The problem of
Fourier restriction estimates for smooth hypersurfaces
of finite
type in
is by now very well understood for a large class of hypersurfaces, including
all analytic ones. In this article, we take up the study of more general
Fourier restriction estimates, by studying a prototypical model class of
two-dimensional surfaces for which the Gaussian curvature degenerates in
one-dimensional subsets. We obtain sharp restriction theorems in the range given by
Tao in 2003 in his work on paraboloids. For high-order degeneracies this
covers the full range, closing the restriction problem in Lebesgue spaces
for those surfaces. A surprising new feature appears, in contrast with the
nonvanishing curvature case: there is an extra necessary condition. Our approach
is based on an adaptation of the bilinear method. A careful study of the
dependence of the bilinear estimates on the curvature and size of the support is
required.