Uniform Fourier restriction for convex curves

TL;DR

This paper extends Fourier restriction estimates to all convex curves in the plane, removing smoothness constraints and establishing uniform bounds, thereby broadening the applicability of Fourier analysis techniques.

Contribution

It generalizes previous Fourier restriction results to convex curves without requiring $ ext{C}^2$ regularity, using an affine invariant measure.

Findings

Established uniform Fourier restriction estimates for convex curves.

Removed the $ ext{C}^2$ regularity condition on curves.

Proved results on Lebesgue points of the Fourier transform on convex curves.

Abstract

We extend the estimates for maximal Fourier restriction operators proved by M\"{u}ller, Ricci, and Wright in \cite{MR3960255} and Ramos in \cite{MR4055940} to the case of arbitrary convex curves in the plane, with constants uniform in the curve. The improvement over M\"{u}ller, Ricci, and Wright and Ramos is given by the removal of the $\mathcal{C}^2$ regularity condition on the curve. This requires the choice of an appropriate measure for each curve, that is suggested by an affine invariant construction of Oberlin in \cite{MR1960918}. As corollaries, we obtain a uniform Fourier restriction theorem for arbitrary convex curves, and a result on the Lebesgue points of the Fourier transform on the curve.