We continue our research on Fourier restriction for hyperbolic surfaces, by studying local perturbations of the hyperbolic paraboloid \(z=xy\) which are of the form \(z=xy+h(y),\) where h(y) is a smooth function which is flat at the origin. The case of perturbations of finite type had already been handled before, but the flat case imposes several new obstacles. By means of a decomposition into intervals on which \(|h'''|\) is of a fixed size \({\lambda },\) we can apply methods devised in preceding papers, but since we lose control on higher order derivatives of h we are forced to rework the bilinear method for wave packets that are only slowly decaying. Another problem lies in the passage from bilinear estimates to linear estimates, for which we need to require some monotonicity of \(h'''.\)

通过研究双曲抛物面\（z = xy \）的局部扰动，其形式为\（z = xy + h（y），\），其中h（y）为在原点处平坦的平滑函数。之前已经处理过有限类型摄动的情况，但是平坦的情况强加了一些新的障碍。通过分解为\（| h'''| \）具有固定大小\（{\ lambda}，\）的间隔，我们可以应用先前论文中设计的方法，但是由于我们失去了对更高阶的控制h的导数我们被迫对仅缓慢衰减的波包重新采用双线性方法。另一个问题在于从双线性估计到线性估计的过渡，为此我们需要要求\（h'''。\）的单调性。