Restriction of Fourier transforms to curves: An endpoint estimate with affine arclength measure

TL;DR

This paper establishes endpoint Fourier restriction estimates for curves in higher-dimensional space using affine arclength measure, employing multilinear interpolation techniques for vector-valued functions.

Contribution

It provides new endpoint restriction estimates for a broad class of curves, utilizing an innovative multilinear interpolation approach with symmetries.

Findings

Proved endpoint restriction estimates for curves in R^d

Developed an interpolation method for multilinear operators with symmetries

Extended restriction theory to affine arclength measures on curves

Abstract

Consider the Fourier restriction operator associated to a curve in $R^d$, $d\ge 3$. We prove for various classes of curves the endpoint restricted strong type estimate with respect to affine arclength measure on the curve. An essential ingredient is an interpolation result for multilinear operators with symmetries acting on sequences of vector-valued functions.