Research

## 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.

## 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 $\mathcal{R}f(x)$.