Restrictions of higher derivatives of the Fourier transform

Abstract

We consider several problems related to the restriction of $({\nabla }^k)\hat{f}$ to a surface $\Sigma \subset {\mathbb{R}}^d$ with nonvanishing Gauss curvature. While such restrictions clearly exist if $f$ is a Schwartz function, there are few bounds available that enable one to take limits with respect to the $L_p({\mathbb{R}}^d)$ norm of $f$. We establish three scenarios where it is possible to do so: $\bullet$ When the restriction is measured according to a Sobolev space $H^{-s}(\Sigma )$ of negative index. We determine the complete range of indices $(k,s,p)$ for which such a bound exists. $\bullet$ Among functions where $\hat{f}$ vanishes on $\Sigma$ to order $k-1$, the restriction of $({\nabla }^k)\hat{f}$ defines a bounded operator from (this subspace of) $L_p({\mathbb{R}}^d)$ to $L_2(\Sigma )$ provided $1\le p\le \frac{2d+2}{d+3+4k}$. $\bullet$ When there is _a priori_ control of $\hat{f}{|}_{\Sigma }$ in a space $H^{\ell }(\Sigma )$, $\ell >0$, this implies improved regularity for the restrictions of $({\nabla }^k)\hat{f}$. If $\ell$ is large enough then even $\Vert \nabla \hat{f}{\Vert }_{L_2(\Sigma )}$ can be controlled in terms of $\Vert \hat{f}{\Vert }_{H^{\ell }(\Sigma )}$ and $\Vert f{\Vert }_{L_p({\mathbb{R}}^d)}$ alone. The techniques underlying these results are inspired by the spectral synthesis work of Y. Domar, which provides a mechanism for $L_p$ approximation by "convolving along surfaces", and the Stein-Tomas restriction theorem. Our main inequality is a bilinear form bound with similar structure to the Stein--Tomas $T^{\ast }T$ operator, generalized to accommodate smoothing along $\Sigma$ and derivatives transverse to it. It is used both to establish basic $H^{-s}(\Sigma )$ bounds for derivatives of $\hat{f}$ and to bootstrap from surface regularity of $\hat{f}$ to regularity of its higher derivatives.

52 pages including 10 figures. Version 2 corrects a bibliographic entry. Version 3 adds some material to the introduction and corrects typos