Sharp estimates for the Fourier transform of surface-carried measures and maximal operators associated with hypersurfaces in $\mathbb{R}^4$ with vanishing Gaussian curvature

TL;DR

This paper establishes sharp Fourier transform estimates and determines the boundedness of maximal operators for hypersurfaces in four-dimensional space with zero Gaussian curvature, advancing harmonic analysis in this geometric setting.

Contribution

It provides the first sharp uniform Fourier transform estimates and exact boundedness exponents for maximal operators on hypersurfaces with vanishing Gaussian curvature in .

Findings

Sharp uniform Fourier transform estimates derived

Exact boundedness exponents for maximal operators determined

Results depend on the hypersurfaces' geometric heights

Abstract

In this paper, we study problems related to harmonic analysis on hypersurfaces in $\mathbb{R}^4 $ with zero Gaussian curvature and given as graphs of polynomial functions. We derive sharp uniform estimates with respect to the direction of frequencies for the Fourier transform of measures supported on such hypersurfaces. Additionally, we study the $L^p$-boundedness problem of maximal operators associated with hypersurfaces. We determine the exact value of the boundedness exponent in terms of the heights of these hypersurfaces.