Improved decay of conical averages of the Fourier transform

TL;DR

This paper establishes a stronger lower bound on how quickly conical averages of Fourier transforms decay for cones of dimension four or higher, utilizing advanced harmonic analysis techniques.

Contribution

It introduces an improved lower bound for decay rates of conical Fourier averages, employing a weighted broad restriction inequality and a narrow decoupling inequality.

Findings

Enhanced decay bounds for conical Fourier averages in dimensions ≥ 4

Application of weighted broad restriction inequality

Use of narrow decoupling techniques for cones

Abstract

An improved lower bound is given for the decay of conical averages of Fourier transforms of measures, for cones of dimension $d \geq 4$. The proof uses a weighted version of the broad restriction inequality, a narrow decoupling inequality for the cone, and some techniques of Du and Zhang originally developed for the Schr\"odinger equation.