In this talk I will discuss some sharp forms of adjoint Fourier
restriction inequalities on the sphere. These were unknown until the
recent works of Christ-Shao and Foschi for the 2-d sphere (2012 and 2013).
We extend the sharp inequalities to several other cases, completely
characterizing the complex-valued extremizers. We also consider some sharp
multilinear weighted restriction inequalities, with weights related to the
$k$-fold convolution of the surface measure. This is a joint work with
Diogo Oliveira e Silva (HCM - Bonn).