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 work analyzes hypersurfaces $S\subset \mathbb{R}^4$ of vanishing Gaussian curvature, represented as graphs $S=\{(x,\phi(x))\}$ with $x\in\mathbb{R}^3$, by harnessing the Newton polyhedron of $\phi$ and adapted coordinates. It proves the existence of adapted coordinates for polynomial $\phi$ with $\det(D^2\phi)=0$, verifies Arnold’s conjecture that the leading asymptotics of the oscillatory integral are governed by the height $h(\phi)$ and the principal face, and derives sharp decay estimates $|\widehat{\rho d\sigma}(\xi)| \lesssim (1+|\xi|)^{-1/h}$ (up to a possible $\log$ factor). These oscillatory estimates translate into $L^p$–bounds for the associated maximal operator $\mathcal{M}$, showing boundedness for $p>\max\{h(\phi),2\}$ and, when $h(\phi)\ge 2$, exact bounds $p(S)=h(\phi)$; for $h(\phi)<2$, results depend on the Hessian rank at the base point. The work also establishes that the uniform oscillation and contact indices equal $1/h(\phi)$, extending the quantitative link between decay, geometry, and operator theory. Together, these results advance sharp three-dimensional oscillatory and maximal-operator phenomena for surface-carried measures in the vanishing-curvature regime and validate several conjectures in this setting.

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.

Theorems & Definitions (21)

• Proposition 2.1
• Theorem 3.1
• Theorem 3.2
• proof
• Corollary 3.3
• Corollary 3.4
• proof : Proof of Corollary \ref{['esseen']}
• Theorem 4.1
• Corollary 4.2
• proof : Proof of Theorem \ref{['th:osc']}