Extremizers for Fourier restriction inequalities: convex arcs

TL;DR

This paper proves the existence of extremizers for Fourier restriction inequalities on convex planar arcs under certain curvature conditions, and shows that extremizing sequences converge to extremizers.

Contribution

It establishes the existence and convergence of extremizers for Fourier restriction inequalities on convex arcs with specific curvature assumptions.

Findings

Existence of extremizers for the inequality.

Convergence of extremizing sequences to extremizers.

Applicability to convex arcs without colinear tangents.

Abstract

We establish the existence of extremizers for a Fourier restriction inequality on planar convex arcs without points with colinear tangents whose curvature satisfies a natural assumption. More generally, we prove that any extremizing sequence of nonnegative functions has a subsequence which converges to an extremizer.