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.
We have not been able to recognize your IP address
44.192.65.228
as that of a subscriber to this journal.
Online access to the content of recent issues is by
subscription, or purchase of single articles.
Please contact your institution's librarian suggesting a subscription, for example by using our
journal-recommendation form.
Or, visit our
subscription page
for instructions on purchasing a subscription.