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

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 $3.054$.