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:

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.

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 $1 \leq p \leq \frac {2d+2}{d+3+4k}$.

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 $\|\nabla \hat {f}\|_{L_2(\Sigma )}$ can be controlled in terms of $\|\hat {f}\|_{H^\ell (\Sigma )}$ and $\|f\|_{L_p(\mathbb {R}^d)}$ alone.

The proofs are based on three main tools: the spectral synthesis work of Y.Domar, which provides a mechanism for $L_p$ approximation by “convolving along surfaces in spectrum”, a new bilinear oscillatory integral estimate valid for ordinary $L_p$ functions, and a convexity-type property of the quantity $\|(\nabla ^k) \hat {f}\|_{H^{-s}(\Sigma )}$ as a function of $k$ and $s$ that allows one to employ the control of $\|\hat {f}\|_{H^\ell (\Sigma )}$.