on the work of Christ and Shao [6, 7], Foschi [12] recently obtained the sharp form of (1.2) in the Stein-Tomas endpoint case (d, p, q) = (3, 4, 2), showing that the constant functions are global extremizers.Here we extend this paradigm to other suitable triples (d, p, q). In fact, defining, our first result is the following:Theorem 1. Let (d, p, q) = (d, 2k, q) with d, k ∈ N and q ∈ R + ∪ {∞} satisfying: