  In this article, we study the problem of obtaining Lebesgue space
inequalities for the Fourier restriction operator associated to rectangular
pieces of the paraboloid and perturbations thereof. We state a conjecture for
the dependence of the operator norms in these inequalities on the sidelengths
of the rectangles, prove that this conjecture follows from (a slight
reformulation of the) restriction conjecture for elliptic hypersurfaces, and
prove that, if valid, the conjecture is essentially sharp. Such questions arise
naturally in the study of restriction inequalities for degenerate
hypersurfaces; we demonstrate this connection by using our positive results to
prove new restriction inequalities for a class of hypersurfaces having some
additive structure.
