Some Sharp Restriction Inequalities on the Sphere

In this paper, we find the sharp forms and characterize the complex-valued extremizers of the adjoint Fourier restriction inequalities on the sphere parallel to <(f sigma) over cap> L-p (R-d) less than or similar to parallel to L-q (Rd) (Sd-1, sigma) the cases (d, p, q) = (d, 2k, q) with d, k E N and gER. U (c)c) satisfying: (a) k= 2, q >= 2, and 3 < d < 7; (b) k= 2, q > 4, and d > 8; (c) k > 3, q > 2k, and di 2. We also prove a sharp multilinear weighted restriction inequality, with weight related to the k-fold convolution of the surface measure.