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

TL;DR

The paper advances the Fourier restriction problem and Bochner–Riesz bounds in $\mathbb{R}^4$ by merging Katz–Zahl's planebrush with the Wang–Wu decoupling–incidence framework, achieving the new range $p > 2 + \frac{200}{251}$. It also derives a Kakeya maximal estimate at dimension $d_0 \approx 3.0543$ via a two-ends Furstenberg argument and a self-improving incidence theory. The methods hinge on a refined incidence analysis of shaded, two-ends, $\delta$-tube families and a careful wave-packet/decomposition approach that translates oscillatory behavior into geometric incidence bounds. Together, these results connect geometric incidence geometry with harmonic-analytic restriction phenomena, pushing existing thresholds toward the conjectured limits. The work also comments on implications for positive-definite Hörmander operators and the broader Bochner–Riesz landscape in higher dimensions.

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 $3.054$.

Theorems & Definitions (38)

• Conjecture 1.1
• Theorem 1.1
• Corollary 1.1
• Definition 1.1
• Conjecture 1.2
• Conjecture 1.3: Kakeya maximal estimate at dimension $d$
• Theorem 1.2
• Definition 1.2
• Definition 2.1: Refinement
• Definition 2.2: Shading