Sharp Fourier restriction to monomial curves

TL;DR

This paper investigates the bounds of Fourier restriction operators on monomial curves, identifying extremizing sequences and introducing new methods potentially applicable to broader manifolds.

Contribution

It establishes lower bounds for operator norms and characterizes extremizing sequences' precompactness, introducing novel techniques for Fourier analysis on manifolds.

Findings

Lower bounds for Fourier restriction operator norms established

Extremizing sequences are precompact modulo symmetry under certain conditions

New analytical tools introduced for studying Fourier restriction on manifolds

Abstract

We establish lower bounds for the operator norms of the Fourier restriction/extension operators associated to monomial curves with affine arclength measure. Furthermore, we prove that the set of all extremizing sequences of such an operator is precompact modulo the operator's symmetry group if and only if the operator norm is strictly larger than this threshold. For the proof, we introduce a number of new ingredients, some of which may be applicable to analogous questions on more general manifolds.