  The problem of $L^p(R^3)\to L^2(S)$ Fourier restriction estimates for smooth
hypersurfaces S of finite type in R^3 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 $L^p(R^3)\to L^q(S)$ Fourier restriction
estimates, by studying a prototypical class of two-dimensional surfaces with
strongly varying curvature conditions. Our approach is based on an adaptation
of the so-called bilinear method. We discuss several new features arising in
the study of this problem.
