# Improved Fourier restriction estimates in higher dimensions

@article{Hickman2019ImprovedFR, title={Improved Fourier restriction estimates in higher dimensions}, author={John L. Hickman and Keith M. Rogers}, journal={Cambridge Journal of Mathematics}, year={2019} }

We consider Guth's approach to the Fourier restriction problem via polynomial partitioning. By writing out his induction argument as a recursive algorithm and introducing new geometric information, known as the polynomial Wolff axioms, we obtain improved bounds for the restriction conjecture, particularly in high dimensions. Consequences for the Kakeya conjecture are also considered.

## 28 Citations

A note on Fourier restriction and nested Polynomial Wolff axioms

- Mathematics
- 2020

This note records an asymptotic improvement on the known $L^p$ range for the Fourier restriction conjecture in high dimensions. This is obtained by combining Guth's polynomial partitioning method…

Improved bounds for the Kakeya maximal conjecture in higher dimensions

- Mathematics
- 2019

We adapt Guth's polynomial partitioning argument for the Fourier restriction problem to the context of the Kakeya problem. By writing out the induction argument as a recursive algorithm, additional…

New bounds for Stein's square functions in higher dimensions

- Mathematics
- 2021

We improve the L(R) bounds on Stein’s square function to the best known range of the Fourier restriction problem when n ≥ 4. Applications including certain local smoothing estimates are also…

Fourier restriction implies maximal and variational Fourier restriction

- MathematicsJournal of Functional Analysis
- 2019

The Stein--Tomas inequality under the effect of symmetries

- Mathematics
- 2021

. We prove new Fourier restriction estimates to the unit sphere S d − 1 on the class of O ( d − k ) × O ( k )-symmetric functions, for every d ≥ 4 and 2 ≤ k ≤ d − 2. As an application, we establish…

Low-Dimensional maximal restriction principles for the Fourier transform

- MathematicsIndiana University Mathematics Journal
- 2022

Following the ideas from a paper by the same author, we prove abstract maximal restriction results for the Fourier transform. Our results deal mainly with maximal operators of convolution-type and…

Fourier restriction for smooth hyperbolic 2-surfaces

- Mathematics
- 2020

We prove Fourier restriction estimates by means of the polynomial partitioning method for compact subsets of any sufficiently smooth hyperbolic hypersurface in threedimensional euclidean space. Our…

Factorisation in Restriction theory and near extremisers

- Mathematics
- 2021

We give an alternative argument to the application of the so-called MaureyNikishin-Pisier factorisation in Fourier restriction theory. Based on an induction-on-scales argument, our comparably simple…

Fourier restriction above rectangles

- Mathematics
- 2019

In this article, we study the problem of obtaining Lebesgue space inequalities for the Fourier restriction operator associated to rectangular pieces of the paraboloid and perturbations thereof. We…

## References

SHOWING 1-10 OF 48 REFERENCES

On the multilinear restriction and Kakeya conjectures

- Mathematics
- 2005

We prove d-linear analogues of the classical restriction and Kakeya conjectures in Rd. Our approach involves obtaining monotonicity formulae pertaining to a certain evolution of families of…

Restriction estimates using polynomial partitioning II

- Mathematics
- 2016

We improve the estimates in the restriction problem in dimension $n \ge 4$. To do so, we establish a weak version of a $k$-linear restriction estimate for any $k$. The exponents in this weak…

An Incidence Theorem in Higher Dimensions

- MathematicsDiscret. Comput. Geom.
- 2012

The polynomial ham sandwich theorem is used to prove almost tight bounds on the number of incidences between points and k-dimensional varieties of bounded degree in Rd.

New Kakeya estimates using the polynomial Wolff axioms

- Mathematics
- 2019

We obtain new bounds for the Kakeya maximal conjecture in most dimensions $n<100$, as well as improved bounds for the Kakeya set conjecture when $n=7$ or $9$. For this we consider Guth and Zahl's…

New bounds for Kakeya problems

- Mathematics
- 2001

We establish new estimates on the Minkowski and Hausdorff dimensions of Kakeya sets and we obtain new bounds on the Kakeya maximal operator.

Bounds on Oscillatory Integral Operators Based on Multilinear Estimates

- Mathematics
- 2010

We apply the Bennett–Carbery–Tao multilinear restriction estimate in order to bound restriction operators and more general oscillatory integral operators. We get improved Lp estimates in the Stein…

A trilinear restriction estimate with sharp dependence on transversality

- Mathematics
- 2016

Abstract:We improve the Bennett-Carbery-Tao trilinear restriction estimate for subsets of the paraboloid in three dimensions, giving the sharp bound with respect to the transversality. The main…

Sharp estimates for oscillatory integral operators via polynomial partitioning

- MathematicsActa Mathematica
- 2019

The sharp range of $L^p$-estimates for the class of H\"ormander-type oscillatory integral operators is established in all dimensions under a positive-definite assumption on the phase. This is…

Polynomial Wolff axioms and Kakeya‐type estimates in R4

- Mathematics
- 2018

We establish new linear and trilinear bounds for collections of tubes in R4 that satisfy the polynomial Wolff axioms. In brief, a collection of δ ‐tubes satisfies the Wolff axioms if not too many…

BOUNDS ON ARITHMETIC PROJECTIONS, AND APPLICATIONS TO THE KAKEYA CONJECTURE

- Mathematics
- 1999

Let A, B, be finite subsets of a torsion-free abelian group, and let G ⊂ A × B be suchth at # A, #B,#{a + b :( a, b) ∈ G }≤ N. We consider the question of estimating the quantity #{a − b :( a, b) ∈…