Fourier Restriction to Smooth Enough Curves

For a surface $S \subset \mathbb{R}^{d}$ equipped with some measure $\mu$, the Fourier restriction problem asks for which $p$ and $q$ there is a bounded linear operator
\mathcal{R} \colon L^{p}\: (\mathbb{R}^{d}\hspace{.5em}) \rightarrow L^{q}\: (S, \mu)
such that $\mathcal{R}f = \hat{f}\restriction_{S}$ for all Schwartz functions $f$. In 1985, Drury was the first person to prove an optimal result for a curve in 3 or more dimensions. He found the optimal range of $p$ and $q$ for the moment curve $\gamma(t)=(t, t^{2}, \dots, t^{d})$ with the affine arclength measure. Since then, many other people have contributed to this field.

Building on many of those prior results, we prove Fourier restriction estimates to arbitrary compact $C^{N}$ curves for any $N>d$, with $p$ and $q$ in the Drury range, using a power of the affine arclength measure as a mitigating factor. In particular, we make no nondegeneracy assumption on the curve.

I gave a 20-minute talk about this paper at the Ohio River Analysis Meeting in 2022.

Fourier Restriction and Maximal Operators on the Moment Curve

In the Fourier restriction problem, the pointwise relationship between $\mathcal{R}f$ and $f$ is not always clear when $f$ is not a Schwartz function. Recently, Müller, Ricci, and Wright initiated the study of maximal restriction theorems to analyze this relationship.

In this paper, we prove a maximal restriction theorem for certain $r$ maximal restriction operators on the moment curve. A corollary of this theorem is that for $f \in L^{p}(\mathbb{R}^{d})$, with $p$ in the optimal (Drury) range, almost every $x$ on the moment curve with respect to arclength measure is a Lebesgue point for $\hat{f}$ and the regularized value of $\hat{f}$ at $x$ coincides with $\mathcal{R}f(x)$.