# 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.

## 25 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

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

- Mathematics
- 2020

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.

- Mathematics
- 2019

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…

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…

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