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 $1 \leq p \leq \frac{2d+2}{d+3+4k}$. $\bullet$ 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 $\|\nabla \hat{f}\|_{L_2(\Sigma)}$ can be controlled in terms of $\|\hat{f}\|_{H^\ell(\Sigma)}$ and $\|f\|_{L_p(\mathbb R^d)}$ alone. The 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 $T^*T$ 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.
Highlights
Questions regarding the fine properties of the Fourier transform of a function in Lp(Rd) have long played a central role in the development of classical harmonic analysis
While the Hausdorff–Young theorem guarantees that for 1 ≤ p ≤ 2, the Fourier transform of f ∈ Lp belongs to its dual space Lp/(p−1), it does not provide guidance on whether fmay be defined on a given measure-zero subset Σ ⊂ Rd
In this paper we investigate the possibility of defining the surface trace of higher order gradients of the Fourier transform of an Lp function, with a focus on uniform estimates in the style of (1)
Summary
On we assume that Σ is a closed smooth embedded (d − 1)-dimensional submanifold of Rd with nonvanishing principal curvatures It will turn out (see Theorem 1.6 below) that for a certain range of p and k, the space ΣLkp contains precisely the functions f ∈ Lp whose Fourier transform vanishes on Σ to order k − 1. It is worth noting that the main inequality used to derive (9) and (10) is valid for all functions in Lp, not just those whose Fourier restriction vanishes on Σ The formulation of this inequality, which may be of independent interest, is given in (29) below and the sharp range of p for which it holds is found in Theorem 1.16.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
More From: Transactions of the American Mathematical Society, Series B
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.