Restriction and decay for flat hypersurfaces

In the first part we consider restriction theorems for hypersurfaces $\Gamma$ in ${\mathbf R}^n$, with the affine curvature $K_{\Gamma}^{1/(n+1)}$ introduced as a mitigating factor. Sjölin, [19], showed that there is a universal restriction theorem for all convex curves in ${\mathbf R}^2$. We show that in dimensions greater than two there is no analogous universal restriction theorem for hypersurfaces with non-negative curvature.

In the second part we discuss decay estimates for the Fourier transform of the density $K_{\Gamma}^{1/2}$ supported on the surface and investigate the relationship between restriction and decay in this setting. It is well-known that restriction theorems follow from appropriate decay estimates; one would like to know whether restriction and decay are, in fact, equivalent. We show that this is not the case in two dimensions. We also go some way towards a classification of those curves/surfaces for which decay holds by giving some sufficient conditions and some necessary conditions for decay.