# 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}
}```
• Published 28 July 2018
• Mathematics
• Cambridge Journal of Mathematics
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.

## Figures from this paper

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
Abstract We give an abstract argument that an a priori Fourier restriction estimate for a certain choice of exponents automatically implies maximal and variational Fourier restriction estimates.
Extremizability of Fourier restriction to the paraboloid
Abstract In this article, we prove that nearly all valid, scale-invariant Fourier restriction inequalities for the paraboloid in R 1 + d have extremizers and that L p -normalized extremizing
Low-dimensional maximal restriction principles for the Fourier transform.
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
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
Sharp estimates for oscillatory integral operators via polynomial partitioning
• Mathematics
Acta 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

## 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
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
• Mathematics, Computer Science
Discret. 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
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
• Mathematics
Acta 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
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) ∈
Negative results for Nikodym maximal functions and related oscillatory integrals in curved space
• Mathematics
• 1997
We expand on counterxamples of Bourgain showing how Nikodym maximal estimates and oscillatory integral estimates can break down in the non-Euclidean case. Our examples show the role that the