Restriction and Kakeya maximal estimates in $\mathbb{R}^4$

TL;DR

This paper advances Fourier restriction and Bochner--Riesz bounds in four-dimensional space by integrating geometric and decoupling techniques, and also establishes a Kakeya maximal estimate at a specific dimension.

Contribution

It combines the planebrush and decoupling-incidence methods to improve bounds in Fourier analysis and extends Kakeya maximal estimates in $\

Findings

New bounds for Fourier restriction in $\

Kakeya maximal estimate at dimension 3.054 in $\

Extended the range for restriction problem to $p > 2 + rac{200}{251}$ in $\

Abstract

By combining the planebrush argument of Katz and Zahl \cite{katz21} with the decoupling-incidence method of Wang and Wu \cite{WangWu2024}, we derive new bounds for the Fourier restriction problem and the Bochner--Riesz problem, extending the range to $p > 2 + \frac{200}{251}$ in $\mathbb{R}^4$. Moreover, leveraging the two-ends Furstenberg estimate in the plane, we also obtain a Kakeya maximal estimate in $\mathbb{R}^4$ at dimension $3.054$.