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

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.