# On the restriction theorem for paraboloid in $\mathbb R^4$

## TL;DR

This paper shows that recent advances in Kakeya maximal operator estimates in four dimensions can improve the known bounds for the restriction problem for the paraboloid, introducing a new trilinear estimate.

## Contribution

It connects Kakeya maximal operator breakthroughs to restriction estimates, providing a novel trilinear estimate and improving the threshold in four-dimensional paraboloid restriction problem.

## Key findings

- Improved the restriction threshold from 14/5 to a higher value.
- Introduced a new trilinear estimate for the paraboloid.
- Linked Kakeya maximal operator results to restriction problem improvements.

## Abstract

We prove that recent breaking by Zahl of the $\frac32$ barrier in Wolff's estimate on the Kakeya maximal operator in $\mathbb R^4$ leads to improving the $\frac{14}{5}$ threshold for the restriction problem for the paraboloid in $\mathbb R^4$. One of the ingredients is a new trilinear estimate. The proofs are deliberately presented in a nontechnical and concise format, so as to make the arguments more readable and focus attention on the key tools.